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THE
SENSATIONS OF TONE
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PRIN'TKD BY
BromawooDS asd co., NKW^rnKnr uguAiiK
LONDON
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ON THE
SENSATIONS OF TONE
AS A PHTSIOLOGICAL BASIS FOR THE
THEOEY OF MUSIC
BY
HERMANN L R HELMHOLTZ, M.D.
FORKION MEUBEB OF THB BOTAL SOCUSTIB8 OP LONDON AND BDINBUROH
FOH1EB8LT PBOFBSSOB OF PHTSIOLOnT IN TUB UNIYBBSITT OF HBIDBLBBBG, AND
MOW PBOFE380B OF PHYSICS IN THE UNIYBBSITT OF BKBLIN
SECOND ENGLISH EDITION
Translatedj thorcmghly Revised atid CorrecjUd, refudered conformable to the Fourth
(and Uut) German Edition of 1877, tf>ith numero^is additional Notes and a
New additional Appendix bringing down information to 1886,
arid espedaUy adapted to tJie use of Musical Students
ALEXANDER J. ELLIS
B.A. P.B.a F.S.A. F.CPA P.CP.
TWICB FBB91DENT OF THB PHILOLOGICAL 80CIETT, MEMBKB OF THB UATHBMATICAL SOCUTT,
POBHEBLT 8CUOLAB OF TBINITT COLLBOB, CAUBBIDOB,
AUTHOB OF ' BABLlr ENGLISH FBOMUNCUTION* AND * ALGEBBA IDENTIFIED WITH OBOMETBT '
LONDOSr
LONGMANS, GREEN, AND CO.
1885
3
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TEANSLATOE'S NOTICE
TO THB
SECOND ENGLISH EDITION.
In preparing a new edition of this translation of Professor Helmhol^z's great work on
the Sensations of Tone, which was originally made from the third German edition
of 1870, and was finished 'in June 1875, my first care was to make it exactly
conform to tiiefov/rth Qennan edition of 1877 (the last which has appeared). The
nomerons alterations nmde in the fourth edition are specified in the Author's pre-
bce. In order that no merely verbal changes might escape me, every sentence
of my translation was carefully re-read with the German. This has enabled me
to correct several misprints and mistranslations which had escaped my previous
very careful revision, and I have taken the opportunity of improving the language
in many places. Scarcely a page has escaped such changes.
Professor Helmholtz's book having taken its place as a work which all candidates
for musical degrees are expected to study, my next care was by supplementary
notes or brief insertions, always carefully ^stinguished from the Author's by being
inclosed in [ ], to explain any difficulties which the student might feel, and to shew
him how to acquire an insight into the Author's theories, which were quite strange
to musicians when they appeared in the first German edition of 1863, ^^^ ui the
twenty-two years which have since elapsed have been received as essentially valid
by those competent to pass judgment.
For this purpose I have contrived the Harmonical, explained on pp. 466-469,
by which, as shewn in numerous footnotes, almost every point of theory can be
illustrated ; and I have arranged for its being readily procurable at a moderato
charge. It need scarcely be said that my interest in this instrument is purely
scientific.
My own Appendix has been entirely re- written, much has been rejected and the
rest condensed, but, as may be seen in the Contents, I have added a considerable
imount of information about points hitherto little known, such as the Determi-
nation and History of Musical Pitch, Non-Harmonic scales. Tuning, &o., and in
especial I have given an account of the work recently done on Beats and Com-
binational Tones, and on Vowel Analysis and Synthesis, mostly since the fourth
German edition appeared.
Finally, I wish gratefully to acknowledge the assistance, sometimes very great,
which I have received from Messrs. D. J. Blaikley, B. H. M. Bosanquet, Colin
Brown, A. Cavaill6-Coll, A. J. Hipkins, W. Huggins, F.E.S., Shuji Isawa, H.
Wird Poole, B. S. Bockstro, Hermann Smith, Steinway, Augustus Stroh, and
James Paul White, as will be seen by referring to their names in the Index.
ALEXANDER J. ELLIS.
25 Argyll Road, Kexri>:gton:
J^^y l8«5. Digitized by GoOglC
AUTHOR'S PREFACE
TO THE
FIRST GERMAN EDITION.
In laying before the Public the result of eight years' labour, I must first pay a
debt of gratitude. The following investigations could not have been accomplished
without the construction of new instruments, which did not enter into the inventory
of a Physiological Institute, and which far exceeded in cost the usual resources of
a German philosopher. The means for obtaining them have come to me from
unusual sources. The apparatus for the artificial construction of vowels, described
on pp. 121 to 126, 1 owe to the munificence of his Majesty King Maximilian of
Bavaria, to whom German science is indebted, on so many of its fields, for ever-
ready sympathy and assistance. For the construction of my Harmonium in
perfectly natural intonation, described on p. 316, 1 was able to use the Soemmering
prize which had been awarded me by the Senckenberg Physical Society (die
Senckenbergische naturforschende Gesellschaft) at Frankfurt-on-the-Main. While
publicly repeating the expression of my gratitude for this assistance in my investi-
gations, I hope that the investigations themselves as set forth in this book will
prove far better than mere words how earnestly I have endeavoured to make a
worthy use of the means thus placed at my command.
H. HELMHOLTZ.
Hbidelbebg : October 1862.
AUTHOR'S PREFACE
TO THE
THIRD GERMAN EDITION.
Thb present Third Edition has been much more altered in some parts than the
second. Thus in the sixth chapter I have been able to make use of the new
physiological and anatomical researches on the ear. This has led to a modification
of my view of the action of Corti's arches. Again, it appears that the peculiar
articulation between the auditory ossicles called ' hammer ' and * anvil * might easily
cause within the ear itself the formation of harmonic upper partial tones for simple
tones which are sounded loudly. By this means that peculiar series of upper partial
tones, on the existence of which the present theory of music is essentially founded,
receives a new subjective value, entirely independent of external alterations in
the quality of tone. To illustrate the anatomical descriptions, I have been able
to add a series of new woodcuts, principally from Henle's Manual of Anatomy,
with the author's permission, for which I here take the opportunity of publicly
thanking him.
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PREFACE. vii
I have made many changes in re-editing the section on the History of Music,
and hope that I have improved its connection. I must, however, request the
reader to regard this section as a mere compilation from secondary sources ; I
have neither time nor preliminary knowledge sufficient for original studies in this
extremely difficult field. The older history of music to the commencement of
Discant, is scarcely more than a confused heap of secondary subjects, while we
can only make hypotheses concerning the principal matters in question. Of
coarse, however, every theory of music must endeavour to bring some order into
this chaos, and it cannot be denied that it contains many important facts.
For the representation of pitch in just or natural intonation, I have abandoned
the method originally proposed by Hauptmann, which was not sufficiently clear in
involved cases, and have adopted the system of Herr A. von Oettingen [p. 276],
as had already been done in M. G. Gu6roult's French translation of this book.
[A comparison of the Third with the Second editions, shewing the changes and additions
individually, is here omitted.]
If I may be allowed in conclusion to add a few words on the reception expe-
rienced by the Theory of Music here propounded, I should say that pubHshed
objections almost exclusively relate to my Theory of Consonance, as if this were
the pith of the matter. Those who prefer mechanical explanations express their
regret at my having left any room in this field for the action of artistic invention
and esthetic inclination, and they have endeavoured to complete my system by
new numerical speculations. Other critics with more metaphysical procUvities
have rejected my Theory of Consonance, and with it, as they imagine, my whole
Theory of Music, as too coarsely mechanical.
I hope my critics will excuse me if I conclude from the opposite nature of
their objections, that I have struck out nearly the right path. As to my Theory
of Consonance, I must claim it to be a mere systematisation of observed facts
(with the exception of the functions of the cochlea of the ear, which is moreover
an hypothesis that may be entirely dispensed with). But I consider it a mistake
to make the Theory of Consonance the essential foundation of the Theory of
Music, and I had thought that this opinion was clearly enough expressed in my book.
The essential basis of Music is Melody. Harmony has become to Western Euro-
peans during the last three centuries an essential, and, to our present taste,
indispensable means of strengthening melodic relations, but finely developed
music existed for thousands of years and still exists in ultra-European nations,
without any harmony at all. And to my metaphysico-esthetical opponents I must
reply, that I cannot think I have undervalued the artistic emotions of the human
mind in the Theory of Melodic Construction, by endeavouring to estabUsh the
physiological facts on which esthetic feeling is based. But to those who think I
have not gone far enough in my physical explanations, I answer, that in the first
place a natural philosopher is never bound to construct systems about everything he
knows and does not know ; and secondly, that I should consider a theory which
claimed to have shewn that all the laws of modem Thorough Bass were natural
necessities, to stand condemned as having proved too much.
Musicians have found most fault with the manner in which I have characterised
the Minor Mode. I must refer in reply to those very accessible documents, the
musical compositions of a.d. 1500 to a.d. 1750, during which the modern Minor
was developed. These will shew how slow and fluctuating was its development,
and that the last traces of its incomplete state are still visible in the works of
Sebastian Bach and Handel.
Heidklbrbg : May 1870.
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AUTHOE'S PEEFACE
TO THB
FOURTH GERMAN EDITION.
In the essential conceptions of masical relations I have fonnd nothing to alter in
this new edition. In this respect I can but maintain what I have stated in the
chapters containing them and in my preface to the third [German] edition. In
details, however, much has been remodelled, and in some parts enlarged. As a
guide for readers of former editions, I take the liberty to enumerate the following
places containing additions and alterations.*
P. i6dt note *.— On the French Bystem of counting vibrations.
P. i8a. — Appunn and Preyer, limits of the highest audible tones.
Pp- 59& to 656. — On the circumstances under which we distinguish compound sensations.
P. 76a, b, c. — Cknnparison of the upper partial tones of the strings on a new and an old
grand pianoforte.
P. 83, note f.— Herr Clement Neumann's observations on the vibrational form of violin
strings.
Pp. 89a to 936.— The action of blowing organ-pipes.
P. 1 10&. — Distinction of Ou from U.
Pp. 1 1 16 to ii6a. — The various modifications in the sounds of vowels.
P. 145a. — The ampulla and semicircular canals no longer considered as parts of the organ
of hearing.
P. 1 476. — Waldeyer*s and Preyer's measurements adopted.
Pp. 1506 to I Sid, — On the parts of the ear which perceive noise.
P. 159&.— Eoenig's observations on combinational tones with tuning-forks.
P. I76<2, note. — ^Preyer's observations on deepest tones.
P. 179c. — Preyer's observation on the sameness of the quality of tones at the highest pitches.
Pp. 2036 to 204a.— Beats between upper partials of the same compound tone condition the
preference of musical tones with harmonic upper partials.
Pp. 328c to 3296.— Division of the Octave into 53 degrees. Bosanquet's harmonium.
Pp. 338c to 3396.— Modulations through chords composed of two major Thirds.
P. 365, note f.— Oettingen and Biemann's theory of the minor mode.
P. 372. — Improved electro-magnetic driver of the siren.
P. 373a.— Theoretical formulaa for the pitch of resonators.
P. 3744;. — Use of a soap-bubble for seeing vibrations.
Pp. 389^2 to 3966.— Later use of striking reeds. Theory of the blowing of pipes.
Pp. 403c to 405&. — Theoretical treatment of sympathetic resonance for noises.
P. 417^.— A. Mayer's experiments on the audibility of vibrations.
P. 428c, (2.— Against the defenders of tempered intonation.
P. 429.— Plan of Bosanquet's Harmonium.
H. HELMHOLTZ.
* [The pages of this edition are substituted first edition of this translation are mostly
for the German throughout these prefaces, pointed out in footnotes as they arise.— Trans-
BsBLiM : April 1877.
^ [The pages of this c
the German throug
and omissions or alterations as respects the lator.]
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CONTENTS.
\* All passages and notes in [ ] are due to the Translator, and the Author is in no
way responsible for their contents.
Tbanblatob*8 NonoB to thb Second Emolibh Edition, p. v.
Authob's Pbbvacb to thb Fibst Gbbuan Edition, p. vi.
AuTHOB*B Pbxfjlcb TO THB Thibd Gebuan Edition, pp. vl-vii.
Authob's Pbxface to thb Foubth Gbbuan Edition, p. viii.
CONTBNTS, p. iz.
List or Fioubbs, p. zv.
List of Passages in Musical Notes, p. zvi.
List of Tables, p. zvii.
INTBODUCTION, pp. i-6.
Belation of Musical Science to Acoustics, i
Distinction between Physical and Physiological Acoustics, 3
Plan of the Investigation, 4
PART L (pp. 7-151.)
ON THE COMPOSITION OF VIBRATIONS.
Upper Pa/rtial T&nea^ and QuaUties of Tone.
CHAPTER I. On the Sensation of Sound in General, pp. 8-25.
Distinction between Noise and Musical Tone, 8
Musical Tone due to Periodic, Noise to non-Periodic Motions in the air, 8
General Property of Undulatory Motion : while Waves continually advance, the Particles
of the Medium through which they pass execute Periodic Motions, 9
Differences in Musical Tones due to Force, Pitch, and Quality, 10
Force of Tone depends on Amplitude of Oscillation, Pitch on the length of the Period of
Oscillation, 10- 14
Simple relations of Vibrational Numbers for the Consonant Intervals, 14
Vibrational Numbers of Consonant Intervals calculated for the whole Scale, 17
Quality of Tone must depend on Vibrational Form, 19
Conception of and Graphical Representation of Vibrational Form, 20
Harmonic Upper Partial Tones, 22
Terms explained : Tone, Musical Tone, Simple Tone, Partial Tone, Compound Tone, Pitch
of Compound Tone, 23
CHAPTER n. On the Composition op Vibrations, pp. 25-36.
Comi>osition of Waves illustrated by waves of water, 25
The Heights of Superimposed Waves of Water are to be added algebraically, 27
Corresponding Superimposition of Waves of Sound in the air, 28
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X CONTENTS.
A Composite Mass of Musical Tones will give rise to a Periodic Vibration when their Pitch
Numbers are Multiples of the same Number, 30
Every such Composite Mass of Tones may be considered to be composed of Simple
Tones, 33
This Composition corresponds, according to G. S. Ohm, to the Composition of a Musical
Tone from Simple Partial Tones, 33
CHAPTEE in. Analysis op Musical Tones by Sympathetic Resonance,
pp. 36-49.
Explanations of the Mechanics of Sympathetic Vibration, 36
Sympathetic Resonance occurs when the exciting vibrations contain a Simple Vibration
corresponding to one of the Proper Vibrations of the Sympathising Body, 38
Difference in the Sympathetic Resonance of Tuning-forks and Membranes, 40
Description of Resonators for the more accurate Analysis of Musical Tones, 43
Sympathetic Vibration of Strings, 45
Objective existence of Partial Tones, 48
CHAPTER IV. On the Analysis of Musical Tones by the Ear, pp. 49-65.
Methods for observing Upper Partial Tones, 49
Proof of G. S. Ohm's Law by means of the tones of Plucked Strings, of the Simple Tones
of Tuning-forks, and of Resonators, 51
Difference between Compound and Simple Tones, 56
Seebeck's Objections against Ohm*s Law, 58
The Difficulties experienced in perceiving TJpper Partial Tones analytically depend upon a
peculiarity common to all human sensations, 59
We practise observation on sensation only to the extent necessary for clearly apprehend-
ing the external world, 62
Analysis of Compound Sensations, 63
CHAPTER V. On the Differences in the Quality op Musical Tones,
pp. 65-119.
Noises heard at the beginning or end of Tones, such as Consonants in Speech, or during
Tones, such as Wind- rushes on Pipes, not included in the Musical Quality of Tone,
which refers to the uniformly continuous musical sound, 65
Limitation of the conception of Musical Quality of Tone, 68
Investigation of the Upper Partial Tones which are present in different Musical Qualities
of Tone, 69
1. Musical Tones without Upper Partials, 69
2. Musical Tones with Inharmonic Upper Partials, 70
3. Musical Tones of Strings, 74
Strings excited by Striking, 74
Theoretical Intensity of the Partial Tones of Strings, 79
4. Musical Tones of Bowed Instruments, 80
5. Musical Tones of Flute or Flue Pipes, 88
6. Musical Tones of Reed Pipes, 95
7. Vowel Qualities of Tone, 103
Results for the Character of Musical Tones in general, 1 18
CHAPTER VI. On the Apprehension of Qualities of Tone, pp. 119-151.
Does Quality of Tone depend on Difference of Phase ? 1 19
Electro-magnetic Apparatus for answering this question, 121
Artificial Vowels produced by Tuning-forks, 123
How to produce Difference of Phase, 125
Musical Quality of Tone independent of Difference of Phase, 126
Artificial Vowels produced by Organ Pipes, 128
The Hypothesis that a Series of Sympathetical Vibrators exist in the ear, explains its
peculiar apprehension of Qualities of Tone, 129
Description of the parts of the internal ear which are capable of vibrating sympa-
thetically, 129
Damping of Vibrations in the Ear, 142
Supposed Function of the Cochlea, 145 ^.^.^.^^^ ^^ GoOqIc
CONTENTS. xi
PART 11. (pp. 152-233.)
ON THE INTERRUPTIONS OP HARMONY.
Combinational Tones and Beats, Consonance and Dissonance.
CHAPTER VII. Combinational Tones, pp. 152-159.
Combinational Tones arise when Vibrations which are not of infinitesimal magnitude are
combined, 152
Description of Combinational Tones, 153
Law determining their Pitch Numbers, 154
Combinational Tones of different orders, 155
Difference of the strength of Combmational Tones on different instruments, 157
Occasional Generation of Ck>mbinational Tones in the ear itself, 158
CHAPTER VllI, On the Beats op Simple Tones, pp. 159-173.
Interference of two Simple Tones of the same pitch, 160
Description of the Polyphonic Siren, for experiments on Interference, 161
Beinforcement or Enfeeblement of Sound, due to difference of Phase, 163
Interference gives rise to Beats when the Pitch of the two Tones is slightly different, 164
Law for the Number of Beats, 165
Visible Beats on Bodies vibrating sympathetically, 166
Limits of Bapidity of Audible Beats, 167
CHAPTER IX. Deep and Deepest Tones, pp. 174-179.
Former Investigations were insufficient, because there was a possibility of the ear being
deceived by Upper Partial Tones, as is shewn by the number of Beats on the Siren, 174
Tones of less than thirty Vibrations in a second fall into a Drone, of which it is nearly
or quite impossible to determine the Pitch, 175
/> Beats of the Higher Upper Partials of one and the same Deep Compound Tone, 17S
CHAPTER X, Beats of the Uppeb Partial Tones, pp. 179-197.
Any two Partial Tones of any two Compound Tones may beat if they are sufficiently
near in pitch, but if they are of the same pitch there will be consonance, 179
Series of the different Consonances, in order of the Distinctness of their Delimitation, 183
Number of Beats which arise from Mistuning Consonances, and their effect in producing
Roughness, 184
Disturbance of any Consonance by the adjacent Consonances, 186
Order of Consonances in respect to Harmoniousness, 1S8
CHAPTER XI. Beats due to Combinational Tones, pp. 197-21 1.
The Differential Tones of the first order generated by two Partial Tones are capable of
producing very distinct beats, 197
Differential Tones of higher orders produce weaker beats, even in the case of simple gene-
rating tones, 199
Influence of Quality of Tone on the Harshness of Dissonances and the Harmoniousness
of Consonances, 205
CHAPTER Xn. Chobds, pp. 211-233.
Consonant Triads, 211
Major and Minor Triads distinguished by their Combinational Tones, 214
Belative Harmoniousness of Chords in different Inversions and Positions, 218
Betrospeot on Preceding Investigations, 226
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xii CONTENTS.
PART IIL (pp. 234-371.)
THE RELATIONSHIP OF MUSICAL TONES.
Scalea and Tonality,
CHAPTER Xin. Gbnbeal View op the Different Principles op Musicax*
Style in the Development of Music, pp. 234-249.
DiSerenoe between the Physioal and the Esthetioal Method, 234
Scales, Keys, and Harmonic TisBues depend upon esthetio Principles of Style as well fts
Physical Causes, 235
Illustration from the Styles of Architecture, 235
Three periods of Music have to be distinguished, 236
1. Homophonic Music, 237
2. Polyphonic Music, 244
3. Harmonic Music, 246
CHAPTER XIV. The Tonality op Homophonic Music, pp. 250-290.
Esthetical Reason for Progression by Intervals, 250
Tonal Relationship in Melody depends on the identity of two partial tones, 253
The Octave, Fifth, and Fourth were thus first discovered, 253
Variations in Thirds and Sixths, 255
Scales of Five Tones, used by Chinese and Gaels, 258
The Chromatic and Enharmonic Scales of the Gre^s, 262
The Pythagorean Scales of Seven tones, 266
The Greek and Ecclesiastical Tonal Modes, 267
Early Ecclesiastical Modes, 272
The Rational Construction of the Diatonic Scales by the principle of Tonal Relationship in
the first and second degrees gives the five Ancient Melodic Scales, 272
Introduction of a more Accurate Notation for Pitch, 276
Peculiar discovery of natural Thirds in the Arabic and Persian Tonal Systems, 2S0
The meaning of the Leading Note and consequent alterations in the Modem Scales, 285
CHAPTER XV. The Consonant Chords of the Tonal Modes, pp. 290-
309*
Chords as the Representatives of compound Musical Tones with peculiar qualities, 290
Reduction of all Tones to the closest relationship in the popular harmonies of the Major
Mode, 292 *
Ambiguity of Minor Chords, 294
The Tonic Chord as the centre of the Sequence of Chords, 296
Relationship of Chords of the Scale, 297
The Major and Minor Modes are best suited for Harmonisation of all the Ancient Modes,
298
Modem Renmants of the old Tonal Modes, 306
CHAPTER XVI. The System op Keys, pp. 310-330.
Relative and Absolute Character of the different Keys, 310
Modulation leads to Tempering the Intonation of the Intervals, 312
Hauptmann*s System admits of a Simplification which makes its Realisation more Practi-
cable, 315
Description of an Harmonium with Just Intonation, 316
Disadvantages of Tempered Intonation, 322
Modulation for Just Intonation, 327
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CONTENTS. xiii
CHAPTER XVn. Of Discords, pp. 330-350.
Ennmeration of the Dissonant Intervals in the Scale, 331
Dissonant Triads, 338
Chords of the Seventh, 341 "^
CfOnoeption of the Dissonant Note in a Discord, 346
Discords as representatives of compoond tones, 347
CHAPTER XVin. Laws op Progression of Parts, pp. 350-362.
The Masical Connection of the Notes in a Melody, 350
Conseqaent Bales for the Progression of Dissonant Notes, 353
Besolation of Discords, 354
Chordal Sequences and Resolution of Chords of the Seventh, 355
Prohihition of Consecutive Fifths and Octaves, 359
Hidden Fifths and Octaves, 361
False Relations, 361
CHAPTER XIX. EsTHETicAL Relations, pp. 362-371.
Review of Besults obtained, 362
Law of Unconscious Order in Works of Art, 366
The Law of Melodic Succession depends on Sensation, not on Consciousness, 368
And similarly for Consonance and Dissonance, 369
Conclusion, 371
APPENDICES, pp. 327-556.
I. On an Electro-Magnetic Driving Machine for the Siren, 372
II. On the Size and Construction of Resonators, 372
III. On the Motion of Plucked Strings, 374
rv. On the Production of Simple Tones by Resonance, 377
V. On the Vibrational Forms of Pianoforte Strings, 380
VL Analysis of the Motion of Violin Strings, 384
Vn. On the Theory of Pipes, 388
A. Influence of Resonance on Reed Pipes, 388
B. Theory of the Blowing of Pipes, 390
I. The Blowing of Beed Pipes, 390
n. The Blowing of Flue Pipes, 394
[Additions by Translator, 396]
Vni. Practical Directions for Performing the Experiments on the Composition of Vowels,
398
IX. On the Phases of Waves caused by Resonance, 400
X. Belation between the Strength of Sympathetic Resonance and the Length of Time
required for the Tone to die away, 405
XI. Vibrations of the Membrana Basilaris in the Cochlea, 406
XIL Theory of Combinational Tones, 41 1
XIIL Description of the Mechanism employed for opening the several Series of Holes in
the Polyphonic Siren, 413
XrV. Variation in the Pitch of Simple Tones that Beat, 414
XV. Calculation of the Intensity of the Beats of Different Intervals, 41 5
XVL On Beats of Combinational Tones, and on Combinational Tones in the Siren and
Harmonium, 418
XVn. Plan for Justly-Toned Instruments with a Single Manual, 421
X7IIL Just Intonation in Singing, 422
XIX. Plan of Mr. Bosanquet*s Manual, 429
:XX. Additions by the Translator, 430-556
*»* See separate Tables of Contents prefixed to each Section.
[Sect. A. On Temperament, 430
[Sect. B. On the Determination of Pitch Numbers, 441
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xiv CONTENTS.
[App. XX- Additions by the Translator— con^ntt«(f.
*«* See separate Tables of Contents prefixed to each Section.
[Sect. G. On the Caloolation of Cents from Interval Batlos, 446
[S£CT. D. Musical Intervals, not exceeding an Octave, arranged in order of Width,
451
[Sect. E. On Musical Duodenes, or the Development of Just Intonation for
Harmony, 457
[Sect. F. Experimental Instruments for exhibiting the effects of Just Intonation,
466
[Sect. G. On Tuning and Intonation, 483
[Sect. H. The History of Musical Pitch in Europe, 493
[Sect. E. Non-Harmonic Scales, 514
[Sect. L. Becent Work on Beats and Combinational Tones, 527
[Sect. M. Analysis and Synthesis of Vowel Sounds, 538
[Sect. N. Miscellaneous Notes, 544
[INDEX, 557-576]
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XV
LIST OF FIGUEES.
13.
14.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30-
31.
32.
Seebeck*B Siren, I ic 33
3, 4. Cagniard de la Tour's Siren, 126
Tuning-fork tracing its Curve, 206
Gtirve traced in Phonautograph, 2od 34.
Curve of Simple Vibration, 216
Curve of Motion of Hammer moved by
Water-wheel, 21c
Curve of Motion of Ball struck up on 35.
its descent, 21c
Beprodaotion of fig. 7i 23(2 36.
Curve shewing the Composition of a
simple Note and its Octave in two
different phases, 30&, c 37-
Curve shewing the Composition of a
simple note and its Twelfth in two
different phases, 326 38.
Tuning-fork on Resonance Box, 40^
Forms of Vibration of a Circular Mem- 39.
brane, 40c, d
Pendulum excited by a membrane
covering a bottle, 42a 40.
a. Spherical Besonator, 436
b. Cylindrical Besonator, 43c 41.
Forms of Vibration of Strings, 46a, b 42.
Forms of Vibration of a String de- 43.
fleeted by a Point, 54^, 6
Action of such a String on a Sounding-
board, S4C 44.
Bottle and Blow-tube for producing a
simple Tone, 60c
Sand figures on circular elastic plates, 45.
71C
The Vibration Microscope, 816 46.
Vibrations as seen in the Vibration
Microscope, 826
Vibrational Forms for the middle of a 47*
Violin String, 836
Crumples on the vibrational form of a
violin string, 846 48.
Gradual development of Octave on a ' 49.
violin string bowed near the bridge, 50.
856 . s«-
An open wooden and stoppea metal 52.
organ flue-pipe, 88
Free reed or Harmonium vibrator, 956 53-
Free and striking reed on an organ
pipe partly in section, 96a, b 54-
Membranous double reed, 97a
Reproduction of fig. 12, 120a, 6 55-
Fork with electro-magnetic exciter, and
sliding resonance box with a lid 56.
(artificial vowels), 121b 57-
Fork with electro-magnet to serve as
interrupter of the current (artificial
vowels), 122&
Appearance of figures seen through the
vibration microscope by two forks
when the phase changes but the
tuning is correct, I26d
The same when the tuning is slightly
altered, 127a
Construction of the ear, general view,
with meatus auditorius, labyrinth,
cochlSa, and Eustachian tube, 129c
The three auditory ossicles, hammer,
anvil, and stirrup, in their relative
positions, 130c
Two views of the hammer of the ear,
1316
Left temporal bone of a newly- born
child with the auditory ossicles in
sitUy 13IC
Right drumskin with hammer seen
from the inside, 131c
Two views of the right anvil, 1330
Three views of the right stirrup, 134a
A, left labyrinth from without. B,
right labyrinth from within. C,
left labyrinth from above, 1366, c
Utriculus and membranous semicircular
canals (left side) seen from without,
137*
Bony cochlea (right side) opened in
front, 137c, d
Transverse section of a spire of a
cochlSa which has been softened
in hydrochloric acid, 138a, b
Max Schnitzels hairs on the internal
surface of the epithiUum in the
ampulla, 138c, d
Expansion of the cochlean nerve, 139c
Corti's membrane, 140a, 6, c
Corti's rods or arches separate, 140^2
Corti's rods or arches in situ, 141 6, c
Diagram of the law of decrease of sym-
pathetic resonance, 144c, d
Interference of similarly disposed
waves, 1606
Interference of dissimilarly disposed
waves, 1 60c
Lines of silence of a tuning-fork,
1616
The Polyphonic Siren, 162
Diagram of origin of beats, 165a
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XVI
LIST OP PASSAGES IN MUSICAL NOTES.
58. Phonautographic representation of
beats, i66a
59. Identical with fig. 52 bat now taken to
shew the intensity of beats excited
by tones making different intervals,
172c
60. A and B. Diagram of the comparative
ronghness of intervals in the first
and second octaves, 1936, c
61. Diagram of the roughness of dissonant
intervals, 333a
62. Reproduction of fig. 24 A, p. 3856
63. Diagram of the motion of a violin
string, 387c
64. Diagram of the arrangements for the
experiments on the composition of
vowels, 3996, c
65. Mechanism for opening the several
series of holes in the Polyphonic
Siren, 414a
66. Section, Elevation, and Plan of Mr.
Bosanqaet*s Manual, 429
In Additions by Translator.
67. Perspective view of Mr. Colin Brown's
Fingerboard, 47 id
68. Perspective view, 69 plan, 70 section
of Mr. H. W. Poole's Keyboard, 475
LIST OF PASSAGES IN MUSICAL NOTES.
The small octave, i$d
The once and twice accented octave, i6a, b
The great octave, 166
The first 16 Upper Partials of C66, 22c
The first 8 Upper Partials of C132, 50a
Prof. Helmholtz's Vowel Resonances, 1 106
First differential tones of the usual har-
monic interval, 1546
Differential tones of different orders of the
usual harmonic intervals, 1556, c
Smnmational tones of the usual harmonic
intervals, 156a
Examples of beating partials, i8oc
Coincident partials of the principal con-
sonant intervals, iSyi
Coincident converted into beating partials
by altering pitch of upper tone, i86c
Examples of intervals in which a pair of
partials beat 33 times in a second, 192a
Major Triads with their Combinational
Tones, 215a
Minor Triads with their Combinational
Tones, 2156
Consonant Intervals and their Combina-
tional Tones, 218c
The most Perfect Positions of the Major
Triads, 219c
The less Perfect Positions of the Major
Triads, 220c
The most Perfect Positions of the Minor
Triads, 22 1&
The less Perfect Positions of the Minor
Triads, 221c
The most Perfect Positions of Major
Tetrads within the Compass of Two
Octaves, 223c
Best Positions of Minor Tetrads with their
false Combinational Tones, 224a
Ich bin spatzieren gegangen^ 2386
Sic canta commat 2396
Palestrina's StabcU Mater, first 4 bars,
247c
Chinese air after Barrow, 260a
Cockle ShellSt older form, 260&
Blythe, blythe, and merry are we, 261a
Chinese temple hymn after Bitschurin, 26 1 6
Bnus of Balqvhidder, 261c
Five forms of Closing Chords, 291c
Two complete closes, 293c
Mode of the Fourth, three forms of com-
plete cadence, 302^
Concluding bars of S. Bach's Chorale, Was
mevn Qott toiU, das gescheh* alUeit, 304&
End of S. Bach's Hymn, Vem redemptor
gentium, 305a
Doric cadence from And with His stripes
toe are healed, in Handel's Messiah, 307a
Doric cadence from Hear, Jacobus Qod, in
Handel's Samson, 3076
Examples of False Minor Triad, 340a
Examples of Hidden Fifths, 36ic2
Example of Duodenals, 465c
Mr.'H. W. Poole's method of fingering and
treatment of the harmonic Seventh, 477.-1
Mr. H. W. Poole's Double Diatonic or Di-
chordal Scale in 0 with accidentals, 478a
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XVll
LIST OF TABLES.
Pitch Nambers of Notes in Just Major
Scftle, 17a
[Scale of Harmonical, 17c, d]
[Analogies of notes of the piano and colours
of the Spectrum, iSd']
Pitch of the different forms of vibration
of a circular membrane, 41c
Relative Pitch Numbers of the prime and
proper tones of a rod free at both ends,
56a
Proper Tones of circular elastic plates, 72a
Proper Tones of Bells, 72c
Proper Tones of Stretched Membranes, 736
Theoretical Intensity of the Partial Tones
of Strings, 79c
[Velocity in Sound in tubes of different
diameters — Blaikley, qod]
[Ptrtials of Bb Clarinet— Blaikley, 99c]
[Harmonics oi Eb horn, 99^]
[Compass of Begisters of male and female
voices— Behnke, loi^
Vowel trigram— Du Bois Baymond, senior,
1056
Vowel Resonances according to Helmholtz
and Bonders, 1096
[Vowel Resonances according to (i) Reyher,
(2)Hellwag, (3) Fldrcke, (4) Bonders after
Helmholtz, (5) Bonders after Merkel,
(6) Hehnholtz, (7) Merkel, (8) Koenig,
(9) Trautmann, logd]
Willis's Vowel Resonances, 117c
[Relative force of the partials for producing
different vowels, 124^]
Relation of Strength of Resonance to
Alterations of Phase, 125a
Bifference of pitch, Ac, necessary to reduce
sympathetic vibration to ^ of that pro-
duced by perfect unisonance, 143a
Nambers from which fig. 52 was constructed,
145a
Measurements of the basilar membrane in
a new-born child, 145c
Alteration of size of Corti's rods as they
approach the vertex of the cochlga, i^S'l
[Preyer's distinguishable and undistinguish -
able intervals, I47d]
First differential tones of the usual har-
monic intervals, 1 54a
'Differential tones of different orders of the
usual harmonic intervals, I55<f]
Different intervals which would give 33
beats of their primes, 1720
[Pitch numbers of Appunn's bass reeds,
1776]
[Experiments on audibility of very deep
tones, 177c]
Coincident partials for the principal con-
sonances, 183a
Pitch numbers of the primes which make
consonant intervals with a tone of 300
vib., 184c
Beating partials of the notes in the last
table with a note of 301 vib., and number
of beats, 184^
Disturbance of a consonance by altering
one of its tones by a Semitone, 185c
Influence of different consonances on each
other, 1876
[Upper partials of a just Fifth, iSSd]
[Upper partials of an altered Fifth, 189c]
[Comparison of the upper partials of a
Fourth and Eleventh, major Sixth and
major Thirteenth, minor Sixth and
minor Thirteenth, 189^ and 1906, c]
[Comparison of the upper partials of a
major and a minor Third, 190^]
[Comparison of the upper partials of all
the usual consonances, pointing out
those which beat, 191 6, c]
[Comparison of the upper partials of
septimal consonances, involving the
seventh partial, and pointing out which
beat, 195c, d]
[General Table of the first 16 harmonics of
C66, shewing how they affect each other
in any combination, 197c, d]
Table of partials of 200 and 301, shewing
their differential tones, 198c
Table of possible triads, shewing consonant,
dissonant, and septimal intervals, 2126, c
Table of consonant triads, 214a
[The first 16 harmonics of C, 21 4(2]
[Calculation of the Combinational Tones of
the Major Triads, 21 ^d]
[Most of the first 40 harmonics of ^,, b , 2 1 5cl
[Calculation of the Combinational Tones of
the Minor Triads, 215^]
[Calculation of the Differential Tones of
the Major Triads in their most Perfect
Positions, 219^2]
[Calculation of the Combinational Tones
of the Major Triads in the less Perfect
Positions, 22od]
[Calculation of the Combinational Tones of
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XVUl
LIST OF TABLES.
the Minor Triads in the most and less
Perfect Positions of the Minor Triads,
22 Id, d']
[Calculation of the false Combinational
Tones of Minor Tetrads in their best
positions, 224^]
Ecclesiastical Modes, 2450, d
Partial Tones of the Tonic, 257a
[Pentatonic Scales, 259c, d]
[Tetraohords i to 8, with intervals in
cents, 263d']
Greek Diatonic Scales, 267c
[Greek Diatonic Scales with the intervals
in cents, 268c]
[Greek Diatonic Scales reduced to begin-
ning with e, with the intervals in cents,
268^1
Greek modes with the Greek Ecclesiastical
and Helmholtzian names, 269a
Later Greek Scale, 270a
Tonal Keys, 270c
Ecclesiastical Scales of Ambrose of Milan,
2716, c
The Five Melodic Tonal Modes, 2726
[The Seven Ascending and Descending
Scales, compared with Greek, with inter-
vals in cents, 274c, d] .
[The different scales formed by a dif-
ferent choice of the intercalary tones,
277c', d']
The Five Modes with variable intercalary
tones, 278a, b
[J. Curwen's characters of the tones in
the major scale, 2796, c]
[Arabic Scale in relation to the major
Thirds, 28id']
Arabic Scales, 282&-283C
[Prof. Land's account of the 12 Arabic
Scales, 284 note]
Five Modes as formed from three chords
each, 293d, 294a
The same with double intercalary tones,
297c, d
The same, final form, 2986, c
Trichordal Relations of the Tonal Modes,
sogd
[Thirds and Sixths in Just, Equal, and
Pythagorean Intonation compared, 313c]
[Combinational Tones of Just, Equal, and
Pythagorean Intonation compared, 314^]
The Chordal System of Prof. Heimholtz's
Just Harmonium, 316c
[Duodenal y statement of the tones on Prof.
Heimholtz's Just Harmonium, 317c, d]
The Chordal System of the minor keys
on Prof. Heimholtz's Just Harmonium,
318a, 6, d
[Table of the relation of the Cycle of 53 to
Just Intonation, 3296, c]
Tabular Expression of the Diagram, fig.
bi,332]
'Table of Koughnesi^, 333^'.
Measurements of Glass Resonators, 373c
Measurements of resonance tubes men-
tioned on p. 55a, 37 7d
Table of tones of a conical pipe of zinc,
calculated from formula 393c [with sub-
sidiary tables, 393d, and 394c]
Table of Mayer's observations on numbers
of beats, 418a
Table of four stops for a single manoal
justly intoned instrument, 421c
Table of five stops for the same, 422a
In the Additions by Translator.
Table of Pythagorean Intonation, 4336, e
Table of Meantone Intonation, 4346
Table of Equal Intonation, 437c, d
Synonymity of Eqyal Temperament, 4386
Synonymity of Mr. Bosanquet's Notes in
Fifths, 439«
Notes of Mr. Bosanquet's Cycle of 53 in
order of Pitch, 4396, c, d
Expression of Just Intonation in the Cyde
of 1200, p. 440
Principal Table for calculation of cents,
450a, Auxiliary Tables, 451a
Table of Intervals not exceeding one Octave,
4536
Unevenly numbered Harmonics up to the
63rd, 457a
Number of any Interval not exceeding a
Tritone, contained in an Octave, 457c
Harmonic Duodene or Unit of Modulation,
461a
The Duodenarlum, 463a
Fingerboard of the Harmonical, first four
Octaves, with scheme, 4676, fifth Octave,
468d
Just Harmonium scheme, 470a
Just English Concertina scheme, 4706
Mr. Colin Brown's Voice Harmonium
Fingerboard and scheme, 471a
Rev. Henry Liston's Organ and scheme,
473*
Gen. Perronet Thompson's Organ scheme,
473^
Mr. H. Ward Poole's 100 tones, 474c
Mr. H. W. Poole's scheme for keys of F,
C. G, 476^*
Mr. Bosanquet's Generalised Keyboard,
480
Expression of the degrees of the 53 divi<
sion by multiples of 2, 5 and 7, p. 481c
Typographical Plan of Mr. J. Paul White'i
Fingerboard, 4826
Specimens of tuning in Meantone Tern
perament, 484c
Specimens of tuning in Equal Tem^Kra
ment, 4856
Pianoforte Tuning- Fourths and Fifths,
485d
Cornu and Mercadirr's observaliou on
Violin Intonation, 4S6f lo 4X76
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LIST OF TABLES.
XIX
Scheme for Tuning in Eqaal Temperament,
4896
Proof of rule for Tuning in Equal Tempera-
ment, 490c, d
Proof of rule for Tuning in Meantone Tem-
perament, 492a
Historical Pitches in order from Lowest to
Highest, 495a to 504a
Classified Index to the last Table, 5046 to
5116
Effects of the length of the foot in differ-
ent countries on the pitch of organs,
Non-harmonic scales, 514c to 519c
Vowel sound ' Oh ! ' Analysis at Tarious
pitches by Messrs. Jenkin <& Ewing, $^gd
to 541 &
Vowel sounds *oo,' 'awe,' *ah,' analysis
at various pitches by Messrs. Jenkin
and Ewing, p. ^ic, d
Mean and actual Compass of the Human
Voice, 545a, 6, c
True Tritonic, False Tritonic, Zarlino*s,
Meantone and Equal Temperaments,
compared, 548a
Presumed Characters of Major and Minor
Keys, 551a, 6
Corrigenda.
v. loid, note, line la from bottom, /<»• i. Upper thick read x. Lower thick.
P. 1396 and eUevhtrt, cochlean/or cochlear U intentional,
P. aSad, note, line 10 from bottom, after 70*6 cents, omit the remainder of the paragraph,
and read For the possible origin of Villoteau's error see in/Hi p. 5ao6 to saod'.
P. 339^', note Xt liae >7 txova. bottom, /or No. 6 read No. 7.
P. 356c, lines 15 and x6 from bottom of text, for o* b — c + ** b read a»b + c — tf» b .
P. 356*1, line 4 from bottom of text, tat c — e^ — g read c + *, — jr.
P. 477, milsic, line a. bar a, dele the reference number 8, and the corresponding note below.
P. 478d', last words of lines 7 and 5 from bottom. /or lightly and bad read tightly and best.
The passage trill thertifore read These 34 lerers are a qoarter of an inch wide, and can
play a pianoforte with hammers half the common width, with single strings, but
larger and tightly strained, so as to yield the maximum tone, tension nearly to
breaking point giving the best tone.'
P. 501, col. 1, for 300 cents read 330 cents.
P. 5X9e, No. X30, /or reosen read riosen, and for additional information on Japanese Scales
generally^ see p. 556.
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INTEODUCTION.
In the present work an attempt will be made to connect the boundaries
of two sciences, which, although drawn towards each other by many
natural aflSnities, have hitherto remained practically distinct — I mean the
boundaries of physical and physiological acoustics on the one side, and of
Jmiisical science and esthetics on the other. The class of readers addressed
will, consequently, have had very different cultivation, and will be affected
by very different interests. It will therefore not be superfluous for the
author at the outset distinctly to state his intention in undertaking the
work, and the aim he has sought to attain. The horizons of physics,
philosophy, and art have of late been too widely separated, and, as a
consequence, the language, the methods, and the aims of any one of these
studies present a certain amount of diflSculty for the student of any other %
of them ; and possibly this is the principal cause why the problem here
undertaken has not been long ago more thoroughly considered and advanced
towards its solution.
It is true that acoustics constantly employs conceptions and names
bcMTowed from the theory of harmony, and speaks of the * scale,' * intervals,'
•consonances,' and so forth; and similarly, manuals of Thorough Bass
generally begin with a physical chapter which speaks of * the numbers of
vibrations,' and fixes their * ratios ' for the different intervals ; but, up to
the present time, this apparent connection of acoustics and music has been
wholly external, and may be regarded rather as an expression given to the
feeling that such a connection must exist, than as its actual formulation.
Physical knowledge may indeed have been useful for musical instrument
makers, but for the development and foundation of the theory of harmony %
it has hitherto been totally barren. And yet the essential facts within the
field here to be explained and turned to accoimt, have been known from the
earliest times. Even Pythagoras (fl. circa b.c. 540-510) knew that when
strings of different lengths but of the same make, and subjected to the
same tension, were used to give the perfect consonances of the Octave,
Fifth, or Fourth, their lengths must be in the ratios of i to 2, 2 to 3, or
3 to 4 respectively, and if, as is probable, his knowledge was partly derived
from the Egyptian priests, it is impossible to conjecture in what remote
antiquity this law was first known. Later physics has extended the law of
Pythagoras by passing from the lengths of strings to the number of vibra-
^ons, and thus making it applicable to the tones of all musical instruments,
and the numerical relations 4 to 5 and 5 to 6 have been added to the above
Digitized by Vj£>OQ IC
t PLAN OF THE WORK. introd.
for the less perfect consonances of the major and mmor Thirds, but I am
not aware that any real step was ever made towards answering the ques-
tion : What haue musical consonances to do with the ratios of the first six
numbers ? Musicians, as well as philosophers and physicists, have generally
contented themselves with saying in effect that human minds were in some
unknown manner so constituted as to discover the numerical relations of
musical vibrations, and to have a peculiar pleasure in contemplating simple
ratios which are readily comprehensible.
Meanwhile musical esthetics has made unmistakable advances in those
points which depend for their solution rather on psychological feeling than
on the action of the senses, by introducing the conception of movement in
^ the examination of musical works of art. E. Hanslick, in his book * on the
Beautiful in Music' (Ueber das mvsikalisch Schone), triumphantly attacked
the false standpoint of exaggerated sentimentality, from which it was
fashionable to theorise on music, and referred the critic to the simple
elements of melodic movement. The esthetic relations for the structure of
musical compositions, and the characteristic differences of individual forms
of composition, are explained more fully in Vischer's * Esthetics ' (Aesthetik).
In the inorganic world the kind of motion we see, reveals the kind of moving
force in action, and in the last resort the only method of recognising and
measuring the elementary powers of nature consists in determining the
motions they generate, and this is also the case for the motions of bodies
or of voices which take place under the influence of human feelings. Hence
^ the properties of musical movements which possess a graceful, dallying, or
a heavy, forced, a dull, or a powerful, a quiet, or excited character, and so
on, evidently chiefly depend on psychological action. In the same way
questions relating to the equilibrium of the separate parts of a musical
composition, to their development from one another and their connection
as one clearly intelligible whole, bear a close analogy to similar questions in
architecture. But all such investigations, however fertile they may have
been, cannot have been otherwise than imperfect and uncertain, so long as
they were without their proper origin and foundation, that is, so long as
there was no scientific foundation for their elementary rules relating to the
construction of scales, chords, keys and modes, in short, to all that is
usually contained in works on ' Thorough Bass.' In this elementary region
^ we have to deal not merely with unfettered artistic inventions, but with the
natural power of immediate sensation. Music stands in a much closer
connection with pure sensation than any of the other arts. The latter
rather deal with what the senses apprehend, that is with the images of
outward objects, collected by psychical processes from immediate sensation.
Poetry aims most distinctly of all at merely exciting the formation of
images, by addressing itself especially to imagination and memory, and it
is only by subordinate auxiliaries of a more musical kind, such as rhythm,
and imitations of sounds, that it appeals to the immediate sensation of
hearing. Hence its effects depend mainly on psychical action. The pUisti^;
arts, although they make use of the sensation of sight, address the eye
almost in the same way as poetry addresses the ear. Their main purpose
8 to excite in us the image of an external object of determinate form and
colour. The spectator is essentially intended to interest himself in this
Digitized by V^jOOQlC
iNTROD. PLAN OF THE WORK. 3
image, and enjoy its beauty ; not to dwell upon the means by which it was
created. It must at least be allowed that the pleasure of a connoisseur or
virtuoso in the constructive art shewn in a statue or a picture, is not an
essential element of artistic enjoyment.
It is only in painting that we find colour as an element which is directly j ^ - ^*
appreciated by sensation, without any intervening act of the intellect. On
the contrary, in mvmc, the sensations of tone are the material of the art.
So far as these sensations are excited in music, we do not create out of
them any images of external objects or actions. Again, when in hearing a
concert we recognise one tone as due to a violin and another to a clarinet,
our artistic enjoyment does not depend upon our conception of a violin or
clarinet, but solely on our hearing of the tones they produce, whereas the ^
artistic enjoyment resulting from viewing a marble statue does not depend
on the white light which it reflects into the eye, but upon the mental image
of the beautiful human form which it calls up. In this sense it is clear that '
music has a more immediate connection with pure sensation than any other
of the fine arts, and, consequently, that the theory of the sensations of
hearing is destined to play a much more important part in musical esthetics,
than, for example, the theory of chiaroscuro or of perspective in painting.
Those theories are certainly useful to the artist, as means for attaining the
most perfect representation of nature, but they have no part in the artistic
effect of his work. In music, on the other hand, no such perfect represen-
tation of nature is aimed at ; tones and the sensations of tone exist for
themselves alone, and produce their effects independently of anything behind %
them.
This theory of the sensations of hearing belongs to natural science, and
comes in the first place under jyhysiological acoustics. Hitherto it is the
physical part of the theory of sound that has been almost exclusively treated
at length, that is, the investigations refer exclusively to the motions produced
by solid, liquid, or gaseous bodies when they occasion the sounds which the
ear appreciates. This physical acoustics is essentially nothing but a section
of the theory of the motions of elastic bodies. It is physically indifferent
whether observations are made on stretched strings, by means of spirals of
bra«8 wire, (which vibrate so slowly that the eye can easily follow their
motions, and, consequently, do not excite any sensation of sound,) or by
means of a violin string, (where the eye can scarcely perceive the vibrations ^
which the ear readily appreciates). The laws of vibratory motion are pre-
cisely the same in both cases ; its rapidity or slowness does not affect the
laws themselves in the slightest degree, although it comjiels the observer to
apply different methods of observation, the eye for one and the ear for
the other. In physical acoustics, therefore, the phenomena of hearing are
taken into consideration solely because the ear is the most convenient and
handy means of observing the more rapid elastic vibrations, and the physicist
is compelled to study the peculiarities of the natural instrument which he is
employing, in order to control the correctness of its indications. In this
way, although physical acoustics as hitherto pursued, has, undoubtedly,
collected many observations and much knowledge concerning the action of
the ear, which, therefore, belong to physiological acoustics, these results wore
not the principal object of its investigations ; they were merely secondary
Digitized by ^^0Ogle
4 PLAN OF THE WOKK. introd.
and isolated facts. The only justification for devoting a separate chapter
to acoustics in the theory of the motions of elastic bodies, to which it
essentially belongs, is, that the application of the ear as an instrument
of research influenced the nature of the experiments and the methods of
observation.
But in addition to a physical there is a physiological theory of acoiisticsy
the aim of which is to investigate the processes that take place within the
ear itself. The section of this science which treats of the conduction of the
motions to which sound is due, from the entrance of the external ear to the
expansions of the nerves in the labyrinth of the inner ear, has received
much attention, especially in Germany, since ground was broken by
H Johannes Mueller. At the same time it must be confessed that not many
results have as yet been established with certainty. But these attempts
attacked only a portion of the problem, and left the rest untouched.
Investigations into the processes of each of our organs of sense, have in
general three different parts. First we have to discover how the agent
reaches the nerves to be excited, as light for the eye and sound for the ear.
This may be called the physical part of the corresponding physiological
investigation. Secondly we have to investigate the various modes in which
the nerves themselves are excited, giving rise to their various scfisations^
and finally the laws according .to which these sensations result in mental
images of determinate external objects, that is, in perceptions. Hence we
have secondly a specially physiological investigation for sensations, and
H thirdly, a specially psychological investigation for perceptions. Now whilst
the physical side of the theory of hearing has been already frequently '
attacked, the results obtained for its physiological and psychologix:al sections
are few, imperfect, and accidental. Yet it is precisely the physiological part
in especial — the theory of the sensations of hearing — to which the theory
of music has to look for the foundation of its structure.
In the present work, then, I have endeavoured in the first place to collect
and arrange such materials for a theory of the sensations of hearing as ah*eady
existed, or as I was able to add from my own personal investigations. Of
course such a first attempt must necessarily be somewhat imperfect, and be
hmited to the elements and the most interesting divisions of the subject
discussed. It is in this light that I wish these studies to be regarded.
IF Although in the propositions thus collected there is little of witirely new
discoveries, and although even such apparently new facts and observations
as they contain are, for the most part, more properly speaking the imme-
diate consequences of my having more completely carried out known
theories and methods of investigation to their legitimate consequences, and
of my having more thoroughly exhausted their results than had heretofore
been attempted, yet I cannot but think that the facts frequently receive new
importance and new illumination, by being regarded from a fresh point of
view and in a fresh connection.
The First Part of the following investigation is essentially physical and
physiological. It contains a general investigation of the phenomenon of
harmonic upper partial tones. The nature of this phenomenon is established,
and its relation to quality of tone is proved. A series of qualities of tone are
analysed in respect to their harmonic upper partial tones, and it results
Digitized by V^jOOQlC
ixTBOD. PLAN OF THE WORK. 5
that these upper partial tones are not, as was hitherto thought, isolated
phenomena of small importance, but that, with very few exceptions, they
determine the qualities of tone of almost all instruments, and are of the
greatest importance for those qualities of tone which are best adapted for
musical purposes. The question of how the ear is able to perceive these '
harmonic upper partial tones then leads to an hypothesis respecting the
mode in which the auditory nerves are excited, which is well fitted to reduce
all the facts and laws in this department to a relatively simple mechanical
conception.
The Second Part treats of the disturbances produced by the simultaneous
production of two tones, namely the combinational tones and beats* The
physiologico-phy^sical investigation shews that two tones can be simul- H
taneously heard by the ear without mutual disturbance, when and onl)^
when they stand to each other in the perfectly determinate and well-known co^'*^'^
relations of intervals which form musical consonance. We are thus imme- ,
diately introduced into the field of music proper, and are led to discover
the physiological reason for that enigmatical numerical relation announced
by Pythagoras. The magnitude of the consonant intervals is independent
of the quality of tone, but the harmoniousness of the consonances, and the
distinctness of their separation from dissonances, depend on the quality of
tone. The conclusions of physiological theory here agree precisely with the
musical rules for the formation of chords ; they even go more into par-
ticulars than it was possible for the latter to do, and have, as I believe, the
authority of the best composers in their favour. f
In these first two Parts of the book, no attention is paid to esthetic
considerations. Natural phenomena obeying a blind necessity, are alone
treated. The Third Part treats of the construction of musical scab's and
Kott's. Here we come at once upon esthetic ground, and the diflfcrcnce« of ^
national and individual tastes begin to appear. Modern music has especially |
developed the principle of tonality, which connects all the tones in a piece
of music by their relationship to one chief tone, called the tonic. On
admitting this principle, the results of the preceding investigations furnish
a method of constructing our modern musical scales and modes, from which
all arbitrary assumption is excluded.
I was unwilling to separate the physiological investigation from its
musical consequences, because the correctness of these consequences must %
be to the physiologist a verification of the correctness of the physical and
physiological views advanced, and the reader, who takes up my book for its
musical conclusions alone, cannot form a perfectly clear view of the meaning
and bearing of these consequences, unless he has endeavoured to get at
least some conception of their foundations in natural science. But in order
to facilitate the use of the book by readers who have no special knowledge
of physics and mathematics, I have transferred to appendices, at the end
of the book, all special instructions for performing the more complicated
experiments, and also all mathematical investigations. These appendices
are therefore especially intended for the physicist, and contain the proofs
of my assertions.* In this way I hope to have consulted the interests of
both classes of readers.
* ^The additional Appendix XX. by the Translator is intended especially for the use of t •
mn?ica: %tr\dentsi.— Translator.] Digitized by V^OOQ IC
««v»t«
6 PLAN OF THE WORK. intkod.
It is of course impossible for any one to understand the investigations
thoroughly, who does not take the trouble o( becoming acquainted by per-
sonal observation with at least the fundamental phenomena mentioned.
Fortunately with the assistance of common musical instruments it is easy
for any one to become acquainted with harmonic upper partial tones, com-
binational tones, beats, and the like.* Personal observation is better than
the exactest description, especially when, as here, the subject of investiga-
tion is an analysis of sensations themselves, which are always extremely
difficult to describe to those who have not experienced them.
In my somewhat unusual attempt to pass from natural philosophy into
the theory of the arts, I hope that I have kept the regions of physiology
f and esthetics sufficiently distinct. But I can scarcely disguise from myself,
that although my researches are confined to the lowest grade of musical
grammar, they may probably appear too mechanical and unworthy of the
dignity of art, to those theoreticians who are accustomed to summon the
enthusiastic feelings called forth by the highest works of art to the scientific
investigation of its basis. To these I would simply remark in conclusion,
that the following investigation really deals only with the analysis of
actually existing sensations — that the physical methods of observation
employed are almost solely meant to facilitate and assure the work of this
analysis and check its completeness — and that this analysis of the sensations
would suffice to furnish all the results required for musical theory, even
independently of my physiological hypothesis concerning the mechanism of
f hearing, already mentioned (p. 5^), but that I was unwilling to omit that
hypothesis because it is so well suited to furnish an extremely simple con-
nection between all the very various and very complicated phenomena
which present themselves in the course of this investigation.")-
* [But the use of the flarmontcaZ, described London, Macmillan, 1873. Such readers will
in App. XX. sect. F. No. i, and invented for also find a clear exposition of the physical
the purpose of illustrating the theories of this relations of sound in J. Tyndall, On Somuit
work, is recommended as greatly superior for a course of eight lectures, London, 1867, (the
students and teachers to any other instrument. last or fourth edition 1883) Longmans, Green,
— Translator.] & Co. A German translation of this work,
f Headers unaccustomed to mathematical entitled Der Scliall, edited by H. Helmholtz
and physical considerations will find an and G. Wiedemann, was published at Bruns-
abridged account of the essential contents of wick in 1874.
this book in Sedley Taylor, Sound aiid Music^
*^* [The marks f in the outer margin of each page, separate the page into
4 sections, referred to as a, 6, c, dj placed after the number of the page. If aiiy
section is in double columns, the letter of the second column is accented, as
p. 13d',]
Digitized by V^jOOQlC
PART I.
ON THE COMPOSITION OF VIBRATIONS.
UPPER PARTIAL TONES, AND QUALITIES OF TONE.
CHAPTER L
ON THE SENSATION OF SOUND IN GENERAL.
Sensations result from the action of an external stimulus on the sensitive apparatus
of our nerves. Sensations differ in kind, partly with the organ of sense excited,
and partly with the nature of the stimulus employed. Each organ of sense pro-
duces peculiar sensations, which cannot be excited by means of any other ; the
eye gives sensations of light, the ear sensations of soimd, the skin sensations of
touch. Even when the same sunbeams which excite in the eye sensations of light,
impinge on the skin and excite its nerves, they are felt only as heat, not as light. %
In the same way the vibration of elastic bodies heard by the ear, can also be felt
by the skin, but in that case produce only a whirring fluttering sensation, not
soimd. ^ The sensation of sound is therefore a species of reaction against external
stimulus, peculiar to the ear, and excitable in no other organ of the body, and is
completely distinct from the sensation of any other sense.
As our problem is to study the laws of the sensation of hearing, our first
business will be to examine how many kinds of sensation the ear can generate, and
what differences in the external means of excitement or sound, correspond to these
differences of sensation.
The first and principal difference between various sounds experienced by our ear,
is that between noises and mtisical tones. The soughing, howling, and w^histling
of the wind, the splashing of water, the rolling and rumbling of carriages, are
examples of the first kind, and the tones of all musical instruments of the second.
Noises and musical tones may certainly intermingle in very various degrees, and H
pass insensibly into one another, but their extremes are widely separated.
The nature of the difference between musical tones and noises, can generally
be determined by attentive aural observation without artificial assistance. We
perceive that generally, a noise is accompanied by a rapid alternation of different
kinds of sensations of sound. Think, for example, of the rattling of a carriage
over granite paving stones, the splashing or seething of a waterfall or of the waves
of the sea, the rustling of leaves in a wood. In all these cases we have rapid,
irregular, but distinctly perceptible alternations of various kinds of sounds, which
crop up fitfully. When the wind howls the alternation is slow, the sound slowly
and gradually rises and then falls again. It is also more or less possible to separate
restlessly alternating sounds in case of the greater number of other noises. We
shall hereafter become acquainted with an instrument, called a resonator, which
will materially assist the ear in making this separation. On the otlier liand, a
musical tone strikes the ear as a perfectly undisturbed, uniform sound which
Digitized by V^OOQIC
8 » NOISE AND MUSICAL TONE. part i.
rRmains unaltered as long as it exists, and it presents no alternation of various
kinds of constituents. To this then corresponds a simple, regular kind of sensation,
whereas in a noise many various sensations of musical tone are irregularly mixed
up and as it were tumbled about in confusion. We can easily compound noises
out of musical tones, as, for example, by simultaneously striking all the keys con-
tained in one or two octaves of a pianoforte. This shews us that musical tones
are the simpler and more regular elements of the sensations of hearing, and that
we have consequently first to study the laws and peculiarities of this class of
sensations.
Then comes the further question : On what difference in the external means of
excitement does the difference between noise and musical tone depend ? The
normal and usual means of excitement for the human ear is atmospheric vibration.
The irregularly alternating sensation of the ear in the case of noises leads us to
^ conclude that for these the vibration of the air must also change irregularly. For
musical tones on the other hand we anticipate a regular motion of the air, con-
tinuing uniformly, and in its turn excited by an equally regular motion of the
sonorous body, whose impulses were conducted to the ear by the air.
Those regular motions which produce musical tones have been exactly investi-
gated by physicists. They are oscillations, vibrations, or swings, that is, up and
down, or to and fro motions of sonorous bodies, and it is necessary that these
oscillations should be regularly periodic. By a periodic motion we mean one which
constantly returns to the same condition after exactly equal intervals of time. The
ength of the equal intervals of time between one state of the motion and its next
exact repetition, we call the length of the oscillation vibration or swing, or the
period of the motion. In what manner the moving body actually moves during
one period, is perfectly indifferent. As illustrations of periodical motion, take the
motion of a clock pendulum, of a stone attached to a string and whirled round in
^ a circle with uniform velocity, of a hammer made to rise and fall uniformly by its
connection with a water wheel. All these motions, however different be their
form, are periodic in the sense here explained. The length of their periods, which
in the cases adduced is generally from one to several seconds, is relatively long in
comparison with the much shorter periods of the vibrations producing musical
tones, the lowest or deepest of which makes at least 30 in a second, while in other
cases their number may increase to several thousand in a second.
Our definition of periodic motion then enables us to answer the question pro-
posed as follows :-iT/i6 sensation of a musical tone is due to a rapid periodic
motion of the sonorous body ; tJie sensation of a noise to non-periodic motions.
The musical vibrations of solid bodies are often visible. Although they may
be too rapid for the eye to follow them singly, we easily recognise that a sounding
string, or tuning-fork, or the tongue of a reed-pipe, is rapidly vibrating between two
fixed limits, and the regular, apparently immovable image that we see, notwith-
^ Ktanding the real motion of the body, leads us to conclude that the backward and
forward motions are quite regular. In other cases we can feel the swinging motions
of sonorous sohds. Thus, the player feels the trembling of the reed in the mouth-
piece of a clarinet, oboe, or bassoon, or of his own lips in the mouthpieces of
trumpets and trombones.
The motions proceeding from the sounding bodies are usually conducted to our
ear by means of the atmosphere. The particles of air must also execute periodi-
cally recurrent vibrations, in order to excite the sensation of a musical tone m our
ear. This is actually the case, although in daily experience sound at first seems
to be some agent, which is constantly advancing through the air, and propagating
itself further and further. We must, however, here distinguish between the motion
of the individual particles of air — which takes place periodically backwards and
forwards within very narrow hmits — and the propagation of the sonorous tremor.
The latter is constantly advancing by the constant attraction of fresh particles into
its spiicre of tremor.
Digitized by VjOOQIC
CHAP. 1. PROPAGATION OP SOUND. 9
This is a peculiarity of all so-called undulatory moHons. Suppose a stone to
be thrown into a piece of calm water. Bound the spot struck there forms a little
ring of wave, which, advancing equally in all directions, expands to a constantly
increasing circle. Corresponding to this ring of wave, soimd also proceeds in the
air from the excited point and advances in all directions as far as the limits of the
mass of air extend. The process in the air is essentially identical with that on the
surface of the water. The principal difference consists in the spherical propagation
of sound in all directions through the atmosphere which fills all surrounding space,
whereas the waves of the water can only advance in rings or circles on its surface.
The crests of the waves of water correspond in the waves of sound to spherical
shells where the air is condensed, and the troughs to shells where it is rarefied.
On the free surface of the water, the mass when compressed can slip upwards and
so form ridges, hut in the interior of the sea of air, the mass must be condensed,
as there is no unoccupied spot for its escape. II
The waves of water, therefore, continually advance without returning. But
we must not suppose that the particles of water of which the waves are composed
advance in a similar manner to the waves themselves. The motion of the particles
of water on the surface can easily be rendered visible by floating a chip of wood
upon it. This will exactly share the motion of the adjacent particles. Now, such
a chip is not carried on by the rings of wave. It only bobs up and down and
finally rests on its original spot. The adjacent particles of water move in the same
manner. When the ring of wave reaches them they are set bobbing ; when it has
passed over them they are still in their old place, and remain there at rest, while
the ring of wave continues to advance towards fresh spots on the surface of the
water, and sets new particles of water in motion. Hence the waves which pass
over the surface of the water are constantly built up of fresh particles of water.
What really advances as a wave is only the tremor, the altered form of the surface,
while the individual particles of water themselves merely move up and down %
transiently, and never depart far from their original position.
The same relation is seen still more clearly in the waves of a rope or chain.
Take a flexible string of several feet in length, or a thin metal chain, hold it at one
end and let the other hang down, stretched by its own weight alone. Now, move
the hand by which you hold it quickly to one side and back again. The excursion
which we have caused in the upper end of the string by moving the hand, will run
down it as a kind of wave, so that constantly lower parts of the string will make a
sidewards excursion while the upper return again into the straight position of rest.
But it is evident that while the wave runs down, each individual particle of the
string can have only moved horizontally backwards and forwards, and can have
taken no share at all in the advance of the wave.
The experiment succeeds still better with a long elastic line, such as a thick
piece of india-rubber tubing, or a brass- wire spiral spring, from eight to twelve feet
in length, fastened at one end, and slightly stretched by being held with the hand «|
at the other. The hand is then easily able to excite waves which will run very
regularly to the other end of the line, be there reflected and return. In this case
it is also evident that it can be no part of the line itself which runs backwards and
forwards, but that the advancing wave is composed of continually fresh particles
of the line. By these examples the reader will be able to form a mental image of
the kind of motion to which sound belongs, where the material particles of the
body merely make periodical oscillations, while the tremor itself is constantly
propagated forwards.
Now let us return to the surface of the water. We have supposed that one of
its points has been struck by a stone and set in motion. This motion has spread
out in the form of a ring of wave over the surface of the water, and having reached
the cliip of wood has set it bobbing up and down. Hence by means of the wave,
the motion which the stone first excited in one point of the surface of the water
has been communicated to the chip which was at another point bf the same surface.
Digitized by V^jOOQ IC
lo FORCE, PITCH, AND QUALITY. part i.
The process which goes on in the atmospheric ocean about us, is of a precisely
similar nature. For the stone substitute a sounding body, which shakes the air ;
for the chip of wood substitute the human ear, on which impinge the waves of air
excited by the shock, setting its movable parts in vibration. The waves of air
proceeding from a sounding body, transport the tremor to the human ear exactly
in the same way as the water transports the tremor produced by the stone to the
floating chip.
In this way also it is easy to see how a body which itself makes periodical
oscillations, will necessarily set the particles of air in periodical motion. A falling
stone gives the surface of the water a single shock. Now replace the stone by a
regular series of drops falling from a vessel with a smaU orifice. Every separate
drop will excite a ring of wave, each ring of wave will advance over the surface of
the water precisely like its predecessor, and will be in the same way followed by
V its successors. In this manner a regular series of concentric rings wiU be formed
and propagated over the surface of the water. The number of drops which fall
into the water in a second will be the number of waves which reach our floating
chip in a second, and the number of times that this chip will therefore bob up and
down in a second, thus executing a periodical motion, the period of which is equal
to the interval of time between the falling of consecutive drops. In the same way
for the atmosphere, a periodically oscillating sonorous body produces a similar
periodical motion, first in the mass of air, and then in the drumskin of our ear,
and the period of these vibrations must be the same as that of the vibration in the
sonorous body.
Having thus spoken of the principal division of sound into Noise and Musical
Tones, and then described the general motion of the air for these tones, we pass
on to the peculiarities which distinguish such tones one from the other. We are
acquainted with three points of difference in musical tones, confining our attention
H in the first place to such tones as are isolatedly produced by our usual musical
instruments, and excluding the simultaneous sounding of the tones of difi'erent
instruments. Musical tones are distinguished : —
1. By their force ^
2. By their ^Jtte^,
3. By their qiiality.
It is unnecessary to explain what we mean by the force and pitch of a tone.
By the quality of a tone we mean that peculiarity which distinguishes the musical
tone of a violin from that of a flute or that of a clarinet, or thafc of the human
voice, when all these instruments produce the same note at the same pitch.
We have now to explain what peculiarities of the motion of sound correspond
to these three principal differences between musical tones.
First, We easily recognise that the force of a musical tone increases and dimi-
nishes with the extent or so-caUed amjditude of the oscillations of the particles of
H the sounding body. When we strike a string, its vibrations are at first sufficiently
large for us to see them, and its corresponding tone is loudest. The visible
vibrations become smaller and smaller, and at the same time the loudness
diminishes. The same observation can be made on strings excited by a violin
bow, and on the reeds of reed-pipes, and on many other sonorous bodies. The
same conclusion results from the diminution of the loudness of a tone when we
increase our distance from the sounding body in the open air, although the pitch
and quality remain unaltered ; for it is only the amplitude of tlie oscillations of
the particles of air which diminishes as their distance from the sounding body
increases. Hence loudness must depend on this amphtude, and none other of the
properties of sound do so.*
* Mechanically the force of the oscillations no measure can be found for the intensity of
for tones of different pitch is measured by the sensation of sound, that is, for the loudness
their vis viva^ that is, by the square of the of sound, which will hold all pitches. [See
greatest velocity attained by the oscillating the addition to a footnote on p. 75^^, referring
particles. But the ear has different degrees of especially to this passage. - TraaaUitor.]
sensibility for tones of different pitch, so that Digitized by V^jOOQIC
CHAP. I.
PITCH AND THE SIEEN.
II
The second essential difference between different musical tones consists in
their pitch. Daily experience shews us that musical tones of the same pitch can
be produced upon most diverse instruments by means of most diverse mechanical
contrivances, and with most diverse degrees of loudness. All the motions of the
air thus excited must be periodic, because they would not otherwise excite in us
the sensation of a musical tone. But the sort of motion within each single
period may be any whatever, and yet if the length of the periodic time of two
musical tones is the same, they have the same pitch. Hence : Pitch depends
solely an the length of time in which each single inhration is executed, or, which
comes to the same thing, on the number of vibrations completed in a given time.
We are accustomed to take a second as the unit of time, and shall consequently
mean by the pitch number [or frequency] of a tone, the number of vibrations which
the particles of a sounding body perform in one second of time.* It is self-evident
that we find the periodic time or vibrational period, that is length of time which ^
is occupied in performing a single vibration backwards and forwards, by dividing
one second of time by the pitch number.
Musical tones are said to be higher, the greater- their pitch numbers, that is,
the shorter their vibrational periods.
The exact determination of the pitch number for such elastic bodies as produce
andible tones, presents considerable diflBculty, and physicists had to contrive many
comparatively complicated processes in order to solve this problem for each
particular case. Mathematical theory and numerous experiments had to render
matual assistance.f It is consequently very convenient for the demonstration of
the fundamental facts in this department of knowledge, to be able to apply a
peculiar instrument for producing musical tones — the so-called siren — which is
constructed in such a manner as to determine the pitch number of the tone
produced, by a direct observation. The principal parts of the simplest form of
the siren are shewn in fig. i , after Seebeck. ^
A is a thin disc of cardboard or tinplate, which can be set in rapid rotation
about its axle b by means of a string f f, which passes over a larger wheel. On
the margin of the disc there is punched a set of holes at equal intervals : of these
there are twelve in the figure ; one or
more similar series of holes at equal
distances are introduced on concentric
circles, (there is one such of eight holes
in the figure), c is a pipe which is
.directed over one of the holes. Now,
' on setting the disc in rotation and blow-
ing through the pipe c, the air will pass
freeely whenever one of the holes comes
under the end of the pipe, but will be
checked whenever an unpierced portion m
of tlie disc comes under it. Each hole
of the disc, then, that passes the end of the pipe lets a single puff of air escape.
Supposing the disc to make a single revolution and the pipe to be directed to the
Pig. I.
♦ IThe pitch number yf&s called the •vibra-
tional namber * in the first edition of this trans-
lation. The pitch number ol a note is commonly
called the pilch of the note. By a convenient
abbreviation we often write a' 440, meaning
the note a' having the pitch number 440 ; or
say that the pitch of a' is 440 vib. that is, 440
doable vibrations in a second. The second
^^^tm frequency f which I have introduced into
the text, as it is much used by acousticians,
properly represents Dhe number of times tliat
•»y peruxUcully recurring event liappens in
otic iuo7td of titne^ and, applied to double
vibrations, it means the same a;» pitch number.
The pitch of a musical instrument is the pitch
of the note by which it is tuned. But as pitch
is properly a sensation, it is necessary here
to distinguish from this sensation the pitch
number or frequency of vibration by which it
is measured. The larger the pitch number,
the higher or sharper the pitch is said to be.
The lower the pitch number the deeper or
flatter the pitch. These arc all metaphorical
expressions which must be taken strictly in
this sense. — Translator.]
f [An account of the more exact modern
methods is given in App. XX. sect. B. —
Translator.]
Digitized by V^.OOQIC
13
PITCH AND THE SIREN.
PART I.
outer circle of holes, we have twelve puffs corresponding to the twelve holes ; but
if the pipe is directed to the inner circle we have only eight puffs. If the disc is
made to revolve ten times in one second, the outer circle would produce 1 20 puffs
in one second, which would give rise to a weak and deep musical tone, and the
inner circle eighty puffs. Generally, if we know the number of revolutions which
the disc makes in a second, and the number of holes in the series to which the
tube is directed, the product of these two numbers evidently gives the number of
puffs in a second. This number is consequently far easier to determine exactly
than in any other musical instrument, and sirens are accordingly extremely well
adapted for studying all changes in musical tones resulting from the alterations
and ratios of the pitch numbers.
The fonn of siren here described gives only a weak tone. I have placed it first
because its action can be most readily understood, and, by changing the disc, it
Pm.
Fig. 3.
Fig. 4.
can be easily applied to experiments of very different descriptions. A stronger tone
is produced in the siren of Cagniard de la Tour, shewn in figures 2, 3, and 4, above.
Here s s is the rotating disc, of which the upper surface is shewn in fig. 3, and
the side is seen in figs. 2 and 4. It is placed over a windchest A A, which is
connected with a bellows by the pipe B B. The cover of the windchest A A,
which lies immediately under the rotating disc s s, is pierced with precisely the
same number of holes as the disc, and the direction of the holes pierced in the
cover of the chest is obhque to that of the holes in the disc, as shewn in fig. 4,
which is a vertical section of the instrument through the line n n in fig. 3. This
position of the holes enables the wind escaping from A A to set the disc s s in
rotation, and by increasing the pressure of tlie bellows, as much as 50 or 60
rotations in a second can be produced. Since all the holes of one circle are blown
through at the same time in this siren, a mucli more powerful tone is produced
than in Seeheck's, fig. i (p. iic). To record tlie revolutions, a counter z z is
Digitized by V^ O OQ IC
CHAP. I. PITCH AND INTERVAL. 13
introdaced, connected with a toothed wheel which works in the screw t, and
advances one tooth for each revolution of the disc s s. By the handle h this
counter may be moved slightly to one side, so that the wheelwork and screw may
be connected or disconnected at pleasure. If they are connected at the beginning
of one second, and disconnected at the beginning of another, the hand of the
counter shows how many revolutions of the disc have been made in the corre-
sponding number of seconds.*
Dove t introduced into this siren several rows of holes through which the wind
might be directed, or from which it might be cut off, at pleasure. A polyphonic
siren of this description with other peculiar arrangements will be figured and
described in Chapter VIII., fig. 56.
It is clear that when the pierced disc of one of these sirens is made to revolve
with a uniform velocity, and the air escapes through the holes in puffs, the motion
of the air thus produced must be periodic in the sense already explained. The If
holes stand at equal intervals of space, and hence on rotation foUow each other at
equal intervals of time. Through every hole there is poured, as it were, a drop of
air into the external atmospheric ocean, exciting waves in it, which succeed each
other at uniform intervals of time, just as was the case when regularly falling
drops impinged upon a surface of water (p. loa). Within each separate period,
each individual puff will have considerable variations of form in sirens of different
construction, depending on the different diameters of the holes, their distance from
each other, and the shape of the extremity of the pipe which conveys the air ; but
m every case, as long as the velocity of rotation and the position of the pipe remain
unaltered, a regularly periodic motion of the air must result, and consequently the
sensation of a musical tone must be excited in the ear, and this is actually the
case.
It results immediately from experiments with the siren that two series of the
same number of holes revolving with the same velocity, give musical tones of the f
same pitch, quite independently of the size and form of the holes, or of the pipe.
We even obtain a musical tone of the same pitch if we allow a metal point to
strike in the holes as they revolve instead of blowing. Hence it follows firstly that
the pitch of a tone depends only on the number of puffs or swings, and not on
their form, force, or method of production. Further it is very easily seen witli
this instrument that on increasing the velocity of rotation and consequently the
number of puffs produced in a second, the pitch becomes sharper or higher. The
same result ensues if, maintaining a uniform velocity of rotation, we first blow into
a series with a smaller and then into a series with a greater number of holes.
The latter gives the sharper or higher pitch.
With the same instrument we also very easily find the remarkable relation
which the pitch numbers of two musical tones must possess in order to form a
consonant interval. Take a series of 8 and anotlier of 16 holes on a disc, and
blow into both sets while the disc is kept at uniform velocity of rotation. Two %
tones will be heard which stand to one another in the exact relation of an Octave.
Increase the velocity of rotation ; both tones will become sharper, but both will
continue at the new pitch to form the interval of an Octave. J '\ Hence we conclude
that a mtisical tone which is an Octave higher than another, makes exactly twice
as many vibrations in a given time as the latter.
* See Appendix I. names of all the intervals usually distinguished
t [Pronounce Doli-veh, in two syllables.— are also given in App. XX. sect. D., with the
Translator.] corresponding ratios and cents. These names
t [When two notes have different pitch were in the first place derived from the ordinal
numbers, there is said to be an interval number of the note in the scales, or succes-
bctween them. This gives rise to a sensa- sions of continually sharper notes. The Octave
tion, very differently appreciated by different is the eighth note in the major scale. An octave
individuals, but in all cases the interval is is a set of notes lying within an Octave. Ob-
measurcd by the ratio of the pitch number s^ serve that in this translation aU names of in-
and, for some purposes, more conveniently by tervals commence with a capital letter, to
other numbers called cents ^ derived from these prevent ambiguity, as almost all such words
ratios, as explained in App. XX. sect. C. The are also used in other senses. — Translator,']
Digitized by V^jOOQ l€
14 PITCH AND INTEEVAL. pakt i.
The disc shewn in fig. i, p. iic, has two circles of 8 and 12 holes respectively.
Each, blown successively, gives two tones which form with each other a perfect
Fifth, independently of the velocity of rotation of the disc. Hence, two musical
tones stand in the relation of a so-called Fifth when the higJier tone makes three
vibrations in the same time as the lower makes two.
If we obtain a muFdcal tone by blowing into a circle of 8 holes, we require a
circle of 16 holes for its Octave, and 12 for its Fifth. Hence the ratio of the
pitch numbers of the Fifth and the Octave is 12 : 16 or 3 : 4. But the interval
between the Fifth and the Octave is the Fourth, so that we see that when two
musical tones form a Fourth, the higher makes four vibrations while the lower
tnakes three.
The polyphonic siren of Dove has usually four circles of 8, 10, 12 and 16 holes
respectively. The series of 16 holes gives the Octave of the series of 8 holes, and
1[ the Fourth of the series of 1 2 holes. The series of 1 2 holes gives the Fifth of the
series of 8 holes, and the minor Third of the series of 10 holes. While the series of
10 holes gives the major Third of the series of 8 holes. The four series con-
sequently give the constituent musical tones of a major chord.
By these and similar experiments we find the following relations of the pitch
numbers : —
1 : 2 Octave
2 : 3 Fifth
3 : 4 Fourth
4 : 5 major Third
5 : 6 minor Third
When the fundamental tone of a given interval is taken an Octave higher, the
interval is said to be inverted. Thus a Fourth is an inverted Fifth, a minor Sixth
f an inverted major Third, and a major Sixth an inverted minor Third. The corre-
sponding ratios of the pitch numbers are consequently obtained by doubling the
smaller number in the original interval.
From 2 : 3 the Fifth, we thus have 3 : 4 the Fourth
„ 4:5 the major Third ... 5 : 8 the minor Sixth
„ 5:6 the minor Third, 6 : 10=3 : 5 the major Sixth.
These are all the consonant intervals which lie within the compass of an
Octave. With the exception of the minor Sixth, which is really the most imperfect
of the above consonances, the ratios of their vibrational numbers are all expressed
by means of the whole numbers, i, 2, 3, 4, 5, 6.
Comparatively simple and easy experiments with the siren, therefore, corrobo-
rate that remarkable law mentioned in the Introduction (p. id), according to which
the pitch numbers of consonant musical tones bear to each other ratios expressible
If by small whole numbers. In the course of our investigation we shall employ the
same instrument to verify more completely the strictness and exactness of this
law.
Long before anything was known of pitch numbers, or the means of countmg
them, Pythagoras had discovered that if a string be divided into two parts by a
bridge, in such a way as to give two consonant musical tones when struck, the
lengths of these parts must be in the ratio of these whole numbers. If the bridge
is so placed that f of the string lie to the right, and ^ on the left, so that the two
lengths are in the ratio of 2 : i, they produce the interval of an Octave, the greater
length giving the deeper tone. Placing the bridge so that f of the string lie on
the right and f on the left, the ratio of the two lengths is 3:2, and the interval
is a Fifth.
These measurements had been executed with great precision by the Greek
musicians, and had given rise to a system of tones, contrived with considerable
art. For these measurements they used a peculiar instrument, the motiochord.
Digitized by V^jOOQlC
CHAP. I. PITCH NUMBERS IN JUST MAJOR SCALE. 15
consisting of a sounding board and box on which a single string was stretched
with a scale below, so as to set the bridge correctly.*
It was not till much later that, through the investigations of Galileo (1638),
Newton, Euler (1729), and Daniel Bemouilli (1771), the law governing the
motions of strings became known, and it was thus found that the simple ratios of
the lengths of the strings existed also for the pitch numbers of thp tones they pro-
duced, and that they consequently belonged to the musical intervals of the tones
of all instruments, and were not confined to the lengths of strings through which
the law had been first discovered.
This relation of whole numbers to musical consonances was from all time
looked upon as a wonderful mystery of deep significance. The Pythagoreans
themselves made use of it in their speculations on the harmony of the spheres.
From that time it remained partly the goal and partly the starting point of the
strangest and most venturesome, fajitastic or philosophic combinations, till in ^
modem times the majority of investigators adopted the notion accepted by Euler
himself, that the human mind had a peculiar pleasure in simple ratios, because it
could better understand them and comprehend their bearings. But it remained
uninvestigated how the mind of a listener not versed in physics, who perhaps was
not even aware that musical tones depended on periodical vibrations, contrived to
recognise and compare these ratios of the pitch numbers. To shew what pro-
cesses taking place in the ear, render sensible the difference between consonance
and dissonance, will be one of the principal problems in the second part of this
work.
Calculation of the Pitch Numbebs fob all the Tones of the
Musical Scale.
By means of the ratios of the pitch numbers already assigned for the consonant
intervals, it is easy, by pursuing these intervals throughout, to calculate the ratios f
for the whole extent of the musical scale.
The major triad or chord of three tones, consists of a major Third and a Fifth,
Hence its ratios are :
C:E: G
I : f : ^
or 4:5:6
If we associate with this triad that of its dominant G : B : Dy and that of its
sub- dominant F : A : C, each of which has one tone in common with the triad of
the tonic C : E : G, we obtain the complete series of tones for the major scale of
C, with the following ratios of the pitch numbers :
C : D : E : F : G : A : B :c
T*9-fi •4-3 •5-15.9
[or 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48]
In order to extend the calculation to other octaves, we shall adopt the following
notation of musical tones, marking the higher octaves by accents, as is usual in
Germany ,t as follows :
I. The unaccented or small octave (the 4-foot octave on the organj ): —
m
zzso
rz2i
c d e f g a b
* r Ab the monochord is very liable to error, below the letters, which are typographically
these results were happy generalisations from inconvenient. Hence the German notation is
necessarily imperfect experiments.— Tratw- retained. — Translator.]
lator.] J [The note C in the small octave was
t [English works use strokes above and once emitted by an organ pipe 4 feet in length :
Digitized by V^jOOSlC
i6 PITCH NUMBERS IN JUST MAJOR SCALE.
2. The oncC'Ctccented octave (2-foot) : —
PABT I.
i
..^
ra"
^ c' ci' e' /
3. The twice-accented octave (i-foot) : —
-^
1221
nr22i
(Z"
/'
r
6"
And so on for higher octaves. Below the small octave lies the great octave,
written with unaccented capital letters ; its G requires an organ pipe of eight feet
^ in length, and hence it is called the 8-foot octave.
4. Great or Z-foot octa/ve : —
D
E
F
O
A
Below this follows the 16-foot or contra-octave ; the lowest on the pianoforte
and most organs, the tones of which may be represented by C^ D, E, F^ G, A, B^,
with an inverted accent. On great organs there is a still deeper, 32 -foot octave, tlie
tones of which may be written C,, Dj, E„ F,, G,, A,, B^„ with two inverted accents,
but they scarcely retain the character of musical tones. (See Chap. IX.)
Since the pitch numbers of any octave are always twice as great as those for
f the next deeper, we find the pitch numbers of the higher tones by multiplying
those of the small or unaccented octave as many times by 2 as its symbol has
upper accents. And on the contrary the pitch numbers for the deeper octaves are
found by dividing those of the great octave, as often as its symbol has lower
accents.
Thus C"=2X2XC=2X2X2(7
C = ixixC = ixix^c.
For the pitch of the musical scale German physicists have generally adopted
that proposed by Scheibler, and adopted subsequently by the German Association
of Natural Philosophers {die deutsche Naturforscherversammlung) in 1834. This
makes the once-accented a' execute 440 vibrations in a second.* Hence results the
thus B^dos (L'Art du Facteurd'Orgues, 1766)
I made it 4 old French feet, which gave a
' note a full Semitone flatter than a pipe of
4 English feet. But in modern organs not even
80 much as 4 English feet are used. Organ
builders, however, in all countries retain the
names of the octaves as here given, which
must be considered merely to determine the
place on the staff, as noted in the text, inde-
pendently of the precise pitch. — Translator,]
* The Paris Academy has lately fixed the
pitch number of the same note at 435. This
is called 870 by the Academy, because French
physicists have adopted the inconvenient
habit of counting the forward motion of a
swinging body as one vibration, and the back-
ward as another, so that the whole vibra-
tion is counted as two. This method of
counting has been taken from the seconds
pendulimi, which ticks once in going forward
and once again on returning. For symmetrical
backward and forward motions it would be
indifferent by which method we counted, but
for non-symmetrical musical vibrations which
are of constant occurrence, the French method
of counting is very inconvenient. The number
440 gives fewer fractions for the first (just]
major scale of C, than a' « 435. The difference
of pitch is less than a comma. [The practical
settlement of pitch has no relation to such
arithmetical considerations as are here sug-
gested, but depends on the compass of the
human voice and the music written for it at
different times. An Abstract of my History
of Musical Pitch is given in Appendix XX.
sect. H. Scheibler's proposal, named in the
text, was chosen, as he tells us (Der Tonmesser^
'834, p. 53), as being the mean between the
limits of pitch within which Viennese piano-
fortes at that time rose and fell by heat and
cold, which he reckons at '^ vibration either
way. That this proposal had no reference to the
Digitized by V^jOOQlC
CHAP. I.
PITCH NUMBERS IN JUST MAJOR SCALE.
17
following table for the scale of C major, which will serve to determine the pitch
of all tones that are defined by their pitch numbers in the following work.
Unaccented
Octave
Once-
Twice-
Thrice-
Four-times
OontraOotaTe
Great Ootare
accented
accented
aooented
accented
Notes
C,UiB,
CtoB
Octave
Octave
Octave
Octave
x6foot
8 foot
£ to &
c'toy
c"toft"
e"' to V"
&"' to y"
afoot
I foot
ifoot
ifoot
c
33
66
132
264
528
1056
1 188
2II2
D
37-125
74-25
148-5
297
594
2376
E
4125
825
165
330
660
1320
2640
F
44
88
176
352
704
1408
2816
0
49-5
99
198
396
IV
1584
3168
A
55
no
220
440
880
1760
3520
B
61-875
12375
247-5
495
990
1980
3960*
The lowest tone on orchestral instruments is the E, of the double bass, making
41^ vibrations in a second. f Modem pianofortes and organs usually go down to C^ %
expression of the jast major scale in \7h0le
nambers, is shewn by the fact that he
proposed it for an equally tempered scale,
for which he calculated the pitch numbers
to four places of decimals, and for which, of
course, none but the octaves of a' are ex-
pressible by whole numbers.— TmtuZa^.]
* [As it is important that students should
be able to hear the exact intervals and pitches
spoken of throughout this book, and as it is
quite impossible to do so on any ordinary in-
strument, I have contrived a specially-tuned
harmonium, called an Harmonical, fully de-
scribed in App. XX. sect. F. No. i, which
Messrs. Bfoore & Moore, 104 Bishopsgate Street,
will, in the interests of science, supply to order,
for the moderate sum of 1655. The follow-
ing are the pitch numbers of the first four
octaves, the tuning of the fifth octave will be
explained in App. XX. sect. F. The names of
the notes are in the notation of the latter part
of Chap. XIV. below. Bead the sign D, as
*D one,' E^\> as * one E flat,' and ^Bb as
* seven B flat.' In playing observe that D, is
on the ordinary D)> or C% digital, and that
^B\> is on the ordinary Gb or ^ digital, and
that the only keys in which chords can be
played are C major and C minor, with the
minor chord D^FA^ and the natural chord of
the Ninth OE^ O ^B b2>. The mode of measuring
intervals by ratios and cents is fully explained
hereafter, and the results are added for con-
venience of reference. The pitches of cf 528,
a' 440, a'' b 422*4 and ^l/b 462, were taken from
forks very carefully tuned by myself to these
numbers of vibrations, by means of my unique
series of forks described in App. XX., at the ^
end of sect. B.
Scale of the Haruonicaii.
Pitch Nambers.
fiatios
Cents
Notes
8 foot
4fbot
afoot
I foot
Note to Note
C to Note
Note to Note
CtoNote
C
66
132
264
528
9:10
I : I
182
0
A
73i
146!
293i
586I
80:81
9:10
22
182
D
74i
H^
297
594
15:16
8:9
112
204
^b
79i
1581
316J
633!
24:25
5:6
70
316
^.
82}
165
330
660
15:16
4:5
112
386
F
88
176
352
704
8T9
3:4
204
498
G
99
198
396
792
15:16
2:3
112
702
A^b
io5f
211J
422f
844!
24:25
5:8
70
814
^1
no
220
440
880
20:21
3:5
85
884
'Bb
ii5i
231
462
924
35-36
4:7
49
969
B«b
ii8|
237!
475i
950I
24:25
5:9
70
IOI8
Bi
123I
247i
495
990
15:16
8:15
112
1088
C
132
264
528
1056
—
1:2
—
1200
t [The following account of the actual tones
^>Md is adapted from my History of Musical
Translator.]
Pitch. C„t commencement of the 32-foot oc-
tave, the lowest tone of very large organs, two
Digitized by V^OOQ IC
i8
COMPASS OP INSTRUMENTS.
PART I,
with 33 vibrations, and the latest grand pianos even down to A,, with 27J vibra-
tions. On larger organs, as already mentioned, there is also a deeper Octave reach-
ing to Cf, with 1 6^ vibrations. But the musical character of all these tones below E,
is imperfect, because we are here near to the limit of the power of the ear to combine
vibrations into musical tones. These lower tones cannot therefore be used musically
^ except in connection with their higher octaves to which they impart a character
of greater depth without rendering the conception of the pitch indeterminate.
Upwards, pianofortes generally reach a"" with 3520, or evenc' with 4224 vibra-
tions. The highest tone in the orchestra is probably the five-times accented d* of the
piccolo flute with 4752 vibrations. Appunn and W. Preyer by means of small
tuning-forks excited by a violin bow have even reached the eight times accented c^*"
with 40,960 vibraticHis in a second. These high tones were very painfully unplea-
sant, and the pitch of those which exceed the boundaries of the musical scale was
^ very imperfectly discriminated by musical observers.* More on this in Chap. IX.
The musical tones which can be used with advantage, and have clearly dis-
tinguishable pitch, have therefore between 40 and 4000 vibrations in a second,
extending over 7 octaves. Those which are audible at all have from 20 to 40,000
vibrations, extending over about 1 1 octaves. This shews what a great variety of
different pitch numbers can be perceived and distinguished by the ear. In this
y/^ I respect the ear is far superior to the eye, which likewise distinguishes light of dif-
1 ferent periods of vibration by the sensation of different colours, for tlie compass of
the vibrations of light distinguishable by the eye but slightly exceeds an Octave.-h
Force and pitch were the two first differences which we found between musical
tones ; the third was quality of tonCj which we have now to investigate. When
of Tone,* (liber die Qrenzen der Tontcahmeh-
mvngt 1876, p. 20), are in the South Kensing-
ton Museum, Scientific Collection. I have
several times tried them. I did not myself
find the tones painful or cutting, probably
because there was no beating of inharmonic
upper partials. It is best to sound them with
two violin bows, one giving the octave of the
other. The tones can be easily heard at a
distance of more than 100 feet in the gallery
of the Museum. — Translator,']
t [Assuming the undulatory theory, which
attributes the sensation of light to the vibra-
tions of a supposed luminous * ether,' resem-
bling air but more delicate and mobile, then
the phenomena of * interference ' enables us
to calculate the lengths of waves of light in
empty space, <&c., hence the numbers of vibra-
tions in a second, and consequently the ratios
of these numbers, which will then clearly
resemble the ratios of the pitch numbers that
measure musical intervals. Assuming, then,
that the yellow of the spectrum answers to the
tenor c in music, and Fraunhofer*8 * line A '
corresponds to the Q below it. Prof. Helm-
holtz, in his Physiological Optics^ (Ha^id-
buck der physiologtschen Optik, 1867, p. 237),
gives the f oUowing analogies between the notes
of the piano and the colours of the spectrum : —
Fj , end of the Red. /« , Violet.
G,Red.
Octaves below the lowest tone of the Violon-
cello. A,^i the lowest tone of the largest
pianos. C., commencement of the 16-foot
octave, the lowest note assigned to the Double
Bass in Beethoven's Pastorid Symphony. £^,
the lowest tone of the German four-stringed
Double Bass, the lowest tone mentioned in
the text. F„ the lowest tone of the English
four-stringed Double Bass. 0„ the lowest tone
of the Italian three-stringed Double Bass. ^,,
the lowest tone of the English three-stringed
Double Bass. C, commencement of the 8-foot
octave, the lowest tone of the Violoncello,
written on the second leger line below the bass
stafiF. G, the tone of the third open string of
the Violoncello, c, commencement of the
4 -foot octave ' tenor C,* the lowest tone of the
Vi61a, written on the second space of the bass
staff, d^ the tone of the second open string of
the Violoncello. /, the tone signified by the
bass or J^-clef. ^, the lowest tone of the
Violin, a, the tone of the highest open string
[ of the Violoncello. c\ commencement of the
2-foot octave, * middle C,' written on the leger
line between the bass and treble staves, the tone
signified by the tenor or C-clef . d', the tone of the
third open string of the Violin. (/', the tone
signified by the treble or G-clef . a', the tone of
the second open string of the Violin, the * tuning
note * for orchestras, c'^ commencement of the
I -foot octave, the usual * tuning note ' for pianos.
e't the tone of the first or highest open string of
the Violin, c"', commencement of the ^-foot
octave, g*^, the usual highest tone of the
Flute, c'*, commencement of the |-foot octave,
e'*, the highest tone on the Violin, being the
double Octave harmonic of the tone of the
highest open string, a''', the usual highest
tone of large pianos, d", the highest tone of
the piccolo flute, e^", the highest tone reached
by Appunn 's forks, see next note. — Translator.}
♦ [Copies of these forks, described in Prof.
Preyer's essay • On the Limits of the Perception
Gt , Red.
A, Bed.
AZ , Orange-red.
Bt Orange.
c, Yellow.
c9 , Green.
dy Greenish-blue.
dt , Cyanogen-blue.
e, Indigo-blue.
/, Violet.
g, Ultra-violet.
g%^ fi
a, „
at J
6, end of the solar
spectrum.
The scale there-
fore extends to
about a Fourth
beyond the oc-
tave. — Transla-
tor.^
Digitized by VjOOQlC
CHAP. I. QUALITY OF TONE AND FORM OF VIBRATION. 19
we hear notes of the same force and same pitch sounded successively on a piano-
forte, a violin, clarinet, oboe, or trumpet, or by the human voice, the character of
the musical tone of each of these instruments, notwithstanding the identity of force
and pitch, is so different that by means of it we recognise with the greatest ease
which of these instruments w^as used. Varieties of quality of tone appear to he
infinitely numerous. Not only do we know a long series of musical instruments
which could each produce a note of the same pitch ; not only do different individual
instruments of the same species, and the voices of different individual singers shew
certain more delicate shades of quality of tone, which our ear is able to distinguish ;
but notes of the same pitch can sometimes be sounded on the same instrument with
several qualitative varieties. In this respect the * bowed ' instruments (i.e. those
of the violin kind) are distinguished above all other. But the human voice is still
richer, and human speech employs these very qualitative varieties of tone, in order
to distinguish different letters. The different vowels, namely, belong to the class ^
of sustained tones which can be used in music, while the character of consonants
mainly depends upon brief and transient noises.
On inquiring to what external physical difference in the waves of sound the
different qualities of tone correspond, we must remember that the amplitude of
the vibration determines the force or loudness, and the period of vibration the
pitch. Quality of tone can therefore depend upon neither of these. The only
possible hypothesis, therefore, is that the quality of tone should depend upon the
manner in which the motion is performed within the period of each single vibra-
tion. For the generation of a musical tone we have only required that the motion
should be periodic, that is, that in any one single period of vibration exactly the
same state should occur, in the same order of occurrence as it presents itself in any
other single period. As to the kind of motion that should take place witliin any
single period, no hypothesis was made. In this respect then an endless variety of
motions might be possible for the production of sound. %
Observe instances, taking first such periodic motions as are performed so slowly
that we can follow them with the eye. Take a pendulum, which we can at any
time construct by attaching a weight to a thread and setting it in motion. The
pendulum swings from right to left with a uniform motion, uninterrupted by jerks.
Near to either end of its path it moves slowly, and in the middle fast. Among
sonorous bodies, which move in the same way, only very much faster, we may
mention tuning-forks. When a tuning-fork is struck or is excited by a violin bow,
and its motion is allowed to die away slowly, its two prongs oscillate backwards
and forwards in the same way and after the same law as a pendulum, only they
make many hundred swings for each single swing of the pendulum.
As another example of a periodic piotion, take a hammer moved by a water-
wheel. It is slowly raised by the millwork, then released, and falls down suddenly,
is then again slowly raised, and so on. Here again we have a periodical backwards
and forwards motion ; but it is manifest that this kind of motion i» totally different ^
from that of the pendulum. Among motions which produce musical sounds, that of
a violin string, excited by a bow, would most nearly correspond with the hammer's,
as will be seen from the detailed description in Chap. V. The string clings for a
time to the bow, and is carried along by it, then suddenly releases itself, like the
hammer in the mill, and, like the latter, retreats somewhat with much greater
Telocity than it advanced, and is again caught by the bow and carried forward.
Again, imagine a ball thrown up vertically, and caught on its descent with a
blow which sends it up again to the same height, and suppose this operation to be
performed at equal intervals of time. Such a ball would occupy the same time in
rising as in falling, but at the lowest point its motion would be suddenly interrupted,
whereas at the top it would pass through gradually diminishing speed of ascent
into a gradually increasing speed of descent. This then would be a third kind of
alternating periodic motion, and would take place in a manner essentially different
from the other two* ^
Digitized by VjOOQ IC
20
FORM OF VIBRATION.
PAUT t.
To render the law of such motions more comprehensible to the eye than is
possible by lengthy verbal descriptions, mathematicians and physicists are in the
habit of applying a graphical method, which must be frequently employed in this
work, and should therefore be well understood.
To render this method intelligible suppose a drawing point b, fig. 5, to be
fastened to the prong A of a tuning-fork in such a manner as to mark a surface
of paper B B. Let the tuning-fork be moved with a uniform velocity in the direc-
tion of the upper arrow, or else the paper be drawn under it in the opposite
direction, as shewn by the lower arrow. When the fork is not sounding, the point
will describe the dotted straight line d c. But if the prongs have been first set in
vibration, the point will describe the undulating hne d c, for as the prong vibrates,
the attached point b will constantly move backwards and forwards, and hence be
sometimes on the right and sometimes on the left of the dotted straight line d c, ad
is shewn by the wavy line in the figure. This wavy line once drawn, remains as a
permanent image of the kind of motion performed by the end of the fork during
its musical vibrations. As the point b is moved in the direction of the straight
line d c with a constant velocity, equal sections of the straight line d c will corre>
spond to equal sections of the time during which the motion lasts, and the distance
of the wavy line on either side of the straight line will shew how far the point b
lias moved from its mean position to one side or the other during those sections of
time.
In actually performing such an experiment as this, it is best to wrap the paper
over a cylinder which is made to rotate uniformly by clockwork. The paper is
wetted, and then passed over a turpentine flame which coats it with lampblack,
on which a fine and somewhat smooth steel point will easily trace delicate lines.
Via. 6.
Fig. 6 is tlie copy of a drawing actually made in this way on the rotating cylinder
of Messrs. Scott and Koenig's PJionautograph.
Fig. 7 shews a portion of this curve on a larger scale. It is easy to see the
meaning of such a curve. The drawing point has passed with a uniform velocity
in the direction e h. Suppose that it has described the section e g in jV ^^ ^
second. Divide e g into 12 equal parts, as in the figure, then the point has been
j^jf of a second in describing the length of any such section horizontally, and
the curve shews us on what side and at what distance from the position of
rest the vibrating point will be at the end of -j-^, y^, and so on, of a second,
or, generally, at any given short interval of time since it left the point e.
We see, in the figure, that after j^ir ^^ & second it had reached the height i,
and that it rose gradually till tlie end of y^^ of a second ; then, however, it began
to descend gradually till, at the end of j^ = ^V second, it had reached its mean
Digitized by V^jOOQIC
CHAP. I.
FORM OP VIBRATION.
91
position f, and then it oontinued descending on the opposite side till the end of
tI^f of a second and so on. We can also easily detennine where the vibrating
point was to be found at the end of any fraction of this hundred-and-twentieth of
a second. A drawing of this kind consequently shews immediately at what point of
its path a vibrating particle is to be found at any given instant, and hence gives a
complete image of its motion. If the reader wishes to reproduce the motion of the
vibrating point, he has only to cut a narrow vertical sUt in a piece of paper, and
place it over fig, 6 or fig. 7, so as to shew a very small portion of the curve through
the vertical slit, and draw the book slowly but uniformly under the slit, from right
to left ; the white or black point in the sht will then appear to move backwards and
forwards in precisely the same manner as the original drawing point attached to
the fork, only of course much more slowly.
We are not yet able to make all vibrating bodies describe their vibrations
Fio,
H
directly on paper, although much progress has recently been made in the
methods required for this purpose. But we are able ourselves to draw such
curves for all sounding bodies, when the law of their motion is known, that is,
when we know how far the vibrating point will be from its mean position at any
given moment of time. We then set off on a horiEontal line, such as e f, fig. 7,
lengths corresponding to the interval of time, and let fall perpendiculars to it on ^
either side, making their lengths equal or proportional to the distance of the vibrat-
ing point from its mean position, and then by joining the extremities of these per-
pendiculars we obtain a curve such as the vibrating body would have drawn if it
had been possible to make it do so.
Thus fig. 8 represents the motion of the hammer raised by a water-wheel, or of
a point in a string excited by a vioUn bow. For the first 9 intervals it rises slowly
and uniformly, and during the loth it falls suddenly down.
Fig. 8.
Fig.
10
Fig. 9 represents the motion of the ball which is struck up again as soon as it f
comes down. Ascent and descent are performed with equal rapidity, whereas in
fig. 8 the ascent takes much longer time. But at the lowest point the blow suddenly
changes the kind of motion.
Physicists, then, having in their mind such curvilinear forms, representing the
law of the motion of sounding bodies, speak briefly of the form of vihration of a
sounding body, and assert that the quality of tone depends on the form of vibration.
This assertion, which has hitherto been based simply on the fact of our knowing
that the quality of the tone could not possibly depend on the periodic time of a
vibration, or on its ampUtude (p. loc), will be strictly examined hereafter. It
will be shewn to be in so far correct that every different quality of tone requires a
different form of vibration, but on the other hand it wiU also appear that different
forms of vibration may correspond to the same quality of tone.
On exactly and carefully examining the effect produced on the ear by different
forms of vibration, as for example that in fig. 8, corresponding nearly to a violin
Digitized by V^jOOQlC
22- COMPOUND AND PARTIAL TONES. pabt i.
string, we meet with a strange and unexpected phenomenon, long known indeed to
indi\ddual musicians and physicists, but commonly regarded as a mere curiosity,
its generality and its great significance for all matters relating to musical tones not
having been recognised. The ear when its attention has been properly directed to
the effect of the vibrations which strike it, does not hear merely that one musical
tone whose pitch is determined by the period of the vibrations in the manner
already explained, but in addition to this it becomes aware of a whole series of
higher musical tones, which we will call the harmonic upper partial tones^ and
sometimes simply the upper partials of the whole musical tone or note, in contra-
distinction to the fundarmntal or prime partial tone or simply the prime^ as it may
be called, which is the lowest and generally the loudest of all the partial tones, and
by the pitch of which we judge of the pitch of the whole compound musical tone
itself. The series of these upper partial tones is precisely the same for all com-
H pound musical tones which correspond to a uniformly periodical motion of the air.
It is as follows : —
The first upper partial tone [or second partial tone] is the upper Octave of the
prime tone, and makes double the number of vibrations in the same time. If we
call the prime 0, this upper Octave will be c.
The second upper partial tone [or third partial tone] is the Fifth of this Octave,
or g, making three times as many vibrations in the same time as the prime.
The third upper partial tone [or fourth partial tone] is the second higher Octave,
or c\ making four times as many vibrations as the prime in the same time.
The fourth upper partial tone [or fifth partial tone] is the major Third of this
second higher Octave, or e\ with five times as many vibrations as the prime in tlie
same time.
The fifth upper partial tone [or sixth partial tone] is the Fifth of the second
higher Octave, or g\ making six times as many vibrations as the prime in the
^ same time.
And thus they go on, becoming continually fainter, to tones making 7, 8, 9,
&c., times as many vibrations in the same time, as the prime tone. Or in musical
notation
^^
-^
'^ c g c' e' g' »6't) c" d" «" "/' g" 'V' '6"t> h"
9
c' e' g' »6't? c" d" e" "/' g" »V
3
456 7 8 9 10 II 12 13
198
464330396 462 528594660726 792858
c"
omfnanjnmberof, ^ 3 4 5 6 7 » 9 lO II 12 13 I4 15 16
Pitch number 66 132 198 264330396 462 528594660726 792858 924 990 105^^*
where the figures [in the first line] beneath shew how many times the corresponding
pitch number is greater than that of the prime tone [and, taking the lowest note
to have 66 vibrations, those in the second hne give the pitch numbers of all the
H other notes].
The whole sensation excited in the ear by a periodic vibration of the air we
* [This diagram has been slightly altered to This slightly flattens each note, and slow beats
introduce all the first i6 harmonic partials can be prodaced in every case (except, of
of C 66, (which, excepting ii and 13, are course, 11 and 13, which are not on the
given on the Harmonical as harmonic notes,) instrument) up to 16. It should also be ob-
and to shew the notation, symbolising, both in served that the pitch of the beat is very nearly
letters and on the staff, the 7th, nth, and that of the upper (not the lower) note in each
13th harmonic partials, which are not used in case. The whole of these 16 harmonics of C66
general music. It is easy to shew on the (except the nth and 13th) can be played
Harmonical that its lowest note, G of this at once on the Harmonical by means of the
scries, contains all these partials, after the harmonical bar, first without and then with
theory of the beats of a disturbed unison the 7th and 14th. The whole series will be
has been explained in Chap. VIII. Keep found to sound like a single fine note, and the
down the note C, and touch in sucoessioi;! the 7th and 14th to materially increase its rich-
notes c, <7, c\ e', g\ A-c, but in touching the latter ness. The relations of the partials in this case
press the finger-key such a little way down may be studied from the tables in the footnotes
that the tone of the note is only just audible. to Chap. X. - Translator.]
Digitized by V^jOOQlC
CHAP. I.
DEFINITION OF TERMS EMPLOYED.
23
have called a musical tone. We now find tbat this is compound^ containing a
series of different tones, which we distinguish as the constitutents or partial tones
of the compound* The first of these constituents is the prvme partial tone of the
compound, and the rest its harmonic upper partial tones. The number which
shews the order of any partial tone in the series shews how many times its
vibrational number exceeds that of the prime tone.* Thus, the second partial
tone makes twice as many, the third three times as many vibrations in the same
lime as the prime tone, and so on.
6. S. Ohm was the first to declare that there is only one form of vibration
which will give rise to no harmonic upper partial tones, and which will therefore
consist solely of the prime tone. This is the form of vibration which we have
described above as peculiar to the pendulum and tuning-forks, and drawn in figs. 6
and 7 (p. 10). We will call tliese pendular vibratio^is, or, since they cannot be
analysed into a compound of different tones, siynple vibrations. In what sense not ^
merely other musical tones, but all other forms of vibration, may be considered
as compound, will be shewn hereafter (Chap. IV.). The terms simple ox pendular
vibration;^ wiU therefore be used as synonymous. We have hitherto used the
expression tone and musical tone indifferently. It is absolutely necessary to dis-
tinguish in acoustics $rst, a musical tofie, that is, the impression made by ani/
periodical vibration of the air ; secondly, a simple tone, that is, the impression
produced by a simple or pendular vibration of the air ; and thirdly, a cofnpou?id
tone, that is, the impression produced by the simultaneous action of several simple
tones with certain definite ratios of pitch as already explained. A musical tone
may be either simple or compound. For the sake of brevity, tone will be used in
* [The ordinal number of a partial tone
in general, must be distinguished from the
ordinal number of an iqrper partial tone in
particular. For the same tone the former
number is alvr-ays greater by unity than the
latter, because the partials in general include
the prime, which is reckoned as the first, and
the upper partials exclude the prime, which
being the loicest partial is of course not an
upper partial at all. Thus the partials gene-
rally numbered 23456789 are the
same as the upper partials numbered i 2 3
45678 respectively. As even the
Author has occasionally failed to carry out
this distinction in the original German text,
and other writers have constantly neglected it,
too much weight cannot be here laid upon it.
The presence or absence of the word upjyer
before the word partial must always be care-
fully observed. It is safer never to speak of
an upper partial by its ordinal number, but to
call the fifth upper partial the sixth partial,
omitting the word upper and increasing the ^
ordinal number by one place. And so in
other cases. — Translator,]
t The law of these vibrations may be
popularly explained by means of the construc-
tion in fig. 10. Suppose a point to describe
the circle of which c is the centre with a
uniform velocity, and that an observer stands
at a considerable distance in the prolongation
of the line e h, so that he does not see the
surface of the circle but only its edge, in
which case the point will appear merely to
move up and down along its diameter a b.
This up and down motion would take place
exactly according to the law of pendular
vibration. To represent this motion graphi-
FlG.
cally by means of a curve, divide the length
e g, supposed to correspond to the time of a
single period, into as many (here 12) equal
parts as the circumference of the circle, and
draw the perpendiculars i, 2, 3, &c., on the
dividing points of the line e g, in order, equal
in length to and in the same direction with,
those drawn in the circle from the correspond-
ing points I, 2, 3, <&c. In this way we obtain
ihe curve drawn in fig. 10, which agrees in
form with that drawn by the tuning-fork,
tg. 6, p. 206, but is of a larger size. Mathe-
matioidly expressed, the distance of the vibrat-
ing point from its mean position at any time
is equal to the sine of an arc proportional to
the corresponding time, and henoe the form of
simple vibrations are also called the sin&-
vibrati/yiis [and the above cui*ve is also known
as the curve of sinesl.
Digitized by VjOOQIC
24
DEFINITION OP TERMS EMPLOYED.
PART I.
the general senae of a musical tone, leaving the context or a prefixed qualification
to determine whether it is simple or compound. A compound tone will often be
briefly called a note, and a simple tone will also be frequently cuXleA tk partial, when
used in connection with a compound tone ; otherwise, the full expression simple
tone will be employed. A note has, properly speaking, no single pitch, as it is
made up of various partials each of which has its own pitch. By the pitch of a
note or compound tone then we shall therefore mean the pitch of its lowest partial
or prime tone. By a chord or combination of tones we mean several musical tones
(whether simple or compound) produced by different instruments or different parts
of the same instrument so as to be heard at the same time. The facts here adduced
shew us then that every musical tone in which harmonic upper partial tones can
be distinguished, although produced by a single instrument, may really be con-
sidered as in itself a chord or combination of various simple tones.*
If ♦ [The above paragraph relating to the
Engliedi terms used in this translation, neces-
sarily differs in many respects from the original,
in which a jnstification is given of the use
made by the Author of certain Oerman ex-
pressions. It has been my object to employ
terms which should be thoroughly English,
and should not in any way recall the German
words. The word tone in English is extremely
ambiguous. Prof. Tyndall {Lectures on Sounds
2nd ed. 1869, p. 117) has ventured to define a
tone as a simple tone^ in agreement with Prof.
Helmholtz, who in the present passage limits
the Grerman word Ton in the same way. But
I felt that an English reader could not be
safely trusted to keep this very peculiar and
important class of musical tones, which he
has very rarely or never heard separately,
invariably distinct from those musical tones
% with which he is familiar, unless the word
tone were uniformly qualified by the epithet
simple. The only exception I could make was
in tiie case of a partial tone, which is received
at once as a new conception. Even Prof.
Helmholtz himself has not succeeded in using
his word Ton consistently for a simple tone
only, and this was an additional warning to
me. English musicians have been also in
the habit of using tone to signify a certain
musical interval, and semitone for half of that
interval, on the equally tempered scale. In
this case I write Tone and Semitone with
capital initials, a practice which, as already
explained (note, p. ijd',) I have found con-
venient for the names of all intervals, as
Thirds, Fifths, Ao. Prof. Hehnholtz uses the
word Klang for a musical tone, which gene-
f rally, but not always, means a compound tone.
Prof. Tyndall (ibid,) therefore proposes to use
the English word clang in the same sense.
But dang has already a meaning in English,
thus de&ied by Webster: *a sharp shrill
sound, made by striking together metallic
substances, or sonorous bodies, as the clang
of arms, or any like sound, as the clang of
trumpets. This word implies a degree of
harshness in the sound, or more harshness
than dink,* Interpreted scientifically, then,
clang according to this definition, is either
noise or one of those musical tones with in-
harmonic upper partials, which will be sub-
sequently explained. It is therefore totally
unadapted to represent a tus^^iual tone in
generiU, for which the simple word tone seems
eminently suited,' being of course originally
the tone produced by a stretched string. The
common word note, properly the mark by
which a musical tone is written, will also, in
accordance with the general practice of musi-
cians, be used for a musical tone, which is
generally compound, without necessarily im-
plying that it is one of the few reco^iised
tones in our musical scale. Of oouree, if
dang could not be used. Prof. Tyndall's
suggestion to translate Prof. Helmholtz's
Klangfarbe by clangtiwt (ibid,) fell to the
ground. I can find no valid season for sup-
planting the time-honoured expression gualtiy
of tone. Prof. Tyndall (ibid,) quotes Dr.
Young to the effect that * this quality of sound
is sometimes called its register, colour, ot
timbre.' Register has a distinct meaning in
vocal music which must not be disturbed.
Timbre, properly a kettledrum, then a helmet,
then the coat of arms surmounted with a
helmet, then the official stamp bearing that
coat of arms (now used in France for a
postage label), and then the mark which
declared a thing to be what it pretends to be,
Bums*s ' guinea's stamp,' is a foreign word,
often odiously mispronounced, and not worth
preserving. Colour I have never met with
as applied to music, except at most as a
passing metaphorical expression. But the
difference of tones in quality is familiar to
our language. Then as to the Partial Tones,
Prof. Helmholtz uses TheilUSne and Partial-
tone, which are aptly Englished by partial
simple tones. The words simple and tone,
however, may be omitted when partials is
employed, as partials are necessarily both
tones and simple. The constilv^nt tones of a
chord may be either simple or compound.
The Qrundton or fundamental tone of a
compound tone tiien becomes its prims tone,
or briefly its prime. The Grundton or root of
a chord will be further explained liereafter.
Upper partial (simple) tones, that is, the
partials exclusive of the prime, even when
hoArmonic, (that is, for the most part, belong-
ing to the first six partial tones,) must be
distinguished from the sounds usually called
harmonics when produced on a violin or harp
for instance, for such harmonics are not neces-
sarily simple tones, but are more generally
compounds of soms of the complete series of
partial tones belonging to the musical tone of
the whole string, selected by damping the
remainder. The fading harmonics heard in
listening to the sound of a pianoforte string,
struck and undamped, as the sound dies away,
are also compound and not simple partial
tones, but as they have the successive partials
for their successive primes, they have the
Digitized by V^jOOQlC
CHAPS. I. II. COEXISTENCE OF DISTINCT WAVES OF SOUND. 25
Now, since quality of tone, as we have seen, depends on the form of vibration,
which also determines the occurrence of upper partial tones, we have to inquire
how &r differences in quality of tone depend on different force or loudness of upper
partials. This inquiry will be found to give a means of clearing up our concep-
tions of what has hitherto been a perfect enigma, — ^the nature of quality of tone.
And we must then, of course, attempt to explain how the ear manages to analyse
every musical tone into a series of partial tones, and what is the meaning of this
analysis. These investigations will engage our attention in the following chapters.
CHAPTER II.
OK THE COMPOSITION OF VIBBATIONS.
At the end of the last chapter we came upon the remarkable fact that the human
ear is capable, under certain conditions, of separating the musical tone produced
by a single musical instrument, into a series of simple tones, namely, the prime
I«rtial tone, and the various upper partial tones, each of which produces its own
separate sensation. That the ear is capable of distinguishing from each other
tones proceeding from different sources, that is, which do not arise from one and
the same sonorous body, we know from daily experience. There is no difficulty
during a concert in following the melodic progression of each individual instru-
ment or voice, if we direct our attention to it exclusively ; and, after some practice,
most persons can succeed in following the simultaneous progression of several
united parts. This is true, indeed, not merely for musical tones, but also for
noises, and for mixtures of music and noise. When several persons are speaking
at once, we can generally listen at pleasure to the words of any single one of them, IT
and even understand those words, provided that they are not too much overpowered
by the mere loudness of the others. Hence it foUows, first, that many different
trains of waves of sound can be propagated at the same time through the same
mass of air, without mutual disturbance ; and, secondly, that the human ear is
capable of again analysing into its constituent elements that composite motion of
the air which is produced by the simultaneous action of several musical instru-
ments. We will first investigate the nature of the motion of the air when it is
produced by several simultaneous musical tones, and how such a compound motion
is distinguished from that due to a single musical tone. We shall see that the ear
has no decisive test by which it can in all cases distinguish between the effect of a
pitch of those partials. But these fading meaning uppery but the English preposition
harvKmics are not regular compound tones of over is equivalent to the German preposition
the kind described on p. 22a, because the lower fiber. Compare Obergahn, an ' upper tooth/ f
partials are absent one after another. Both i.e. a tooth in the upper jaw» with Uebermhn^
sets of harmonics serve to indicate the exist- an * overtooth/ i.e. one grown over another,
enoe and place of the partials. But they are a projecting tooth. The continual recurrence
no more those upper partial tones themselves, of such words as clang, clangtifU, overtone^
than the original compound tone of the string would combine to give a strange un-English
is its own prime. Great confusion of thought appearance to a translation from the German,
having, to my own knowledge, arisen from Chi the contrary I have endeavoured to put it
eonfounding such ^rmomc« with upper partial into as straightforward English as possible.
t€ne$, I have generally avoided using ^e am- But for those acquainted with the original and
biguons substantive ^rmonic. Properly speak- with Prof. Tyndall's work, this explanation
ing the harmonics of any compound tone are seemed necessary. Finally I would caution
other oompound tones of which the primes are the reader against using overtones for partial
partials of the original compound tone of tones in general, as almost every one who
which they are said to be harmonics. Prof. adopts Prof. Tyndall's word is in the habit of
Hehnholtz's term OberWne is merely a con- doing. Indeed I have in the course of this
traction for OberpartialtOne, but the casual translation observed, that even Prof. Helmholtz
resemblance of the sounds of ober and over, has himself has been occasionally misled to em-
led Prof. Tyndall to the erroneous translation ploy OberUitie in the same loose manner. See
overtones. The German obcr is an adjective my remarks in note, p. 23c.— TransZato-.l
Digitized by VjOOvIc
26. COMPOSITION OF WAVES. part i.
motion of the air caused by several diflferent musical tones arising from different
sources, and that caused by the musical tone of a single sounding body. Hence
the ear has to analyse the composition of single musical tones, under proper con-
ditions, by means of the same febculty which enabled it to analyse the compositicHi
of simultaneous musical tones. We shall thus obtain a clear conception of what
is meant by analysing a single musical tone into a series of partial simple tones^
and we shall perceive that this phenomenon depends upon one of the most
essential and fundamental properties of the human ear.
We begin by examming the motion of the air which corresponds to several
simple tones acting at the same time on the same mass of air. To illustrate this
kind of motion it will be again convenient to refer to the waves formed on a calni
surface of water. We have seen (p. 9a) that if a point of the surface is agitated by a
stone thrown upon it, the agitation is propagated in rings of waves over the surface
If 1,0 more and more distant points. Now, throw two stones at the same time on to
different points of the surface, thus producing two centres of agitation. Each will
give rise to a separate ring of waves, and the two rings gradually expanding, will
finally meet. Where the waves thus come together, the water will be set in
motion by both kinds of agitation at the same time, but this in no wise prevents
both series of waves from advancing further over the surface, just as if each were
alone present and the other had no existence at all. As they proceed, those
parts of both rings which had just coincided, again appear separate and unaltered
in form. These httle waves, caused by throwing in stones, may be accompanied
by other kinds of waves, such as those due to the wind or a passing steamboat.
Our circles of waves will spread out over the water thus agitated, with the same
quiet regularity as they did upon the calm surface. Neitlier wiU the greater waves
be essentially disturbed by the less, nor the less by the greater, provided the waves
never break ; if that happened, their regular course would certainly be impeded.
% Indeed it is seldom possible to survey a large surface of water from a high
point of sight, without perceiving a great multitude of different systems of waves,
mutually overtopping and crossing each other. This is best seen on the surface of
the sea, viewed from a lofty cHff, when there is a lull after a stiff breeze. We first
see the great waves, advancing in far-stretching ranks from the blue distance, here
and there more clearly marked out by their white foaming crests, and following
one another at regular intervals towards the shore. From the shore they rebound,
in different directions according to its sinuosities, and cut obliquely across the
advancing waves. A passing steamboat forms its own wedge-shaped wake of
waves, or a bird, darting on a fish, excites a small circular system. The eye of the
spectator is easily able to pursue each one of these different trains of waves, great
and small, wide and narrow, straight and curved, and observe how each passes
over the surface, as undisturbedly as if the water over which it flits were not
agitated at the same time by other motions and other forces. I must own that
IT whenever I attentively observe this spectacle it awakens in me a peculiar kind of
intellectual pleasure, because it bares to the bodily eye, what the mind's eye grasps
only by the help of a long series of complicated conclusions for the waves of the
invisible atmospheric ocean.
We have to imagine a perfectly similar spectacle proceeding in the interior of a
baU-room, for instance. Here we have a number of musical instruments in action*
speaking men and women, rustling garments, gliding feet, clinking glasses, and so
on. All these causes give rise to systems of waves, which dart through the mass
of air in the room, are reflected from its walls, return, strike the opposite wall, are
again reflected, and so on till they die out. We have to imagine that from the
mouths of men and from the deeper musical instruments there proceed waves of
from 8 to 12 feet in length [c to jP], from the lips of the women waves of 2 to 4
feet in length [c" to &], from the rustling of the dresses a fine small crumple of
wave, and so on ; in short, a tumbled entanglement of the most different kinds of
motion, complicated beyond conception.
Digitized by VjOOQlC
CHAP. XI. ALGEBRAICAL ADDITION OF WAVES. 27
And jet, as the ea«r is able to distinguish all the separate constitaent parts of
this confused whole, we are forced to conclude that all these different systems of
wave coexist in the mass of air, and leave one another mutually undisturbed.
Bat how is it possible for them to coexist, since every individual train of waves has
at any particular point in the mass of air its own particular degree of condensa-
tion and rarefaction, which determines the velocity .of the particles of air to this
side or that ? It is evident that at each point in the mass of air, at each instant
of time, there can be only one single degree of condensation, and that the particles
of air can be moving with only one single determinate kind of motion, having only
one single determinate amount of velocity, and passing in only one single deter-
minate direction.
What happens imder such circumstances is seen directly by the eye in the
waves of water. If where the water shews large waves we throw a stone in, the
waves thus caused will, so to speak, cut into the larger moving surface, and this ^
sarface will be partly raised, and partly depressed, by the new waves, in such a
way that the fresh crests of the rings wiU rise just as much above, and the troughs
sink just as much below the curved surfaces of the previous larger waves, as they
would have risen above or sunk below the horizontal surface of calm water.
Hence where a crest of the smaller system of rings of waves comes upon a crest
of the greater system of waves, the surface of the water is raised by the sum of
the two heights, and where a trough of the former coincides with a trough of the
latter, the aurfiace is depressed by the sum of the two depths. This may be
expressed more briefly if we consider the heights of the crests above the level of
the surface at rest, as positive magnitudes, and the depths of the troughs as negative
magnitudes, and then form the so-caUed algebraical sum of these positive and
negative magnitudes, in which case, as is well known, two positive magnitudes
(heights of crests) must be added, and similarly for two negative magnitudes (depths
of troughs) ; but when both negative and positive concur, one is to be subtracted H
from the other. Performing the addition then in this algebraical sense, we can
express our description of the surface of the water on which two systems of waves
concur, in the following simple manner : The distance of the surface of the water
at any point from its position of rest is at any moment equal to the [algebraical]
sum of the distances at which it would have stood had each wave acted separately
at the same place and at tJie same time.
The eye most clearly ^and easily distinguishes the action in such a case as has
been just adduced, where a smaller circular system of waves is produced on a large
rectilinear system, because the two systems are then strongly distinguished from
each other both by the height and shape of the waves. But with a little attention
the eye recognises the same fact even when the two systems of waves have but
slightly different forms, as when, for example, long rectilinear waves advancing
towards the shore concur with those reflected from it in a slightly different
direction. In this case we observe those well-known comb-backed waves where H
the crest of one system of waves is heightened at some points by the crests of the
other system, and at others depressed by its troughs. The multipHcity of forms
is here "extremely great, and any attempt to describe them would lead us too
far. The attentive observer will readily comprehend the result by examining
any disturbed surface of water, without further description. It will suffice for our
purpose if the first example has given the reader a clear conception of what is
meant by adding waves together,*
Hence although the surface of the water at any instant of time can assume
only one single form, while each of two different systems of waves simultaneously
attempts to impress its own shape upon it, we are able to suppose in the above
*^ The velocities and displacements of the addition of waves as is spoken of in the text, I
particles of water are also to be added accord- is not perfectly correct, unless the heights of 1
ing to the law of the so-called parallelogram the waves are infinitely small in comparison I
of forces. Strictly sijeaking, such a simple with their lengths. \
Digitized by V^jOOQlC
28 ALGEBRAICAL ADDITION OP WAVES. part i.
sense that the two systems coexist and are superimposed, by considering the
actual elevations and depressions of the sur&ce to be suitabW. separated into two
parts, each of which belongs to one of the systems alone.
In the same sense, then, there is also a superimposition of different systems of
sound in the air. By each train of waves of sound, the density of the air and the
velocity and position of the particles of air, are temporarily altered. There are
places in the wave of sound comparable with the crests of the waves of water, in
which the quantity of the air is increased, and the air, not having free space to
escape, is condensed; and other places in the mass of air, comparable to the
l^oughs of the waves of water, having a diminished quantity of air, and hence
diminished density. It is true that two different degrees of density, produced by
two different systems of waves, cannot coexist in the same place at the same time ;
nevertheless the condensations and rarefactions of the air can be (algebraically)
il added, exactly as the elevations and depressions of the surface of the water in the
former case. Where two condensations are added we obtain increased condensation,
where two rarefactions are added we have increased rarefaction ; while a concur-
rence of condensation and rarefaction mutually, in whole or in part, destroy or
neutralise each other.
The displacements of the particles of air are compounded in a similar manner.
If the displacements of two different systems of waves are not in the same direc-
tion, they are compounded diagonally ; for example, if one system would drive a
particle of air upwards, and another to the right, its real path will be obliquely
upwards towards the right. For our present purpose there is no occasion to enter
more particularly into such compositions of motion in different directions. We
are only interested in the effect of the mass of air upon the ear, and for this we
are only concerned with the motion of the air in the passages of the ear. Now the
passages of our ear are so narrow in comparison with the length of the waves of
H sound, that we need only consider such motions of the air as are parallel to the
axis of the passages, and hence have only to distinguish displacements of the
particles of air outwards and inwards, that is towards the outer air and towards
the interior of the ear. For the magnitude of these displacements as well as for
their velocities with which the particles of air move outwards and inwards, the
same (algebraical) addition holds good as for the crests and troughs of waves of
water.
Hence, when several sonorous bodies in the surrounding atmosphere, simul-
taneously excite different systems of waves of sound, the changes of density of the
air, and the displacements and velocities of the particles of the air within the
passages of the ear, are each equal to the (algebraical) sum of the corresponding
changes of density, displacements, and velocities, which each system of waves
wotUd have separately produced, if it had acted independently ; * and in this sense
we can say that all the separate vibrations which separate waves of sound would
% have produced, coexist undisturbed at the same time within the passages of our ear.
After having thus in dnswer to the first question explained in what sense it is
possible for several different systems of waves to coexist on the same surface of
water or within the same mass of air, we proceed to determine the means possessed
by our organs of sense, for analysing this composite whole into its original consti-
tuents.
I have already observed that an eye which surveys an extensive and disturbed
surface of water, easily distinguishes the separate systems of waves from each
other and follows their motions. The eye has a great advantage over the ear in
being able to survey a large extent of surface at the same moment. Hence the
eye readily sees whether the individual waves of water are rectilinear or curved,
and whether they have the same centre of curvature, and in what direction they
* The eame is true for the whole moss of according to the law of the parallelogram of
external air, if only the addition of the dis* forces,
placements in different directions is made
Digitized by VjOOQIC
CHAP. iL EYE AND EAR CONTEASTED. 29
are adTancing. All these observations assist it in determining whether two systems
of waves are connected or not, and hence in discovering their corresponding parts.
Moreover, on the surface of the water, waves of unequal length advance with
unequal velocities, so that if they coincide at one moment to such a degree as to
be difficult to distinguish, at the next instant one train pushes on and the other
lags behind, so that they become again separately visible. In this way, then, the
observer is greatly assisted in referring each system to its point of departure, and
in keeping it distinctly visible during its further course. For the eye, then, two
systems of waves having different points of departure can never coalesce; for
example, such as arise from two stones thrown into the water at different points.
If in any one place the rings of wave coincide so closely as not to be easily
separable, they always remain separate during the greater part of their extent.
Hence the eye could not be easily brought to confuse a compound with a simple
undulatory motion. Yet this is precisely what the ear does under similar circum- ^
stances when it separates the musical tone which has proceeded from a single
source of sound, into a series of simple partial tones.
But the ear is much more unfavourably situated in relation to a system of waves
of sound, than the eye for a system of waves of water. The ear is affected only
by the motion of that mass of air which happens to be in the immediate neigh-
bourhood of its tympanum within the aural passage. Since a transverse section
of the aural passage is comparatively smaU in comparison with the length of waves
of sound (which for serviceable musical tones varies from 6 inches to 32 feet),* it
eorresponds to a single point of the mass of air in motion. It is so smaU that
distinctly different degrees of density or velocity could scarcely occur upon it,
because the positions of greatest and least density, of greatest positive and nega-
tive velocity, are always separated by half the length of a wave. The ear is
therefore in nearly the same condition as the eye would be if it looked at one point
of the surface of the water through a long narrow tube, which would permit of ^
seeing its rising and fEkUing, and were then required to undertake an analysis
of the compound waves. It is easily seen that the eye would, in most cases,
completely fail in the solution of such a problem. The ear is not in a condition
to discover how the air is moving at distant spots, whether the waves which strike
it are spherical or plane, whether they interlock in one or more circles, or in what
direction they are advancing. The circumstances on which the eye chiefly depends |
for forming a judgment, are all absent for the ear.
If, then, notwithstanding all these difficulties, the ear is capable of distin-
guishing musical tones arising from different sources — and it really shews a
marvellous readiness in so doing — it must employ means and possess properties
altogether different from those employed or possessed by the eye. But whatever
these means may be — and we shall endeavour to determine them hereafter — ^it
is clear that the analysis of a composite mass of musical tones must in the first
place be closely connected with some determinate properties of the motion of the f
air, capable of impressing themselves even on such a very minute mass of air as
that contained in the aural passage. If the motions of the particles of air in this
passage are the same on two different occasions, the ear will receive the same
sensation, whatever be the origin of those motions, whether they spring from one
or several sources.
We have already explained that the mass of air which sets the tympanic
membrane of the ear in motion, so far as the magnitudes here considered are
concerned, must be looked upon as a single point in the surrounding atmosphere.
Are there, then, any peculiarities in the motion of a single particle of air which
would differ for a single musical tone, and for a combination of musical tones ?
We have seen that for each single musical tone there is a corresponding periodical
* [These are of course rather more than flue organ pipes. See Chap. V. sect. 5, and
twice the length of the corresponding open compare p. 26d,— Translator,]
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so
COMPOSITION OF. SIMPLE WAVES.
PART I.
motion of the aii^, and that its pitch is determined by the length of the periodic
time, but that the kind of motion during any one single period is perfectly arbitrary,
and may indeed be infinitely various. If then the motion of the air lying in the
aural passage is not periodic, or if at least its periodic time is not as short as that
of an audible musical tone, this fact will distinguish it from any motion which
belongs to a musical tone ; it must belong either to noises or to several simultaneous
musical tones. Of this kind are really the greater number of cases where the dif-
ferent musical tones have been only accidentally combined, and are therefore not
designedly framed into musical chords; nay, even where orchestral music is per-
formed, the method of tempered tuning which at present prevails, prevents an
accurate fulfilment of the conditions under which alone the resulting motion of
the air can be exactly periodic. Hence in the greater number of cases a want
of periodicity in the motion might furnish a mark for distinguishing the presence
^ of a composite mass of musical tones.
But a composite mass of musical tones may also give rise to a purely periodic
motion of tJie air, namely, when all tfie mtisical tones which intermingle, have
pitch numbers which are all midtiples of one and the same old number, or, which
Fm. IX.
B e
comes to the same thing, when all tliese musical tones, so far as their pitch is
concerned, may be regarded as the upper partial tones of the same prime tone. It
^ was mentioned in Chapter I. (p. 22a, b) that the pitch numbers of the upper partial
tones are multiples of the pitch number of the prime tone. The meaning of this
rule will be clear from a particular example. The curve A, fig. 11, represents a
pendular motion in the manner explained in Chapter I. (p. 216), as produced in the
air of the aural passage by a tuning-fork in action. The horizontal lengths in the
curves of fig. 11, consequently represent the passing time, and the vertical heights
the corresponding displacements of the particles of air in the aural passage. Now
suppose that with the first simple tone to which the curve A corresponds, there is
sounded a second simple tone, represented by the curve B, an Octave higher than
the first. This condition requires that two vibrations of the curve B should be
made in the same time as one vibration of the curve A. In A, the sections of the
curve do3 and h 8^ are perfectly equal and similar. The curve B is also divided
into equal and similar sections e c and c cj by the points e, c, cj. We could cer-
tainly halve each of the sections e c and c c^, and thus obtain equal and similar
sections, each of which would then correspond to a single period of B. But by
Digitized by V^jOOQlC
CHAP. II. COMPOSITION OF SIMPLE WAVES. 31
taking sections consisting of two periods of B, we divide >B into larger sections,
each of which is of the same horizontal length, and hence corresponds to the same
duration of time, as the sections of A.
If, then, both simple tones are heard at once, and the times of the points e and
do, c and 8, e^ and 8, coincide, the heights of the portions of the section of carve
e € have to be [algebraically] added to heights of the section of curve do8, and
similarly for the sections c C| and 88|. The result of this addition is shewn in tlie
carve 0. The dotted line is a duplicate of the section d^fi in the curve A. Its
object is to make the composition of the two sections immediately evident to the
eye. It is easily seen that the curve C in every place rises as much above or sinks
as much below the curve A, as the curve B respectively rises above or sinks
beneath the horizontal line. The heights of the curve 0 are consequently, in ac-
cordance with the rule for compounding vibrations, equal to the [algebraical] sum
of the corresponding heights of A and B. Thus the perpendicular C| in C is the f
sum of the perpendiculars a, and b, in A and B ; the lower part of this perpen-
dicular C], from the straight hne up to the dotted curve, is equal to the perpen-
dicular a|, and the upper part, from the dotted to the continuous curve, is equal to
the perpendicular bj. On the other hand, the height of the perpendicular Cq is
equal to the height aa diminished by the depth of the fail bj. And in the same
way all other points in the curve C are found.*
It is evident that the motion represented by the curve C is also periodic, and
that its periods have the same duration as those of A. Thus the addition of the
section do^ of A and e c of B, must give the same result as the addition of tlie
perfectly equal and similar sections S S^ and c ci, and, if we supposed both curves
to be continued, the same would be the case for all the sections into which they
would be divided. It is also evident that equal sections of both curves could not
continually coincide in this way after completing the addition, unless the curves thus
added could be also separated into exactly equal and similar sections of the same ^
length, as is the case in fig. 1 1 , where two periods of B last as long or have the
same horizontal length as one of A. Now the horizontal lengths of our figure
represent time, and if we pass from the curves to the real motions, it results that
the motion of air caused by the composition of the two simple tones, A and B, is
also periodic, just because one of these simple tones makes exactly twice as many
vibrations as the other in the same time.
It is easily seen by this example that the peculiar form of the two curves A
and B has notliing to do with the fact that their sum C is also a periodic curve.
Whatever be the form of A and B, provided that each can be separated into equal
and similar sections which have the same horizontal lengths as the equal and
similar sections of the other — no matter whether these sections correspond to one
or two, or three periods of the individual curves— then any one section of the cune
A compounded with any one section of the curve B, will always give a section
of the curve C, which will have the same length, and will be precisely equal and f
similar to any other section of the curve C obtained by compounding any other
section of A with any other section of B.
When such a section embraces several periods of the corresponding curve (as in
fig. II, the sections e c and c ci each consist of two periods of the simple tone B,)
then the pitch of this second tone B, is that of an upper partial tone of a prime
(as tlie simple lone A in fig. 11), whose period has the length of that principal
section, in accordance with the rule above cited.
In order to give a slight conception of the multiplicity of forms producible by
comparatively simple compositions, I may remark that the compound curve would
* [Readers not ased to geometrical con- sponding perpendiculars in A and 6 in proper
stractions are strongly recommended to trace directions, and joining the extremities of the
the two enrves A and B, and to construct the lengths thus found by a curved line. In this
curve C from them, by drawing a number of way only can a clear conception of the com-
perpendienlars to a straight line, and then position of vibrations be rendered sufficiently
MMkig off upon them the lengths of the corre- familiar for subFeqnent u»e.—TraiisJator.]
Digitized by VjOOQlC
32
DIFFERENCE OF PHASE.
PABT I.
receive another form if the carves B, fig. ii, were displaced a little with respect to
the curve A before the addition were commenced. Let B be displaced by being
slid to the right mitil the point e falls under dj in A, and the composition will then
give the curve D with narrow crests and broad troughs, both sides of the crest
being, however, equally steep ; whereas in the curve 0 one side is steeper than the
other. If we displace the curve B still more by sliding it to the right till e fialls
under d,, the compound curve would resemble the reflection of G in a mirror :
that is, it would have the same form as C reversed as to right and left ; the steeper
inclination which in C lies to the left would now he to the right. Again, if we
displace B till e fedls under dj we obtain a curve similiar to D, fig. ii, but reversed
as to up and down, as may be seen by holding the book upside-down, the crests
being broad and the troughs narrow.
Pro
All these curves with their various transitional forms are periodic curves.
Other composite periodic curves are shewn at G, D, fig. 12 above, where they are
compounded of the two curves A and B, having their periods in the ratio of i to 3.
The dotted curves are as before copies of the first complete vibration or period
of the curve A, in order that the reader may see at a glance that the compound
curve is always as much higher or lower than A, as B is higher or lower than the
horizontal Hne. In G, the curves A and B are added as they stand, but for D the
curve B has been first slid half a wave's length to the right, and then the addition
f has been effected. Both forms differ £rom each other and firom all preceding ones.
G has broad crests and broad troughs, D narrow crests and narrow troughs.
In these and similar cases we have seen that the compound motion is per£actly
and regularly p^odic, that is, it is exactly of the same kind as if it proceeded
from a single musical tone. The curves compounded in these examples correspond
to the motions of single simple tones. Thus, the motions shown in fig. 11 (on
p. 30&, c) might have been produced by two tuning-forks, of which one sounded an
Octave higher than the other. But we shall hereafter see that a flute by itself
when gently blown is sufficient to create a motion of the air corresponding to that
shown in 0 or D of fig. 11. The motions of fig. 12 might be produced by two
tuning-forks of which one sounded the twelfth of the other. Also a single closed
organ pipe of the narrower kind (the stop called Quintaten^) would give nearly the
same motion as that of G or D in fig. 12.
* [The names of the stops on German
organs do not always agree with those on
English organs. I find it best, therefore, ncyt
to translate them, but to give their ezplaiui.
Digitized by V^jOOQlC
CHAP. II. ANALYSIS INTO SIMPLE VIBRATIONS. 33
Here, then, the motion of the air in the aural passage has no property by which
the composite* musical tone can be distinguished from the single musical tone.
If the ear is not assisted by other accidental circumstances, as by one tuning-fork
beginning to sound before the other, so that we hear them struck, or, in the other
case, the rustling of the wind against the mouthpiece of the^flute or lip of the
organ pipe, it has no means of deciding whether the musical tone is simple or
composite.
Now, in what relation does the ear stand to such a motion of the air ? Does
it analyse it, or does it not ? Experience shews us that when two tuning-forks, an
Octave or a Twelfth apart in pitch, are sounded together, the ear is quite able to
distinguish their simple tones, although the distinction is a Httle more diMcult
with these than with other intervals. But if the ear is able to analyse a compo-
site musical tone produced by two tuning-forks, it cannot but be in a condition to
carry out a similar analysis, when the same motion of the air is produced by a ^
single flute or organ pipe. And this is really the case. The single musical tone
of such instruments, proceeding from a single source, is, as we have already men-
tioned, analysed into partial simple tones, consisting in each case of a prime tone,
and one upper partial tone, the latter being different in the two cases.
The analysis of a single musical tone into a series of partial tones depends,
then, upon the same property of the ear as that which enables it to distinguish
different musical tones from each other, and it must necessarily effect both analyses
by a rule which is independent of the fact that the waves of sound are produced
by one or by several musical instruments.
The rule by which the ear proceeds in its analysis was first laid down as
generally true by G. S. Ohm. Part of this rule has been already enunciated in
the last chapter (p. 23a), where it was stated that only that particular motion of
the air which we have denominated a simple vibration^ for which the vibrating
particles swing backwards and forwards according to the law of pendular motion, ^
is capable of exciting in the ear the sensation of a single simple tone. Every
motion of tlie air, then, which corresponds to a composite viass of musical tones,
is, according to Ohm's law, capable of being ayvaly^ed into a sum of simple pen-
dular vibrations, and to each such siwjle simple vibration corresponds a simple
tone, sensible to the ear, and liaving a pitch determined by the periodic time of the
corresponding motion of the air.
The proofs of the correctness of this law, the reasons why, of all vibrational
forms, only that one which we have called a simple vibration plays such an
important part, must be left for Chapters IV. and VI. Our present business is
only to gain a clear conception of what the rule means.
The simple vibrational form is inalterable and always the same. It is only its
amplitude and its periodic time which are subject to change. But we have seen
in figs. II and 12 (p. 306 and p. 326) what varied forms the composition of only two
simple vibrations can prodiice. The number of these forms might be greatly in- ^
creased, even without introducing fresh simple vibrations of different periodic
times, by merely changing the proportions which the heights of the two simple
tions from E. J. Hopkins's TJie Organ, its in other cases, • a pipe for sounding the Twelfth
' History and Construction, 1870, pp. 444-448. in addition to the fundamental tone.' It seems
In this case Mr. Hopkins, following other to be proporly the English stop • Twelfth,
aathoritiea, prints the word ' quintato^i/ and Octave Quint, Duodecitna,^ No. 6ii,p. 141 of
deiines it, in 16 feet tone, as * double stopi)ed Hopkins. — 2Va?w/ator.j
diapason, of rather small scale, producing the * [The reader must distinguish between
Twelfth of the fondamental sound, as well as single and simple musical tones. A single tone
the ground-tone itself, that is, somiding the may be a compound tone inasmuch as it may
16 and s\ ^^* tones,' which means sounding the be compounded of several simple musical tones,
notes beginning with C,, simultaneously witn the but it is single because it is produced hy one
notes beginning with Q, which is called the sounding body. A composite musical toue is
5^ foot tone, because according to the organ- necessarily compound, but it is called composite
makers' theory (not practice) the length of the because it is made up of tones (simple or com-
G pipe is ^ of the length of the C pipe, and i^of pound) produced by several sounding bodies. —
16185^. |.See p. I5<i', note J.j And smiilar ly. Translator.]
Digitized by V^OOQ IC
34 ANALYSIS INTO SIMPLE VIBRATIONS. paet i.
vibrational curves A and B bear to each other, or displacing the curve B by other
distances to the right or left, than those already selected in the figures. By these
simplest possible examples of such compositions, the reader will be able to form
some idea of the enormous variety of forms which would result from using more
than two simple forms of vibration, each form representing an upper partial tone
of the same prime, and hence, on addition, always producing fresh periodic curves-
We should be able to make the heights of each single simple vibrational curve
greater or smaller at pleasure, and displace each one separately by any amount in
respect to the prime, — or, in physical language, we sliould be able to alter their
amplitudes and the difference of their phases ; and each such alteration of ampli-
tude and difference of phase in each one of the simple vibrations would produce a fresh
change in the resulting composite vibrational form. [See App. XX. sect. M. No. 2.]
The multiplicity of vibrational forms which can be thus produced by the corn-
el position of simple pendular vibrations is not merely extraordinarily great : it is so
great that it cannot be greater. The French mathematician Fourier has proved
the correctness of a mathematical law, which in reference to our present subject
may be thus enunciated: Any given regular periodic form of vibration can
always be produced by the addition of simple vibrations^ having pitch numbers
which are once, twice, thrice, four times, dc, as great as the pitch numbers of the
given motion.
The amplitudes of the elementary simple vibrations to which the height of our
wave-curves corresponds, and the difference of phase, that is, the relative amount
of horizontal displacement of the wave-curves, can always be found in every given
case, as Fourier has shewn, by pecuhar methods of calculation, (wliich, however,
do not admit of any popular explanation,) so that any given regularly periodic
motion can always be exhibited in one single way, and in no other way whatever^
as the sum of a certain number of pendular vibrations,
^ Since, according to the results already obtained, any regularly periodic motion
corresponds to some musical tone, and any simple pendular vibration to a simple
musical tone, these propositions of Fourier may be thus expressed in acoustical
terms :
Any vibrational motion of the air in the entrance to the ear, correspondijig to a
musical tone, may be always, and for each case only in one single way, exhibited as
the sum of a number of simple vibratioyial motions, corresponding to the partials
of this musical tone.
Since, according to these propositions, any form of vibration, no matter what
shape it may take, can be expressed as the sum of simple vibrations, its analysis
into such a sum is quite independent of the power of the eye to perceive, by looking
at its representative curve, whether it contains simple vibrations or not, and if it
does, what they are. I am obUged to lay stress upon this point, because I have by
no means unfrequently found even physicists start on the false hypothesis, that the
^ vibrational form must exhibit little waves corresponding to the several audible
upper partial tones. A mere inspection of the figs. 11 and 12 (p. 306 and p. 326)
will snfiice to shew that although the composition can be easily traced in the parts
where the curve of the prime tone is dotted in, this is quite impossible in those
parts of the curves C and D in each figure, where no such assistance has been
provided. Or, if we suppose that an observer who had rendered himself thoroughly
familiar with the curves of simple vibrations imagined that he could trace the com-
position in these easy cases, he would certainly utterly fail on attempting to dis-
cover by his eye alone the composition of such curves as are shewn in figs. 8
and 9 (p. 21c). In these will be found straight lines and acute angles. Perhaps
it will be asked how it is possible by compounding such smooth and uniformly
rounded curves as those of our simple vibrational forms A and B in figs. 1 1 and
12, to generate at one time straight lines, and at another acute angles. The
answer is, that an infinite number of simple vibrations are required to generate
curves with such discontinuities as are there shewn. But when a great many
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CHAP.n. ANALYSIS INTO SIMPLE VIBRATIONS. 35
such cnrves are combined, and are so chosen that in certain places they all bend
in the same direction, and in others in opposite directions, the curvatures mutually 1
strengthen each other in the first case, finally producing an infinitely great curva-
ture, that is, an acute angle, and in the second case they mutually weaken each
other, 80 that ultimately a straight line results. Hence we can generally lay it
down as a rule that the force or loudness of the upper partial tones is the greater,
the sharper the discontinuities of the atmospheric motion. When the motion
alters uniformly and gradually, answering to a vibrational curve proceeding in
smoothly curved forms, only the deeper partial tones, which lie nearest to the
prime tone, have any perceptible intensity. But where the motion alters by jumps,
and hence the vibrational curves shew angles or sudden changes of curvature, the
upper partial tones will also have sensible force, although in all these cases the
amplitudes decrease as the pitch of the upper partial tones becomes higher.*
We shall become acquainted with examples of the analysis of given vibrational ^
forms into separate partial tones in Chapter V . ^'
The theorem of Fourier here adduced shews first that it is mathematically
possible to consider a musical tone as a sum of simple tones, in the meaning we
have attached to the words, and mathematicians have indeed always found it
convenient to base their acoustic investigations on this mode of analysing vibrations.
But it by no means follows that we are obliged to consider the matter in this way. . t«^>f**»^^**
We have rather to inquire, do these partial constituents of a musical tone, such as I
the mathematical theory distinguishes and the ear perceives, really exist in the 1
mass of air external to the ear ? Is this means of analysing forms of vibration '
which Fourier's theorem prescribes and renders possible, not merely a mathematical
fiction, permissible for facihtating calculation, but not necessarily having any
corresponding actual meaning in things themselves ? What makes us hit upon
pendular vibrations, and none other, as the simplest element of all motions pro-
ducing sound ? We can conceive a whole to be split into parts in very different
and arbitrary ways. • Thus we may find it convenient for a certain calculation to ^
consider the number 12 as the sum 8-I-4, because the 8 may have to be cancelled,
but it does not follow that 12 must always and necessarily be considered as merely
the sum of 8 and 4. In another case it might be more convenient to consider 1 2
as the sum of 7 and 5. Just as little does the mathematical possibility, proved by
Fourier, of compounding all periodic vibrations out of simple vibrations, justify
us in concluding that this is the only permissible form of analysis, if we cannot in
addition establish that jhis analysis has^jjag jan-^asential meaning^in nature. That
this is indeed the case, that this analysis has a meaning in nature independently
of theory, is rendered probable by the fact that the ear really effects the same
anal^B, and also by^the circumstance already named, that this kind of analysis
has been Tound so mucb,.more advantageous in mathematical investigations' than
^y^other. Those modes of regarding phenomena that correspond to the most
intimate constitution of the mjktter under investigation are, of course, also always
fcHoie which lead to the most suitable and evident theoretical treatment. But it
woiJd notlbe ad^'isable to begin the investigation with the functions of the ear,
because these are very intricate, and in themselves require much explanation.
In the next chapter, therefore, we shall inquire whether the analysis of compound
into simple vibrations has an actually sensible, meaning in the external world,
independently of the action of the ear, and we shall really be in a condition to
shew that certain mechanical effects depend upon whether a certain partial tone
* SupjKjsing n to be the number of the a sudden jump, and hence the curve has an
order of a partial tone, and n to be very large, „«„♦« ««„i«. ^\ „„ ' „,v. *i, x
Ax^ >i_ i«x J t \\ A- 1 * acute angle; 3) as , when the curvature
then the amplitude of the upper partial tones ^ '^' n.n.n
decreases: i) as A, when the amplitude of the "-^f" suddenly ; 4) when none of the diflferen-
n tial quotients are discontinuous, they must
vibrations themselves makes a sudden jump; ^^^^.^^^33 ^t least as fast as e'^
2) as — -, when their differential quotient makes
•^ D 2
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36 MECHANICS OF SYMPATHETIC KESONANCE. pabt i.
is or is not contained in a composite mass of musical tones. The existence
of partial tones will tbus acquire a meaning in nature, and our^kno^jSge of
their mechanical effects wilTin turn shed a new light on their relations to the
Buman ear.
CHAPTER III.
ANALYSIS OF MUSICAL TONES BY SYMPATHETIC RESONANCE.
We proceed to shew that the simple partial tones contained in a composite mass
of musical tones, produce peculiar mechanical effects in nature, altogether inde-
pendent of the human ear and its sensations, and also altogether independent of
% merely theoretical considerations. These effects consequently give a peculiar objec-
tive significance to this peculiar method of analysing vibrational forms.
Such an effect occurs in the phenomenon of sympathetic resonance. This
phpnomenon is always found in those bodies which when once set in motion by
any impulse, continue to perform a long series of vibrations before they come to
rest. When these bodies are struck gently, but periodically, although each blow
may be separately quite insufficient to produce a sensible motion in the vibratory
body, yet, provided the periodic time of the gentle blows is precisely the same as
the periodic time of the body's ovm vibrations, very large and powerful oscilla-
tions may result. But if the periodic time of the regular blows is different from
the periodic time of the oscillations, the resulting motion will be weak or quite
insensible.
Periodic impulses of this kind generally proceed from another body which is
already vibrating regularly, and in this case the swings of the latter in the course
f of a little time, call into action the swings of the former. Under these circum-
stances we have the process called sympathetic oscillation or sympathetic resonayice.
The essence of the mechanical effect is independent of the rate of motion, which
may be fast enough to excite the sensation of sound, or slow enough not to produce
anything of the kind. Musicians are well acquainted with sympathetic resonance.
When, for example, the strings of two violins are in exsrct unison, and one string is
bowed, the other will begin to vibrate. But the nature of the process is best seen
in instances where the vibrations are slow enough for the eye to follow the whole
of their successive phases.
Thus, for example, it is known that the largest church-bells may be set in motion
by a man, or even a boy, who pulls the ropes attached to them at proper and regular
intervals, even when their weight of metal is so great that the strongest man could
scarcely move them sensibly, if he did not apply his strength in determinate
periodical intervals. When such a bell is once set in motion, it continues, like a
f struck pendulum, to oscillate for some time, until it gradually returns to rest, even
if it is left quite by itself, and no force is employed to arrest its motion. The
motion diminishes gradually, as we know, because the friction on the axis and the
resistance of the air at every swing destroy a portion of tlie existing moving force.
As the bell swings backwards and forwards, the lever and rope fixed to its axis
rise and fall. If when the lever falls a boy clings to the lower end of the bell-rope,
his weight will act so as to increase the rapidity of the existing motion. This
increase of velocity may be very small, and yet it will produce a corresponding
increase in the extent of the bell's swings, which again will continue for a while,
until destroyed by the friction and resistance of the air. But if the boy clung to the
bell-rope at a wrong time, while it was ascending, for instance, the weight of his
body would act in opposition to the motion of the bell, and the extent of swing
would decrease. Now, if the boy continued to cling to the rope at each swing so
long as it was falling, and then let it ascend freely, at every avdng the motion of
the bell would be only increased in speed, and its swings would gi-adually become
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CHAP. ra. MECHANICS OF SYMPATHETIC RESONANCE. 37
greater and greater, until by their increase the motion imparted on every oscillation
of the bell to the walls of the belfry, and the external air would become so great
as exactly to be covered by the power exerted by the boy at each swing.
The success of this process depends, therefore, essentially on the boy's applying
his force only at those moments when it will increase the motion of the bell. That
is, he must employ his strength periodically, and the periodic time must be equal
to that of the bell's swing, or he will not be successful. He would just as easily
bring the swinging bell to rest, if he clung to the rope only during its ascent, and
thus let his weight be raised by the bell.
A similar experiment which can be tried at any instant is the following. Con-
struct a pendulum by hanging a heavy body (such as a ring) to the lower end of a
thread, holding the upper end in the hand. On setting the ring into gentle pen-
dular vibration, it will be found that this motion can be gradually and considerably
increased by watching the moment when the pendulum has reached its greatest IT
departure from the vertical, and then giving the hand a very small motion in the
opposite direction. Thus, when the pendulum is furthest to the right, move the
hand very slightly to the left ; and when the pendulum is furthest to the left, move
the hand to the right. The pendulum may be also set in motion from a state of
rest by giving the hand similar very slight motions having the same periodic time
as the penduhma's own swings. The displacements of the hand may be so small
under these circumstances, that they can scarcely be perceived with the closest
attention, a circumstance to which is due the superstitious application of this
httle apparatus as a divining rod. If namely the observer, without thinking of
his hand, follows the swings of the pendulum with his eye, the hand readily follows
the eye, and involuntarily moves a httle backwards or forwards, precisely in the
same time as the pendulum, after this has accidentally begun to move. These
involuntary motions of the hand are usually overlooked, at least when the observer
is not accustomed to exact observations on such unobtrusive influences. By this 1|
means any existing vibration of the pendulum is increased and kept up, and any
accidental motion of the ring is readily converted ' into pendular vibrations,
which seem to arise spontaneously without any co-operation of the observer,
and are hence attributed to the influence of hidden metals, running streams, and
so on.
K on the other hand the motion of the hand is intentionally made in the con-
trary direction, the pendulum soon comes to rest.
The explanation of the process is very simple. When the upper end of the
thread is fastened to an immovable support, the pendulum, once struck, continues
to swing for a long time, and the extent of its swings diminishes very slowly. We
can suppose the extent of the swings to be measured by the angle which the thread
makes with the vertical on its greatest deflection from it. If the attached body
at the point of greatest deflection lies to the right, and we move the hand to the
left, we manifestly increase the angle between the string and the vertical, and con- ^F
fiequently also augment the extent of the swing. By moving the upper end of the
string in the opposite direction we should decrease the extent of the swing.
In this case there is no necessity for moving the hand in the same periodic time
as the pendulum swings. We might move the hand backwards and forwards only
at every third or fifth or other swing of the pendulum, and we should still produce
large swings. Thus, when the pendulum is to the right, move the hand to the
left, and keep it still, till the pendulum has swung to the left, then again to the
right, and then once more to the left, and then return the hand to its first position,
afterwards wait till the pendulum has swung to the riglil, then to the left, and
again to the right, and then recommence the first motion of the hand. In this
way three complete vibrations, or double excursions of the pendulum, will corre-
spond to one left and right motion of the hand. In tlie same way one left and
right motion of the hand may be made to correspond with seven or more swings
of the pendulum. The meaning of this process is always that the motion of the
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38 MECHANICS OF SYMPATHETIC EESONANCE. pabt i.
hand must in each case be made at such a time and in such a direction as to be
opposed to the deflection of the pendulum and consequently to increase it.
By a slight alteration of the process we can easily make two, four, six, &c.,
swings of the pendulum correspond to one left and right motion of the hand ; for
a sudden motion of the hand at the instant of the pendulum's passage through the
vertical has no influence on the size of the swings. Hence when the pendulum
lies to the right move the hand to the left, and so increase its velocity, let it swing
to the left, watch for the moment of its passing the vertical line, and at that instant
return the hand to its original position, allow it to reach the right, and then again
the left and once more the right extremity of its arc, and then recommence the
first motion of the hand.
We are able then to communicate violent motion to the pendulum by very
small periodical vibrations of the hand, having their periodic time exactly as great,
% or else two, three, four, &c., times as great as that of the pendular oscillation. We
have here considered that the motion of the hand is backwards. This is not
necessary. It may take pla(5e continuously in any other way we please. When it
moves continuously there will be generally portions of time during which it will
increase the pendulum's motion, and others perhaps in which it will diminish tlie
same. In order to create strong vibrations in the pendulum, then, it will be
necessary that the increments of motion should be permanently predominant, and
should not be neutralised by the sum of the decrements.
Now if a determinate periodic motion were assigned to the hand, and we wished
to discover whether it would produce considerable vibrations in the pendulum, we
could not always predict the result without calculation. Theoretical mechanics
would, however, prescribe the following process to be pursued : Analyse the periodic
motion of the hand into a sum of simple peyidular vibrations of the Jiand — exactly
in the same way as was laid down in the last chapter for the periodic motions of
If the particles of air, — then, if the periodic tims of one of these vibrations is equal
to the periodic time of tJie pendulum's own oscillations, the pendulum will be set
ifito violent motion, but not otherwise. We might compound small pendular
motions of the hand out of vibrations of other periodic times, as much as we liked,
but we should fail to produce any lasting strong swings of the pendulum. Hence
the analysis of the motion of the hand into pendular swings has a real meaning in
nature, producing determinate mechanical effects, and for the present purpose no
other analysis of the motion of the hand into any other partial motions can be
substituted for it.
In the above examples the pendulum could be set into sympathetic vibration,
when the hand moved periodically at the same rate as the pendulum ; in this case
the longest partial vibration of the hand, corresponding to the prime tone of a
resonant vibration, was, so to speak, in unison with the pendulum. When three
swings of the pendulum went to one backwards and forwards motion of the hand,
IF it was the third partial swing of the hand, answering as it were to the Twelfth of
its prime tone, which set the pendulum in motion. And so on.
The same process that we have thus become acquainted with for swings of long
periodic time, holds precisely for swings of so short a period as sonorous vibrations.
Any elastic body which is so fastened as to admit of continuing its vibrations for
some length of time when once set in motion, can also be made to vibrate sym-
pathetically, when it receives periodic agitations of comparatively small amounts,
ha\dng a periodic time corresponding to that of its own tone.
Gently touch one of the keys of a pianoforte without striking the string, so as
to raise the damper only, and then sing a note of the corresponding pitch forcibly
dh-ecting the voice against the strings of the instrument. On ceasing to sing, the
note will be echoed back from the piano. It is easy to discover that this echo is
caused by the string which is in unison with the note, for directly the hand is
removed from the key, and the damper is allowed to fall, the echo ceases. The
sympatlietic vibration of the string is still better shown by putting little paper
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CHAP.m. DIFFERENT EXTENT OF SYMPATHETIC RESONANCE. 39
riders upon it, which are jerked off as soon as the string vibrates. The more
exactly the singer hits the pitch of the string, the more strongly it vibrates. A
very httle deviation from the exact pitch fails in exciting sympathetic vibration.
In this experiment the somiding board of the instrument is first struck by the
vibrations of the air excited by the human voice. The sounding board is well
known to consist of a broad flexible wooden plate, which, owing to its exten-
sive surface, is better adapted to convey the agitation of the strings to the air,
and of the air to the strings, than the small surface over which string and air are
themselves directly in contact. The sounding board first communicates the agita-
tions which it receives from the air excited by the singer, to the points where the
string is fastened. The magnitude of any single such agitation is of course infini-
tesimally small. A very large number of such effects must necessarily be aggre-
gated, before any sensible motion of the string can be caused. And such a con-
tinuous addition of effects really takes place, if, as in the preceding experiments with ^
the bell and the pendulum, the periodic time of the small agitations which are com-
municated to the extremities of the string by the air, through the intervention of the
sounding board, exactly corresponds to the periodic time of the string*s Own vibra-
tions. When this is the case, a long series of such vibrations will really set the
string into motion which is very violent in comparison with the exciting cause.
In place of the human voice we might of course use any other musical instru-
ment. Provided only that it can produce the tone of the pianoforte string accu-
rately and sustain it powerfully, it will bring the latter into sympathetic vibration.
In place of a pianoforte, again, we can employ any other stringed instrument
having a sounding board, as a violin, guitar, harp, &c., and also stretched mem-
branes, bells, elastic tongues or plates, &c., provided only that the latter are so
fastened as to admit of their giving a tone of sensible duration when once made
to sound.
When the pitch of the original sounding body is not exactly that of the sym- %
pathising body, or that which is meant to vibrate in sympathy with it, the latter
will nevertheless often make sensible sympathetic vibrations, which will diminish
in amplitude as the difference of pitch increases. But in this respect different
sounding bodies shew great differences, according to the length of time for which
they continue to sound after having been set in action before communicating their
whole motion to the air.
Bodies of small mass, which readily communicate their motion to the air, and
quickly cease to sound, as, for example, stretched membranes, or violin strings, are
readily set in sympathetic vibration, because the motion of tlie air is conversely
readily transferred to them, and they are also sensibly moved by sufficiently strong
agitations of the air, even when the latter have not precisely the same periodic
time as the natural tone of the sympathising bodies. The limits of pitch capable
of exciting sympathetic vibration are consequently a little wider in this case. By
the comparatively greater influence of the motion of the air upon liglit elasti( ^
bodies of this kind which offer but little resistance, their natural periodic time can
be slightly altered, and adapted to that of the exciting tone. Massive elastic
bodies, on the other hand, which are not readily movable, and are slow in com-
municating their sonorous vibrations to the air, such as bells and plates, and con-
tinue to sound for a long time, are also more difficult to move by the air. A much
longer addition of effects is required for this purpose, and consequently it is also
necessary to hit the pitch of their own tone with much greater nicety, in order to
make them vibrate sympathetically. Still it is well known that bell-shaped glasses
can be put into violent motion by singing their proper tone into them ; indeed it is
related that singers with very powerful and pure voices, have sometimes been able
to crack them by the agitation thus caused. The principal difficulty in this experi-
ment is in hitting the pitch with sufficient precision, and retaining the tone at that
exact pitch for a sufficient length of time.
Tuning-forks are the most difficult bodies to set in sympathetic vibration. To
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40 INFLUENCE OF PAETIALS ON SYMPATHETIC EESONANCE. part i.
effect this they may be fastened on sounding boxes which have been exactly tuned to
their tone, as shewn in fig. 13. If we have two such forks of exactly the same
pitch, and excite one by a violin bow,
the other will begin to vibrate in sym- ^^^' '3-
pathy, even if placed at the further
end of the same room, and it will con-
tinue to sound, after the first has been
damped. The astonishing nature of
such a case of sympathetic vibration
will appear, if we merely compare the
heavy and powerful mass of steel set
in motion, with the Hght yielding mass
of air which produces the effect by such
% small motive powers that they could
not stir the lightest spring which was
not in tune with the fork. With such
forks the time required to set them
in full swing by sympathetic action,
is also of sensible duration, and the
slightest disagreement in pitch is sufficient to produce a sensible diminution in
the sympathetic effect. By sticking a piece of wax to one prong of the second
fork, sufficient to make it vibrate once in a second less than the first — a difference
of pitch scarcely sensible to the finest ear — tlie sympathetic vibration wiU be
wholly destroyed.
After having thus described the phenomenon of sympathetic vibration in
general, we proceed to investigate the influence exerted in sympathetic resonance
by the different forms of wave of a musical tone.
% First, it must be observed that most elastic bodies which have been set into
sustained vibration by a gentle force acting periodically, are (with a few exceptions
1^02^^
Fio. 14.
to be considered hereafter) always made to swing in pendular vibrations. But they
are in general capable of executing several kinds of such vibration, with difierent
periodic times and with a different distribution over the various parts of the
vibrating body. Hence to the different lengths of the periodic times correspond
different simple tones producible on such an elastic body. These are its so-called
proper tones. It is, however, only exceptionally, as in strings and tlie narrower
kinds of organ pipes, that these proper tones correspond in pitch with the har-
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CHAP, in INFLUENCE OF PAETIALS ON SYMPATHETIC RESONANCE. 41
monic upper partial tones of a musical tone already mentioned. They are for the
most part inharmonic in relation to the prime tone.
In many cases the vibrations and their mode of distribution over the vibrating
bodies can be rendered visible by strewing a little fine sand over the latter. Take, for
example, a membrane (as a bladder or piece of thin india-rubber) stretched over a
circular ring. In fig. 14 are shewn the various forms which a membrane can
assume when it vibrates. The diameters and circles on the surface of the mem-
brane, mark those points which remain at rest during the vibration, and are known
as nodal lines. By these the surface is divided into a number of compartments
which bend alternately up and down, in such a way that while those marked ( -f )
rise, those marked (--) fall. Over the figures a, b, c, are shewn the forms of a
section of the membrane during vibration. Only those forms of motion are drawn
which correspond with the deepest and most easily producible tones of the mem-
brane. The number of circles and diameters can be increased at pleasure by ^
taking a sufficiently thin membrane, and stretching it with sufficient regularity,
and in this case the tones would continually sharpen in pitch. By strewing sand
on the membrane the figures are easily rendered visible, for as soon as it begins
to vibrate the particles of sand collect on the nodal lines.
In the same way it is possible to render visible the nodal lines and forms of
vibration of oval and square membranes, and of differently-shaped plane elastic
plates, bars, and so on. These form a series of very interesting phenomena dis-
covered by Chladni, but to pursue them would lead us too far from our proper
subject. It will suffice to give a few details respecting the simplest case, that of a
circular membrane.
In the time required by the membrane to execute 100 vibrations of the form a,
fig. 14 (p. 40c), the number of vibrations executed by the other forms is as
follows : —
Form of Vibration
a without nodal lines .
b with one circle ....
c with two circles
d with one diameter .
e with one diameter and one circle
f with two diameters .
Pitch Number
ICO
229*6
359-9
159
292
214
Cents*
Notes nearly
1439
2217
805
1858
I3«7
c
d' +
b'b¥
ab
dl +
The prime tone has been here arbitrarily assumed as c, in order to note the inter-
vals of the higher tones. Those simple tones produced by the membrane which are
shghtly higher than those of the note written, are marked ( + ) ; those lower, by
( — ). In this case there is no commensurable ratio between the prime tone and
the other tones, that is, none expressible in whole numbers.
Strew a very thin membrane of this kind with sand, and sound its prime tone
strongly in its neighbourhood ; the sand will be driven by the vibrations towards %
the edge, where it collects. On producing another of the tones of the membrane,
the sand collects in the corresponding nodal lines, and we are thus easily able to
determine to which of its tones the membrane has responded. A singer who
knows how to hit the tones of the membrane correctly, can thus easily make the
* [Cents are hundredths of an equal Semi-
tone, and are exceedingly valuable as measures
of any, especially unusual, musical intervals.
They are fully explained, and the method of
calculating them from the Interval Batios is
given in App. XX. sect. C. Here it need only
be said that the number of hundreds of cents
is the number of equals that is, pianoforte
Semitones in the interval, and these may be
counted on the keys of any piano, while the
units and tens shew the number of hundredths
of a Semitone in excess. Wlierevor cents are
spoken of in the text, (as in this table), they
must be considered as additions by the transla-
tor. In the present case, they give the inter-
vals exactly, and not roughly as in the column
of notes. Thus, 1439 cents is sharper than 14
Semitones above c, that is, sharper than d' by
39 hundredths of a Semitone, or about ^ of a
Semitone, and 1858 is flatter than 19 Semitones
above c, that is, flatter than g by 42 hun-
dredths of a Semitone, or nearly ^ a Semitone.
— Translator.]
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42 INFLUENCE OF PARTIALS ON SYMPATHETIC RESONANCE, part i.
sand arrange itself at pleasure in one order or the other, by singing the correspond-
ing tones powerfully at a distance. But in general the simpler figures of the deeper
tones are more easily generated than the complicated figures of the upper tones.
It is easiest of all to set the membrane in general motion by sounding its prime
tone, and hence such membranes have been much used in acoustics to prove the
existence of some determinate tone in some determinate spot of the surrounding
air. It is most suitable for this purpose to connect the membrane with an inclosed
mass of air. A, fig. 1 5 , is a glass bottle,
having an open mouth a, and m place
of its bottom b, a stretched membrane,
consisting of wet pig's bladder, al-
lowed to dry after it has been stretched
and fastened. At c is attached a
H single fibre of a silk cocoon, bearing a
drop of sealing-wax, and hanging down
like a pendulum against the membrane.
As soon as the membrane vibrates, the little pendulum is violently agitated. Such
a pendulum is very convenient as long as we have no reason to apprehend any con-
fusion of the prime tone of the membrane with any other of its proper tones. There
is no scattering of sand, and the apparatus is therefore always in order. But to decide
with certainty what tones are really agitating the membrane, we must after all
place the bottle with its mouth downwards and strew sand on the membrane.
However, when the bottle is of the right size, and the membrane uniformly
stretched and fastened, it is only the prime tone of the membrane (slightly altered
by that of the sympathetically vibrating mass of air in the bottle) which is easily
excited. This prime tone can be made deeper by increasing the size of the mem-
brane, or the volume of the bottle, or by diminishing the tension of the membrane
% or size of the orifice of the bottle.
A stretched membrane of this kind, whether it is or is not attached to the bot-
tom of a bottle, will not only be set in vibration by musical tones of the same pitch
as its own proper tone, but also by such musical tones as contain the proper tone
of the membrane among its upper partial tones. Generally, given a number of
interlacing waves, to discover whether the membrane will vibrate sympathetically,
we must suppose the motion of the air at the given place to be mathematically
analysed into a sum of pendular vibrations. If there is one such vibration among
them, of which the periodic time is the same as that of any one of the proper tones
of the membrane, the corresponding vibrational form of the membrane will be super-
induced. But if there are none such, or none sufficiently powerful, the membrane
will remain at rest.
In this case, then, we also find that the analysis of the motion of the air into
pendular vibrations, and the existence of certain vibrations of this kind, are dcci-
^ sive for the sympathetic vibration of the membrane, and for this purpose no other
similar analysis of the motion of the air can be substituted for its analysis into
pendular vibrations. The pendular vibrations into which the composite motion of
the air can be analysed, here shew themselves capable of producing mechanical
I effects in external nature, independently of the ear, and independently of mathe-
1 matical theory. Hence the statement is confirmed, that the theoretical view which
I first led mathematicians to this method of analysing compoimd vibrations, is
I founded in the nature of the thing itself.
As an example take the following description of a single experiment : —
A bottle of the shape shewn in fig. 1 5 above was covered with a thin vulcan-
ised india-rubber membrane, of which the vibrating surface was 49 millimetres
(i*93 inches)* in diameter, the bottle being 140 millimetres (5-51 inches) high, and
♦ [As 10 inches are exactly 254 milh*metres the calculation of one set of measures from
and 1 00 metres, that is, 1 00,000 millimetres are the other. Ilonpfhly we may assume 25 mm.
3tj37 inches, it is easy to form little tables for to be i inch. But whenever dimensions are
Digitized by V^ O OQ IC
CHAP, m.
EESONATOKS.
43
Fig. i6 a.
having an opening at the brass mouth of 13 millimetres {'51 inches) in diameter.
When blown it gave /'Jl, and the sand heaped itself in a circle near the edge of the
membrane. The same circle resulted from my giving the same tone f'% on an
harmonium, or its deeper Octave /Jl, or the deeper Twelfth B. Both F% and Z>
gave the same circle, but more weakly. Now the /'J of the membrane is the prime
tone of the harmonium tone/'jl, the second partial tone of/iJ, the third of -B, the
fourth oiF% and fifth of D.* All these notes on being sounded set the membrane
in the motion due to its deepest tone. A second smaller circle, 19 millimetres
(•75 inches) in diameter was produced on the membrane by h' and the same more
faintly by 6, and there was a trace of it for the deeper Twelfth e, that is, for simple
tones of which vibrational numbers were \ and \ that of 5'.t
Stretched membranes of this kind are very convenient for these and similar
experiments on the partials of compound tones. They have the great advantage
of being independent of the ear, but they f
are not very sensitive for the fainter simple
tones. Their sensitiveness is far inferior to
that of the res'ondtors which I have intro-
duced. These are hollow spheres of glass
or metal, or tubes, with two openings as
shewn in figs. 16 a and 16 b. One opening
(a) has sharp edges, the other (b) is funnel-
shaped, and adapted for insertion into the
ear. This smaller end I usually coat with
melted sealing wax, and when the wax has
cooled down enough not to hurt the finger
on being touched, but is still soft, I press the opening into the entrance of my
ear. The sealing wax thus moulds itself to the shape of the inner surface of this
opening, and when I subsequently use the resonator, it fits easily and is air-tight. %
Such an instrument is very like the resonance bottle already described, fig. 15
„ , ^ (p. 42a) , for which the observer's
Pig. i6b. vr t /»
own tympanic membrane has
been made to replace the for-
mer artificial membrane.
The mass of air in a reso-
nator, together with that in the
aural passage, and vnth the
tympanic membrane or drumskin itself, forms an elastic system which is capable
of vibrating in a peculiar manner, and, in especial, the prime tone of the sphere,
which is much deeper than any other of its proper tones, can be set into very
powerful sjrmpathetic vibration, and then the ear, which is in immediate connec-
tion with the air inside the sphere, perceives this augmented tone by direct action.
If we stop one ear (which is best done by a plug of sealing wax moulded into the ^
form of the entrance of the ear), J and apply a resonator to the other, most of the
tones produced in the surrounding air will be considerably damped; but if the
proper tone of the resonator is sounded, it brays into the ear most powerfully.
given in the text in mm. (that is, millimetres),
they will be reduced to inches and decimals of
to mch.~Translator,]
* [As the instrument was tempered, we
should have, approximately, for /g the partials
/t , /5 , <fcc. ; for B the partials B, 6, fU , Ac ;
tor Ft the partials F%,fU,cU,fU, Ac. ; and
for D the partials D, d, o, d\ fU , Ac. To
prevent confusion I have reduced the upper
partials of the text to ordinary partials, as
suggested in p. 236', note.— Tranalaior.]
t [Here the partials of b arc 6, b\ Ac, and
of e aro e, e\ ^, Ac, ao that both b and e
contain b^,— Translator,]
X [For ordinary purposes this is quite
enough, indeed it is generally unnecessary to
stop the other ear at all. But for such experi-
ments as Mr. Bosanquet had to make on beats
(see App. XX. section L. art. 4, b) he was
obliged to use a jar as the resonator, conduct
the sound from it through first a glass and
then an elastic tube to a semicircular metal tube
which reached from ear to ear, to each end of
which a tube coated with india-rubber, could be
screwed into the ear. By this means, when
proper care was taken, all sound but that
coming from the resonance jar was perfectly
excluded. — TranslaforJ] t
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44 RESONATORS. part i.
Hence any one, even if he has no ear for music or is quite unpractised in detecting
musical sounds, is put in a condition to pick the required simple tone, even if com-
paratively faint, from out of a great number of others. The proper tone of the
resonator may even be sometimes heard cropping up in the whistling of the >*'ind,
the rattling of carriage wheels, the splashing of water. For these purposes such
resonators are incomparably more sensitive than tuned membranes. When the
simple tone to be observed is faint in comparison with those which a^scompany it,
it is of advantage to alternately apply and withdraw the resonator. We thus easily
feel whether the proper tone of the resonator begins to sound when the instrument
is apphed, whereas a uniform continuous tone is not so readily perceived.
A properly tuned series of such resonators is therefore an important instrument
for experiments in which individual faint tones have to be distinctly heard, although
accompanied by others which are strong, as in observations on the combinational
% and upper partial tones, and a senes of other phenomena to be hereafter described
relating to chords. By their means such researches can be carried out even by
ears quite untrained in musical observation, whereas it had been previously
impossible to conduct them except by trained musical ears, and much strained
attention properly assisted. These tones were consequently accessible to the
observation of only a very few individuals ; and indeed a large number of physi-
cists and even musicians had never succeeded in distinguishing them. And again
even the trained ear is now able, with the assistance of resonators, to caiTy the
analysis of a mass of musical tones much further than before. Without their help,
indeed, I should scarcely have succeeded in making the observations hereafter
described, with so much precision and certainty, as I have been enabled to attain
at present.*
It must be carefully noted that the ear does not hear tlie required tone with
augmented force, unless that tone attains a considerable intensity within the mass
^ of air inclosed in the resonator. Now the mathematical theory of the motion of
the air shews that, so long as the amplitude of the \'ibration8 is sufficiently small,
the inclosed air will execute pendular oscillations of the same periodic time as
those in the external air, and none other, and that only those pendular oscillations
whose periodic time corresponds with that of the proper tone of the resonator,
have any considerable strength ; the intensity of the rest diminishing as the differ-
ence of their pitch from that of the proper tone increases. All this is independent
of the connection of the ear and resonator, except in so far as its tympanic mem-
brane forms one of the inclosing walls of the mass of air. Theoretically this
apparatus does not differ from the bottle with an elastic membrane, in fig. 15
(p. 42a), but its sensitiveness is amazingly increased by using the drumskin of the ear
for the closing membrane of the bottle, and thus bringing it in direct connection
with the auditory ner\'es themselves. Hence we cannot obtam a powerful tone in
the resonator except when an analysis of the motion of the external air into
^ pendular vibrations, would shew that one of them has the same periodic time aa
the proper tone of the resonator. Here again no other analysis but that into
pendular vibrations would give a correct result.
It is easy for an observer to convince himself of the above-named properties of
resonators. Apply one to the ear, and let a pffice of harmonised music, in which
the proper tone of the resonator frequently occurs, be executed by any instruments.
As often as this tone is struck, the ear te which the instrument is held, will hear
it violently contrast with all the other tones of the chord.
This proper tone will also often be heard, but more weakly, when deeper
musical tones occur, and on investigation we find that in such cases tones have
been struck which include the proper tone of the resonator among their upper
partial tones. Such deeper musical tones are called the harmonic uncUr tones of
the resonator. They are musical tones whose periodic time is exactly 2, 3, 4, 5,
and so on, times as great as that of the resonator. Thus if the proper tone of
* See Appendix II. for tlie measurefl and different forms of these Krsonators.
Digitized by VjOOQlC
CHAP. III. SYMPATHETIC RESONANCE OF STRINGS. 45
the resonator is c", it will be heard when a musical instrument sounds c', /, c, A\),
F, D, (7, and so on.* In this case the resonator is made to sound in sympathy
with one of the harmonic upper partial tones of the compound musical tone which
is vibrating in the external air. It must, however, be noted that by no means all
the haimonic upper partial tones occur in the compound tones of every instrument,
and that they have very different degrees of intensity in different instruments. In
the musical tones of vioHns, pianofortes, and hannoniums, the first five or six are
generally very distinctly present. A more detailed account of the upper partial
tones of strings will be given in the next chapter. On the harmonium the un-
evenly numbered partial tones (i, 3, 5, &c.) are generally stronger than the evenly
numbered ones (2, 4, 6, &c.). In the same way, the upper partial tones are clearly
heard by means of the resonators in the singing tones of the human voice, but
differ in strength for the different vowels, as wiU be shewn hereafter. II
Among the bodies capable of strong sympathetic vibration must be reclconed
stretched strings which are connected with a sounding board, as on the pianoforte.
The principal mark of distinction between strings and the other bodies which
vibrate sympathetically, is that different vibrating forms of strings give simple
tones corresponding to the harvwiiic upper partial tones of the prime tone, whereas
the secondary simple tones of membranes, bells, rods, &c., are iV^harmonic with the
prime tone, and the masses of air in resonators have generally only very high
upper partial tones, also chiefly mharmonic with the prime tone, and not capable
of being much reinforced by the resonator.
The vibrations of strings may be studied either on elastic chords loosely
stretched, and not sonorous, but swinging so slowly that their motion may be
followed with the hand and eye, or else on sonorous strings, as those of the piano-
forte, guitar, monochord, or violin. Strings of the first kind are best made of thin ^
spirals of brass wire, six to ten feet in length. They should be gently stretched,
and both ends should be fastened. A string of this construction is capable of
making very large excursions with great regularity, which are easily seen by a large
audience. The swings are excited by moving the string regularly backwards and
forwards by the finger near to one of its extremities.
A string maybe first made to vibrate as in fig. 17, a (p. 466), so that its appear-
ance when displaced from its position of rest is always that of a simple half wave.
The string in this case gives a single simple tone, the deepest it can produce, and
no other harmonic secondary tones are audible.
But the string may also during its motion assume the forms fig. 17, b, c, d.
In this case the form of the string is that of two, three, or four half waves of a
simple wave-curve. In the vibrational form b the string produces only the upper
Octave of its prime tone, in the form c the Twelfth, and in the form d the second
Octave. The dotted lines shew the position of the string at the end of half its ^
periodic time. In b the point p remains at rest, in c two points y^ and y^ remain
at rest, in d three points S,, h^, 3,. These points are called nodes. In a swingmg
spiral wire the nodes are readily seen, and for a resonant string they are shewn by
httle paper riders, which are jerke^ofi* from the vibrating parts and remain sitting
on the nodes. When, then, the string is divided by a node into two swinging
sections, it produces a simple tone having a pitch number double that of the prime
[The c" occurs as the 2nd, 3rd, 4th,
6th, 7th, 8th partials of these notes.
the 7th being rather flat. The partials are
in fact :—
d c"
f f c"
c d f d'
A\) a\> et> a'\) d'
F f d f a'
D d a d' fl
C c f d d
a d'
f b'\> c". Translator.]
Digitized by CjOOQI
46
SYMPATHETIC RESONANCE OF STRINGS.
PABTL
tone. For three sections the pitch number is tripled, for fonr sections quadrupled,
and so on.
To bring a spiral wire into these different forms of vibration, we move it
periodically with the finger near one extremity, adopting the period of its slowest
swings for a, twice that rate for b, three times for c, and four times for d. Or else
we just gently touch one of the nodes nearest the extremity with the finger, and pluck
the string half-way between this node and the nearest end. Hence when y^ in c,
or S, in d, is kept at rest by the finger, we pluck the string at c The other nodes
then appear when the vibration commences.
f
For a sonorous string the vibrational forms of fig. 17 above are most purely
produced by applying to its sounding board the handle of a tuning-fork which has
been struck and gives the simple tone corresponding to the form required. K only
a determinate number of nodes are desired, and it is indifferent whether the indi-
vidual points of the string do or do not execute simple vibrations, it is sufficient to
touch the string very gently at one of the nodes and either pluck the string or rub
it with a violin bow. By touching the string with the finger all those simple vibra-
tions are damped which have no node at that point, and only those remain which
allow the string to be at rest in that place.
The number of nodes in long thin strings may be considerable. They cease to
be formed when the sections which lie between the nodes are too short and stiff to
^ be capable of sonorous vibration. Very fine strings consequently give a greater
number of higher tones than thicker ones. On the violin and the lower pianoforte
strings it is not very difficult to produce tones with 10 sections ; but with extremely
fine wires tones with 16 or 20 sections can be made to sound. [Also compare p. 78^.]
The forms of vibration here spoken of are those in which each point of the
string performs pendular oscillations. Hence these motions excite in the ear the
sensation of only a single simple tone. In all other vibrational forms of the
strings, the oscillations are not simply pendular, but take place according to a differ-
ent and more compUcated law. This is always the case when the string is plucked
in the usual way with the finger (as for guitar, harp, zither) or is struck with a
hammer (as on the pianoforte), or is rubbed with a violin bow. The resulting motions
may then be regarded as compounded of many simple vibrations, which, w^hen
taken separately, correspond to those in fig. 17. The multiphcity of such com-
posite forms of motion is infinitely great, the string may indeed be considered
as capable of assuming any given form (provided we confine ourselves in all cases
Digitized by V^jOOQlC
CHAP.m. SYMPATHETIC EESONANCE OF STKINGS. 47
to very small deviations from the position of rest), because, according to what was
said in Chapter 11., any given form of wave can be compounded out of a number
of simple waves such as those indicated in fig. 17, a, b, c, d. A plucked, struck,
or bowed string therefore allows a great number of harmonic upper partial tones to
be heard at the same time as the prime tone, and generally the number increases
with the thinness of the string. The peculiar tinkling sound of very fine metallic
strings, is clearly due to these very high secondary tones. It is easy to distinguish
tbe upper simple tones up to the sixteenth by means of resonators. Beyond the
sixteenth they are too close to each other to be distinctly separable by this means.
Hence when a string is sympathetically excited by a musical tone in its neigh-
bom-hood, answering to the pitch of the prime tone of the string, a whole series of
different simple vibrational forms will generally be at the same time generated in
the string. For when the prime of the musical tone corresponds to the prime of
the string aU the harmonic upper partials of the first correspond to those of the ^
second, and are hence capable of exciting the corresponding vibrational forms in
the string. Generally the string will be brought into as many forms of sympa-
thetic vibration by the motion of the air, as the analysis of that motion shews that
it possesses simple vibrational forms, having a periodic time equal to that of some
vibrational form, that the string is capable of assuming. But as a general rule
when there is one such simple vibrational form in the air, there are several such,
and it will often be difficult to determine by which one, out of the many possible
simple tones which would produce the effect, the string has been excited. Conse-
quently the usual unweighted strings are not so convenient for the determination
of the pitch of any simple tones which exist in a composite mass of air, as the
membranes or the inclosed air of resonators.
To make experiments with the pianoforte on the sympathetic vibrations of
strings, select a fiat instrument, raise its lid so as to expose the strings, then press
down the key of the string (for d suppose) which you wish to put into sympathetic m
vibration, but so slowly that the hammer does not strike, and place a little chip of
wood across this d string. You will find the chip put in motion, or even throvm
o£f, when certain other strings are struck. The motion of the chip is greatest when
one of the wnder tcmts of d (p. 44^^) is struck, as c, i^, 0, A)^, F,, D,, or 0^. Some,
but much less, motion also occiurs when one of the upper partial tones of d is
struck, as d\ g", or d", but in this last case the chip will not move if it has been
placed over one of the corresponding nodes of the string. Thus if it is laid across
the middle of the string it will be still for d' and d'\ but will move iotg". Placed
at one third the length of the string from its extremity, it will not stir for g", but
will move for d' or d". Finally the string d will also be put in motion when an
under tone of one of its upper partial tones is struck ; for example, the note/, of which
the third partial tone d' is identical with the second partial tone of d. In this case
also the chip remains at rest when put on to the middle of the string c', which is
its node for c". In the same way the string d wiQ move, with the formation of -r
two nodes, for g', g, or S^, all which notes have g" as an upper partial tone, which
is also the third partial of c'.*
Observe that on the pianoforte, where one end of the strings is commonly
concealed, the position of the nodes is easily found by pressing the string gently
on both sides and striking the key. If the finger is at a node the corresponding
upper partial tone will be heard purely and distinctly, otherwise the tone of the
string is dull and bad.
As long as only one upper partial tone of the string d is excited, the corre-
sponding nodes can be discovered, and hence the particular form of its vibration
determined. But this is no longer possible by the above mechanical method when
* [These experiments oan of coarse not be struck and damped. And this sounding of c',
conducted on the usual upright cottage piano. although unstruck, is itself a very interesting
Bat the experimenter can at least hear the phenomenon. But of course, as it depends on
tone of c\ if c, F, C, <&c., are struck and the ear, it does not establish the results of the
immediately damped, or if c", g", c'" are text. — TraTislator.]
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rH^
^
48 OBJECTIVE EXISTENCE OF PAKTIALS. part r.
two upper partial tones are excited, such as c" and g", as would be the case if both
these notes were struck at once on the pianoforte, because the whole string of d
would then be in motion.
Although the relations for strings appear more complicated to the eye, their
sympathetic vibration is subject to the same law as that which holds for resonators,
membranes, and other elastic bodies. The sympathetic vibration is always deter-
mined by the analysis of whatever sonorous motions exist, into simple pendular
vibrations. If the periodic time of one of these simple vibrations corresponds to
the periodic time of one of the proper tones of the elastic body, that body, whether
it be a string, a membrane, or a mass of air, will be put into strong sympathetic
vibration.
These facts give a real objective value to the analysis of sonorous motion into
simple pendular vibration, and no such value would attach to any other analysis.
^ Every individual single system of waves formed by pendular vibrations exists as
an independent mechanical imit, expands, and sets in motion other elastic bodies
having the corresponding proper tone, perfectly undisturbed by any other simple
tones of other pitches which may be expanding at the same time, and which may
^•oceed either from the same or any other source of sound. Each single simple
^Tone, then, can, as we have seen, be separated from the composite mass of tones,
by mechanical means, namely by bodies, which will vibrate sympathetically with
it. Hence every individual partial tone exists in the compoxmd musical tone
produced by a single musical instrument, just as truly, and in the same sense, as the
different colours of the rainbow exist in the white light proceeding from the sun
or any other luminous body. Light is also only a vibrational motion of a peculiar
elastic medium, the luminous ether, just as sound is a vibrational motion of the
air. In a beam of white Ught there is a species of motion which may be repre-
sented as the sum of many oscillatory motions of various periodic times, each of
^ which corresponds to one particular colour of the solar spectrum. But of course
each particle of ether at any particular moment has only one determinate velocity,
and only one determinate departure from its mean position, just hke each particle
of air in a space traversed by many systems of sonorous waves. The really exist-
ing mction of any particle of ether is of course only one and individual ; and our
theoretical treatment of it as compound, is in a certain sense arbitrary. But the
undulatory motion of hght can also be analysed into the waves corresponding to
the separate colours, by external mechanical means, such as by refraction in a
prism, or by transmission through fine gratings, and each individual simple wave
of light corresponding to a simple colour, exists mechanically by itself, indepen-
dently of any other colour.
We must therefore not hold it to be an illusion of the ear, or to be mere
imagination, when in the musical tone of a single note emanating from a musical
instrument, we distinguish many partial tones, as I have found musicians inclined
m to think, even when they have heard those partial tones quite distinctly with their
own ears. If we admitted this, we should have also to look upon the colours of
the spectrum which are separated from white light, as a mere illusion of the eye.
The real outward existence of partial tones in nature can be estabHshed at any
moment by a sympathetically vibrating membrane which casts up the sand strewn
upon it.
Finally I would observe that, as respects the conditions of sympathetic vibra-
tion, I have been obliged to refer frequently to the mechanical theory of tlie
motion of air. Since in the theory of sound we have to deal with well-known
mechanical forces, as the pressure of the air, and with motions of material
particles, and not with any hypothetical explanation, theoretical mechanics have
an unassailable authority in this department of science. Of course those readers
who are unacquainted with mathematics, must accept the results on faith. An
experimental way of examining the problems in question will be described in the
next chapter, in which the laws of the analysis of musical tones by the ear have
Digitized by V^OOQIC
CHAPS, in. IV. METHODS OP OBSERVING PARTIAL TONES. 49
to be established. The experimental proof there given for the ear, can also be
carried oat in precisely the same way for membranes and masses of air which
vibrate sympathetically, and the identity of the laws in both cases will result from
those investigations.*
CHAPTER IV.
ON THE ANALYSIS OF MUSICAL TONES BY THE EAR.
It was frequently mentioned in the preceding chapter that musical tones could be
resolved by the ear alone, unassisted by any peculiar apparatus, into a series of
partial tones corresponding to the simple pendular vibrations in a mass of air, that ^
is, into the same constituents as those into which the motion of the air is resolved
by the sympathetic vibration of elastic bodies. We proceed to shew the correctness
of this assertion.
Any one who endeavours for the first time to distinguish the upper partial
tones of a musical tone, generally finds considerable difficulty in merely hearing
them.
The analysis of our sensations when it cannot be attached to corresponding
differences in external objects, meets with peculiar difficulties, the nature and
significance of which will have to be considered hereafter. The attention of the
observer has generally to be drawn to the phenomenon he has to observe, by
peculiar aids properly selected, until he knows precisely what to look for ; after he
Las once succeeded, he wiU be able to throw aside such crutches. Similar diffi-
culties meet us in the observation of the upper partials of a musical tone. I shall
first give a description of such processes as will most easily put an untrained f
observer into a position to recognise upper partial tones, and I will remark in
passing that a musically trained ear will not necessarily hear upper partial tones
with greater ease and certainty than an untrained ear. Success depends rather
upon a peculiar power of mental abstraction or a peculiar mastery over attention,
than upon musical training. But a musically trained observer has an essential
advantage over one not so trained in his power of figuring to himself how the
simple tones sought for, ought to sound, whereas the untrained observer has con-
tmnally to hear these tones sounded by other means in order to keep their effect
fresh in his mind.
First we must note, that the unevenly numbered partials, as the Fifths, Thirds,
Sevenths, &c., of the prime tones, are usually easier to hear than the even ones,
which are Octaves either of the prime tone or of some of the upper partials which
lie near it, just as in a chord we more readily diFtinguish whether it contains
Fifths and Thirds than whether it has Octaves. The second, fourth, and eighth H
partials are higher Octaves of the prime, the sixth partial an Octave above the
third partial, that is, the Twelfth of the prime ; and some practice is required for
distinguishing these. Among the uneven partials which are more easily dis-
tinguished, the first place must be assigned, from its usual loudness, to the third
partial, the Twelfth of the prime, or the Fifth of its first higher Octave. Then
follows the fifth partial as the major Third of the prime, and, generally very faint,
the seventli partial as the minor Seventhf of the second higher Octave of the
prime, as will be seen by their following expression in musical notation, for the
compound tone c.
• Optical means for rendering visible weak f i^^ Taore correctly s?i6-minor Seventh ;
sympathetic motions of sonorous masses of as the real minor Seventh, formed by taking
air, are described in App. II. These means two Fifths down and then two Octaves ap, is
are valuable for demonstrating the facts to sharper by 27 cents, or in the ratio of 63 : 64.
hearers unaccustomed to the observing and — Translator.']
distinguishing musical tones.
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so METHODS OF OBSERVING PARTIAL TONES. part i.
^^
^^^^
:?2Z
-h-
I2«.>'345 6 78
c & g' d' e" g" ''h"\} c'"
[Cents, o 1 200 1902 2400 2786 3102 3369 3600] ♦
In commencing to observe upper partial tones, it is advisable just before pro-
ducing the musical tone itself which you wish to analyse, to sound the note you
wish to distinguish in it, very gently, and if possible in the same quality of tone
as the compound itself. The pianoforte and harmonium are well adapted for
these experiments, because they both have upper partial tones of considerable
power.
II First gently strike on a piano the note g\ as marked above, and after letting
the digital t rise so as to damp the string, strike the note c, of which g' is the
third partial, with great force, and keep your attention directed to the pitch of the
f which you had just heard, and you will hear it again in the compound tone of
c. Similarly, first stroke the fifth partial e" gently, and then c strongly. These
upper partial tones are often more distinct as the sound dies away, because they
appear to lose force more slowly than the prime. The seventh and ninth partialB
h"\^ and d^" are mostly weak, or quite absent on modem pianos. If the same ex-
periments are tried with an harmonium in one of its louder stops, the seventh
partial will generally be well heard, and sometimes even the ninth.
To the objection which is sometimes made that the observer only imagines he
hears the partial tone in the compound, because he had just heard it by itself, I
need only remark at present that if e!' is first heard as a partial tone of c on a
good piano, tuned in equal temperament, and then e" is struck on the instrument
IF itself, it is quite easy to perceive that the latter is a httle sharper. This follows
from the method of tuning. But if there is a difference in pitch between the two
tones, one is certainly not a continuation of the mental effect produced by the
other. Other facts wliich completely refute the above conception, will be subse-
quently adduced.
A still more suitable process than that just described for the piano, «an be
adopted on any stringed instrument, as the piano, monochord, or violin. It con-
sists in first producing the tone we wish to hear, as an harmonic, [p. 2$d, note] by
touching the corresponding node of the string when it is struck or rubbed. The
resemblance of the tone first heard to the corresponding partial of the compound
is then much greater, and the ear discovers it more readily. It is usual to place a
di\aded scale by the string of a monochord, to facilitate the discovery of the nodes.
Those for the third partial, as shewn in Chap. III. (p. 45^), divide the string into
three equal parts, those for the fifth into five, and so on. On the piano and violin
II the position of these points is easily found experimentally, by touching the string
gently with the finger in the neighbourhood of the node, which has been approxi-
matively determined by the eye, then striking or bowing the string, and moving
the finger about till the required harmonic comes out strongly and purely. By
then sounding the string, at one time with the finger on the node, and at another
without, we obtain the required upper partial at one time as an harmonic, and at
another in the compound tone of the whole string, and thus learn to recognise the
f existence of the first as part of the second, with comparative ease. Using thin
' strings which have loud upper partials, I have thus been able to recognise the
* [The cents, (see p. 4ie2, note) reckoned piano or org&n, are best called digitals or
from the lowest note, are assigned on the finger-keys, on the analogy of pedals and foot-
supposition that the harmonics are perfect, keys on the organ. The word key having
as on the Harmonica!, not tempered as on another musical sense, namely, the scale *n
the pianoforte. See also diagram, p. 22c — which a piece of music is written, will without
Translator.} prefix be confined to this meaning. — Tran^-
t [The keys played by the fingers on a lator.]
Digitized by VjOOQlC
CHAP. IV. METHODS OP OBSEBVING PARTIAL TONES. 51
partials separately, up to the sixteenth. Those which lie still higher are too near
to each other in pitch for the ear to separate them readily.
In such experiments I recommend the following process. Touch the node of
the string on the pianoforte or monochord with a camel* s-hair pencil, strike the
note, and immediately remove the pencil from the string. If the pencil has been
pressed tightly on the string, we either continue to hear the required partial as an
harmonic, or else in addition hear the prime tone gently sounding with it. On
repeating the excitement of the string, and continuing to press more and more
lightly with the camel's-hair pencil, and at last removing the pencil entirely, the
prime tone of the string wiU be heard more and more distinctly with the harmonic
till we have finally the full natural musical tone of the string. By this means
we obtain a series of gradual transitional stages between the isolated partial and
the compound tone, in which the first is readily retained by the ear. By applying
this last process I have generally succeeded in making perfectly untrained ears f
recognise the existence of upper partial tones.
It is at first more difficult to hear the upper partials on most wind instruments
and in the human voice, than on stringed instruments, harmoniums, and the more
penetrating stops of an organ, because it is then not so easy first to produce the
upper partial softly in the same quality of tone. But still a little practice suffices
to lead the ear to the required partial tone, by previously touching it on the piano.
The partial tones of the himian voice are comparatively most difficult to distinguish
for reasons which will be given subsequently. Nevertheless they were distin-
gnished even by Eameau* without the assistance of any apparatus. The process
is as follows : —
Get a powerful bass voice to sing ^ to the vowel 0, in sore [more like aw
in SOU) than o in so], gently touch h'\} on the piano, which is the Twelfth, or
third partial tone of the note e|;>, and let its sound die away while you are listening
to it attentively. The note h'\^ on the piano will appear really not to die away, %
but to keep on sounding, even when its string is damped by removing the finger
from the digital, because the ear unconsciously passes from the tone of the piano
to the partial tone of the same pitch produced by the singer, and takes the latter
for a continuation of the former. But when the finger is removed from the key,
and the damper has fallen, it is of course impossible that the tone of the string
should have continued soimding. To make the experiment for g" the fifth partial,
or major Third of the second Octave above «(;>, the voice should sing to the vowel
kmfatlier.
The resonators described in the last chapter furnish an excellent means for
this purpose, and can be used for the tones of any musical instrument. On apply,
ing to the ear the resonator corresponding to any given upper partial of the com-
pound c, such as g', this g' is rendered much more powerful when c is sounded.
Now hearing and distinguishing g' in this case by no means proves that the ear
alone and without this apparatus would hear g' as part of the compound c. But %
the increase of the loudness of g' caused by the resonator may be used to direct
the attention of the ear to the tone it is required to distinguish. On gradually
removing the resonator from the ear, the force of g' will decrease. But the
attention once directed to it by this means, remains more readily fixed upon
it, and the observer continues to hear this tone in the natural and unchanged
compound tone of the given note, even with his unassisted ear. The sole office
of the resonators in this case is to direct the attention of the ear to the required
tone.
By frequently instituting similar experiments for perceiving the upper partial
tones, the observer comes to discover them more and more easily, till he is finally
able to dispense with any aids. But a certain amount of undisturbed concentration
is always necessary for analysing musical tones by the ear alone, and hence the
use of resonators is quite indispensable for an accurate comparison of different
* Nouveau Syatmne de Musique thiorique. Paris : 1726. Preface.
Digitized by
<^i:iPgle
52 PEOOF OF OHM'S LAW. pairt i.
qualities of tones, especially in respect to the weaker upper partials. At least, I
must confess, that my own attempts to discover the upper partial tones in the
human voice, and to determine their differences for different vowels, were most
unsatisfactory until I applied the resonators.
We now proceed to prove that the human ear really does analyse musical
tones according to the law of simple vibrations. Since it is not possible to insti-
tute an exact comparison of the strength of our sensations for different simple
tones, we must confine ourselves to proving that when an analysis of a composite
tone into simple vibrations, effected by theoretic calculation or by sympathetic
resonance, shews that certain upper partial tones are absent, the ear also does
not perceive them.
The tones of strings are again best adapted for conducting this proof, because
they admit of many alterations in their quality of tone, according to the manner
II and the spot in which they are excited, and also because the theoretic or experi-
mental analysis is most easily and completely performed for this case. Thomas
Young* first shewed that when a string is plucked or struck, or, as we may add,
bowed at any point in its length which is the node of any of its so-called
liarmonics, those simple vibrational forms of the string which have a node in that
point are not contained in the compound vibrational form. Hence, if we attack
the string at its middle point, all the simple vibrations due to the evenly numbered
partials, each of which has a note at that point, will be absent. This gives the
sound of the string a peculiarly hollow or nasal twang. If we excite the string at
^ of its length, the vibrations corresponding to the third, sixth, and ninth partials
will be absent ; if at ^, then those corresponding to the fourth, eighth, and twelfth
partials will fail ; and so on.f
This result of mathematical theory is confirmed, in the first place, by analys-
ing the compound tone of the string by sympathetic resonance, either by the
f resonators or by other strings. The experiments may be easily made on the
pianoforte. Press down the digitals for the notes c and c', without allowing the
hammer to strike, so as merely to free them from their dampers, and then pluck
the string c with the nail till it sounds. On damping the c string the higher &
will echo the sound, except in the particular case when the c string has been
plucked exactly at its middle point, which is the point where it would have to be
touched in order to give its first harmonic when struck by the hammer.
If we touch the c string at i or f its length, and strike it with the hammer,
we obtain the harmonic g* ; and if the damper of the g^ is raised, this string echoes
the sound. But if we pluck the c string with the nail, at either 1 or § its length,
g' is not echoed, as it will be if the c string is plucked at any other spot.
In the same way observations with the resonators shew that when the c string
is plucked at its middle the Octave c' is missing, and when at ^ or ^ its length the
Twelfth g' is absent. The analysis of the sound of a string by the sympathetic
% resonance of strings or resonators, consequently fully confirms Thomas Young's
law.
But for the vibration of strings we have a more direct means of analysis than
that furnished by sympathetic resonance. If we, namely, touch a vibrating string
gently for a moment with the finger or a camePs-hair pencil, we damp all those
simple vibrations which have no node at the point touched. Those vibrations,
however, which have a node there are not damped, and hence will continue to
sound without the others. Consequently, if a string has been made to speak in
any way whatever, and we wish to know whether there exists among its simple
vibrations one corresponding to the Twelfth of the prime tone, we need only touch
one of the nodes of this vibrational form at ^ or f the length of the string, in
order to reduce to silence all simple tones which have no such node, and leave the
Twelfth sounding, if it were there. If neither it, nor any of the sixth, ninth,
* London. Philosophical Transactio7iSt 1800, vol. i. p. 137.
t See Appendix III.
Digitized by VjOOQIC
CHAP. IV. PEOOF OF OHM'S LAW. 53
twelfth, &c., of the partial tones were present, giving corresponding harmonics,
the string will be reduced to absolute silence by this contact of the finger.
Press down one of the digitals of a piano, in order to free a string from its
damper. Pluck the string at its middle point, and immediately touch it there.
The string will be completely silenced, shewing that plucking it in its middle
excited none of the evenly numbered partials of its compound tone. Pluck it at ^ or ^
its length, and immediately touch it in the same place ; the string will be silent,
proving the absence of the third partial tone. Pluck the string anywhere else
than in the points named, and the second partial will be heard when the middle is
touched, the third when the string is touched at ^ or § of its length.
The agreement of this kind of proof with the results from sympathetic reso-
nance, is well adapted for the experimental establishment of the proposition based
in the last chapter solely upon the results of mathematical theory, namely, that
sympathetic vibration occurs or not, according as the corresponding simple H
vibrations are or are not contained in the compound motion. In the last described
method of analysing the tone of a string, we are quite independent of the theory
of sympathetic vibration, and the simple vibrations of strings are exactly charac-
terised and recognisable by their nodes. If the compound tones admitted of being
analysed by sympathetic resonance according to any other vibrational forms except
those of simple vibration, this agreement could not exist.
If, after having thus experimentally proved the correctness of Thomas Young's
law, we try to analyse the tones of strings by the unassisted ear, we shall continue
to find complete agreement.* If we pluck or strike a string in one of its nodes,
ail those upper partial tones of the compound tone of the string to which the node
belongs, disappear for the ear also, but they are heard if the string is plucked at
any other place. Thus, if the string c be plucked at ^ its length, the partial tone
g* cannot be heard, but if the string be plucked at only a little distance from this
point the partial tone g^ is distinctly audible. Hence the ear analyses the sound f
of a string into precisely the same constituents as are found by sympathetic reso-
nance, that is, into simple tones, according to Ohm*s definition of this conception.
These experiments are also well adapted to shew that it is no mere play of imagina-
tion when we hear upper partial tones, as some people believe on hearing them for
the first time, for those tones are not heard when they do not exist.
The following modification of this process is also very well adapted to make
the upper partial tones of strings audible. First, strike alternately in rhythmical
sequence, the third and fourth partial tone of the string alone, by damping it in the
corresponding nodes, and request the listener to observe the simple melody thus
produced. Then strike the undamped string alternately and in the same rhythmical
sequence, in these nodes, and thus reproduce the same melody in the upper partials,
which the listener will then easily recognise. Of course, in order to hear the
third partial, we must strike the string in the node of the fourth, and conversely.
The compound tone of a plucked string is also a remarkably striking example %
of the power of the ear to analyse into a long series of partial tones, a motion
which the eye and the imagination are able to conceive in a much simpler manner.
A string, which is pulled aside by a sharp point, or the finger nail, assumes the
form fig. 18, A (p. S4a), before it is released. It then passes through the series of
forms, fig. 18, B, C, D, E, F, till it reaches G, which is the inversion of A, and
then returns, through the same, to A again. Hence it alternates between the forms
A and G. AU these forms, as is clear, are composed of three straight lines, and
on expressing the velocity of the individual points of the strings by vibrational
curves, these would have the same form. Now the string scarcely imparts any
perceptible portion of its own motion directly to the air. Scarcely any audible
tone results when both ends of a string are fastened to immovable supports, as
metal bridges, which are again fastened to the walls of a room. The sound of
* See Brandt in Poggcndorff's Annalen der Physik, vol. cxii. p. 324, where this fact ia
proved.
Digitized by VjOOQ IC
54
PROOF OF OHM'S LAW.
PART I.
Fio. x8.
the string reaches the air through that one of its extremities which rests upon
a bridge standing on an elastic sounding board. Hence the sound of the string
essentially depends on the motion of this
extremity, through the pressure which it
exerts on the sounding board. The magni-
tude of this pressure, as it alters periodically
with the time, is shewn in fig. 19, where
the height of the line h h corresponds to
the amount of pressure exerted on the bridge
by that extremity of the string when the
string is at rest. Along h h suppose
lengths to be set off corresponding to con-
secutive intervals of time, the vertical
% heights of the broken line above or below
h h represent the corresponding augmenta-
tions or diminutions of pressure at those
times. The pressure of the string on the
sounding board consequently alternates, as
the figure shews, between a higher and a
lower value. For some time the greater
pressure remains unaltered ; then the lower
suddenly ensues, and likewise remains for a
time unaltered. The letters a to g in fig. 19
correspond to the times at which the string
assumes the forms A to G in fig. 18. It is this alteration between a greater and
a smaller pressure which produces the sound in the air. We cannot but feel
astonished that a motion produced by means so simple and so easy to comprehend,
f should be analysed by the ear into such a complicated sum of simple tones. For
the eye and the understanding the action of the string on the sounding board can
be figured with extreme simplicity. What has the simple broken line of fig. 19
to do with wave*curv6S, which, in the course of one of their periods, shew
1-I--I-U:
1) c d
4...|.,..|-+.H.-4-|-4-.
f g t
Fig.
h a
19.
3, 4, 5, up to 16, and more, crests and troughs ? This is one of the most striking
examples of the different ways in which eye and ear comprehend a periodic
motion.
There is no sonorous body whose motions under varied conditions can be so
^ completely calculated theoretically and contrasted with observation as a string.
The following are examples in which theory can be compared with analysis by
ear: —
I have discovered a means of exciting simple pendular vibrations in the air. A
tuning-fork when struck gives no harmonic upper partial tones, or, at most, traces
of them when it is brought into such excessively strong vibration that it no longer
exactly follows the law of the pendulum.* On the other hand, tuning-forks have
some very high inharmonic secondary tones, which produce that peculiar sharp
* [On all ordinary tuning-forks between a
and d" in pitch, I have been able to hear the
Beoond partial or Octave of the prime. In
Bome low forks this Octave is so powerful that
on pressing the handle of the fork against the
table^ the prime quite disappears and the
Octave only is heard, .and this has often
proved a source of embarrassment in tuning
the forks, or in counting beats to determine
pitch numbers. But the prime can always be
heard when the fork is held to the ear or over
a properly tuned resonance jar, as described in
this paragraph. I tune such jars by pouring
water in or out until the resonance is strongest,
and then I register the height of the water
and pitch of the fork for future use on a slip
of paper gummed to the side of the jar. I
have found that it is not at all necessary to
Digitized by V^jOOQlC
CHAP. IV. PBOOP OF OHM'S LAW. 55
tinkling of the fork at the moment of being struck, and generally become rapidly
inaudible. If the tuning-fork is held in the fingers, it imparts very little of its
tone to the air, and cannot be heard unless it is held close to the ear. Instead of
holding it in the fingers, we may screw it into a thick board, on the under side of
which some pieces of india-rubber tubing have been fastened. When this is laid
upon a table, the india-rubber tubes on which it is supported convey no sound to
the table, and the tone of the tuning-fork is so weak that it may be considered in-
audible. Now if the prongs of the fork be brought near a resonance chamber * of
a bottle-form of such a size and shape that, when we blow over its mouth, the air
it contains gives a tone of the same pitch as the fork's, the air within this chamber
vibrates sympathetically, and the tone of the fork is thus conducted with great
strength to the outer air. Now the higher secondary tones of such resonance
chambers are also inharmonic to the prime tone, and in general the secondary
tones of the chambers correspond neither with the harmonic nor the inharmonic H
secondary tones of the forks ; this can be determined in each particular case by
producing the secondary tones of the bottle by stronger blowing, and discovering
those of the forks with the help of strings set into sympathetic vibration, as will
be presently described. If, then, only one of the tones of the fork, namely the
prime tone, corresponds with one of the tones of the chamber, this alone will be
reinforced by sympathetic vibration, and this alone will be communicated to the
external air, and thus conducted to the observer's ear. The examination of the
motion of the air by resonators shews that in this case, provided the tuning-fork be
not set into too violent motion, no tone but the prime is present, and in such case
the unassisted ear hears only a single simple tone, namely the common prime of
the tuning-fork and of the chamber, without any accompanying upper partial tones.
The tone of a tuning-fork can also be purified from secondary tones by placing
its handle upon a string and moving it so near to the bridge that one of the proper
tones of the section of string lying between the fork and the bridge is the same as ^
that of the tuning-fork. The string then begins to vibrate strongly, and conducts
the tone of the tuning-fork with great power to the sounding board and surround-
ing air, whereas the tone is scarcely, if at all, heard as long as the above-named
section is not in unison with the tone of the fork. In this way it is easy to find
the lengths of string which correspond to the prime and upper partial tones of the
fork, and accurately determine the pitch of the latter. If this experiment is con-
ducted with ordinary strings which are uniform throughout their length, we shield
the ear from the inharmonic secondary tones of the fork, but not firom the harmonic
upper partials, which are sometimes faintly present when the fork is made to
vibrate strongly. Hence to conduct this experiment in such a way as to create
purely pendular vibrations of the air, it is best to weight one point of the string, if
only so much as by letting a drop of melting sealing-wax fall upon it. This causes
the upper proper tones of the string itself to be inharmonic to the prime tone, and
hence there is a distinct interval between the points where the fork must be placed f
to bring out the prime tone and its audible Octave, if it exists.
In most other cases the mathematical analysis of the motions of sound is not
nearly far enough advanced to determine with certainty what upper partials will
be present and what intensity they will possess. In circular plates and stretched
membranes which are struck, it is theoretically possible to do so, but their inhar-
ptit the fork into excessively strong vibration of Chap. VII., and Prof. Preyer's in App. XX.
in order to make the Octave sensible. Thus, sect. L. art. 4, c. The conditions according
taking a fork of 232 and another of 468 vibra- to Eoenig that tuning-forks should have no
tions, after striking them both, and letting the upper partials are given in App. XX. sect. L.
deeper fork spend most of its energy until I art. 2, a. — Tranakdcyr.]
could not see the vibrations with the eye at all, * Either a bottle of a proper size, which
the beats were heard distinctly, when I pressed can readily be more accurately tuned by pour-
both on to a table, and continued to be heard ing oil or water into it, or a tube of i>asteboard
even after the forks themselves were separately quite closed at one end, and having a small
inaudible. See also Prof. Helmhoitz's experi- round opening at the other. See the proper
ments on a fork of 64 vibrations at the close sizes of such resonance chambers in App. IV.
Digitized by V^OOQIC
St5
PROOF OF OHM'S LAW.
PAST I.
monic secondary tones are so numerous and so nearly of the same pitch that most
observers would probably fail to separate them satisfactorily. On elastic rods, how-
ever, the secondary tones are very distant from each other, and are inharmonic, so
that they can be readily distinguished from each other by the ear. The following
are the proper tones of a rod which is free at both ends ; the vibrational number
of the prime tone taken to be c, is reckoned as i : —
Pitch Number
Cent«»
Notation
Prime tone
Second proper tone
Third proper tone
Fourth proper tone
I -0000
27576
5-4041
I3-344*
0
1200+556
2400+521
3600 + 886
C
f +0-2
f 4 0-I
a" -o-i
The notation is adapted to the equal temperament, and the appended fractions
" are parts of the interval of a complete tone.
Where we are unable to execute the theoretical analysis of the motion, we can,
at any rate, by means of resonators and other sympathetically vibrating bodies,
analyse any individual musical tone that is produced, and then compare this
analysis, which is determined by the laws of sympathetic vibration, with that
effected by the unassisted ear. The latter is naturally much less sensitive than
one armed with a resonator ; so that it is frequently impossible for the unarmed
ear to recognise amongst a number of other stronger simple tones those which the
resonator itself can only faintly indicate. On the other hand, so far as my ex-
perience goes, there is complete agreement to this extent : the ear recognises with-
out resonators the simple tones which the resonators greatly reinforce, and perceives
no upper partial tone which the resonator does not indicate. To verify this con-
clusion, I performed numerous experiments, both with the human voice and the
harmonium, and they all confirmed it.f
If By the above experiments the proposition enunciated and defended by G. S.
Ohm must be regarded as proved, viz. that the human ear perceives pendular vibra-
tions alone as simple tones, and resolves all other periodic motions of the air into
a series of pendular vibrations ^ hearing the series of simple tones which correspond
tvith these simple vibrations.
Calling, then, as already defined (in pp. 23, 24 and note), the sensation excited
in the ear by any periodical motion of the air a musical tone, and the sensation
excited by a simple pendular vibration a simple tone, the rule asserts that the
sensation of a musical tone is compoufided out of the sensations of several simple
tones. In particular, we shall henceforth call the sound produced by a single
sonorous body its (simple or compound) tone, and the sound produced by several
musical instruments acting at the same time a composite tone, consisting generally
of several (simple or compound) tones. If, then, a single note is sounded on a
m * [For cents see note p. ^id. As a Tone is
200 ct., O'l Tone " 20 ct., these would give for
the Author's notation / + 40 ct., /' + 20 ct., a'"
- 10 ct., whereas the column of cents shews
that they are more accurately / + 56 ct., /' +
21 ct., a'" —14 ct. For convenience, the cents
for Octaves are separated, thus 1200+556 in
place of 1756, but this separation is quite
unnecessary. The cents again shew the inter-
vals of the inharmonic partial tones without
any assumption as to the value of the prime.
By a misprint in all the German editions,
followed in the first English edition, the second
proper tone was made/— 0*2 in place of / +
©•2. — Translator.]
t [In my * Notes of Observations on Musi-
cal Beats,' Proceedings of the Royal Society,
May 1880, vol. xxx. p. 531, largely cited in
App. XX. sect. B. No. 7, 1 showed Uiat I was
able to determine the pitch numbers of deep
reed tones, by the beats (Chap. VIII.) that their
upper partials made with the primes of a set of
Scheibler's tuning-forks. The correctness of
the process was proved by the fact that the
results obtained ftrom different partials of the
same reed tone, which were made to beat with
different forks, gave the same pitch numbers
for the primes, within one or two hundredths of
a vibration in a second. I not only employed
such low partials as 3, 4, 5 for one tone, and
4, 5, 6 for others, but I determined the pitch
number 31*47, by partials 7, 8, 9, 10, 11, 12,
13, and the pitch number 15*94 by partials 25
and 27. The objective reality of these ex-
tremely high upper partials, and their inde-
pendence of resonators or resonance jars, was
therefore conclusively shewn. On the Har-
monical the beats of the i6th partial of C 66,
with c'", when slightly flattened by pressirg the
note lightly down, are very cleai.— Translator, j
Digitized by.V^jOOQlC
CHAP. IV.. DIFFICULTIES IN OBSERVING PAETIALS. 57
masical instnunent, as a violin, trmnpet, organ, or by a singing voice, it mast be
called in exact language a tone of the instrument in question. This is also the
ordinary language, but it did not then imply that the tone might be compound.
When the tone is, as usual, a compound tone, it will be distinguished by this term,
or the abridgment, a compound ; while tone is a general term which includes both
simple and compound tones.* The prime tone is generally louder than any of the
upper partial tones, and hence it alone generally determines the pitch of the com-
pound. The tone produced by any sonorous body reduces to a single simple tone
in very few cases indeed, as the tone of tuning-forks imparted to the air by reso-
nance chambers in the manner already described. The tones of wide-stopped
oigan pipes when gently blown are almost free from upper partials, and are accom-
panied only by a rush of wind.
It is well known that this union of several simple tones into one compound
tone, which is naturally effected in the tones produced by most musical instruments, ^
is artificially imitated on the organ by peculiar mechanical contrivances. The
tones of organ pipes are comparatively poor in upper partials. When it is desirable
to use a stop of incisive penetrating quality of tone and great power, the wide pipes
{principal register and weitgedackt t) are not sufficient ; their tone is too soft, too
defective in upper partials ; and the narrow pipes (getgen-register and quintaten i)
are also unsuitable, because, although more incisive, their tone is weak. For such
occasions, then, as in accompanying congregational singing, recourse is had to the
compound stops. § In these stops every key is connected with a larger or smaller
series of pipes, which it opens simultaneously, and which give the prime tone and
a certain number of the lower upper partials of the compound tone of the note in
question. It is very usual to connect the upper Octave with the prime tone, and
after that the Twelfth. The more complex compounds (comet §) give the first six
partial tones, that is, in addition to the two Octaves of the prime tone and its
Twelfth, the higher major Third, and the Octave of the Twelfth. This is as much ^
of the series of upper partials as belongs to the tones of a major chord. But
to prevent these compound stops from being insupportably noisy, it is necessary
to reinforce the deeper tones of each note by other rows of pipes, for in all natural
tones which are suited for musical purposes the higher partials decrease in force as
they rise in pitch. This has to be regarded in their imitation by compound stops.
These compound stops were a monster in the path of the old musical theory, which
'was acquainted only with the prime tones of compounds; but the practice of
organ-builders and organists necessitated their retention, and when they are
suitably arranged and properly applied, they form a very effective musical apparatus.
* [Here, again, as on pp. 23, 24, 1 have, in toned diapason, eight feet.* Hopkins, Organ,
the translation, been necessarily obliged to p. 445. * A manual stop of eight feet, produ-
deriate slightly from the original. Klang, as oing a pungent tone very like that of the
here defined, embraces Ton as a particular Gamba, except that the pipes, being of larger
case. I use tone for the general term, and scale, speak quicker and produce a fuller tone.
compound tone and simple tone for the two Examples of the stop exist at Doncaster, ir
particular cases. Thus, as presently mentioned the Temple Church, and in the Exchange
in the text, the tone produced by a tuning-fork Organ at Northampton.' Ibid. p. 138. For
held over a proper resonance chamber we know, quintaten^ see supra, p. 33 J, note. — Translator.]
on analysis, to be simple^ but before analysis it § [As described in Hopkins, Organ, p. 142,
is to us only a (musical) tone like any other, these are the sesquialtera * of five, four, three,
and hence in this case the Author's Klang or two ranks of open metal pipes, tuned
becomes the Author's Ton, I believe that the in Thirds, Fifths, and Octaves to the Diapa-
language used in my translation is best adapted son.' The mixture, consisting of five to two
for the constant accurate distinction between ranks of open metal pipes smaller than the
compound and simple tones by English readers, last, is in England the second, in Germany the
u I leave nothing which runs counter to old first, compound stop (p. 143). The Furniture ot
habits, and by the use of the words simple and five to two sets of small open pipes, is variable,
compound, constantly recall attention to this i) The Comet, mounted has five ranks of very
newly discovered and extremely important rela- large and loudly voiced pipes, 2) the echo, ia
tion.— Tran^Zator.] similar, but light and delicate, and is inclosed
t [Principal — double open diapason. Oross- in a box. In German organs the comet is also
96(2acfct— double stopped diapason. Hopkins, a pedal reed stop of four and two feet (ibid.). —
Orqan, p. 444- ^.—Translator.'] Translator."]
X [* Geigen Principal— violin or crisp-
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S8 DIFFICULTIES IN OBSERVING PARTIALS. pabt i.
The nature of the case at the same time fully justifies their use. The musician is
bound to regard the tones of all musical instruments as compounded in the same
way as the compound stops of organs, and the important part this method of com-
position plays in the construction of musical scales and chords will be made evident
in subsequent chapters.
We have thus been led to an appreciation of upper partial tones, which differs
considerably from that previously entertained by musicians, and even physicists,
and must therefore be prepared to meet the opposition which will be raised. The
upper partial tones were indeed known, but almost only in such compound tones as
those of strings, where there was a favourable opportunity for observing them ;
but they appear in previous physical and musical works as an isolated accidental
phenomenon of small intensity, a kind of curiosity, which was certainly occasion-
ally adduced, in order to give some support to the opinion that nature had pre-
If figured the construction of our major chord, but which on the whole remained
almost entirely disregarded. In opposition to this we have to assert, and we shall
prove the assertion in the next chapter, that upper partial tones are, with a few
exceptions already named, a general constituent of all musical tones, and that a
certain stock of upper partials is an essential condition for a good musical quality
of tone. Finally, these upper partials have been erroneously considered as weak,
because they are difficult to observe, while, in point of fact, for some of the best
musical qualities of tone, the loudness of the first upper partials is not far inferior
to that of the prime tone itself.
There is no difficulty in verifying this last fact by experiments on the tones of
strings. Strike the string of a piano or monochord, and immediately touch one of
its nodes for an instant with the finger ; the constituent partial tones having this
node will remain with unaltered loudness, and the rest will disappear. We might
also touch the node in the same way at the instant of striking, and thus obtain the
f corresponding constituent partial tones from the first, in place of the complete
compound tone of the note. In both ways we can readily convince ourselves that the
first upper partials, as the Octave and Twelfth, are by no means weak and difiicult
to hear, but have a very appreciable strength. In some cases we are able to assign
numerical values for the intensity of the upper partial tones, as. will be shewn in
the next chapter. For tones not produced on strings this d posteriori proof is not
so easy to conduct, because we are not able to make the upper partials speak
separately. But even then by means of the resonator we can appreciate the in-
tensity of these upper partials by producing the corresponding note on the same
or some other instrument until its loudness, when heard through the resonator,
agrees with that of the former.
The difiiculty we experience in hearing upper partial tones is no reason for
considering them to be weak ; for this difficulty does not depend on their intensity,
but upon entirely different circumstances, which could not be properly estimated
% mitil the advances recently made in the physiology of the senses. On this diffi-
culty of observing the upper partial tones have been founded the objections which
A. Seebeck ♦ has advanced against Ohm's law of the decomposition of a musical
tone ; and perhaps many of my readers who are unacquainted with the physiology
of the other senses, particularly with that of the eye, might be inclined to adopt
Seebeck's opinions. I am therefore obliged to enter into some details concerning
this difference of opinion, and the peculiarities of the perceptions of our senses,
on which the solution of the difficulty depends.
Seebeck, although extremely accomplished in acoustical experiments and
observations, was not always able to recognise upper partial tones, where Ohm's
law required them to exist. But we are also bound to add that he did not apply
the methods already indicated for directing the attention of his ear to the upper
partials in question. In other cases when he did hear the theoretical upper
* In Poggendorff's Annalen der Physik, vol. Ix. p. 449, vol. Ixiii. pp. 353 and 368.— OAm,
ibid, vol. lix. p. 513, and vol. Ixii. p. i.
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CHAP. IV. DIFFICULTIES IN OBSERVING PARTIALS. 59
paiiials, they were weaker than the theory required. He concluded that the defi-
nition of a simple tone as given by Ohm was too limited, and that not only pen-
dular vibrations, but other vibrational forms, provided they were not too widely
separated from the pendular, were capable of exciting in the ear the sensation of
a single simple tone, which, however, had a variable quality. He consequently
asserted that when a musical tone was compounded of several simple tones, part
of the intensity of the upper constituent tones went to increase the intensity of
the prime tone, with which it fused, and that at most a small remainder excited in
the ear the sensation of an upper partial tone. He did not formulate any deter-
minate law, assigning the vibrational forms which would give the impression of
a simple and those which would give the impression of a compound tone. The
experiments of Seebeck, on which he founded his assertions, need not be here
described in detail. Their object was only to produce musical tones for which
either the intensity of the simple vibrations corresponding to the upper partials %
could be theoretically calculated, or in which these upper partials could be
rendered separately audible. For the latter purpose the siren was used. We have
just described how the same object can be attained by means of strings. Seebeck
shews in each case that the simple vibrations corresponding to the upper partials
have considerable strength, but that the upper partials are either not heard at all,
or heard with difficulty in the compound tone itself. This fact has been already
mentioned in the present chapter. It may be perfectly true for an observer who
has not appUed the proper means for observing upper partials, while another, or
even the first observer himself when properly assisted, can hear them perfectly well.*
Now there are many circumstances which assist us first in separating the
musical tones arising from different sources, and secondly, in keeping together the
partial tones of each separate source. Thus when one musical tone is heard for
some time before being joined by the second, and then the second continues after
the first has ceased, the separation in sound is facilitated by the succession of time. ^
We have already heard the first musical tone by itself, and hence know imme-
diately what we have to deduct from the compound effect for the effect of this first
tone. Even when several parts proceed in the same rhythm in polyphonic music,
the mode in which the tones of different instruments and voices commence, the
nature of their increase in force, the certainty with which they are held, and the
manner in which they die off, are generally slightly different for each. Thus the
tones of a pianoforte commence suddenly with a blow, and are consequently
strongest at the first moment, and then rapidly decrease in power. The tones of
brass instruments, on the other hand, commence sluggishly, and require a small
but sensible time to develop their full strength. The tones of bowed instruments
are distinguished by their extreme mobility, but when either the player or the
instrument is not unusually perfect they are interrupted by little, very short,
pauses, producing in the ear the sensation of scraping, as will be described more
in detail when we come to analyse the musical tone of a violin. When, then, such ^f
instruments are sounded together there are generally points of time when one or
the other is predominant, and it is consequently easily distinguished by the ear.
But besides all this, in good part music, especial care is taken to facihtate the
separation of the parts by the ear. In polyphonic music proper, where each part
has its own distinct melody, a principal means of clearly separating the progres-
sion of each part has always consisted in making them proceed in different rhythms
and on different divisions of the bars ; or where this could not be done, or was at
any rate only partly possible, as in four-part chorales, it is an old rule, contrived
for this purpose, to let three parts, if possible, move by single degrees of the scale,
and let the fourth leap over several. The small amount of alteration in the pitch
makes it easier for the listener to keep the identity of the several voices distinctly
in mind.
^* [Here from * Upper partial tones,* p. 94, to * former analysis,* p. 100 of the ist English
edition are omitted, in accordance with the 4th German edition. — Translator,]
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6o
FUSION OF PARTIALS INTO A COMPOUND.
PABT I.
Fm. s
AH these helps £a.il in the resolution of musical tones into their constituent
partials. When a compound tone commences to sound, all its partial tones
commence with the same comparative strength; when it swells, all of them
generally swell uniformly ; when it ceases, all cease simultaneously. Hence no
opportunity is generally given for hearing them separately and independently. In
precisely the same manner as the naturally connected partial tones form a single
source of sound, the partial tones in a compound stop on the organ fuse into one, as
all are struck with the same digital, and all move in the same melodic progression
as their prime tone.
Moreover, the tones of most instruments are usually accompanied by charac-
teristic irregular noises, as the scratching and rubbing of the violin bow, the rush
of wind in flutes and organ pipes, the grating of reeds, &c. These noises, with
which we are already familiar as characterising the instruments, materially
H facilitate our power of distinguishing them in a composite mass of sounds. The
partial tones in a compound have, of course, no such characteristic marks.
Hence we have no reason to be surprised that the resolution of a compound
tone into its partials is not quite so easy for the ear to accomplish, as the resolu-
tion of composite masses of the musical sounds of many instruments into their
proximate constituents, and that even a trained musical ear requires the applica-
tion of a considerable amount of attention when it undertakes the former problem.
It is easy to see that the auxiliary circumstances already named do not always
suffice for a correct separation of musical tones. In imiformly sustained musical
tones, where one might be considered as an upper partial of another, our
judgment might readily make default. This is reaUy the case. G. S. Ohm
proposed a very instructive experiment to shew this, using the tones of a violin.
But it is more suitable for such an experiment to use simple tones, as those of a
stopped organ pipe. The best instrument, however, is a glass bottle of the form
H shewn in fig. 20, which is easily procured and
prepared for the experiment. A Uttle rod c
supports a guttapercha tube a in a proper
position. The end of the tube, which is
directed towards the bottle, is softened in warm
water and pressed flat, forming a narrow chink,
through which air can be made to rush over
the mouth of the bottle. When the tube is
fastened by an india-rubber pipe to the nozzle
of a bellows, and wind is driven over the bottle,
it produces a hollow obscure soimd, like the
vowel 00 in too, which is freer from upper
partial tones than even the tone of a stopped
pipe, and is only accompanied by a slight
H noise of wind. I find that it is easier to keep
the pitch unaltered in this instrument while
the pressure of the wind is slightly changed,
than in stopped pipes. We deepen the tone by
partially shading the orifice of the bottle with
a little wooden plate; and we sharpen it by
pouring in oil or melted wax. We are thus able to make any required little
alterations in pitch. I tuned a large bottle to bj;} and a smaller one to b^ and
united them with the same bellows, so that when used both began to speak at the
same instant. When thus united they gave a musical tone of the pitch of the
deeper bj;}, but having the quahty of tone of the vowel oa in toad, instead of 00 in
too. When, then, I compressed first one of the india-rubber tubes and then the
other, so as to produce the tones alternately, separately, and in connection, I was
at last able to hear them separately when sounded together, but I could not
continue to hear them separately for long, for the upper tone gradually fused with
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CHAP. nr. SEPAEATION OF THE PABTIALS. 6i
the lower. This fusion takes place even when the upper tone is somewhat stronger
than the lower. The alteration in the quality of tone which takes place during
this fusion is characteristic. On producing the upper tone first and then letting
the lower sound with it, I found that I at first continued to hear the upper tone
with its full force, and the under tone sounding helow it in its natural quality of
00 in too. But by degrees, as my recollection of the sound of the isolated upper
tone died away, it seemed to become more and more indistinct and weak, while
the lower tone appeared to become stronger, and sounded like oa in toad. This
weakening of the upper and strengthening of the lower tone was also observed by
Ohm on the violin. As Seebeck remarks, it certainly does not always occur, and
probably depends on the liveliness of our recollections of the tones as heard
separately, and the greater or less uniformity in the simultaneous production of
the tones. But where the experiment succeeds, it gives the best proof of the
essential dependence of the result on varying activity of attention. With the tones If
produced by bottles, in addition to the reinforcement of the lower tone, the altera-
tion in its quality is very evident and is characteristic of the nature of the process.
This alteration is less striking for the penetrating tones of the violin.*
This experiment has been appealed to both by Ohm and by Seebeck as a
corroboration of their different opinions. When Ohm stated that it was an
' illusion of the ear ' to apprehend the upper partial tones wholly or partly as a
reinforcement of the prime tone (or rather of the compound tone whose pitch is
determined by that of its prime), he certainly used a somewhat incorrect expression,
although he meant what was correct, and Seebeck was justified in replying that
the ear was the sole judge of auditory sensations, and that the mode in which it
apprehended tones ought not to be called an * illusion.' However, our experiments
just described shew that the judgment of the ear differs according to the liveliness
of its recollection of the separate auditory impressions here fused into one whole,
and according to the intensity of its attention. Hence we can certainly appeal from %
the sensations of an ear directed without assistance to external objects, whose
interests Seebeck represents, to the ear which is attentively observing itself and
is suitably assisted in its observation. Such an ear really proceeds according to
the law laid down by Ohm.
Another experiment should be adduced. Eaise the dampers of a pianoforte so
that all the strings can vibrate freely, then sing the vowel a ia father, art, loudly
to any note of the piano, directing the voice to the sounding board ; the sym-
pathetic resonance of the strings distinctly re-echoes the same a. On singing oe
in toe, the same oe is re-echoed. On singing a in fare, this a is re-echoed. For ee
in see the echo is not quite so good. The experiment does not succeed so well if
the damper is removed only from the note on which the vowels are sung. The
vowel character of the echo arises from the re-echoing of those upper partial tones
which characterise the vowels. These, however, will echo better and more
clearly when their corresponding higher strings are free and can vibrate sym- ^
pathetically. In this case, then, in the last resort, the musical effect of the
resonance is compounded of the tones of several strings, and several separate
partial tones combine to produce a musical tone of a peculiar quality. In addition
to the vowels of the human voice, the piano will also quite distinctly imitate the
quality of tone produced by a clarinet, when strongly blown on to the sounding
board.
Finally, we must remark, that although the pitch of a compound tone is, for
* [A very convenient form of this ezpezi- The tone is also brighter and unaccompanied
mont, useful even for lecture purposes, is to by any windrush. By pressing the handle of
employ two tuning-forks, tuned as an Octave, the deeper fork on the table, we can excite its
say & and c", and held over separate resonance other upper partials, and thus produce a third
jars. By removing first one and then the other, quality of tone, which can be readily appre-
or letting both sound together, the above effects ciated ; thus, simple c', simple cf + simple c",
can be made evident, and they even remain compound c'. — Translator,]
when the Octave is not tuned perfectly true.
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62 ANALYSIS OP COMPOUND SENSATIONS. pabt i.
musical purposes, determined by that of its prime, the influence of the upper
partial tones is by no means unfelt. They give the compound tone a brighter and
higher effect. Simple tones are dull. When they are compared with compound
tones of the same pitch, we are inclined to estimate the compound as belonging to
a higher Octave than the simple tones. The difference is of the same kind as that
heard when first the vowel oo in too and then a in tar are sung to the same note.
It is often extremely difficult to compare the pitches of compound tones of different
qualities. It is very easy to make a mistake of an Octave. This has happened
to the most celebrated musicians and acousticians. Thus it is well known that
Tartini, who was celebrated as a violinist and theoretical musician, estimated all
combinational tones (Chap. XI.) an Octave too high, and, on the other hand,
Henrici * assigns a pitch too low by an Octave to the upper partial tones of
tuning-forks.f
I The problem to be solved, then, in distinguishing the partials of a compound
tone is that of analysing a given aggregate of sensations into elements which no
longer admit of analysis. We are accustomed in a large number of cases where
sensations of different kinds or in different parts of the body, exist simultaneously,
to recognise that they are distinct as soon as they are perceived, and to direct our
attention at will to any one of them separately. Thus at any moment we can be
separately conscious of what we see, of what we hear, of what we feel, and dis-
tinguish what we feel in a finger or in the great toe, whether pressure or a gentle
touch,or warmth. So also in the field of vision. Indeed, as I shall endeavour to shew
in what follows, we readily distinguish our sensations from one another when we
have a precise knowledge that they are composite, as, for example, when we have
become certain, by frequently repeated and invariable experience, that our present
sensation arises from the simultaneous action of many independent stimuli, each
of which usually excites an equally well-known individual sensation. This induces
1 f us to think that nothing can be easier, when a number of different sensations are
, simultaneously excited, than to distinguish them individually from each other, and
1 that this is an innate faculty of our minds.
Thus we find, among others, that it is quite a matter of course to hear sepa-
rately the different musical tones which come to our senses collectively, and expect
that in every case when two of them occur together, we shall be able to do the
like.
The matter is very different when we set to work at investigating the more un-
usual cases of perception, and at more completely understanding the conditions under
which the above-mentioned distinction can or cannot be made, as is the case in the
physiology of the senses. We then become aware that two different kinds or grades
must be distinguished in our becoming conscious of a sensation. The lower grade of
this consciousness, is that where the influence of the sensation in question makes
itself felt only in the conceptions we form of external things and processes, and assists
^ in determining them. This can take place without our needing or indeed being able
to ascertain to what particular part of our sensations we owe this or that relation
of our perceptions. In this case we will say that the impression of the sensation in
question is perceived synthetically. The second and higher grade is when we
immediately distinguish the sensation in question as an existing part of the sum
of the sensations excited in us. We will say then that the sensation is perceived
analytically .X The two cases must be carefully distinguished firom each other.
♦ Poggd. Ann., vol. xcix. p. 506. The with wahrgenommeny and then restricting the
same difficulty is mentioned by Zamminer meaning of this very common German word.
{Die Musik und die musikalischen Instru- It appeared to me that it would be clearer to
merUe, 1855, p. 1 11) as well known to musicians. an English reader not to invent new words
f [Here the passage from ' The problem or restrict the sense of old words, but to
to be solved,* p. 626, to * from its simple use perceived in both cases, and distinguish
tones,* p. 656, is inserted in this edition from the them (for percipirt and apperdpirt respectively)
4th German edition.— TraTt^Za^r.] by the adjuncts synthetically and analytically^
i [Prof. Helmholtz uses Leibnitz's terms the use of which is clear from the explanations
percipirt and apperdpirt ^ alternating the latter given in the iBxt,— Translator.]
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CHAP. IV. ANALYSIS OP COMPOUND SENSATIONS. 63
Seebeck and Ohm are agreed that the upper partials of a musical tone are
perceiyed synthetically. This is acknowledged by Seebeck when he admits that
their action on the ear changes the force or quality of the sound examined. The
dispute turns upon whether in all cases they can be perceived analytically in their
individual existence ; that is, whether the ear when unaided by resonators or other
physical auxiliaries, which themselves alter the mass of musical sound heard by the
observer, can by mere direction and intensity of attention distinguish whether, and
if so in what force, the Octave, the Twelfth, &c , of the prime exists in the given
mnsical sound.
In the first place I will adduce a series of examples which shew that the
difficulty felt in analysing musical tones exists also for other senses. Let us
begin with the comparatively simple perceptions of the sense of taste. The
ingredients of our dishes and the spices with which we flavour them, are not so
complicated that they could not be readily learned by any one. And yet there are if
very few people who have not themselves practically studied cookery, that are able
readily and correctly to discover, by the taste alone, the ingredients of the dishes
placed before them. How much practice, and perhaps also peculiar talent, belongs
to wine tasting for the purpose of discovering adulterations is known in all wme-
growing countries. Similarly for smell ; indeed the sensations of taste and smell
may unite to form a single whole. Using our tongues constantly, we are scarcely
aware that the peculiar character of many articles of food and drink, as vinegar or
wine, depends also upon the sensation of smell, their vapours entering the back
part of the nose tlirough the gullet. It is not till we meet vdth persons in whom
the sense of smell is deficient that we learn how essential a part it plays in
tasting. Such persons are constantly in fault when judging of food, as mdeed any
one can learn from his own experience, when he suffers from a heavy cold in the
head without having a loaded tongue.
When our hand glides unawares along a cold and smooth piece of metal we %
are apt to imagine that we have wetted our hand. This shews that the sensation
of witness to the touch is compounded out of that of unresisting ghding and cold,
which in one case results from the good heat-conducting properties of metal, and
in the other from the cold of evaporation and the great specific heat of water.
We can easily recognise both sensations in wetness, when we think over tlie
matter, but it is the above-mentioned illusion which teaches us that the peculiar
feeling of wetness is entirely resolvable into these two sensations.
The discovery of the stereoscope has taught us that the power of seeing the
depths of a field of view, that is, the different distances at which objects and
their parts lie from the eye of the spectator, essentially depends on the simul-
taneous synthetical perceptions of two somewhat different perspective images of
the same objects by the two eyes of the observer. If the difference of the two
images is sufficiently great it is not difficult to perceive them analytically as
separate. For example, if we look intently at a distant object and hold one of H
our fingers slightly in front of our nose we see two images of our finger against
the background, one of which vanishes when we close the right eye, the other
belonging to the left. But when the differences of distance are relatively small,
and hence the differences of the two perspective images on the retina are so also,
great practice and certainty in the observation of double images is necessary to
keep them asnnder, yet the synthetical perception of their differences still exists,
and makes itself felt in the apparent relief of the surface viewed. In this case
also, as well as for upper partial tones, the ease and exactness of the analytical
perception is far behind that of the synthetical perception.
In the conception which we form of the direction in which the objects viewed
fieem to he, a considerable part must be played by those sensations, mainly muscular,
which enable us to recognise the position of our body, of the head with regard to
the body, and of the eye with regard to the head. If one of these is altered, for
example, if the sensation of the proper position of the eye is changed by pressing t
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64 ANALYSIS OF COMPOUND SENSATIONS. pabt l
a finger against the eyeball or by injury to one of the muscles of the eye, our per-
ception of the position of visible objects is also changed. But it is only by such
occasional illusions that we become aware of the fact that muscular sensations form
part of the aggregate of sensations by which our conception of the position of a
visible object is determined.
t^Vrv The phenomena of mixed colours present considerable analogy to those of com-
Jl^wjiaWI P^^'i^d musical tones, only in the case of colour the number of sensations reduces to
^^'^^ three, and the analysis of the composite sensations into their simple elements is still
more difficult and imperfect than for musical tones. As early as 1686 R. Waller
mentions in the Philosophical Transactions the reduction of all colours to the
mixture of three fundamental colours, as something already well known. This
view could in earlier times only be founded on sensations and experiments arising
from the mixture of pigments. In recent times we have discovered better methods,
% by mixing light of different colours, and hence have confirmed the correctness of
that hj^othesis by exact measurements, but at the same time we have learned that
this confirmation only succeeds within a certain limit, conditioned by the fact that no
kind of coloured light exists which can give us the sensation of a single one of the
fundamental colours with exclusive purity. Even the most saturated and purest
colours that the external world presents to us in the prismatic spectrum, may by
the development of secondary images of the complementary colours in the eye
be still freed as it were from a white veil, and hence cannot be considered as abso-
lutely pure. For tliis reason we are unable to shew objectively the absolutely pure
fundamental colours from a mixture of which all other colours without exception
can be formed. We only know that among the colours of the spectrum scarlet-red,
yellow-green, and blue-violet approach to them nearer than any other objective
colours.* Hence we are able to compound out of these three colours almost all the
colours that usually occur in different natural bodies, but we cannot produce the
^ yellow and blue of the spectrum in that complete degree of saturation which they
reach when purest within the spectrum itself. Our mixtures are always a little
whiter than the corresponding simple colours of the spectrum. Hence it follows
that we never see the simple elements of our sensations of colour, or at least see
them only for a very short time in particular experiments directed to this end, and
consequently cannot have any such exact or certain image in our recollection, as
would indisputably be necessary for accurately analysing every sensation of colour
into its elementary sensations by inspection. Moreover we have relatively rare
opportunities of observing the process of the composition of colours, and hence of
recognising the constituents in the compound. It certainly appears to me very
characteristic of this process, that for a century and a half, from Waller to Goethe,
every one relied on the mixtures of pigments, and hence believed green to be a
mixture of blue and yellow, whereas when sky-blue and sulphur-yellow beams of
Hght, not pigments, are mixed together, the result is white. To this very cir-
f cumstance is due the violent opposition of Goethe, who was only acquainted with
the colours of pigments, to the assertion that white was a mixture of variously
coloured beams of hght. Hence we can have Httle doubt that the power of dis-
tinguishing the different elementary constituents of the sensation is originally
absent in the sense of sight, and that the little which exists in highly educated
observers, has been attained by specially conducted experiments, through which of
course, when wrongly planned, error may have ensued.
On the other hand every individual has an opportunity of experimenting on the
* [In his Physiological Optics^ p. 227, E^^ hence I translate span-griln by * yellow-
Prof. Helmholtz calls scarlet-red or vermilion green.' Maxwell's blae or third colour was
the part of the spectrum before reaching between the lines F and Gy but twice as far
Fraunhofer's line C He does not use span- from the latter as the former. This gives the
grUnt ( - ChrUn-span or verdigris, literally colour which Prof. H. in his Optics calls * cya-
' Spanish green ') in his Optics, but talks of nogen blue,' or Prussian blue. The violet
green-yellow between the lines E and 6, and proper does not begin till after the line O. It
he says, on p. 844, that Maxwell took as one of is usual to speak of these three colours, vaguely,
the fundamental colours * a green near the line as Bed, Green, and Blue.— Translator.]
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CHAPS. IV. V. ANALYSIS OP COMPOUND SENSATIONS. ^5
composition of two or more musical sounds or noises on the- most extended scale,
and the power of analysing even extremely involved compounds of musical tones,
into the separate parts produced by individual instruments, can readily be acquired
by any one who directs his attention to the subject. But the ultimate simple
elements of the sensation of tone, simple tones themselves, are rarely heard alone.
Even those instruments by which they can be produced, as tuning-forks before
resonance chambers, when strongly excited, give rise to weak harmonic upper
partials, partly within and partly without the ear, as we shall see in Chapters V.
and VII. Hence in this case dso, the opportunities are very scanty for impress-
ing on our memory an exact and sure image of ^ese simple elementary tones.
But if the constituents to be added are only indefinitely and vaguely known, the
analysis of the sum into those parts must be correspondingly uncertain. If we do
not know with certainty how much of the musical tone mider consideration is to
be attributed to its prime, we cannot but be uncertain as to what belongs to the ^
partials. Consequently we must begin by making the individual elements which
have to be distinguished, individually audible, so as to obtain an entirely fresh
recollection of the corresponding sensation, and the whole business requires un-
disturbed and concentrated attention. We are even without the ease that can be
obtained by frequent repetitions of the experiment, such as we possess in the
analysis of musical chords into their individual tones. In that case we hear the
individual tones sufficiently often by themselves, whereas we rarely hear simple
tones and may almost be said never to hear the building up of a compound from its
simple tones.
The results of the preceding discussion may be summed up as follows : —
I.) The upper partial tones corresponding to the simple vibrations of a com-
pound motion of the air, are perceived synthetically, even when they are not always
perceived analytically.
2.) But they can be made objects of analytical perception without any other If
help than a proper direction of attention.
3.) Even in the case of their not being separately perceived, because they fuse
into the whole mass of musical sound, their existence in our sensation is established
by an alteration in the quality of tone, the impression of their higher pitch being
eharacteristically marked by increased brightness and acuteness of quality.
In the next chapter we shall give details of the relations of the upper partials
to the quality of compound tones.
CHAPTER V.
ON THE DIFFEBENCEB IN THE QUALITY OF MUSICAL TONES.
TowABDS the close of Chapter I. (p. 21 j), we found that differences in the quality
of musical tones must depend on the form of the vibration of the air. The T
reasons for this assertion were only negative. We had seen that force depended
on amplitude, and pitch on rapidity of vibration : nothing else was left to distin-
guish quality but vibrational form. We then proceeded to shew that the existence
and force of the upper partial tones which accompanied the prime depend also on
the vibrational form, and hence we could not but conclude that musical tones of
the same quality would always exhibit the same combination of partials, seeing
that the peculiar vibrational form which excites in the ear the sensation of a certain
quality of tone, must always evoke the sensation of its corresponding upper partials.
The question then arises, can, and if so, to what extent can the differences of
musical quality be reduced to the combination of different partial tones with dif-
ferent intensities in different musical tones? At the conclusion of last chapter
(p. &od)j we saw that even artificially combined simple tones were capable of fusing
into a musical tone of a quality distinctly different from that of either of its con-
stituents, and that consequently the existence of a new upper partial really altered
Digitized by^DOQlC
66 CONCEPTION OF MUSICAL QUALITY AND TONE, pabt l
the quality of a tone. By this means we gained a clue to the hitherto enigmatical
nature of quality of tone, and to the cause of its varieties.
There has been a general inclination to credit quaUty with all possible pecu-
liarities of musical tones that were not evidently due to force and pitch. This was
correct to the extent that quality of tone was merely a negative conception. But
very slight consideration will suffice to shew that many of these peculiarities of
musical tones depend upon the way in which they begin and end. The methods of
attacking and releasing tones are sometimes so characteristic that for the human
voice they have been noted by a series of different letters. To these belong the ex-
plosive consonants B, D, G, and P, T, E. The effects of these letters are produced
by opening the closed, or closing the open passage through the mouth. For B
and P the closure is made by the lips, for D and T by the tongue and upper teeth,*
for G and E by the back of the tongue and soft palate. The series of the mediae
V B, D, G is distinguished from that of the tenues P, T, E, by the glottis being suffi-
ciently narrowed, when the closure of the former is released, to produce voice, or at
least the rustle of whisper, whereas for the latter or termes the glottis is wide open,t
and cannot sound. The mediae are therefore accompanied by voice, which is
capable of commencing at the beginning of a syllable an instant before the open-
ing of the mouth, and of lasting at the end of a syllable a moment after the closure
of the mouth, because some air can be still driven into the closed cavity of the
mouth and the vibration of the vocal chords in the larynx can be still maintained.
On account of the narrowing of the glottis the influx of air is more moderate, and
the noise of the wind less sharp for the mediae than the tenueSy which, being spoken
with open glottis, allow of a great deal of wind being forced at once from the chest.t
At the same time the resonance of the cavity of the mouth, which, as we shall
more clearly understand further on, exercises a great influence on the vowels,
varies its pitch, corresponding to the rapid alterations in the magnitude of its volume
f and oriflce, and this brings about a corresponding rapid variation in the quality of the
speech sound.
, As with consonants, the differences in the quality of tone of struck strings,
also partly depends on the rapidity with which the tone dies away. When tlie
strings have little mass (such as those of gut), and are fEistened to a very mobile
sounding board (as for a violin, guitar, or zither), or when the parts on which they
rest or which they touch are but slightly elastic (as when the violin strings, for
example, are pressed on the finger board by the soft point of the finger), their
vibrations rapidly disappear after striking, and the tone is dry, short, and without
ring, as in the pizzicato of a violin. But if the strings are of metal wire, and
hence of greater weight and tension, and if they are attached to strong heavy
bridges which cannot be much shaken, they give out their vibrations slowly to the
* [This is true for German, and most Con- examples, it seemed better in the present ease,
tinental languages, and for some dialectal where the author was speaking especially of
% English, especially ^n Cumberland, Westmore- the phenomena of speech to which he waa
land, Yorkshire, Lancashire, the Peak of Derby- personidly accustom^, to leave the text un-
shire, and Ireland, but even then only in con- altered and draw attention to English peculiari-
nection with the trilled B. Throughout Eng- ties in footnotes. — Translator,]
land generally, the tip of the tongue is quite % [Observe again that this description of
free from the teeth, except for TH in thin and the rush of wind accompanying P, T, K,
then, and for T and D it only touches the hard although true for German habits of speech, is
palate, seldom advancing so far as the root of not true for the usual English habits, which
the gamB.^Translator,] require the windrush between the opening of
f [This again is true for German, but not the mouth and sounding of the vowel to be
for English, French, or Italian, and not even entirely suppressed. The English result ia a
for the adjacent Slavonic languages. In these gliding vowel sound preceding Uie true vowel on
languages the glottis is quite closed for both commencing a syllable, and following the vowel
the mediae and the tenues in ordinary speech, on ending one. The difference between English
but the voice begins for the mediae before P and German Pis precisely the same (as I have
releasing the closure of the lips or tongue and verified by actual observation) as that between
palate, and for the tenues at the moment of the simple Sanscrit tenuis P, and the postaspi.
release. Although in giving vowel sounds, Ac, rated Sanscrit Ph, as now actually pronounced
I have generally contented myself with trans- by cultivated Bengalese. Beemy Early English
lating the same into English symbols and Pronunciationt'p, 11^6^061,1,— Translator:]
Digitized by V^jOOQlC
CHAP. V. CONCEPTION OF MUSICAL QUALITY AND TONE. 67
air and the sounding board ; their vibrations continue longer, their tone is more
durable and fuller, as in the pianoforte, but is comparatively less powerful and
penetrating than that of gut strings, which give up their tone more readily when
struck with the same force. Hence the pizzicato of bowed instruments when well
executed is much more piercing than the tone of a pianoforte. Pianofortes with
their strong and heavy supports for the strings have, consequently, for the same
thickness of string, a less penetrating but a much more lasting tone than those
instruments of which the supports for the strings are lighter.
It is very characteristic of brass instruments, as trumpets and trombones,
that their tones commence abruptly and sluggishly. The various tones in these
instruments are produced by exciting different upper partials through different
styles of blowing, which serve to throw the column of air into vibrating portions
of different numbers and lengths similar to those on a string. It always requires
a certain amount of effort to excite the new condition of vibration in place of the ^
old, but when once established it is maintained with less exertion. On the other
hand, the transition from one tone to another is easy for wooden wind instruments,
as the flute, oboe, and clarinet, where the length of the colmnn of air is readily
changed by application of the fingers to the side holes and keys, and where the
style of blowing has not to be materially altered.
These examples will suffice to shew how certain characteristic peculiarities in
the tones of several instruments depend on the mode in which they begin and end.
When we speak in what follows of musical quaUty of tone, we shall disregard
these peculiarities of beginning and ending, and confine our attention to the
peculiarities of the musical tone which continues uniformly.
But even when a musical tone continues with uniform or variable intensity,
it is mixed up, in the general methods of excitement, with certain noises, which
express greater or less irregularities in the motion of the air. In wind instruments
where the tones are maintained by a stream of air, we generally hear more or less ^
whizzing and hissing of the air which breaks against the sharp edges of the
mouthpiece. In strings, rods, or plates excited by a violin bow, we usually hear
a good deal of noise from the rubbing. The hairs of the bow are naturally full of
many minnte irregularities, the resinous coating is not spread over it with absolute
evenness, and there are also little inequalities in the motion of the arm which
holds the bow and in the amount of pressure, all of which influence the motion
of the string, and make the tone gf a bad instrument or an unskilful performer
rough, scraping, and variable. We shall not be able to explain the nature of the
motions of the air and sensations of the ear which correspond to these noises till
we have investigated the conception of heats. Those who listen to music make
themselves deaf to these noises by purposely withdrawing attention from them, but
a slight amoimt of attention generally makes them very evident for all tones pro-
duced by blowing or rubbing. It is well known that most consonants in human
speech are characterised by the maintenance of similar noises, as F, V ; S, Z ; TH f
in thin and in then ; the Scotch and German guttural CH, and Dutch G. For
some the tone is made still more irregular by trilling parts of the mouth, as for
B and L. In the case of B the stream of air is periodically entirely interrupted by
trilling the uvula * or the tip of the tongue ; and we thus obtain an intermitting
sound to which these interruptions give a peculiar jarring character. In the case
of L the soft side edges of the tongue are moved by the stream of air, and, without
completely interrupting the tone, produce inequalities in its strength.
Even the vowels themselves are not free from such noises, although they are
kept more in the background by the musical character of the tones of the voice.
Donders first drew attention to tiiese noises, which are partly identical with those
which are produced when the corresponding vowels are indicated in low voiceless
* [In the northern parts of Germany and of There are also many other trillg, into which,
France, and in Noitiiumberland, but not other- as into other phonetio details, it is not neces-
wise in England, except as an organic defect, sary to enter.— Translator.]
Digitized by V^jOOQ IC
68 CONCEPTION OP MUSICAL QUALITY AND TONE, pabt i.
speech. They are strongest for ee m see, the French u in vu (which is nearly the
same as the Norfolk and Devon oo in too), and for oo in too. For these vowels they
can be made audible even when speaking aloud.* By simply increasing their force
the vowel ee in. see becomes the consonant y in yon, and the vowel oo in too the
consonant w in wan,\ For a in a/rt, a mat, em met, there, and o in more, the
noises appear to me to be produced in the glottis alone when speaking gently, and
to be absorbed into the voice when speaking aloud.j: It is remarkable that in
speaking, the vowels a in art, a in at, and e in met, there, are produced with less
musical tone than in singing. It seems as if a feeling of greater compression in
the larynx caused the tuneful tone of the voice to give way to one of a more jarring
character which admits of more evident articulation. The greater intensity thus
given to the noises, appears in this case to facilitate the characterisation of the
peculiar vowel quality. In singing, on the contrary, we try to favour the musical
IF part of its quality and hence often render the articulation somewhat obscure.§
Such accompanying noises and little inequalities in the motion of the air,
furnish much that is characteristic in the tones of musical instruments, and in the
vocal tones of speech which correspond to the different positions of the mouth ;
but besides these there are numerous peculiarities of quality belonging to the
musical tone proper, that is, to the perfectly regular portion of the motion of the
air. The importance of these can be better appreciated by listening to musical
instruments or human voices, from such a distance that the comparatively weaker
noises are no longer audible. Notwithstanding the absence of these noises, it is
generally possible to discriminate the different musical instruments, although it
must be acknowledged that under such circumstances the tone of a French horn
may be occasionally mistaken for that of the singing voice, or a violoncello may
be confused with an harmonium. For the human voice, consonants first disappear
at a distance, because they are characterised by noises, but M, N, and the vowels
f can be distinguished at a greater distance. The formation of M and N in so far
resembles that of vowels, that no noise of wind is generated in any part of the
cavity of the mouth, which is perfectly closed, and the sound of the voice escapes
through the nose. The mouth merely forms a resonance chamber which alters the
quality of tone. It is interesting in calm weather to listen to the voices of men
who are descending from high hills to the plain. Words can no longer be recog-
nised, or at most only such as are composed of M, N, and vowels, as Mamma, No,
Noon, But the vowels contained in the spoken words are easily distinguished.
Wanting the thread which connects them into words and sentences, they form a
strange series of alternations of quality and singular inflections of tone.
In the present chapter we shall at first disregard all irregular portions of the
motion of the air, and the mode in which sounds commence or terminate, directing
our attention solely to the musical part of the tone, properly so called, which
corresponds to a uniformly sustained and regularly periodic motion of the air,
f and we shall endeavour to discover the relations between Ijie quality of the sound
* [At the Com6die Francaise I have heard the important phonetio observations In the
M. Gk>t pronounce the word oui and Mme. iezi,— Translator.]
Provost-Ponsin pronounce the last syllable of § [These observations must not be con-
liachis entirely without voice tones, and yet sidered as exhausting the subject of the dif-
make them audible throughout the theatre. — ference between the singing and the speak-
TranslatorJ] ing voice, which requires a peculiar stadj
f [That this is not the whole of the pheno- here merely indicated. See my PronundattKm
menon is shewn by the words ye^ woo. The for Sinaers (Curwen) and Speech in Sena
whole subject is discussed at length in my (Novello). The difference between English aji^
Early English Pronunciation, pp. 1092- 1094, German habits of speaking and singing must
and 1149-11 $1,— Translator.] also be borne in mind, and allowed for by
X [By * speaking gently * (leise) seems to the reader. The English vowels given in the
be meant either speaking absolutely without text are not the perfect equivalents of ProL
voice, that is with an open glottis, or in a Helmholtz's German sounds. The noises
whisper, with the glottis nearly closed. For which accompany the vowels are not nearly
voice the glottis is quite closed, and this is so marked in English as in German, but they
indicated by * speaking aloud * {b&itn la/ulen differ very mudi locally, even in England.—
Spre^ien). It would lead too far to discuss Translator.]
Digitized by V^jOOQlC
CHAP. V. I. TONES WITH NO UPPEE PARTIALS. 69
and lis composition out of individual simple tones. The peculiarities of quality
of sound belonging to this division, we shall briefly call its musical quality.
The object of the present chapter is, therefore, to describe the different com-
position of musical tones as produced by different instruments, for the purpose of
shewing how different modes of combining the upper partial tones correspond to
characteristic varieties of musical quality. Certain general rules will result for
the arrangement of the upper partials which answer to such species of musical
quality as are called, soft^ pisrcingt braying^ hollow or poor^ full or rich, dull,
bright, crisp, pungent, and so on. Independently of our inmiediate object (the
determination of the physiological action of the ear in the discrimination of
musical quality, which is reserved for the following chapter), the results of this
investigation are important for the resolution of purely musical questions in later
chapters, because they shew us how rich in upper partials, good musical qualities
of tone are found to be, and also pomt out the peculiarities of musical quality 5
fftvoured on those musical instruments, for which the quality of tone has been to
some extent abandoned to the caprice of the maker.
Since physicists have worked comparatively little at this subject I shall be
forced to enter somewhat more minutely into the mechanism by which the tones
of several instruments are produced, than will be, perhaps, agreeable to many of
my readers. For such the principal results collected at the end of this chapter will
suffice. On the other hand, I must ask indulgence for leaving many large gaps
in this almost unexplored region, and for confining myself principally to instru-
ments sufficiently well known for us to obtain a tolerably satisfactory view of the
source of their tones. In this inquiry lie rich materials for interesting acoustical
work. Bat I have felt bound to confine myself to what was necessary for the
continuation of the present investigation.
I . Musical Tones without Upper Partials. ^
We begin with such musical tones as are not decomposable, but consist of a
single simple tone. These are most readily and purely produced by holding a
struck tuning-fork over the mouth of a resonance tube, as has been described in
the last chapter (p. 54^).* These tones are uncommonly soft and free from all
shrillness and roughness. As already remarked, they appear to he comparatively
deep, so that such as correspond to the deep tones of a bass voice produce the
impression of a most remarkable and unusual depth. The musical quahty of such
deep simple tones is also rather dull. The simple tones of the soprano pitch
sound bright, but even those corresponding to the highest tones of a soprano voice
are very soft, without a trace of that cutting, rasping shrillness which is displayed
by most instruments at such pitches, with ihe exception, perhaps, of the flute, for
which the tones are very nearly simple, being accompanied with very few and
faint upper partials. Among vowels, the 00 in too comes nearest to a simple tone,
but even this vowel is not entirely free from upper partials. On comparing the il
musical quahty of a simple tone thus produced with that of a compound tone in
which the first harmonic upper partial tones are developed, the latter will be found
to be more tuneful, metallic, and brilliant. Even the vowel 00 in too, altliough
the dullest and least tuneful of all vowels, is sensibly more brilHant and less dull
than a simple tone of the same pitch. The series of the first six partials of a
compound tone may be regarded musically as a major chord with a very predominant
fundamental tone, and in fact the musical quality of a compound tone possessing
these partials, as, for example, a fine singing voice, when heard beside a simple tone,
very distinctly produces the agreeable effect of a consonant chord.
Since the form of simple waves of known periodic time is completely given
when their amphtude is given, simple tones of the same pitch can only differ
in force and not in musical quality. In fact, the difference of quahty remains
* On possible sources of disturbance, see Appendix IV.
Digitized by VjOOQIC
70 TONES WITH INHARMONIC UPPER PARTIALS. pabt i.
perfectly indistinguishable, whether the simple tone is conducted to the external
air in the preceding methods by a tmiing-fork and a resonance tube of any given
material, glass, metal, or pasteboard, or by a string, provided only that we guard
against any chattering in the apparatus.
Simple tones accompanied only by the noise of rushing wind can also be pro-
duced, as already mentioned, by blowing over the mouth of bottles with necks
(p. 6oc). If we disregard the friction of the air, the proper musical quality of such
tones is really the same as that produced by tuning-forks.
2. Musical Tones with Inharmonic Upper Partials.
Nearest to musical tones without any upper partials are those with secondary
tones which are inharmonic to the prime, and such tones, therefore, in strictness,
^ should not be reckoned as musical tones at all. They are exceptionally used in
artistic music, but only when it is contrived that the prime tone should be so much
more powerful than the secondary tones, that the existence of the latter may be
ignored. Hence they are placed here next to the simple tones, because musically
they are available only for the more or less good simple tones which they represent.
The first of these are tuning-forks themselves, when they are struck and applied
to a sounding board, or brought very near the ear. The [inharmonic] upper partials
of tuning-forks lie very high. In those which I have examined, the first made
from 5*8 to 6*6 as many vibrations in the same time as the prime tone, and hence
lay between its third diminished Fifth and major Sixth. The pitch numbers of
these high upper partial tones were to one another as the squares of the odd
numbers. In the time that the first upper partial would execute 3 x 3=9 vibra-
tions, the next would execute 5 x 5=25* and the next 7 x 7=^49, and so on. Their
pitch, therefore, increases with extraordinary rapidity, and tiiey are usually all
^ inharmonic with the prime, though some of them may exceptionally become
harmonic. If we call the prime tone of the fork c, the next succeeding tones are
nearly a"[>, dT^ &'^.* These high secondary tones produce a bright inharmonic
clink, which is easily heard at a considerable distance when the fork is first struck,
whereas when it is brought close to the ear, the prime tone alone is heard. The
ear readily separates the prime from the upper tones and has no inclination to fuse
them. The high simple tones usually die off rapidly, while the prime tone remains
audible for a long time. It should be remarked, however, that the mutual relations
of the proper tones of tuning-forks differ somewhat according to the form of the
fork, and hence the above indications must be looked upon as merely approximate.
In theoretical determinations of the upper partial tones, each prong of the fork
may be regarded as a rod fixed at one end.
The same relations hold for straight elastic rods, which, as already mentioned,
when struck, give rather high inharmonic upper partial tones. When such a rod
f is firmly supported at the two nodal lines of its prime tone, the continuance of
that tone is favoured in preference to the other higher tones, and hence the latter
disturb the effect very slightly, more especially as they rapidly die away after the
rod has been struck. Such rods, however, are not suitable for real artistic music,
* [On oaloolating the number of cents (as henoe it is called dT* in the text. The interval
in App. XX. sect. C.), we find that the first to the next tone is 25 : 49 or 1165 cents,
tone mentioned, which vibrates from 5*8 to Adding this to the former numbers the interval
6*6 as fait as the prime, makes an interval with the prime must be between 5977 and
with it of from 3043 to 3267 ot., so that if 6201 cents, or between b^ + 77 and d"-^, for
the prime is called c, the note lies between which in the text c^U ia selected. The inde-
^'bi-43, and af'—^Sf where g"b and a" are terminacy arises from the difficulty of finding
the third diminished Fifth and major Sixth of the pitch of the first inharmonic upper partiaL
the prime c mentioned in the text. This Prof. The intervals between that and the next upper
Helmholtz calls a'"bt or 3200 cents. Then the partials are 9 : 25 or 1769 ct., 9 : 49 or 2934
interval between this partial and the next is ct., 9 : 81 or 3699 ct., and so on. The word
9 : 25 or 1769 ct., and hence the interval 'inhannonic* has been inserted- in the text,
with the prime is between 4812 and 5036 as tuning-forks have also generally harmonic
cents, or lies between 0^^+12 and d'^ + 36, and upper purtials. See p. 54^^, note.— Translator.]
Digitized by V^jOOQlC
CHAP. V. 2. TONES WITH INHARMONIC UPPER PARTIALS. 71
although they have lately been introduced for military and dance music on account
of their penetrating qualities of tone. Glass rods or plates, and wooden rods, were
formerly used in this way for the glass harmonicon and the straw-fiddle or wood
harmonicon. The rods were inserted between two pairs of intertwisted strings,
which grasped them at their two nodal lines. The wooden rods in the German
straw-fiddle were simply laid on straw cylinders. They were struck with hammers
of wood or cork.
The only effect of the material of the rods on the quality of tone in these
cases, consists in the greater or less length of time that it allows the proper tones
at different pitches to continue. These secondary tones, including the higher ones,
usually continue to sound longest in elastic metal of fine uniform consistency,
because its greater mass giyes it a greater tendency to continue in any state of
motion which it has once assumed, and among metals the most perfect elasticity
is found in steel, and the better alloys of copper and zinc, or copper and tin. In ^
slightly alloyed precious metals, their greater specific gravity lengthens the dura-
tion ,of the tone, notwithstanding their inferior elasticity. Superior elasticity
appears to favour the continuance of the higher proper tones, because imperfect
elasticity and friction generally seem to damp rapid more quickly than slow vibra- ^
tions. Hence I think that I may describe the general characteristic of what is
usually called a metallic quality of tone, as the comparatively continuous and
uniform maintenance of higher upper partial tones. The quality of tone for glass
is similar ; but as it breaks when violently agitated, the tone is always weak and
FlO. 31.
soft, and it is also comparatively high, and dies rapidly away, on account of the
smaller mass of the vibrating body. In wood the mass is small, the internal
structure comparatively rough, being full of countless interstices, and the elasticity
also comparatively imperfect, so that the proper tones, especially the higher ones,
rapidly die away. And for this reason the straw-fiddle or wood harmonicon is per-
haps more satisfactory to a musical ear, than harmonicons formed of steel or glass
rods or plates, with their piercing inharmonic upper partial tones, — at least so &r
as simple tones are suitable for music at all, of which I shall have to speak later on.*
For all of these instruments which have to be struck, the hammers are made
of wood or cork, and covered with leather. This renders the highest upper
partials much weaker than if only hard metal hammers were employed. Greater ^
hardness of the striking mass produces greater discontinuities in the original
motion of the plate. The influence exerted by the manner of striking will be
considered more in detail, in reference to strings, where it is also of much impor-
tance.
According to Ghladni's discoveries, elastic plates , cut in circular, oval, square,
oblong, triangular, or hexagonal forms, will sound in a great number of different
vibrational forms, usually producing simple tones which are mutually inharmonic.
Fig. 21 gives the more simple vibrational forms of a circular plate. Much more
complicated forms occur when several circles or additional diameters appear as
nodal lines, or where both circles and diameters occur. Supposing the vibrational
form A to give the tone c, the others give the following proper tones : —
* [In Java the principal music is produced the rods are laid on the edges of boat-shaped
by harmonicons of metal or wooden rods and vessels, like old fashion cheese-trays, and kept
kettle-shaped gongs. The wooden harmonicons in position by nails passing loosely through
are frequent 8d(so in Asia and Africa. In Java holes. See App. XX. sect. K,— Translator,] j
Digitized by V^jOOQ IC
72
TONES WITH INHARMONIC UPPER PARTIALS.
PAKF I.
Number
of Nodal
Circles
Namber of Diameters
o
I
2 1 3
4
5
O
I
2
9"» +
b'b
C
d*
c"
i^'-y"«
This shews that many proper tones of nearly the same pitch are produced by a
plate of this kind. When a plate is struck, those proper tones which have no
node at the point struck, will all sound together. To obtain a particular deter-
minate tone it is of advantage to support the plate in points which lie in the nodal
lines of that tone ; because those proper tones which have no node in those poinds
will then die off more rapidly. For example, if a circular plate is supported at
^ 3 points in the nodal circle of fig. 2 1 , G (p. 71c), and is struck exactly in its middle,
the simple tone called ^ in the table, which belongs to that form, will be heard,
and all those other proper tones which have diameters as some of their nodal
lines * will be very weak, for example c, d\ c"^ g'\ h^ in the table. In the same
way the tone g'% with two nodal circles, dies off inmiediately, because the points
of support fall on one of its ventral segments, and the first proper tone which can
sound loudly at the same time is that corresponding to three nodal circles, one of
its nodal lines being near to that of No. 2. But this is 3 Octaves and more than
a whole Tone higher than the proper tone of No. 2, and on account of this great
interval does not disturb ilie latter. Hence a disc thus struck gives a tolerably
good musical tone, whereas plates in general produce sounds composed of many in-
harmonic proper tones of nearly the same pitch, giving an einpty tin-kettle sort of
qualityi which cannot be used in music. But even when the disc is properly sup-
ported the tone dies away rapidly, at least in the case of glass plates, because
\ contact at many points, even when nodal, sensibly impedes the freedom of vibra-
tion.
The sound of helli is also accompanied by inharmonic secondary tones, which,
however, do not lie so close to one another as those of flat plates. The vibrations
which usually arise have 4, 6, 8, 10, &c., nodal lines extending from the vertex of
the bell to its margin, at equal intervals from each other. The corresponding
proper tones for glass bells which have approximatively the same thickness
throughout, are nearly as the squares of the numbers 2, 3, 4, 5, so that if we call
the lowest tone c, we have for the
Nnmber of nodal lines .
4
6
8
10
12
Tones ../...
Cento . . ...
c
0
1404
2400
ft-
3173
3804
The tones, hoT/aver, vary with the greater or less thickness of the wall of the
H bell towards the margin, and it appears to be an essential point in the art of
casting bells, tr make the deeper proper tones mutually harmonic by giving the
bell a certaiTi empirical form. According to the observations of the organist
01eitz,t the bell cast for the cathedral at Erfurt in 1477 has the following proper
tones: E, €, ^ h^ e\ g%, h', c"#. The [former] bell of St. Paul's,. London, gave
a and c%. Hemony of Ziitphen, a master in the seventeenth century, required a
good bell to hava three Octaves, two Fifths, one major and one minor Third. The
deepest tone is not the strongest. The body of the bell when struck gives a
deeper tone than the ' sound bow,' but the latter gives the loudest tone. Probably
other vibrational forms of bells are also possible in which nodal circles are formed
* Provided that the supported points do
not happen to belong to a system of diameters
juaking equal angles with eaoh other.
t 'Historical Notes on the Great Bell
And the other Bells in Erfurt Cathedral'
{Qeschichlliches fiber die grosse Glocke und,
die ilbrigen Oloc/cen des Domes eu Erfurt),
Erfurt, 1 867. -See also Sohafhautl in the
Kunst' und GewerbeblaU fUr das KOnigreich
Bayem, 1868, liv. 325 to 350; 385 to 427.
Digitized by V^jOOQlC
CHAP. V. 2. TONES WITH INHARMONIC UPPER PARTIALS.
73;
parallel to the margin. But these seem to be produced with difficulty and have
not yet been examined.
If a bell is not perfectly symmetrical in respect to its axis, if, for example, the
wall is a little thicker at one point of its circumference than at another, it will
give, on being struck, two different tones of very nearly the same pitch, which will
' beat * together. Four points on the margin will be found, separated from each
other by quarter-circles, in which only one of these tones can be heard without
accompanying beats, and four others, half-way between the pairs of the others,
where the second tone only sounds. If the bell is struck elsewhere both tones are
heard, producing beats, and such beats may be perceived in most bells as their
tone dies gradually away.
Stretched membranes have also inharmonic proper tones of nearly the same
pitch. For a circular membrane, of which the deepest tone is c, these are, in a
vacuum and arranged in order of pitch, as follows : — f
Nmnber of Nodal Lines
Tone
Diameters
Circles
0
I
2
o
I
o
0
O
0
I
I
2
C
ab
d% +0-1 •
d' +0-2
^ — 0'2
6'b + o-i
These tones rapidly die out. If the membranes sound in air,t or are associated
with an air chamber, as in the kettledrum, the relation of the proper tones may
be altered. No detailed investigations have yet been made on the secondary tones
of the kettledrum. The kettledrum is used in artistic music, but only to mark «-
certain accents. It is tuned, indeed, but only to prevent ii^jury to the harmony,
not for the purpose of filling up chords.
The common character of the instruments hitherto described is, that, when
struck they produce inharmonic upper partial tones. If these are of nearly the
same pitch as the prime tone, their quaUty of sound is in tbe highest degree un-
musical, bad, and tinkettly. If the secondary tones are of very different pitch
from the prime, and weak in force, the quality of sound is more musical, as for
example in tuning-forks, harmonicons of rods, and bells ; and such tones are applic-
able for marches and other boisterous music, principally intended to mark time.
But for really artistic music, such instruments as these have always been rejected,
as they ought to be, for the inharmonic secondary tones, although they rapidly die
away, always disturb the harmony most unpleasantly, renewed as they are at every
fresh blow. A very striking example of this was furnished by a company of bell-
rmgers, said to be Scotch, that lately travelled about Germany, and performed all «-
kinds of musical pieces, some of which had an artistic character. The accuracy
and skill of the performance was undeniable, but the musical effect was detestable,
on account of the heap of false secondary tones which accompanied the music,
although care was taken to damp each bell as soon as the proper duration of its
note had expired, by placing it on a table covered with cloth.
Sonorous bodies with inharmonic partials, may be also set in action by violin
bows, and then by properly damping them in a nodal line of the desired tone, the
secondary tones which lie near it can be prevented from interfering. One simple
tone then predominates distinctly, and it might consequently be used for musical
purposes. But when the violin bow is applied to any bodies with inharmonic
upper partial tones, as tuning-forks, plates, bells, we hear a strong scratching
* [These decimals represent tenths of a
tone, or 20 cents for the first place. As there
can be no sounds in a vacuum, these notes
are merely used to conveniently symbolise
numbers of vibrations in a second.— Trar^s-
lator.]
t See /. Bourgelt L'Institut, xxzviii., 1870,
pp. 189, 19a
Digitized by VjOOQlC
74 MUSICAL TONES OF STRINGS. paet i.
sound, which on investigation with resonators, is found to consist mainly of these
same inharmonic secondary tones of such bodies, not sounding continuously but
only in short irregular fits and starts. Intermittent tones, as I have already noted,
produce the effect of grating or scratching. It is only when the body excited by
the violin bow has harmonic upper partials, that it can perfectly accommodate itse^
to every impulse of the bow, and give a really musical quality of tone. The
reason of this is that any required periodic motion such as the bow aims at pro-
ducing., can be compounded of motions corresponding to harmonic upper partial
tones, but not of other, inharmonic vibrations.
3. M^8^cal Tones of Strings.
We now proceed to the analysis of musical tones proper, which are characterised
H by harmonic upper partials. These may be best classified according to their mode
of excitement : i. By striking. 2. By bowing. 3. By blowing against a sharp
edge. 4. By blowing against elastic tongues or vibrators. The two first classes
comprehend stringed instruments alone, as longitudinally vibrating rods, the only
other instruments producing harmonic upper partial tones, are not used for musical
purposes. The third class embraces flutes and the flute or flue pipes of organs ;
the fourth all other wind instruments, including the human voice.
Strings excited by Striking. — Among musical instruments at present in use,
this section embraces the pianoforte, harp, guitar, and zither ; among physical,
the monochord, arranged for an accurate examination of the laws controlling the
vibrations of strings ; the pizzicato of bowed instruments must also be placed in
this category. We have idready mentioned that the musical tones produced by
strings whidi are struck or plucked, contain numerous upper partial tones. We
have the advantage of possessing a complete theory for the motion of plucked
% strings, by which the force of their upper partial tones may be determined. In
the last chapter we compared some of the conclusions of this theory with the
results of experiment, and found them agree. A similarly complete theory maybe
formed for the case of a string which has been struck in one of its points by a
hard sharp edge. The problem is not so simple when soft elastic hammers are
used, such as those of the pianoforte, but even in this case it is possible to assign
a theory for the motion of the string which embraces at least the most essential
features of the process, and indicates the force of the upper partial tones.*
The force of the upper partial tones in a struck string, depends in general
on: —
1. The nature of the stroke.
2. The place struck.
3. The density, rigidity, and elasticity of the string.
First, as to the nature of the stroke. The string may be plucked, by drawing
% it on one side with the finger or a point (the plectrum, or the ring of the zither-
player), and then letting it go. This is a usual mode of exciting a string in a great
number of ancient and modem stringed instruments. Among the modem, I need
only mention the harp, guitar, and zither. Or else the string may be struck with
a hammer-shaped body, as in the pianoforte.f I have already remarked that the
strength and number of the upper partial tones increases with the number and
abruptness of the discontinuities in the motion excited. This fact determines the
various modes of exciting a string. When a string is plucked, the finger, before
quitting it, removes it from its position of rest throughout its whole length. A
discontinuity in the string arises only by its forming a more or less acute angle at
the place where it wraps itself about the finger or point. The angle is more acute
for a sharp point than for the finger. Hence the sharp point produces a shriUer
tone with a greater number of high tinkling upper partials, than the finger. But
* See Appendix V. be struck by a hammer-shaped body. See
t [I have here omitted a few words in pp. 77c and jSd'.^Trafislator.]
which, by an oversight, the spinet was said to ,
Digitized by VjOOQ IC
CHAP. V. 3. MUSICAL TONES OF STBINGS. 75
in each ease the intensity of the prime tone exceeds that of any upper partial. If
the string is struck with a sharp-edged metallic hammer which rehounds instantly,
only the one single point struck is directly set in motion. Immediately after the
blow the remainder of the string is at rest. It does not move until a wave of de-
flection rises, and runs backwards and forwards over the string. This limitation
of the original motion to a single point produces the most abrupt discontinuities,
and a corresponding long series of upper partial tones, having intensities,* in most
oases equalling or even surpassing that of the prime. When the hammer is soft
and elastic, the motion has time to spread before the hammer rebounds. When
thus struck the point of the string in contact with such a hammer is not set in
motion with a jerk, but increases gradually and continuously in velocity during the
contact. The discontinuity of the motion is consequently much less, diminishing
as the softness of the hammer increases, and the force of the higher upper partial
tones is correspondingly decreased. y^
We can easily convince ourselves of the correctness of these statements by
opening the lid of any pianoforte, and, keeping one of the digitals down with a
weight, so as to free the string from the damper, plucking the string at pleasure
with a finger or a point, and striking it with a metaJhc edge or the pianoforte ham-
mer itself. The qualities of tone thus obtained will be entirely different. When
the string is struck or plucked with hard metal, the tone is piercing and tinkling,
and a little attention enables us to hear a multitude of very high partial tones.
These disappear, and the tone of the string becomes less bright, but softer, and
more harmonious, when we pluck the string with the soft finger or strike it with
the soft hammer of the instrument. We also readily recognise the different loud-
ness of the prime tone. When we strike with metal, the prime tone is scarcely
heard and the quality of tone is correspondingly iJpoT, The peculiar quality of
tone conmionly termed poverty, as opposed to richness, arises firom the upper
partials being comparatively too strong for the prime tone. The prime tone is f
heard best when the string is plucked with a soft finger, which produces a rich and
yet harmonious quality of tone. The prime tone is not so strong, at least in the
middle and deeper octaves of the instrument, when the strings are struck with the
pianoforte hammer, as when they are plucked with the finger.
This is the reason why it has been found advantageous to cover pianoforte ham-
mers with thick layers of felt, rendered elastic by much compression. The outer
layers are the softest and most yielding, the lower are firmer. The sur&ce of the
hanuner comes in contact with the string without any audible impact ; the lower
layers give the elasticity which throws the hammer back from the string. If you
remove a pianoforte hammer and strike it strongly on a wooden table or against a
wall, it rebounds from them like an india-rubber ball. The heavier the hammer
and the thicker the layers of felt— as in the hammers for the lower octaves — the
longer must it be before it rebounds from the string. The hammers for the upper
octaves are lighter and have thinner layers of felt. Clearly the makers of these ^
instruments have here been led by practice to discover certain relations of the
elasticity of the hanuner to the best tones of the string. The make of the hammer
has an immense influence on the quality of tone. Theory shews that those upper
partial tones are especially favoured whose periodic time is nearly equal to twice
* When intensity is here mentioned, it is .as the pitoh number. Messrs. Preeoe and
always measured objeetiyely, by the via viva, Stroh, Proc. R, S,, vol. xxviii. p. 366, think
or mechanical equivalent of work of the eorre- that * loudness does not depend upon amplitude
sponding motion. [Mr. Bosanquet {Academy, of vibration only, but upon the quantity of air
Dee. 4, 1875, p. 580, col. i) points out that put in vibration; and, therefore, there exists
p. lod, note, and Chap. IX., paragraph 3, shew an absolute physical magnitude in acoustics
this measure to be inadmissible, and adds : analogous to that of quantity of electricity or
'if we admit that in similar organ pipes quantity of heat, and which may be called
similar proportions of the wind supplied are quantity of sound,* and they illustrate this by
employed in the production of tone, the me- the effect of differently sized discs in their
chanical energy of notes of given intensity automatic phonograph there described. See
varies inversely as the vibration number,* i.e. also App. XX. sect. M. No. 2,— Translator.'}
Digitized by V^jOOQ IC
76 MUSICAL TONES OP STRINGS. pabt i.
the time during which the hammer lies on the string, and that, on the other hand,
those disappear whose periodic time is 6, lo, 14, &c., times as great.*
It will generally be advantageous, especially for the deeper tones, to eliminate
from the series of upper partials, those which lie too close to each other to give a
good compound tone, that is, from about the seventh or eighth onwards. Those
with higher ordinal numbers are generally relatively weak of themselves. On ex-
amining a new grand pianoforte by Messrs. Steinway of New York, which was
remarkable for the evenness of its quality of tone, I find that the damping result-
ing from the duration of the stroke falls, in the deeper notes, on the ninth or tenth
partials, whereas in the higher notes, the fourth and fifth partials were scarcely to
be got out with the hammer, although they were distinctly audible when the string
was plucked by the nail.f On the other hand upon an older and much used grand
piano, which originally sliewed the principal damping in the neighbourhood of the
f seventh to the fifth partial for middle and low notes, the ninth to the thirteenth
partials are now strongly developed. This is probably due to a hardening of the
hammers, and certainly can only be prejudicial to the quality of tone. Observa-
tions on these relations can be easily made in the method recommended on p. 526, c.
Put the point of the finger gently on one of the nodes of the tone of which you
wish to discover the strength, and then strike the string by means of the digital.
By moving the finger till the required tone comes out most purely and sounds the
longest, the exact position of the node can be easily found. The nodes which lie
near the striking point of the hammer, are of course chiefly covered by the damper,
but the corresponding partials are, for a reason to be given presently, relatively
weak. Moreover the fifth partial speaks well when the string is touched at two-
fifths of its length from the end, and the seventh at two-sevenths of that length.
These positions are of course quite free of the damper. Generally we find all the
partials which arise from the method of striking used, when we keep on striking
f while the finger is gradually moved over the length of the string. Touching the
shorter end of the string between the striking point and the further bridge will thus
bring out the higher partials from the ninth to the sixteenth, which are musically
undesirable.
The method of calculating the strength of the individual upper partials, when
the duration of the stroke of the hammer is given, will be found further on.
Secondly as to the pUice struck. In the last chapter, when verifying Ohm's
law for the analysis of musical tones by the ear, we remarked that whether strings
are plucked or struck, those upper partials disappear which have a node at the
point excited. Conversely ; those partials are comparatively strongest which have
a maximum displacement at that point. Generally, when the same method of
striking is successively applied to different points of a string, the individual upper
partials increase or decrease with the intensity of motion, at the point of excite-
ment, for the corresponding simple vibrations of the string. The composition of
^ the musical tone of a string can be consequently greatly varied by merely changing
the point of excitement.
Thus if a string be struck in its middle, the second partial tone disappears,
* [The following paragraph on p. 123 of several times. I got out the 7th and 9th
the iBt English edition has been omitted, harmonic of c, but on aocoant of difficul-
and the passage from * It will generally be ties due to the over-stringing and over-barring
advantageous,' p. 76a, to * found farther on/ of the instrument and other circumstanceti
p. 76c, has been inserted, both in accordance I did not pursue the investigation. Mr. A. J.
with the 4th German edition. — Translator,] Hipkins informs me that on another occasion
f [As Prof. Helmholtz does not mention he got out of the & string, struck at ^ the
the striking distance of the hammer, I obtained length, the 6th, 7th, 8th, and 9th har>
permission from Messrs. Steinway A Sons, at monies, as in the experiments mentioned in
their London house, to examine the c, & and the next footnote, ' the 6th and 7th beautifully
c" strings of one of their grand pianos, and strong, the 8th and 9th weaker but clear and
found the striking distance to be ^, ^, and unmistakable.* He struck with the hammer
^ of the length of the string respectively, always. Observe the 9th harmonic of a string
I did not measure the other strings, but I struck with a pianoforte hammer at its node,
observed that the striking distances varied or J its length.— jfranskitor.]
Digitized by V^jOOQlC
CHAP. V. 3.
MUSICAL TONES OF STEINGS.
77
because it has a node at that point. But the third partial tone comes out forcibly,
because as its nodes lie at ^ and f the length of the string from its extremities,
the string is struck half-way between these two nodes. The fourth partial has its
nodes at i, } (=^), and ^ the length of the string from its extremity. It is not
heard, because the point of excitement corresponds to its second node. The sixth,
eighth, and generally the partials with even numbers disappear in the same way, but
the fifth, seventh, ninth, and the other partials with odd numbers are heard. By
this disappearance of tlie evenly numbered partial tones when a string is struck at its
middle, the quality of its tone becomes peculiar, and essentially different from that
usually heard from strings. It sounds somewhat hollow or nasal. The experi-
ment is easily made on any piano when it is opened and the dampers are raised.
The middle of the string is easily found by trying where the finger must be laid
to bring out the first upper partial clearly and purely on striking the digital.
If the string is struck at ^ its length, the third, sixth, ninth, &c., partials f
vanish. This also gives a certain amount of hollowness, but less than when the
string is struck in its middle. When the point of excitement approaches the end
of the string, the prominence of the higher upper partials is favoured at the
expense of the prime and lower upper partial tones, and the sound of the string
becomes poor and tinkling.
In pianofortes, the point struck is about | to | the length of the string from
its extremity, for the middle part of the instrument. We must therefore assume
that this place has been chosen because experience has shewn it to give the finest
musical tone, which is most suitable for harmonies. The selection is not due to
theory. It results from attempts to meet the requirements, of artistically trained
eai's, and from the technical experience of two centuries.* This gives particular
* [As my friend Mr. A. J. Hipkins, of
Broadwoods', author of a paper on the ' History
of the Pianoforte,* in the Journal of the Society
of Arts (for March 9, 18H3, with additions on
Sept. 21, 1883), has paid great attention to the
archieology of the pianoforte, and from his
position at Messrs. Broadwoods' has the best
means at his disposal for making experiments,
I requested him to favour me with his views
upon the subject of the striking place and
harmonics of pianoforte strings, and he has
obliged me with the following observations : —
'Harpsichords and spinets, which were set
in vibration by quill or leather plectra, had
no fixed point for plucking the strings. It
was generally from about ^ to | of the vibra-
ting length, and although it had been observed
by Huyghens and the Antwerp harpsichord-
maker Jan Couchet, that a difference of quality
of tone could be obtained by varying the
plucking place on the same string, which led
to the so-called lute stop of the i8th century,
no attempt appears to have been made to gain
a uniform striking place throughout the scale.
Thus in the latest improved spinet, a Hitch-
cock, of early i8th century, in my possession,
the striking place of the c*8 varies from ^ to
l« and in the latest improved harpsichord, a
Rirkman of 1773, also in my possession, the
striking distances vary from i to y^ and for
the lute stop from jt to ^ of the string, the
longest distances in the bass of course, but
all without apparent rule or proportion. Nor
was any attempt to gain a uniform striking
place made in tiie first pianofortes. Stein of
Augsburg (the favourite pianoforte-maker of
Mozart, and of Beethoven in his virtuoso
time) knew nothing of it, at least in his early
instruments. The great length of the bass
strings as carried out on the single belly-
bridge copied from the harpsichord, made a
reasonable striking place for that part of the
scale impossible.
* John Broadwood, about the year 1788, wan ^
the first to try to equalise the scale in tension
and striking place. He called in scientific
aid, and assisted by Signor Gavallo and the
then Dr. Gray of the British Museum, he
produced a divided belly -bridge, which shorten-
ing the too great length of the bass strings,
permitted the establishment of a striking
place, which, in intention, should be propor-
tionate to the length of the string throughout.
He practically adopted a ninth of the vibrating
length of the string for his striking place,
allowing some latitudie in the treble. This
division of the belly-bridge becEme universally
adopted, and with it an approximately rational
striking place.
• Carl Kiitzing {Das Wissenschaftliche der
Fortepiano-Baukunst, 1844, p. 41) was enabled
to propound from experience, that i of the
length of the string was the most suitable m
distance in a pianoforte for obtaining the best
quality of tone from the strings. The love of
noise or effect has, however, inclined makers to
shorten distances, particularly in the trebles.
Kiitzing appears to have met with ^th in some
instances, and Helmholtz has adopted that
very exceptional measure for his table on
p. 79c. I cannot say I have ever met with a
striking place of this long distance from the
wrestplank-bridge. The present head of the
firm of Broadwood (Mr. Henry Fowler Broad -
wood) has arrived at the same conclusions as
Eutzing with respect to the superiority of the
^th distance, and has introduced it in his
pianofortes. At Ath the hammers have to be
softer to get a like quality of tone ; an equal
system of tension being presupposed.
'According to Young's law, which Helm-
holtz by experiment confirms, the impact of
Digitized by V^jOOQlC
78 MUSICAL TONES OF STRINGS. pabt i.
interest to the mvestigation of the composition of musical tones for this point of
excitement. An essential advantage in the choice of this position seems to be
that the seventh and ninth partial tones disappear or at least become very weak.
These are the first in the series of partial tones which do not belong to the major
chord of the prime tone. Up to the sixth partial we have only Octaves, Fifths,
and major Thirds of the prime tone ; the seventh is nearly a minor Seventh, the
ninth a major Second of the prime. Hence these will not fit into the major
chord. Experiments on pianofortes shew that when the string is struck by the
hammer and touched at its nodes, it is easy to bring out the six first partial tones
(at least on the strings of the middle and lower octaves), but that it is either not
possible to bring out the seventh, eighth, and ninth at all, or that we obtain at
best very weak and imperfect results. The difficulty here is not occasioned by the
incapacity of the string to form such short vibrating sections, for if instead of striking
% the digital we pluck the string nearer to its end, and damp the corresponding
nodes, the seventh, eighth, ninth, nay even the tenth and eleventh partial may be
clearly and brightly produced. It is only in the upper octaves that the strings are
too short and stiff to form the high upper partial tones. For these, several instru-
ment-makers place the striking point near^ to the extremity, and thus obtain a
brighter and more penetrating tone. The upper partii^ of these strings, which
their stiffness renders it difficult to bring out, are thus &voured as against the
prime tone. A similarly brighter tone, but at the same time a thinner and poorer
one, can be obtained from the lower strings by placing a bridge nearer the striking
point, so that the hammer falls at a point less than 4 of the effective length of the
string from its extremity.
While on the one hand the tone can be rendered more tinkling, shrill, and
acute, by striking the string with hard bodies, on the other hand it can be rendered
duller, Uiat is, the prime tone may be made to outweigh the upper partials, by
f striking it with a soft and heavy hammer, as, for example, a little iron hammer
covered with a thick sheet of india-rubber. The strings of the lower octaves then
produce a much fuller but duller tone. To compare the different qualities of tone
thus produced by using hammers of different constructions, care must be taken
always to strike the string at the same distance from the end as it is struck by the
proper hammer of the instrument, as otherwise the results would be mixed up with
the changes of quality depending on altering the striking point. These circum-
stances are of course well known to the instrument-makers, because they have
the hammer abolishes the node of the striking diately after production, they last mnoh longer
place, and with it the partial belonging to it and are much brighter.
throughoat the string. I do not find, however, * I do not think the treble strings are from
that the hammer striking at the ^th elimi- shortness and stiiffness incapable of forming
nates the 8th partial. It is as audible, when high proper tones. If it were so the notes
touched as an harmonic, as the 9th and higher would be of a very different quality of tone to
parti^. It is easy, on a long string of say that which they are found to have. Owing to
m from 25 to 45 inches, to obtain the series of the very acute pitch of these tones our ears
upper partials up to the fifteenth. On a cannot follow Uiem, but their existence is
string of 45 inches I have obtained as far as proved by the fact that instrument-makers
the 23rd harmonic, the diameter of the wire often bring their treble striking place very
being 1-17 mm. or 'o; inches, and the tension near the wrestplank-bridge in order to secure
being 71 kilogrammes or 156*6 lbs. The a brilliant tone effect, or ring, by tiie pre-
partials diminish in intensity with the re- ponderance of these harmonics,
duction of the vibrating length; the 2nd is *The clavichord differs entirely from
stronger than the 3rd, and the 3rd than the hammer and plectrum keyboard instruments
4th, &c. Up to the 7th a good harmonic note in the note being started from the end, the
can always be brought out. After the 8th, as tangent (brass pin) which stops the string
Helmholtz says, the higher partials are all being also the means of exciting the sound,
comparatively weak and become gradually But the thin brass wires readily break up into
fainter. To strengthen them we may use a segments of short recurrence, the bass wires,
narrower harder hammer. To hear them which are most indistinct, being helped in the
with an ordinary hammer it is necessary to latest instruments by lighter octave strings^
excite them by a firm blow of the hand upon which serve to make the fundamental tones
the finger-key and to continue to hold it down, apparent.' See also the last note, p. 76<f , and
They sing out quite clearly and last a very App. XX. sect. '^.—Translators]
sensible time. On removing the stop imme-
Digitized by
Google
CHAP. V. 3.
MUSICAL TONES OP STBINGS.
79
themselves selected heavier and softer hammers for the lower, and lighter and
harder for the upper octaves. But when we see that they have not given more
than a certain weight to the hammers and have not increased it sufficiently to
reduce the intensity of the upper partial tones still furtlier, we feel convinced that
a musically trained ear prefers that an instrument to be used for rich combinations
of harmony should possess a quality of tone which contains upper partials with a
certain amount of strength. In this respect the composition of the tones of
pianoforte strings is of great interest for the whole theory of music. In no other
instrument is there so wide a field for alteration of quality of tone; in no other,
then, was a musical ear so unfettered in the choice of a tone that would meet its
wishes.
As I have already observed, the middle and lower octaves of pianoforte strings
generally allow the six first partial tones to be clearly produced by striking the
digital, and the three first of them are strong, the fifth and sixth distinct, but much f
weaker. The seventh, eighth, and ninth are eliminated by the position of the
striking point. Those higher than the ninth are always very weak. For closer
comparison I subjoin a table in which the intensities of the partial tones of a string
for di£ferent methods of striking have been theoretically calculated firom the
fonnulae developed in the Appendix V. The effect of the stroke of a hammer
depends on the length of time for which it touches the string. This time is given
in the table in fractions of the periodic time of the prime tone. To this is added
a calculation for strings plucked by the finger. The striking point is always
assumed to be at | of the length of the string from its extremity.
Theoretical Intensity of the Partial Tones of Strings.
striking point at | of the length of the string
Stmck by a hammer which touches the string for
Nnniber of
the Partial
Tone
Excited by
Plucking
0
e"
1 tV
: the periodic tiix
e of the prime to
le
Struck by a
perfect hard
Hammer
I
100
100
100
100
100
100
2
8i-2
997
189-4
249
2857
3247
3
561
8-9
107-9
242-9
357-0
504-9
4
31-6
2-3
17-3
1 18-9
259-8
504-9
5
13^
1*2
0
26-1
108-4
3247
6
a-8
001
o-s
1-3
i8-8
loo-o
7
0
0
0
0
0
0
If
For easier comparison the intensity of the prime tone has been throughout
assumed as 100. I have compared the calculated intensity of the upper partials
with their force on the grand pianoforte already mentioned, and found that the
first series, under ^, suits for about the neighbourhood of 0". In higher parts of ^
the instrument the upper partials were much weaker than in this column. On
striking the digital for c", I obtained a powerful second partial and an almost in-
audible third. The second column, marked -^, corresponded nearly to the region of
g\ the second and third partials were very strong, the fourth partial was weak.
The third column, inscribed -^, corresponds with the deeper tones from & down-
wards ; here the four first partials are strong, and the fifth weaker. In the next
column, under 1^, the third partial tone is stronger than the second ; there was
no corresponding note on the pianoforte which I examined. With a perfectly hard
hammer the third and fourth partials have the same strength, and are stronger
than all the others. It results from the calculations in the above table that piano-
forte tones in the middle and lower octaves have their fundamental tone weaker
than the first, or even than the two first upper partials. This can also be con-
firmed by a comparison with the effects of plucked strings. For the latter the
second partial is weaker than the first; and it will be found that the prime
Digitized by VjOOQlC
8o MUSICAL TONES OF BOWED INSTRUMENTS. part i.
tone is much more distinct in tlie tones of pianoforte strings when plucked by the
finger, them when struck by the hammer.
Although, as is shewn by the mechanism of the upper octaves on pianofortes,
it is possible to produce a compound tone in which the prime is predominant,
makers have preferred arranging the method of striking the lower strings in such
a way as to preserve the five or six first partials distinctly, and to give the second
and third greater intensity than the prime.
Tliirdly, as regards the thickness and material of the strings. Very rigid
strings will not form any very high upper partials, because they cannot readily
assume inflections in opposite directions within very short sections. This is easily
observed by stretching two strings of different thicknesses on a monochord and
endeavouring to produce their high upper partial tones. We always succeed much
better with the thinner than with the thicker string. To produce very high upper
^ partial tones, it is preferable to use strings of extremely fine wire, such as gold lace
makers employ, and when they are excited in a suitable manner, as for example by
plucking or striking with a metal point, these high upper parfcials may be heard in
the compound itself. The numerous high upper pai-tials which he close to each
other in the scale, give that peculiar high inharmonious noise which we are
accustomed to call ' tinkUng.' From the eighth partial tone upwards these simple
tones are less than a whole Tone apart, and from the fifteenth upwards less than a
Semitone. They consequently form a series of dissonant tones. On a string of
the finest iron wire, such as is used in the manufJEUsture of artificial flowers, 700
centimetres (22*97 feet) long, I was able to isolate the eighteenth partial tone. The
peculiarity of the tones of the zither depends on the presence of these tinkling
upper partials, but the series does not extend so flEbr as that just mentioned, because
the strings are shorter.
Strings of gut are much lighter than metal strings of the same compactness,
^ and hence produce higher partial tones. The difference of their musical quality
depends partly on this circumstance and partly on the inferior elasticity of the gut,
which damps their partials, especially their higher partials, much more rapidly.
The tone of plucked cat-gut strings {guitar, harp) is consequently much lesa
tinkling than that of metal strings.
4. Mtisical Tones of Bowed Instruments,
No complete mechanical theory can yet be given for the motion of strings
excited by the violin-bow, because the mode in which the bow afifects the motion
of the string is unknown. But by applying a peculiar method of observation,
proposed in its essential features by the I^Vench physicist Lissajous, I have found
it possible to observe the vibrational form of individual points in a violin string,
and from this observed form, which is comparatively very simple, to calculate the
^ whole motion of the string and the intensity of the upper partial tones.
Look through a hand magnifying glass consisting of a strong convex lens, at
any small bright object, as a grain of starch reflecting a flame, and appearing as a
fine point of Ught. Move the lens about while the point of light remains at rest,
and the point itself will appear to move. In the apparatus I have employed, which
is shewn in fig. 22 opposite, this lens is fastened to the end of one prong of the
tuning-fork G, and marked L. It is in faxst a combination of two achromatic
lenses, like those used for the object-glasses of microscopes. These two lenses
may be used alone as a doublet, or be combined with others. When more
f magnifying power is required, we can introduce behind the metal plate A A, which
I carries the fork, the tube and eye-piece of a microscope, of which the doublet then
forms the object-glass. This instrument may be called a vibration microscope.
. When it is so arranged that a fixed luminous point may be clearly seen through it,
and the fork is set in vibration, the doublet L moves periodically up and down in
pendular vibrations. The observer, however, appears to see the luminous point
Digitized by V^jOOQlC
CHAP. V. 4. MUSICAL TONES OF BOWED INSTRUMENTS.
8i
itself vibrate, and, since the separate vibrations succeed each other so rapidly that
the impression on the eye cannot die away daring the time of a whole vibration,
the path of the luminous point appears as a fixed straight line, increasing in length
with the excursions of the fork.*
The grain of starch which reflects the Hght to be seen, is then fastened to the
resonant body whose vibrations we intend to observe, in such a way that the grain
moves backwards and forwards horizontally, while the doublet moves up and down
vertically. When both motions take place at once, the observer sees the real
horizontal motion of the luminous point combined with its apparent vertical motion,
and the combination results in an apparent curvilinear motion. The field of vision
in the microscope then shews an apparently steady and unchangeable bright
Frn. 22.
/
curve, when either the periodic times of the vibrations of the grain of starch and ^
of the tuning-fork are exactly equal, or one is exactly two or three or four times as
great as the other, because in this case the luminous point passes over exactly the
same path every one or every two, three, or four vibrations. If these ratios of the
vibrational numbers are not exactly perfect, the curves alter slowly, and the effect
to the eye is as if they were drawn on the surface of a transparent cylinder which
slowly revolved on its axis. This slow displacement of the apparent curves is not
disadvantageous, as it allows the observer to see them in different positions. But
if the ratio of the pitch numbers of the observed body and of the fork differs too
* The end of the other prong of the fork
is thickened to counterbalance the weight of
the doublet. The iron loop B which is clamped
on to one prong serves to alter the pitch of
the fork slightly; we flatten the pitch by
moving the loop towards the end of the prong.
E is an electro-magnet by which the fork is
kept in constant uniform vibration on passing
intermittent electrical currents through its
wire coils, as will be described more in detail
in Chapter VI.
Digitized by G(90gle
82 MUSICAL TONES OP BOWED INSTRUMENTS. part i.
much from one expressible by small whole numbers, the motion of the curve is too
rapid for the eye to follow it, and all becomes confusion.
If the vibration microscope has to be used for observing the motion of a violin
string, the luminous point must be attached to that string. This is done by first
blackening the required spot on the string with ink, and when it is dry, rubbing it
over with wax, and powdering this with starch so that a few grains remain sticking.
The violin is then fixed with its strings in a vertical direction opposite the micro-
scope, so that the luminous reflection from one of the grains of starch can be
clearly seen. The bow is drawn across the strings in a direction parallel to the
prongs of the fork. Every point in the string then moves horizontally, and on
setting the fork in motion at the same time, the observer sees the peculiar
vibrational curves already mentioned. For the purposes of observation I used the
a' string, which I tuned a little higher, as V^, so that it was exactly two Octaves
"' higher than the tuning-fork of the microscope, which sounded B^.
In fig. 23 are shewn the resulting vibrational curves as seen in the vibration
microscope. The straight horizontal lines in the figures, atoa, btob, ctoe
yiQ. 23.
\ shew the apparent path of the observed luminous point, before it had itself been
set in vibration ; the curves and zigzags in the same figures, shew the apparent
path of the luminous point when it also was made to move. By their side, in A,
B, C, the same vibrational forms are exhibited according to the methods used in
Chapters I. and 11., the lengths of the horizontal line being directly proportional
to the corresponding lengths of tvme^ whereas in figures a to a, b to b, c to c, the
horizontal lengths are proportional to the eoccursions of the vibrating microscope.
^ A, and a to a, shew the vibrational curves for a tuning-fork, that is for a simple
pendular vibration ; B and b to b those of the middle of a violin string in unison
with the fork of the vibration microscope ; G and c, c, those for a string which was
tuned an Octave higher. We may imagine the figures a to a, b to b, and c to c, to
be formed from the figures A, B, G, by supposing the surface on which these are
drawn to be wrapped round a transparent cylinder whose circumference is of the
same length as the horizontal line. The curve drawn upon the surface of the
cylinder must then be observed from such a point, that the horizontal line which
when wrapped round the cylinder forms a circle, appears perspectively as a single
straight line. The vibrational curve A wiU then appear in the forms a to a, B in
the forms b to b, G in the forms c to c. When the pitch of the two vibrating
bodies is not in an exact harmonic ratio, this imaginary cylinder on which the
vibrational curves are drawn, appears to revolve so that the forms a to a, &c., are
assumed in succession.
It is now easy to rediscover the forms A, B, G, from the forms a to a, b to b«
Digitized by V^OOQIC
CHAP. Y. 4. MUSICAL TONES OF BOWED INSTRUMENTS. 83
and c to c, and as the former give a more intelligible image of the motion of the
string than the latter, the cnrves, which are seen as if they were traced on the
snrface of a cylinder, will be drawn as if their trace had been unrolled from the
cylinder into a plane figure like A, B, G. The meaning of our vibrational curves
will then precisely correspond to the similar curves in preceding chapters. When
four vibrations of the violin string correspond to one vibration of the fork (as in
our experiments, where the fork gave ^ and the string &]>, p. 82a), so that
four waves seem to be traced on the surface of the imi^ginary cylinder, and when
moreover they are made to rotate slowly and are thus viewed in different positions,
it is not at aJl difGicult to draw them from inunediate inspection as if they had
been rolled off on to a plane, for the middle jags have then nearly the same
appearance on the cylinder as if they were traced on a plane.
The figures 23 B and 23 C (p. Sib), inmiediately give the vibrational forms for
the middle of a violin string, when the bow bites well, and the prime tone of the f
string is folly and powerfully produced. It is easily seen that these vibrational
forms are essentially different from that of a simple vibration (fig. 23, A). When
the point is taken nearer the ends of the string the vibrational figure is shewn in
fig. 24, A, and the two sections afi, Py, of any wave, are to one another as the two
sections of the string which lie on either side of the observed point. In the figure
Fig. 24.
this ratio is 3 : i, the point being at ^ the length of the string from its extremity.
Close to the end of the string the form is as in fig. 24, B. The short lengths of
line in the figure have been made fEunt because the corresponding motion of the ^
Imninous point is so rapid that they often become invisible, and the thicker lengths
are alone seen.*
These figures shew that every point of the string between its two extremities
vibrates with a constant velocity. For the middle point, the velocity of ascent is
equal to that of descent. If tiie violin bow is used near the right end of the
string descending, the velocity of descent on the right half of the string is less
than that of ascent, and the more so the nearer to the end. On the left half of
the string the converse takes place. At the place of bowing the velocity of descent
appears to be equal ta that of the violin bow. During the greater part of each
vibration the string here clings to the bow, and is carried on by it ; then it suddenly
detaches itself and rebounds, whereupon it is seized by other points in the bow and
again carried fbrward.f
Our present purpose is chiefly to determine the upper partial tones. The
vibrational forms of the individual points of the string being known, the intensity f
of each of the partial tones can be completely calculated. The necessary mathe-
matical formula are developed in Appendix YI. The following is the result of the
calculation. When a string excite4 by a violin bow speaks well, all the upper
partial tones which can be formed^by a string of its degree of rigidity, are present,
and their intensity diminishes d^ their pitch increases. The amplitude and the
intensity of the second partial is one-fourth of that of the prime tone, that of the
* [Dr. Hnggins, FJft.S., on ezperimentmg, string has been given by Herr Qem. Neumann
finds it probable that under the bow, the in the Proceedings {SitnmgsberiokU) of the
reUtive vdooity of descent to that of the J. and R. Academy at Vienna, mathematical
lebonnd of tile string, or ascent, is influenced and physical class, vol. Izi. p. 89. He fastened
by &e tension of tiie hairs of the bow. — bits of wire in the fonn of a oomb to the bow
TramUUor.l itself. On looking through this grating. at
t These facts suffice to determine the the string the observer sees a system of
complete motion of bowed strings. Bee rectilinear zigaag lines. The conclusions as
Appendix YL A much simpler method of to the mode of motion of the string agree
obeerving the vibrational ftnn of a violin with those given above.
Digitized by
Cdogle
84 MUSICAL TONES OP BOWED INSTRUMENTS. pabt i.
third partial a ninth, that of the fourth a sixteenth, and so on. This is the same
scale of intensity as for the partial tones of a string plucked in its middle, with
this exception, that in the latter case the evenly numbered partials all disappear,
whereas they are all present when the bow is used. The upper partials in the
compound tone of a violin are heard easily and will be found to be strong in sound
if they have been first produced as so-called harmonics on the string, by bowing
lightly while gently touching a node of the required partial tone. The strings of
a violin will allow the harmonics to be produced as high as the sixth partial tone
with ease, and with some difficulty even up to the tenth. The lower tones speak
best when the string is bowed at from one-tenth to one-twelfth the length of the
vibrating portion of the string from its extremity. For the higher harmonics
where the sections are smaller, the strings must be bowed at about one-fourth or
one-sixth of their vibrating length from the end.*
^ The prime in the compound tones of bowed iostruments is comparatively more
powerful than in those produced on a pianoforte or guitar by striking or plucking
the strings near to their extremities ; the first upper partials are comparatively
weaker ; but the higher upper partials from the sixth to about the tenth are much
more distinct, and give these tones their cutting character.
The fimdamental form of the vibrations of a violin string just described, is,
when the string speaks well, tolerably independent of the place of bowing, at least
in all essential features. It does not in any respect alter, like the vibrational form
of struck or plucked strings, according to the position of the point excited. Yet
there are certain obser- ^^^ ^^
vable differences of the
vibrational figure which
depend upon the bowing
point. Little crumples are
m usually perceived on the
lines of the vibrational
figure, as in fig. 25, which
increase in breadth and height the further the bow is removed from the extremity
of the string. When we bow at a node of one of the higher upper partials
which is near the bridge, these crumples are simply reduced by the absence of
that part of the normal motion of the string which depends on the partial tones
having a node at that place. When the observation on the vibrational form is
made at one of the other nodes belonging to the deepest tone which is elimi-
nated, none of these crumples are seen. Thus if the string is bowed at |th,
or |ths, or fths, or |ths, &c., of its length from the bridge, the vibrational
figure is simple, as in fig. 24 (p. 836). But if we observe between two nodes,
the crumples appear as in fig. 25. Variations in the quality of tone partly
depend upon this condition. When the violin bow is brought too near the
finger board, the end of which is Jtth the length of the string from the bridge,
^ the 5th or 6th partial tone, which is generally distinctly audible, will be absent.
The tone is thus rendered duller. The usual place of bowing is at about Vv^
of the length of the string ; for piano passages it is somewhat further from
the bridge and for forte somewhat nearer to it. If the bow is brought near the
bridge, and at the same time but lightly pressed, another alteration of quality
occurs, which is readily seen on the vibrational figure. A mixtiure is formed of
* (The position of the finger for prodndng near the not, out of 165 mm. the actual
the harmonio is often slightly different from half length of the strings. These differences
that theoretici^y assigned. Dr. Hnggins, most therefore be due to some imperfec-
F.B.S., kindly tried for me the position of tions of the strings themselves. Dr. Huggins
the Octave harmonio on the four strings of finds that there is a space of a quarter of
his Stradivari, a mark with Chinese white an inch at any point of which the Octave
being made under his finger on the finger harmonic may be brought out, but the quality
board. Besult, ist and 4th string exact, of tone is best at the points named above.—
2ud string 3 mm., and 3rd string 5 mm. too Translator.]
Digitized by V^OOQIC
CHAP. V. 4. MUSICAL TONES OF BOWED INSTRUMENTS. 85
the 4>rimd tone and first harmonic of the string. By light and rapid bowing,
namely at about ^^th of the length of the string from the bridge, we sometimes
obtain the upper Octave of the prime tone by itself, a node being formed in the
middle of the string. On bowing more firmly the prime tone immediately sounds.
Intermediately the higher Octave may mix with it in any proportion. This is
immediately recognised in the vibrational figure. Fig. 26 gives the corresponding
series of forms. It is seen how a firesh crest appears on the longer side of the
front of a wave, jutting out at first slightly, then more strongly, till at length the
crest of the new waves are as high as those of the old, and Uien the vibrational
number has doubled, and the pitch has passed into the Octave above. The quality
of the lowest tone of the string is rendered softer and brighter, but less full and
powerful when the intermixture commences. It is interesting to observe the
Tibrational figure while httle changes are made in the style of bowing, and note
how the resulting slight changes of quality are immediately rendered evident by '
very distinct changes in the vibrational figure itself.
The vibrational forms just described may be maintained in a uniformly steady
and unchanged condition by carefuUy uniform bowing. The instrument has then
an uninterrupted and pure musical quality of tone. Any scratching of the bow is
inmiediately shewn by sudden jumps, or discontinuous displacements and changes
in the vibrational figure. If the scratching continues, the eye has no longer time
to perceive a regular figure. The scratching noises of a violin bow must therefore
be regarded as irregular interruptions of the normal vibrations of the string,
making them to recommence from a new starting point. Sudden jumps in the
Fio. 26.
vibrational figure betray every little stumble of the bow which the ear alone would
scarcely observe. Inferior bowed instruments seem to be distinguished from good
ones by the frequency of such greater or smaller irregularities of vibration. On
the string of my monochord, which was only used for the occasion as a bowed
instmment, great neatness of bovring was required to preserve a steady vibrational
figure lasting long enough for the eye to apprehend it ; and the tone was rough in
quality, accompanied by much scratching. With a very good modem violin made
by Bausch it was easier to maintain the steadiness of the vibrational figure for
some time ; but I succeeded much better with an old Italian violin of Guadanini,
which was the first one on which I could keep the vibrational figure steady enough %
to count the crumples. This great uniformity of vibration is evidently the reason
of the purer tone of these old instruments, since every little irregularity is imme-
diately felt by the ear as a roughness or scratchiness in the quality of tone.
An appropriate structure of the instrument, and wood of the most perfect
elasticity procurable, are probably the important conditions for regular vibrations
of the string, and when these are present, the bow can be easily made to work
uniformly. This allows of a pure flow of tone, undisfigured by any roughness*.
On the other hand, when the vibrations are so uniform the string can be more
vigorously attacked with the bow. Good instruments consequently allow of a much
more powerful motion of the string, and the whole intensity of their tone can be
communicated to the air without diminution, whereas the friction caused by any
imperfection in the elasticity of the wood destroys part of the motion. Much of
the advantages of old violins may, however, also depend upon their age, and espe-
cially their long use, both of which cannot but act favourably on the elasticity of
Digitized by V^OOQ IC
86 MUSICAL TONES OP BOWED INSTRUMENTS. past i.
the wood. But the art of bowing is evidently the most important condition of all.
How delicately this must be cultivated to obtain certainty in producing a very
perfect quality of tone and its different varieties, cannot be more clearly demon-
strated than by the observation of vibrational figures. It is also well known that
great players can bring out full tones from even indifferent instruments.
The preceding observations and conclusions refer to the vibrations of the strings
of the instrument and the intensity of their upper partial tones, solely in so feu: as
they are contained in the compound vibrational movement of the string. But
partial tones of different pitches are not equally well communicated to the air, and
hence do not strike the ear of the listener with precisely the same degrees of
intensity as those they possess on the string itself. They are communicated to
the air by means of the sonorous body of the instrument. As we have had
already occasion to remark, vibrating strings do not directly communicate any
f sensible portion of their motion to the air. The vibrating strings of the violin,
in the first place, agitate the bridge over which they are stretched. This stands
on two feet over the most mobile part of the ' belly ' between the two '/ holes.'
One foot of the bridge rests upon a comparatively firm support, namely, the * sound-
post,' which is a solid rod inserted between the two plates, back and belly, of the
instrument. It is only the other leg which agitates the elastic wooden plates, and
through them the included mass of air.*
An inclosed mass of air, like that of the violin, vi61a, and violoncello, bounded
by elastic plates, has certain proper tones which may be evoked by blowing
across the openings, or * f holes.* The violin thus treated gives c' according to
Savart, who examined instruments made by Stradivari (Stradiuarius).t Zam-
miner found the same tone constant on even imperfect instruments. For the
violoncello Savart found on blowing over the holes F, and Zamminer G.t Ac-
cording to Zamminer the sound-box of the vi61a (tenor) is tuned to be a Tone
^ deeper than that of the violin. § On placing the ear against the back of a violin
and playing a scale on the pianoforte, some tones will be found to penetrate the
ear with more force than others, owing to the resonance of the instrument. On a
* [Thia aoooont is not quite suffioieni. agitation transmitted by the rod.* In short.
Neither leg of the bridge rests exactly on the touoh rod acts as a soond-post to the
the sound-post, becaase it is found that this finger. The place of least vibration of the
position materially injures the quality of tone, belly is exactly over the sound-post and of the
The sound-post is a little in the rear of the back at the point under the sound-post. On
leg of the bridge on the «" string side. The removing the sound-post, or covering its ends
position of the sound-post with regard to the with a sheet of india-rubber, which did not
bridge has to be adjusted for each individual diminish the support, the tone was poor and
instrument. Dr. William Huggins, F.B.S., in thin. But an external wooden clamp oonneet-
his paper * On the Function of the Sound-post, ing belly and back in the places where the
and on the Proportional Thickness of the sound-post touches them, restored the tone.-^
Strings of the Violin,' read May 24, 1883, Translator.']
Proceedings of the Royal Society^ vol. xxxv. f [Zanmiiner, Die Miisih^ 1855, voL L
«|T pp. 241-248, has experimentally investigated p. 37, says d of 256 vlb. — Translator.]
^' the whole action of the sound-post, and finds % [Zamminer, ibid. p. 41, and adds that
that its main function is to convey vibrations judging from the violin the resonance shoold
from the belly to the back of the violin, in be Fff . — Translator,]
addition to those conveyed by the sides. The § [The passage referred to has not been
(apparently ornamental) cuttings in the bridge found. But Zamminer says, p. 40, * The
of the violin, sift the two sets of vibrations, length of the box of a violin is 13 Par. inches,
set up by the bowed string at right angles to and of the vi61a 14 inches 5 lines. Exactly
each other and ' allow those only or mainly to in inverse ratio stand the pitch nombers
pass to the feet which would be efficient in 470 (a misprint for 270 most probably) and
setting the body of the instrument into vibra- 241, which were found by blowing over the
tion.* As the peculiar shape of the instru- wind-holes of the two instruments.* Now the
ment rendered strewing of sand unavailable, ratio 13 : 14^3 gives 182 cents, and the ratio
Pr. Huggins investigated the vibrations by 241 : 270 gives 197 cents, which are very
means of a * touch rod,* consisting of ' a small nearly, though not * exactly ' the same. Hiis,
round stick of straight grained deal a few however, makes the resonance of the violin
inches long; the forefinger is placed on one 270 vib. and not 256 vib., and agrees with the
end and the other end is put lightly in contact next note. I got a good resonance with a fork
with the vibrating surface. The finger soon of 268 vib. from Dr. Huggins's violoncoUo by
becomes very sensitive to small differences of Nicholas about ^.d. if<)2.— Translator.]
Digitized by V^jOOQlC
OBAP. T. 4. MUSICAL TONES OF BOWED INSTBUMENTS.
«7
violin made by Bansoh two tones of greatest resonance were thus discovered, one
between d and c% [between 264 and 280 vib.], and the other between a' and &]>
[between 440 and 466 vib.]. For a vi61a (tenor) I found the two tones about a
Tone deeper, which agrees with Zanuniner's calculation.*
The consequence of this peculiar relation of resonance is that those tones of
the strings which lie near the proper tones of the inclosed body of air, must be
proportionably more reinforced. This is clearly perceived on botii the violin and
violoncello, at least for the lowest proper tone, when the corresponding notes are
produced on the strings. They sound particularly fuU, and the prime tone of these
compounds is especially prominent. I think that I heard this also for al on the
violin, which corresponds to its higher proper tone.
Since the lowest tone on the violin is ^, the only upper partials of its musical
tones which can be somewhat reinforced by the resonance of the higher proper
tone of its inclosed body of air, are the higher octaves of its three deepest notes. «p
But the prime tones of its higher notes will be reinforced more than their upper
partials, because these prime tones are more nearly of the same pitch as the
proper tones of the body of air. This produces an effect similar to that of the con-
struction of the hanmier of a piano, which favours the upper partials of the deep
notes, and weakens those of the higher notes. For the violoncello, where the lowest
string gives C, the stronger proper tone of the body of air is, as on the violin, a
Fourtti or a Fifth higher than the pitch of the lowest string. There is consequently
a similar relation between the fi&voured and unfavoured partial tones, but all of
* [Throogh the kindness of Dr. Haggins,
F JI.S., the Bev. H. B. Haweis, and the violin-
makers, Messrs. Hart, Hill ft Withers, I was
in 1880 enabled to examine the pitch of the
resonance of some fine old violins bv Duiflo-
pmgoar (Swiss Tyrol* Bologna, and Lyons
1510-1530)* Amati (Cremona 1596-1684), Bug-
gieri (Cremona 1608- 1720), Stradivari (Cre-
mona 1644^1737), Giuseppe Gameri (known as
'Joseph,' Cremona 16S3-1745), Lnpot (France
i750-i82O>. The method adopted was to hold
timing-forks, of which the exact pitch had
been determined by Scheibler's forks, in saoces-
sion over the widest part of the/ hole on the
^ string side of the violin (furthest from the
somid-post) and observe what fork excited the
maiimnm resonance. My forks form a series
proceeding by 4 vib. in a second, and hence I
oould only tell the pitch within 2 vib., and it
was often extremely difficult to decide on the
fork which gave the best resonance. By far
the strongest resonance lay between 268 and
272 vib.» but one early Stradivari (1696) had a
fine resonance at 264 vib. There was also a
secondary but weaker maximum resonance at
about 252 vib. The 256 vib. was generally
decidedly inferior. Hence we may take 270
vib. as the primary maximum, and 252 vib. as
the secondary. The first corresponds to the
highest English concert pitch c''«540 vib.,
now used in London, and agrees with the
lower resonance of Bausch's instrument men-
tioned in the text. The second, which is 120
cents, or rather more than an equal Semitone
flatter, gives the pitch which my researches
shew was common over all Europe at the
time (see App. XX. sect. H.). But although
ihe low pitch was prevalent, a high pitch, a
great Semitone (117 ct.) hi^er, was also in
use as a * ohambcff pitch.* A violin of Mazzini
of Brescia (1560.1640) belonging to the eldest
dau^ter of Mr. Yemon Lushington, Q.C., had
the same two maximum resonances, the higher
being decidedly the superior. I did not ex-
amine for the higher or a* pitches named in
the text. Mr. Healey (of the Science and Art
Department, South Kensington) thought his
violin ^supposed to be an Amati) sounded best
at the low pitch c" » 504. Subsequently, I ex-
amined a fine instrument, bearing inside it the
label * Petrus Guamerius Cremonensis fecit,
MantuBB sub titulo S.Theresiie, anno 1701,* in ^
the possession of Mr. A. J. Hipkins, who knew ■<
it to be genuine. I tried this with a series
of forks, proceeding by differences of about
4 vibrations from 240 to 56a It was surprising
to find that every fork was to a certain extent
reinforced, that is, in no case was the tone
quenched, and in no case was it reduced in
strength. But at 260 vib. there was a good,
and at 364 a better resonance; perhaps 262
may therefore be taken as the best. There
was no secondary low resonance, but there
were two higher resonances, one about 472,
(although 468 and 476 were also good,) and
another at 520 (although 524 and 528 were
also good). As this sheet was passing through
the press I had an opportunity of trying the
resonance of Dr. Huggins's Stradivari of 1708,
figured in Grovels Dictionary of MusiCt iii.
728, as a specimen of the best period of Stradi- «
vari's work. The result was essentially the
same as the last ; every fork was more or less
reinforced ; there was a subordinate maximum
at 252 vib. ; a better at from 260 to 268 vib. ;
very slight maxima at 312, 348, 384, 412, 420,
428 (the last of which was the best, but was
only a fair reinforcement), 472 to 480, but 520
was decidedly best, and 540 good. No one
fork was reinforced to the extent it would have
been on a resonator properly tuned to it, but
no one note was deteriorated. Dr. Huggins says
that ' the strong feature of this violin is the
great equality of all four strings and the per-
sistenoe of the same fine quality of tone
throughout the entire range of the instru-
ment.' — Tianslator.]
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83
MUSICAL TONES OP FLUTE OR FLUE PIPES. pabt i.
Fig. 27.
/a
I
them are a Twelfth lower than on the violin. On the other hand, the most
favoured partial tones of the vi61a (tenor) corresponding nearly with h\ do not
lie between the first and second strings, but
between the second and third ; and this seems
to be connected with the altered quality of
tone on this instrument. Unfortunately this
influence cannot be expressed numerically.
The maximum of resonance for the proper
tones of the body of air is not very marked ;
were it otherwise there would be much more
inequaHty in the scale as played on these
bowed instruments, inmiediately on passing
the pitch of the proper tones of their bodies of
f air. We must consequently conjecture that
their influence upon the relative intensity of
the individual partials in the musical tones of
these instruments is not very prominent.
5. Musical Tones of Flute or Flue Pipes.
In these instruments the tone is produced
by driving a stream of air against an opening,
generally furnished with sharp edges, in some
\ hollow space filled with air. To this class
belong the bottles described in the last chapter,
and shewn in fig. 20 (p. 60c), and especially
flutes and the majority of organ pipes. For
flutes, the resonant body of air is included in
H its own cylindrical bore. It is blown with the
mouth, which directs the breath against the
somewhat sharpened edges of its mouth hole.
The construction of organ pipes will be seen
from the two adjacent figures. Fig. 27, A,
shews a square wooden pipe, cut open long-
wise, and B the external appearance of a round
tin pipe. B B in each shews the tube which
incloses the sonorous body of air, a b is the
mouth where it is blown, terminating in a sharp
lip. In fig. 27, A, we see the air chamber or
throat E into which the air is first driven from
the bellows, and whence it can only escape
through the narrow slit c d, which directs it
H against the edge of the Hp. The wooden pipe
A as here drawn is open, that is its extremity
is uncovered, and it produces a tone with a
wave of air tivice as long as the tube B B.
The other pipe, B, is stopped, that is, its upper
extremity is closed. Its tone has a vrmefour
times the length of the tube B B, and hence an
Octave deeper than an open pipe of the same
length.*
Any air chambers can be made to give a
musical tone, just like organ pipes, flutes, the bottles previously described, ihe
windchests of vioUns, Ac, provided they have a sufticiently narrow opening,
0
* [These relations are only approximate,
as is explained below. The mode of excite-
ment by the lip of the pipe makes them
inexact. Also they take no notice of the
* scale * or diameters and depths of the pipes,
or of the force ol the wind, or of the tempera-
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CHAP. V. 5. MUSICAL TONES OP FLUTE OB FLUE PIPES.
89
famished with somewhat projectmg sharp edges, by directing a thin flat stream of
air across the opening and against its edges.*
The motion of air that takes place in the inside of organ pipes, corresponds to
a system of plane waves which are reflected backwards and forwards between the ^
two ends of the pipe. At the stopped end of a cylindrical pipe the reflexion of
every wave that strikes it is very perfect, so that the reflected wave has the same
intensity as it had before reflexion. In any train of waves moving in a given
direction, the velocity of the oscillating molecules in the condensed portion of the
wave takes place in ihe same direction as that of the propagation of the waves, and
in the rarefied portion in the opposite direction. But at the stopped end of a pipe
its cover does not allow of any forward motion of the molecules of air in the
direction of the length of the pipe. Hence the incident and reflected wave at this
place combine so as to excite opposite velocities of oscillation of the molecules of
air, and consequently by their superposition the velocity of the molecules of air at ^
the cover is destroyed. Hence it follows that the phases of pressure in both will
agree, because opposite motions of oscillation and opposite propagation, result in
accordant pressure.
Hence at the stopped end there is no motion, but great alteration of pressure.
The reflexion of the wave takes place in such a manner that the phase of conden*
sation remains unaltered, but the direction of the motion of oscillation is reversed.
The contrary takes place at the open end of pipes, in which is also included the
opening of their mouths. At an open end where the air of the pipe communi-
cates freely with the great outer mass of air, no sensible condensation can take
place. In the explanation usually given of the motion of air in pipes, it is assumed
that both condensation and rarefEMstion vanish at the open ends of pipes, which is
approximatively but not exactly correct. If there were exactly no alteration of
density at that place, there would be complete reflexion of every incident wave
at the open ends, so that an equally large reflected wave would be generated with ^
an opposite state of density, but the direction of oscillation of the molecules of
air in both waves would agree. The superposition of such an incident and such a
tore of the air. The following are adapted
from the roles given by M. GavaU16-Goll, the
celebrated Frendi oigan-builder, in Comptea
Rwdu9t i860, p. 176, supposing the tempera-
ture to be 59* F. or 15** C, and the pressure of
the wind to be about 3^ inohes, or 8 centi-
metres (meaning that it will support a column
of water of that height in the wind gauge).
The pitch numbers, for double vibrations, are
found by dividing 20,080 when the dimensions
are given in inches, and 510,000 when in
millimetres by the following numbers : (i) for
cylindrical open pipes, 3 times the length
added to 5 times the diameter ; (2) for cyUndri-
cal slopped pipes, 6 times the lexigth added to
10 times the diameter; (3) for square open
pipcSf 3 times the length added to 6 times the
depth (clear internal distance from mouth to
back) ; {4) for square stopped pipes, 6 times the
length added to 12 times the depth.
This rule is always sufficiently accurate for
cutting organ pipes to their approximate
length, and piercing them to bring out the
Octave harmonic, and has long been used for
these purposes in M. Cavaill6-Ck>irB factory.
The rule is not so safe for the square wooden
as for the cylindrical metal pipes. The pitch
of a pipe of known dimensions ought to be
first ascertained by other means. Then this
pitch number multiplied by the divisors in (3)
and (4) should be used in place of the 20,080
or 510,000 of the rule, for all similar pipes.
As to strength of wind, as pressure varies
from 2} to 3^ inches, the pitch number
increases by about 1 in 300, but as pressure
varies from 3i to 4 inches, the pitch number
increases by about i in 440, the whole increase
of pressure from 2| to 4 inches increases the
pitch number by i m 180.
Por temperature, I found by numerous
observations at very different temperatures'
that the following practical rule is sufficient
for reducing the pitch number observed at one
temperature to that due to another. It is not
quite accurate, for the air blown from the
bellows is often lower than the external tem-
perature. Let P be the pitch number observed f
at a given temperature, and d the difference of
temperature in degrees Fahr. Then the pitch
number is P x ^i ± '00104CQ according as the
temperature is higher or lower. The practical
operation is as follows : supposing P » 528, and
(2=14 increase of temperature. To 528 add
4 in 100, or 21*12, giving 549*12, Divide by
1000 to 2 places of decimals, giving *55..
Multiply hj d- 14, giving 7*70. Adding this to
528, we get 535*7 for the pitch number at the
new temperature. — Translator,]
* [Here the passage from 'These edges,'
p. 140, to * resembling a violin,* p. 141 of the
1st English edition, has been omitted, and the
passage from * The motion of air,' p. 89a,
to * their comers are rounded off,' p. 936, haa
been inserted in accordance with the 4th
Grerman edition. — Translator.]
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90 MUSICAL TONES OP FLUTE OR FLUE PIPES. pabt i.
reflected wave would indeed leave the state of density unaltered at the open ends,
but would occasion great velocity in the oscillating molecules of air.
In reality the assumption made explains the essential phenomena of organ pipes.
Consider first a pipe with two open ends. On our exciting a wave of condensation
at one end, it runs forward to the other end, is there reflected as a wave of rare-
faction, runs back to the first end, is here again reflected with another alteration of
phase, as a wave of condensation, and then repeats the same path in the same way
a second time. This repetition of the same process therefore occurs, after the
wave in the tube has passed once forwards and once backwards, that is twice through
the whole length of the tube. The time required to do this is equal to double the
length of the pipe divided by the velocity of sound. This is the duration of the
vibration of the deepest tone which the pipe can give.
Suppose now that at the time when the wave begins its second forward and
f backward journey, a second impulse in the same direction is given, say by a vibra-
ting tuning-fork. The motion of the air wiU then receive a reinforcement, which
will constantly increase, if the fresh impulses take place in the same rhythm as the
forward and backward progression of the waves.
Even if the returning wave does not coincide with the first following similar
impulse of the tuniag-fork, but only with the second or third or fourth and so on,
the motion of the air wiU be reinforced after every forward and backward passage.
A tube open at both ends will therefore serve as a resonator for tones whose
pitch number is equal to the velocity of sound (332 metres) * divided by twice the
length of the tube, or some multiple of that number. That is to say, the tones of
strongest resonance for such a tube wiU, as in strings, form the complete series of
harmonic upper partials of its prime.
The case is somewhat different for pipes stopped at one end. If at the open
end, by means of a vibrating tuning-fork, we excite an impulse of condensation
% which propagates itself along the tube, it will run on to the stopped end, will be
there reflected as a wave of condensation, return, will be again reflected at the
open end with altered phase as a wave of rarefaction, and only after it has been
again reflected at the stopped end with a similar phase, and then once more at the
open end with an altered phase as a condensation, will a repetition of the process
ensue, that is to say, not till after it has traversed the length of the pipe four times.
Hence the prime tone of a stopped pipe has twice as long a period of vibration as an
I open pipe of the same length. That is to say, the stopped pipe wiU be an Octave
deeper than the open pipe. If, then, after this double forward and backward passage,
the first impulse is renewed, there will arise a reinforcement of resonance.
Partials f of the prime tone can also be reinforced, but only those which are
unevenly numbered. For since at the expiration of half the period of vibration,
the prime tone of the wave in the tube renews its path with an opposite phase of
density, only such tones can be reinforced as have an opposite phase at the expira-
^ tion of half the period of vibration. But at this time the second partial has just
completed a whole vibration, the fourth partial two whole vibrations, and so on.
* [This is 1089-3 '^t in a second, which before the Physical Society, and published
is the mean of several observations in free in the Philosophical Moffoeine tor Dec. 1883,
air ; it is usual, however, in England to take pp. 447-455, and Oct. 1884, pp. 328-334, as the
the whole number 1090 feet, at freezing. At means of many observations on the velocity
60° F. it is about 1 120 feet per second. Mr. D. of sound in dry air at 32^ F., in tubes, obtained
J. Blaikley (see note p. gjd), in two papers read
for diameter '45
pitch various, velocity 1064*26
pitch 260 vib. , velocity 1 062' 1 2
•75
107253
1072-47
1-25
1078-71
1078-73
2-08
1081-78
1082-51
3*47 English inches.
1083-13 „ feet.
1084-88 „
The velocity in tubes is therefore always less note p. 23c,) but it is precisely the latter which
than in free hix,— Translator,] are not excited in the present case. This is
t [The original says * upper partials' only mentioned as a warning to those who
(Obert&ne), but the upper partials which are faultily use the faulty expression * overtones *
unevenly numbered are the ist, 3rd, 5th, &c.» indifferently for both partials and upper
and these are really the 2nd, 4th, 6th, &c., (that partials.— ZVa9w2a(or.]
is, the evenly numbered) partial tones, (see foot-
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cHAP.v.s. MUSICAL TONES OF FLUTE OR FLUE PIPES. 91
These therefore have the same phases, and cancel their effect on the return of the
wave with an opposite phase. Hence the tones of strongest resonance in stopped
pipes correspond with the series of unevenly numbered partials of its fundamental
tone. Supposing its pitch number is n, then ^n is the Twelfth of n, that is the
Fifth of 2n the higher Octave, and ^n is the major Third of 4n the next higher
Octave, and jn the [sub] minor Seventh of the same Octave, and so on.
Now although the phenomena follow these rules in the principal points, certain
deviations from them occur because there is not precisely no change of pressure
at the open ends of pipes. From these ends the motion of sound communicates
itself to the unindosed air beyond, and the waves which spread out from the open
ends of the tubes have relatively very little alteration of pressure, but are not
entirely without some. Hence a part of every wave which is incident on the open
end of the pipe is not reflected, but runs out into the open air, while the remainder
or greater portion of the wave is reflected, and returns into the tube. The re- %
flexion is the more complete, the smaller are the dimensions of the opening of \\
the tube in comparison with the wave-length of the tone in question.
Theory* also, agreeing with experiment, shews that the phases of the reflected
part of the wave are the same as they would be if the reflexion did not take place
at the siirfB.ce of the opening itself but at another and somewhat different plane.
H^ice what may be called the redticed length of the pipe, or that answering to the
pitch, is somewhat different from the real length, and the difference between the
two depends on the form of the mouth, and not on the pitch of the notes pro-
duced unless they are so high and hence their wave-lengths so short, that the
dimensions of the opening cannot be neglected in respect to them.
For cylindrical pipes of circular section, with ends cut at right angles to the
length, the distance of the plane of reflexion from the end of the pipe is theoreti-
cally determined to be at a distance of 0*7854 the radius of the circle.f For a
wooden pipe of square section, of which the sides were 36 mm. (1*4 inch) internal 11
measure, I found the distance of the plane of reflexion 14 mm. (-55 inch).j:
Now since on account of the imperfect reflexion of waves at the open ends of
organ pipes (and respectively at their mouths) a part of the motion of the air must
escape into i^e free air at every vibration, any oscillatory motion of its mass of air
must be speedily exhausted, if there are no forces to replace the lost motion. In .
fact, on ceasing to blow an organ pipe scarcely any after sound is observable. I
Nevertheless the wave is frequently enough reflected forward and backward for its
pitch to become perceptible on tapping against the pipe.
The means usuaUy adopted for keeping them continually sounding, is blowing.
In order to understand the action of this process, we must remember that when
* See my paper in CreUe's Journal for tion of the plug. [The Bameness of the pitch
MathemaHcSt vol. Ivii. ia detennined by seeing that each makes the
t Mr. Bosanqaet (Proe. Mtta. Aasn. 1877-8, same number of beats with the same fork.]
p. 65) is reported as saying:* Lord Bay leigh and The nodal sorfaoe lay 137 mm. (5-39 inoh)
himself had gone fully into the matter, and had from the end of the pipe, while a quarter of 1l
oome to the condusion that this correction was a wave-length was 151 mm. (5*94 inoh). At the
much less than Helmholtz supposed. Lord Bay- mouth end of the pipe, on tiie other hand,
leigh adopted the figure *6 of the radius, whilst 83 mm. (3*27 inoh) jrere wanting to complete
he himself adopted -55.' See papers by Lord the theoretical length of the pipe. [The addi-
Rayleigh and Mr. Bosanquet in PhUoaophical tional piece being half the length of the wave,
MoMmne. Mr. Blaikley by a new process the pitdi of the pipe before and after the
finds *576, which lies between the other two, addition of this piece remains the same, by
see hispaper in Phil. Miig. May 1879, p. 342. which property the length of the additions
X The pipe was of wood, made by Marloye, piece is found. The length of the pipe from
the additional length being 302 mm. (11*9 in.), the bottom of the mouth to the open end was
corresponding exactly with half the length of 205 mm. = 8-07 inch ; the node, as determined,,
wave of the pipe. The position of the nodal was 137 nun. » 5*39 inch from the open end,
plane in the inside of the pipe was determined and 68 mm. « 2*68 inch from the bottom of the
by inserting a wooden plug ox the same diameter mouth. These lengths had to be increased by
•s of the mtemal opening of the pipe at its 14 mm. » *55 in. and 83 mm. » 3*27 in. respeo-
quarter length
— 2Van«fcUor.]
pipe . __ _ _ _
open end, until the pitch of the pipe, which tively, to xnake up each to the quarter length
nad now beoome a closed one, was exactly the of the wave 151 mm. » 5*95 inch.~2Van«2ator.]
nme as that of the open pipe before the inser-
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93 MUSICAL TONES OP FLUTE OB FLUE PIPES. pabt i..
air is blown out of snch a slit as that which lies below the lip of the pipe, it breaks
through the air which lies at rest in &ont of the slit in a thin sheet like a blade or
lamina, and hence at £rst does not draw any sensible part of that air into its own
motion. It is not tiU it reaches a distance of some centimetres [a centimetre is
nearly four-tenths of an inch] that the outpouring sheet splits up into eddies or
vortices, which effect a mixture of the air at rest and the air in motion. This
blade-shaped sheet of air in motion can be rendered visible by sending a stream of
air impregnated with smoke or clouds of salammoniac through the mouth of a
pipe from which the pipe itself is removed, such as is commonly found among
physical apparatus. Any blade-shaped gas flame which comes from a split burner
is also an example of a similar process. Burning renders visible the limits between
the outpouring sheet of gas and the atmosphere. But the flame does not render
the continuation of the stream visible.
IF Now the blade-shaped sheet of air at the mouth of the organ pipe is wafiied to
one side or the other by every stream of air which touches its surface, exactly as
this gas flame is. The consequence is that when the oscillation of the mass of air
in the pipe * causes the air to enter through the ends of the pipe, the blade-shaped
stream of air arising from the mouth is also inclined inwards, and drives its whole
mass of air into the pipe.f During the opposite phase of vibration, on the other
hand, when the air leaves the ends of the pipe the whole mass of this blade of air
is driven outwards. Hence it happens that exactly at the times when the air in
^ the pipe is most condensed, more air still is driven in from the bellows, whence
the condensation, and consequently also the equivalent of work of the vibration of
the air is increased, while at the periods of rarefaction in the pipe the wind of the
bellows pours its mass of air into the open space in front of the pipe. We must
remember also that the blade-shaped sheet of air requires time in order to traverse
the width of the mouth of the pipe, and is during this time exposed to the action .
If of the vibrating column of air in the pipe, and does not reach the lip (that is the
line where the two paths, inwards and outwards, intersect) until the end of this
time. Every particle of air that is blown in, consequently reaches a phase of
vibration in the interior of the pipe, which is somewhat later than that to which
it was exposed ui traversing the opening. If the latter motion was inwards, it
encounters the following condensation in the interior of the pipe, and so on.
This mode of exciting the tone conditions also the peculiar quality of tone of
these organ pipes. We may regard the blade-shaped stream of air as very thin in
comparison with the amplitude of the vibrations of air. The latter often amount
to lo or i6 millimetres ('39 to '63 inches), as may be seen by bringing small
flames of gas close to this opemng. Gonse.quently the alternation between the
periods of time for which the whole blast is poured iato the interior of the pipe,
and those for which it is entirely emptied outside, is rather sudden, in fact almost
instantaneous. Hence it follows % that the oscillations excited by blowing are of
^ a similar kind ; namely, that for a certain part of each vibration the velocity of the
particles of air in the mouth and in free space, have a constant value directed out-
wards, and for a second portion of the same, a constant value directed inwards.
With stronger blowing that directed inwards will be more intense and of shorter
duration ; with weaker blowing, the converse may take place. Moreover, the pres-
sure in the mass of air put in motion in the pipe must also alternate between two
constant values with considerable rapidity. The rapidity of this change will,
however, be moderated by the circumstance thai the blade-shaped sheet of air is
not infinitely thin, but requires a short time to pass over the lip of the pipe, and
* [It has, however, not been explained how side the pipe is very small. A candle flame
that * oscillation ' commences. This will be held at the end of the pipe only pulsates ;
alluded to in the additions to App. VII. sect. B. held a few inches from the lip, along the edge
— Translator.] of the pipe, it is speedily extinguished.— Trans-
t [The amount of air which enters as com- latorJ]
pared with that which passes ov«r the lip out- % ^^ Appendix VII. [especially sect. B, II.].
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CHAP. V. 5. MUSICAL TONES OP FLUTE OR FLUE PIPES. 93
that secondly the higher upper partials, whose wave-lengths only slightly exceed the
diameter of the pipe, are as a general rule imperfectly developed.
The kind of motion of the air here described is exactly the same as that shewn
in fig. 23 (p. 826), B and 0, fig. 24 (p. 836), A and B, for the vibrating points of i
a violin string. Organ-builders have long since remarked the similarity of the
quality of tone, for the narrower cylindrical-pipe stops when strongly blown, as
shewn by the names : Geigenprincipal, Vidla di Gamba, Violoncello, Violon-bcLSS.*
That these conclusions firom the mechanics of blowing correspond with the
£Gu;ts in nature, is shewn by the experiments of Messrs. Toepler & Boltzmann,t who
rendered the form of the oscillation of pressure in the interior of the pipe optically
observable by the interference of light passed through a node of the vibrating mass
of air. When the force of the wind was small they found almost a simple vibration
(the smaller the oscillation of the air-blade at the lip, the more completely the dis-
continuities disappear). But when the force of the wind was greater they found ^
a very rapid alternation between two different values of pressure, each of which
remained almost unaltered for a fraction of a vibration.
Messrs. Mach and J. Hervert's X experiments with gas flames placed before the
end of an open pipe to make the vibrations visible, shew that the form of motion
just described really occurs at the ends of the pipes. The forms of vibration which
they deduced from the analysis of the forms of the flames correspond with those of
a violin string, except that, for the reason given above, their comers are rounded off.
By using resonators I find that on narrow pipes of this kind the partial tones V
may be clearly heard up to the sixth.
For wide open pipes, on the other hand, the adjacent proper tones of the tube
are all somewhat sharper than the corresponding harmonic tones of the prime, and
hence these tones will be much less reinforced by the resonance of the tube. Wide
pipes, having larger masses of vibrating air and admitting of being much more
strongly blown without jumping up into an harmonic, are used for the great body i[
of sound on the organ, and are hence called prificipalstimfnen,^ For the above
reasons they produce the prime tone alone strongly and fully, with a much weaker
retinue of secondary tones. For wooden ' principal ' pipes, I find the prime tone
lufid its Octave or first upper partial very distinct; the Twelfth or second upper
partial is but weak, and the higher upper partials no longer distinctly perceptible.
For metal pipes the fourth partial was also still perceptible. The quality of tone in
these pipes is faller and softer than that of the geigenprincipaL* When flute or
flue stops of the organ, and the German flute are blown softly, the upper partials
lose strength at a greater rate than the prime tone, and hence the musical quality
becomes weak and soft.
Another variety is observed on the pipes which are conically narrowed at their
* [Oetgenprincipal — violin or crisp-toned eively conical with a bell top. Prom Hopkins
diapason, 8 feet, — violin principal, 4 feet. See on the Organ, pp. 137, 445, (fee. — Translator.]
supra p. gid, note. Violoncello — * crisp-toned f Poggendorff^s Annd., vol. czli. pp. 321- ^
open stop, of small scale, the Octave to the 352.
violone, 8 feet.' Violon-baas— ihis fails in :( Poggendor£f*s ^nnaZ., vol. cxlvii. pp. 590-
Hopkins, but it is probably his *violone — 604.
doable bass, a unison open wood stop, of mnoh § [Literally * principal voices or parts ; '
smaller scale than the Diapason, and formed may probably be best translated ' principal
of pipes that are a little wider at the top than work * or * diapason-work,' including * all the
at the bottom, and famished with ears and open cylindrical stops of Open Diapason
beard at the mouth ; the tone of the Violone measure, or which have their scale deduced
is crisp and resonant, like that of the orches- from that of the Open Diapason ; such stops
tral Double Bass ; and its speech being a little are the chief, most important or ** principal,"
slow, it has the Stopped Bass always drawn as they are also most numerous in an organ,
with it, 16 feet.' Qamba or viol da ganiba — The Unison and Double Open Diapasons,
* bass vioL unison stop, of smaller scale, and Principal, Fifteenth and Octave Fifteenth ;
thinner but more pungent tone than the violin the Fifth, Twelfth, and Larigot ; the Tenth
diapason, 8 feet, . . . one of the most highly and Tierce ; and the Mixture Stops, when of
esteemed and most frequently disposed stops full or proportional scale, belong to the Dia-
ifi Continental organs ; the German gamba is pason-work.' From Hopkins on the Organ,
oBuaUy composed of cylindrical pipes.' In p. 131. — Translator,]
England till very recently it was made ezclu-
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Google
94 MUSICAL TONES OF FLUTE OR FLUE PIPES. part i.
upper end, in the aaUcional^ geffuhom, and spitzflote stops.* Their upper opening
has generally haU the diameter of the lower section ; for the same length the
saUeional pipe has the narrowest, and the spitzflote the widest section. These pipes
liave, I find, the property of rendering some higher partial tones, from the Fifth
to the Seventh, comparatively stronger than the lower. The quality of tone is
consequently poor, but peculiarly bright.
The narrower stopped cylindrical pipes have proper tones corresponding to the
unevenly numbered partials of the prime, that is, the third partial or Twelfth, the
fifth partial or higher major Third, and so on. For the wider stopped pipes, as for
the wide open pipes, the next adjacent proper tones of the mass of air are distinctly
higher than the corresponding upper partials of the prime, and consequently these
upper partials are very slightly, if at all, reinforced. Hence wide stopped pipes,
especially when gently blown, give the prime tone almost alone, and they were
IT therefore previously adduced as examples of simple tones (p. 6oc), Narrow stopped
pipes, on the other hand, let the Twelfth be very distinctly heard at tlie same
time with the prime time ; and have hence been called quintaten (qmntam tenentes),f
When these pipes are strongly blown they also give the fifth partial, or liigher
major Third, very distinctly. Another variety of quality is produced by the
rohrfldte.t Here a tube, open at both ends, is inserted in the cover of a. stopped
pipe, and in the examples I examined, its length was that of an open pipe giving
the fifth partial tone of the prime tone of the stopped pipe. The fif h partial tone
is thus proportionably stronger than the rather weak third partial on these pipes,
and the quality of tone becomes peculiarly bright. Compared with open pipes the
quality of tone in stopped pipes, where the evenly numbered partial tones are
absent, is somewhat hollow ; the wider stopped pipes have a dull quality of tone,
especially when deep, and are soft and powerless. But their softness ofiers a very
effective contrast to the more cutting qualities of the narrower open pipes and the
f noisy con^ownd stops^ of which I have already spoken (p. 576), and which, as is
well known, form a compound tone by uniting many pipes corresponding to a prime
and its upper partial tones.
Wooden pipes do not produce such a cutting windrush as metal pipes. Wooden
sides also do not resist the agitation of the waves of sound so well as metal ones, and
hence the vibrations of higher pitch seem to be destroyed by friction. For these
reasons wood gives a softer, but duller, less penetrating quality of tone than metal.
It is characteristic of all pipes of this kind that they speak readily, and hence
admit of great rapidity in musical divisions and figures, but, as a little increase of
force in blowing distinctly alters the pitch, their loudness of tone can scarcely be
changed. Hence on the organ forte and pianx) have to be produced by stops, which
regulate the introduction of pipes with various qualities of tone, sometimes more,
sometimes fewer, now the loud and cutting, now the weak and soft. The means of
expression on this instrument are therefore somewhat limited, but, on the other
f hand, it clearly owes part of its magnificent properties to its power of sustaining
tones with unsJtered force, undistiurbed by subjective excitement.
* [ScMdonal^^ t9i&SlJ Double Duloiana, 16 oonioal bodies, 8 feet.* * This stop is fotmd of
feet and 8 feet, octave siJicional, 4 feet.' The 8, 4, and 2 feet length in Gennan oigans. In
Duleiana is'desoribed as * belonging to the Flate- England it has hitherto been made oUefly as a
work, . . . the pipes mooh smaller in scale than 4-feet stop ; i.e. of principal pitch* The pipes
those of the open diapason . • . tone peculiarly of the Spitz-flute are sUghtly oonicaJ, beong
soft and gentle ' {Hopkins, p. 113). Gwishom^ about ^ narrower at top than at the mouth,
Utorally * chamois horn ; * in Hopkins, * €k>at- and the tone is therefore rather softer than
horn, a unison open metal stop, more conical that of the cylindrical stop, but of very pleas*
than the Spitz-FlOte, 8 feet.' * A member of ing quality ' {ibid, p. 140) TranskUor:\
the Flute-work and met with of 8, 4, or 2 feet t [See supra p. 33d, note.— Tfonslotor.]
length in Continental organs. The pipes of this t [BohrflOie — ' Double Stopped Diapason of
stop are only | the diameter at the top that they metal pipes with chimneys, 16 feet, Beed>flute«
are at the mouth ; and the tone is consequently Metal Stopped Diapason, with reeds, tubes or
light, but very clear and travelling ' (i6td. chimneys, 8 feet. Stoppied Metal Flute, with
p. 140). SpiUflOte — ^* Spire or taper flute, a reeds, tubes or chimneys, 4 feet' (Hopkins,
unison open metal stop formed of pipes with pp. 444, 445)^ — TranslatcrJ]
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CHAP^ V. 6.
MUSICAL TONES OP REED PIPES.
95
6. Musical Tones of Beed Pipes.
The mode of producing the tones on these instruments resembles that used for I
the siren : the passage for the air being alternately closed and opened, its stream is |
separated into a series of individual pulses. This is effected on the siren, as we
have already seen, by means of a rotating disc pierced with holes. In reed infftm-
ments, elastic plates or tongues are employed which are set in vibration and thus
alternately close and open the aperture in which they are fastened. To these
belong —
I. The reed pipes of organs and the vibrators of harmoniums. Their tongues,
sdiewn in perspective in fig. 28, A, and in section in fig. 28, B, are thin oblong
metal plates, z z, fastened
''•*• *•• to a brass block, a a, in
which there is a hole, b b, IT
behind the tongue and of
the same shape. When the
tongue is in its position of
rest, it closes the hole com-
pletely, with the exception
of a very fine chink all round
its margin. When in motion
it oscillates between the po-
sitions marked Z| and z^ in fig. 28, B. In the position z^ there is an aperture for
the stream of air to enter, in the direction shewn by the arrow, and this is closed
when the tongue has reached the other extreme position z^. The tongue shewn /
is a free vibrator or anche libre, such as is now universally employed. These
tongues are slightly smaller than the corresponding opening, so that they can bend
inwards without touching the edges of the hole.^ Formerly, striking vibrators 1[
or reeds were employed, which on each oscillation struck against their fi-ame.
Bat as these produced a harsh quality of tone and an uncertain pitch they have
gone out of use.f
* [The quality of tone prodaoed by the free
reed can be greatly modified by oomparatively
slight changes. If the reed is quite flat, the
end not turning up, as it does in fig. 28, above,
no tone can be produced. If the size of the
slit round the edges be enlarged, by forcing a
thin plate of steel between the spring and tiie
flange, and then withdrawing it, the quality of
tone is permanently changed. Another change
is produced by curving the middle part up and
then down in a curve of contrary flexure.
Another change results from curving the ends
of the reed up as in 'American organs*— a
species of harmonium. One of the earliest free
reed instruments is the Chinese ' shdng,* which
Mr. Hermann Smith thus describes from his
own specimen. See also App. XX. sect. E. 'The
body of the instrument is in the form and size
of a teacup with a tightly fitting cover, pierced
with a series of holes, arranged in a circle, to
reoeive a set of small pipe-like canes, 17 in
number, and of various lengths, of which 13
are capable of sounding and 4 are mute, but
necessary for structure. The lower end of each
pipe is fitted with a little free reed of very
delicate workmanship, about half an indi long,
and stamped in a thin metal plate, having its
tip slightly loaded with beeswax, which is also
Qsed for keeping the reed in position. One
peculiarity to be noticed is that the reed is
quite level with the face of the plate, a condi-
tion in which modern free re^s would not
speak. But this singular provision is made to ,
ensure speaking either by blowing or suction. '
The comers of the reeds are rounded off, and
thus a little space is left between the tip of the
reed and the frame for the passage of air, an
arrangement quite adverse to the speaking of
harmonium reeds. In each pipe the integrity
of the column of air is broken by a hole in
the side, a short distance above the cup. By
this strange contrivance not a single pipe will
sound to the wind blown into the cup from
a flexible tube, until its side hole has been
covered by the finger of the player, and then
the pipe gives a note correspondmg to its full IT
specJcing length. Whatever be the speaking
length of the pipe the hole is placed at a short
distance above the cup. Its position has no
relation to nodal distance, and it effects its
purpose by breaking up the air column and
preventing it from furnishing a proper recipro-
cating relation to the pitch of the reed.* Hie
instrument thus described is the 'sing' of
Barrow {Travels in China, 1S04, where it is
well figured as *a pipe, with unequal reeds
or bamboos'), and 'le petit cheng' of Pdre
Amiot (Mimoires concemani VMstoire . . .
des Chmoia . . ., 1780, vol. vi., where a ' cheng '
of 24 pipes is figured. — Translator,]
f [It will be seen by App. VII. to this
edition, end of sect. A., ih&i Prof. Helmholtz
has somewhat modified his opinion on this
point, in consequence of the information I
Digitized by V^jOOQlC
96
MUSICAL TONES OF REED PIPES.
PART I.
The mode in which tongues are fastened in the reed stops of organs is shewn
in fig. 29, A and B below. A bears a resonant cup above; B is a longitu-
dinal section ; p p is the air chamber
into which the wind is driven ; the
tongue 1 is fastened in the groove r,
which fits into the block s ; d is the
tuning wire, which presses against the
tongue, and being pushed down shortens
it and hence sharpens its pitch, and,
conversely, flattens the pitch when pulled
up. Slight variations of pitch are thus
easily produced.*
I 2 . The tongues of clarinets , oboes , and
^ bassoons are constructed in a somewhat
similar manner and are cut out of elastic
reed plates. The clarinet has a single
wide tongue which is fastened before the
corresponding opening of the mouth-
piece like the metal tongues previously
described, and would strike the frame if
its excursions were long enough. But its
obtained from some of the principal English
organ-boilders, which I here insert from p. 711
of the first edition of this translation: — Mr.
Willis tells me that he never uses free reeds,
that no power can be got from them, and that
he looks upon them as * artificial toys.'
Messrs. J. W. Walker & Sons say that they
m have also never used free reeds for the forty or
'' more years that they have been in business,
and consider that free reeds have been super-
seded by striking reeds. Mr. Thomas Hill
informs me that free reeds had been tried by
his father, by M. Cavaill^-Coll of Paris, and
others, in every imaginable way, for the last
' thirty or forty years, and were abandoned as
' utterly worthless.' But he mentions that
Schulze (of Paulenzelle, Schwartzburg) told
him that he never saw a striking reed till
he came over to England in 1851, and that
Walcker (of Ludwigsburg, Wuertemberg) had
little experience of them, as Mr. Hill learnt
from him about twenty years ago. Mr. Hill
adds, however, that both these builders speedily
abandoned the free reed, after seeing the
English practice of voicing striking reeds.
This is corroborated by Mr. Hermann Smith's
^ statement (1875) t^a* Schulze, in 1862, built
the great organ at Doncaster with 94 stops,
of which only the Trombone and its Octave
had free reeds (see Hopkins on the Organ^
p. 530, for an account of this organ) ; and
that two years ago he built an organ of 64
stops and 4,052 pipes for Sheffield, with not
one free reed; also that Walcker built the
great organ for Ulm cathedral, with 6,500
pipes and 100 stops, of which 34 had reeds,
and out of them only 2 had free reeds ; and
that more recently he built as large a one for
Boston Music Hall, without more free reeds ;
and again that Cavaill6-Coll quite recently
built an organ for Mr. Hopwood of Kensington
of 2,252 pipes and 40 stops, of which only one
— the Musette —had free reeds. He also says
that Lewis, and probably most of the London
organ -builders not previously mentioned, have
never used the free reed. The harshness of the
Fio. 29.
striking reed is obviated in the English method
of voicing, according to Mr. H. Smith, by so
curving and manipulating the metal tongue,
that instead of coming with a discontinooos
* flap ' from the fixed extremity down on to the
sUt of the tube, it 'rolls itself' down, and
hence gradually covers the aperture. The art
of curving the tongue so as to produce this
effect is very difficult to acquire ; it is entirely
empirical, and depends upon the keen eye and
fine touch of the ' artist,' who notes lines and
curves imperceptible to the uninitiated observer,
and foresees their influence on the production
of quality of tone. Consequently, when an
organ-builder has the misfortune to lose his
* reed-voicer,' he has always great difficulty in
replacing him. — Translator.]
* [It should be observed that fig. 29, A,
shews tkfree reed, and fig. 29, B, a striking reed ;
and that the tuning wire is right in fig. 29, B,
because it presses the reed against the edges of
its groove and hence shortens it, but it is wrong
in fig. 29, A, for the reed being free would strike
against the wire and rattle. For free reeds a
clip is used which grasps the reed on both sides
and thus limits its vibrating length.
Fig. 28, p. 956, shews the vibrator of aA
harmonium, not of an organ pipe. The figures
are the same as in all the German editions. —
Translator.]
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OHAP. V. 6.
TONES OF REED PIPES.
97
Pio.
excursions are small, and the pressure of the lips brings it just near enough to
make the chink sufficiently small without allowing it to strike. For the oboe and
bassoon two reeds or tongues of the same kind are placed opposite each other at the
end of the mouthpiece. They are separated by a narrow chink, and by blowing are
pressed near enough to close the chink whenever they swing inwards.
3. Membranous tongues, — The peculiarities of these tongues are best studied
on those artificially constructed. Gut the end of a wooden or gutta-percha tube
obliquely on both sides, as shewn in fig. 30,
leaving two nearly rectangular points standing
between the two edges which are cut obUquely.
Then gently stretch strips of vulcanised india-
rubber over the two obUque edges, so as to leave
a small sUt between them, and fasten them with
a thread. A reed mouthpiece is thus constructed ^
which may be connected in any way with tubes
or other air chambers. When the membranes
bend inwards the slit is closed ; when outwards,
it is open. Membranes which are fastened in |
this obliqiie manner speak much better than those which are laid at right angles >
to the axis of the tube, as Johannes Miiller proposed, for in the latter case they
require to be bent outwards by the air before they can begin to open and shut
alternately. Membranous tongues of the kind proposed may be blown either in
the direction of the arrows or in the opposite direction. In the first case they open
the slit when they move towards the air chamber, that is, towards the further end
of the conducting tube. Tongues of this kind I distinguish as striking inwards, i^
When blown they always give deeper tones than they would do if allowed to
vibrate freely, that is, without being connected with an air chamber. The tongues
of organ pipes, harmoniums, and wooden wind instruments already mentioned, ^
are likewise always arranged to strike inwards. But both membranous and metal
tongues may be arranged so as to act against the stream of aur, and hence to open
when they move towards the outer opening of the instrument. I then say that they
strike (mtwards. The tones of tongues which strike outwards are always sharper ^
than those of isolated tongues.
Only two kinds of membranous tongues have to be considered as musical in-
struments : the human lips in brass instruments, and the human larynx in singing.
The lips must be considered as very shghtly elastic membranous tongues,
loaded with much inelastic tissue containing water, and they would consequently
vibrate very slowly, if they could be brought to vibrate by themselves. In brass
instruments they form membranous tongues which strike outwards, and conse-
quently by the above rule produce tones sharper than their proper tones. But as
they offer very slight resistance, they are readily set in motion, by the alternate
pressure of the vibrating column of air, when used with brass instruments.* m
* [Mr. D. J. Blaikley (manager of Messrs.
fioosey A Ck).*B Military Musioal Instrament
Manoiactory, who has studied all such instm-
ments theoretically as well as practically, and
read many papers npon them, to some of which
I shall have to refer) finds that this statement
does not represent his own sensations when
playing the horn. * The lips,' he says, * do not
▼ibrate throoghoat their whole length, bat only
through a certain length determined by the
diameter of the cap of the mouthpiece. Pro-
bably also the vibrating length can be modified
by the mere pinch, at least this is the sensa-
tion I experience when sounding high notes on a
largemouthpiece. The compass (about 4 octaves)
possible on a given mouthpiece is much greater
than that of any one register of the voice, and
the whole range of brass instruments played
thus with the hps is about one octave greater
than the whole range of the human voioe from
basso profundo to the highest soprano. That
the lips, acting as the vocal chords do, can
themselves vibrate rapidly when supported by
the rim of a mouthpiece, may be proved, for if
such a rim, unconnected witii any resonating
tube, be held against the lips, various notes of
the scale can be produced very faintly, the dif-
ficulty being to maintain steadiness of pitch
{Philos. Mag,, Aug. 1878, p. 2). The qffke of
the air in the tube in relation to the lips (leav-
ing out of consideration its work as a resonant
body, intensifying and modifying the tone) is
to act as a pendulum governor in facilitating
the maintenance (not the origination) of a
Digitized by V^jOOQlC
98 TONES OF EEED PIPES. pakt i.
In tlie larynx, the elastic vocal chords act as membranous tongues. They are
^ stretched across the windpipe, from front to back, like the india-rubber strips in
fig, 30 (p. 97a), and leave a small slit, the glottis, between them. They have the
advantage over all artificially constructed tongues of allowing the width of their slit,
their tension, and even their form to be altered at pleasure with extraordinary
rapidity and certainty, at the same time that the resonant tube formed by the
opening of the mouth admits of much variety of form, so that many more qualities
of tone can be thus produced than on any instrument of artificial construction. If
the vocal chords are examined from above with a laryngoscope, while producing a
tone, they will be seen to make very large vibrations for the deeper breast voice,
shutting the glottis tightly whenever they strike inwards.
The pitch of the various reeds or tongues just mentioned is altered in very
different manners. The metal tongues of the organ and harmonium are always
H intended to produce one single tone apiece. On the motion of these comparatively
heavy and stiff tongues, the pressure of the vibrating air has very small influence,
and their pitch within the instrument is consequently not much different from that
of the isolated tongues. There must be at least one tongue for each note on such
instruments.
In wooden wind instruments, a single tongue has to serve for the whole series
of notes. But the tongues of these instruments are made of light elastic wood,
which is easily set in motion by the alternating pressure of the vibrating column
of air, and swings sympathetically with it. Such instruments, therefore, in
addition to those very high tones, which nearly correspond to the proper tones of
their tongues, can, as theory and experience alike shew, also produce deep tones of
a very different pitch,* because the waves of air which arise in the tube of the in-
strument excite an alternation in the pressure of air adjacent to the tongue itself
sufficiently powerful to make it vibrate sensibly. Now in a vibrating column of
II air the alteration of pressure is greatest where the velocity of the particles of air is
smallest ; and since the velocity is always null, that is a minimum, at the end of a
closed tube, such as a stopped organ pipe, and the alteration of pressure in that
place is consequently a maximum, the tones of these reed pipes must be the same as
those which the resonant tube alone would produce, if it were stopped at the place
where the tongue is placed, and were blown as a stopped pipe. In musical practice,
then, such tones of the instrument as correspond to the proper tones of the tongue
are not used at all, because they are very high and screaming, and their pitch can-
not be preserved with sufficient steadiness when the tongue is wet. The only
tones produced are considerably deeper than the proper tone of the tongue, and
have their pitches determined by the length of the column of air, which corresponds
to the proper tones of the stopped pipe.
The clarinet has a cylindrical tube, the proper tones of which correspond to
the third, fifth, seventh, &c., partial tone of the prime. By altering the style of
% blowing, it is possible to pass from the prime to the Twelfth or the higher major
Third. The acoustic length of the tube may also be altered by opening the side
periodic vibration of the lips. Prof. Helmholtz which he produced a tone of 40 vib., the tone
does not say above what produces the alternate was, even at that depth, remarkably rich and
pressure, and I can conceive no source for it but fine, owing to the large and deep cup extinguiah-
a periodic vibration of the lips of a time suited ing the beating upper partials. Mr. Blaikley
to the particular note required.' The depth of also drew my attention to the fact that where
the cup is also important:—* The shallower and the tube opens out into the cup, there must
more *' cup-like " the cup,' says Mr. Blaikley, be no sharp shoulder, but that the edge must
* the greater the strength of the upper partials. be carefully rounded ofif, otherwise there is a
Compare the deep and narrow cup of the great loss of power to the blower. In the ease
French horn with weak upper partials, and of the French horn the cup is very long and
the wide and shallow cup of the trumpet with almost tapers into the tube.— SVon^^tor.}
strongupperpurtials.'— (MS. communications.) * See Helmholtz, Verhandlungen des na-
Mr. Blaikley kindly sounded for me the same turhisiorischen medidnischen Vereins bii Hei-
instrument with different mouthpieces or cups, delberg^ July 26, 1861, in the Heidelberger
to shew the great difference of quality they Jahrbiicher. Poggendorff's Annalen^ 1861.
produce. In the great bass bombardon on [Reproduced in part in App. VII. sect. B., I.j
Digitized by V^OOQIC
CHAP. V. 6.
TONES OF REED PIPES.
99
holes of the clarinet, in which case the vibrating column of air is principally that
between the mouthpiece and the uppermost open side hole.*
The oboe (hautbois) and bassoon (fagotto) have conical tubes which are closed up
to the vertex of their cone, and have proper tones that are the same as those of
open tubes of the same length. Hence the tones of both of these instruments
nearly correspond to those of open pipes. By overblowing they give the Octave,
Twelfth, second Octave, and so on, of the prime tone. Intermediate tones are
produced by opening side holes.
The older horns and trumpets consist of long conical bent tubes, without keys
or side holes.f They can produce such tones only as correspond to the proper
tones of the tube, and these again are the natural harmonic upper partials of the
prime. But as the prime tone of such a long tube is very deep, the upper partial
tones in the middle parts of the scale lie rather close together, especially in the
extremely long tubes of the hom,t so that they give most of the degrees of the scale. ^
* [Mr. D. J. Blaikley obligingly furnished me
with the Bobstanoe of the following remarks on
clarinets, and repeated his experiments before
me in May 1884. The ordinary form of the
clarinet is not wholly cylindrical. It is slightly
eonstricted at the mouthpiece and provided
with a spreading bell at the other end. The
modification of form by key and finger holes
also must not be neglected. On a cylindrical
pipe played with the lips, the evenly numbered
partials are quite inaudible. When a clarinet
mouthpiece was added I found traces of the
4th and 6th partials beating with my forks.
But on the clarinet with the bell, the 2nd,
4th, and 6th partials were distinct, and I could
obtain beats from them with my forks. Mr.
Blaikley brought them out (i) by bead and
diaphragm resonators tuned to them (fig. 15,
p. 42a), which I also witnessed, (2) by an irre-
gularly-shaped tubular resonator sunk gra-
dually in water, on which I also heard them.
(3) by beats with an harmonium with a con-
stant blast, which I also heard. On the cylin-
drical tube all the unevenly numbered partials
are in tune when played as primes of inde-
pendent harmonic notes. On the clarinet
only the 3rd partial, or 2nd proper tone, can
be used as the prime of an independent har-
monic tone. The 3rd, 4th, and 5th proper
tones of the instrument, are sufficiently near
in pitch to the 5th, 7th, and 9th partials of
the fundamental tone for these latter to be
greatly strengthened by resonance, but the
agreement is not close enough to allow of the
higher proper tones being used as the primes
of independent harmonic compound tones,
fience practically only the 3rd harmonics,
or Twelfths, are used on the clarinet. The
following table of the relative intensity of the m
partials of a Bb clarinet was given by Mr.
Blaikley in the Proc. of ttu Uus. Assn. for
1877-8, p. 84:--
PAKTIilLg— Bb OLAHIHETa
.
Pitch
I
a
3
4
5
6
7
8, Ac.
f
/
^
/
V
mf
P
...
6b
/
/
P
mf
mf
PP
a
/
/
P
mf
...
mf
PP
9
/
8
/
...
mf
mf
P
PP
f
/
•0
/
...
fnf
P
mf
PP
«b
/
t
mf
...
P
P
mf
PP
d
/
^
mi
P
mf
P
P
PP
Where/ means forte, mf mezzoforte,
t [Such brass tubes are first worked unbent
from cylindrical brass tubes, by putting solid
steel cores of the required form inside, and then
drawing them through a hole in a piece of
lead, which yields enough for the tube to pass
through, bat presses the brass firmly enough
against the core to make the tube assume the
proper form. Afterwards the tube is filled
with lead, and then bent into the required coils,
after which the lead is melted out. The in-
ttroments are also not conical in the strict
Bense of the word, but * approximate in form
to the hyperboHe cone, where the axis of the
instrument is an asymptote, and the vertex is
at a great or even an infinite distance from
the bdl end.' From information furnished by
Mr. BUdkley.— 5rV»fwtotor.]
X The tube of the Waldhom [foresthom,
Notes . . . e'b / ^
Just cents . . o, 204, 386,
Harmonic cents . o, 204, 386,
Hannonics, No. . 8, 9, 10,
p piano, |7p pianissimo.— TVanaZator.]
hunting horn of the Germans, answering to
our French horn] is, according to Zamminer
[p. 3 1 2], 13*4 feet long. Its proper prime tone ^
is E,b. This and the next Eb are not used,
but only the other tones, Bb, eb, gr, 6b, i'b-,
«'b, /, gr', a'b + , 6'b, Ac. [Mr. Blaikley
kindly sounded for me the harmonics 8, 9, 10,
II, 12, 13, 14 on an E^b French horn. The
result was almost precisely 320, 360, 400, 440,
480, 520, 560 vib., that is the exact harmonics
for the prime tone 40 vib. to which it was
tuned, the pitch of English military musical
instruments being as nearly as possible cf 269,
efb 3i9'9f a' 452*4. This scale was not com-
pleted because the 15th and i6th harmonics
600 and 640 vib. would have been too high for
me to measure. Expressed in cents we may
compare this scale with just intonation thus :--
a'b 6'b c" d"b d" «"b
498, 702, 884, 996, 1088, 1200
551, 702, 841. 9&9. 1088, 1200 T
II, 12, 13, 14, Digii5^dbyV6lOOyiL
H 2 ^
loo TONES OF REED PIPES. pabt i.
The trumpet is restricted to these natural tones. But by introducing the hand
into the bell of the French horn and thus partly closing it, and by lengthening
the tube of the trombone,* it was possible in some degree to supply the missing
tones and improve the faulty ones. In later times trumpets and horns have been
frequently supplied with keys f to supply the missing tones, but at some expense
of power in the tone and the brilliancy in its quaUty. The vibrations of the air
in these instruments are unusually powerful, and require the resistance of firm,
smooth, unbroken tubes to preserve their strength. In the use of brass inatra-
ments, the different form and tension of the lips of the player act only to determine
which of the proper tones of the tube shall speak ; the pitch of the individual
tones is almost t entirely independent of the tension of the lips.
On the other hand, in the larynx the tension of the vocal chords, which here
form the membranous tongues, is itself variable, and determines the pitch of
^ the tone. The air chambers connected with the larynx are not adapted for
materially altering the tone of the vocal chords. Their walls are so yielding that
they cannot allow the formation of vibrations of the air within them sufficiently
powerful to force the vocal chords to oscillate with a period which is different firom
that required by their own elasticity. The cavity of the mouth is also far too
short, and generally too widely open for its mass of air to have material influence
on the pitch.
In addition to the tension of the vocal chords (which can be increased not
only by separating the points of their insertion in the cartilages of the larynx, but
also by voluntarily stretching the muscular fibres within them), their thickness
seems also to be variable. Much soft watery inelastic tissue lies underneath the
elastic fibrils proper and the muscular fibres of the vocal chords, and in the breast
voice this probably acts to weight them and retard their vibrations. The head
voice is probably produced by drawing aside the mucous coat below the chords,
^ thus rendering the edge of the chords sharper, and the weight of the vibrating
part less, while the elasticity is unaltered.§
Hence the Fourth a'b was 53 cents (33 : 32) trombone can be altered at will, and chosen
too sharp, and the Sixth c" was 43 cents to make its harmonics produce a jnst scale.
(40 : 39) too fiat, and they were conseqaently Some trumpets also are made with a sdiort
unusable without modification by the hand. slide worked by two fingers one way, and
The minor Seventh <2"b was too fiat by 27 cents returning to its position by a spring. Snch
(64 : 63), but unless played in (intended) instruments are sometimes used by first-rate
unison against the just form, it produces a players, such as Harper, the late celebrated
better effect. * In trumpets, strictly so called,' trumpeter, and his son. But, as Mr. Blaiklej
says Mr. Blaikley, * a great portion of the length informed me, an extremely small percentage
is cylindrical and the bell curves out hyper- of the trumpets sold have slides. At ptesent
boliciJly, the two lowest partials are not the piston brass instruments have nearly driven
required as a rule and are not strictly in all slides, except the trombone, out of the field,
tune, so the series of partials may be taken — Translator.]
as about 75, 1-90, 3, 4i 5i 6, 7, 8, Ac, all the f P"*^© ^^7^ *ro nearly obsolete, and have
upper notes being brought into tune by modi- been replaced by pistons which open vahres,
m ficationsintheformof thebellinagoodinstru- and thus temporarily increase the length of
^ ment.' The length of the French horn varies the tube, so as to make the note blown i, 2,
with the * crook * which determines its pitch, or 3 Semitones fiatter. These can also be
The following contains the length in English used in combination, but are then not so trae.
inches for each crook, as given by Mr. Blaikley: This is tantamount to an imperfect slide
Bb (alto) 108, Afi 114J, Ab I2ij^, O I28|, action. Instruments of this kind are now
F 144^, ^H 153* Eb 102, DQ 171^, C 192}, much used in all military bands, and are
Bb (basso) 2i6|, hence the length varies from made of very different sizes and pitohea. —
9 ft. to 18 ft. I inch. By a curious error in Translator.]
all the German editions, Zamminer is said to X (^^^ ^y no means * quite.* It is possible
make the length of the .^b Waldhom 27 feet, to blow out of tune, and to a small extent
or the length of the wave of the lowest note, temper the harmonics. — Translator.]
in place of his 13*4 feet. Zamminer, however, § [On the subject of the registers of the
says that the instrument is named from the human voice and its production generally, see
Octave above the lowest note, and that hence Lennox Browne and Emil Behnke, Voice, Song^
the wave-length of this Octave is the length of and Speech (Sampson Low, London, 1883,
the horn. — Translator.] pp. 322). This work contains not merely
* [A large portion of the trombone is com- accurate drawings of the larynx in the different
posed of a double narrow cylindrical tube on registers, but 4 laryngoscopic photographs
which another slides, so that the length of the from Mr. Behnke's own larynx. A register
Digitized by V^jOOQlC
OHAP. y. 6.
TONES OP REED PIPES.
lOI
We now proceed to investigate the quality of tone produced on reed pipes, !
which is onr proper subject. The sound in these pipes is excited by intermittent
pulses of air, which at each swing break through the opening that is closed by
ihe tongue of the reed. A freely vibrating tongue has far too small a surface to
communicate any appreciable quantity of sonorous motion to the surrounding air ; /
and it is as little able to excite the air inclosed in pipes. The sound seems to be
reaUy produced by pulses of air, as in the siren, where the metal plate that opens
and closes the orifice does not vibrate at all. By the alternate opening and closing
of a passage, a continuous influx of air is changed into a periodic motion, capable
of affecting the air. Like any other periodic motion of the air, the one thus
produced can also be resolved into a series of simple vibrations. We have already
remarked that the number of terms in such a series will increase with the discon-
tinuity of the motion to be thus resolved (p. Z4^)' ^^^ ^^ motion of the air which
passes through a siren, or past a vibrating tongue, is discontinuous in a very high ^
degree, since the individual pulses of air must be generally separated by complete
pauses during the closures of the opening. Free tongues without a resonance
tube, in which all the individual simple tones of the vibration which they excite
in the air are given off freely to the surrounding atmosphere, have consequently
always a very sharp, cutting, jarring quahty of tone, and we can really hear with
either armed or unarmed ears a long series of strong and clear partial tones up
to the i6th or 20th, and there are evidently still higher partials present, although
it is difficult or impossible to distinguish them from each other, because tliey do
not lie so much as a Semitone apart.* This whirring of dissonant partial tones
makes the musical quality of free tongues very disagreeable.f A tone thus pro-
duced also shews that it is really due to puffs of air. I have examined the vibra-
ting tongue of a reed pipe, like that in fig. 28 (p. 956), when in action with the
vibration microscope of Lissajous, in order to determine the vibrational foim of
the tongue, and I found that the tongue performed perfectly regular simple vibra- f
tions. Hence it would communicate to the air merely a simple tone and not a
compound tone, if the sound were directly produced by its own vibrations.
The intensity of the upper partial tones of a free tongue, unconnected with a
resonance tube, and their relation to the prime^ are greatly dependent on the
18 defined as 'a series of tones produced by
the same mechanism * (p. 163). The names of
the registers adopted are those introduced
by the late John Gurwen of the Tonio Sol-fa
movement. They depend on the appearance of
the glottis and vocal chords, and are as follows :
I. Lower thick, 2. Upper thick (both * chest
voice '), 3. Lower thin (' high chest * voice in
men), 4. Upper thin (* falsetto* in women),
5. Small (* head voice ' in women). The extent
of the registers are stated to be (p. 171)
1. lower thick. 3. upper thick. Z. lower thin.
/Mew ^toa, 6 to/, /toe"
lWoMiN« toe', d'to/, /toe"
L lower thick. 3. upper thick. 8. lower thin.
WoaCKK 0»LY,
<i"to/'.
4. upper thin.
/'to/"
A. small.
The mechanism is as follows (pp. 163- 171) : —
1 . Upper thick. The hindmost points of the
pyramids (arytenoid cartilages) close together,
an elliptical slit between the vocal ligaments
(or chords), which vibrate through their whole
length, breadth, and thickness fully, loosely,
and visibly. The lid (epiglottisj is low.
2. Upper thick. The elliptical chink dis-
appears and becomes linear. The lid (epiglottis)
rises ; the vocal ligaments are stretched.
3. Lower thin. The lid (epiglottis) is more
raised, so as to shew the cushion below it, the
whole larynx and the insertions of the vocal
ligaments in the shield (thyroid) cartilage.
The vocal ligaments are quite still, and their
vibrations are confined to the thin inner edges.
The vocal ligaments are made thiwner and
transparent, as shown by illumination from
below. Male voices oease here.
4. Upper thin. An elliptical slit again forms
between the vooal ligaments. When this is
used by men it gives the falsetto arising from
the upper thin being carried below its true
place. This slit is gradually reduced in size
as the contralto and soprano voices ascend. mr
5. Small. The back part of the glottis '
contracts for at least two-thirds of its length,
the vocal ligaments being pressed together so
tightly that scarcely any trace of a slit remains,
and no vibrations are visible. The front part
opens as an oval chink, and the edges of this
vibrate so markedly that the outline is blurred.
The drawings of the two last registers (pp. 168-
169) were made from laryngoscopic examina-
tion of a lady.
Beference should be made to the book
itself for full explanations, and the reader
should especially consult Mr. Behnke*s admir-
able little work The Mechanism of the Human
Voice (Curwen, 3rd ed., 1881, pp. 125). — Trans-
lator,]
* [See footnote f p. S6d^.— Translator,]
f [The cheap little mouth harmonicons ex-
hibit this dleci very well.— IVaYuIa^or.]
Digitized by V^jOOQlC
I02 TONES OF REED PIPES. pabt i.
nature of the tongue, its position with respeot to its frame, the tightness with
which it closes, &c. Striking tongues which produce the most discontinuous pulses
of air, also produce the most cutting quality of tone.* The shorter the puff of air,
and hence the more sudden its action, the greater number of high upper partials
should we expect, exactly as we find in the siren, according to Seebeck's investi-
gations. Hard, unyielding material, like that of brass tongues, will produce
pulses of air which are much more disconnected than those formed by soft and
yielding substances. This is probably the reason why the smging tones of the
human voice are softer than all others which are produced by reed pipes. Never-
theless the number of upper partial tones in the human voice, when used in
emphatic forte, is very great, and they reach distinctly and powerfully up to the
four- times accented [or quarter-foot] Octave (p. 26a). To this we shall have to
return.
% The tones of tongues are essentially changed by the addition of resonance
tubes, because they reinforce and hence give prominence to those upper partial
tones which correspond to the proper tones of these tubes.f In this case the
resonance tubes must be considered as closed at the point where the tongue is
inserted.^:
A brass tongue such as is used in organs, and tuned to h^, was appUed to one
of my larger spherical resonators, also tuned to ^, instead of to its usual resonance
tube. After considerably increasing the pressure of wind in the bellows, the
tongue spoke somewhat flatter than usual, but with an extraordinarily fuU, beautiful,
soft tone, £rom which almost all upper partials were absent. Very httle wind was
used, but it was under high pressure. In this case the prime tone of the compound
was in unison with the resonator, which gave a powerful resonance, and conse-
quently the prime tone had also great power. None of the higher partial tones
could be reinforced. The theory of the vibrations of air in the sphere further
IF shews that the greatest pressure must occur in the sphere at the moment that the
tongue opens. Hence arose the necessity of strong pressure in the bellows to over-
come the increased pressure in the sphere, and yet not much wind really passed.
If instead of a glass sphere, resonant tubes are employed, which admit of a
greater number of proper tones, the resulting musical tones are more complex.
In the clarinet we have a cylindrical tube which by its resonance reinforces tlie
uneven partial tones.§ The conical tubes of the oboe, bassoon, trumpet, and
French horn, on the other hand, reinforce all the harmonic upper partial tones of
the compound up to a certain height, determined by the incapacity of the tubes
to resound for waves of sound that are not much longer than the width of the
opening. By actual trial I found only unevenly numbered partial tones, distinct to
the seventh inclusive, in the notes of the clarinet,§ whereas on other instruments,
which have conical tubes, I found the evenly numbered partials also. I have not yet
had an opportunity of making observations on the further differences of quality in
f the tones of individual instruments with conical tubes. This opens rather a wide
field for research, since the quality of tone is altered in many ways by the style of
blowing, and even on the same instrument the different parts of the scale, when
they require the opening of side holes, shew considerable differences in quality.
On wooden wind instruments these differences are striking. The opening of side
holes is by no means a complete substitute for shortening the tube, and the reflec-
tion of the waves of sound at the points of opening is not the same as at the free
open end of the tube. The upper partials of compound tones produced by a tube
limited by an open side hole, must certainly be in general materially deficient in
harmonic purity, and this will also have a marked influence on their resonance.**
* [Bat see footnote f p. 95(2'.~ TVan^- p. 89, 1. 2, but was cancelled in the 4th
laUyr^ Gennan edition. — TraiMlaJUiT:\
f [A line has been here cancelled in the 1 See Appendix VIL
translation which had been accidentally left | [But see note * p. 996.— Tmnslo/or/
standing in the Grerman, as it refers to a re- ^* [The theory of side holes is ezoessiveljr
mark on the passage which formerly followed complicated and has not been as yet worked
Digitized by V^OOQIC
CHAP. Y. 7- yOWEL QUALITIES OF TONE. 103
7. Vowel Qualities of Totie.
We have hitherto discussed cases of resonance, generated in such air chambers
as were capable of reinforcing the prime tone principally, but also a certain
(number of the harmonic upper partial tones of the compound tone produced. The
case, however, may also occur in which the lowest tone of the resonance chamber
applied does not correspond with the prime, but only with some one of the upper
partials of the compound tone itself, and in these cases we find, in accordance with
the principles hitherto developed, that the corresponding upper partial tone is
really more reinforced than the prime or other partials by the resonance of the
chamber, and consequently predominates extremely over all the other partials in
the series. The quahty of tone thus produced has consequently a peculiar cha-
racter, and more or less resembles one of the vowels of the human voice. For the j
vowels of speech are in reality tones produced by membranous tongues (the vocal Hj
chords), with a resonance chamber (the mouth) capable of altering in length, jf '
width, and pitch of resonance, and hence capable also of reinforcing at different f
times different partials of the compound tone to which it is applied.*
In ocder to understand the composition of vowel tones, we must in the first
place bear in mind that the source of their sound lies in the vocal chords, and
that when the voice is heard, these chords act as membranous tongues, and like
all tongues produce a series of decidedly discontinuous and sharply separated
pulses of air, which, on being represented as a sum of simple vibrations, must
consist of a very large number of them, and hence be received by the ear as a very
long series of partials belonging to a compound musical tone. With the assistance •
of resonators it is possible to recognise very high partials, up to the sixteenth,
when one of the brighter vowels is sung by a powerful bass voice at a low pitch,
and, in the case of a strained forte in the upper notes of any human voice, we can
hear, more clearly than on any other musical instrument, those high upper partials V
that belong to the middle of the four-times accented Octave (the highest on
modem pianofortes, see note, p. i8c2), and these high tones have a peculiar relation
to the ear, to be subsequently considered. The loudness of such upper partials,
especially those of highest pitch, differs considerably in different individuals. For
catting bright voices it is greater than for soft and dull ones. The quality of tone
in cutting screaming voices may perhaps be referred to a want of sufficient
smoothness or straightness in the edges of the vocal chords, to enable them to
close in a straight narrow slit without striking one another. This circumstance
would give the larynx more the character of striking tongues, and the latter have
a much more cutting quality than the free tongues of the normal vocal chords.
Hoarseness in voices may arise from the glottis not entirely closing during the
vibrations of the vocal chords. At any rate, when alterations of this kind are
made in artificial membranous tongues, similar results ensue. For a strong and
yet soft quality of voice it is necessary that the vocal chorda should, even when H
most strongly vibrating, join rectilinearly at the moment of approach with perfect
tightness, effectually closing the glottis for the moment, but without overlapping
oat Bcientifically. * The general principles,* edited with additional letters by W. S. Broad-
writes Mr. Blaikley, * are not difiicolt of com- wood, and pablished by Budall, Oarte, A Co.,
prehension ; the difficulty is to determine qaan- makers of his flutes, tiee also Victor Mahillon,
titatively the values in each p|articular case/ 6tude sur le doigti de la FlAte Boehm^ 1882,
The paper by Schafhautl (writing under the and a paper by M. Aristide Cavaill6-Goll, in
name of Pellisov), 'Theorie gedeckter oylin- I/'&^ Afttf icaZ for 1 1 Jan. 1883 —Tran«2ator.]
drischer und conischer Pfeifen und der Quer- * The theory of vowel tones' was first enun
lldten,* Schweiger, Joum, Ixviii. 1833, is dis- elated by Wheatstone in a criticism, unfortu-
figured by misprints so that the formula are nately little known, on Willises experiments,
unintelligible, and the theory is also extremely The latter are described in the Transactions
hazardous. JBut they are the only papers I of the Cambridqe Philosophical Society, vol.
have found, and are referred to by Theobald iii. p. 231, and Poggendorff^s Annaleyi der
. Boehm, Veber den Fldtenbau, Mains, 1847. Physik, vol. xxiv. p. 397. Wheatstone's re-
An English version of this, by himself, made port upon them is contained in the London
ior Mr. Budall in 1847, b&s recently been and }yestmin8i4^ lieview for October 1S37.
Digitized by V^OOQ IC
I04 VOWEL QUALITIES OP TONE. pabt i.
or striking against each other. If they do not close perfectly, the stream of air
ivill not be completely interrupted, and the tone cannot be powerful. If they
overlap, the tone must be cutting, as before remarked, as those arising from,
striking tongues. On examining the vocal chords in action by means of a
laryngoscope, it is marvellous to observe the accuracy with which they close even
when making vibrations occupying nearly the entire breadth of the chords them-
selves.*
There is also a certain difference in the way of putting on the voice in speak-
ing and in singing, which gives the speaking voice a much more cutting quality
of tone, especially in the open vowels, and occasions a sensation of much greater
pressure in the larynx. I suspect that in speaking the vocal chords act as striking
tongues.t
When the mucous membrane of the larynx is affected with catarrh, the
f laryngoscope sometimes shews little flakes of mucus in the glottis. When these
are too great they disturb the motion of the vibrating chords and make them irre-
gular, causing the tone to become unequal, jarring, or hoarse. It is, however, re-
markable what comparatively large flakes of mucus may lie in the glottis withoujt
producing a very striking deterioration in the quality of tone.
It has already been mentioned that it is generally more difficult for the un-
assisted ear to recognise the upper partials in the human voice, than in the tones
of musical instruments. Besonators are more necessary for this examination
than for the analysis of any other kind of musical tone. The upper partials of the
human voice have nevertheless been heard at times by attentive observers. Bameau
had heard them at the beginning of last century. And at a later period 8eiler of
Leipzig relates that while listening to the chant of the watchman during a sleepless
night, he occasionally heard at first, when the watchman was at a distance, the
Twelfth of the melody, and afterwards the prime tone. The reason of this difiiculty
% is most probably that we have all our lives remarked and observed the tones of
the human voice more than any other, and always with the sole object of grasping
it as a whole and obtaining a clear knowledge and perception of its manifold changes
of quality.
We may certainly assume that in the tones of the human larynx, as in all
other reed instruments, the upper partial tones would decrease in force as they
increase in pitch, if they could be observed without the resonance of the cavity of
the mouth. In reality they satisfy this assumption tolerably well, for those vowels
iriiich are spoken with a wide funnel-shaped cavity of the mouth, as A [a in art], or
jL[ain bat lengthened, which is nearly the same as a in bare]. But this relation is
materially altered by the resonance which takes place in the cavity of the mouth.
The more this cavity is narrowed, either by the lips or the tongue, the more dis-
tinctly marked is its resonance for tones of determinate pitch, and the more there-
fore does this resonance reinforce those partials in the compound tone produced by
% the vocal chords, which approach the favoured pitch, and the more, on the contrary,
win the others be damped. Hence on investigating the compound tones of the
human voice by means of resonators, we find pretty uniformly that the first six to
eight partials are clearly perceptible, but with very different degrees of force accord-
ing to the different forms of the cavity of the mouth, sometimes screaming loudly
into the ear, at others scarcely audible.
Under these circumstances the investigation of the resonance of the cavity of
the mouth is of great importance. The easiest and surest method of finding the
tones to which the air in the oral cavity is tuned for the different shapes it assumes
* [Probably these observations were made f [The German habit of
on the * upper thick * register, because the vowels with the ' check * or Arabic
chords are then more visible. It is evident which is very marked, and instantly charac-
that these theories do not apply to the lower terises his nationality, is probably what is
thick, upper thin, and small registers, and here alluded to, as occasioning a sensation of
scarcely to the lower thin, as described above, much greater pressure. This does not apply
footnote p. loic— Translator.] in the least to English speakers.— -Traf»2ator^
Digitized by V^jOOQlC
CHAP. V. 7*
VOWEL QUALITIES OF TONE.
"5
in the production of vowels, is that which is used for glass bottles and other spaces
fiUed with air. That is, tuning-forks of different pitches have to be struck and
held before the opening of the air chamber — ^in the present case the open mouth
— and the louder the proper tone of the fork is heard, the nearer does it corre-
spond with one of the proper tones of the included mass of air.* Since the shape
of the oral cavity can be altered at pleasure, it can always be made to suit the
tone of any given tuning-fork, and we thus easily discover what shape the mouih
must assume for its included, mass of air to be tuned to a determinate pitch.
Having a series of tuning-forks at command, I was thus able to obtain the
following results : —
The pitch of strongest resonance of the oral cavity depends solely upon the
vowel for pronouncing which the mouth has been arranged, and alters considerably
for even slight alterations in the vowel quality, such, for example, as occur in the
different dialects of the same language. On the other hand, the proper tones of f
the cavity of the mouth are nearly independent of age and sex. I have in general
fonnd the same resonances in men, women, and children. The want of space in
the oral cavity of women and children can be easily replaced by a great closure of its
opening, which will make the resonance as deep as in the larger oral cavities of men.f
The vowels can be arranged in three series, according to the position of the
parts of the mouth, which may be written thus, in accordance with Du Bois-
Beymond the elder J : —
E I
tJ
U
The vowel A [a in father, or Scotch a in man\ forms the common origin of
all three series. With this vowel corresponds a funnel-shaped resonance cavity, 1
* [See note * p. 876, on determining violin
resonaaoe. One diffioodty in the ease of the
mouth is that there is a constant tendency to
varythe shape of the oral cavity. Another, as
shewn at the end of the note cited, is that
the same irregular cavity, such as that of the
znouth, often more or less reinforces a large
numbor of different tones. As it was impor-
tant for my phonetic researches, I have niade
many attempts to determine mv own vowel
resonances, but have hitherto failed in all my
attempts. —Translaior.}
t [Easily tried by more or less covering
the top of a tumbler with the hand, till it
resounds to any fork from d to d" or higher.
—Translaior.]
t Norddeuische ZeitBchrift, edited by de
la Motte Fouqu6, 181 2. Kadmus oder cUlge-
fneine Alphabetik, von F. H. du Bois-Beymond,
Berlin, 1862, p. 152. [This is the arrange-
ment usually adopted. But in 1867 Mr.
Melville Bell, an orthoepical teacher of many
years* standing, who had been led profession-
aUy to pay great attention to the shapes of the
mouth necessary to produce certain sounds, in
his Visible Speech; the Science of Universal
Alphabetics (London: Simpkin, Marshall &
Co., 4to., pp. X. 126, with sixteen lithographic
tables), proposed a more elaborate method of
classifying vowels by the shape of the mouth.
He conmienced with 9 positions of the tongue,
consisting of 3 in which the middle, or as he
terms it, * front ' of the tongue was raised,
highest for ea in seat^ not so l]dgh for a in sate,
and lowest for a in sat\ 3 others in which the
p' back, instead of the middle, of the tongue
was raised, highest for 00 in snood, lower for o
in node, and lowest for aw in gna/wed (noiie of
which three are determined by the position of
the tongue alone), and 3 intermediate positimis,
where the whole tongue is raised almost evenly
at three different elevations. These 9 Ungual
positions might be accompanied with the
ordinary or with increased distension of the
pharynx, giving 9 primary and 9 *wide'
vowels. And each of the 18 vowels, thus
produced, could be * rounded,* that is, modified
by shading the mouth in various degrees with
the lips. He thus obtains 36 distinct vowel
cavities, among which almost all those used
for vowel qualities in different nations may be
placed. Subsequent research has shewn how
to extend this arrangement materially. See ^
my Early English Pronunciation, part iv.,
1874, p. 1279. Also see generally my Pro-
nunciationfor Singers (Curwen, 1877, pp. 246)
and Speech in Song (NoveUo, 1878, pp. 140).
German vowels differ materially in qualit7
from the English, and consequently complete
agreement between Prof. Helmholtz's obser-
vations and those of any Englishman, who
repeats his experiments, must not be expected.
I have consequently thought it better in this
place to leave his German notation untrans-
lated, and merely subjoin in parentheses the
nearest English sounds. For the table in the
text we may assume A to » a in faXher, or else
Scotch a in man (different sounds), E to *- erin
there, I to » i in machine, O to » o in more, U
to -* tt in sure ; and 0 to » sii in French peu
or else in peuple (different sounds), and tf to
« tt in French pu,— Translator.]
Digitized by V^jOOQlC
xo6 VOWEL QUALITIES OF TONE. pabt i,
enlargmg with tolerable uniformity from the larynx to the lips. For the vowels of
the lower series, 0 [o in more] and U [oo in poor], the opening of the mouth is
contracted by means of the Hps, more for U than for 0, while the cavity is enlarged
as much as possible by depression of the tongue, so that on the whole it becomes
like a bottle without a neck, with rather a narrow mouth, and a single unbroken
cavity.* The pitch of such a bottle-shaped chamber is lower the larger its cavity
and the narrower its mouth. Usually only one upper partial with strong resonance
can be clearly recognised ; when other proper tones exist they are comparatively
very high, or have only weak resonance. In conformity with these results, obtained
with glass bottles, we find that for a very deep hollow U [oo in poor nearly], where
the oral cavity is widest and the mouth narrowest, the resonance is deepest and
answers to the unaccented/. On passing from U to 0 [o in more nearly] the
resonance gradually rises ; and for a full, ringing, pure 0 the pitch is 6^« The
m position of the mouth for 0 is peculiarly favourable for resonance, the opening of the
mouth being neither too large nor too small, and the internal cavity sufficiently
spacious. Hence if a b'\} tuning-fork be struck and held before the mouth while 0
is gently uttered, or the O-position merely assumed without really speaking, the tone
of the fork will resound so fully and loudly that a large audience can hear it. The
usual a' tuning-fork of musicians may also be used for this purpose, but then it will be
necessary to make a somewhat duller O, if w^e wish to bring out the full resonance.
On gradually bringing the shape of the mouth from the position proper to 0,
through those due to 0* [nearly o in cot, with rather more of the 0 sound], and A**
[nearly au in caught, with rather more of the A sound] into that for A [Scotch a
in man, with rather more of an 0 quahty in it than English a in father], the
resonance gradually rises an Octave, and reaches h'*\}. This tone corresponds with
the North German A ; the somewhat brighter A [a in father] of the English and
Itahans rises up to d'*\ or a major Third higher. It is particularly remarkable what
„ little differences in pitch correspond to very sensible varieties of vowel quality in
the neighbourhood of A ; and I should therefore recommend philologists who wish
to define the vowels of different languages to fix them by the pitch of loudest
resonance.t
For the vowels already mentioned I have not been able to detect any second
proper tone, and the analogy of the phenomena presented by ai'tificial resonance
chambers of similar shapes would hardly lead us to expect any of sensible loudness.
* [This depressed position of the tongne able to discriminate vowel sounds, is frequently
answers better for English aw in saw than for not acute for differences of pitch. The deter-
either o in more or oo in poor. For the o the mination of the pitch even under favour-
tongue is slightly more raised, especially at the able circumstances is not easy, especially, as it
back, while for English oo the back of the will be seen, for the higher pitches. Without
tongue is almost as high as for k, and greatly mechanical appliances even good ears are
impedes the oral cavity. If, however, the deceived in the Octave. The differences of
longue be kept in the position for aw by sound- pitch noted by Helmholtz, Donders, Merkel,
^ ing this vowel, and, while sounding it steadily, and Koenig, as given on p. I09d, probably point
'< the lips be gradually contracted, the sound to fundamental differences of pronunciation,
will be found to pass through certain obscure and shew the desirability of a very extensive
qualities of tone till it suddenly comes out series of experiments being carried out with
clearly as a sound a little more like aw than o special apparatus, by an operator with an
in more (really the Danish aa), and then again extremely acute musical ear, on speakers of
passing through other obscure phases, comes various nationalities and also on various
out again clearly as a deep sound, not so bright speakers of the same nationaUty. Chreat difli-
as our 00 in poor, but more resembling the culty will even then be experienced on account
Swedish o to which it will reach if the tongue of the variability of the same speaker in his
be slightly raised into the A position. It is vowel quality for differences of pitch and
necessary to bear these facts in mind when expression, the want of habit to maintain the
following the text, where U is only almost, not position of the mouth unmoved for a sufficient
quite = oo in poor, which is the long sound of u length of time to complete an observation
in pull^ and is duller than oo in pool or French satisfactorily, and, worst of all, the involuntary
ou in poule.— Translator.] tendency of the organs to accommodate them-
f [Great difficulties lie in the way of carry- selves to the pitch of the fork presented. Com-
ing out this recommendation. The ear of pare note * p. 105c. — Translator.]
philologists and even of those who are readily
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CHAP. V. 7. VOWEL QUALITIES OF TONK 107
Experiments hereafter desoribed shew that the resonance of this single tone is
sufficient to characterise the vowels above mentioned.
The second series of vowels consists of A, A, E, I. The lips are drawn so far
apart that they no longer contract the issuing stream of air, but a firesh constric-
tion is formed between the front (middle) parts of the tongue and the hard palate,
the space immediately above the larynx being widened by depressing the root of
the tongue, and hence causing the larynx to rise simultaneously. The form of the
oral cavity consequently resembles a bottle with a narrow neck. The belly of the
bottle is behind, in the pharynx, and its neck is the narrow passage between the
upper surface of the tongue and the hard palate. In the above series of letters,
A, E, I, these changes increase until for I the internal cavity of the bottle is greatest
and the neck narrowest. For A [the broadest French ij broader than e in therCj
and nearly as broad as a in bat lengthened, with which the name of their city is
pronounced by the natives of Bath], the whole channel is, however, tolerably wide, IF
80 that it is quite easy to see down to the larynx when the laryngoscope is used.
Indeed this vowel gives the very best position of the mouth for the application of
this instrument, because the root of the tongue, which impedes the view when A
is uttered, is depressed, and the observer can see over and past it.
When a bottle with a long narrow neck is used as a resonance chamber, two
simple tones are readily discovered, of which one can be regarded as the proper
tone of the belly, and the other as that of the neck of the bottle. Of course the
air in the belly cannot vibrate quite independently of that in the neck, and both
proper tones in question must consequently be different, and indeed somewhat
deeper than they would be if belly and neck were separate and had their resonance
examined independently. The neck is approximately a short pipe open at both
ends. To be sure, its inner end debouches into the cavity of the bottle instead 0£
the open air, but if the neck is very narrow, and the belly of the bottle very wide,
the latter may be looked upon in some respect as an open space with regard to the %
vibrations of the air inclosed in the neck. These conditions are best satisfied for
I, in which the length of the channel between tongue and palate, measured from
the upper teeth to the back edge of the bony palate, is about 6 centimetres [2*36
inches]. An open pipe of this length when blown would give e""^ while the
observations made for determining the tone of loudest resonance for I gives nearly
d'^^', which is as close an agreement as could possibly have been expected in such
an irregularly shaped pipe as that formed by the tongue and palate.
In accordance with these experiments the vowels A, E, I have each a higher
and a deeper resonance tone. The higher tones continue the ascending series of
the proper tones of the vowels U, 0, A. By means of tuning-forks I found for A
a tone between g'^' and a'"\}, and for E the tone &'''[>• ^ ^^ ^o ^^^^ suitable for
I, but by means of the whistling noise of the air, to be considered presently
Cp, 108ft), its proper tone was determined with tolerable exactness to be d'"'.
The deeper proper tones which are due to the back part of the oral cavity are %
rather more difficult to discover. Tuning-forks may be used, but the resonance is
comparatively weak, because it must be conducted through the long narrow neck
of the air chamber. It must further be remembered that this resonance only
occurs during the time that the corresponding vowel is gently whispered, and dis-
appears as soon as the whisper ceases, because the form of the chamber on which
the resonance depends then immediately changes. The tuning-forks after being
struck must be brought as close as possible to the opening of the air chamber
which lies behind the upper teeth. By this means I found d" for A and/ for E.
For I, direct observation with tuning-forks was not possible ; but from the upper
partial tones, I conclude that its proper tone is as deep as that of U, or near/.
Hence, when we pass from A to I, these deeper proper tones of the oral cavity sink,
and the higher ones rise in pitch.*
♦ [Mr. Graham Bell, the inventor of the mentioned (p. I05<Z, note), was in the habit of
Telephone, son of the Mr. Melville Bell already bringing out this fact by placing his mouth in
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to8 VOWEL QUALITIES OF TONE. pabt i.
For the third series of vowels from A through 0 [French eu in peu, or the
deeper eu in peuple], towards tJ [French u in pu, which is rather deeper than the
German sound], we have the same internal positions of the mouth as in the last-
named series of vowels. For tl the mouth is placed in nearly the same position
as for a vowel lying between E and I, and for 0 as for an E which inclines towards
A. In addition to the constriction between the tongue and palate as in the second
series, we have also a constriction of the lips, which are made into a sort of tube,
forming a front prolongation of that made by the tongue and palate. The air
chamber of the mouth, therefore, in this case also resembles a bottle with a neck,
but the neck is longer than for the second series of vowels. For I the neck was
6 centimetres (2*36 inches) long, for t7, measured from the front edge of the upper
teeth to the commencement of the soft palate, it is 8 centimetres (3*15 inches).
The pitch of the higher proper tone corresponding to the resonance of the neck
% must be, therefore, about a Fourth deeper than for I. If both ends were free, a pipe
of this length would give b'", according to the usual calculation. In reahty it
resounded for a fork lying between ^'" and a"^, a divergence similar to that
found for I, and also probably attributable to the back end of the tube debouching
into a wider but not quite open space. The resonance of the back space has to be
observed in the same way as for the I series. For 0 it is /, the same as for E,
and for O it is/, the same as for I.
The fact that the cavity of the mouth for different vowels is tuned to different
pitches was first discovered by Bonders,* not with the help of tuning-forks, but by
the whistling noise produced in the mouth by whispering. The cavity of the
mouth thus reinforces by its resonance the corresponding tones of the windrush,
which are produced partly in the contracted glottis,t and partly in the forward
contracted passages of the mouth. In this way it is not usual to obtain a complete
musical tone ; this only happens, without sensible change of the vowel, for tJ and
f U, when a real whistle is produced. This, however, would be a fault in speaking.
We have rather only such a degree of reinforcement of the noise of the air as
occurs in an organ pipe, which does not speak well, either from a badly-constructed
lip or an insufficient pressure of wind. A noise of this kind, although not brought
up to being a complete musical tone, has nevertheless a tolerably determinate
pitch, which can be estimated by a practised ear. But, as in all cases where tones
of very different qualities have to be compared, it is easy to make a mistake in the
Octave. However, after some of the important pitches have been determined by
the required positions and then tapping against Chr. Hellwag, De FormaUone Loqudae DisM,^
a finger placed just in front of the upper teeth, Tubingtiet 17 10. — Fldrcke, Neus Berlitur
for the higher resonance, and placed against Monatsschrift, Sept. 1803, Feb. 1804. — Olivier
the neck, just above the larynx, for the lower. Ortho - epo - graphischea Elementar - Werk^
He obligingly performed the experiment several 1S04, part iii. p. 21.
times privately before me, and the successive t In whispering, the vocal chords are kept
alterations and differences in their direction close, but the air passes through a small
m were striking. The tone was dull and like triangular opening at the back part of the
a wood hannonica. Considerable dexterity glottis between the arytenoid cartilages. [Ac-
seemed necessary to produce the effect, and I cording to Gzermak (SiUungiberichte^ Wiener
could not succeed in doing so. He carried out Akad., Math.-Naturw. 01. April 29, 1858,
the experiment much further than is suggested p. 576) the vocal chords as seen through tiie
in the text, embracing the whole nine positions laryngoscope are not quite close for whisper,
of the tongue in his father's vowel scheme, but are nicked in the middle. Merkel (Die
and obtaining a double resonance in each case. Funktionen des menschlichen Schlund- und
This fact is stated, and the various vowel Kehlkopfea, . . . nach eigenen pharyngo- und
theories appreciated in Mr. Graham Bell's laryngoakopischen Untersuchungent Leipzig,
paper on * Vowel Theories * read before the 1862, p. 77) distinguishes two kinds of whisper-
American National Academy of Arts and ing: (i) the loud, in which the opening between
Sciences, April 15, 1879, &nd printed in the the chords is from | to ) of a line wide, pro-
Ameriean Journal of Otology , vol. i. July duoing no resonant viorations, and that between
1879. — Translator.] the arytenoids is somewhat wider; (2) the
* Arehiv fUr die Holldndischen BeitrOge gentle, in which the vowel is commenced as in
fUr Natur- tmd HeUkunde von Bonders und loud speaking, with dosed glottis, and, after it
Berlin, vol. i. p. 157. Older incomplete obser- has begun, the back part of the glottis is
vations of the same circumstance in Samuel opened, while the chords remain close and
Beyher's Matliesis Mosaica, Kiel, 161 9.^ motionless.— TramlatorJ]
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CHAP. V. 7.
VOWEL QUALITIES OF TONE.
109
tuning-forks, and others, as tJ and 0, by allowing the whisper to pass into a
regular whistle, the rest are easily determined by arranging them in a melodio
progression with the first. Thus the series : —
aearA
A
B
I
[a in father]
[a in ma€]
\t in ikwe\
[t in machine]
forms an ascending minor chord of g in the second Inversion J, [with the Fifth in
the bass,] and can be readily compared with the same melodic progression on the
pianoforte. I was able to determine the pitch for clear A, A, and E by tuning-
forks, and hence to fix that for I also.*
* The statements of Bonders differ slightly
from mine, partly because they have reference
to Dutch pronunciation, while mine refer to the
North German vowels; and partly because
Bonders, not having been assisted by tuning- «r
forks, was not always able to determine with
certainty to what Octave the noises he heard
should be assigned.
Pitch accord.
Fitch aooord-
Vowel
inirto
Inifto
Dondora
Helmholts
U
r
/
0
d'
6'b
A
6'b
6"b
0
fl'?
c'"J
u
a"
fif^'toa'^b
E
c'"J
6'"b
I
/'"
d""
[The extreme divergence of results obtained
by different investigators shews the inherent
^^oultias of the determination, which (as
already indicated) arise partly from different
values attributed to the vowels, partly from the
difficulty of retaining the form of the mouth
steadily for a suflcient time, partly from the
wide range of tones which the same cavity of
the mouth will more or less reinforce, partly
from the dif&culty of judging of absolute pitch
in general, and especially of the absolute pitch
of a scarcely musical whisper, and other causes.
In C. L. MerkePs Physioloaie der mensch-
lichen Sprache (Leipzig, 1866), p. 47, a table is
given of the results of Beyher, Hellwag,
Fl&rcke, and Bonders (the latter differing ma-
terially from that just given by Prof. Helm-
hoHz), and on MerkePs p. 109, he adds his last
results. These are reproduced in the following
table with the notes, and their pitch to the f
nearest vibration, taking a' 440, and supposing
equal temperament. To these I add the re*
suits of Bonders, as just given, and of Helm-
holtz, both with pitches similarly assumed.
Koenig {Comptee HendtUy April 25, 1870) also
gives his pitches with exact numbers, reckoEed
as Octaves of the 7th harmonic of c' 256, and
hence called 6)>, although they are nearer the
a of this standard. Reference should also be
made to Br. Eoenig*s paper on ' Manometrio
Flames* translated in the Philosophical Maga-
tine, 1873, vol. xlv. pp. 1-18, 105-114. Lastly,
Br. Moritz Trautmann (Anglia, vol. i. p. 590)
very confidently gives results utterly different
from all the above, which I subjoin with the
pitch as before. I give the general form of
TABI.K OF YOWRL BeAONANCKS.
Obserrtr.
V
0
A
A 1 B
I
V
0
I. Reyher .
c 131
flWi56
a 990 )
c'a62r
dIiS6
/'349
c"sa3
2. Hellwag
cist
CI139
aaao
6247
0^262
6b 233
j;r92o8
3. Flttrcke .
C131
^196
c'a6a
^^392
d'440
e"S23
^^392
•'SSo
4, Bonden ac- .
cording to i
Helmholts .[
/'349
<f 294
6'b 466
/"i«;68
c'TT 1109
/^i397
a" 880
^196?
+drs^
S. Bonders ac-x
cording to I
Merkel . )
y65 J
t;.7i75
ei6s
6047
c's6a
/"698
0*440
^196
6. Helmiioltz.
6'b 466
6"b93«
^'"1568
6*" 1976
if"' 2349
^1568
cTt 1 109
Ou,/'349
+f 587
Jt-C,349
V^
+/I7S
■t/'349
7. Merkel . .
di47
0*,yi96
A\/I320
^"587
or 0' 440
ly^^^^
a' 440
/»370
A', 6 247
E',«^659
or<f 294
8.KoenSg, 7th
harmonics .
6b 234
6'b 448
6''b896
i,T^J792
ft'"b3584
0. Trautmann .
regs
O\c'"io47
/'" 1397
-F?
EV 1760:/"" 2794
6-1976
*,iriS68
I
O',a"88o
E',c""2093l
iO'/i'"i76o|
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no
VOWEL QUALITIES OP TONE,
PABT I.
For U it is also by no means easy to find the pitch of the resonance by a fork,
as the smaUness of the opening makes the resonance weak. Another phenomenon
has guided me in this ease. If I sing the scale from c upwards, uttering the vowel
U for each note, and taking care to keep the quality of the vowel correct, and not
allowing it to pass into 0,* I feel the agitation of the air in the mouth, and even
en the drums of both ears, where it excites a tickling sensation, most powerfully
when the voice reaches /. As soon as / is passed the quality changes, the strong^
agitation of the air in the mouth and the tickling in the ears cease. For the note
/ the phenomenon in this case is the same as if a spherical resonance chamber
were placed before a tongue of nearly the same pitch as its proper tone. In this
case also we have a powerful agitation of the air within the sphere and a sudden
alteration of quality of tone, on passing from a deeper pitch of the mass of air
through that of the tongue to a higher. The resonance of the mouth for U is thus
^ fixed at/ with more certainty than by means of tuning-forks. But we often meet
with a U of higher resonance, more resembling 0, which I will represent by the
French Ou. Its proper tone may rise as high as /.f The resonance of the
cavity of the mouth for different vowels may then be expressed in notes as follows :
S j; h*0 h'h
d" f
E^*^
f gfe
'% 9"
f
:tr— tr=
U Ou
0
1 E
f
I ±:
0 n
^ The mode in which the resonance of the cavity of the mouth acts upod the
quality of the voice, is then precisely the same as that which we discovered to
exist for artificially constructed reed pipes. All those partial tones are reinforced
which coincide with a proper tone of the cavity of the mouth, or have a pitch
sufficiently near to that of such a tone, while the other partial tones will be more
or less damped. The damping of those partial tones which are not strengthened
is the more striking the narrower the opening of the mouth, either between the
lips as for U, or between the tongue and palate as for I and IT.
These differences in the partial tones of the different vowel sounds can be easily
and clearly recognised by means of resonators, at least within the once and twice
accented Octaves [264 to 1056 vib.] . For example, apply a 6't> resonator to the
ear, and get a bass voice, that can preserve pitch well and form its vowels with
purity, to sing the series of vowels to one of the harmonic under tones of 6% such
as b\}, e\}, P|>, G\}i E\}. It will be found that for a pure, full-toned 0 the &'(> of
f the resonator will bray violently into the ear. The same upper partial tone is
still very powerful for a clear A and a tone intermediate between A and 0, but is
weaker for A, E, 0, and weakest of all for U and I. It will also be found that
the resonance of 0 is materially weakened if it is taken too duU, approaching U»
the vowel at the head of each column, and
when the writer distinguishes different forms
I add them immediately before the resonance
note. Thus we have Helmholtz*8 Ou between
U and O ; MerkePs O between O and A, his
obscure A\ K' and clear A', E' ; Trautmann*s
O' = Italian open O, and (as he says) English
a in all (which is, however, slightly different),
O' ordinary o in Berliner ohne, E' Berlin
Schnee, E' French pire (the same as X ?), 0'
Berlin schOn, French peu, & French leur. Of
course this is far from exhausting the list of
vowels in actual use.— Tmnalator.l
* [That is, according to the previous direc-
tions, to keep the tongue altogether depressed,
in the position for aw in gnaw, which is not
natural for an Englishman, so that for English
00 in ^ we may expect the result to be ma-
terially different.— 7*ran«Zator.]
t [Prof. Helmholtz may mean the Swedish
o, see note * p. io6d. The following words im-
mediately preceding the notes, which oooar
in the 3rd (German edition, appear to have
been accidentally omitted in the 4th. They
are, however, retained as they seem neoessaiy.
— Translator.}
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CHAP. V. 7. VOWEL QUALITIES OF TONE. iii
or too open, becoming A". But if the b"\} resonator be used, which is an Octave
higher, it is the vowel A that excites the strongest sympathetic resonance ; while
O, which was so powerful with the b]} resonator, now produces only a slight effect.
For the high upper partials of A, E, I, no resonators can be made which are
capable of sensibly reinforcing them. We are, then, driven principally to observa-
tions made with the unassisted ear. It has cost me much trouble to determine these
strengthened partial tones in the vowels, and I was not acquainted with them when
my previous accounts were published.* They are best observed in high notes of
women's voices, or the falsetto of men's voices. The upper partials of high notes
in that part of the scale are not so nearly of the same pitch as those of deeper notes,
^ and heneer they are more readily distinguished. On b]), for example, women's
voices could easily bring out all the vowels, with a full quality of tone, but at
higher pitches the choice is more limited. When b^ is sung, then, the Twelfth/"
is heard for the broad A, the double Octave 6'"b for E, the high Third ^i"" for I, f
*all clearly, the last even piercingly. [See table on p. 124, note.] f
Further, I should observe, that the table of notes given on the preceding page,
relates only to those kinds of vowels which i^pear to me to have the most cha-
racteristic quality of tone, but that in addition to these, all intermediate stages
ure possible, passing insensibly from one to the other, and are actually used partly
in dialects, partly by particular individuals, partly in peculiar pitches while singing,
or to give a more decided character while whispering.
It is easy to recognise, and indeed is sufficiently well known, that the vowels
with a single resonance from U through 0 to clear A can be altered in continuous
succession. But I wish further to remark, since doubts have been thrown on the
deep resonance I have assigned to U, that when I apply to my ear a resonator
tuned to /, and, singing upon f or B\} b,9 the fundamental tone, try to find the
vowel resembling U which has the strongest resonance, it does not answer to a
dull U, but to a U on the way to O.t %
Then again transitions are possible between the vowels of the A — 0 — U series
and those of the A — 0 — tT series, as well as between the last named and those of
the A — E— I series. I can begin on the position for U, and gradually transform
the cavity of the mouth, already narrowed, into the tube-like forms for 0 and Q,
in which case the high resonance becomes more distinct and at the same time
higher, the narrower the tube is made. If we make this transition while applying
a resonator between b]} and b']} to the ear, we hear the loudness of the tone
increase at a certain stage of the transition, and then diminish again. The higher
the resonator, the nearer must the vowel approach to 0 or tT. With a proper
position of the mouth the reinforced tone may be brought up to a whistle. Also
in a gentle whisper, where the rustle of the air in the larynx is kept very weak, so
that with vowels having a narrow opening of the mouth it can be scarcely heard, a
strong fricative noise in the opening of the mouth is often required to make the
vowel audible. That is to say, we then make the vowels more like their related ^
consonants, Enghsh W and German J [English Y].
Generally speaking the vowels § with double resonance admit of numerous
modifications, because any high pitch of one of the resonances may combine with
any low pitch of the other. This is best studied by applying a resonator to the
ear and trying to find the corresponding vowel degrees in the three series which
reinforce its tone, and then endeavouring to pass from one of these to the other in
such a way that the resonator should have a reinforced tone throughout.
♦ Gelekrte Ameigen der Bayerischen % [An U Bound verging towards O is gene-
Akademieder Wisseruchaften, June 18, 1859. rally conceived to be dfiiZ/cr, not brighter ^ by
f [The passage * In these experiments * English writers, but here V is taken as the
to ' too deep to be sensible,' pp. 166-7 of the dullest vowel. This remark is made merely
1st English edition, is here cancelled, and to prevent confusion with English readers. —
p. 1 1 1&, * Further, I should observe,' to p. i i6a, Translator J]
' high tones of A, E, I,' inserted in its place § [Misprinted Consonanten in the German,
from the 4th German edition.— TratwZa/or.] —Translator.'] '
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112 VOWEL QUALITIES OF TONE. pabt i.
Thus the resonator V\} answers to 0, to an Ao and to an E which resembles A,
and these sounds may pass continuously one into the other.
The resonator / answers to the transition Ou — 0 — E. The resonator d" to
Oa — ^Ad — A,. In a similar manner each of the higher tones may be connected
with various deeper tones. Thus assuming a position of the mouth which would
give e^'^ for whistling, we can, without changing*the pitch of the fricative sound in
the mouth, whisper a vowel inclining to 0 or inclining to tT, by allowing the
fricative sound in the larynx to have a higher or deeper resonance in the back part
of the mouth.*
In comparing the strength of the upper partials of different vowels by means of
resonators, it is further to be remembered, that the reinforcement by means of the
resonance of the mouth affects the prime tone of the note produced by the voice,
as well as the upper partials. And as it is especially the vibrations of the prime,
f which by their reaction on the vocal chords retain these in regular vibratory motion,
the voice speaks much more powerfully, when the prime itself receives such a
reinforcement. This is especially observable in those parts of the scale which
the singer reaches with difficulty. It may also be noted with reed pipes having
metal tongues. When a resonance pipe is applied to them tuned to the tone of tl^e
tongue, or a little higher, extraordinarily powerful and rich tones are produced, by
means of strong pressure but little wind, and the tongue oscillates in large ex-
cursions either way. The pitch of a metal tongue becomes a little flatter than
before. This is not perceived with the human voice because the singer is able to
regulate the tension of the vocal chords accordingly. Thus I find distinctly that
at b]}, the extremity of my falsetto voice, I can sing powerfully an 0, an A, and an
A on the way to 0, which have their resonance at this pitch, whereas U, if it is
not made to come very near 0, and I, are dull and uncertain, with the expenditure
of more air than in the former case. Begard must be had to this circumstance in
f experiments on the strength of upper partials, because those of a vowel which speaks
powerfully, may become proportionally too powerful, as compared with those of a
vowel which speaks weakly. Thus I have found that the high tones of the soprano
voice which lie in the reinforcing region of the vowel A at the upper extremity of the
doubly-accented [or one-foot] Octave, when sung to the vowel A, exhibit their
higher Octave more strongly than is the case for the vowels E and I, which do not
speak so well although the latter have their strong resonance at the upper end of
the thrice-accented [or six-inch] Octave.
It has been already remarked (p. 39c) that the strength and amplitude of
sympathetic vibration is affected by the mass and boundaries of the body which
vibrates sympathetically. A body of considerable mass which can perform its
vibrations as much as possible without any hindrance from neighbouring bodies,
and has not its motion damped by the internal friction of its parts, after it has
once been excited, can continue to vibrate for a long time, and consequently, if it
f has to be set in the highest degree of sympathetic vibration, the oscillations of the
exciting tone must, for a comparatively long time, coincide with those proper
vibrations excited in itself. That is to say, the highest degree of sympathetic
resonance can be produced only by using tones which he within very narrow limits
of pitch. This is the case with tuning-forks and bells. The mass of air in the
cavity of the mouth, on the other hand, has but slight density and mass, its walls,
so far as they are composed of soft parts, are not capable of offering much resist-
ance, are imperfectly elastic, and when put in vibration have much internal friction
to stop their motion. Moreover the vibrating mass of air in the cavity of the
mouth conmiunicates through the orifice of the mouth with the outer air, to whicli
it^rapidly gives off large parts of the motion it has received. For this reason a
* This appears to me to meet the objec- my attention to the habit of using sach devia.
tions whioh were made by Herr G. Engel, in tions from the usual qualities of vowels in
Beichart's and Du Bois-Reymond's Archiv., syllables which are briefly uttered.
1869, pp. 317-319* Herr J. Stockhausen drew
Digitized by
Google
CHAP. V. 7. MODIFICATIONS OP VOWEL QUALITIES. 113
vibratory motion once excited in the air filling the cavity of the mouth is very
rapidly extinguished, as any one may easily observe by filliping his cheek with a
finger when the mouth is put into different vowel positions. We thus very easily
distinguish the pitch of the resonance for the various transitional degrees from 0
towards U in one direction and towards A in the other. But the tone dies away
rapidly. The various resonances of the cavity of the mouth can also be made
audible by rapping the teeth. Just for this reason a tone, which oscillates approxi-
mately in agreement with the few vibrations of such a brief resonance tone, will be
reinforced by sympathetic vibration to an extent not much less than another tone
which exactly coincides with the first ; and the raoge of tones which can thus
be sensibly reinforced by a given position of the mouth, is rather considerable.*
This is confirmed by experiment. When I apply a b]} resonator to the right,
and an /' resonator to the left ear and sing the vowel 0 on ^, I find a reinforce-
ment not only of the 4th partial b]} which answers to the proper tone of the ^
cavity of the mouth, but also, very perceptibly, though considerably less, of /',
the 6th partial, also. If I then change 0 into an A, until /' finds its strongest
resonance, the reinforcement of b^} does not entirely disappear although it becomes
much less.
The position of the mouth from 0 to 0^ appears to be that which is most
favourable for the length of its proper tone and the production of a resonance
limited to a very narrow range of pitch. At least, as I have before remarked, for
this position the reinforcement of a suitable tuning-fork is most powerful, and tap-
ping the cheek or the lips gives the most distinct tone. If then for 0 the rein-
forcement by resonance extends to the interval of a Fifth, the intervals will be stiU
greater for the other vowels. With this agree experiments. Apply any resonator
to the ear, take a suitable under tone, sing the different vowels to this under tone, and
let one vowel melt into another. The greatest reinforcements by resonance take
place on that vowel or those vowels, for which one of the characteristic tones in f
the diagram p. 1006 coincides with the proper tone of the resonator. But more or
less considerable reinforcement is also observed for such vowels as have their charac-
teristic tones at moderate differences of pitch from the proper tone of the resonator,
and the reinforcement will be. less the greater these differences of pitch.
By this means it becomes possible in general to distinguish the vowels from
each other even when the note to which they are sung is not precisely one of the
harmonic under tones of the vowels. From the second partial tone onwards, the
intervals are narrow enough for one or two of the partials to be distinctly reinforced
by the resonance of the mouth. It is only when the proper tone of the cavity of
the mouth falls midway between the prime tone of the note sung by the voice and
its higher Octave, or is more than a Fifth deeper than that prime tone, that the
characteristic resonance will be weak.
Now in speaking, both sexes choose one of the deepest positions of their voice.
Men generally choose the upper half of the great (or eight-foot) Octave ; and If
women the upper half of the smaU (or four-foot) Octave.f With the exception of
U, which admits of fluctuations in its proper tone of nearly an Octave, all these
pitches of the speaking voice have the corresponding proper tones of the cavity
of the mouth situated within sufficiently narrow intervals from the upper partials of
the speaking tone to create sensible resonance of one or more of these partials,
and thus characterise the vowel.t To this must be added that the speaking voice,
probably through great pressure of the vocal hgaments upon one another, converting
* On this subject see Appendix X., and of certain of its partials with exact pitches
the corresponding investigation in the text in but in their coming near enough to thc^
Part I. Chap. VI. therein referred to. pitches to receive reinforcement, and that the
t [That is both use their * lower thick ' character of a vowel quality of tone, like that
register, as described in the note p. loid^ but of all qualities of tone, depends not on the
are an Octave apart.— Translator.] absolute pitch, but on the relative force of the
t [Observe here that the quality of the upper partials. As Prof. Helmholtz's theory
vowel tone is not made to consist in the identity has often been grievously misunderstood, I
Digitized by VjOt)Ql€
114 MODIFICATIONS OF VOWEL QUALITIES. parti.
them into striking reedi, has a jarring quality of tone, that is, possesses stronger
upper partials than the singing voice.
In singing, on the other hand, especially at higher pitches, conditions are less
favourable for the characterisation of vowels. Every one knows that it is generally
much more difficult to understand words when sung than when spoken, and that
the difficulty is less with male than with female voices, each having been equally well
cultivated. Were it otherwise, ' books of the words ' at operas and concerts would
be unnecessary. Above /, the characterisation of U becomes imperfect even if it
is closely assimilated to 0. But so long as it remains the only vowel of indetermi-
nate sound, and the remainder allow of sensible reinforcement of their upper partials
in certain regions, this negative character wiU distinguish U. On the other hand
a soprano voice in the neighbourhood of/' should not be able to clearly distinguish
U, 0, and A ; and this agrees with my own experience. On singing the three vowels
^ in ixomediate succession, the resonance /'' for A will, however, be still somewhat
clearer in the cavity of the mouth when tuned for b"\}j than when it is tuned to b^^
for 0. The soprano voice will in this case be able to make the A clearer, by eleva-
ting the pitch of the cavity of the mouth towards t2''' and thus making it approach
to/''. The 0, on the other hand, can be separated from U by approaching 0«, and
giving the prime more decisive force. Nevertheless these vowels, if not sung in
immediate succession, will not be very clearly distinguished by a listener who is
unacquainted with the mode of pronouncing the vowels that the soprano singer
uses.*
A further means of helping to discriminate vowels, moreover, is found in com-
mencing them powerfully. This depends upon a general relation in bodies excited
to sympathetic vibration. Thus, if we excite sympathetic vibration in a suitable
body with a tone somewhat different from its proper tone, by commencing it suddenly
with great power, we hear at first, in addition to the exciting tone which is rein-
^ forced by resonance, the proper tone of the sympathetically vibrating body.f Bat
the latter soon dies away, while the first remains. In the case of tuning-forks with
laige resonator, we can even hear beats between the two tones. Apply a b');} resonator
to the ear, and commence singing the vowel 0 powerfully on g, of which the upper
partials g^ and d" have only a weak lasting resonance in the cavity of the mouth*
and you may hear immediately at the commencement of the vowel, a short sharp
beat between the b^ of the cavity of the mouth and of the resonator. On selecting
another vowel, this 51> vanishes, which shews that the pitch of the cavity of the
mouth helps to generate it. In this case then also the sudden commencement of
the tones g^ and df^ belonging to the compound tone of the voice, excites the inter-
mediate proper tone &t> of the cavity of the mouth, which rapidly hdea. The
same thing may be observed for other pitches of the resonator used, when we sing
notes, powerfully commenced, which have upper partials that are not reinforced by
the resonator, provided that a vowel is chosen with a characteristic pitch which
answers to the pitch of the resonator. Hence it results that when any vowel in
any pitch is powerfully commenced, its characteristic tone becomes audible as a
1 short beat. By this means the vowel may be distinctly characterised at the
jmoment of commencement, even when it becomes intermediate on long con-
. tinuance. But for this purpose, as already remarked, an exact and energetic com-
mencement is necessary. How much such a commencement assists in rendering
the words of a singer intelligible is well known. For this reason also the vocal-
isation of the briefly uttered words of a reciting parlando, is more distinct than
that of sustained song4
draw particular attention to the point in this may make in the vowels in English, German,
place. See also the table which I have added French and Italian, at different pitches, so as
in a footnote on p. i24d,— Translator.'] to remain intelligible.— TrntuZotor.]
♦ [In my Pronunciation for Singers (Cur- f See the mathematical statement of this pro-
wen, 1877), and my Speech in Song (Novello, cess in App. IX., remarks on equations 4 to 46.
1S78) I have endeavoured to give a popular I The facts here adduced meet, I think, the
explanation cf the alterations which a singer objections brought against my vowel theory by
Digitized by V^OOQIC
CHAP. V. 7. CHARACTERISTICS OF VOWELS. 115
Moreover vowels admit of other kinds of alterations in their qnalities of tone,
conditioned by alterations of their characteristic tones within certain limits. Thus
the resonating capability of the cavity of the month may undergo in general altera-
tions in strength and definition, which would render the character of the various
vowels and their difference from one another in general more or less conspicuous
or obscure. Flaccid sofi; walls in any passage with sonorous masses of air, are
generally prejudicial to the force of the vibrations. Partly too much of the motion
is given off to the outside through the soft masses, partly too much is destroyed by
friction within them. Wooden organ pipes have a less energetic quality of tone
than metal ones, and those of pasteboard a still duller quality, even when the
mouthpiece remains unaltered. The walls of the human throat, and the cheeks,
are, however, much more yielding than pasteboard. Hence if the tone of the voice
with all its partials is to meet with a powerful resonance and come out unweakened,
these most flaccid parts of the passage for our voice, must be as much as possible f
thrown out of action, or else rendered elastic by tension, and in addition the passage
must be made as short and wide as possible. The last is effected by raising the
larynx. The sofli wall of the cheeks can be almost entirely avoided, by taking care
that the rows of teeth are not too far apart. The lips, when their co-operation is
not necessary, as it is for 0 and tT, may be held so far apart that the sharp firm
edges of the teeth define the orifice of the mouth. For A the angles of the mouth
can be drawn entirely aside. For 0 they can be firmly stretched by the muscles
above and below them (levator anguli oris and triangukms menti), which then feel
like stretched cords to the touch, and can be thus pressed against the teeth, so that
this part of the margin of the orifice of the mouth is also made sharp and capable
of resisting.
In the attempt to produce a clear energetic tone of the voice we also become
aware of the tension of a large number of muscles lying in front of the throat,
both those which he between the under jaw and the tongue-bone and help to form ^
the floor of the cavity of the mouth {mylohyoideu8y geniohyoideus, and perhaps
also hiventer), and likewise those which run down near the larynx and air tubes, and
draw down the tongue-bone (st&mokyoidetis, stemothyroideus and thyrohyoidetts).
Without the coxmteraction of the latter, indeed, considerable tension of the former
would be impossible. Besides this a contraction of the skin on both sides of the
larynx which takes place at the commencement of the tone of the voice, shews that
the omohyoideus muscle, which runs obliquely down from the tongue-bone back-
wards to the shoulder-blade, is also stretched. Without its co-operation the muscles
arising from the under jaw and breast-bone would draw the larynx too far forwards.
Now the greater part of these muscles do not go to the larynx at all, but only to
the tongue-bone, from which the larynx is suspended. Hence they cannot directly
assist in the formation of the voice, so far as this depends upon the action of the
larynx. The action of these muscles, so far as I have been able to observe it on
myself, is also much less when I utter a dull guttural A, than when I endeavour to ^
change it into a ringing, keen and powerfully penetrating A. Ringing and keen,
applied to a quality of tone, imply many and powerful upper partials, and the
stronger they are, of course the more marked are the differences of the vowels
which their own differences condition. A singer, or a declaimer, will occasionally
interpose among his bright and rich tones others of a duller character as a contrast.
Sharp characterisation of vowel quality is suitable for energetic, joyful or vigorous
frames of mind ; indifferent and obscure quality of tone for sad and troubled, or taci-
turn states. In the latter case speakers like to change the proper tone of the vowels,
by drawing the extremes closer to a middle Ad (say the short German E [the final
HeiT E. y. Qnanten (Poggendorff*s Annal., article, pp. 724-741, with espeolal reference to
▼oL cUt. pp. 272 and 522), so far as they do not it. In oonseqaence of the new matter added
rest upon misconceptions. [In the ist edition by Prof. HehxLholtz in his 4th German edition
of this translation, during the printing of which here followed, this article is omitted from the
V. Quanten*8 first paper appeared, I added an present edition. — Translator.] j
Digitized by V^OQ^lC
ii6 VOWEL QUALITIES OF TONE. parti.
English obscare A in idea]), and heiice select somewhat deeper tones in place of the
high tones of A, E, I.
A pecuhar circumstance must also be mentioned which distinguishes the
human voice from all other instruments and has a peculiar relation to the human
ear. Above the higher reinforced partial tones of I, in the neighbourhood of e"''
up to g'^" [2640 to 3168 vib.] the notes of a pianoforte have a peculiar cutting
effect, and we might be easily led to believe that the hammers were too hard, or
that their mechanism somewhat differed from that of the adjacent notes. But the
phenomenon is the same on all pianofortes, and if a very small glass tube or sphere
is applied to the ear, the cutting effect ceases, and these notes become as soft and weak
as the rest, but another and deeper series of notes now becomes stronger and more
cutting. Hence it foUows that the human ear by its own resonance favours the tones
between c"" and ^"", or, in other words, that it is timed to one of these pitches.*
^ These notes produce a feeling of pain in sensitive ears. Hence the upper partial
tones which have nearly this pitch, if any such exist, are extremely prominent
and affect the ear powerfully. This is generally the case for the human voice when
it is strained, and will help to give it a screaming effect. In powerful male voices
singing forte, these partial tones sound Uke a clear tinkling of little bells, accom-
panying the voice, and are most audible in choruses, when the singers shout a
little. Every individual male voice at such pitches produces dissonant upper partials.
When basses sing their high e\ the 7th partial tonef is d"", the 8th e"", the
9th/'"#, and the loth g'"% Now, if e"" and/'"# are loud, and d"" and sr""Jf,
though weaker, are audible, there is of course a sharp dissonance. If many voices
are sounding together, producing these upper partials with small differences of
pitch, the result is a very peculiar kind of tinkling, which is readily recognised a
second time when attention has been once drawn to it. I have not noticed any
difference of effect for different vowels in this case, but the tinkling ceases as soon
^ as the voices are taken piano; although the tone produced by a chorus will of
course still have considerable power. This kind of tinkling is peculiar to htmian
voices ; orchestral instruments do not produce it in the same way either so sensibly
or so powerfully. I have never heard it from any other musical instrument so
clearly as from human voices.
The same upper partials are heard also in soprano voices when they sing forte ;
in harsh, uncertain voices they are tremulous, and hence shew some resemblance
to the tinkling heard in the notes of male voices. But I have heard them brought
out with exact purity, and continue to sound on perfectly and quietly, in some
steady and harmonious female voices, and also in some excellent tenor voices. In
the melodic progression of a voice part, I then hear these high upper partials of
the four-times accented Octave, falling and rising at different times within the
compass of a minor Third, according as different upper partials of the notes snng
enter the region for which our ear is so sensitive. It is certainly remarkable that
^ it should be precisely the human voice which is so rich in those upper partials for
which the human ear is so sensitive. Madame E. Seiler, however, remarks that
dogs are also very sensitive for the high e"" of the violin.
This reinforcement of the upper partial tones belonging to the middle of the
four-times accented Octave, has, however, nothing to do with the characterisation
of vowels. I have mentioned it here, merely because these high tones are readily
remarked in investigations into the vowel qualities of tone, and the observer must
not be misled to consider them as peculiar characteristics of individual vowels.
They are simply a characteristic of strained voices.
The humming tone heard when singing with closed mouth, Ues nearest to U.
* I have lately found that my right ear is applying a short paper tuhe to the entrance of
most sensitive for /"", and my left for c"". my ear, this chirp is rendered extraordinarily
When I drive air into the passage leading to the weak.
tympanmn, the resonance descends to c""Z and f [The first six partial tones are «', c", 6",
g'"U . The chirp of the cricket corresponds pre- e"', g'^'U , &"', the seventh is 27 cents flatter than
ciaely to the higher resonance, and on merely d'"*,— Translator.]
Digitized by VjOOQIC
CHAP. v. 7.
VOWEL QUALITIES OP TONE.
117
This hum is ased in uttering the conson^its M, N and N^. The size of the exit
of the air (the nostrils) is in this case much smaller in comparison with the
resonant chamber (the internal nasal cavity) than the opening of the lips for U in
comparison with the corresponding resonant chamber in the mouth. Hence, in
humming, the peculiarities of the U tone are much enhanced. Thus although
upper partials are present, even up to a considerably high pitch, yet they decrease
in strength as they rise in pitch much faster than for U. The upper Octave is
tolerably strong in humming, but all the higher partial tones are weak. Humming
in the N-position differs a little horn that in the M-position, by having its upper
partials less damped than for M. But it is only at the instant when the cavity of
the mouth is opened or closed that a clear difference exists between these conso-
nants. We cannot enter into the details of the composition of the sound of the
other consonants, because they produce noises which have no constant pitch, and
are not musical tones, to which we have here to confine our attention. f
The theory of vowel sounds here explained may be confirmed by experiments
with artificial reed pipes, to which proper resonant chambers are attached. This
was first done by Willis, who attached reed pipes to cylindrical chambers of variable
length, and produced different tones by increasing the length of the resonant tube.
The shortest tubes gave him I, and then E, A, 0, up to U, until the tube exceeded
the length of a quarter of a wave. On further increasing the length the vowels
returned in converse order. His determination of the pitch of the resonant pipes
agrees well with mine for the deeper vowels. The pitch foxmd by WilUs for the
higher vowels was relatively too high, because in this case the length of the wave
was smaller than the diameter of the tubes, and consequently the usual calcula-
tion of pitch firom the length of the tubes alone was no longer applicable. The
vowels E and I were also far from accurately resembling those of the voice, because
the second resonance was absent, and hence, as WiUis himself states, they could
not be well distinguished.* f
Vowel
In the Word
Pitch,
Willis
Pitch,
Helmholtx
Length of Tube
in Inches
0
No
c"
c"
47
A»
Nought
e"b
e"b
3-8
Paw
9"
f
305
A
Part ^
d'"b
d"'b
2*2
Pad
r
1-8
£
Pay ^
d""
h"'b
i-o
Pet
*■ fjntf
d'"
0-6
I
See
r'
d""
038 (?)
The vowels are obtained much more clearly and distinctly with properly tuned
resonators, than with cylindrical resonance chambers. On applying to a reed pipe
which gave i|>, a glass resonator tuned to l^, I obtained the vowel U ; changing H
the resonator to one tuned for h*);}, I obtained 0 ; the h"\} resonator gave a rather
close A, and the d'" resonator a clear A. Hence by tuning the applied chambers
in the same way we obtain the same vowels quite independently of the form of the
chamber and nature of its walls. I also succeeded in producing various grades of
* [Probably the first treatise on phonology
in which Willises experiments were given at
length, and the above table cited, with Wheat-
stone's article from the London and Westmin-
ster Bemew^ which was kindly brought under
my notice by Sir Charles Wheatstone himself,
was my Alphabet of Nature j London , 1 845. The
table includes U exemplified by but^ boot^ with
an indefinite length of pipe. The word pad is
misprinted paa in all the Oerman editions of
Helmholtz (even the 4th, which appeared after
the correction in my translation), and as he
therefore could not separate its A from that in
part, he gives no pitch. It is really the nearest
English representative of the German . The
sounds in noughtt paw, which Sir John Her-
schel, when citing Willis (Art. * Sound,' in
Erusyc. Metropol., par. 375), could not distin-
guish, were probably meant for the broad
Italian open O, or English o in more, and the
English aw in maw respectively. The length
of the pipe in inches is here added from Willis's
paper. I have heard Willis's pxperiments
repeated by Wheatstone. — jTraiw/ator.l
Digitized by V^jOOQlC'
ii8 VOWEL QUALITIES OF TONE. part !•
A, 0, E, and I with the same reed pipe, by applying glass spheres into whose external
opening glass tubes were inserted from 6 to lo centimetres (2*36 to 3*94 inches) in
length, in order to imitate the doable resonance of the oral cavity for these
vowels.
Willis has also given another interesting method for producing vowels. If a
toothed wheel, with many teeth, revolve rapidly, and a spring be applied to its
teeth, the spring wiU be raised by each tooth as it passes, and a tone will be pro-
duced having its pitch number equal to the number of teeth by which it has been
struck in a second. Now if one end of the spring is well £a>stened, and the spring
be set in vibration, it will itself produce a tone which will increase in pitch as the
spring diminishes in length. If then we turn the wheel with a constant velocity,
and allow a watch spring of variable length to strike against its teeth, we shall
obtain for a long spring a quality of tone resembling U, and as we shorten the
f spring other qualities in succession like 0, A, E, I, the tone of the spring here
playing the part of the reinforced tone which determines the vowel. But this
imitation of the vowels is certainly much less complete than that obtaaned by reed
pipes. The reason of this process also evidently depends upon our produciag
compound tones in which certain upper partials (which in this case correspond with
the proper tones of the spring itse^) are more reinforced than others.
WiUis himself advanced a theory concerning the nature of vowel tones which
differs from that I have laid down in agreement with the whole connection of all
other acoustical phenomena. Willis imagines that the pulses of air which produce
the vowel qualities, are themselves tones which rapidly die away, corresponding to
the proper tone of the spring in his last experiment, or the short echo produced by
a pulse or a little explosion of air in the mouth, or in the resonance chamber of a
reed pipe. In fact something like the sound of a vowel will be heard if we only
tap against the teeth with a little rod, and set the cavity of the mouth in the posi-
f tion required for the different vowels. Willis's description of the motion of sound
for vowels is certainly not a great way from the truth ; but it only assigns the
mode in which the motion of the air ensues, and not the corresponding reaction
which this produces in the ear. That this kind of motion as well as all others
is actually resolved by the ear into a series of partial tones, according to the laws
of sympathetic resonance, is shewn by the agreement of the analysis of vowel
qualities of tone made by the unarmed ear and by the resonators. This will
appear still more clearly in the next chapter, where experiments will be described
shewing the direct composition of vowel qualities from their partial tones.
Vowel qualities of tone consequently are essentially distinguished from the
tones of most other musical instruments by the fact that the loudness of their
partial tones does not depend solely upon their numerical order but preponder-
antly upon the absolute pitch of those partials. Thus when I sing the vowel A to
the note ^,* the reinforced tone b"\} is the 12th partial of the compound tone ;
% and when I sing the same vowel A to the note 6^, the reinforced tone is still &'^,
but is now the 2nd partial of the compound tone sung.f
From the examples adduced to shew the dependence of quality of tone from
the mode in which a musical tone is compounded, we may deduce the following
general rules : —
1. Simple Tones f like those of tuning-forks applied to resonance chambers and
wide stopped organ pipes, have a very soft, pleasant sound, free from aU roughness,
but wanting in power, and dull at low pitches.
2. Musical Tones, which are accompanied by a moderately loud series of the
* [Eb has for 2nd partial «b, for 3rd 6b, t [See App. XX. sec. M. No. i, for Jen-
and hence for 6th b'b, and for 12th, b"b. — kin and Ewing^s analysis of vowel sounds by
TransUUor.] means of the Phonograph,— rransfattor.]
Digitized by VjOOQlC
CHAPB.v.vi. APPREHENSION OF QUALITIES OF TONE. 119
lower partial tones, up to about the sixth partial, are more harmonious and
musical. Compared with simple tones they are rich and splendid, while they are
at the same time perfectly sweet and soft if the higher upper partials are absent.
To these belong the musical tones produced by the pianoforte, open organ pipes,
the softer piano tones of the human voice and of the French horn. The last-
namad tones form the transition to musical tones with high upper partials ; while
the tones of flutes, and of pipes on the flue-stops of organs with a low pressure
of wind, approach to simple tones.
3. If only the unevenly numbered partials are present (as in narrow stopped
organ pipes, pianoforte strings struck in their middle points, and clarinets), the
quality of tone is hollow, and, when a large number of such upper partials are
present, ncuai. When the prime tone predominates the quality of tone is rich ;
but when the prime tone is not sufficiently superior in strength to the upper
partialfl, the quality of tone is poor. Thus the quality of tone in the wider open ^
organ pipes is richer than that in the narrower ; i^trings struck with pianoforte
hammers give tones of a richer quality than when struck by a stick or plucked
by the finger ; the tones of reed pipes with suitable resonance chambers have a
richer quality than those without resonance chambers.
4* When partial tones higher than the sixth or seventh are very distinct, the
quality of tone is cutting and rough. The reason for this will be seen hereafter to
lie in the dissonances which they form with one another. The degree of harshness
may be very different. When their force is inconsiderable the higher upper partials
do not essentially detract from the musical applicability of the compound tones ;
on the contrary, they are useful in giving character and expression to the music.
The most important musical tones of this description are those of bowed instru-
ments and of most reed pipes, oboe (hautbois), bassoon (febgotto), harmonium, and
the human voice. The rough, braying tones of brass instruments are extremely
penetrating, and hence are better adapted to give the impression of great power ^
than similar tones of a softer quality. They are consequently little suitable for
artistic music when used alone, but produce great effect in an orchestra. Why
high dissonant upper partials should make a musical tone more penetrating will
appear hereafter.
CHAPTER VI.
ON THE APPBEHENSION OF QUALITIES OF TONE.
Up to this point we have not endeavoured to analyse given musical tones further
than to determine the differences in the number and loudness of their partial tones.
Before we can determine the function of the ear in apprehending qualities of tone, T
we must inquire whether a determinate relative strength of the upper partials
suffices to give us the impression of a determinate musical quality of tone or
whether there are not also other perceptible differences in qualil^ which are
independent of such a relation. Since we deal only with musical tones, that is
with such as are produced by exactly periodic motions of the air, and exclude all
irregular motions of the air which appear as noises, we can give this question a
more definite form. If we suppose the motion of the air corresponding to the
given musical tone to be resolved into a sum of pendular vibrations of air, such
individual pendular vibrations wiU not only differ from each other in force or
amplitude for different forms of the compound motion, but also in their relative
position, or, according to physical terminology, in their difference of phase. For
example, if we superimpose the two pendular vibrational curves A and B, fig. 31
(p. i2oa), first with the point e of B on the point do of A, and next with the point
e of B on the point d* of A^ we obtain the two entirely distinct vibrational curves
Digitized by V^OOQIC
I20
DOES QUALITY DEPEND ON PHASE?
PART I.
C and D. By farther displacement of the initial point e so as to place it on d^ or
d, we obtain other forms, which are the inversions of the forms C and D, as has
been already shewn (supra, p. 32a). If, then, musical quality of tone depends solely
on the relative force of the partial tones, aU the various motions C, D, &c., must
Fig. 31.
B
IT
make the same impression on the ear. But if the relative position of the two
^ waves, that is the difference of phase, produces any effect, they must make different
impressions on the ear.
Now to determine this point it was necessary to compoxmd various musical
tones out of simple tones artificially, and to see whether an alteration of quality
ensued when force was constant but phase varied. Simple tones of great purity,
which can have both their force and phase exactly regulated, are best obtained
from tuning-forks having the lowest proper tone reinforced, as has been already
described (p. 54^^), by a resonance chamber, and communicated to the air. To set
the tuning-forks in very uniform motion, they were placed between the limbs of a
little electro-magnet, as shewn in fig. 32, opposite. Each tuning-fork was screwed
mto a separate board d d, which rested upon pieces of india-rubber tubing e e that
were cemented below it, to prevent the vibrations of the fork from being directly
communicated to the table and hence becoming audible. The limbs b b of the
electro-magnet are surrounded with wire, and its pole f is directed to the fork.
f There are two clamp screws g on the board d d which are in conductive connection
with the coils of the electro-magnet, and serve to introduce other wires which
conduct the electric current. To set the forks in strong vibration the strength of
these streams must alternate periodically. These are generated by a separate
apparatus to be presently described (fig. 33, p. 1226, c).
When forks thus arranged are set in vibration, very little indeed of their tone
is heard, because they have so little means of communicating their vibrations to
the surrounding air or adjacent soHds. To make the tone strongly audible, the
resonance chamber i, which has been previously tuned to the pitch of the fork,
must be brought near it. This resonance chamber is fastened to another board k,
which slides in a proper groove made in the board d d, and thus allows its opening
to be brought very near to the fork. In the figure the resonance chamber is shewn
at a distance from the fork in order to exhibit the separate parts distinctly ; when
in use, it is brought as close as possible to the fork. The mouth of the resonance
chamber can be closed by a hd 1 attached to a lever m. By pulling tlie string n
Digitized by V^jOOQlC
CHAP. VI.
ARTIFICIAL VOWELS.
121
the lid is withdrawn from the opening and the tone of the fork is communicated
to the air with great force. When the thread is let loose, the lid is brought over
the mouth of the chamber by the spring p, and the tone of the fork is no longer
heard. By partial opening of the mouth of the chamber, the tone of the fork can
be made to receive any desired intermediate degree of strength. The whole of
the strings which open the various resonance chambers belonging to a series of
such forks are attached to a keyboard in such a way that by pressing a key the
corresponding chamber is opened.
At first I had eight forks of this kind, giving the tones B{} and its first seven
harmonic upper partials, namely l^,f, b]}, d", /', a"[>,* and h"\}. The prime
tone I^ corresponds to the pitch in which bass voices naturally speak. Afterwards
I had forks made of the pitches d"',f"', a'"b* and h'"\^, and assumed l^ for the
prime of the compound tone.
To set the forks in motion, intermittent electrical currents had to be conducted H
through the coils of the electro-magnet, giving as many electrical shocks as the
FW. 32.
f
lowest forks made vibrations in a second, namely 120. Every shock makes the
iron of the electro-magnet b b momentarily magnetic, and hence enables it to
attract the prongs of the fork, which are themselves rendered permanently magnetic.
The prongs of the lowest fork B^ are thus attracted by the poles of the electro- 1;
magnet, for a Very short time, once in every vibration ; the prongs of the second
for 6|>, which moves twice as fast, once every second vibration, and so on. The
vibrations of the forks are therefore both excited and kept up as long as the electric
currents pass through the apparatus. The vibrations of the lower forks are very
powerful, those of the higher proportionally weaker.
The apparatus shewn in fig. 33 (p. 1226, c) serves to produce intermittent currents
of exactly determinate periodicity. A tuning-fork a is fixed horizontally between
the limbs b b of an electro-magnet ; at its extremities are fastened two platinum
wires c c, which dip into two little cups d filled half with mercury and half with
alcohol, forming the upper extremities of brass columns. These columns have clamp-
ing screws i to receive the wires, and stand on two boards f, g, which turn about
an axis, as at f, and which can each be somewhat raised or lowered by a thumb-
and a'"b, in the justly intoned scale of e\>, —
Translator.]
Digitized by V^ O OQ IC
* [These being 7th harmonics Vb and
Vb are 27 cents flatter than the a"\>
122
ARTIFICIAL VOWELS.
PABTI.
screw, as at g, so as to make the points of the platinum wires c c exactly toach
the mercury below the alcohol in the cups d. A third clamping screw e is in con-
ductive connection with the handle of the tuning-fork. When the fork vibrates,
and an electric current passes through it from i to e, the current will be broken
every time that the end of the fork a rises above the surface of the mercury in the
cup d, and re-made every time the platinum wire dips again into the mercury.
This intermittent current being at the same time conducted through the electro-
magnet b b, fig. 33, the latter becomes magnetic every time it passes, and thus
keeps up the vibrations of the fork a, which is itself magnetic. Generally only
one of the cups d is used for conducting the current. Alcohol is poured over the
mercury to prevent the latter from being burned by the electrical sparks which
arise when the stream is interrupted. This method of interrupting the current
was invented by Neef , who used a simple vibrating spring in place of the tamng-
% fork, as may be generally seen in the induction apparatus so much used for medical
purposes. But the vibrations of a spring communicate themsedves to all adjacent
Fio. 33.
bodies and are for our purposes both too audible and too irregular. Hence the
necessity of substituting a tuning-fork for the spring. The handle of a well worked
symmetrical tuning-fork is extremely little agitated by the vibrations of the fork
and hence does not itself agitate the bodies connected with it, so powerfully as the
IT fixed end of a straight spring. The tuning-fork of the apparatus in fig. 33 must
be in exact unison with the prime tone B\}. To effect this I employ a little clamp
of thick steel wire h, placed on one of the prongs. By slipping this towards the
free end of the prong the tone is deepened, and by shpping it towards the handle
of the fork, the tone is raised.*
When the whole apparatus is in action, but the resonance chambers are closed,
all the forks are maintained in a state of uniform motion, but no sound is heard,
beyond a gentle humming caused by the direct action of the forks on the air. But
on opening one or more resonance chambers, the corresponding tones are heard
with sufficient loudness, and are louder as the hd is more widely opened. By this
means it is possible to form, in rapid succession, different combinations of the prime
♦ This apparatas was made by Fesael in in Appendix VIII. [This apparatus was ex-
Cologne. More detailed descriptions of its hibited by R. Koenig (see Appendix II.) in the
separate parts, and instructions for the ex- International Exhibition of 1872 in London,
periments to be made by its means, are given — Translator.]
Digitized by V^ O OQ IC
CHAP. VI. ARTIFICIAL VOWELS. 123
tone with one or more harmonic upper partials having different degrees of loudness,
and thus produoe tones of different qualities.
Among the natural musical tones which appear suitable for imitation with forks,
the vowels of the human voice hold the first rank, because they are accompanied by
comparatively little extraneous noise and shew distinct differences of quality which
are easy to seize. Most vowels also are characterised by comparatively low upper
partials, which can be reached by our forks ; E and I alone somewhat exceed these
limits. The motion of the very high forks is too weak for this purpose when in-
fluenced only by such electrical currents as I was able to use without disturbance
from the noise of the electric sparks.
The first series of experiments was made with the eight forks S\} to b']}. With
these U, 0, 0, and even A could be imitated ; the last not very well because of my
not possessing the upper partials &" and d"\ which lie immediately above its
characteristic tone ^'l^, and are sensibly reinforced in the natural sound of this f
vowel. The prime tone JB\} of this series, when sounded alone, gave a very dull
U, much duller than could be produced in speech. The sound became more like
U when the second and third partial tones l^ and/' were allowed to sound feebly
at the same time. A very fine 0 was produced by taking V\} strong, and £(>, /', d'^
more feebly ; the prime tone ^ had then, however, to be somewhat damped. On
Budteily changing the pressure on the keys and hence the position of the hds
before the resonance chambers, so as to give B[} strong, and all the upper partials
weak, the apparatus uttered a good dear U after the 0.
A or rather A® [nearly 0 in not] was produced by making the fifth to the eighth
partial tones as loud as possible, and keeping the rest under.
The vowels of the second and third series, which have higher characteristic tones,
could be only imperfectly imitated by bringing out their reinforced tones of the lower
pitch. Though not very clear in themselves they became so by contrast on alterna-
tion with U and 0. Thus a passably clear A was obtained by giving loudness f
chiefly to the fourth and fifth tones, and keeping down the lower ones, and a sort
of E by reinforcing the third, and letting the rest sound feebly. The difference
between 0 and these two vowels lay principally in keeping the prime tone Bj[} and
its Octave l\} much weaker for A and E than for 0.*
To extend my experiments to the brighter vowels, I afterwards added the forks
^"'»/"» ^'"b> ^'"b» ^^^ ^^^ upper ones of which, however, gave a very fidnt tone,
and I chose h\} as the prime tone in place of ^. With these I got a very good A
and A, and at least a much more distinct E than before. But I could not get up
to the high characteristic tone of I.
In this higher series of forks, the prime tone 5t>, when sounded alone, repro-
duced U. The same prime 5t> with moderate force, accompanied with a strong
Octave 6t>, and a weaker Twelfth /", gave 0, which has the characteristic tone 6t>.
A was obtained by taking 5t>, b'[}, and f' moderately strong, and the characteristic
tones b"\} and d'" very strong. To change A into A it was necessary to increase %
somewhat the force of b^ and/'' which were adjacent to the characteristic tone
d", to damp b'% and bring out (i'" and/" as strongly as possible. For E the two
deepest tones of the series, bj;} and b'\}, had to be kept moderately loud, as being
adjacent to the deeper characteristic tone/', while the highest/''', a"'b, &'"[> had
to be made as prominent as possible. But I have hitherto not succeeded so well
with this as with the other vowels, because the high forks were too weak, and
because perhaps the upper partials which he above the characteristic tone 6"'[>
could not be entirely dispensed with.f
* The statementB in the MUnchener gelehrte above results will serve to shew their relations
Anzeigen for June 20, 1859, mast be corrected more clearly. In the first line are placed the
accordingly. At that time I did not know the notes of the forks and the numbers of the
higher upper partials of £ and I, and hence corresponding partials. The letters pp, p, mf,
made the O too dull to distinguish it from the /, ff below them are the usual musical indica-
unperfect E. tions of force, pianissiino^ piano, mezzo forte^
t [The following tabular statement of the forte^ fortissimo. Where no such mark is
Digitized by V^jOOQlC
124
QUALITY INDEPENDENT OF PHASE.
FABT I.
In precisely the same way as the vowels of the human voice, it is possible to
imitate the quality of tone produced by organ pipes of different stops, if they have
not secondary tones which are too high, but of course the whizzing noise, formed
by breaking the stream of air at the lip, is wanting in these imitations. The
tuning-forks are necessarily limited to the imitation of the purely musical part of
the tone. The piercing high upper partials, required for imitating reed instru-
ments, were absent, but the nasahty of the clarinet was given by using a series
of unevenly numbered partials, and the softer tones of the horn by the full chorus
of all the forks.
But though it was not possible to imitate every kind of quality of tone by the
present apparatus, it sufficed to decide the important question as to the effect of
altered difference of phase upon quality of tone. As I particularly observed at the
beginning of this chapter, this question is of fundamental importance for the
IT theory of auditory sensation. The reabder who is unused to physical investigations
must excuse some apparently dif&cult and dry details in the explanation of the
experiments necessary for its decision.
The simple means of altering the phases of the secondary tones consists in
bringing the resonance chambers somewhat out of tune by narrowing their
apertures, which weakens the resonance, and at the same time alters the phase.
If the resonance chamber is tuned so that the simple tone which excites its
strongest resonance coincides with the simple tone of the corresponding fork, then,
as the mathematical theory shews,* the greatest velocity of the air at the mouth
of the chamber in an outward direction, coincides with the greatest velocity of the
ends of the fork in an inward direction. On the other hand, if the chamber is
tuned to be slightly deeper than the fork, the greatest velocity of the air slightly
precedes, and if it is timed slightly higher, that greatest velocity slightly lags
behind the greatest velocity of the fork. The more the tuning is altered, the
^ greater will be the difference of phase, tiU at last it reaches the duration of a
quarter of a vibration. The magnitude of the difference of phase agrees during
this change precisely with the strength of the resonance, so that to a certain degree
we are able to measure the former by the latter. If we represent the strength of
the sound in the resonance chamber when in unison with the fork by lo, and
divide the periodic time of a vibration, like the circumference of a circle, into 360
added the partial is not mentioned in the text, ones, but the whole are now numbered as par-
For the second series of experiments the forks tials of 6 b.
of corresponding pitches are kept under the old
f -I
Firet )
Forks)-
z
a
}'
^\
!''
6
ah
8
lO
xa
»4
16
u
f
PP
PP
.3
0
mf
P
P
f
P
^
A°
P
P
P
P
ff
//
ff
ff
>
A
P
PP
?
f
f
£
P
PP
f
P
P
P
P
P
Second)
Forks r
X
a
/"
A
i-
6
ah
2
U
f
%
0
mf
f
P
1
A
A
E
mf
mf
mf
mf
f
P
/
f,
ff
ff
See Appendix XX. sect. M. No. 2, for
Messrs. Preece and Stroh's new method of
vowel synthesis. — Translator.l
See the first part of Appendix IX.
Digitized by
Google
CHAP. VI.
QUALITY INDEPENDENT OF PHASE.
125
degrees, the relation between the strength of the resonance and the difference of
phase is shewn by the following table : —
strength of.
Difference of Phase in angular
Beeonance
degrees
10
o<^
9
35*^ 54'
8
SO"" 12'
7
60*^40'
6
68<' 54'
5
75° 31'
4
8o<>48'
3
84° 50'
2
87° 42'
I
89° 26'
This table shews that a comparatively slight weakening of resonance by
altering the timing of the chamber occasions considerable differences of phase,
but that when the weakening is considerable there are relatively slight changes
of phase. We can take advantage of this circmnstance when compomiding the
vowel somids by means of the tmiing-forks to produce every possible alteration of
phase. It is only necessary to let the lid shade the mouth of the resonance
chamber till the strength of the tone is perceptibly diminished. As soon as we
have learned how to estimate roughly the amount of diminution of loudness, the
above table gives us the corresponding alteration of phase. We are thus able to
alter the vibrations of the tones in question to any amount, up to a quarter of the
periodic time of a vibration. Alterations of phase to the amount of half the
periodic time are produced by sending the electric current through the electro-
magnets of the corresponding fork in an opposite direction, which causes the ends
of the fork to be repelled instead of attracted by the electro-magnets on the ^
passage of the current, and thus sets the fork vibrating in the contrary direction.
This counter-excitement of the fork, however, by repelling currents, must not be
continued too long, as the magnetism of the fork itself would otherwise gradually
diminish, whereas attracting currents strengthen it or maintain it at a maximum.
It is well known that the magnetism of masses of iron that are violently agitated
is easily altered.
After a tone has been compounded, in which some of the partials have been
weakened and at the same time altered in phase by the half-shading of the
apertures of their corresponding resonance chambers, we can re-compound the
same tone by an equal amount of weakening in the same partials, but without
shading the aperture, and therefore without change of phase, by simply leaving
the months of the chambers wide open, and increasing their distances from the
exciting forks, until the required amount of enfeeblement of sound is attained.
For example, let us first sound the forks Bj;} and Vp, with fully opened resonance %
chambers, and perfect accord. They will vibrate as shewn by the vibrational
forms fig. 31, A and B (p. 120a), with the points e and do coincident, and produce
at a distance the compound vibration represented by the vibrational curve C. But
by closing the resonance chamber of the fork Bj;} we can make the point e on the
curve B coincide with the points between d© and dj on the curve A. To make e
coincide with d,, the loudness of B[} must be made about three-quarters of what
it would be if the mouth of the chamber were unshaded. The point e can be made
to coincide with d4 by reversing the current in the electro-magnets and folly
opening the mouth of the resonance chamber ; and then by imperfectly opening
the chamber of J3|;> the point e can be made to move towards 8. On the other
hand, an imperfect opening of the chamber h\} will make e recede from coincidence
with 8 (which is the same thing as coincidence with do) or with d4 , towards d4 or
dj respectively. The proportions of loudness may be made the same in all these
Digitized by V^jOOQlC
126 QUALITY INDEPENDENT OF PHASE. pakt i.
cases, without any alteration of phase, by removing the corresponding chambers
to the proper distance from its forks without shading its mouth.
In this manner every possible difference of phase in the tones of two chambers
can be produced. The same process can of course be applied to any required
number of forks. I have thus experimented upon numerous combinations of tone
with varied differences of phase, and I have never experienced the slightest dif-
ference in the quality of tone. So far as the quality of tone was concerned, I
found that it was entirely indifferent whether I weakened the separate partial
tones by shading the mouths of their resonance chambers, or by moving the
chamber itself to a sufficient distance &om the fork. Hence the answer to the
proposed question is: the quality of the musical portion of a compound tone
depends solely on the nv/nvher and relative strength of its partial simple tones^
and in no respect on their differences of phase.*
f The preceding proof that quality of tone is independent of difference of
phase, is the easiest to carry out experimentally, but its force lies solely in the
theoretical proposition that phases alter contemporaneously with strength of tone
. V when the mouths of the resonance chambers are shaded, and this proposition is
%^^^S»»^^j the result of mathematical theory alone. We cannot make vibrations of air
^ directly visible. But by a slight change in the experiment it may be so conducted
as to make the alteration of phase immediately visible. It is only necessary to
put the tuning-forks themselves out of tune with their resonance chambers, by
attaching little lumps of wax to the prongs. The same law holds for the phases
of a tuning-fork kept in vibration by an electric current, as for the resonance
chambers themselves. The phase gradually alters by a quarter period, while the
strength of the tone of the fork is reduced from a maximum to nothing at all, by
putting it out of tune. The phase of the motion of the air retains the same
relation to the phase of the vibration of the fork, because the pitch, which is
f determined by the number of interruptions of the electrical current in a second, is
not altered by the alteration of the fork. The change of phase in the fork can be
observed directly by means of Lissajou's vibration microscope, already described
and shewn in fig. 22 (p. Sod). Place the prongs of the fork and the microscope of
this instrument horizontally, and the fork to be examined vertically ; powder the
upper end of one of its prongs with a little starch, direct the microscope to one of
the grains of starch, and excite both forks by means of the electrical currents of
the interrupting fork (fig. 33, p. 122b). The fork of Lissajou's instrument is in
unison with the interrupting fork. The grain of starch vibrates horizontally, the
object-glass of the microscope vertically, and thus, by the composition of these
two motions, curves are generated, just as in the observations on violin strings
previously described.
When the observed fork is in unison with the interrupting fork, the curve
becomes an obHque straight Une (fig. 34, i), if both forks pass through their
% RG
VI
position of rest at the same moment. As the phase alters, the straight line passes
through a long oblique ellipse (2, 3), tiU on the difference of phase becoming a
quarter of a period, it develops into a circle (4) ; and then as the difference of
phase increases, it passes through oblique ellipses (5, 6) in another direction, till it
reaches another straight Une (7), on the difference becoming half a period.
If the second fork is the upper Octave of the interrupting fork, the curves
* [The experiments of Koenig with the modification. Moreover Koenig contends that
wave-siren, explained in App. XX. sect. L. the 'apparent exception' of p. 127c, is an
art. 6, shew that this law requires a slight ' actual ' one (ibid.), — Translator.]
Digitized by V^jOOQlC
CHAP. VI. QUALITY INDEPENDENT OP PHASE. 127
i» ^f 3» 4» 5> ui fig. 35, shew the series of forms. Here 3 answers to the case when
both forks pass through their position of rest at the same time ; 2 and 4 differ from
that position by ^, and i and 5 by ^ of a wave of the higher fork.
If we now bring the forks into the most perfect possible unison with the
interrupting fork, so that both vibrate as strongly as possible, and then alter their
timing a little by putting on or removing pieces of wax, we also see one figure of the
microscopic image gradually passing into another, and can thus easily assure our- f
selves of the correctness of the law already cited. Experiments on quality of tone
are then conducted by first bringing all the forks as exactly as possible to the
pitches of the harmonic upper partial tones of the interrupting fork, next removing
the resonance chambers to such distances from the forks as will give the required
relations of strength, and finally putting the forks out of tune as much as we please
by sticking on lumps of wax. The size of these lumps should be previously so
regulated by microscopical observation as to produce the required difference of
phase. This, however, at the same time weakens the vibrations of the forks, and
hence the strength of the tones must be restored to its former state by bringing the
resonance chambers nearer to the forks. u
The result in these experiments, where the forks are put out of tune, is the
same as in those where the resonance chambers were put out of tune. There is
no perceptible alteration of quality of tone. At least there is no alteration so
marked as to be recognisable after the expiration of the few seconds necessary %
for resetting the apparatus, and hence certainly no such change of quality as
would change one vowel into another.
An apparent exception to this rule must here be mentioned. If the forks JB\}
and H} are not perfectly tuned as Octaves, and are brought into vibration by rub-
bing or striking, an attentive ear will observe very weak beats which appear like
small changes in the strength of the tone and its quality. These beats are cer-
tainly connected with the successive entrance of the vibrating forks on varying
difference of phase. Their explanation will be given when combinational tones are
considered, and it will then be shewn that these slight variations of quality are
referable to changes in the strength of one of the simple tones.
Hence we are able to lay down the important law that differences in musical
quaiity of tone depend solely on the presence and strength of partial tones, and in
no respect on the differences in phase under which these partial tones enter into
composition. It must be here observed that we are speaking only of musical ^
quality as previously defined. When the musical tone is accompanied by un-
musical noises, such as jarring, scratching, soughing, whizzing, hissing, these
motions are either not to be considered as periodic at all, or else correspond to
high upper partials, of nearly the same pitch, which consequently form strident
dissonances. We were not able to embrace these in our experiments, and hence
we must leave it for the present doubtful whether in such dissonating tones
difference of phase is an element of importance. Subsequent theoretic considera-
tions will lead us to suppose that it really is.
If we wish only to imitate vowels by compound tones without being able to
distinguish the differences of phase in the individual constituent simple tones, we
can effect our purpose tolerably well with organ pipes. But we must have at least
two series of them, loud open and soft stopped pipes, because the strength of tone
cannot be increased by additional pressure of wind without at the same time
changing the pitch. I have had a double row of pipes of this kind made by Herr
Digitized by V^jOOQlC
128
APPEEHENSION OF QUALITY OP TONE.
PART I.
A
Appunn in Hanau, giving the first sixteen partial tones of B\}. All these pipes
stand on a common windchest, which also contains the valves by which they can
be opened or shut. Two larger valves cut off the passage from the windchest to
the bellows. While these valves are closed, the pipe valves are arranged for the
required combination of tones, and then one of the main valves of the windchest
is opened, allowing all the pipes to sound at once. The character of the vowel is
better produced in this way by short jerks of sound, than by a long continued
sound. It is best to produce the prime tone and the predominant upper partial
tones of the required vowels on both the open and stopped pipes at once, and to
open only the weak stopped pipes for the next adjacent tones, so that the strong
tone may not be too isolated.
The imitation of the vowels by this means is not very perfect, because, among
other reasons, it is impossible to graduate the strength of tone on the different pipes
^ so delicately as on the tuning-forks, and the higher tones especially are too scream-
ing. But the vowel sounds thus composed are perfectly recognisable.
We proceed now to consider the part played by the ear in the apprehension of
quality of tone. The assumption formerly made respecting the function of the ear,
was that it was capable of distinguishing both the pitch number of a musical tone
(which gives the pitch), and also the /orm of the vibrations (on which the difference
of quality depends). This last assertion was based simply on the exclusion of all
other possible assumptions. As it could be proved that sameness of pitch always
required equal pitch numbers, and as loudness visibly depended upon the ampli-
tude of the vibrations, the quality of tone must necessarily depend on something
which was neither the number nor the amplitude of the vibrations. There was
nothing left us but form. We can now make this view more definite. The ex-
periments just described shew that waves of very different forms (as fig. 31,
0, D, p. 1 20a, and fig. 12, C, D, p. 22b), may have the same quality of tone, and
m indeed, for every case, except the simple tone, there is an infinite number of forms
of wave of this kind, because any alteration of the difference of phase alters the
form of wave without changing the quality of tone. The only decisive character
of a quality of tone, is that the motion of the air which strikes the ear when re-
solved into a sum of pendulum vibrations gives the same degree of strength to the
^ame simple vibration.
' Hence the ear does not distinguish the different forms of waves in themselves,
/as the eye distinguishes the different vibrational curves. The ear must be said
/ rather to decompose every wave form into simpler elements according to a definite
^^^'W. It then receives a sensation from each of these simpler elements as from an
harmonious tone. By trained attention the ear is able to become conscious of each
of these simpler tones separately. And what the ear distinguishes as different
qualities of tone are only different combinations of these simpler sensations.
The comparison between ear and eye is here very instructive. When the
«r vibrational motion is rendered visible, as in the vibration microscope, the eye is
capable of distinguishing every possible different form of vibration one from
another, even such as the ear cannot distinguish. But the eye is not capable of
directly resolving the vibrations into simple vibrations, as the ear is. Hence the
eye, assisted by the above-named instrument, really distinguishes the form ofvibra-
tioUy as such, and in so doing distinguishes every different form of vibration. The
ear, on the other hand, does Tiot distinguish every different form of vibration, but
only such as when resolved into pendular vibrations, give different constituents.
But on the other hand, by its capability of distinguishing and feeling these very
constituents, it is again superior to the eye, which is quite incapable of so doing.
This analysis of compound into simple pendular vibrations is an astonishing
property of the ear. The reader must bear in mind that when we apply the term
* compound ' to the vibrations produced by a single musical instrument, the * com-
position ' has no existence except for our auditory perceptions, or for mathematical
theory. In reality, the motion of the particles of the air is not at all compound.
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CHAP. VI. SYMPATHETICALLY VIBKATING PARTS OF THE EAR. 129
it is quite simple, flowing from a single source. When we turn to external nature
for an analogue of such an analysis of periodical motions into simple motions, we
find none but the phenomena of sympathetic vibration. Li reality if we suppose
the dampers of a pianoforte to be raised, and allow any musical tone to impinge
powerfully on its sounding board, we bring a set of strings into sympathetic vibra-
tion, namely all those strings, and only those, which correspond with the simple
tones contained in the given musical tone. Here, then, we have, by a purely me-
chanical process, a resolution of air waves precisely similar to that performed by the
ear. The air wave, quite simple in itself, brings a certain number of strings into
sympathetic vibration, and the sympathetic vibration of these strings depends on
the same law as the sensation of harmcHiic upper partial tones in the ear.*
There is necessarily a certain difference between the two kinds of apparatus,
because the pianoforte strings readily vibrate with their upper partials in S3nnpathy,
and hence separate into several vibrating sections. We will disregard this pecu- ^
liarity in making our comparison. It would besides be easy to make an instrument
in which the strings would not vibrate sensibly or powerfully for any but their
prime tones, by simply loading the strings slightly in the middle. This would make
their higher proper tones inharmonic to their primes.
Now suppose we were able to connect every string of a piano with a nervous fibre
in such a manner that this fibre would be excited and experience a sensation every
time the string vibrated. Then every musical tone which impinged on the instru-
ment would excite, as we know to be really the case in the ear, a series of sensa-
tions exactly corresponding to the pendular vibrations into which the original
motion of the air had to be resolved. By this means, then, the existence of each
partial tone would be exactly so perceived, as it really is perceived by the ear.
The sensations of simple tones of different pitch would under the supposed con-
ditions fisll to the lot of different nervous fibres, and hence be produced quite
separately, and independently of each other. ^
Now, as a matter of fact, later microscopic discoveries respecting the internal
construction of the ear, lead to the hypothesis, that arrangements exist in the ear
similar to those which we
have imagined. The end of
every fibre of the auditory
nerve is connected with small
elastic parts, which we cannot
but assume to be set in sym-
pathetic vibration by the
waves of sound.
The construction of the
ear may be briefly described
as follows: — The fine ends
of the fibres of the auditory ^
nerves are expanded on a deli-
cate membrane in a cavity
filled with fluid. Owing to
its involved form this cavity
is known as the labyrinth of the ear. To conduct the vibrations of the air with
sufficient force into the fluid of the labyrinth is the office of a second portion of
the ear, the tympdrmm or drum and the parts within it. Fig. 36 above is a
^ [Baise the dampers of a piano, and utter
the Yowel A (ah) sharply and loudly, directing it
well on to the sound board, pause a second and
the vowel will be echoed from the strings. Be-
^h
Fig. 36.
damp, raise the dampers and cry U {00) as be-
fore, and that will also be echoed. He-damp,
raise the dampers and cry I (ee), and that
again will be echoed. The other vowels may
be tried in the same way. The echo, though
imperfect, is always true enough to surprise
a hearer to whom it is new, even if the pitch of
the vowel is taken at hazard. It will be im-
proved if the vowels are sung loudly to notes
of the piano. The experiment is so easy to
make and so fundamental in character, that
it should be witnessed by every student. -
Translator.]
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130 SYMPATHETICALLY VIBRATING PARTS OF THE EAR. parti.
diagrammatic section, of the size of life, shewing the cavities belonging to the
auditory apparatus. A is the labyrinth, B B the cavity of the tympdwwm or drum,
D the funnel-shaped entrance into the meatus or external auditory passage, nar-
rowest in the middle and expanding slightly towards its upper extremity. This
medtiis, in the ear or passage, is a tube formed partly of cartilage or gristle and
partly of bone, and it is separated &om the tymp&num or drum, by a thin circular
membrane, the membrdna tymp&nl or d/mmshin* c c, which is rather laxly stretched
in a bony ring. The drum (tym'p&num) B lies between the outer passage
(meatus) and the labyrinth. Th^ drum is separated from the labyrinth by bony
walls, pierced with two holes, closed by membranes. These are the so-called
windows [fenes'trae) of the labyrinth. The upper one o, called the oval window
(fenes'tra dvdUs), is connected with one of the ossicles or little bones of the ear
called the stirrup. The lower or round window r (fenes'tra rotun'da) has no
% connection with these ossicles.
The drum of the ear is consequently completely shut off from the external
passage and from the labyrinth. But it has free access to the upper part of the
pharynx or throat, through the so-called Eustachianf tube E, which in Oermany
is termed a trumpet, because of the trumpet-like expansion of its pharyngeiJ
extremity and the narrowness of its opening into the drum. The end which opens
into the drum is formed of bone, but the expanded pharyngeal end is formed of thin
flexible cartilage or gristle, split along its upper side. The edges of the split are
closed by a sinewy membrane. By closing the nose and mouth, and either con-
densing the air in the mouth by pressure, or rarefying it by suction, air can be
respectively driven into or drawn out of the drum through this tube. At the
entrance of air into the drum, or its departure &om it, we feel a sudden jerk in
the ear, and bear a dull crack. Air passes from the pharynx to the drum, or from
the drum to the pharynx only at the moment of making the motion of swallowing.
f When the air has entered the drum it remains there, even after nose and mouth
are opened again, until we make another motion
of swallowing. Then the air leaves the drum,
as we perceive by a second cracking in the ear,
and the cessation of the feeling of tension in the
drumskin which had remained up till that time.
These experiments shew that the tube is not
usually open, but is opened only during swallow-
ing, and this is explained by the fact that the
muscles which raise the velum paldtl or soft
palate, and are set in action on swallowing, arise
partly from the cartilaginous extremity of the tube.
Hence the drum is generally quite closed, and
filled with air, which has a pressure equal to
m that of the external air, because it has from
time to time, that is whenever we swallow, the
means of equalising itself with the same by free
communication. For a strong pressure of the
air, the tube opens even without the action of
swallowing, and its power of resistance seems to
be very different in different individuals.
In two places, this air in the drum is like-
wise separated from the fluid of the labyrinth
merely by a thin stretched membrane, which closes the two windows of the
Mm
Ossfdes of the ear in xnntaal oonnectSon.
seen from the front, and taken from the
right side of the head, which has been
tamed a little to the right round a
rertioal axis. M hammer or maUem,
J anril or imut. S stirrup or tiapm.
Mcp head, Mc neck, ICl long prooeea or
procei'sus grd'eUiiy Mm handle or fmiiiil'-
brUan of the hammer.— -Jc body, Jb short
process, Jl long prooeas, Jpl orUcnlar
process or ot orbieOUtre or proett^tus Umti-
e^UarUf of the anrll.— Sop head or espir'-
iUum of the stirmp.
* [In common parlance the drumskin of
the ear, or tympanic membrafie, is spoken of
as the drum itself. Anatomists as well as
drummers distinguish the membranous cover
(drumskin) which is struck, from the hollow
cavity (drum) which contains the resonant air.
The quantities of the Latin words are marked,
as I have heard musicians give them incor-
rectly.— Translator.']
t [Generally pronounced yoo-staV-ki-an,
but sometimes yoO'Stai'-shl'&n,'— Translator,',
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rHAP. VI. SYMPATHETICALLY VIBRATING PARTS OF THE EAR. 131
Fio. 38.
labyrinth, already mentioned, namely, the oval window (o, fig. 36, p. 129c) and
the round window (r). Both of these membranes are in contact on their outer
side with the air of the drum, and on their inner side with the water of the laby-
rinth. The membrane of the round window is free, but that of the oval window
is connected with the drumskin of the ear by a series of three Httle bones or
auditory ossicles, jointed together. Fig. 37 shews the three ossicles in their natural
connection, enlarged four diameters. They are the hammer (mal'leus) M, the anvil
(incus) J, and the stirrup (sta'pes*) S. The hammer is attached to the drumskin,
and the stirrup to the membrane of the oval window.
The hammer, shewn separately in fig. 38, has a thick, rounded upper extremity,
the head cp, and a thinner lower extremity, the handle m. Between these two is
a contraction c, the neck. At the
back of the head is the surface of the
joint, by means of which it fits on to ^
the anvil. Below the neck, where
the handle begins, project two pro-
cesses, the long 1, also called pro-
cessus Folidnu^ and pr, grdcllis, and
the short b, also called pr, hre'vis.
The long process has the proportion-
ate length shewn in the figure, in
children only ; in adults it appears to
be absorbed down to a little stump.
Bight bammer A from the front. B from behind cph«id^ J^ jg directed forwards, and is COVCrod
c nock, b short, 1 long priKjess, ni handle. • Surface of i . i *
the joint. by the bands which fasten the hammer
in front. The slwrt process b, on the other hand, is directed towards the drumskin,
and presses its upper part a little forwards. From the point of this process b to
the point of the handle m the hammer is attached to the upper portion of the ^
Fia. 39.
Fio. 40.
Lrft temporal bone of a newly-born child, ^ith thfi anditorj Right drumskin with the hammer, seen from the
MBicles in tUu. Sta, spina tympftnica anterior. Stp, intdde. The inner layer of the fold of muooui
xplna tympinica postllrior. Mcp, head of the hammer. membrane belonging to the hammer (see
Mb ».hort. Ml long process of hammer. J anvil. S below) is removed. Btp, spina tympftnica
stirmp. post. Mop, head of the hammer. Ml, long
process of hammer, ma, ligftroen'tum mallM
ant. I chorda tympAnl. a Eustachian tube.
* Tendon of the M. tensor tympftnl, cut
through close to its insertion.
drumskin, in such a manner that the point of the handle draws the drumskin
considerably towards the inner part of the ear.
Fig. 39 above shews the hammer in its natural position as seen from
without, after the drumskin has been removed, and fig. 40 shows the hammer
lying against the drumskin as seen from within. The hammer is fastened along
* {Stapes is asaally called stdi'-p^ez. It is
not a classical word, and is usually received as
a contraction for stdVipfa or foot-rest, also not
classical.- Transluior.]
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132 SYMPATHETICALLY VIBBATING PARTS OF THE EAR parti.
the upper margin of the drumskin by a fold of mucous membrane, within which
run a series of rather stiff bundles of tendinous fibres. These straps arise in a
line which passes from the processus Folianus (fig. 38, 1), above the contraction of
the neck, towards the lower end of the surface of the joint for the anvil, and in
elderly people is developed into a prominent ridge of bone. The tendinous bands
or ligaments are strongest and stiffest at the front and back end of this line of
insertion. The front portion of the ligament, lig. mallei anterius (fig. 40, ma),
surrounds the processus Folianus, and is attached partly to a bony spine (figs. 39
and 40, Stp) of the osseous ring of the drum, which projects close to the neck of
the hammer, and partly to its under edge, and partly falls into a bony fissure
which leads towards the articulation of the jaw. The back portion of the same
hgament, on the other hand, is attached to a sharp-edged bony ridge projecting
inwards firom the drumskin, and parallel to it, a little above the opening, through
f which a traversing nerve, the chorda tymp&nl (fig. 40, 1, 1, p. 131c), enters the bone.
This second bundle of fibres may be called the lig. mall§! posterius. In fig. 39
(p. 1310) the origin of this ligament is seen as a httle projection of the ring to
which the drumskin is attached. This projection bounds towards the right the
upper edge of the opening for the drumskin, which begins to the left of Stp, exactly
at the place where the long process of the anvil makes its appearance in the figure.
These two ligaments, front and back, taken together form a moderately tense
sinewy chord, round which the hammer can turn as on an axis. Hence even when
the two other ossicles have been carefcdly removed, without loosening these two
Hgaments, the hammer will remain in its natural position, although not so stiffly
as before.
The middle fibres of the broad ligamentous band above mentioned pass outwards
towards the upper bony edge of the drumskin. They are comparatively short, and
are known as lig. maJlei externum. Arising above the line of the axis of the
^ hammer, they prevent the head from turning too far inwards, and the handle with
the drumskin from turning too far outwards, and oppose any down-dragging of the
ligament forming the axis. The first effect is increased by a ligament (lig. mallei
superius) which passes from the processus Folianus, upwards, into the small sht,
between the head of the hammer and the wall of the drum, as shewn in fig, 40
(p. 131c).
It must be observed that in the upper part of the channel of the Eustachian
tube, there is a miiscUfor tightening the drumskin (m. tensor tymp&nl), the tendon
of which passes obliquely across the cavity of the drum and is attached to the
upper part of the handle of the hammer (at*, fig. 40, p. 131c). This muscle
must be regarded as a moderately tense elastic band, and may have its tension
temporarily much increased by active contraction. The effect of this muscle is
also principally to draw the handle of the hammer inwards, together with the
drumskin. But since its point of attachment is so close to the ligamentous axis,
^ the chief part of its pull acts on this axis, stretching it as it draws it inwards.
Here we must observe that in the case of a rectilinear inextensible cord, which
is moderately tense, such as the ligamentous axis of the hammer, a slight force
which pulls it sideways, suffices to produce a very considerable increase of tension.
This is the case with the present arrangement of stretching muscles. It should
also be remembered that quiescent muscles not excited by innervation, are always
stretched elastically in the living body, and act like elastic bands. This elastic
tension can of course be considerably increased by the innervation which brings
the muscles into action, but such tension is never entirely absent from the majority
of oiu" muscles.
The anvil, which is shown separately in fig. 41, resembles a double tooth with
two fangs ; the surface of its joint with the hammer (at *, fig. 41), replacing the
masticating surface. Of the two roots of the tooth which are rather widely
separated, the upper, directed backwards, is called the short process b ; the other,
thinner and directed downwards, the long process of the anvil 1. At the tip of
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CHAP. VI. SYMPATHETICALLY VIBRATING PARTS OF THE EAR. 133
the latter is the knob which articulates with the stirrup. The tip of the short
process, on the other hand, by means of a short ligament and an imperfectly
yj^ ^ g developed joint at its under surface, is con-
nected with the back wall of the cavity of
the drum, at the spot where this passes
backwards into the air cavities of the mastoid
process behind the ear. The joint between
anvil and hammer is a curved depression of
a rather irregular form, like a saddle. In
its action it maybe compared with the joints
of the well-known Breguet watchkeys, which
have rows of interlocking teeth, offering
scarcely any resistance to revolution in one
Bight BXkru. A medial surface. B front yiew. o direction, but allowing no rcvolution what- ^
body, b short, 1 long process, pi proc. lenticu- ^«^y :„ +1^0 nfhfir TntArlnokinff tfipth of
lAris or os orbiculftre. • Articulation with the ®^®^ "^ '"^ OlOer. inieriOCKmg TOCin OI
head of the hammer. •* Surface resting on the tluS kind are dcveloped UPOU the Under side
wall of the drum. i?xT-''xi-i.i. j -i
of the jomt between hammer and anvil.
The tooth on the hammer projects towards the drumskin, that of the anvil lies
inwards ; and, conversely, towards the upper end of the hollow of the joint, the
anvil projects outwards, and the hammer inwards. The consequence of this
arrangement is that when the hammer is drawn inwards by the handle, it bites
the anvil firmly and carries it with it. Conversely, when the drumskin, with the
hammer, is driven outwards, the anvil is not obliged to foUow it. The interlocking
teeth of the surfaces of the joint then separate, and the surfaces glide over each
other with very Uttle friction. This arrangement has the very great advantage of
preventing any possibility of the stirrup's being torn away from the oval window,
when the air in the auditory passage is considerably rarefied. There is also no
danger from driving in the hammer, as might happen when the air in the auditory ^
passage was condensed, because it is powerfully opposed by the tension of the
drumskin, which is drawn in like a funnel.
When air is forced into the cavity of the drum in the act of swallowing, the
contact of hammer and anvil is loosened. Weak tones in the middle and upper
regions of the scale are then not heard much more weakly than usual, but stronger
tones are very sensibly damped. This may perhaps be explained by supposing that
the adhesion of the articulating surfaces suffices to transfer weak motions from one
bone to the other, but that strong impulses cause the surfaces to sHde over one
another, and hence the tones due to such impulses must be enfeebled.
Deep tones are damped in this case, whether they are strong or weak, perhaps
because these always require larger motions to become audible.*
Another important effect on the apprehension of tone, which is due to the above
arrangement in the articulation of hammer and anvil, will have to be considered in
relation to combinational tones. [See p. 158&.] ^
Since the attachment of the tip of the short process of the anvil lies sensibly
inwards and above the ligamentous axis of the hammer, the head of the hammer
separates from the articulating surface between hammer and anvil, when the head
is driven outwards, and therefore the handle and drumskin are driven inwards.
The consequence is that the ligaments holding the anvil against the hammer, and
on the tip of the short process of the anvil, are sensibly stretched, and hence the
tip is raised from its osseous support. Consequently in the normal position of the
ossicles for hearing, the anvil has no contact with any other bone but the hammer,
and both bones are held in position only by stretched ligaments, which are tolerably
tight, so that only the revolution of the hammer about its ligamentous axis remains
comparatively free.
The third ossicle, the stirrup, shewn separately in fig. 42, has really a most
striking resemblance to the implement after which it has been named. The foot B
* On this point see Part II. Chapter IX.
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134 SYMPATHETICALLY VIBKATING PARTS OF THE EAIl. parti.
is fastened into the membrane of the oval window, and fills it all up, with the
exception of a narrow margin. The head op, has an articulating hole for the tip
of the long process of the anvil
(processus lenticol&ris, or os ^ ^ ^^- ^'- ^
orbiculare). The joint is sur-
rounded by a lax membrane. f^^^^\
When the drumskin is normally ^^^"^
drawn inwards, the anvil presses ^^
on the stirrup, so that no tighter _£_ p
ligamentous fiebstening of the ^
loint ifl TififtAB«ii.TV l?.v«rv in Right stirrup : seen, -4 from within, 5 from front, C from behind. '
JOmi IS necessary. J^jVery m- ^iq^^^ cp, head or capitulum. a Front, p back llmb.
crease in the push on the hammer
arising from the drumskin also occasions an increiftse in the push of the stirrup
^ against the oval window ; but in this action the upper and somewhat looser
margin of its foot is more displaced than the under, so that the head rises slightly ;
this motion again causes a slight elevation of the tip of the long process in the
anvil, in the direction conditioned by its position, inwards and underneath the
ligamentous axis of the hammer.
The excursions of the foot of the stirrup are always very small, and according
to my measurements* never exceed one-tenth of a millimetre ('00394 or about
^^^ of an inch). But the hammer when freed fr'om anvil and stirrup, with its
handle moving outwards, and sliding over the atticulating surface of the anvil, can
make excursions at least nine times as great as it can execute when acting in
connection with anvil and stirrup.
The first advantage of the apparatus belonging to the drum of the ear, is tliat
the whole sonorous motion of the comparatively wide surface of the drumskin (ver-
tical diameter 9 to 10 millimetres, [or 0*35 to 0*39 inches,] just over one- third of an
^ inch ; horizontal diameter, 7*5 to 9 millimetres, [or 0*295 ^ ^'35 ii^ches,] that is
about five-sixths of the former dimensions) is collected and transferred by the
ossicles to the relatively much smaller surface of the oval window or of the foot of
the stirrup, which is only 1*5 to 3 millimetres [o*o6 to 0*12 inches] in diameter.
The surface of the drumskin is hence 15 to 20 times larger than that of the oval
window.
Li this transference of the vibrations of air into the labyrinth it is to be observed
that though the particles of air themselves have a comparatively large amplitude of
vibration-, yet their density is so small that they have no very great moment of inertia,
and consequently when their motion is impeded by the drumskin of the ear, they
are not capable of presenting much resistance to such an impediment, or of exert-
ing any sensible pressure against it. The fluid in the labyrinth, on the other hand,
is much denser and heavier than the air in the auditory passage, and for moving it
rapidly backwards and forwards as in sonorous oscillations, a feir greater exertion of
^ pressure is required than was necessary for the air in the auditory passage. On
the other hand the amplitude of the vibrations performed by the fluid in the laby-
rinth are relatively very small, and extremely minute vibrations will in this case
suffice to give a vibratory motion to the terminations and appendages of the nerves,
which lie on the very limits of microscopic vision.
The mechanical problem which the apparatus within the drum of the ear had
to solve, was to transform a motion of great amplitude and little force, such as im-
pinges on the drumskin, into a motion of small amplitude and great force, such as
had to be conimunicated to the fluid in the labyrinth.
A problem of this sort can be solved by various kinds of mechanical apparatus,
such as levers, trains of pulleys, cranes, and the like. The mode in which it is
solved by the apparatus in the drum of the ear, is quite unusual, and very peculiar.
* Helmholtz, * Mechanism of the Auditory attempt is made to prove the correctness of
Ossicles/ in Pfiucger's Archiv fUr Physio- the account of this mechanism given in the
hijie vol. i. pp. 34 43. In this paper an text.
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CHAP. VI. SYMPATHETICALLY VIBRATING PARTS OF THE EAR. 135
A leverage is certainly employed, but only to a moderate extent. The tip of
the handle of the hammet, on which the poll of the drumskin first acts» is about
once and a half as far from the axis of rotation as that point of the anvil which
presses on the stirrup (see fig. 39, p. 1 31c). The handle of the hammer consequently
forms the longer arm of a lever, and the pressure on the stirrup will be once and a
half as great as that which drives in the hammer.
The chief means of reinforcement is due to the form of the drumskin. It has
been already mentioned that its middle or navel (umbilicus) is drawn inwards by
the handle, so as to present a funnel shape. But the meridian lines of this funnel
drawn from the navel to the circumference, are not straight lines ; they are sHghtiy
convex on the outer side. A diminution of pressure in the auditory passage in-
creases this convexity, and an augmentation diminishes it. Now the tension caused
in an inextensible thread, having the form of a flat arch, by a force acting perpen-
dicular to its convexity, is very considerable. It is well known that a sensible force ^
must be exerted to stretch a long thin string into even a tolerably straight horizon-
tal line. The force is indeed very much greater than the weight of the string which
pulls the string from the horizontal position.* In the case of the drumskin, it is
not gravity which prevents its radial fibres from straightening themselves, but partly
the pressure of the air", and partly the elastic pull of the circular fibres of the mem-
brane. The latter tend to contract towards the axis of the funnel-shaped mem-
brane, and hence produce the inflection of the radial fibres towards this axis. By
means of the variable pressure of air during the sonorous vibrations of the at-
mosphere this pull exerted by the circular fibres is alternately strengthened and
weakened, and produces an effect on the point where the radial fibres are attached
to the tip of the handle of the hammer, similar to that which would happen if we
could alternately increase and diminish the weight of a string stretched horizontally,
for this would produce a proportionate increase and decrease in the pull exerted by
the hand which stretched it. ^
In a horizontally stretched string such as has been just described, it should be
further remarked that an extremely small relaxation of the hand is followed by a
considerable fall in the middle of the string. The relaxation of the hand, namely,
takes place in the direction of the chord of the arc, and easy geometrical con-
siderations shew that chords of arcs of the same length and different, but always
very small curvature, differ very slightly indeed from each other and from the
lengths of the arcs themselves.f This is also the case with the drumskin. An ex-
tremely little yielding in the handle of the hammer admits of a very considerable
change in the curvature of the drumskin. The consequence is that, in sonorous
vibrations, the parts of the drumskin which lie between the inner attachment of
this membrane to the hammer and its outer attachment to the ring of the drum,
are able to follow the oscillations of the air with considerable freedom, while the
motion of the air is transmitted to the handle of the hammer with much diminished
amplitude but much increased force. After this, as the motion passes from the ^
handle of the hammer to the stirrup, the leverage already mentioned causes
a second and more moderate reduction of the amplitude of vibration with corre-
sponding increase of force.
We now proceed to describe the innermost division of thes)rgan of hearing*
called the labyrinth. Fig. 43 (p. 134c) represents a cast of its cavity, as seen from
different positions. Its middle portion, containing the oval window Fv (fenestra
vestibull) that receives the foot of the stirrup, is called the vestibule of the labyrinth.
* [The following qaatrain, said to have Into a horizontal line,
been unconsciously produced by Vince, as a So as to make it truly straight. — TratuUUor.]
coroUary^ to one of the propositions in his . rj^e amount of difference varies as the
• Mechamcs/ will serve to impress the fact 1^ ^^ ^^e depth of the arc. If the length
on a non-mathematical nader :— ^^ ^j^^ ^ ^^ j ^^j t^,, distance of its middle
Hence no force, however great, from the chord be «, tl
Can stretch a cord, however fine, the arc by the length
3 2
Digitized by VjOOQIC
136
LABYRINTH OF THE EAR.
PAKT I.
From the vestibule proceeds forwards and underwards, a spiral caiial, the snail-
shell or cochlea, at the entrance to which hes the rouTid window Fc (fenestra
cochleae), which is turned towards the cavity of the drum. Upwards and back-
wards, on the other hand, proceed three semicircular canals from the vestibule, the
horizontal, front vertical ^ and back vertical semicircular canals, each of which
debouches with both its mouths in the vestibule, and each of which has at one
end a bottle- shaped enlargement, or ampulla (ha, vaa, vpa). The aquaeductus
vestlbuli shewn in the figure, Av, appears (from Dr. Fr. E. Weber's investigations)
to form a communication between tlie water of the labyrinth, and the spaces for
lymph within the cranium. The rough places Tsf and • are casts of canals which
introduce nerves.
The whole of this cavity of the labyrinth is filled with fluid, and surrounded by
the extremely hard close mass of the petrous bone, so that there are only two
'^ yielding spots on its walls, the two windows, the oval Fv, and the round Fc. Into
the first, as already described, is fastened the foot of the stirrup, by a narrow
membranous margin. The second is closed by a membrane. When the stirrup
is driven against the oval window, the whole mass of fluid in the labyrinth is
necessarily driven against the round window, as the only spot where its walls can
give way. If, as Politzer did, we put a finely drawn glass tube as a manometer
into the romid window, without in other respects injuring^he labyrinth, the water
in this tube will be driven upwards as soon as ar^strong pressure of air acts on the
Fig. 43.
Rp vaa
A
\
1.. ^
V^
F ^
^
r--
Tpa
A, left labyrinth from without. B, right labyrinth from within. C, left labyrinth from abore. Fc, fenestra
cochleae or rrjuud window. Pr, fenestra TestTbail, or oval window. Re, recessns elliptTcus. Rs, recessus
sphaerlcus. h, horizontal t<euiicircnlar canal, ha, ampulla of the same, raa, ampulla of the front rertical
semicircular canal, ypa, ampulla of the back Tertical semicircular canal, tc, common limb of the two rertioal
semicircular canals. Ar, cast of the aquaeductus restlbfill. Tgf, traotus splrftlis forftmlnteus. * Cast of the
little canals which debouch on the pj^rftmis vestlbtUI.
outside of the drumskin and causes the foot of the stirrup to be driven into the oval
window. ;
The terminations of the auditory nerve are spread over fine membranous
^ formations, which lie partly floating and partly expanded in the hollow of the bony
labyrinth, and taken together compose the membranous labyrinth. This last has
on the whole the same shape as the bony labyrinth. But its canals and ca\4ties
are smaller, and its interior is divided into two separate sections; first the
utricUlus with the semicircular canals, and second the saccUlus with the mem-
branous cochin. Both the utriculus and the sacculus lie in the vestibule of the
bony labyrinth ; the utriculus opposite to the recessus ellipttcus (Ke, fig. 43 above),
the sacculus opposite to the recessus sphaericus (Rs). These are floating bags
filled with water, and only touch the wall of the labyrinth at the point where the
nerves enter them.
The form of the utriculus with its membranous semicircular canals is shewn in
fig. 44. The ampullae project much more in the membranous than in the bony semi-
circular canals. According to the recent investigations of Riidinger, the mem-
branous semicircular canals do not float in the bony ones, but are fastened to tlie
convex side of the latter. lu each ampulla there is a pad -like prominence directed
Digitized by V^jOOQlC
CH.4P. VI.
COCHLEA OF THE EAR.
137
inwards, into which fibriles of the auditory nerve enter ; and on the utriculus
there is a place which is flatter and thickened. The peculiar manner in which
the nerves terminate in this place will be described
hereafter. Whether these, and the whole apparatus
of the semicircular canals, assist in the sensation of
hearing, has latterly been rendered very doubtful. [See
p. 151^-]
In the inside of the utriculus is found the auditory
sand, consisting of little crystals of lime connected by
means of a mucous mass with each other and with the
thickened places where the nerves are so abundant.
In the hollow of the bony vestibule, near the utriculus,
and fastened to it, but not communicating with it, lies
the sacculus, also provided with a similar thickened^
utricaina and inembmiions aemicir- gp^^ f^n Qf nervos. A narrow caual counccts it with
cQur canals (left side) seen from *^
witboQt. TH front, rp back rertioai, the caual of the membrauous cochlea. As to the cavity
h horizontal semicircular canal. • ,, 1 « •■ /> 'x n j '^ •
of the cochlea, we see by ng. 43 opposite, that it is
exactly similar to the shell of a garden snail ; but the canal of the cochlea is
divided into two almost completely separated galleries, by a transverse partition,
partly bony and partly membranous. These galleries communicate only at the
vertex of the cochlea through a small opening, the hSlicotrema, bounded by the
hamulus or hook-shaped termination of its central axis or viddl'dltis. Of the two
galleries into which the cavity of the bony cochlea is divided, one communicates
directly with the vestibule and is hence called the vestibule gallery (scala vestibuli).
The other gallery is cut off from the vestibule by the membranous partition, but
jtist at its base, where it begins, is the round window, and the yielding membrane,
which closes this, allows the fluid in the gallery to exchange vibrations with the
air in the drum. Hence this is called the drum gallery (scala tymp&ni). 5!
Finally, it must be observed that the membranous partition of the cochlea is
not a single membrane, but a membranous canal (ductus cochlearis). Its inner
margin is turned towards the central axis or
m6di6lus, and attached to the rudimentary
bony partition (l&mina spirahs). A part of
the opposite external surface is attached to the
inner surface of the bony gallery. Fig. 45
shews the bony parts of a cochlea which has
been laid open, and flg. 46 (p. 138a), a trans-
verse section of the canal (which is imperfect
on the left hand at bottom). In both figures
Ls denotes the bony part of the parUtion, and
in fig. 46 V and b are the two unattached parts
of the membranous canal. The transverse ^
section of this canal is, as the figure shews,
nearly triangular, so that an angle of the
triangle near Lis is attached to the edge of
the bony partition. The commencement of
the ductus cochlearis at the base of tlie
cochlea, communicates, as already stated, by
means of a narrow membranous canal with
the sacculus in the vestibule. Of the two un-
attached strips of its membranous walls, the
one facing the vestibule gallery is a soft mem-
brane, incapable of offering much resistance — Beissner's membrane (membrana vesti-
bularis, V, fig. 46, p. 138a) ; but the other, the membrana bdslldris (b), is a firm,
tightly stretched, elastic membrane, striped radially, corresponding to its radial
fibres. It splits easily in the direction of these fibres, shewing that it is but loosely
Digitized by V^jOOQ IC
Pig. 45-
Bony cochlea (right side) opened in front. Md,
uodlSIud. Ls, lAmTna splr&Iis. H, h&nin u.4.
?ec, fenestra cochleae, t Section of the partition
of the cochlea, tt Upper extremity of the Mime.
138
COCHLEA OF THE EAR.
FABT I,
connected in a direction transverse to them. The terminations of the nerves of
the cochlea and their appendages, are attached to the membrana b&silaris, as is
shewn by the dotted hnes in fig. 46.
When the drumskin is driven inwards by increased pressure of air in the auditory
passage, it also forces the auditory ossicles inwards, as already explained, and as a
consequence the foot of the stiiTup ^
Transyerse section of a spire of a cochlea which has been
softened in hydrochloric acid. Ls, lAmTna spIHUis. Lis,
limbos Iftmlnae splr&lis. Sr, sc&la TestlbOlI. St, scala
tAmpftnl. Dc, ductus oochleArls. Lsp, ITgftmentum
splr&le. T, membrflna TentibulAris. b, membrftna b&sUAris.
e, outer wall of the ductus oochleftrls. * its fillet. The
dotted lines shew sections of the membrftna tectAria and
the auditory rods.
penetrates deeper into the oval window
The fluid of the labyrinth, being sur-
rounded in all other places by firm
bony walls, has only one means of
escape, — the round window with its
yielding membrane. To reach it, the
fluid of the labyrinth must either pass
% through the hSlicdtrema, the narrow
opening at the vertex of the cochlea,
flowing over from the vestibule gallery
into the drum gallery, or, as it would
probably not have sufficient time to do
this in the case of sonorous vibrations,
press the membranous partition of the
cochlea against the drum gallery. The
converse action must take place when
the air in the auditory passage is rare-
fied.
Hence the sonorous vibrations of the air in the outer auditory passage are
finally transferred to the membranes of the labyrinth, more especially those of the
cochlea, and to the expansions of the nerves upon them.
51 The terminal expansions of these nerves, as I have already mentioned, are con-
nected with very small elastic appendages, which appear adapted to excite the
nerves by their vibrations.
The nerves of the vestibule terminate in the thickened places of the bags of
the membranous labyrinth, already mentioned
(p. 137a), where the tissue has a greater and
almost cartilaginous consistency. One of these
places provided with nerves, projects hke a fillet
into the inner part of the ampulla of each semi-
circular canal, and another lies on each of the little
bags in the vestibule. The nerve fibres here enter
between the soft cylindrical cells of the fine cuticle
(Spithglium) which covers the internal surface of
the fillets. Projecting from the internal surface
^ of this epithelium in the ampullae. Max Schultze
discovered a number of very peculiar, stiff, elastic
hairs, shewn in fig. 47. They are much longer
than the vibratory hairs of the ciliated epithelium
(their length is ^y of a Paris line, [or '00355
English inch,] in the ray fish), brittle, and running
to a very fine point. It is clear that fine stiff
hairs of this kind are extremely well adapted for
moving sympathetically with the motion of the
fluid, and hence for producing mechanical irri-
tation in the nerve fibres which He in the soft
epithelium between their roots.
According to Max Schultze, the corresponding
thickened fillets in the vestibules, where the nerves tenninate, have a similar soft
epithelium, and have short hairs which are easily destroyed. Close to these
Digitized by VjOOQlC
Pio. 47.
CHAP. VI.
COCHLEA OP THE EAR.
139
¥1G. 48.
surfjAces, which are covered with nerves, lie the calcareous concretions, called
auditory stones (6t6lith8), which in fishes form connected convexo-concave solids,
aliewing on their convex side an impression of the nerve fillet. In human heings,
on the other hand, the otoliths are heaps of little crystalline bodies, of a longish
angular form, lying close to the membrane of the little bags, and apparently
attached to it. These otoUths seem also extremely well suited for producing a
mechanical irritation of the nerves whenever the fluid in the labyrinth is suddenly
agitated. The fine light membrane, with its interwoven nerves, probably instantly
follows the motion of the fluid, whereas the heavier crystals are set more slowly in
motion, and hence also yield up their motion more slowly, and thus partly drag
and partly squeeze the adjacent nerves. This would satisfy the same conditions
of exciting nerves, as Heidenhain^s tetdnomdtor. By this instrument the nerve
which acts on a muscle is exposed to the action of a very rapidly oscillating ivory
hanuner, which at every blow squeezes without bruising the nerve. A powerful
and continuous excitement of the nerve is thus produced, which is shewn by a ^
powerful and continuous contraction of the corresponding muscle. The above parts
of the ear seem to be well suited to produce similar mechanical excitement.
The construction of the cochlea is much more complex. The nerve fibres enter
through the axis or modiolus of the cochlea into the bony part of the partition,
and then come on to the membranous part. Where they reach this, peculiar
formations were discovered quite recently (1851) by the Marchese Corti, and have
been named after him. On these the nerves terminate.
The expansion of the cochlean nerve is shewn in fig. 48. It enters through
the axis (2) and sends out its fibres in a radial direction from the axis through the
bony partition (1,3, 4),
as far as its margins.
At this point the nerves
pass under the com-
mencement of the mem- %
brana basilaris, pene-
trate this in a series of
openings, and thus reach
the ductus cochlearis
and those nervous,
elastic formations which
lie on the inner zone
(Zi) of the membrane.
The margin of the
bony partition (a to b,
flg* 49* P* 140a), and the
inner zone of the mem-
brana basilaris (a a') are shewn after Hensen. The under side of the figure *
corresponds with the scala tymp&ui, the upper with the ductus cochlearis. Here h ^
at the top and k at the bottom, are the two plates of the bony partition, between
which the expansion of the nerve b proceeds. The upper side of the bony parti-
tion bears a fillet of close hgamentous tissue (Z, fig. 49, also shewn at Lis, fig. 46,
p. 138a), which, on account of the toothlike impressions on its upper side, is called
the toothed layer {zo'na denticula'ta), and which carries a peculiar elastic pierced
membrane, Corti'B membrane, M.C. fig. 49. This membrane is stretched parallel
to the membrana basilaris as far as the bony wall on the outer side of the duct,
and is there attached a little above the other. Between these two membranes
lie the parts in and on which the nerve fibres terminate.
Among these Cortl's arches (over g in fig. 49) are relatively the most solid
formations. The series of these contiguous arches consists of two series of rods
* [As the engraving would have been too the left side consequently corresponds to the
wide for the page if placed in its proper hori- upper, and its right to the wndcrsidc.— jTm/w-
zontal position^ it has been printed vertically; lator.] . ,, i OOOIC
14©
COCHLEA OF THE EAR.
PABT I.
W'
OT fibres, an external and an internal. A single pair of these is shewn in fig. 50,
A, below, and a short series of them in fig. 50, B, attached to the membrana
basilaris, and at t also connected with the pierced ^,0. ^
tissue, into which fit the terminal cells of the nerves
(fig. 49, c), which will be more folly described pre-
sently. These formations are shewn in fig. 51,
(p. 1416, c), as seen from the vestibule gallery ; a is
the denticulated layer, c the openings for the
nerves on the internal margin of the membrana
basilaris, its external margin being visible at u u ;
d is the inner series of Corti's rods, e the outor ;
over these, between e and x is seen the pierced
membrane, against which lie the terminal cells of
% the nerves.
The fibres of the first, or outer series, are flat,
somewhat S-shaped formations, having a swelling
at the spot where they rise from the membrane to
which they are attached, and ending in a kind of
articulation which serves to connect them with the
second or inner seiies. In fig. 51, p. 141, at d
will be seen a great number of these ascending
fibres, lying beside each other in regular succession.
In the same way they may be seen all along the
membrane of the cochlea, close together, so that
there must be many thousands of them. Their
sides lie close together, and even seem to be con-
nected, leaving however occasional gaps in the line
% of connection, and these gaps are probably tra-
versed by nerve fibres. Hence the fibres of the
first series as a whole form a stiff layer, which
endeavours to erect itself when the natural fasten-
ings no longer resist, but allows the membrane on
which they stand to crumple up between the at-
tachments d and e of Corti's arches.
The fibres of the second, or inner series, which
form the descending part of the arch e, fig. 50,
below, are smooth, flexible, cylindrical threads
with thickened ends. The upper extremity forms
a kind of joint to connect them with the fibres
of the first series, the lower extremity is enlarged in a bell shape and is attached
closely to the membrane at the base. In the microscopic preparations they gene-
H Fro. 50.
A B
A, external and internal rod in connection seen In profile. B, membrana basilaris (b) with the
terminal fiiscTcQll of nerres (n), and the internal and external rodft (i and e). i internal,
3 external cells of the floor, 4' attachments of the cells of the corer. * * cpith^lum.
rally appear bent in various ways ; but there can be no doubt tliat in their natural
condition they are stretched with some degree of tension, so that they pull down
Digitized by V^jOOQlC
CHAP. VI.
COCHLEA OF THE EAR.
141
the upper jointed ends of the fibres of the first series. The fibres of the first
series arise from the inner margin of the membrane, which can be relatively little
agitated, but the fibres of the second series are attached nearly in the middle of the
membrane, and this is precisely the place where its vibrations will have the greatest
excursions. When the pressure of the fluid in the drum gallery of the labyrinth
is increased by driving the foot of the stirrup against the oval window, the mem-
brane at the base of the arches will sink downwards, the fibres of the second series
be more tightly stretched, and perhaps the corresponding places of the fibres of the
first series be bent a little downwards. It does not, however, seem probable that
the fibres of the first series themselves move to any great extent, for their lateral
connections are strong enough to make them hang together in masses like a
membrane, when they have been released from their attachment in anatomical
preparations. On reviewing the whole arrangement, there can be no doubt that
Corti*s organ is an apparatus adapted for receiving the vibrations of the membrana f
rrTrr
basilaris, and for vibrating of itself, but our present knowledge is not sufficient to
determine with accuracy the manner in which these vibrations take place. For
this purpose we require to estimate the stability of the several parts and the degree
of tension and flexibility, with more precision than can be deduced from such
observations as have hitherto been made on isolated parts, as they casually group
themselves under the microscope.
Now Corti's fibres are wound round and covered over with a multitude of very
dehcate, frail formations, fibres and cells of various kinds, partly the finest ter-
minational runners of nerve fibres with appended nerve cells, partly fibres of liga-
mentous tissue, which appear to serve as a support for fixing and suspending the
nerve formations.
The connection of these parts is best shewn in fig. 49 opposite. They are
grouped like a pad of soft c^lls on each side of and within Corti's arches. The
most important of them appear to be the cells c and d, which are furnished with
Digitized by V^ O OQ IC
H
142 DAMPING OF THE VIBRATIONS IN THE EAR. parti.
hairs, precisely resembling the ciliated cells in the ampullae and utriculus. They
appear to be directly connected by fine varicose nerve fibres, and constitute the
most constant part of the cochlean formations ; for with birds and reptiles, where
the structure of the cochlea is much simpler, and even Corti's arches are absent,
these little ciliated cells are always to be found, and their hairs are so placed as to
strike against Corti*s membrane during the vibration of the membrana basilaris.
The cells at a and a', fig. 49 (p. 140), which appear in an enlarged condition at b
and n in fig. 51 (p. 141), seem to have the character of an epithelium. In fig. 51
there will also be observed bundles and nets of fibres, which may be partly merely
supporting fibres of a ligamentous nature, and may partly, to judge by their appear-
ance as strings of beads, possess the character of bundles of the finest fibriles of
nerves. But these parts are all so frail and delicate that there is still much
doubt as to their connection and office,
f The essential result of our description of the ear may consequently be said to
consist in having found the terminations of the auditory nerves everywhere con-
nected with a pecuhar auxiliary apparatus, partly elastic, partly firm, which may be
put in sympathetic vibration under the influence of external vibration, and will then
probably agitate and excite the mass of nerves. Now it was shewn in Chap. HI.,
that the process of sympathetic vibration was observed to differ according as the
bodies put into sympathetic vibration were such as when once put in motion con-
tinued to sound for a long time, or soon lost their motion, p. 39c. Bodies which,
like timing-forks when once struck, go on sounding for a long time, are susceptible
of sympathetic vibration in a high degree notwithstanding the difficulty of putting
their mass in motion, because they admit of a long accumulation of impulses in
themselves minute, produced in them by each separate vibration of the exciting
tone. But precisely for this reason there must be the exactest agreement between
the pitches of the proper tone of the fork and of the exciting tone, because other-
% wise subsequent impulses given by the motion of the air could not constantly recur
in the same phase of vibration, and thus be suitable for increasing the subsequent
effect of the preceding impulses. On the other hand if we take bodies for which
the tone rapidly dies away, such as stretched membranes or thin light strings, we
find that they are not only susceptible of S3rmpathetic vibration, when vibrating
air is allowed to act on them, but that this sympathetic vibration is not so limited
to a particular pitch, as in the other case, and they can therefore be easily set in
motion by tones of different kinds. For if an elastic body on being once struck
and allowed to sound freely, loses nearly the whole of its motion after ten vibra-
tions, it will not be of much importance that any fresh impulses received after the
expiration of this time, should agree exactly with the former, although it would be
of great importance in the case of a sonorous body for which the motion generated
by the first impulse would remain nearly unchanged up to the time that the second
impulse was apphed. In the latter case the second impulse could not increase the
^ amount of motion, unless it came upon a phase of the vibration which had
pi'eeisely the same direction of motion as itself.
The coimection between these two relations can be calculated independently of
the nature of the body put into sympathetic vibration,* and as the results are im-
portant to enable us to form a judgment on the state of things going on in the ear,
a short table is annexed. Suppose that a body which vibrates sympathetically has
been set into its state of maximum vibration by means of an exact unison, and
that the exciting tone is then altered till the sympathetic vibration is reduced to
^ of its former amount. The amount of the required difference of pitch is given in
the first column in terms of an equally tempered Tone, [which is I of an Octave].
Now let the same sonorous body be struck, and let its sound be allowed to die
away gradually. The number of vibrations which it has made by the time that its
intensity is reduced to ^ of its original amount is noted, and given in the second
column.
* The mode of calculation in explained in Appendix X.
Digitized by VjOOQIC
CHAP. VI. DAMPING OF THE VIBRATIONS IN THE EAR.
143
IMfference of Pitch, in terms of an eqnally tempered Tone, neces-
maj to rednoe the intensity of sympathetic Tlbnition to ^ of that
luroduoed by perfect uuisonance
Number of vibrations after which
the intensity of tone in a sonorous
body whooe sound is allowed to die
out, reduces to t^ of its original
amount
1. One eighth of a Tone
2. One quarter of a Tone
3. One Semitone
4. Three quarters of a Tone
5. A whole Tone
6. A Tone and a quarter
7. A tempered minor Third or a Tone and a half. .
8. A Tone and three quarters
9. A tempered major Third or two whole Tones .
3800
19*00
950
6-33
3-8o
3*17
271
2-37
Now, although we are not able exactly to discover how long the ear and its
individual parts, when set in motion, will continue to sound, yet well-known «|
experiments allow us to form some sort of judgment as to the position which
the parts of the ear must occupy in the scale exhibited in this table. Thus, there
cannot possibly be any parts of the ear which continue to sound so long as a
tuning-fork, for that would be patent to the commonest observation. But even if
there were any parts in the ear answering to the first degree of our table, that is
requiring 38 vibrations to be reduced to -jJ^ of their force, — we should recognise
this in the deeper tones, because 38 vibrations last ^ of a second tor A, ^ for a,
^ for a\ &c., and such a long endurance of sensible sound would render rapid
musical passages impossible in the unaccented and once-accented Octaves. Such
a state of things would disturb musical effect as much as the strong resonance of
a vaulted room, or as raising the dampers on a piano. When making a shake, we
can readily strike 8 or 10 notes in a second, so that each tone separately is struck
from 4 to 5 times. If, then, the sound of the first tone had not died off in our ear
before the end of the second sound, at least to such an extent as not to be sensible ^r
when the latter was sounding, the tones of the shake, instead of being individually
distinct, would merge into a continuous mixture of both. Now shakes of this kind,
with 10 tones to a second, can be clearly and sharply executed throughout almost
the whole scale, although it must be owned that from A downwards, in the great
and contra Octaves they sound bad and rough, and their tones begin to mix. Yet
it can be easily shewn that this is not due to the mechanism of the instrument.
Thus if we execute a shake on the harmonium, the keys of tlie lower notes are
just as accurately constructed and just as easy to move as those of the higher
ones. Each separate tone is completely cut off with perfect certainty at the
moment the valve &lls on the air passage, and each speaks at the moment the valve
is rfdsed, because during so brief an interruption the tongues remain in a state of
vibration. Similarly for the violoncello. At the instant when the finger which
makes the shake &lls on the string, the latter must commence a vibration of a
different periodic time, due to its length ; and the instant that the finger is m
removed, the vibration belonging to the deeper tone must return. And yet the
shake in the bass is as imperfect on the violoncello as on any other instrument.
Bans and shakes can be relatively best executed on a pianoforte because, at the
moment of striking, the new tone soimds with great but rapidly decreasing inten-
sity. Hence, in addition to the inharmonic noise produced by the simultaneous
continuance of the two tones, we also hear a distinct prominence given to each
separate tone. Now, since the difficulty of shaking in the bass is the same for all
instruments, and for individual instruments is demonstrably independent of the
manner in which the tones are produced, we are forced to conclude that the
difficulty lies in the ear itself. We have, then, a plain indication that the vibrating
parts of the ear are not damped with sufficient force and rapidity to allow of
successfully effecting such a rapid alternation of tones.
Nay more, this fact further proves that there must be different parts of the ear
vrhich are set in vihration by tones of different 2)itch and which receive the sejisntion
Digitized by V^OOQIC
144 DAMPING OF THE VIBRATIONS IN THE EAE. parti.
of these tones. Thus, it might be supposed that as the vibratory mass of the -whole
ear, the drumskin, auditory ossicles, and fluid in the labyrinth, were vibratinf? at
tlie same time, the inertia of this mass was the cause why the sonorous vibrations
in the ear were not immediately extinguished. But this hypothesis would not
suffice to explain the &ct observed. For an elastic body set into sympathetic
vibration by any tone, vibrates sympathetically in the pitch number of the exciting
tone ; but as soon as the exciting tone ceases, it goes on sounding in the pitch
number of its own proper tone. This fact, which is derived from theory, may be
perfectly verified on tuning-forks by means of the vibration microscope.
If, then, the ear vibrated as a single system, and were capable of continuing
its vibration for a sensible time, it would have to do so with its own pitch number,
which is totally independent of the pitch number of the former exciting tone.
The consequence is that shakes would be equally difficult upon both high and
^' low tones, and next that the two tones of the shake would not mix with each
other, but that each would mix with a third tone, due to the ear itself. We became
acquainted with such a tone in the last chapter, the high /'", p. 1 1 6a. The result,
then, under these circumstances would be quite different from what is observed.
Now if a shake of lo notes in a second, be made on ^, of which the vibra-
tional number is no, this tone would be struck every Ji of a second. We may
justly assume that the shake would not be clear, if the intensity of the expiring
tone were not reduced to ^^ of its original amount in this ^ of a second. In this
case, after at least 22 vibrations, the parts of the ear which vibrate sympathetically
with A must descend to at least ^V ^^ their intensity of vibration as their tone
expires, so that their power of sympathetic vibration cannot be of the first degree
in the table on p. 143a, but may belong to the second, third, or some other higher
degree. That the degree cannot be any much higher one, is shewn in the first
place by the fact that shakes and runs begin to be difficult even on tones which do
^ not lie much lower. This we shall see by observations on beats subsequently de-
tailed. We may on the whole assume that the parts of the ear which vibrate
sympathetically have an amount of damping power corresponding to the third
degree of our table, where the intensity of sympathetic vibration with a Semitone
difference of pitch is only ^ of what it is for a complete unison. Of course there
can be no question of exact determinations, but it is important for us to be able
to form at least an approximate conception of the influence of damping on the
sympathetic vibration of the ear, as it has great significance in tlie relations
of consonance. Hence when we hereafter speak of individual parts of the ear
vibrating sympathetically with a determinate tone, we mean that they are set into
strongest motion by that tone, but are also set into vibration less strongly by tones
of nearly the same pitch, and that this sympathetic vibration is still sensible for
the interval of a Semitone. Fig. 52 may serve
to give a general conception of the law by which
If the intensity of the sympathetic vibration de-
creases, as the difference of pitch increases. The
horizontal line a b c represents a portion of the
musical scale, each of the lengths a b and b c
standing for a whole (equally tempered) Tone.
Suppose that the body which vibrates sympa-
thetically has been tuned to the tone b and that
the vertical line b d represents the maximum
of intensity of tone which it can attain when excited by a tone in perfect unison
with it. On the base line, intervals of ^^^ of a whole Tone are set off, and the ver-
tical lines drawn through them shew the corresponding intensity of the tone in the
body which vibrates sympathetically, when the exciting tone differs from a unison
by the corresponding interval. The following are the numbers from which fig. 52
was constructed : —
Digitized by
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CHAP. VI. THEOBY OP THE FUNCTION OP THE COCHLEA.
145
Difleronce of Pitch
Intensity
of Sympathetio Vibration
Difference of Pitch
Intensity
of Sympathetic VibraUon
0*0
O'l
0-2
0-3
0-4
Semitone
lOO
74
41
24
15
10
0-6
Whole Tone
7-2
5-4
4-2
3*3
2-7
Now we cannot precisely ascertain what parts of the ear actually vibrate sym-
pathetically with individual tones.* We can only conjecture what they are at
present in the case of human beings and mammals. The whole construction of
the partition of the cochlea, and of Corti's arches which rest upon it, appears most
suited for executing independent vibrations. We do not need to require of them
the power of continuing their vibrations for a long time without assistance. ^
But if these formations are to serve for distinguishing tones of dififerent pitch,
and if tones of different pitch are to be equally well perceived in all parts of the
scale, the elastic formations in the cochlea, which are connected with different
nerve fibres, must be differently tuned, and their proper tones must form a regu-
larly progressive series of degrees through the whole extent of the musical scale.
According to the recent anatomical researches of V. Hensen and G. Hasse, it
is probably the breadth of the membrana basilaris in the cochlea, which deter-
mines the tuning.f At its commencement opposite the oval window, it is
comparatively narrow, and it continually increases in width as it approaches the
apex of the cochlea. The following measurements of the membrane in a newly
bom child, from the line where the nerves pass through on the inner edge, to the
attachment to the ligamentum spirale on the outer edge, are given by V. Hensen :
Place of Section
Breadth of Membnuie or Length of Trans-
verse Fibres
Millimetres
Inches
0-2625 mm. [« 0-010335 i»-J from root .
0*8626 mm. [=0*033961 in.J from root .
Middle of the first spire
End of first spire
Middle of second spire ......
End of second spire
At the hamalos
0-04125
0-0825
0-169
0-3
0-4125
0-45
0-495
•00162
•00325
•00665
•OII81
•01624
•01772
•01949
The breadth therefore increases more than twelvefold from the beginning to
the end.
Corti's rods also exhibit an increase of size as they approach the vertex of the
cochlea, but in a much less degree than the membrana basilaris. The following
are Hensen's measurements : —
at the round window
at the hunolus
nun.
inch
mm.
inch
Length of inner rod .
Length of outer rod ....
Span of the arch ....
0-048
0-048
0*019
000189
000189
0-00075
fit
0-C0337
0-00386
0-00335
* [Here the passage, *The particles of
anditory sand,' to 'used for musical tones,*
on pp. 217-18 of the ist English edition has
been cancelled, and the passage *W6 can only
oonjectnre,' to * without assistance,' on p. 145a
added in its place from the 4th German edition.
— Translator.']
t In the 1st [German] edition of this book
(1863), which was written at a time when the
more delicate anatomy of the cochlea was just
beginning to be developed, I supposed thai the
different degrees of stiffness and tension in
Corti's rods themselves might furnish the
reason of their different tuning. By Hensen's
measures of the breadth of the membrana
basilaris {Ztitschrift filr wissenseh. Zoologie,
voL ziii. p. 492) and Hasse's proof that Corti's
rods are absent in birds and amphibia, far more
definite foundations for forming a judgment
have been furnished, than I then possessed.
Digitized by V^jOOQlC
146 THEORY OP THE FUNCTION OF THE COCHLEA. part u
Hence it follows, as Henle has also proved, that the greatest increase of breadth
falls on the outer zone of the basilar membrane, beyond the line of the attach-
ment of the outer rods. This increases from 0*023 mm. [='000905 in.] to 0*41
nun. [= -016 142 inch] or nearly twentyfold.
In accordance with these measures, the two rows of Corti's rods are almost
parallel and upright near to the roimd window, but they are bent much more
strongly towards one another near the vertex of the cochlea.
It has been already mentioned that the membrana basilaris of the cochlea
breaks easily in the radial direction, but that its radial fibres have considerable
tenacity. This seems to me to furnish a very important mechanical relation,
namely that this membrane in its natural connection admits of being tightly
stretched in the transverse direction from the modiolus to the outer wall of the
cochlea, but can have only little tension in the direction of its length, because it
^ could not resist a strong puU in this direction.
Now the mathematical theory of the vibration of a membrane with different ten-
sions in different directions shews that it behaves very differently from a membrane
which has the same tension in all directions.* On the latter, vibrations produced
in one part, spread uniformly in all directions, and hence if the tension were uniform
it would be impossible to set one part of the basilar membrane in vibration, without
producing nearly as strong vibrations (disregarding individual nodal lines) in all other
parts of the membrane.
But if the tension in direction of its length is infinitesimally small in com-
parison with the tension in direction of the breadth, then the radial fibres of
the basilar membrane may be approximatively regarded as forming a system of
stretched strings, and the membranous connection as only serving to give a ful-
crum to the pressure of the fluid against these strings. In that case the laws of
their motion would be the same as if every individual string moved independently
4|[ of all the others, and obeyed, by itself, the influence of the periodically alternating
pressure of the fluid of the labyrinth contained in the vestibule gallery. Conse-
quently any exciting tone would set that part of the membrane into sympathetic
vibration, for which the proper tone of one of its radial fibres that are stretched
and loaded with the various appendages already described, corresponds most nearly
with the exciting tone ; and thence the vibrations will extend with rapidly dimi-
nishing strength on to the adjacent parts of the membrane. Fig. 52, on p. 14402,
might be taken to represent, on an exaggerated scale of height, a longitudinal sec-
tion of that part of the basilar membrane in which the proper tone of the radial
fibres of the membrane are nearest to the exciting tone.
The strongly vibrating parts of the membrane would, as has been explained in
respect to all bodies which vibrate sympathetically, be more or less limited, accord-
ing to the degree of damping power in the adjacent parts, by friction against the
fluid in the labyrinth and in the soft gelatinous parts of the nerve fillet.
^ Under these circumstances the parts of the membrane in unison with higher
tones must be looked for near the round window, and those with the deeper, near
the vertex of the cochlea, as Hensen also concluded from his measurements. That
such short strings should be capable of corresponding with such deep tones, must
be explained by their being loaded in the basilar membrane with all kinds of solid
formations ; the fluid of both galleries in the cochlea must also be considered as
weighting the membrane, because it cannot move without a kind of wave motion
in that fluid.
The observations of Hasse shew that Corti's arches do not exist in the cochlea
of birds and amphibia, although the other essential parts of the cochlea, as the
basilar membrane, the ciliated cells in coimection with the terminations of the
nerves, and Corti's membrane, which stands opposite the ends of these ciliae, are
all present. Hence it becomes very probable that Corti's arches play only a
secondary part in the function of the cochlea. Perhaps we might look for the effect
* See Appendix XI.
Digitized by VjOOQIC
CHAP. VI. THEORY OF THE FUNCTION OF THE COCHLEA. 147
of Corti's arches in their power, as relatively firm objects, of transmitting the
vibrations of the basilar membrane to small limited regions of the upper part of
the relatively thick nervous fillet, better than it could be done by the immediate
communication of the vibrations of the basilar membrane through the soft mass
of this fillet. Close to the outside of the upper end of the arch, connected with
it by the stiffer fibriles of the membrana reticularis, are the ciliated cells of the
nervous fillet (see c in fig. 49, p. 140). In birds, on the other hand, the ciliated cells
form a thin stratum upon the basilar membrane, and this stratum can readily
receive limited vibrations from the membrane, without communicating them too
&r sideways.
According to this view Corti's arches, id the last resort, will be the means of
transmitting the vibrations received from the basilar membrane to the terminal
appendages of the conducting nerve. In this sense the reader is requested here-
after to understand references to the vibrations, proper tone, and intonation of ^
Corti's arches ; the intonation meant is that which they receive through their
connection with the corresponding part of the basilar membrane.
According to Waldeyer there are about 4500 outer arch fibres in the human
cochlea. If we deduct 300 for the simple tones which lie beyond musical limits,
and cannot have their pitch perfectly apprehended, there remain 4200 for the
seven octaves of musical instruments, that is, 600 for every Octave, 50 for every
Semitone [that is, i for every 2 cents] ; certainly quite enough to explain the
power of distinguishing small parts of a Semitone.* According to Prof. W.
Preyer's investigations,t practised musicians can distinguish with certainty a
difference of pitch arising from half a vibration in a second, in the doubly
accented Octave. This would give 1000 distinguishable degrees of pitch in the
Octave, from 500 to 1000 vibrations in the second. Towards the Umits of the
scale the power to distinguish differences diminishes. The 4200 Corti'd arches
appear then, in this respect, to be enough to apprehend distinctions of thiscr
amount of delicacy. But even if it should be found that many more than
4200 d^rees of pitch could be distinguished in the Octave, it would not prejudice
our assumption. For if a simple tone is struck having a pitch between those of
two adjacent Corti's arches, it would set them both in sympathetic vibration, and
that arch would vibrate the more strongly which was nearest in pitch to the
proper tone. The smallness of the interval between the pitches of two fibres still
distinguishable, will therefore finally depend upon the delicacy with which the
different forces of the vibrations excited can be compared. And we have thus
also an explanation of the fact that as the pitch of an external tone rises con-
tinuously, our sensations also alter continuously and not by jumps, as must be the
case if only one of Corti's arches were set in sympathetic motion at once.
To draw farther conclusions from our hypothesis, when a simple tone is pre-
sented to the ear, those Corti's arches which are nearly or exactly in unison with
it will be strongly excited, and the rest only slightly or not at aJl. Hence every ir
simple tone of determinate pitch will be felt only by certain nerve fibres, and
♦ [A few lines of the ist Engligh edition at rib. a difference of orlitenralof
have here been cancelled, and replaced by S^ -300 vib. ro cents 1t/?a5 per-
others from the 4th German edition.— Traw- ^ '00° , '5«> ». '9 ,. jceived.
l^ilfyr 1 but on the other hand
« Jh7 (On the limits of the perception of ,^ .^ ^'^- 6 cents vaa
tone), June 1876. Rearranged in English by "° !^' " l^ " ^^
the Translator in the Proceedings of the r^ '^ " '° '» P ,,
^T^u iT^^!^ *!?' H ^^ v^* PP- V?^' the intervals perceii^d. or not ^r;;jiv^! blSfg
under the title of 'On the Sensitiveness of the ^^^ ^^^^y^^ ^^^^ diiferS^t And geL"^
EartoPitchandChangeofPitehm Music/ On ^^^j throughout the scale a difference of fvib.
p. II of this arrangement It is stated that. .^ ^^ j^^^^l ^^^ ^<^« .
includmg Delezenne's results. ^^^^ q^ ^ . a ^y, ■^
•t Tib. a difference of or Interral of __ j *._„ '* . \^ J I . ,^.
^^ J -r t. T fi ». ~-Tra7islator.]
Digitized by ^0©gie
148
THEORY OP THE FUNCTION OP THE COCHLEA. pabt i.
1.T.O \>
7.1 h
4
simple tones of different pitch will excite different fibres. When a compound
musical tone or chord is presented to the ear, all those elastic bodies will be
excited, which have a proper pitch corresponding to the various individual simple
tones contained in the whole mass of tones, and hence by properly directing
attention, all the individual sensations of the individual simple tones can be
perceived. The chord must be resolved into its individual compound tones, and
the compound tone into its individual harmonic partial tones.
This also explains how it is that the ear resolves a motion of the air into
pendular vibrations and no other. Any particle of air can of course execute only
one motion at one time. That we considered such a motion mathematically as a
sum of pendular vibrations, was in the first instance merely an arbitrary assump-
tion to facilitate theory, and had no meaning in nature. The first meaning in
nature that we found for this resolution came from considering sympathetic
^vibration, when we discovered that a motion which was not pendular, could
produce sympathetic vibrations in bodies of those different pitches, which cor-
responded to the harmonic upper partial tones. And now our hypothesis has also
reduced, tjie phenomenon of hearing to that of sympathetic xJliration^and thus
furnished a reason why an originally simple periodic vibifttipn^ of the au: pro-
duces a sum of different sensations, and hence also a^tpeaxs as compound to our
gerceptionp.
The sensation of different pitch would consequently be a sensation in different
nerve fibres. The sensation of a quality of tone would depend upon the power of
a given compound tone to set in vibration not only those of Corti*s arches which
correspond to its prime tone, but also a series of other arches, and hence to excite
sensation in several different groups of nerve fibres.
Physiologically it should be observed that the present assumption reduces
sensations which differ qualitatively according to pitch and quality of tone, to a
^ difference in the nerve fibres which are excited. This is a step similar to that
taken in a wider field by Johannes Miiller in his theory of the specific energies of
sense. He has shewn that the difference in the sensations due to various senses,
does not depend upon the actions which excite them, but upon the various nervous
arrangements which receive them. We can convince ourselves experimentally
that in whatever manner the optic nerve and its expansion, the retina of the eye,
may be excited, by light, by twitching, by pressure, or by electricity, the result is
never anything but a sensation of light, and that the tactual nerves, on the contrary,
never give us sensations of hght or of hearing or of taste. The same solar rays
which are felt as light by the eye, are felt by the nerves of the hand as heat ; the
same agitations which are felt by the hand as twitterings, are tone to the ear.
Just as the ear apprehends vibrations of different periodic time as tones of
different pitch, so does the eye perceive luminiferous vibrations of different periodic
time as different colours, the quickest giving violet and blue, the mean green and
^ yellow, the slowest red. The laws of the mixture of colours led Thomas Young
to the hypothesis that there were three kinds of nerve fibres in the eye, with
different powers of sensation, for feeling red, for feeling green, and for feeling
violet. In reality this assumption gives a very simple and perfectly consistent
explanation of all the optical phenomena depending on colour. And by this means
the qualitative differences of the sensations of sight are reduced to differences in
/the nerves which receive the sensations. For the sensations of each individual
fibre of the optic nerve there remains only the quantitative differences of greater or
less irritation.
The same result is obtained for hearing by the hypothesis to which our
investigation of quality of tone has led us. The qualitative difference of pitch
and quality of tone is reduced to a difference in the fibres of the nerve receiving
the sensation, and for each individual fibre of the nerve there remains only the
quantitative difierences in the amount of excitement.
The processes of irritation within the nerves of the muscles, by which their
contraction is determined, have hitherto been
more accessi
Digitized '
?,^^t3d?gl^^°8^*^
CHAP. VI. THEORY OF THE FUNCTION OF THE COCHLEA. 149
investigation than those which take place in the nerves of sense. In those of the
muscle, indeed, we find only quantitative differencea of- more or less excitement,
and TU) ^nftlifrfitivf^ jifTftrflt^(*<i>g at all. In them we are able to establish, that during
excitement the electrically active particles of the nerves undergo determinate (i
changes, and that these changes ensue in exactly the same way whatever be the \\
excitement which causes them. But precisely the same changes also take place in 1 1
an excited nerve of sense, although their consequence in this case is a sensation,
while in the other it was a motion ; and hence we see that the mechanism of the
process of irritation in the nerves of sense must be m every respect_similar to that j
in the nerves of motion. The two* hypotheses just explained really reduce the -2.Z2-
processes in the nerves of man's two principal senses, notwithstanding their
apparently involved qualitative ^dilbiences of sensations, to the same simple
gfiliprnft ^K^^}\ Msrh\oh wa Ara fftyniliflj in the nerves of motioiu, Nerves have been r^^^J*^
often and not unsuitably compared to telegraph wires. Such a wire conducts one;1f
kind of electric current and no other ; it may be stronger, it may be weaker, it may
move in either direction ; it has no other quahtative differences. Nevertheless,
according to the different kinds of apparatus with which we provide its termina-
tions, we can send telegraphic despatches, ring bells, explode mines, decompose
water, move magnets, magnetise iron, develop hght, and so on. So with the
nerves. The condition of excitement which can be produced in them, and is con-
ducted by them, is, so far as it can be recognised in isolated fibres of a nerve,
everywhere the same, but when it is brought to various parts of the brain, or
the body, it produces motion, secretions of glands, increase and decrease of the
quantity of blood, of redness and of warmth of individual organs, and also sensa-
tions of light, of hearing, and so forth. Supposing that every qualitatively
different action is produced in an organ of a different kind, to which also separate
fibres of nerve must proceed, then the actual process of irritation in individual
nerves may always be precisely the same, just as the electrical current in the tele- ^
graph wires remains one and the same notwithstanding the various kinds of
effects which it produces at its extremities. On the other hand, if we assume that
the same fibre of a nerve is capable of conducting different kinds of sensation, we
should have to assume that it admits of various kinds of processes of irritation,
and this we have been hitherto unable to establish.
In this respect then the view here proposed, like Young's hypothesis for the
difference of colours, has a still wider signification for the physiology of the
nerves in general.
Since the first publication of this book, the theory of auditory sensation here
explained, has received an interesting confirmation from the observations and
experiments made by V • Hensen * on the auditory apparatus of the Crustaceae.
These animals have bags of auditory stones (otoUths), partly closed, partly
opening outwards, in which these stones float freely in a watery fluid and are
supported by hairs of a peculiar formation, attached to the stones at one end, and, f
partly, arranged in a series proceeding in order of magnitude, from larger and
thicker to shorter and thinner. In many crustaceans also we find precisely
similar hairs on the open surface of the body, and these must be considered as
auditory hairs. The proof that these external hairs are also intended for hearing,
depends first on the similarity of their construction with that of the hairs in the
bags of otoliths ; and secondly on Hensen's discovery that the sensation of
hearing remained in the Mysis (opossum shrimp) when the bags of otoUths had
been extirpated, and the external auditory hairs of the antennae were left.
Hensen conducted the sound of a keyed bugle through an apparatus formed on
the model of the drumskin and auditory ossicles of the ear into the water of a
little box in which a specimen of Mysis was fastened in such a way as to allow
the external auditory hairs of the tail to be observed. It was then seen that
certain tones of the horn set certain hairs into strong vibration, and other tones
* Studien ilber das Gehdrorgan der Deca- and lLb\\\\i%i*^ ZeUschrift fUt loissenselmftliche
podcn, Leipzig, 1863. Benrinted from Siebold Zooiooi^, vol. xiii. • r\r^rsi/>
' Digitized by VJwwVlC
ISO THEORY OF THE FUNCTION OF THE COCHLEA. pabt i;
other hairs. Each hair answered to several notes of the horn, and from the
notes mentioned we can approximatively recognise the series of under tones of one
and the same simple tone. The results could not be quite exact, because the
resonance of the conducting apparatus must have had some influence.
Thus one of these hairs answered strongly to djj^ and d% more weakly to g,
and very weakly to O. This leads us to suppose that it was tuned to some pitch
between d'^ and d'% In that case it answered to the second partial of d' to d'jj^
the third of g to g$, the fourth of d to djj^, and the sixth of G to (r$. A second
hair answered strongly to aj{l and the adjacent tones, more weakly to djj^ and Ajj^
Its proper tone therefore seems to have been a$.
By these observations (which through the kindness of Herr Hensen I have
myself had the opportunity of verifying) the existence of such relations as we have
supposed in the case of the human cochlea, have been directly proved for these
f Crustaceans, and this is the more valuable, because the concealed position and
ready destructibility of the corresponding organs of the human ear give little hope
of our ever being able to make such a direct experiment on the intonation of its
individual parts.*
So far the theory which has been advanced refers in the first place only to
the lasting sensation produced by regular and continued periodical oscillations.
But as regards the perception of irregular motions of the mr, that is, of noises, it
is clear that an elastic apparatus for executing vibrations could not remain at
absolute rest in the presence of any force acting upon it for a time, and even a
momentary motion or one recurring at irregular intervals would suffice, if only
powerful enough, to set it in motion. The peculiar advantage of resonance over
proper tone depends- precisely on the fact that disproportionately weak individual
impulses, provided that they succeed each other in correct rhythm, are capable of
producing comparatively considerable motions. On the other hand, momentary
^ but strong impulses, as for example those which result from an electric spark, will
set every part of the basilar membrane into an almost equally powerful initial
motion, after which each part would die off in its own proper vibrational period.
By that means there might arise a simultaneous excitement of the whole of the
nerves in the cochlea, which although not equally powerful would yet be propor-
tionately gradated, and hence could not have the character of a determinate pitch.
Even a weak impression on so many nerve fibres wiU produce a clearer impression
than any single impression in itself. We know at least that small differences of
brightness are more readily perceived on large than on small parts of the circle of
vision, and little differences of temperature can be better perceived by plunging
the whole arm, than by merely dipping a finger, into the warm water.
Hence a perception of momentary impulses by the cochlear nerves is quite
possible, just as noises are perceived, without giving an especially sensible pro-
minence to any determinate pitch.
If If the pressure of the air which bears on the drumskin lasts a little longer, it
will favour the motion in some regions of the basilar membrane in preference to
other parts of the scale. Certain pitches will therefore be especially prominent.
This we may conceive thus : every instant of pressure is considered as a pressure
that will excite in every fibre of the basilar membrane a motion corresponding
to itself in direction and strength and then die off; and all motions in each
fibre which are thus excited are added algebraically, whence, according to cir-
cumstances, they reinforce or enfeeble each other.t Thus a uniform pressure
which lasts during the first half vibration, that is, as long as the first positive
excursion, increases the excursion of the vibrating body. But if it lasts longer
it weakens the effect first produced. Hence rapidly vibrating bodies would be
proportionably less excited by such a pressure, than those for which half a vibra-
tion lasts as long as, or longer than, the pressure itself. By this means such an
* [From here to the end of this chapter is f See the mathematical expression for this
an addition from the 4th German edition,— conception at the end of Appendix XI.
Trajislaior.'
Digitized by V^jOOQiC
CHAP. VI. THEORY OF THE FUNCTION OF THE COCHLEA. 151
impression would acquire a certain, though an ill-defined, pitch. In general the
intensity of the sensation seems, for an equal amount of vis viva in the motion, to
increase as the pitch ascends. So that the impression of the highest strongly
excited fibre preponderates.
A determinate pitch, to a more remarkable extent, may also naturally result, if
the pressure itself which acts on the stirrup of the drum alternates several times
between positive and negative. And thus all transitional degrees between noises
without any determinate pitch, and compound tones with a determinate pitch may
be produced. This actually takes place, and herein lies the proof, on which Herr
8. Exner* has properly laid weight, that such noises must be perceived by those
parts of the ear which act in distinguishing pitch.
In former editions of this work I had expressed a conjecture that the auditory
dliae of the ampullae, which seemed to be but Uttle adapted for resonance, and
those of the Httle bags opposite the otoliths, might be especially active in the ^
perception of noises.
As regards the ciliae in the ampullae, the investigations of Goltz have made it
extremely probable that they, as well as the semicircular canals, serve for a totally
different kind of sensation, namely for the perception of the turning of the head.
Revolution about an axis perpendicular to the plane of one of the semicircular
canals cannot be immediately transferred to the ring of water which lies in the
canal, and on account of its inertia lags behind, while the relative shifting of the
water along the wall of the canal might be felt by the ciliae of the nerves of the
ampullae. On the other hand, if the turning continues, the ring of water itself
will be gradually set in revolution by its friction against the wall of the canal,
and wiU continue to move, even when the turning of the head suddenly ceases.
This causes the illusive sensation of a revolution in the contrary direction, in the
well-known form of giddiness. Injuries to the semicircular canals without injuries
to the brain produce the most remarkable disturbances of equihbrium in the lower ^
animals. Electrical discharges through the ear and cold water squirted into the
ear of a person vdth a perforated drumskin, produce the most violent giddiness.
Under these circumstances these parts of the ear can no longer with any probability
be considered as belonging to the sense of hearing. Moreover impulses of the
stirrup against the water of the labyrinth adjoining the oval window are in reahty
ill adapted for producing streams through the semicircular canals.
On the other hand the experiments of Eoenig with short sounding rods, and
those of Preyer with Appunn's tuning-forks, have established the fact that very
high tones with from 4000 to 40,000 vibrations in a second can be heard, but that
for these the sensation of interval is extremely deficient. Even intervals of a Fifth
or an Octave in the highest positions are only doubtfully recognised and are often
wrongly appreciated by practised musicians. Even the major Third c" — e" [4096 :
5120 vibrations] was at one time heard as a Second, at another as a Fourth or a
Fifth ; and at still greater heights even Octaves and Fifths were confused. ^
If we maintain the hypothesis, that every nervous fibre hiaars in its own peculiar
pitch, we should have to conclude that the vibrating parts of the ear which convey
these sensations of the highest tones to the ear, are much less sharply defined in their
capabilities of resonance, than those for deeper tones. This means that they lose any
motion excited in them comparatively soon, and are also comparatively more easily
brought into the state of motion necessary for sensation. This last assumption
must be made, because for parts which are so strongly damped, the possibility of
adding together many separate impulses is very hmited, and the construction of the
auditory ciliae in the Uttle bags of the otohths seems to me more suited for this
purpose than that of the shortest fibres of the basilar membrane. If this hypo-
thesis is confirmed we should have to regard the auditory ciliae as the bearers of
squeaking, hissing, chirping, crackling sensations of sound, and to consider their
reaction as differing only in degree from that of the cochlear fibres.f
♦ Pflueger, Archiv, fUr Physiologic, vol. f [See App. XX. seet. L. art. 5.— Tmiw-
^- ^^r.] Digitized by ^.jOOgie
PAET 11.
ON THE INTERRUPTIONS OF HARMONY.
COMBINATIONAL TONES AND BEATS/
CONSONANCE AND DISSONANCE.
CHAPTER VII.
COMBIKATIONAD TONBS.
In the first part of this book we had to enunciate and constantly apply the pro-
position that oscillatory motions of the air and other elastic bodies, produced by
several sources of sound acting simultaneously, are always the exact sum of the
individual motions producible by each source separately. This law is of extreme
importance in the theory of sound, because it reduces the consideration of com-
pound cases to those of simple ones. But it must be observed that this law holds
^ strictly only in the case where the vibrations in all parts of the mass of air and of
the sonorous elastic bodies are of infinitesimally small dimensions ; that is to say,
only when the alterations of density of the elastic bodies are so small that they,
may be disregarded in comparison with the whole density of the same body ; and
in the same way, only when the displacements of the vibrating particles vanish as
compared with the dimensions of the whole elastic body. Now certainly in all
practical applications of this law to sonorous bodies, the vibrations are always
very small, and near enough to being infinitesimally small for this law to hold
witii great exactness even for the real sonorous vibrations of musical tones, and by
far the greater part of their phenomena can be deduced from that law in con-
formity with observation. Still, however, there are certain phenomena which
result from the fact that this law does not hold with perfect exactness for vibra-
tions of elastic bodies, which, though almost always very STnall, are far from being
infinitesimally small.f One of these phenomena, with which we are here interested,
IT is the occurrence of Combinational Tones, which were first discovered in 1745 by
Sorge,t a German organist, and were afterwards generally known, although their
pitch was often wrongly assigned, through the Italian violinist Tartini (1754), from
whom they are often called Tartini' s tones. §
These tones are heard whenever two musical tones of different pitches are
* [So much attention has recently been views, before taking up the Appendix. — Trans-
paid to the whole subject of this second part lator,]
—Combinational Tones and Beats— mostly f Helmholtz, on 'Combinational Tones,'
since the publication of the 4th German in Poggendorff's Annalefit vol. xcix. p. 497.
edition, that I have thought it advisable to Monatsberichte of the Berlin Academy, May 22,
give a brief account of the investigations of 1856. From this last an extract is given in
Koenig, Bosanquet, and Preyer in App. XX. Appendix XII.
sect. L., and merely add a few footnotes to t Vorgemach musikalischer Composilion
refer the reader to them where they especially (Antechamber of musical composition),
relate to the statements in the text. But the § [In England they have hence been often
reader should study the text of this second called by Tartini's name, ierzi suoni, or third
part, so as to be familiar with Prof. HelmhoUz's sounds, resulting from the combination of two.
Digitized by V^jOOQlC
CHAP. VII. COMBINATIONAL TONES. IS3
sounded together, loudly and continuously. The pitch of a combinational tone
is generally different from that of either of the generating tones, or of their
harmonic upper partials. In experiments, the combinational are readily distin-
guished &om the upper partial tones, by not being heard when only one generating
tone is sounded, and by appearing simultaneously with the second tone. Combi-
national tones are of two kinds. The first class, discovered by Sorge and Tartini,
I have termed differential tones, because their pitch number is the difference of
the pitch numbers of the generating tones. The second class of summational
tones, having their pitch number equal to the sum of the pitch numbers of the
generating tones, were discovered by myself.
On investigaidng the combinational tones of two compound musical tones, we
find that both the primary and the upper partial tones may give rise to both dif-
ferential and summational tones. In such cases the number of combinational
tones is very great. But it must be observed that generally the differential are H
stronger than the summational tones, and that the stronger generating simple
tones also produce the stronger combinational tones. The combinational tones,
indeed, increase in a much greater ratio than the generating tones, and diminish
also more rapidly. Now since in musical compound tones the prime generally pre-
dominates over the partials, the differential tones of the two primes are generally
heard more loudly than all the rest, and were consequently first discovered. They
are most easily heard when the two generating tones are less than an octave apart,
because in that case the differential is deeper than either of the two generating
tones. To hear it at first, choose two tones which can be held with great force for
some time, and form a justly intoned harmonic interval. First sound the low
tone and then the high one. On properly directing attention, a weaker low tone
will be heard at the moment that the higher note is struck ; this is the required
combinational tone.* For particular instruments, as the harmonium, the com-
binational tones can be made more audible by properly tuned resonators. In this ^
case the tones are generated in the air contained within the instrument. But in
other cases, where they are generated solely within the ear, the resonators are of
little or no use.
A commoner English name is ^mv« ^rmontcj, The differential tones are well heard on the
which is inapplicable, as they are not neces- English concertina, for the same reason as on
sarily graver than both of the generating tones. the harmonium. High notes forming Semi-
Prof. Tyndall calls them resultant Umes, I tones tell well. It is convenient to choose
prefer retaining the Latin expression, first in- close dissonant intervals for first examples in
troduced, as Prof. Preyer informs as (Akiiati- order to dissipate the old notion that the
scJie Untersuchungen,^. II) J hy Gt.XJ. A,Yieih 'grave harmonic' is necessarily the *true
(d. 1836 in Dessau) in Gilbert's Annalen der fundamental bass' of the 'chord.' It is very
Physik 1805, vol. xxi. p. 265, but only for the easy when playing two high generating notes,
tones here distinguished as differential, and as g"' and g"% or the last and a'", to hear at
afterwards used by Scheibler and Prof. Helm- the same time the rattle of the beats (see next
holtz. I shall, however, use ' combinational chapter) and the deep combinational tones
tones ' to express all the additional tones which about F^Z and 0,% , much resembling a thrash-
are heard when two notes are sounded at the ing machine two or three fields off. The beats %
same time. — Translator,'] and the differentials have the same frequency
* [I have found that combinational tones (note p. i id). See infr^, App. XX. sect. L. art.
can be made quite audible to a hundred people 5, /. The experiment can also be made with
at once, by means of two flageolet fifes or 6'' cT' and h"\> h'' on any harmonium. And if
whistles, blown as strongly as possible. I all three notes \>'\> , 6'', <i" are held down to-
choose very closedissonant intervals because the gether, the ear can perceive the two sets of
great depth of the low tone is much more strik- beats of the upper notes as sharp high rattles,
ing, being very far below anything that can be and the beats of the two combinational tones,
touched by the instrument itself. Thus ^"' about the pitch of C, which have altogether a
being sounded loudly on one fife by an assis- different character and frequency. On the
iant, I give/"'S , when a deep note is instantly Harmouical, notes h" d' should beat 66, notes
heard which, if the interval were pure, would 6 "b h" should beat 39*6, and notes ''h''\> 6'"b
be ^, and is sufficiently near to g to be recog- should beat 26*4 in a second, and these should
nised as extremely deep. As a second experi- be the pitches of their combinational notes ;
ment, the &'" being held as before, I give first the two first should therefore beat 26*4 times
f'"% and men e"" in succession. If the inter- in a second, and the two last 13*2 times in a
vals were pure the combinational tones would second. But the tone 26'4 is so difficult to
jump from g to c\ and in reality, the jump is hear that the beats are not distinct.— Trans-
very nearly the same and quite appreciable. latorJ]
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'54
COMBINATIONAL TONES.
PABT II.
The following table gives the first differential tones of the usual harmonic
intervals : —
Interrals
Ratio of the
yibrational
numbers
Difference of
the same
The combinational tone is deeper than
Octave .
Fifth .
Fourth .
Major Third
Minor Third
Major Sixth .
Minor Sixth .
1 : 2
2:3
3:4
4:5
5:6
3:5
5:8
2
3
a Unison
an Octave
a Twelfth
Two Octaves
Two Octaves and a major Third
a Fifth
a major Sixth
or in ordinary musical notation, the generating tones being written as minims and
5f the diHerential tones as crotchets —
Octave. Fifth. Fourth. Major Minor Major Minor
Third. Third. Sixth. Sixth.
t
^.
3a:
^
IZS?
m
^
Wheu the ear has learned to hear the combinational tones of pure intervals
and sustained tones, it will be able to hear them from inharmonic intervals and in
the rapidly fading notes of a pianoforte. The combinational tones from inhar-
^ monic intervals are more difficult to hear, because these intervals beat more or less
strongly, as we shaH have to explain hereafter. The combinational tones arising
from such as fade rapidly, for example those of the pianoforte, are not strong
enough to be heard except at the first instant, and die off sooner than the gene-
rating tones. Combinational tones are also in general easier to hear &om the simple
tones of tuning-forks and stopped organ pipes than from compound tones where a
number of other secondary tones are also present. These compound tones, as has
been already said, also generate a number of differential tones by their harmonic
upper partials, and these easily distract attention from the differential tones of the
primes. Combinational tones of this kind, arising from the upper partials, are
frequently heard from the violin and harmonium.
Example, — Take the major Third c'e', ratio of pitch numbers 4 : 5. First difference i, that
is C. The first harmonic upper partial of c' is c'', relative pitch number 8. Ratio of this and
e', S : 8, difference 3, that is g. The first upper partial of e' is 6", relative pitch number 10 ;
m ratio for this and c', 4 : 10, difference 6, that is g'. Then again c" e" have ratio 8 : 10, difference
2, that is c. Heoce, taking only the first upper partials we have the series of combinational
tones I, Si 6, z 01 Ct g^g't c. Of these the tone 3, or ^, is often easily perceived.
These multiple combinational tones cannot in general be distinctly heard, except
when the generating compound tones contain audible harmonic upper partials.
Yet we cannot assert that the combinational tones are absent, where such partials
are absent ; but in that case they are so weak that the ear does not readily recognise
them beside the loud generating tones and first differential. In the first place
theory leads us to conclude that they do exist in a weak state, and in the next
place the beats of impure intervals, to be discussed presently, also establish their
existence. In this case we may, as Hallstroem suggests,* consider the multiple
combinational tones to arise thus : the first differential tone, or combinational tone
of the first order, by combination with the generating tones themselves, produce
other differential tones, or combinational tones of the second order ; these again
* Poggendorff's X/i;i^/c;i, vol. xxiv.p. 438.
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GHAP. VII.
COMBINATIONAL TONES.
155
produce new ones with the generators and differentials of the first order, and
so on.
Example, — Take two simple tones & and e\ ratio 4:5, difference i, differential tone of the
first order C. This with the generators gives the ratios i : 4 and I : 5, differences 3 and 4,
differential tones of the second order g, and c' once more. The new tone 3, gives with the
generators the ratios 3 : 4 and 3:5, differences i and 2, giving the differential tones of the third
order G and c, and the same tone 3 gives with the differential of the first order i, the ratio 1:3,
difference 2, and hence as a differential of the fourth order c once more, and so on. The dif-
ferential tones of different orders which coincide when the interval is perfect, as it is supposed to
be in this example, no longer exactly coincide when the generating interval is not pure ; and
consequently such beats are heard, as would result from the presence of these tones. More on
this hereafter.
The differential tones of different orders for different intervals are given in the
following notes, where the generators are minims, the combinational tones of the ^
first order crotchets, of the second quavers, and so on. The same tones also occur
with compound generators as combinational tones of their upper paartials.*
Octave.
J
Fifth.
Fourth.
Major Third.
=}=
-A^
3t:
-<s»-
-«s>-
r
■Wr
-^
:J^«e
Minor Third.
Major Sixth.
Minor Sixth.
The series are broken off as soon as the last order of differentials furnishes no
firesh tones. In general these examples shew that the complete series of harmonic
partial tones i, 2, 3, 4, 5, &c., up to the generators tliemselves,t is produced.
The second kind of combinational tones, which I have distinguished Bkastimma-
tional, is generally much weaker in sound than the first, and is only to be heard
♦ [These examples are best calculated by
giving to the notes in the example the numbers
representing the harmonics on p. 22c. Thus
Octave, notes 4 : 8. Diff. 8 — 4 = 4.
Fifth, notes 4 : 6. Diff. 6-4 = 2.
2nd order, 4 — 2 = 2, 6 — 2 = 4,
Fourth, notes 6 : 8. Diff. 8-6 = 2.
2nd order, 8 — 2 = 6,6 — 2 = 4.
3rd order, 6—4 = 2, 6-2 = 4.
Major Third, notes 4 : 5. Diff. 5—4 = 1.
2nd. 4-1=3. S->=4.
3rd. 4-3 = ii 5-3 = 2.
4th. 4-2 = 2,4-1=3.
Minor Third, notes 5 : 6. DifiF. 6-5 -i.
2nd. 5 — 1=4, 6-1 = 5.
3rd. 5-4=1,6-4-2.
4th. 4-1=3,6-2-4.
5th. 6-4-2. 6-33.
Major Sixth, notes 6 : 10. Diff. 10—6 = 4. -vi
2nd. 10-4 = 6,6-4 = 2. ^'
3rd. 10-2 = 8, 6-2 « 4.
4th. 6-4 = 2.
Minor Sixth, notes 5 : 8. Diff. 8-5 = 3.
2nd. 5-3 = 2,8-3 = 5-
3rd. 5-2 = 3,8-2 = 6.
4th. 3-2=1,5-3 = 2.
5th. 5-1=4.8-1=7.
6th. 8-7 = 5-4=1,4-2 = 2,8-4 = 4.
The existence of these differential tones of
higher orders cannot be considered as com-
pletely established. — Translator,']
, t [See App. XX. sect. L. art. 7, for the
influence of such a series on the consonance of
simple tones. It is not to be supposed that all
these tones are audible. Mr. Bosanquet derives
them direct from the generators, see App. XX.
sect. L. art. 5, a, — Translator.]
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156
COMBINATIONAL TONES.
PABT n.
with decent ease under peculiarly favourable circumstances on the harmonium and
polyphonic siren. Scarcely any but the first summational tone can be perceived,
having a vibrational number equal to the sum of those of the generators. Of course
sunmiational tones may also arise from the harmonic upper partials. Since their
vibrational number is always equal to the sum of the other two, they are always
higher in pitch than either of the two generators. The following notes will shew
their nature for the simple intervals : —
In relation to music I will here remark at once that many of these summa-
tional tones form extremely inharmonic intervals with the generators. Were they
not generally so weak on most instruments, they would give rise to intoler-
able dissonances. In reality, the major and minor Third, and the minor Sixth,
sound very badly indeed on the polyphonic siren, where all combinational tones
are remarkably loud, whereas the Octave, Fifth, and major Sixth are very beautiful.
Even the Fourth on this siren has only the ejQfect of a tolerably harmonious chord
of the minor Seventh.
^ It was formerly believed that the combinational tones were purely subjective,
and were produced in the ear itself .f Differential tones alone were known, and these
were connected with the beats which usually result from the simultaneous sounding
of two tones of nearly the same pitch, a phenomenon to be considered in the follow-
ing chapters. It was believed that when these beats occurred with sufficient
rapidity, the individual increments of loudness might produce the sensation of a
new tone, just as numerous ordinary impulses of the air would, and that the
frequency of such a tone would be equal to the frequency of the beats. But this
supposition, in the first place, does not explain the origin of summational tones,
being confined to the differentials ; secondly, it may be proved that under certain
conditions the combinational tones exist objectively, independently of the ear
which would have had to gather the beats into a new tone ; and thirdly, this
supposition cannot be reconciled with the law confirmed by all other experiments,
that the only tones which the ear hears, correspond to pendular vibrations of the
fair. J
And in reaUty a different cause for the origin of combinational tones can be
estabUshed, which has already been mentioned in general terms (p. 152c). When-
ever the vibrations of the air or of other elastic bodies which are set in motion at
the same time by two generating simple tones, are so powerful that they can no longer
be considered infinitely small, mathematical theory shews that vibrations of the
air must arise which have the same frequency as the combinational tones.§
Particular instruments give very powerful combinational tones. The condition
* [The notation of the last 5 bars has been
altered to agree with the diagram of harmonicB
of C on p. 22c.^ Translator.]
f [The result of Mr. Bosanqnet's and Prof.
Preyer's quite recent experiments is to shew
that tliey are so. See App. XX. sect. L. art. 4,
fc, c. — Translator,]
X [For Prof. Preyer's remarks on these
objections, and for other objections, see App.
XX. sect. Ii. art. 5, 6, c — Translator.]
§ [The tones supposed to arise from beats,
and the differential tones thus generated, are
essentially distinct, having sometimes the same
but frequently different pitch numbers. See
App. XX. sect. L. art. 3, d, — Translator,]
Digitized by
Google
CHAP.vn. COMBINATIONAL TONES. 157
for their generation is that the same mass of air should be violently agitated by
two simple tones simultaneously. This takes place most powerfully in the poly-
phonic siren/ in which the same rotating disc contains two or more series of
holes which are blown upon simultaneously from the same windchest. The air
of the windchest is condensed whenever the holes are closed ; on the holes being
opened, a large quantity of air escapes, and the pressure is considerably diminished.
Consequently the air in the windchest, and partly even that in the bellows, as
can be easily felt, comes into violent vibration. If two rows of holes are blown,
vibrations arise in the air of the windchest corresponding to both tones, and each
row of openings gives vent not to a stream of air uniformly supplied, but to a
stream of air already set in vibration by the other tone. Under these circumstances
the combinational tones are extremely powerful, almost as powerful, indeed, as the
generators. Their objective existence in the mass of air can be proved by vibra-
ting membranes tuned to be in unison with the combinational tones. Such f
membranes are set in sympathetic vibration immediately upon both generating
tones being sounded simultaneously, but remain at rest if only one or the other of
them is sounded. Indeed, in this case the summational tones are so powerful
that they make all chords extremely unpleasant which contain Thirds or minor
Sixths. Instead of membranes it is more convenient to use the resonators already
reconmiended for investigating harmonic upper partial tones. Besonators are
also unable to reinforce a tone when no pendular vibrations actually exist in the
air ; they have no effect on a tone which exists only in auditory sensation, and
hence they can be used to discover whether a combinational tone is objectively
present. They are much more sensitive than membranes, and are well adapted
for the clear recognition of very weak objective tones.
The conditions in the harmonium are similar to those in the siren. Here, too,
there is a common windchest, and when two keys are pressed down, we have two
openings which are closed and opened rhythmically by the tongues. In this case f
also the air in the common receptacle is violently agitated by both tones, and aii-
is blown through each opening which has been already set in vibration by the
other tongue. Hence in this instrument also the combinational tones are objectively
present, and comparatively very distinct, but they are far from being as powerful
as on the siren, probably because the windchest is very much larger in proportion
to the openings, and hence the air which escapes during the short opening of an
exit by the oscillating tongue cannot be sufBcient to diminish the pressui-e sensibly.
In the harmonium also the combinational tones are very clearly reinforced by
resonators tuned to be in unison with them, especially the first and second dif-
ferential and the first sunmaational tone.f Nevertheless I have convinced myself, by
particular experiments, that even in this instrument the greater part of the force
of the combinational tone is generated in the ear itself. I arranged the portvents
in the instrument so that one of the two generators was suppUed with air by the
bellows moved below by the foot, and the second generator was blown by tht ^
reserve bellows, which was first pumped full and then cut off by drawing out the
so-called expression-stop, and I then found that the combinational tones were not
much weaker than for the usual arrangement. But the objective portion which
tlie resonators reinforce was much weaker. The noted examples given above
(pp. 154-5-6) will easily enable any one to find the digitals which must be
pressed down in order to produce a combinational tone in unison with a given
resonator.
On the other hand, when the places in which the two tones are struck are
entirely separate and have no mechanical connection, as, for example, if they come
from two singers, two separate wind instruments, or two violins, the reinforcement
* A detailed description of this instrnment apparent reinforcement by a resonator arose
will be given in the next chapter. from imperfect blocking of both ears when
t [The experiments of Bosanquet, App. XX. using it. See also p. 43d', note,— Translator.]
sect. L. art. 4, 6, render it probable that this
Digitized by V^OOQIC
158 COMBINATIONAL TONES. part 11.
of the combinational tones by resonators is small and dubious. Here, then, there
does not exist in the air any clearly sensible pendular vibration corresponding to
the combinational tone, and we must conclude that such tones, which are often
powerfully audible, are really produced in the ear itself. But analogously to the
former cases we are justified in assuming in this case also that the external vibra-
ting parts of the ear, the drumskin and auditory ossicles, are really set in a suffi-
ciently powerful combined vibration to generate combinational tones, so that the
vibrations which correspond to combinational tones may really exist objectively in
the parts of the ear without existing objectively in the external air. A slight rein-
forcement of the combinational tone in this case by the corresponding resonator
may, therefore, arise from the drumskin of the ear communicating to the air in the
resonator those particular vibrations which correspond to the combinational tone.*
Now it so happens that in the construction of the external parts of the ear for
fl conducting sound, there are certain conditions which are peculiarly favourable for
the generation of combinational tones. First we have the unsynmaetrical form of
the drumskin itself. Its radial fibres, which are externally convex, undergo a much
greater alteration of tension when they make an oscillation of moderate amplitude
towards the inside, than when the oscillation takes place towards the outside.
For this purpose it is only necessary that the amplitude of the oscillation should
not be too small a fraction of the minute depth of the arc made by these radial
fibres. Under these circumstances deviations from the simple superposition of
vibrations arise for very much smaller amphtudes than is the case when the vibra-
ting body is symmetrically constructed on both sides.f
But a more important circumstance, as it seems to me, when the tones are
powerful, is the loose formation of the joint between the hammer and anvil (p. 1336).
If the handle of the hammer is driven inwards by the drumskin, the anvil and
stirrup must follow the motion unconditionally. But that is not the case for the
f subsequent outward motion of the handle of the hammer, during which the tettli
of the two ossicles need not catch each other. In this case the ossicles may cli^k.
Now I seem to hear this clicking in my own ear whenever a very strong and deep
tone is brought to bear upon it, even when, for example, it is produced by a tuning-
fork held between the fingers, in which there is certainly nothing that can make
any click at all.
This peculiar feeling of mechanical tingling in the ear had long ago struck me
when two clear and powerful soprano voices executed passages in Thirds, in which
case the combinational tone comes out very distinctly. If the phases of the two
tones are so related that after every fourth oscillation of the deeper and every fifth
of the higher tone, there ensues a considerable outward displacement of the drum-
skin, sufficient to cause a momentary loosening in the joint between the hammer
and anvil, a series of blows will be generated between the two bones, which would
be absent if the connection were firm and the oscillation regular, and these blows
^ taken together would exactly generate the first differential tone of the interval of
a major Third. Similarly for other intervals.
It must also be remarked that the same peculiarities in the construction of a
sonorous body which makes it suitable for allowing combinational tones to be heard
when it is excited by two waves of diff^erent pitch, must also cause a single simple
tone to excite in it vibrations corresponding to its harmonic upper partials ; the
effiect being the same as if this tone then formed summational tones with itself.
This result ensues because a simple periodical force, corresponding to pendular
vibrations, cannot excite similar pendular vibrations in the elastic body on which
it acts, unless the elastic forces called into action by the displacements of the ex-
* [See latter half of Appendix XVI. — are proportional to the first pover of the am -
Translator,] plitude, whereas for symmetrical ones they
f See my paper on combinational tones are proportional to only the second power of
alreajdy cited, and Appendix XII. For unsym- this magnitude, which is very small in both
metrical vibrating bodies the disturbances cases.
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CHAPS. VII. VIII. INTEBFERENCE OF SOUND. 159
cited body from its position of equilibrium, are proportional to these displacements
themselves. This is always the case so long as these displacements are infinitesimal.
But if the amplitude of the oscillations is great enough to cause a sensible devia-
tion from this proportionality, then the vibrations of the exciting tone are increased
by others which correspond to its harmonic upper partial tones. That such har-
monic upper partials are occasionally heard when tuning-forks are strongly ex-
cited, has been already mentioned (p. 54^). I have lately repeated these experi-
ments with forks of a very low pitch. With such a fork of 64 vib. I could, by
means of proper resonators, hear up to the fifth partial. But then the amplitude
of the vibrations was almost a centimetre ['3937 inch]. When a sharp-edged
body, such as the prong of a tuning-fork, makes vibrations of such a length,
vortical motions, differing sensibly from the law of simple vibrations, must arise
in the surrounding air. On the other hand, as the sound of the fork fades, these
upper partials vanish long before their prime, which is itself only very weakly ^
audible. This agrees with our hypothesis that these partials arise from disturb-
ances depending on the size of the amplitude.
Herr B. Eoenig,* with a series of forks having sliding weights by which the pitch
might be gradually altered, and provided also with boxes giving a good resonance
and possessing powerful tones, has investigated beats and combinational tones, and
found that those combinational tones were most prominent which answered to the
difference of one of the tones from the partial tone of the other which was nearest
to it in pitch ; and in this research partial tones as high as the eighth were effec-
tive (at least in the number of beats). f He has unfortunately not stated how far the
corresponding upper partials were separately recognised by resonators. J
Since the human ear easily produces combinational tones, for which the prin-
cipal causes lying in the construction of that organ have just been assigned, it
must also form upper partials for powerful simple tones, as is the case for tuning-
forks and the masses of air which they excite in the observations described. Hence If
we cannot easily have the sensation of a powerful simple tone, without having also
the sensation of its harmonic upper partials.§
The importance of combinational tones in the construction of chords will appear
hereafter. We have, however, first to investigate a second phenomenon of the
simultaneous sounding of two tones, the so-called heats.
CHAPTEB Vm.
ON THE BEATS OF SIMPLE TONES.
We now pass to the consideration of other phenomena accompanying the simul %
taneous sounding of two simple tones, in which, as before, the motions of the air
and of the other co-operating elastic bodies without and within the ear may be con-
ceived as an undisturbed coexistence of two systems of vibrations corresponding to
the two tones, but where the auditory sensation no longer corresponds to the sum
of the two sensations which the tones would excite singly. Beats, which have
now to be considered, are essentially distinguished from combinational tones as
follows: — In combinational tones the composition of vibrations in the elastic
vibrating bodies which are either within or without the ear, undergoes certain dis-
turbances, although the ear resolves the motion which is finally conducted to it,
* PoggendorfF's AnnaLy vol. clvii. pp. 177- sect. L. — Translator,]
236. X [Koenig states that no upper partials
f [Even with this parenthetical correction, could be heard. See Appendix XX. sect. L.
the above is calculated to give an inadequate art. 2, a,--Translator.]
impression of the results of Koenig's paper, § [See App. XX. sect. L. art. i,ii. — Trans-
which is more fully described in Appendix XX. lator,'\
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i6o
INTERFERENCE OF SOUND.
PART n.
into a series of simple tones, according to the usual law. In beats, on the
contrary, the objective motions of the elastic bodies follow the simple law ; but
the composition of the sensations is disturbed. As long as several simple tones of
a sufiQciently dijQferent pitch enter the ear together, the sensation due to each
remains undisturbed in the ear, probably because entirely different bundles of
nerve fibres are affected. But tones of the same, or of nearly the same pitch,
' which therefore affect the same nerve fibres, do not produce a sensation which is
the sum of the two they would have separately excited, but new and peculiar
phenomena aarise which we term interference^ when caused by two perfectly equal
simple tones, and heats when due to two nearly equal simple tones.
We will begin with interference. Suppose that a point in the air or ear
is set in motion by some sonorous force, and that its motion is represented by
the curve i, fig. 53. Let
IT the second motion be ^'®' 53-
precisely the same at the
same time and be repre-
sented by the curve 2, so
that the crests of 2 fall
on the crests of i, and
also the troughs of 2 on
the troughs of i . If both
motions proceed at once,
the whole motion will be
their sum, represented by
3, a curve of the same
kind but with crests twice as high and troughs twice as deep as those of either of
the others. Since the intensity of sound is proportional to the square of the
^ amplitude, we have consequently a tone not of twice but of four times the loudness
of either of the others.
But now suppose the vibrations of the second motion to be displaced by
The curves to be added will stand under one another, as
Pio. 54.
half the periodic time.
4 and 5 in fig. 54, and
when we come to add
to them, the heights of
the second curve will be
still the same as those
of the first, but, being
always in the contrary
direction, the two will
mutually destroy each
other, giving as their
^ sum the straight line 6, or no vibration at all. In this case the crests of 4 are
added to the troughs of 5, and conversely, so that the crests fill up the troughs,
and crests and troughs mutually annihilate each other. The intensity of sound
also becomes nothing, and when motions are thus cancelled within the ear, sensa-
tion also ceases ; and although each single motion acting alone would excite the
corresponding auditory sensation, when both act together there is no auditory
sensation at all. One sound in this case completely cancels what appears to be
an equal sound. This seems extraordinarily paradoxical to ordinary contempla-
tion because our natural consciousness apprehends sound, not as the motion of
particles of the air, but as something really existing and analogous to the sensation
of sound. Now as the sensation of a simple tone of the same pitch shews no oppo-
sitions of positive and negative, it naturally appears impossible for one positive
sensation to cancel another. But the really cancelling things in such a case- are
the vibrational impulses which the two sources of sound exert on the ear. When
it so happens that the vibrational impulses due to one source constantly coincide
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CHAP. VIII. INTEKFEEENCE OP SOUND. i6i
with opposite ones due to the other, and exactly counterbalance each other, no
motion can possibly ensue in the ear, and hence the auditory nerve can experience
no sensation.
The following are some instances of sound cancelling sound : —
Put two perfectly similar stopped organ pipes timed to the same pitch close
beside each other on the same portvent. Each one blown separately gives a
powerful tone ; but when they are blown together, the motion of the air in the
two pipes takes place in such a manner that as the air streams out of one it streams
into the other, and hence an observer at a distance hears no tone, but at most the
rushing of the air. On bringing the fibre of a feather near to the lips of the
pipes, this fibre will vibrate in the same way as if each pipe were blown separately.
Also if a tube be conducted from the ear to the mouth of one of the pipes, the
tone of that pipe is heard so much more powerfully that it cannot be entirely
destroyed by the tone of the other.* ^
Every tuning-fork also exhibits phenomena of interference, because the prongs
move in opposite directions. On striking a tuning-fork and slowly revolving it
about its longitudinal axis close to the ear, it will be found that there are four
positions in which the tone is heard clearly ; and four intermediate positions in
which it is inaudible. The four positions of strong sound are those in which
either one of the prongs, or one of the side surfEkces of the fork, is turned towards
the ear. The positions of no sound lie between the former, almost in planes
which make an angle of 45^ with the sur£a.ces of the prongs, and pass through
the axis of the fork. If in fig. 55, a and b are the ends of the fork seen from
above, c, d, e, f will be the four places of strong sound, and the dotted lines
the places four of silence. The arrows under a
Fio- 55- and b shew the mutual motion of the two prongs.
. , Hence while the prong a gives the air about c an im-
I ^^'' pulse in the direction c a, the prong b gives it an ^
V. .' opposite one. Both impulses only partially cancel
J . ' one another at c, because a acts more powerfully
thanb. But the dotted lines shew the places where
the opposite impulses from a and b are equally
J • — ^ ,steong, and consequently completely cancel each
■ >- -*r — other. If the ear be brought into one of these
'\^ places of silence and a narrow tube be slipped over
one of the prongs a or b, taking care not to touch it,
/' '\, the sound will be immediately augmented, because
/ \ the influence of the covered prong is almost entirely
'' '^ destroyed, and the uncovered prong therefore acts
alone and undisturbed.f
A double siren which I have had constructed is very convenient for the demon-
stration of these relations.^ Fig. 56 (p. 162) is a perspective view of this instru- fl
ment. It is composed of two of Dove's polyphonic sirens, of the kind previously
mentioned, p. 13a ; ao and ai are the two windchests, Cq and c^ the discs attached
to a common axis, on which a screw is introduced at k, to drive a counting
apparatus which can be introduced, as described on p. 12b, The upper box a^
can be turned round its axis, by means of a toothed wheel, in which works a
smaller wheel e provided with the driving handle d §. The axis of the box a^
round which it turns, is a prolongation of the upper pipe g,, which conducts
the wind. On each of the two discs of the siren are four rows of holes, which
* [If a screen of any sort, as the hand, be resonance chamber, the alternation of sound
interposed^ between the mouths of the pipe, and silence, Ac, can be made audible to many
the tone is immediately restored, and then persons at once. — Translator^]
generally remains even if the hand be re- J Constructed by the mechanician Sauer-
moved. — Translator.] wald in Berlin.
t [If instead of bringing the tuning-fork to § [Three turns of the handle cause one
the ear, it be slowly turned before a proper turn of the box round its axis.— Translator .]
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l62
INTERFERENCE OP SOUND.
PABT n«
can be either blown separately or together in any combination at pleasure, and at i
are the studs for opening and closing the series of holes by a peculiar arrange-
ment.* The lower disc has four rows of 8, lo, 12, 1 8 holes, the upper of 9, 12,
15, 16. Hence if we call the tone of 8 holes c, the lower disc gives the tones c, e,
Pio. 56.
gf, 6/ and the upper d, gr, b, d. We are therefore able to produce the following
intervals : —
1. Unison : gg on the two discs simultaneously.
2. Octaves : c d and dd! on tlie two.
* Described in Appendn XIII.
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CHAP.7III. INTERFERENCE OP SOUND. 163
3. Fifths ! c g and g dl either on the lower disc alone or on both discs together.
4. Fourths : d g and g c' on the upper disc alone or on both together.
5. Major Third : c a on the lower alone, and ^ 6 on the upper alone, or gh on
both together.
6. Minor Third : eg on the lower, or on both together ; hd/ on both together.
7. Whole Tone [major Tone] : cd and c'd' on both together [the minor Tone
is produced by d and e on both together].
8. Semitone [diatonic Semitone] : 6 c' on the upper.
When both tones are produced from the same disc the objective combinational
tones are very powerful, as has been already remarked, p. 157a. But if the tones
are produced from different discs, the combinational tones are weak. In the latter
case, (and this is the chief point of interest to us at present) the two tones can
be made to act together with any desired difference of phase. This is effected by
altering the position of the upper box. ^
We have first to investigate the phenomena as they occur in the unison gg.
The effect of the interference of the two tones in this case is complicated by the
fact that the siren produces cq^pound and not simple tones and that the in-
terference of individual partial tones is independent of that of the prime tone
and of one another. In order to damp the upper partial tones in the siren by
means of a resonance chamber, I caused cylindrical boxes of brass to be made,
of which the back hcjves are shewn at hj h, and ho ho fig. 56, opposite. These
boxes are each made in two sections, so that they can be removed, and be again
attached to the windchest by means of screws. When the tone of the siren
approaches the prime tone of these boxes, its quality becomes full, strong and soft,
like a fine tone on the French horn ; otherwise the siren has rather a piercing tone.
At the same time we use a small quantity of air, but considerable pressure. The
circumstances are of the same nature as when a tongue is applied to a resonance
chamber of the same pitch. Used in this way the siren is very well adapted for ^
experiments on interference.
If the boxes are so placed that the puffs of air follow at exactly equal intervals
from both discs, similar phases of the prime tone and of all partials coincide, and
aU are reinforced.
If the handle is turned round half a right angle, the upper box is turned round
I of a right angle, or ^ of the circumference, that is half the distance between
the holes in the series of 12 holes which is in action for g. Hence the difference
in the phase of the two primes is half the vibrational period, the puffs of air in
one box occur exactly in the middle between those of the other, and the two
prime tones mutually destroy each other. But under these circumstances the
difference of phase in the upper Octave is precisely the whole of the vibrational
period ; that is, they reinforce each other, and similarly all the evenly numbered
harmonic upper partials reinforce each other in the same position, and the unevenly
numbered ones destroy each other. Hence in the new position the tone is weaker, ^
because deprived of several of its partials ; but it does not entirely cease ; it rather
jumps up an Octave. If we further turn the handle through another half a right
angle so that the box is turned through a whole right angle, the puffs of the two
discs again agree completely, and the tones reinforce one another. Hence in a
complete revolution of the handle there are four positions where the whole tone of
the siren appears reinforced, and four intermediate positions where the prime tone
and all uneven upper partials vanish, and consequently the Octave occurs in a
weaker form accompanied by the evenly numbered upper partials. If we attend to
the first upper partial, which is the Octave of the prime, by listening to it through
a proper resonator, we find that it vanishes after turning through a quarter of a
right angle, and is reinforced after turning through half a right angle, and hence
for every complete revolution of the handle it vanishes 8 times, and is reinforced
8 times. The third partial, (or second wpper partial,) the Twelfth of the prime
tone, vanishes in the same time 12 times, the fourth partial 16 times, and so on.
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1 64 ORIGIN OP BEATS. pabt ii.
Other compound tones behave likelihose of the siren. When two tones of the
same pitch are sounded together having differences of phase corresponding to half
the periodic time, the tone does not vanish, but jumps up an Octave. When, for
example, two open organ pipes, or two reed pipes of the same construction and
pitch, are placed beside each other on the same portvent, their vibrations generally
accommodate themselves in such a manner that the stream of air enters first one
and then the other alternately ; and while the tone of stopped pipes, which have
only unevenly numbered partials, is then almost entirely destroyed, the tone of the
open pipes and reed pipes falls into the upper Octave. This is the reason why no
reinforcement of tone can be effected on an organ or harmonium by combining
tongues or pipes of the same kind, [on the same portvent].
So far we have combined tones of precisely the same pitch; now let us inquire
what happens when the tones have slightly different pitch. The double siren
% just described is also well fitted for explaining this case, for we can slightly alter
the pitch of the upper tone by slowly revolving the upper box by means of the
handle, the tone becoming flatter when the direction of revolution is the same as
that of the disc, and sharper when it is opposite to the same. The vibrational
period of a tone of the siren is equal to the time required for a hole in the rotating
disc to pass from one hole in the windbox to the next. If, through the rotation of
the box, the hole of the box advances to meet the hole of the disc, the two holes
come into coincidence sooner than if the box were at rest : and hence the vibra-
tional period is shorter, and the tone sharper. The converse takes place when the
revolution is in the opposite direction. These alterations of pitch are easily heard
when the box is revolved rather quickly. Now produce the tones of twelve holes
on both discs. These will be in absolute unison as long as the upper box is at
rest. The two tones constantly reinforce or enfeeble each other according to the
position of the upper box. But on setting the upper box in motion, the pitch of
% the upper tone is altered, while that of the lower tone, which has an inmiovable
windbox, is unchanged. Hence we have now two tones of slightly different pitch
sounding together. And we hear the so-called beats of the tones, that is, the
intensity of the tone will be alternately greater and less in regular succession.* The
arrangement of our siren makes the reason of this readily intelligible. The
revolution of the upper box brings it alternately in positions which as we have
seen correspond to stronger and weaker tones. When the handle has been turned
through a right angle, the windbox passes from a position of loudness through a
position of weakness to a position of strength again. Consequently every complete
revolution of the handle gives us four beats, whatever be the rate of revolution of
the discs, and hence however low or high the tone may be. If we stop the box at
the moment of maximum loudness, we continue to hear the loud tone ; if at a
moment of minimum force, we continue to hear the weak tone.
The mechanism of the instrument also explains the connection between the
f number of beats and the difference of the pitch. It is easily seen that the number
of the puffs is increased by one for every quarter revolution of the handle. But
V every such quarter revolution corresponds to one beat. Hence the number of beats
in a given time is equal to the difference of the numbers of vibrations executed by
the two tones in the same time. This is the general law which determines the
number of beats, for all kinds of tones. This law results immediately from the
construction of the siren ; in other instruments it can only be verified by very
accurate and laborious measurements of the numbers of vibrations.
The process is shewn graphically in fig. 57. Here c c represents the series of
puffs belonging to one tone, and d d those belonging to the other. ' The distance
for c c is divided into 18 parts, the same distance is divided into 20 parts for d d. At
* [The German word Schioebuna, which ' beat.' But it is not asnal to make the dis-
might be rendered ' fluotuation/ implies this : tinotion in English, where the whole pheno-
The loudest portion only is called the StosSy or menoni ailed hea,iB,^Translator,]
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CHAP. VIII. ORIGIN OF BEATS. 165
^» 3» 5. both puffs concur, and the tone is reinforced. At 2 and 4 they are inter-
mediate and mutually enfeeble each other. The number of beats for the whole
distance is 2, because the difference of the numbers of parts, each of which cor-
respond to a vibration, is also 2.
The intensity of tone varies ; swelling from a minimum to a maximum, and
lessening from the maximum to the TniniTnnTn. It is the places of maximum
FIO. 57.
.^/« .... ^. ...'?...,. 4, ... ^^
intensity which are properly called beats, and these are separated by more or less
distinct pauses.
Beats are easily produced on all musical instruments, by striking two notes of ^
nearly the same pitch. They are heard best from the simple tones of tuning-forks
or stopped organ pipes, because here the tone really vanishes in the pauses. A
little fluctuation in the pitch of the beating tone may then be remarked.* For the
compound tones of other instruments the upper partial tones are heard in the
pauses, and hence the tone jumps up an Octave, as in the case of interference
already described. If we have two tuning-forks of exactly the same pitch, it is
only necessary to stick a little wax on to the end of one, to strike both, and bring
them near the same ear or to the surface of a table, or sounding board. To make
two stopped pipes beat, it is only necessary to bring a flnger slowly near to the lip
of one, and thus flatten it. The beats of compound tones are heard by striking
any note on a pianoforte out of tune when the two strings belonging to the same
note are no longer in unison ; or if the piano is in tune it is sufficient to attach a
piece of wax, about the size of a pea, to one of the strings. This puts them suffi-
ciently out of tune. More attention, however, is required for compound tones ^
because the enfeeblement of the tone is not so striking. The beat in this case
resembles a fluctuation in pitch and quality. This is very striking on the siren
according as the brass resonance cylinders (ho ho and h^ h^ of fig. 56, p. 162) are
attached or not. These make the prime tone relatively strong. Hence when beats
are produced by turning a handle, the decrease and increase of loudness in the tone
is very striking. On removing the resonance cylinders, the upper partial tones
are relatively powerful, and since the ear is very uncertain when comparing the
loudness of tones of different pitch, the alteration of force during the beats is
much less striking than that of pitch and quality of tone.
On listening to the upper partials of compound tones which beat, it will be
found that these beat also, and that for each beat of the prime tone there are two
of the second partial, three of the third, and so on. Hence when the upper partials
are strong, it is easy to make a mistake in counting the beats, especially when the
beats of the primes are very slow, so that they occur at intervals of a second or two. IT
It is then necessary to pay great attention to the pitch of the beats counted, and
sometimes to apply a resonator.
It is possible to render beats visible by setting a suitable elastic body into
sympathetic vibration with them. Beats can then occur only when the two
exciting tones lie near enough to the prime tone of the sympathetic body for the
latter to be set into sensible sympathetic vibration by both the tones used. This
is most easily done with a thin string which is stretched on a sounding board
on which have been placed two tuning-forks, both of very nearly the same pitch
as the string. On observing the vibrations of the string through a microscope,
or attaching a fibril of a goosefeather to the string which will make the same
excursions on a magnified scale, the string will be clearly seen to make sympathetic
^ See the explanation of this phenomenon French translator of this work,] in Appen-
which was given me by Mons. G. Gu6roult, [the dix XIV.
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Google
i66 ORIGIN OP BEATS. part n.
vibrations with alternately large and small excursions, according as the tone of the
two forks is at its maximum or minimum.
The same effect can be obtained from the sympathetic vibration of a stretched
membrane. Fig. 58 is the copy of a drawing made by a vibrating membrane of
PfG. 58.
this sort, used in the phonautograph of Messrs. Scott & Eoenig, of Paris. The mem-
brane of this instrument, which resembles the drumskin of the ear, carries a small
stiff style, which draws the vibrations of the membrane upon a rotating cylinder.
In the present case the membrane was set in motion by two organ pipes, that beat.
^ The undulating line, of which only a part is here given, shews that times of strong
vibration have alternated with times of almost entire rest. In this case, then, the
beats are also sympathetically executed by the membrane. Similar drawings
again have been made by Dr. Politzer, who attached the writing style to the
auditory bone (the columella) of a duck, and then produced a beating tone by
means of two organ pipes. This experiment shewed that even the auditory bones
follow the beats of two tones.*
Generally this must always be the case when the pitches of the two tones
struck differ so little from each other and from that of the proper tone of the sym-
pathetic body, that the latter can be put into sensible vibration by both tones at
once. Sympathetic bodies which do not damp readily, such as tuning-forks,
consequently require two exciting tones which differ extraordinarily little in pitch,
in order to shew visible beats, and the beats must therefore be very slow. For
bodies readily damped, as membranes, strings, &c., the difference of the exciting
f tones may be greater, and consequently the beats may succeed each other more
rapidly.
This holds also for the elastic terminal formations of the auditory nerve fibres.
Just as we have seen that there may be visible beats of the auditory ossicles, Gorti's
arches may also be made to beat by two tones sufficiently near in pitch to set the
same Gorti's arches in sympathetic vibration at the same time. If then, as we
have previously supposed, the intensity of auditory sensation in the nerve fibres
involved increases and decreases with the intensity of the elastic vibrations, the
strength of the sensation must also increase and diminish in the same degree as the
vibrations of the corresponding elastic appendages of the nerves. In this case also
the motion of Gorti's arches must still be considered as compounded of the motions
which the two tones would have produced if they had acted separately. According
as these motions are directed in the same or in opposite directions they will rein-
force or enfeeble each other by (algebraical) addition. It is not till these motions
f excite sensation in the nerves that any deviation occurs from the law that each of
the two tones and each of the two sensations of tones subsist side by side without
disturbance.
We now come to a part of the investigation which is very important for the
[ theory of musical consonance, and has also unfortunately been little regarded by
; acousticians. The question is : what becomes of the beats when they grow Caster
and faster ? and to what extent may their number increase without the ear being
unable to perceive them ? Most acousticians were probably inclined to agree with
the hypothesis of Thomas Young, that when the beats became very quick they
gradually passed over into a combinational tone (the first differential). Young
imagined that the pulses of tone which ensue during beats, might have the same
* The beats of two tones are also clearly tones. Even withont using the rotating mirror
shewn by the vibrating flame described at the for observing the flames, we can easily reoog-
end of Appendix II. The flame must be con- nise the alterations in the shape of the flaihe
nected with a resonator having a pitch suffi- which take place isoohronously with the aiidible
ciently near to those of the two generating beats.
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oHAP.viii. LIMITS OP THE FREQUENCY OP BEATS. 167
effect on the ear as elementary pnlses of air (in the siren, for example), and that
just as 30 puffs in a second tlu*ough a siren would produce the sensation of a deep
tone, so would 30 beats in a second resulting from any two higher tones produce
the same sensation of a deep tone. Certainly this view is well supported by the
fact that the vibrational number of the first and strongest combinational tone is
actually the number of beats produced by the two tones in a second. It is, however,
of much importance to remember that there are other combinational tones (my
summational tones), which will not agree with this hypothesis in any respect,*
but on the other hand are readily deduced from the theory of combinational tones
which I have proposed (in Appendix XII.). It is moreover an objection to Young's
theory, that in many cases the combinational tones exist extem^y to the ear, and '
are able to set properly tuned membranes or resonators into sympathetic vibra-
tion,t because this could not possibly be the case, if the combinational tones were
nothing but the series of beats with undisturbed superposition of the two waves. ^
For the mechanical theory of sympathetic vibration shews that a motion of the
air compounded of two simple vibrations of different periodic times, is capable of
putting such bodies only into sympathetic vibration as have a proper tone corre-
sponding to one of the two given tones, provided no conditions intervene by which
the simple superposition of two wave systems might be disturbed ; and the nature
of such a disturbance was investigated in the last chapter.^ Hence we may ;
consider combinational tones as an accessory phenomenon, by which, however, the
course of the two primary wave systems and of their beats is not essentially
interrupted.
Against the old opinion we may also adduce the testimony of our senses, which
teaches us that a much greater number of beats than 30 in a second can be
distinctly heard. To obtain this result we must pass gradually from the slower to
the more rapid beats, taking care that the tones chosen for beating are not too far
apart from each other in the scale, because audible beats are not produced unless ^
the tones are so near to each other in the scale that they can both make the same
elastic appendages of the nerves vibrate sympathetically .§ The number of beats,
however, can be increased without increasing the interval between the tones, if
both tones are taken in the higher octaves.
The observations are best begun by producing two simple tones of the same
pitch, say from the once-accented octave by means of tuning-forks or stopped organ
pipes, and slowly altering the pitch of one. This is effected by sticking more and
more wax on one of the forks ; or more and more covering the mouth of one of
the pipes. Stopped organ pipes are also generally provided with a movable plug
or lid at the stopped end, in order to tune them ; by pulling this out we flatten, by
pushing it in we sharpen the tone.**
When a slight difference in pitch has been thus produced, the beats are heard
at first as long drawn out fluctuations alternately swelling and vanishing. Slow
beats of this kind are by no means disagreeable to the ear. In executing music ^
containing long sustained chords, they may even produce a solemn effect, or else
give a more lively, tremulous or agitating expression. Hence we find in modem
♦ [Prof. Preyer shews, App. XX. sect. L. tion of the following facts, is made with two
art. 4, c2, that summational tones, as snggested * pitch pipes,* each consisting of an extensible
by Appmm, may be considered as differential stopped pipe, which has the compass of the
tones of the second order, ii such are admitted, once-accented octave and is blown as a whistle,
— Translaior.^ the two being connected by a bent tabe with
t [After the experiments of Prof. Preyer a single mouthpiece. By carefully adjusting
and Mr. Bosanqnet, App. XX. sect. L. art. 4, the lengths of the pipes, I was first able to pro-
this mast be considered as due to some error dace complete destruction of the tone by inter-
of observation. — Translator.] ference, the sound returning immediately when
X [See Bosanquet*s theory of * transforma- the mouth of one whistle was stopped by the
tion ' in App. XX. sect. L. art. 5, a.— Trans- finger. Then on gradually lengthening one of
Jator.] the pipes the beats began to be heard slowly,
§ [Eoenig knows no such limitation. See and increased in rapidity. The tone being
App. XX. sect. L. art. 3. — Translator,] nearly simple the beats are well heard. —
** [A cheap apparatus, useful for demonstra- Translator.]
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i68 LIMITS OP THE FREQUENCY OF BEATS. paot ii.
organs and harmoniums, a stop with two pipes or tongues, adjusted to heat. This
imitates the trembling of the human voice and of violins which, appropriately in-
troduced in isolated passages, may certainly be very expressive and effective, but
apphed continuously, as is unfortunately too common, is a detestable malpractice.
f The ear easily follows slow beats of not more than 4 to 6 in a second. The
hearer has time to apprehend all their separate phases, and become conscious of
each separately, he can even count them without difficulty.* But when the interval
between the two tones increases to about a Semitone, the number of beats becomes
20 or 30 in a second, and the ear is consequently unable to follow them sufficiently
well for counting. If, however, we begin with hearing slow beats, and then increase
their rapidity more and more, we cannot fiEul to recognise that the sensational im-
pression on the ear preserves precisely the same character, appearing as a series
of separate pulses of sound, even when their frequency is so great that we have
^ no longer time to fix each beat, as it passes, distinctly in our consciousness and
count it.t
But while the hearer in this case is quite capable of distinguishing that his ear
now hears 30 beats of the same kind as the 4 or 6 in a second which he heard
before, the effect of the collective impression of such a rapid beat is quite different.
In the first place the mass of tone becomes confused, which I principally refer to
the psychological impressions. We actuaUy hear a series of pulses of tone, and
are able to recognise it as such, although no longer capable of following each
singly or separating one from the other. But besides this psychological considera-
tion, the sensible impression is also unpleasant. Such rapidly beating tones are
jarring and rough. The distinctive property of jarring, is the intermittent cha-
racter of the sound. We think of the letter B as a characteristic example of
a jarring tone. It is well known to be produced by interposing the uvula, or else
the thin tip of the tongue, in the way of the stream of air passing out of the mouth,
^ in such a manner as only to allow the air to force its way through in separate pulses,
the consequence being that the voice at one time sounds freely, and at another is
cut off. j:
Intermittent tones were also produced on the double siren just described by
using a Uttle reed pipe instead of the vdnd-conduit of the upper box, and driving
the air through this reed pipe. The tone of this pipe can be heard externally only
when the revolution of the disc brings its holes before the holes of the box and
open an exit for the air. Hence, if we let the disc revolve while air is driven
through the pipe, we obtain an intermittent tone, which sounds exactly like beats
arising firom two tones sounded at once, although the intermittence is produced by
purely mechanical means. Such effects may also be produced in another way on
the same siren, Eemove the lower windbox and retain only its pierced cover,
over which the disc revolves. At the under part apply one extremity of an india-
rubber tube against one of the holes in the cover, the other end being conducted
f by a proper ear-piece to the observer's ear. The revolving disc alternately opens
and closes the hole to which the india-rubber tube has been applied. Hold a
tuning-fork in action or some other suitable musical instrument above and near
* [See App. XX. sect. B. No. 7, for direc- Octave, but become rapidly too fast to be
tions for observing heata.— Translator.] follo^eed. As, however, these are not simple
t [The Harmonical is very convenient for tones, the beats are not perfectly olear. —
this purpose. On the db key is a d, one Translator J]
comma lower than d. These dd^ beat about % [Phonautographio figures of the effect
9, 18, 36, 73 times in 10 seconds in the of r, resemble those of fig. 58, p. i66a. 8ix
different Octaves, the last barely countable, varieties of these figures are given on p. 19 of
Also e'b and e, beat 33, 66, 132, 364 in 10 .Donders's important little tract, on 'The Physio-
seconds in the different Octaves. The two fiirst llogy of Speech Sounds, and especially of those
of these sets of beats can be counted, the two / in the Dutch Language * {De Physioloaie der
last cannot be counted, but will be distinctly / Spraakklanken^ in het bijeonder van ate der
perceived as separate pulses. Similarly the \ nederlandsche taal, Utrecht 1870, pp. 24). —
beats between all consecutive notes, (except F ' Translator,]
and Of B and C), can be counted in the lowest
-^.^v
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CHAP. vin. LIMITS OP THE FREQUENCY OP BEATS. [t6gj
the rotating disc. Its tone will be heard intermittently and the number of
intermissions can be regulated by altering the velocity of the rotation of the
disc.
In both ways then we obtain intermittent tones. In the first case the tone of
the reed pipe as heard in the outer air is interrupted, because it can only escape
from time to time. The intermittent tone in this case can be heard by any number
of listeners at once. In the second case the tone in the outer air is continuous,
but reaches the ear of the observer, who hears it through the disc of the siren,
intermittently. It can certainly be heard by one observer only, but then aU kinds
of musical tones of the most diverse pitch and quality may be employed for the
purpose. The intermission of their tones gives them all exactly the same kind of
roughness which is produced by two tones which beat rapidly together. We thus
come to recognise clearly that beats and intermissions are identical, and that either
when fast enough produces what is termed a jar or rattle. f
Beats produce intermittent excitement of certain auditory nerve fibres. The
refikson why such an intermittent excitement acts so much more unpleasantly than
an equally strong or even a stronger continuous excitement, may be gathered from ^j^ ^
the analogous action of other human nerves. Any powerful excitement of a nerve ^*"
deadens its excitability, and consequently renders it less sensitive to fresh irritants, ^j^ ]
But after the excitement ceases, and the nerve is left to itself, irritabihty is speedily ^'i
re-established in a living body by the influence of arterial blood. Fatigue and re- X^^^
freshment apparently supervene in different organs of the body with different
velocities ; but they are found wherever muscles and nerves have to operate. The
eye, which has in many respects the greatest analogy to the ear, is one of those
organs in which both fatigue and refreshment rapidly ensue. We need to look at
the sun but an instant to find that the portion of the retina, or nervous expansion
of the eye, which was affected by the solar light has become less sensitive for other
light. Immediately afterwards on turning our eyes to a uniformly illuminated ^
surface, as the sky, we see a dark spot of the apparent size of the sun ; or several
such spots with lines between them, if we had not kept our eye steady when look-
ing at the sun but had moved it right and lefk. An instant suffices to produce this
effect ; nay, an electric spark, that lasts an immeasurably short time, is fully
capable of causing this species of fatigue.
When we continue to look at a bright surface, the impression is strongest at
first, but at the same time it blunts the sensibility of the eye, and consequently
the impression becomes weaker, the longer we allow the eye to act. On coming
out of darkness into full daylight we feel blinded ; but after a few minutes, when
the sensibility of the eye has been blunted by the irritation of the light, — or, as we
say, when the eye has grown accustomed to the glare, — this degree of brightness is
found very pleasant. Conversely, in coming from fall daylight into a dark vault,
we are insensible to the weak light about us, and can scarcely find our way about,
yet after a few minutes, when the eye has rested from the effect of the strong hght, ^
we are able to see very well in the semi-dark room.
These phenomena and the like can be conveniently studied in the eye, because
individual spots in the eye may be excited and others left at rest, and the sensations
of each may be afterwards compared. Put a piece of black paper on a tolerably
well-lighted white surface, look steadily at a point on or near the black paper, and
then withdraw the paper suddenly. The eye sees a secondary image of the black
paper on the white surfebce, consisting of that portion of the white surfetce where
the black paper lay, which now appears brighter than the rest. The place in the
eye where the image of the black paper had been formed, has been rested in com-
parison with all those places which had been affected by the white surface, and
on removing the black paper this rested part of the eye sees the white surface in
its first fresh brightness, while those parts of the retina which had been already
fatigued by looking at it, see a decidedly greyer tinge on the whiter surfeuse.
Hence by the continuous uniform action of the irritation of light, this irritation
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■/
I70 LIMITS OP THE PEEQUENOY OP BEATS* pabt n.
itself blunts the sensibility of the nerve, and thus effeotually protects this organ
against too long and too violent excitement.
It is quite different when we allow intermittent light to act on the eye, such as
flashes of light with intermediate pauses. During these pauses the sensibility is
again somewhat re-established, and the new irritation consequently acts much
more intensely than if it had lasted with the same uniform strength. Every one
knows how unpleasant and annoying is any flickering light, even if it is relatively
very weak, coming, for example, from a little flickering taper or rushlight.
The same thing holds for the nerves of touch. Scraping with the nail is far
more annoying to the skin than constant pressure on the same place with the
same pressure of the nail. The unpleasantness of scratching, rubbing, tickling
depends upon the intermittent excitement which they produce in the nerves of
touch,
f A jarring intermittent tone is for the nerves of hearing what a flickering light
is to the nerves of sight, and scratching to the nerves of touch. A much more
intense and unpleasant excitement of the organs is thus produced than would be
occasioned by a continuous uniform tone. This is even shewn when we hear very
weak intermittent tones. If a tuning-fork is struck and held at such a distance
from the ear that its sound cannot be heard, it becomes immediately audible if the
handle of the fork be revolved by the Angers. The revolution brings it alternately
into positions where it can and cannot transmit sound to the ear [p. 161&], and
this alternation of strength is immediately perceptible by the ear. For the same
reason one of the most deUcate means of hearing a very weak, simple tone consists
in sounding another of nearly the same strength, which makes from 2 to 4 beats in
a second with the first. In this case the strength of the tone varies from nothing
to 4 times the strength of the single simple tone, and this increase of strength
combines with the alternation to make it audible,
f Just as this alternation of strength will serve to strengthen ihe impression of
the very weakest musical tones upon the ear, we must conclude that it must also
serve to make the impression of stronger tones much more penetrating and violent,
than they would be if their loudness were continuous.
We have hitherto confined our attention to cases where the number of beats
did not exceed 20 or 30 in a second. We saw that the beats in the middle part of
the scale are still quite audible and form a series of separate pulses of tone. But
this does not furnish a Umit to their number in a second.
The interval V c" gave us 33 beats in a second, and the effect of sounding the two
notes together was very jarring. The interval of a whole tone b^} c" gives nearly
twice as many beats, but these are no longer so cutting as the former. The rule
assigns 88 beats in a second to the minor Third a' c'^ but in reality this interval
scarcely shews any of the roughness produced by beats from tones at closer intervals.
We might then be led to conjecture that the increasing number of beats weakened
^ their impression and made them inaudible. This conjecture would find an analogy
in the impossibiUty of separating a series of rapidly succeeding impressions of
light on the eye, when their number in a second is too large. Think of a Rowing
stick swung round in a circle. If it executes 10 or 15 revolutions in a second, the
eye believes it sees a continuous circle of fire. Similarly for colour-tops, with
which most of my readers are probably familiar. If the top be spun at the rate
of more than 10 revolutions in a second, the colours upon it mix and form a per-
/ fectly unchanging impression of a mixed colour. It is only for very intense light
' that the alternations of the various fields of colour have to take place more quickly,
I 20 to 30 times in a second. Hence the phenomena are quite analogous for ear and
{ eye. When the alternation between irritation and rest is too feist, the alternation
ceases to be felt, and sensation becomes continuous and lasting.
However, we may convince ourselves that in the case of the ear, an increase of
the number of beats in a second is not the only cause of the disappearance of the
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CHAP. VIII. LIMITS OP THE FREQUENCY OF BEATS. 171
corresponding sensation. As we passed from the interval of a Semitcme V c" to
that of a minor Third a' c", we not only increased the number of beats, but the
width of the interval. Now we can increase the nmnber of beats without increasing
the interval by taking it in a higher Octave. Thus taking V d' an Octave higher
we have V c'" with 66 beats, and another Octave would give us V" c"" with as
many as 132 beats, and these are really audible in the same way as the 33 beats
of V cf'y although they certainly become weaker in the higher positions. Never-
theless the 66 beats of the interval h" d" are much more distinct and penetrating
than the same number in the whole Tone h'\}c"t and the ^Z of the interval e'" f"
are still quite evident, while the 88 of the minor Third a' c" are practically in-
audible. My assertion that as many as 132 beats in a second are audible will per-
haps appear very strange and incredible to acousticians. But the experiment is
easy to repeat, and if on an instrument which gives sustained tones, as an organ
or harmonium, we strike a series of intervals of a Semitone each, beginning low %
down, and proceeding higher and higher, we shall hear in the lower parts very
slow beats {Bfi gives 4I, B c gives 8^, b d gives 16^ beat in a second), and as we
ascend the rapidity will increase but the character of the sensation remain un-
altered. And thus we can pass gradually from 4 to 132 beats in a second, and
convince ourselves that though we become incapable of counting them, their cha-
racter as a series of pulses of tone, producing an intermittent sensation, remains
unaltered. It must be observed, however, that the beats, even in the higher parts ' '
of the scale, become much shriller and more distinct, when their number is \
diminished by taking intervals of quarter tones or less. The most penetrating i
roughness arises even in the upper parts of the scale from beats of 30 to 40 in a
second. Hence high tones in a chord are much more sensitive to an error in
tuning amounting to the fraction of a Semitone, than deep ones. While two d
notes which differ from one another by the tenth part of a Semitone, produce about
3 beats in two seconds^* which cannot be observed without considerable attention, f
and, at least, gives notfeeling of roughness, two d' notes with the same error give
3 beats in one second, and two c"' notes 6 beats in one second, which becomes very
disagreeable. The character of the roughness also alters with the number of beats. 1
Slow beats give a coarse kind of roughness, which may be considered as chattering 1
or jarring ; and quicker ones have a finer but more cutting roughness.
Hence it is not, or at least not solely, the large number of beats which renders
them inaudible. The magnitude of the interval is a £a.ctor in the result, and con-
sequently we are able with high tones to produce more rapid audible beats than
with low tones.
Observation shews us, then, on the one hand, that equally large intervals by
no means give equally distinct beats in all parts of the scale. The increasing
number of beats in a second renders the beats in the upper part of the scale less
distinct. The beats of a Semitone remain distinct to the upper limits of the four-
times accented octave [say 4000 vib.], and this is also about the limit for musical ^
tones fit for the combinations of hannony. The beats of a whole tone, which in
deep positions are very distinct and powerfal, are scarcely audible at the upper
limit of the thrice-accented octave [say at 2000 vib.]. The major and minor
Third, on the other hand, which in the middle of the scale [264 to 528 vib.] may
be regarded as consonances, and when justly intoned scarcely shew any roughness,
are decidedly rough in the lower octaves and produce distinct beats.
On the other hand we have seen that distinctness of beating and the roughness
of the combined sounds do not depend solely on the number of beats. For if we
could disregard their magnitudes all the following intervals, which by calculation
should have 33 beats, would be equally rough :
* [Taking c'»264, a tone one-tenth of a second. The figures in the text have bee
Semitone or 10 cents higher miJce 265-5 vibra- altered to these more exact numbers.— 2VaiM-
tions, and these tones beat i} times in a 2a^.]
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172
LIMITS OP THE FREQUENCY OP BEATS.
PABT n.
t^
[528-495=33]
[major, 297-264] and S e' [minor 330-297]
[198-165]
[165-132]
[132-99]
the tones with ^ of its own
between the intensities of vibration o and -^.
the Semitone . • h' c"
the whole Tones . . d W
the minor Third . . e gf
the major Third . . ce
the Fourth . . . Gc
the Fifth . . . CQ [99-66]
and yet we find that these intervals are more and more free from roughness.*
The roughness arising from sounding two tones together depends, then, in a
compound manner on the magnitude of the interval and the number of beats pro-
duced in a second. On seeking for the reason of this dependence, we observe that,
as before remarked, beats in the ear can exist only when two tones are produced
sufficiently near in the scale to set the same elastic appendages of the auditory
nerve in sympathetic vibration at the same time. When the two tones produced
^ are too far apart, the vibrations excited by both of them at once in Corti*s organs
are too weak to admit of their beats being sensibly felt, supposing of course that
no upper partial or combinational tones intervene. According to the assumptions
made in the last chapter respecting the degree of damping possessed by Corti's
organs (p. 144c), it would result, for example, that for the interval of a whole Tone
c dy such of Corti's fibres as have the proper tone c|!, would be excited by each of
intensity; and these fibres will therefore fluctuate
But if we strike the simple tones c
and d$fy it follows from the table there given that Corti's fibres which correspond
to the middle between c and oj)! will alternate between the intensities o and |f.
Conversely the same intensity of beats would for a minor Third amount to only
0*194, and for a major Third to only 0*108, and hence would be scarcely perceptible
beside the two primary tones of the intensity i.
Pig« 59» which we used on p. 144^^ to express the
% intensity of the sympathetic vibration of Corti's
fibres for an increasing interval of tone, may
here serve to shew the intensity of the beats
which two tones excite in the ear when forming
different intervals in the scale. But the parts on
the base line must now be considered to repre-
sent fifths of a wlioU Tone, and not as before of
a Semitone. In the present case the distance of
the two tones from each other is doubly as great as that between either of them
and the intermediate Corti*s fibres.
Had the damping of Corti's organs been equally great at all parts of the scale,
and had the number of beats no influence on the roughness of the sensation, equal
intervals in all parts of the scale would have given equal roughness to the combined
tones. But as this is not the case, as the same intervals diminish in roughness
% as we ascend in the scale, and increase in roughness as we descend, we must either
assume that the damping power of Corti's organs of higher pitch is less than that
of those of lower pitch, or else that the discrimination of the more rapid beats
meets with certain hindrances in the nature of the sensation itself.
At present I see no way of deciding between these two suppositions ; but the
former is possibly the more improbable, because, at least with our artificial musical
instruments, the higher the pitch of a vibrating body, the more difficulty is ex-
perienced in isolating it sufficiently to prevent it from communicating its vibrations
to its environment. Very short, high-pitched strings, little metal tongues or plates,
&c., yield high tones which die off with great rapidity, whereas it is easy to
generate deep tones with correspondingly greater bodies which shall retain their
tone for a considerable time. On the other hand the second supposition is favoured
by the analogy of another nervous apparatus, the eye. As has been already re-
the student shonld listen to the beats of the
primes only.— TrawZator.]
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♦ [All these intervals can be tried on the
Harmonical, but as the tones are compound,
CHAP. vin. LIMITS OF THE FREQUENCY OF BEATS.
173
■f
marked, a series of impressions of light, following each other rapidly and regularly,
excite a uniform and continuous sensation of light in the eye. When the separate
luminous irritations follow one another very quickly, the impression produced by
each one lasts unweakened in the nerves till the next supervenes, and thus the
pauses can no longer be distinguished in sensation. In the eye, the number of
separate irritations cannot exceed 24 in a second without being completely fused
into a single sensation. In this respect the eye is far surpassed by the ear, which
can distinguish as many as 132 intermissions in a second, and probably even that
is not the extreme limit. Much higher tones of sufficient strength would probably
allow us to hear still more.* It lies in the nature of the thing, that different kinds
of apparatus of sensation should shew different degrees of mobility in this respect,
since the result does not depend simply on the mobility of the molecules of the
nerves, but also depends upon the mobility of the auxiliary apparatus through
which the excitement is induced or expressed. Muscles are much less mobile than ^
the eye ; ten electrical discharges in a second directed through them generally
suffice to bring the voluntary muscles into a permanent state of contraction. For
the muscles of the involuntary system, of the bowels, the vessels, &c., the pauses
between the irritations may be as much as one, or even several seconds long, with-
out any intermission in the continuity of contraction.
The ear is greatly superior in this respect to any other nervous apparatus. It
is eminently the organ for small intervals of time, and has been long used as such
by astronomers. It is well known that when two pendulums are ticking near one
another, the ear can distinguish whether the ticks are or are not coincident, within
one hundredth of a second. The eye would certainly fail to determine whether
two flashes of light coincided within ^^ second ; and probably within a much larger
fraction of a second.f
But although the ear shews its superiority over other organs of the body in
this respect, we cannot hesitate to assume that, like every other nervous apparatus, ^
the rapidity of its power of apprehension is limited, and we may even assume that
we have approached very near the limit when we can but faintly distinguish 132
beats in a second.
* [In the two high notes g"" f"'t of the
flageolet fifes (p. 153(2, note), which if justly
intoned should give 198 beats in a second, I
could hear none, though the tones were very
powerful, and the scream was very cutting
indeed* — ^In the case of V d'\ which on the
Harmonical are tuned to make 1056 and 990,
the rattle of the 66 beats, or thereabouts, is
quite distinct, and the differential tone is very
powerful at the same time. — Translator.]
f [The following is an interesting compari-
son between eye and ear, and eye and hand.
The usual method of observing transits is by
counting the pendulum ticks of an astronomi-
cal clock, and by observing the distances of
the apparent positions of a star before and after
passing each bar of the transit instrument at
the moments of ticking, to estimate the moment
at which it had passed each bar. This is done
for five bars and a mean is taken. But a few
years ago a chronograph was introduced at
Greenwich Observatory, consisting of a uni-
formly revolving cylinder in which a point
pricks a hole every second. Electrical com-
munication being established with a knob on
the transit instrument, the observer presses
the knob at the moment he sees a star dis-
appear behind a bar, and an electrical current
causes another point to make a hole between
the seconds holes on the chronograph. By
applying a scale, the time of transit is thus
measured off. A mean, of course, is taken as
before. On my asking Mr. Stone (now Astrono-
mer at Oxford, then chief assistant at Green-
wich Observatory) as to the relative degree of
accuracy of the two methods, he told me that
he considered the first gave results to one-
tenth, and the second to one-twentietii of a
second. It must be remembered that the first
method also required a mental estimation
which had to be performed in less than a m
second, and the result borne in mind, and that
this was avoided by the second plan. On the
other hand in the latter the sensation had to
be conveyed from the eye to the brain, which
issued its orders to the hand, and the hand
had to obey them. Hence there was an endea-
vour at performing simultaneously, several
acts which could only be successive. Any one
will find upon trial that an attempt to merely
make a mark at the moment of hearing an
expected sound, as, for example, the repeated
tick of a common half seconds clock, is liable
to great error. — Translator,]
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174 DEEP AND DEEPEST TONES. part n.
CHAPTEE IX.
DEEP AND DEEPEST TONES.
Beats give us an important means of determining the limit of the deepest tones,
and of accounting for certain peculiarities of the transition from the sensation of
separate pulses of air to a perfectly continuous musical tone, and to this inquiry
we now proceed.
The question : what is the smallest number of vibrations in a second which
can produce the sensation of a musical tone ? has hitherto received very contra-
dictory replies. The estimates of different observers fluctuate between 8 (Savort)
^ and about 30. The contradiction is explained by the existence of certain difScul-
ties in the experiments.
In the first place it is necessary that the strength of the vibrations of the air
for very deep tones should be extremely greater than for high tones, if they are to
make as strong an impression on the ear. Several acousticians have occasionally
started the hypothesis that, caeteris pariJms, the strength of tones of different
heights is directly proportional to the vis viva of the motion of the air, or, which
comes to the same thing, to the amount of the mechanical work applied for pro-
ducing it. But a simple experiment with the siren shews that when equal amounts
of mechanical work are applied to produce deep and high tones under conditions
otherwise alike, the high tones excite a very much more powerful sensation than
the deep ones. Thus, if the siren is blown by a bellows, which makes its disc
revolve with increasing rapidity, and if we take care to keep up a perfectly
uniform motion of the bellows by raising its handle by the same amount the same
^ number of times in a minute, so as to keep its bag equally filled, and drive the
same amount of air under the same pressure through Ifee siren in the same time,
we hear at first, while the revolution is slow, a weaJc deep tone, which continually
ascends, but at the same time gains strength at an extraordinary rate, till when the
highest tones producible on my double siren (near to a", with 880 vibrations in a
second) are reached, their strength is almost insupportable. In this case by £ar
the greatest part of the uniform mechanical work is applied to the generation of
sonorous motion, and only a small part can be lost by the friction of the revolving
disc on its axial supports, and the air which it sets into a vortical motion at the
same time ; and these losses must even be greater for the more rapid rotation than
for the slower, so that for the production of the high tones less mechanical work
remains applicable than for the deep ones, and yet the higher tones appear to our
sensation extraordinarily more powerful than the deep ones. How far upwards
this increase may extend, I have as yet been unable to determine, for the rapidity
^ of my siren cannot be farther increased with the same pressure of air.
The increase of strength with height of tone is of especial consequence in the
deepest part of the scale. It follows that in compound tones of great depth, the
upper partial tones may be superior to the prime in strength, even though in
musical tones of the same description, but of greater height, the strength of the
prime greatly predominates. This is readily proved on my double siren, because
by means of the beats it is easy to determine whether any partial tone which we
hear is the prime, or the second or third partial tone of the compound under
examination. For when the series of 12 holes are open in both windboxes, and
the handle, which moves the upper windbox, is rotated once, we shall have, as
already shown, 4 beats for the primes, 8 for the second partials, and 12 for the
third partials. Now we can make the disc revolve more slowly than usual, by
allowing a well-oiled steel spring to rub against the edge of one isc with different
degrees of pressure, and thus we can easily produce series of puffs which corre-
spond to very deep tones, and then, turning the handle, we can count the beats.
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CHAP. IX. DEEP AND DEEPEST TONES. 175
By allowing the rapidity of the revolution of the discs to increase gradually, we
find that the first audible tones produced make 1 2 beats for each revolution of the
handle, the number of puffs being firom 36 to 40 in the second. For tones with
from 40 to 80 puffs, each revolution of the handle gives 8 beats. In this case,
then, the upper Octave of the prime is the strongest tone. It is not till we have
80 puffs in a second that we hear the four beats of the primes.
It is proved by these experiments that motions of the air, which do not take
the form of pendular vibrations, can excite distinct and powerful sensations of tone,
of which the pitch number is 2 or 3 times the number of the pulses of the air,
and yet that the prime tone is not heard through them. Hence, when we continu-
ally descend in the scale, the strength of our sensation decreases so rapidly that
the sound of the prime tone, although its vis viva is independently greater than that
of the upper partials, as is shewn in higher positions of a musical tone of the
same composition, is overcome and concealed by its own upper partials. Even ^
when the action of the compound tone on the ear is much reinforced, the effect
remains the same. In the. experiments with the siren the uppermost plate of the
bellows is violently agitated for the deep tones, and when I laid my head on it, my
whole head was set into such powerful sympathetic vibration that the holes of the
rotating disc, which vanish to an eye at rest, became again separately visible,
through an optical action similar to that which takes place in stroboscopic discs.
The row of holes in action appeared to stand still, the other rows seemed to move
partly backwards and partly forwards, and yet the deepest tones were no more
distinct than before. At another time I connected my ear by means of a properly
introduced tube with an opening leading to the interior of the bellows. The
agitation of the drum skin of the ear was so great that it produced an intolerable
itching, and yet the deepest tones remained as indistinct as ever.
In order, then, to discover the limit of deepest tones, it is necessary not only to
produce very violent agitations in the air but to give these the form of simple %
pendular vibrations. Until this last condition is fulfilled we cannot possibly say
whether the deep tones we hear belong to the prime tone or to an upper partial tone
of the motion of the air.* Among the instruments hitherto employed the wide-
stopped organ pipes are the most suitable for this purpose. Their upper partial
tones are at least extremely weak, if not quite absent. Here we find that even the
lower tones of the 16-foot octave, C, to E^^ begin to pass over into a droning noise,
so that it becomes difficult for even a practised musical ear to assign their pitch with
certainty ; and, indeed, they cannot be tuned by the ear alone, but only indirectly
by means of the beats which they make with the tones of the upper octaves. We
observe a similar effect on the same deep tones of a piano or harmonium ; they
form drones and seem out of tune, although their musical character is on the
whole better established than in the pipes, because of their accompanying upper
partial tones. In music, as artistically applied in an orchestra, the deepest tone
used is, therefore, the E, of the [4-stringed German] double bass, with 41^ vibra- f
tions in a second, [see p. i8c, note] and I think I may predict with certainty that all
efforts of modem art applied to produce good musical tones of a lower pitch must
fail, not because proper means of agitating the air cannot be discovered, but
because the human ear cannot hear them. The 16-foot C, of the organ, with
33 vibrations in a second, certainly gives a tolerably continuous sensation of
drone, but does not allow us to give it a definite position in the musical scale.
We almost begin to observe the separate pulses of air, notwithstanding the regular
form of the motion. In the upper half of the 32 -foot octave, the perception of the
separate pulses becomes still clearer, and the continuous part of the sensation,
* Thos Savart's instrument, where a rota- tion, and consequently the upper partial tones
ting rod strikes through a narrow slit, is totally must be very strongly developed, and the
unsuitable for making the lowest tone audible, deepest tones, which are heard for 8 to 16
The separate puSs of air are here very short in passages of the rod through the hole in a second,
relation to the whole periodic time of the vibra- can be nothing but upper partials.
Digitized by VjOOQlC
176
DEEP AND DEEPEST TONES*
PART IK
which may be compared with a sensation of tone, continaally weaker, and in Uiq
lower half of the 3 2 -foot octave we can scarcely be said to hear anything but the
individual pulses, or if anything else is really heard, it can only be weak upper
partial tones, from which the musical tones of stopped pipes are not quite free.
I have tried to produce deep simple tones in another way. Strings which are
weighted in their middle with a heavy piece of metal, on being struck give a com-
pound tone consisting of many simple tones which are mutually inharmonic. The
prime tone is separated from the nearest upper partials by an interval of several
Octaves, and hence there is no danger of confusing it with any of them ; besides,
the upper tones die away rapidly, but the deeper ones continue for a very long time.
A string of this kind* was stretched on a sounding-box having a single opening
which could be connected with the auditory passage, so that the air of the sounding-
box could escape nowhere else but into the ear. The tones of a string of customary
m pitch are under these circumstances insupportably loud. But for D„ with 37^
vibrations in a second, there was only a very weak sensation of tone, and even this
was rather jarring, leading to the conclusion that the ear began even here to feel
the separate pulses separately, notwithstanding their regularity. At 5^^, with
29^ vibrations in a second, there was scarcely anything audible left. It appears,
then, that those nerve fibres which perceive such tones begin as early as at this
note to be no longer excited with a uniform degree of strength during the whole
time of a vibration, whether it be the phases of greatest velocity or the phases of
greatest deviation from their mean position in the vibrating formations in the ear
which effect the excitement.f
* It was a thin brass pianoforte string. The
weight was a copper kreutzer piece, [pronounce
kroitser; three kreutzers make a penny at
Heidelberg, where the experiment was pro-
bably tried,] pierced in the middle by a hole
«r through which the wire passed, and then made
'' to grip the wire immovably by driving a steel
point between the hole in the kreutzer and the
string.
f Subsequently I obtained two large tuning-
forks from Herr Koenig in Paris, with sliding
weights on their prongs. By altering the posi-
tion of the weights, the pitch was changed,
and the corresponding number of vibrations
was given on a scale which runs along the
prongs. One fork gave 24 to 35, the other 35
to 61 vibrations. The sliding weight is a plate,
5 centimetres [nearly 2 inches] in diameter,
and forms a mirror. On bringing the ear close
to these plates the deep tones are well heard.
For 30 vibrations I could still hear a weak
drone, for 28 scarcely a trace, although this
arrangement made it easily possible to form
«r oscillations of 9 millimetres [about ^ inch] in
amplitude, quite close to the ear. Prof. W.
Preyer has been thus able to hear down to 24
vib. He has also applied another method
(Pkysioloffische Ahhandlungen^ Physiological
Treatises, Series i, part i, *0n the limits of
the perception of tone,' pp. 1-17) by using very
deep, loaded tongues, in reed pipes, which were
constructed for this purpose by Herr Appunn
of Hanau, and gave from 8 to 40 vib. These
were set into strong vibration by blowing, and
then on interrupting the wind, the dying off
of the vibrations was listened to by laying the
ear against the box. He states that tones were
heard downwards as low as 15 vib. But the
proof that the tones heard corresponded with
the primes of the pipes depends only on the
fact, that the pitch gradually ascended as they
passed over into the tones of from 25 to 32
vib.: which were more audible, but died off more
rapidly. With extensive vibrations, however,
the tongues may have very easily given their
point of attachment longitudinal impulses of
double the frequency, because when they
reached each extremity of their amplitude they
might drive back the point of attachment
through their flexion, whereas in the middle
of the vibration they would draw it forward by
the centrifugal force of their weight. Since
the power of distinguishing pitch for these
deepest tones is extremely imperfect, I do not
feel my doubts removed by the judgment of
the ear when the estimates are not checked by
the counting of beats.
[This check I am fortunately able to supply.
A copy of the instrument used by Prof. Preyer
is in the South Kensington Museum. It con-
sists of an oblong box, in the lower part of
which are the loaded harmonium reeds, not
attached to pipes, but vibrating within the box,
and governed by valves which can be opened
at pleasure. On account of the beats between
tongue and tongue taking place in strongly
condensed air, they are accelerated, and the
nominal pitch, obtained by counting the beats
from reed to reed, is not quite the same
as the actual pitch (see App. XX. sect. B.
No. 6). The series of tones is supposed to
proceed from 8 to 32 vib. by differences of i
vib., from 32 to 64 by differences of 2 vib., and
from 64 to 128 by differences of 4 vibs. In
November 1879, for another purpose, I deter-
mined the pitch of every reed by Scheibler's
forks, (see App. XX. sect. B. No. 7) by means
of the upper partials of the reeds. For Beeds
8, 9, xo, II, I used from the 20th to the 30U1
partial, but I consider only Beed 11 as quite
certain. I found it made 10-97 ^i^- ^y ^he 20th,
and 10-95 by both the 21st and 24th partiids.
From Heed 11 upwards I determined every
pitch, in many cases by several partials, the
result only differing in the second place of
decimals. I give the two lowest Octaves, the
Digitized by VjOOQlC
CHAP. IX.
DEEP AND DEEPEST TONES.
^77
Hence although tones of 24 to 28 vib. have been heard, notes do not begin to
/have a definite pitch till about 40 vibrations are performed in a second. These
facts will agree with the hypothesis concerning the elastic appendages to the audi-
tory nerves, on remembering that tlie deeply intoned fibres of Corti may be set in
sympathetic vibration by still deeper tones, although with rapidly decreasing
strength, so that sensation of tone, but no discrimination of pitch, is possible. If
the most deeply intoned of Corti's fibres lie at greater intervals from each other in
the scale, but at the same time their damping power is so great that every tone
which corresponds to the pitch of a fibre, also pretty strongly affects the neighbour-
ing fibres, there will be no safe distinction of pitch in their vicinity, but it will
proceed continuously without jumps, while the intensity of the sensation must at
the same time become small.
Whilst simple tones ia the upper half of the 16-foot octave are perfectly con-
only pitches of interest for the present pur-
pose, premising that I consider the three lowest
pitches (for which the upper partials lay too
close together) and the highest (which had a H
bad reed) to be very uncertain.
Nominal
Actual
Nominal
Actual
Nominal
Actual
8 9 10 II 12 13 14 15 16
791 889 981 10-95 11-90 12-90 13-93 14-91 15-91
17 18 19 20 21 22 23 24 25
16*90 17-91 18-89 J9'9i 20-91 21-91 22-88 23-97 24-92
26 27 28 29 30 31 32
25-9226-86 2785 28-84 29-77 30-68 31-47
There can therefore be no question as to the
real pitch. At Prof. Preyer's request I ex-
amined this instrument in Oct. 1877, >^Qd he
has printed my notes in his Akustische Unter-
suchungen, pp. 6-8. From these I extract the
following : —
R means Reed, and R 21 "25 means that the
two reeda 21 and 25 were sounded together and
gave beats.
R 21-25, beat 4 in i sec, counted for 20 sec.
Henoe both of their lowest partials must have been
effective.
R 20-24, beat 4 in i sec, counted for 10 sec.
R 19 ••23, beat 4 in i sec, counted for 20 sec.
R 17 ••21, same beats.
R 16 ••20, same beats quite distinctly.
R 15- 19, at first I lost the beats, but afterwards
by getting li 15 well into action before R 19 was
set 00, and keeping on pumping, I got out the 4 in
a second quite distinctly. Hence the lowest partial
of R 15 was effective.
R 15" 17, here also I once heard 4 in a sec, but
this must have been from the Octaves.
R 14*' 16, I was quite unable to distinsrnish
anything in the way of beats, but volleys like a
feu de foie about a second in length, but impossible
to count accurately ; thpy may have been 2 in a
sec. and I counted double. At the same time I
seemed occasionally to hear a low beat, so low and
gentle that I could not count it, and the great
exertion of pumping the bel'ows full enough to
keep these two low reeds in action, prevented
accurate observation.
R 15 decidedly seemed flatter thsn R i^, so
that I could have only heard the lowest partial of
R 15 and the Octave of R 13.
On soundinjj; R 14 and R 15 separately, I
seemed to hear from each a very low tone, in
quality more like a differential tone than any-
thing else. This could also be heard even with
R 13 and R Z2, below the thumps, and even in
Rii.
At R 8 I he^rd only the sighing of the escape
of wind from the reed, 8 times in a second, as well
as I could count, and I also heard beats evidently
arising ftom the higher partials, and also 8 in a
.«iecond.
At R 9 there was the same kind of sishing and
equally rapid beats. But in addition I seemed to
hear a faiui low tone.
At R 10 there was no mistake as to the existence
of such a musical tone.
At R zi and R 12 it was still more distinct.
At R 13 the tene was ven* distinct and was
quite a good musical tone at *R 14, but the sish
was still audible. Was this the lowest partial or
its Octave ?
R 16 gave quite an organ tone, nothing like
a hum or a differentia], but the sish and beats .
remain. I must have heard the lowest partial, ^
and by continual pumping I was able to Keep it
in my ear.
R x8 •• 20 gave beats of 2 in a see. very distinctly.
Up to R 25 the sish could be heard at the
commencement, but it rapidly disappeared. It
feels as if the tone were getting gradually into
practice. ITiis effect continued up to R 22, after
which the sish was scarcely brought out at all.
In fact long before this the sish was made only at
the first moment, and was rather a bubble than a
sish.
In listening to the very low beats, the beats of
the lowest partials as such could not be separated
f^om the general m&<<s of beats, but the 4 in a sec
were ^uite clear from R 15-19. The lowest pair
in which I was distinctly able to hear the bell-like
beat of the lowest partials distinct from the general
crash was R 30-34. But I fancied I heard it at
R 28-32.
Prof. Preyer also, in the same place, details %
his experiments with two enormous tuning-
forks giving 13-7 and 18 6 vib. The former
gave no musical tone at all, though the vibria-
tions were visible for 3 min. and were dis-
tinctly separable by touch. The latter had
*an unmistakable dull tone, without droning
or jarring.' He concludes: 'Less than 15
vib. in a sec. give no musical tone. At from
16 to 24, say then 20 in the sec. the series of
aerial impulses begins to dissolve into a tone,
assuming that there are no pauses between
them. Above 24 begins the musical character
of these bass tones. Herr Appunn,* adds
Prof. Preyer, « informed me that the differen-
tial tone of 27-85 vib., generated by the two
forks of 111-3 and 83-45 \ib., was '* surprisingly
beautiful" and had a "wondrous effect."' —
Translator.]
Digitized by
\^o0gle
178 DEEP AND DEEPEST TONES. part n.
tinaous and musioal, yet for aerial vibrations of a different form, for example when
compound tones are used, discontinuous pulses of sound are still heard even witliin
this octave. For example, blow the disc of the siren with gradually increasing
speed. At first only pulses of air are heard ; but after reaching 36 vibrations in a
second, weak tones sound with them, which, however, are at first upper partials.
As the velocity increases the sensation of the tones becomes continually stronger,
but it is a long time before we cease to perceive the discontinuous pulses of air,
although these tend more and more to coalesce. It is not till we reach 1 10 or 1 1 7^
vibrations in a second (A or ^ of the great octave) that the tone is tolerably con-
tinuous. It is just the same on the harmonium, where, in the cor anglais stop, c
with 132 vibrations in a second stiU jars a little, and in the bassoon stop we observe
the same jarring even in & with 264 vibrations in a second. Generally the same
observation can be made on all cutting, snarling, or braying tones, which, as has
^ been already mentioned, are always provided with a very great number of distinct
upper partial tones.
The cause of this phenomenon must be looked for in the beats produced by the
high upper partials of such compound tones, which are too nearly of the same pitch.
If the 15 th and i6th partials of a compound tone are still audible, they form the
interval of a Semitone, and naturally produce the cutting beats of this dissonance.
That it is really the beats of these tones which cause the roughness of the whole
compound tone, can be easily felt by using a proper resonator. If G, is struck,
having 49^ vibrations in a second, the 15 th partial is f'jj^, the i6th g^\ and the
17th ^"J [nearly], &c. Now when I apply the resonator ^", which reinforces ^"
most, and f% g"% somewhat less, the roughness of the tone becomes extremely
more prominent, and exactly resembles the piercing jar produced when f'^ and
g" are themselves sounded. This experiment succeeds on the pianoforte, as well
as on both stops of the harmonium. It also distinctly succeeds for higher pitches,
f as far as the resonators reach. I possess a resonator for g'", and although it only
slightly reinforces the tone, the roughness of G, with 99 vibrations in a second,
was distinctly increased when the resonator was applied.*
Even the 8th and 9th partials of a compound tone, which are a whole Tone
apart, cannot but produce beats, although they are not so cutting as those from the
higher upper partials. But the reinforcement by resonators does not now succeed
BO well, because the deeper resonators at least are not capable of simultaneously
reinforcing the tones which differ from each other by a whole Tone. For the
higher resonators, where the reinforcement is slighter, the interval between the
tones capable of being reinforced is greater, and thus by means of the resonatoi^
g ' and g'" I succeeded in increasing the roughness of the tones Gio g (having
99 and 198 vibrations in a second respectively), which is due to the 7th, 8th, and
9th partial tones (/', g", a", and /", g"\ a'" respectively). On comparing the
tone of G as heard in the resonators with the tone of the dissonances f* g"
^ and g" a^' as struck directly, the ear felt their close resemblance, the rapidity of
intermittence being nearly the same.
Hence there can no longer be any doubt that motions of the air corresponding
to deep musical tones compounded of numerous partials, are capable of exciting at
one and the same time a continuous sensation of deep tones and a discontinuous
sensation of high tones, and become rough or jarring through the latter .f Herein
lies the explanation of the fact already observed in examining qualities of tone,
that compound tones with many high upper partials are cutting, jarring, or bray-
ing ; and also of the fact that they are more penetrating and cannot readily pass
unobserved, for an inteimittent impression excites our nervous apparatus much
more powerfully than a continuous one, and continually forces itself afresh on our
♦ [The student should now perform the ex- punn's Reed pipes of 3*2 and 64 vib. in the South
periments on the Harmonical indicated on Kensington Museum. Their musical character
p. 22d, note.— Translator,] is quite destroyed by the loud thumping of the
t [This is particularly noticeable on Ap- upper partials. — Translator.]
Digitized by VjOOQIC
CHAPS. IX. X, BEATS OF UPPEE PAETIALS.* 179
perception.* On the other hand simple tones, or compound tones which have only
a few of the lower upper partials, lying at wide intervals apart, must produce per-
fectly continuous sensations in the ear, and make a soffc and gentle impression,
without much energy, even when they are in reality relatively strong.
We have not yet been able to determine the upper limit of the number of inter-
mittences perceptible in a second for high notes, and have only drawn attention to
their becoming more difficult to perceive, and making a slighter impression, as they
became more iiumerous. Hence even when the form of vibration, that is the
quality of tone, remains the same, while the pitch is increased, the quality of tone
will generally appear to diminish in roughness. The part of the scale adjacent to
/ '"J, for which the ear is peculiarly sensitive, as I have already remarked (p. 1 16a),
must be particularly important, as dissonant upper partials which lie in this neigh-
bourhood cannot but be especially prominent. Now f"% is the 8th partial of f%
with 366I vibrations in a second, a tone belonging to the upper tones of a man's and f
the lower tones of a woman's voice, and it is the i6th partial of the unaccented
f^, which lies in the middle of the usual compass of men's voices.f I have already
mentioned that when human voices are strained these high notes are often heard
sounding with them. When this takes place in the deeper tones of men's voices,
it must produce cutting dissonances, and in fact, as I have already observed, when
a powerful bass voice is trumpeting out its notes in full strength, the high upper
partial tones in the four-times-accented octave are heard, in quivering tinkles
(p. 1 1 6c). Hence jarring and braying are much more usual and more powerful in
bass than in higher voices. For compound tones above f% the dissonances of the
higher upper partials in the four-times-accented octave, are not so strong as those
of a whole Tone, and as they occur at so great a height they can scarcely be
distinct enough to be clearly sensible.
In this way we can explain why high voices have in general a pleasanter tone,
and why all singers, male and female, consequently strive to touch high notes. If
Moreover in the upper parts of the scale slight errors of intonation produce muny
more beats than in the lower, so that the musical feeling for pitch, correctness, and
beauty of intervals is much surer for high than low notes.
According to the observations of Prof. W. Preyer the difference in the qualities
of tone of tuning-forks and reeds entirely disappears when they reach a height of
c'' 4224, doubtless for the reason he assigns, namely that the upper partials of the
reeds fall in the seventh and eighth accented octave, which are scarcely audible.
CHAPTER X.
BEATS OF THE UPPER PARTIAL TONES. If
The beats hitherto considered, were produced by two simple tones, without any
intervention of upper partial or combinational tones. Such beats could only arise
when the two given tones made a comparatively small interval with each other.
As soon as the interval increased even to a minor Third the beats became indistinct.
Now it is well known that beats can also arise from two tones which make a much
greater interval with each other, and we shall see hereafter that these beats play
a principal part in settling the consonant intervals of our musical scales, and they
* [In Prof. Tyndall's paper * On the Atmo- throwing the horns slightly out of unison ; bat
sphere as a Vehicle of Sound,* read before though the beats rendered the sound charac-
the Boyal Society, Feb. 12, 1874, in trying the teristic, they did not seem to augment the
distance at which intense sounds could be range.'— Tmiu2ator.]
heard at sea, he says {Philosophical Transac- f fOn the compass of voices see App. XX.
tions for 1874, vol. clxiv. p. 189), ' The influence sect. N. No. I.— TrawsZator.]
of "beats" was tried on June 3 [1J57J] by
Digitized by
\^o(5gle
i8o
BEATS OF UPPER PARTIALS.
PART II.
must consequently be closely examined. The beats heard when the two genera-
ting tones are more than a minor Third apart in the scale, arise from upper partial
and combinational tones.* When the compound tones have distinctly audible upper
partials, the beats resulting from them are generally clearer and stronger than
those due to the combinational tones, and it is much more easy to determine their
source. Hence we begin the investigation of the beats occurring in wider intervals
with those which arise from the presence of upper partial tones. It must not be
forgotten, however, that beats of combinational tones are much more general than
these, as they occur with all kinds of musical tones, both simple and compound,
whereas of course those due to upper partial tones are only found when such partials
are themselves distinct. But since all tones which are useful for musical purposes
are, with rare exceptions, richly endowed with powerful upper partial tones, the
beats due to these upper partials are relatively of much greater practical importance
f^ than those due to the weak combinational tones.
When two compound tones are sounded at the same time, it is readily seen,
from what precedes, that beats may arise whenever any two upper partial tones' lie
sufficiently near to each other, or when the prime of one tone approaches to an upper
partial of the other. The number of beats is of course, as before, the difference of
the vibrational numbers of the two partial tones to which the beats are due.
When this difference is small, and the beats are therefore slow, they are relatively
most distinct to hear and to count and to investigate, precisely as for beats of prime
tones. They are also more distinct when the particular partial tones which gene-
rate them are loudest. Now, for the tones most used in music, partials with a lovr
ordinal number are loudest, because the intensity of partial tones usually diminishes
as their ordinal number increases.
Let us begin, then, with examples like the following, on an organ in its princi-
pal or violin stops,t or upon an harmonium :
^
i
J.
u -i
I
=^3=
zsiz
^
^
H
The minims in these examples denote the prime tones of the notes struck, and
the crotchets the corresponding upper partial tones. If the octave C c in the first
example is tuned accurately, no beats will be heard. But if the upper note is
changed into B as in the second example, or e^ as in the third, we obtain the same
beats as we should from the two tones Bc,ot c d)^, where the interval is a Semitone.
The number of beats (i6^ in a second) is the same in each case, but their intensity
is naturally less in the foi-mer case, because they are somewhat smothered by the
strong deep tone C, and also because c, the second partial of 0, has generally less
force than its prime, j:
In examples 4 and 5 beats will be heard on keyed instruments tuned according
to the usual system of temperament. If the tempered intonation is exact there
will be one beat in a second,§ because the note a" on the instrument does not exactly
* [But as upper partial and combinational
tones are both simple, it is always simple tones
which beat together, and the laws of Chap.
VIII. therefore govern all beats. With a little
practice the bell-like sound of the beating par-
tials may be distinguished amid the confused
beating of harsh reed tones. It only remains
to determine when and how these extra beating
tones arise. — TrarislatorJ]
t [See p. 93, notes * and §. On English
organs the open diapason and keraulophon or
gamba might be used. — Translator.]
X [On tlie Ilarmonical, instead of varying
the Octave in C c by a Semitone np or down,
we can slightly flatten the upper note, by just
pressing it down enough to speak, when the
beats will arise. Or by using the d and d, we
can produce mistuned Octaves as D d^ or I)j d.
And for the Fifth in No. 4 and 5, we can use
d' a" or d! a\ or take this mistuned Fifth lower,
as da' or d a, the true Fifth being <i, a, which
may be contrasted with ii.—Translaior,]
§ [Suppose d'has 297, then equally tempered
a ought to have 445 vibs. The third partial of
d' has therefore 3 x 297 » 891 vib., and the
Octave of a has 2x445 = 890 vib., and these
two tones beat 891 -890 = once in a second. —
Translator.l
Digitized by VjOOQlC
CHAP.x. BEATS OF UPPER PARTIALS. i8i
tbgree with the note al\ which is the third partial tone of the note dl. On the otiier
hand the note a" on the instrument exactly coincides with a!\ the second partial
tone of the note al in the fifth example, so that on instruments exactly tuned in
any temperament the two examples 4 and 5 should give the same number of beats.
Since the first upper partial tone makes exactly twice as many vibrations in a
second as its prime, the c on the instrument in Ex. i , is identical with the first upper
partial of the prime tone 0, provided c makes twice as many vibrations in a second
as 0* The two notes 0, c, cannot be struck together without producing beats, unless
this exact relation is maintained. The least deviation from this exact relation is
betrayed by beats. In the fourth example the beats will not cease till we tune a"
on the instrument so as to coincide with the third partial tone of the note d, and
this can only happen when the pitch number of a!' is precisely three times that of
d'. In the fifth example we have to make the pitch number of a! half as great as
that of a", which is three times that of d' ; that is the pitch numbers of d' and a' ^
must be exactly as 2 : 3, or beats will ensue. Any deviation from this ratio will be
detected at once by beats.
Now we have already shewn that the pitch numbers of two tones which form
an Octave are in the ratio 1:2, and those of two which form a Fifth in that of 2 : 3.
These ratios were discovered long ago by merely following the judgment of the ear
respecting the most pleasant concord of two tpnes. The circumstances just stated
furnish the reason why these intervals when tuned according to these simple ratios
of numbers, and in no other case, will produce an undisturbed concord, whereas
very small deviations from this mathematical intonation will betray themselves by
that restless fluctuation of tone known as beats. The d' and a' of the last example,
if d' tuned as a perfect Fifth below a [that is as dx on the Harmonical], make 293^
and 440 vibrations in a second respectively, and their common upper partial a!'
makes 3 x 293^=2 x 440=880 vibrations in a second. In the tempered intonation
d! makes almost exactly 293! vibrations in a second, and hence its second upper ^
partial (or third partial) tone makes 881 vib. in the same time, and this extremely
small difference is betrayed to the ear by one beat in a second. That imperfect
Octaves and Fifths will produce beats, was a fact long knOwn to organ-builders,
who made use of it practically to obtain the required just or tempered intonation
with greater ease and certainty. Indeed, there is no more sensitive means of
proving the correctness of intervals.
Two musical tones, therefore, which stand in the relation of a^perfect Octave,
a perfect Twelfth, or a perfect Fifth, go on sounding uniformly without disturbance,
and are thus distinguished from the next adjacent intervals, imperfect Octaves and
Fifths, for which a part of the tone breaks up into distinct pulses, and consequently the
two tones do not continue to sound without interruption. For this reason the perfect
Octave, Twelfth, and Fifth will be called consonant intervals in contradistinction to
the next adjacent intervals, which are termed dissonant. Although these names
were given long ago, long before anything was known about upper partial tones and ^
their beats, they give a very correct notion of the essential character of the pheno-
menon which consists in the undisturbed or disturbed coexistence of sounds.
Since the phenomena just described form the essential basis for the construction
of normal musical intervals, it is advisable to establish them experimentally in every
possible form.
We have stated that the beats heard are the beats of those partial tones of both
compounds which nearly coincide. Now it is not always very easy on hearing a
Fifth or an Octave which is slightly out of tune, to recognise clearly with the un-
assisted ear which part of the whole sound is beating. On listening we are apt
to feel that the whole sound is alternately reinforced and weakened. Yet an ear
accustomed to distinguish upper partial tones, after directing its attention on the
common upper partials concerned, will easily hear the strong beats of these par-
ticular tones, and recognise the continued and undisturbed sound of the primes.
Strike the note d', attend to its upper partial a", and then strike a tempered Fifth
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i82 BEATS OF UPPER PARTIALS. partii.
a' ; the beats of a!' will be clearly heard. To an unpractised ear the resonators
already described will be of great assistance. Apply the resonator for a'', and the
above beats will be heard with great distinctness. If, on the other hand, a resonator,
tuned to one of the prime tones d! or a', be employed, the beats are heard much less
distinctly, because the continuous part of the tone is then reinforced.
This last remark must not be taken to mean that no other simple tones beat in
this combination except a^'. On the contrary, there are other higher and weaker
upper partials, and also combinational tones which beat, as we shall learn in the
next chapter, and tliese beats coexist with those already described. But the beats
of the lowest common upper partials are the most prominent, simply bcksause these
beats are the loudest and slowest of all. •
Secondly, a direct experimental proof is desirable that the numerical ratios here
deduced from the pitch numbers are really those which give no beats. This proof
f is most easily given by means of the double siren (fig. 56, p. 162). Bet the discs
in revolution and open the series of 8 holes on the lower and 16 on the upper, thus
obtaining two compound tones which form an Octave. They continue to sound
without beats as long as the upper box is stationary. But directly we begin to
revolve the upper box, thus slightly sharpening or flattening the tone of the upper
disc, beats are heard. As long as the box was stationary, the ratio of the pitch
numbers was exactly 1:2, because exactly 8 pulses of air escaped on one rotation
of the lower, and 16 on one rotation of the upper disc. By diminishing the speed
of rotation of the handle this ratio may be altered as slightly as we please, but how-
ever slowly we turn it, if it move at all, the beats are heard, which shews that the
interval is mistuned.
Similarly with the Fifth. Open the series of 12 holes above, and 18 below, and
a perfectly unbroken Fifth will be heard as long as the upper windbox is at rest.
The ratio of the vibrational numbers, fixed by the holes of the two series, is exactly
^ 2 to 3. On rotating the windchest, beats are heard. We have seen that each
revolution of the handle increases or diminishes the number of vibrations of the
tone due to the 12 holes by 4 (p. 164c). When we have the tone of 12 holes on the
lower discs also, we thus obtain 4 beats. But with the Fifth from 12 and 18 holes
each revolution of the handle gives 12 beats, because the pitch number of the
third partial tone increases on each revolution of the handle by 3x4=12, when
that of the prime tone increases by 4, and we are now concerned witii the beats
of this partial tone.
In these investigations the siren has the great advantage over all other musical
instruments, of having its intervals tuned according to their simple numerical rela-
tions with mechanical certainty by the method of constructing the instrument, and
we are consequently relieved from the extremely laborious and difficult measure-
ments of the pitch numbers which would have to precede the proof of our law on
any other musical instnmient. Yet the law had been already established by such
^ measurements, and the ratios were shewn to approximate more and more closely to
those of the simple numbers, as the degree of perfection increased, to which the
methods of measuring numbers of vibrations and tuning perfectly had been brought
Just as the coincidences of the two first upper partial tones led us to the natural
consonances of the Octave and Fifth, the coincidences of higher upper partials
would lead us to a further series of natural consonances. But it must be remarked
that in the same proportion that these higher upper partials become weaker, the
less perceptible become the beats by which the imperfect are distinguished from
the perfect intervals, and the error of tuning is shewn. Hence the delimitation of
those intervals which depend upon coincidences of the higher upper partials be-
comes continually more indistinct and indeterminate as the upper partials involved
are higher in order. In the following table the first horizontal line and first ver-
tical column contain the ordinal numbers of the coincident upper partial tones,
and at their intersection will be found the name of the con-esponding interval
between the prime tones, and the ratio of the vibrational numbers of the tones
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CHAP. X.
BEATS OF UPPER PAKTIALS.
183
composing it. This numerical ratio always results from the ordinal numbers of the
two coincident upper partial tones.
Ooincideibt
Partial Tones
I
a
3
4
5
6
2 Octaves
and Fifth
I :6
Twelfth
1:3
Octave
I : 2
Fifth
2:3
Minor
Third
5:6
5 ]
2 Octaves A
Major Third
1:5
Major
Tenth
2:5
Major
Sixth
3:5
Major
Third
4:5
4 {
Donble Octave
I :4
Octave
I ; 2
Fourth
3:4
3 {
Twelfth
I :3
Fifth
2:3
•
2 1
Octave
I : 2
The two lowest lines of this table contain the intervals already considered, the
Octave, Twelfth, and Fifth. In the third line from the bottom the 4th partial
gives the intervals of the Fourth and double Octave. The 5th partial determines
the major Third, either simple or increased by one or two Octaves, and the major
Sixth. The 6th partial introduces the minor Third in addition. Here I have
stopped, because the 7th partial tone is entirely eliminated, or at least much
weakened, on instruments such as the piano, where the quality of tone can be
reg^llated within certain limits.* Even the 6th partial is generally very weak, but
an endeavour is made to favour all the partials up to the 5th. We shall return
hereafter to the intervals characterised by the 7th partial, and to the minor Sixth,
which is determined by the 8th. The following is the order of the consonant '
intervals beginning with those distinctly characterised, and then proceeding to
those which have their limits somewhat blurred, so to speak, by the weaker beats
of the higher upper partial tones : —
1. Octave 1:2
2. Twelfth I
3. Fifth 2
4. Fourth 3
5. Major Sixth 3
6. Major Third 4
7. Minor Third 5
The following examples in musical notation shew the coincidences of the upper
partials. The primes are as before represented by minims, and the upper partials
by crotchets. The series of upper partials is continued up to the common tone IT
only.
T
Octave.
I : 2
Twelfth.
I ' 3
Fifth.
2 : 3
p— T— pr-1 p-
Fourth. Maj. Sixth. Maj. Third. Min. Third,
3:4 3:5 4:5 5:6
We have hitherto confined our attention to beats arising from intervals which
differ but slightly from those of perfect consonances. When the difference is
* [But see Mr. Hipkins' remarks and experiments, supra, p. 'J^CJ note.— 2><xiwto/or.]
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1 84
BEATS OF UPPER PARTIALS.
PABT II.
small the beats are slow, and hence easy both to observe and count. Of course
beats continue to occur when the deviation of the two coincident upper partials
increases. But as the beats then become more numerous the overwhelming mass
of sound of the louder primes conceals their real character more easily than the
quicker beats of dissonant primes themselves. These more rapid beats give a
rough effect to the whole mass of sound, but the ear does not readily recognise its
cause, unless the experiments have been conducted by gradually increasing the
imperfection of an harmonic interval, so as to make the beats gradually more and
more rapid, thus leading the observer to mark the intermediate steps between the
numerable rapid beats on the one hand, and the roughness of a dissonance oa
the other, and hence to convince liimself that the two phenomena differ only in
degree.
In the experiments with pairs of simple tones we saw that the distinctness and
^ roughness of their beats depended partly on the magnitude of the interval between
the beating tones, and partly upon the rapidity of the beats themselves, so that for
high tones this increasing rapidity injured the distinctness of even the beats arising
from small intervals, and obliterated them in sensation. At present, as we have
to deal with beats of upper partials, which, when their primes lie in the middle
region, principally belong to the higher parts of the scale, the rapidity of the beats
has a preponderating influence on the distinctness of their definition.
The law determining the number of beats in a second for a given imperfection
in a consonant interval, results immediately from the law above assigned for the
beats of simple tones. When two simple tones, making a small interval, generate
beats, the number of beats in a second is the difference of their vibrational numbers.
Let us suppose, by way of example, that a certain prime tone has the pitch number
300. The pitch numbers of the primes which make consonant intervals with it,
will be as follows : —
Hi
Prime, tone =300 j
Upper Octave =600
„ Fifth «450
„ Fourth =400
„ Major Sixth = 500
„ Major Third =375
„ Minor Third = 360
Lower Octave =150
„ Fifth =200
„ Fourth « 225
„ Major Sixth = 180
„ Major Third » 240
„ Minor Third * 250
Now assume that the prime tone has been put out of tune by one vibration in
a second, so that its pitch number becomes 301, then calculating the vibrational
immber of the coincident upper partial tones, and taking their difference, we find
the number of beats thus : —
Interval upwards
Beating Partial Tones
XanilH»r of '
Beats
Prime
Octave .
Fifth .
Fourth .
Major Sixth
Major Third
Minor Third
300 «
600 =
450 =
400 «
500 =
375 =
360 =
300
600
900
1200
1500
1500
1800
301 =
301 =
301 =
301
301
301
301
301
602
= 903
= 1204
= 1505
« 1505
= 1806
Interral downwariLi
Prime
Octave .
Fifth .
Fourth .
Major Sixth
Major Third
Minor Third
Beating Partial Tones
Number of
Boats
I X 300 =
300
I X 301 = 301
I
2 X 150 =
300
I X 301 = 301
I
3 X 200 =
600
2 X 301 « 602
2
4 X 225 =
900
3 X 301 = 903
3
5 X 180 =
900
3 X 301 = 903
3
5 X 240 =
1200
4 X 301 = 1204
4
6 y 250 ^
1500
5 ^ 301 1 50s
5
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CHAP. X.
BEATS OF UPPER PARTIALS.
185
Hence the number of beats which arise from putting one of the generating
tones out of tune to the amount of one vibration in a second, is always given by
the two numbers which define the interval. The smaller number gives the number
of beats which arise from increasing the pitch number of the upper tone by i.
The larger number gives the number of beats which arise from increasing the
pitch number of the lower tone by i. Hence if we take the major Sixth c a,
having the ratio 3:5, and sharpen a so as to make one additional vibration in a
second, we shall have 3 beats in a second ; but if we sharpen c so as to make one
more vibration in a second, we obtain 5 beats in a second, and so on.
Our calculation and the rule based on it shew that if the amount by which one
of the tones is put out of tune remains constant, the number of the beats increases
according as the interval is expressed in larger numbers. Hence for Sixths and
Thirds the pitKsh numbers of the tones must be much more nearly in the normal •
ratio, if we wish to avoid slow beats, than for Octaves and Unisons. On the other %
hand a sUght imperfection in the tuning of Thirds brings us much sooner to the
hmit where the beats become too rapid to be distinctly separable. If we change
the Unison c" c", by flattening one of the tones, into the Semitone b' c", on
sounding the notes together there results a clear dissonance with 33 beats, the
number which, as before observed, seems to give the maximum of harshness.
But to obtain 33 beats from fifth / c", it is only necessary to alter c" by a quarter
of a Tone. If it is changed by a Semitone, so that/ c" becomes/ b', there result
66 beats, and their clearness is already much injured. To obtain 33 beats the c'^
must not be changed in the Fiffch c" g'^ by more than one-sixth of a Tone, in the
Fourth c" /' by more than one-eighth, in the major Tliird c" e" and major Sixth
e' a" by more than one-tenth, and in the minor Third c" e"|> by more than one-
twelfth. Conversely, if in each of these intervals the pitch number of c" be
altered by 33, so that c" becomes b' or d']}, we obtain the following numbers of
beats: — .%
The Intcrra' of the
becomes
or
anil girea beats
Octave . . . . d'd"
Fifth . . , . d'g"
Fourth .... cV"
Major Third . . . d' d'
Minor Third . . . d' d'b
b'd"
b'g"
b'f"
bfd'
b' d'\>
d"\> d"
d")> g"
d"\>f"
d"\> d'
d"\) d'b
66
99
132
16S
198
Now since 99 beats in a second produce very weak effects even imder favourable
circumstances for simple tones, and 132 beats in a second seem to lie at the hmit
of audibility, we must not be surprised if such numbers of beats, produced by the
weaker upper partials, and smothered by the more powerful prime tones, no longer
produce any sensible effect, and in feict vanish so far as the ear is concerned. Now
this relation is of great importance in the practice of music, for in the table it will ^
be seen that the mistuned Fifth gives the interval h' g", which is much used as an
imperfect consonance under the name of mitior Sixth, In the same way we find
the major Third d'^f as a mistuned Fourth, and the Fourth b'e" as a mistuned
major Third, and so on. That, at least in this part of the scale, the major Third
does not produce the beats of a mistuned Fourth, or the Fourth those of a mis-
tuned major Third, is explained by the great number of beats. In point of fact
these intervals m this part of the scale give a perfectly uninterrupted sound, with-
out a trace of beats or harshness, when they are tuned perfectly.
This brings us to the investigation of those circumstances which affect the I
perfection of the consonance for the different intervals. A consonance has been •
characterised by the coincidence of two of tlie upper partial tones of the compounds •
forming the chord. When this is the case the two compound tones cannot gene-
rate any slow beats. But it is possible that some other two upper partial tones of
these two compounds may be so nearly of the same pitch that they can generate
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i86 DEGREE OF HARMONIOUSNESS OF CONSONANCES, pakt ir.
rapid beats. Cases of this kind occur in the last examples in musical notation
(p. 183^). Among the upper partials of the major Third FA occur f and e\ side
by sidjB ; and among those of the minor Third FA\} will be found a' and «'[>. In
each case there is the dissonance of a Semitone, and these must produce the same
beats as if they had been given directly as simple prime tones. Now although
such beats can produce no very prominent impression, partly on account of their
rapidity, partly on account of the weakness of the tones which generate them, and
partly because the primes and other partial tones are sounding on at the same time
unintermittently, yet they cannot but exert some effect on the harmoniousness of
the interval. In the last chapter we found that in certain qualities of tone, where
the higher upper partials are strongly developed, sensible dissonances may arifle
within a single compound tone (p. 1786). When two such musical tones are
sounded together, there will be not only the dissonances resulting from the higher
% upper partial tones in each individual compound, but also those which arise from
a partial tone of the one forming a dissonance with a partial tone of the other, and
in this way there must be a certain increase in roughness.
An easy method of finding those upper partials in each consonant interval
which form dissonances with each other, may be deduced from what has been already
stated concerning larger imperfections in tuning consonant intervals (p. 185c, d).
We thus found that the major Third might be considered as a mistuned Fourth,
and the Fourth again as a mistuned Third. On raising the pitch of a compound
tone by a Semitone, we raise the pitch of all its upper partial tones by the same
amount. Those upper partials which coincide for the interval of a Fourth, sepa-
rate by a Semitone when by altering the pitch of one generating tone we con-
vert the Fourth into a major Third, and similarly those which coincide for the
major Third differ by a Semitone for the Fourth, as will appear in the following
example : —
Major Third.
r
WmoT Third.
The 4th and 3rd partial in the Fourth of the first example coincide as/. But
if the Fourth B[} sinks, as in the second example, to the major Third A, its 3rd
partial/ sinks also to «', and forms a dissonance with the 4th partial/ of F, which
was unaltered. On the other hand the 5th and 4th tone of the two compounds,
which in the first example formed the dissonance a' 61?, now coincide as a'. In
' the same way the consonant unison a'a^ of the second example appears as the dis-
sonance a'a'\} in the third, and the dissonance c"c"^ in the second becomes the
consonant unison c"c" in the third.
Hence in each consonant interval those upper partials form a dissonancey tvhich
coincide in one of the adjacent consonant intervals,* and in this sense we can say,
that every consonance is disturbed by the proximity of the consonances next
adjoining it in the scale, and that the resulting disturbance is the greater, the
major Third, and f x^«=| a major Tono.
The adjacency of the consonant intervals is
best shewn in fig. 60, A (p. 193), where it
appears that the order may be taken as; i)
Unison, 2) minor Third, 3) major Third,
4) Fourth, 5) Fifth, 6) minor Sixth, 7) major
Sixth, 8) Octave. In the table on p. iSyft,
other intervals, not perfectly consonant, arc
intercalated among these. - Translator.]
♦ [That is, in intervals which differ from
the first by raising or depressing one of its tones
by a Semitone (either |$ or |J), as in the table
on p. 185c, or even a Tone (?). Thus for the
Fifth, 3 X If = 5 a minor Sixth ; and § x g = J a
Fourth' For the Fourth, ^x|^r=?"a major
Third; and ^ivj^-^ a Fifth! For the major
Tliird i/xj';^^* a Fourth; and ^'^5! = ? a
minor Third. For the minor Third 5 > 3k-^ 2 a
Digitized by VjOOQlC
CHAP. X. DEGEEE OF HARMONlOUSNESS OF CONSONANCES. 187
lower and londer the upper partials which by their coincidence characterise the
disturbing interval, or, in other words, the smaller the number which expresses the
ratio of the pitch numbers.
The following table gives a general view of this influence of the different con-
sonances on each other. The partials are given up to the 9th inclusive, and cor-
responding names assigned to the intervals arising from the coincidence of the
higher upper partial tones. The third column contains the ratios of their pitch
numbers, which at the same time furnish the number of the order of the coincident
partial tones. The fourth column gives the distance of the separate intervals from
each other, and the last a measure of the relative strength of the beats resulting
&om the mistuning of the corresponding interval, reckoned for the quality of tone
of the violin.* The degree to which any interval disturbs the adjacent intervals,
increases with this last number. f ^ v^ ^ 1 \ "? I
Interralfl
Notation
Ratio of the
Pitch Numbers
Belative
Distance
Oants In the
Intexrals
Difference
of Cents
Intensity
of Influence
Unison
c
I : I
8T9
0
204
lOO'O
Second
D
8:9
63": 64
204
27
14
Snpersecond
D +
7:8
48 : 49
231
36
1-8
Subminor Third
Eb-
6:7
35:36
267
49
24
Minor Third
Eb
5:6
24:25
316
70
3*3
Major Third
E
4:5
35:36
386
49
50
Sapermajor Third
E +
7:9
27T28
435
63
1-6
Fourth
F
3:4
20 : 21
498
85
, 8-3
Sabminor Fifth .
Gb-
5:7
14: IS
583
119
2-8
Fifth ....
G
2:3
IS : 16
702
112
167
Minor Sixth
Ab
5:8
24:25
814
70
2-5
1
Major Sixth
A
3:5
20 : 21
884
85
67
Sabminor Seventh
Bb-
4:7
35:36
969
49
3-6
Minor Seventh .
Bb
5:9
9 : 10
1018
182
2-2
Octave
c
I : 2
—
1200
—
500
The most perfect chord is the Umson, for which both compound tones have the
same pitch. All its partial tones coincide, and hence no dissonance can occur
except such as is contained in each compound separately (p. 1786). II
It is much the same with the Octave. All the partial tones of the higlier note
of this interval coincide with the evenly numbered partials of the deeper, and re-
inforce them, so that in this case also there can be no dissonance between two upper
partial tones, except such as already exists, in a weaker form, among those of the
/deeper note. A note accompanied by its Octave consequently becomes brighter
Jn quality, because the higher upper partial tones on which brightness of quality
depends, are partly reinforced by the additional Octave. But a similar effect would
also be produced by simply increasing the intensity of the lower note without add-
ing the Octave ; the only difference would be, that in the latter case the reinforce-
ment of the different partial tones would be somewhat differently distributed.
The same holds for the Twelfth and double Octave^ and generally for all those
• See Appendix XV.
f [Two cohimns have bern added, shewing
the c^nts in the intervals named, and in the
intervals between adjacent notes.
App. XX. sect. D. — TranslatoT.]
See also
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i88 DEGREE OF HABMONIOUSNESS OF CONSONANCES, pabt n.
cases in which the prime tone of the higher note coincides with one of the partial
tones of the lower note, although as the interval between the two notes increases
the difference between consonance amd dissonance tends towards obliteration.
The cases hitherto considered, where the prime of one compound tone coincides
with one of the partials of the other, may be termed absolute consonances. The
second compound tone introduces no new element, but merely reinforces a part of
the other.
Unison and Octave disturb the next adjacent intervals considerably, in the sense
assigned to this expression on p. i86^, so that the minor Second C jD\}, and the
major Seventh C B, which differ from the Unison and Octave by a Semitone
respectively, are the harshest dissonances in our scale. Even the major Second
C D, and the minor Seventh C ^, which are a whole Tone apart from the dis-
turbing intervals, must be reckoned as dissonances, although, owing to the greater
% interval of the dissonant partial tones, they are much milder than the others. In
the higher regions of the scale their roughness is materially lessened by the
increased rapidity of the beats. Since the dissonance of the minor Seventh is due
to the second partial tone, which in most musical qualities of tone is much weaker
than the prime, it is still milder than that of the major Second, and hence Ues on
the very boundary between dissonance and consonance.
To find additional good consonances we must consequently go .to the middle of
the Octave, and the first we meet is the Fifth, Immediately next to it within the
interval of a Semitone there are only the intervals 5 : 7 and 5 : 8 in our table, and
these cannot much disturb it, because in all the better kinds of musical tones the
7th and 8th partials are either very weak or entirely absent. The next intervals
with stronger upper partials are the Fourth 3 : 4 and the major Sixth 3:5. But
here the interval is a whole Tone, and if the tones i and 2 of the interval of the
Octave could produce very Uttle disturbing effect in the minor Seventh, the dis-
% turbance by the tones 2 and 3, or by the vicinity of the Fifth to the Fourth and
major Sixth must be insignificant, and the reaction of these two intervals with tlie
tones 3 and 4 or 3 and 5 on the Fifth must be entirely neglected. Hence tlie Fifth
remains a perfect consonance, in which there is no sensible disturbance of closely
adjacent upper partial tones. It is only in harsh qualities of tone (harmonium,
double-bass, violoncello, reed organ pipes) with high upper partial tones, and deep
primes, when the number of beats is small, that we remark that the Fifth is some-
what rougher than the Octave.* Hence the Fifth has been acknowledged as a
consonance from the earliest times and by all musicians. On the other hand the
intervals next adjacent to the Fifth are those which produce the harshest disso-
nances after those next adjacent to the Octave. Of the dissonant intervals next
* [The above discuBBion may jbe rendered numbers of the two prime tones which form
easier by the following considerations, which the Fifth to be 2 and 3, and find those of their
the student should illustrate or hear illus- upper pai'tials thus, assuming C 6r to be the
m trated on the Harmonical. Take the pitch two notes.
Nos. of the Partials
Partials of lower note
Lower note .
Fifth or 2 : 3, upper note
Partials of upper note
Nos. of the Partials
12345678
2 4 6 8 10 12 14 16
C c g d 6' (f 6'b
Q g d! ^ V
3 6 9 12 IS
12345
We see that the principal beating tones The next beating partial tones arc 8 and 9, or
arc 14 and 15, or 6'b h\ the 7th partial of the c d\ the 4th partial of the lower and 3rd of
lower and 5th of the upper; and 15 and 16, the upper note, and these being a whole Tone
or b' c'\ the 5th of the upper and 8th of the apart, the beats are not of importance even
lower note, and that these beats are unimpor- when strong, and with weak upper partials
taut because the 7th and 8th partials are are insignificant. Similarly for the beats of
generally weak ; but if they are strong these 9 and 10, or d' e'y the 3rd partial of the upper
beats being those of a Semitone and of nearly and 5th of the lower note. On referring to the
a Semitone, arc very harsh. On the Har- text it will be seen that the same intervals
monical it will be found that the 12th G g\^ are there compared and in the same order as
faultless, but the 5th C G is decidedly harsh. here. — Translator.]
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CHAP. X. DEGREE OP HARMONIOUSNESS OF CONSONANCES. 189
the Fifth, those in which the Fifth is flattened, that is which lie between the Fifth
and Fourth, and are disturbed firstly by the tones 2 and 3, and secondly by the
tones 3 and 4, are more decidedly dissonant than those in Which the Fifth is
sharpened and which lie between the Fifth and major Sixth, because for the latter
the second disturbance arises from the tone 3 and the weaker tone 5.* The
intervals between the Fifth and Fourth are consequently always considered dis-
sonant m musical practice. But between the Fifth and major Sixth lie^ the
interval of the minor Sixths which is treated as an imperfect consonance, and owes
this preference mainly to its being the inversion of the major Third. On keyed
instruments, as the piano, the same keys will strike notes which at one time
represent the consonance G A\}, and at another the dissonance C Gjj^.f
Next to the Fifth follow the consonances of the Fourth 3 : 4 and the major
Sixth, the chief disturbance of which arises usually from the Fifth. The Fourth
is somewhat further from the Fifth (the interval is 8 : 9) than the major Sixth is ^
(the interval is 9 : 10), and hence the major Sixth is a less perfect consonance
than the Fourth. But close by the Fourth lies the major Third with the 4th and
5th partials coincident, and hence when these partials are strongly developed, the
Fourth may lose its advantage over the major Sixth. It is also well known that
the old theoretical musicians long disputed as to whether the Fourth should be
considered consonant or dissonant. The precedence given to the Fourth over the
major Sixth and major Third, is rather due to its being the inversion of the Fifth
than to its own inherent harmoniousness. The Fourth, the major Sixth and
minor Sixth, are rendered less pleasant by being widened by an Octave (thus
becoming the Eleventh, and major and minor Thirteenth), because they then lie
near the Twelfth, and consequently the disturbance by the characteristic tones of
the Twelfth i and 3, is greater, and hence also the adjacent intervals 2 : 5 for the
Eleventh, and 2 : 7 for the Thirteenth, are more disturbing than are the 4 : 5 for
the Fourth and the 4 : 7 for the Sixth in the lower Octave.t %
* [Taking the scheme in the last note, and supposing O to be altered first to Ob and then
to ^b, we may write the several schemes thus :
No. of Partials of lower note ..12345678
Lower Note C c g d e ^ h'\> t
Fifth wi\z\ [ G g d' g' V
Flattened Worms of the upper note sGb gb d'b g'b b'b
Sharpened ) I Ab ab db a'b c"
No. of Partials of upper note . • i 23 4 5
If the Ob were made sufficiently flat, we 3 of the upper and 5 of the lower note, instead
should have its 5th partial b'b coinciding with of from d'b c' or tones 3 of the upper and 4 of
the 7th partial of C, which, however, is never the lower note, and as the tone 5 is weaker
felt as a consonance, and the interval then the disturbance on the whole is weaker. This
becomes 5 : 7. This, however, never occurs is the case in musical practice. — Translator,]
in musical practice, where the b'b from Ob is f [This is the result of equal temperament,
always sharper than that from C, but this in which i4'b, which is 814 cents above C, is
dissonance is not felt, the gb g or tones 2 of the confounded with G^K , which is only 772 cents H
upper and 3 of the lower note, and & d'b or above C, a difference of 42 cents. The in-
tones 3 of the upper and 4 of the lower note, terval c' a^'b can be played on the Harmonical
producing the chief disturbance. If Ab is and at that pitch will be found good. The
taken sufficiently sharp for its 5th partial cf' interval a* b e', which is the same as that of
to coincide with the eighth of C we have the dg'tU 1 but a major Third lower, will be found
interval 5 : 8 or minor Sixth. Here again we very h&Tsh.—Translator.]
have the disturbance from ab g the tones 2 of % [Treating these intervals as in the pre-
the upper and 3 of the lower note, but the ceding notes we have :
second disturbance is now from db 0^ or tones
No. of Partials .
Lower note «
Fourth or 3 : 4
No. of Partials
No. of Partials ,
Lower note
Elevmth or 3 : 8
No. of Partials
12345678
C c g c' €f g^ b'b c"
F f d f a' d'
i_ 2 3 4_ 5 6_
2345678
c g d d g' 6'b c"
/ f c"
I 2 3
Digitized by VjOOQlC
I90 DEGBEE OF HARMONIOUSNESS OF CONSONANCES, part ii.
Next in the order of the consonances come the major and miwyr Third. The
latter is very imperfectly delimited on instmments which, like the pianoforte, do
not strongly develop the 6th partial of t^e compound tone, because it can then be
imperfectly tuned without producing sensible beats.* The minor Third is Bensibly
exposed to disturbance from the Unison, and the major Third from the Fourth ;
and both mutually disturb each other, the minor Third coming off worse than the
major.f For the harmoniousness of either interval it is necessary that the disturb-
ing beats should be very rapid. Hence in the upper part of the scale these intervals
are pure and good, but in the lower part they are very rough. All antiquity, there-
fore, refused to accept Thirds as consonances. It was not till the time of Franco <^
Cologne (at the end of the twelfth century) that they were admitted as imperfect
consonances, Th^ reason of this may probably be that musical theory was developed
among classical nations and in medieval times principally in respect to men's
voices, and in the lower part of this scale Thirds are £bu: from good. With this we
must connect the fact that the proper intonation of major Thirds was not dis-
covered in early times, and that the Pythagorean Third, with its ratio of 64 : 81,
was looked upon as the normal form till towards the close of the middle ages^
No. of Partiah .
Lower note
Major Sixth or 3 : 5
No. of Partials .
. I 2
. C c
A
I
3
9
a
2
4
c'
5 6 7
e! g[ 6'b
e a'
3 4
8
5
No. of Partials.
Lower note
Major Thi/rteenth or 3 :
No. of Partials.
. I 2
. C c
10
3 4
9 d
a
I
5678
</ ^ h'\> c"
a'
2
9
10
e'
3
No. of Partials.
Lower note
Minor Sixth or 5 : 8
No. of Partials .
I 2
. C c
A\>
I
3
9
ab
2
4
5 6 7
«' 9" 6'b
e'b a'b
3 4
8
c"
c"
5
No. of Partials . . . i 2 3 4 5 6 7 8 9 10 13 16
Lower note . . , C c g d d il V\> d' dT tf' f c"
Minor Thirteenth or s:i^ a\> a'b «"b a'b c'"
No. of Partials ... i 2 345
These diagrams will make the text imme-
diately intelligible, but as the notes refer to
the ordinary notation the fact that / to ^ in
the Fourth is a wider interval than g to a in
the major Sixth is not expressed. It is, how-
ever, readily seen how mach worse is the
minor Sixth with g to ab, and that in all
these cases the disturbance arises from the
2nd and 3rd partials which coincide for the
Fifth. It is also seen how the disturbance is
increased in the Eleventh and Thirteenths
because one of the disturbing tones then
No. of Partials . . . i
Lower note . , . C
Major Third or 4 : 5 . E
No. of Partials . . • i
becomes a prime, and hence Bounds much londcr.
See also the table of partials on p. 197c, d. -
Translator.]
* [As the usual tempered tuning of the
piano makes the minor. Third greatly too flat,
the circumstance mentioned in the text becomes
a great advantage on that instrument. On
the tempered harmonium even e' ^, e" g*' are
very harsh, as compared with the same inter-
vals on the Harmonical. — Translator.']
t [This will be made clearer by the follow-
ing diagrams :
2345678
c g d e' g' b'b c"
eh e' g% h' d"
23 456
No. of Partials .
Lower note
Minor Third or 5
No. of Partials .
12345678
C c g d e' gT h'b d'
Eb eb bb db g' b'b d"b
12345 67
The 6th partial of this Eb is not the same as
the 7th partial of C, although the notation
makes it appear so, but it is sharper in the
ratio of 36 : 35, and hence if the partials were
not so high would be very disturbing. It is
seen that g' ^S are the 6th and 5th partials
for the major Third, and db d the 4th and
5th for the minor Third; the interval being
the same (24 : 25), the disturbance is worse in
the latter case, because the partials are looser
and hence louder. — Translator.]
X [The ordinary major Third on the tem-
pered harmonium is very little flatter than
this, but still it is much less harsh. The
Harmonical does not contain a Pythagorean
major Third, 64 : 81, the nearest approach
being !6b : rf, -63 : 80, but it contains a Py-
thagorean minor Third df, which may be con-
Digitized by V^jOOQIC
CHAP.x. DEGEEE OF HARMONIOUSNESS OP CONSONANCES. 191
The important influence exercised on the harmoniousness of tlie consonances,
especially the less perfect ones, by the rapidity of the weak beats of the dissonajit
upper paurtials, has already been indicated. If we place all the intervals alcove the
same bass note, the number of their beats in a second varies much, and is much
greater for the imperfect than for the perfect consonances. But we can give all the
intervals hitherto considered such a position in the scale that the number of their
beats in a second should be the same. Since we have found that 33 beats in a
second produce about the maximum amount of roughness, I have so chosen the
position of the intervals in the following examples in musical notation, as to give
trasted with the just minor Third dj. The
following arrangement of the consonant in-
tervals will show the beating partials in each
case, and the exact ratios of their intervals.
The number of the partial is subscribed in
each case. The beating interval is inoffensive
for 5 : 6, but its action becomes sensible for
7 : 8, 8 : 9, and 9 : 10, and for 14 : 15, 15 : 16,
24 : 25 the effect is decidedly bad if the tones
are strong enough and the beats slow enough ;
the strength depends on the lowness of the
ordinal numbers of the beating partials, and the
rapidity depends on their position in the scale.
This must be taken into consideration, as in
fig. 60, p. 193. A prefixed *, f , ^, || draws at-
tention to the beatiog partials. The order of
the intervals is that of their relative harmoni- %
ousness as assigned in my paper *0n the
Physical Constitution and Belations of Musical
Chords,' in the Proceedings of the Boyal
Society y June 16, 1864, vol. xiii, p. 392, Table
Vm., here re-arranged.
Cc
Octave or I : 2, cents 1200
CO
Fifth or 2 : 3, cents 702
Ce
Major Tenth or 2 : 5, cents 1386
{■
4 s
4>
6 7
6,
8 9
8,
10
10.
{':
Cg
Twelfth or I : 3, cents 1902
CF
Fourth or 3 : 4, cents 498
CA
Major Sixth or 3 : 5, cents 884
CE J
Major Third or 4 : 5, cents 386 \
CE\, r 5,
Minor Third or 5 : 6, cents 316 \
cAb ~r
Minor Sixth or 5 : 8, cents 814 \ 8,
r 2. 4, 6. % 'lo.
I 3, 6, *9,
12,
12.
ti4, ti6,
t»Si
18.
t20..
f 2, 4, 6, 84 10,
•X Si 10,
12.
•14, *j6,
•is.
18,
9
9>
20,,
20,
/ > 2 3 4 5
I 3,
6
6.
7 8
10
6. •9,
12,
ti5, i8« 21,
ti64 2O4
24, t2;, t3o,,
24a J28,
/ 3i 6, ♦9, 124
I 5i *io.
«5s
i5s
4, 8, 12, ♦id, 20,
5i_ >o,_*i5, 20,_
10, 15, 20, ♦25s
6| 12, 18, *244
18. t2lT J24, 27, 30,,
_t204 t25» 30. _
t24. 28, 32, t36, 40,0
USs 30, t35, 40.
30. t357 Uo* 45» H50i»
30* t36fl t42; 1148,
/ 5i 10*
♦15. 20, t25, t3o. 35,
*i6, t24« t32,
40.
40,
Il45i. llSo„
Ceb f Si 102 i5t 20, ♦25,30. t3S7 40, t45
Minor Tenth or 5 : 1 2, cents 1 5 16 \ 12,
*24,
t36.
tSOia 55ii 6o,j
US, 60.
Cf
3i
6, ♦9.
♦8,
Eleventh or 3 : 8, cents 1698
Ca / 3i 6, ♦9, 12, 15,
Ma.Thrtnth.0r3 : lo.cents 2084 \ ♦iOi
Cab / 5i lOt *i5« 204 25»
Mi. Thrtnth. or 5 : i6,cents 2014 (^ ♦id,
ti5s tiSe
ti6.
121,
|24.
i243
t27. 30 •
ti8. Uh
_ t20^|
t3o/ t3S7
t32.
24s
40.
t27. 30 •
t3o. ,
USb tSO;.
t48.
See note p. 195 for the intervals depending on 7.
The last four of the above intervals are so
rough that they are seldom reckoned as con-
sonances. The order was determined merely
by frequently sounding the intervals in just
intonation on justly intoned reed instruments,
and relates solely to the effect on my own ear.
The greater richness of the major Tenth over
the Twelfth made me prefer the former. The
effect is very much like that of a compound
tone, in which the prime is inaudible ; even
the tones i and 3 are supplied partly by com-
binational tones. Hence when a man's voice
accompanies a woman's at a Third below (that
is really a tenth) the cITect is moro agreeable
than when another woman sings the real
Third below, as long as the Thirds are major ;
the contrary is the case when the Thirds are
minor. In ordinary rules for harmony no dis«
tinction is made between Tenths and Thirds,
Fourths and Elevenths, <&c. The above table
shews that the differences are of extreme im-
portance. The dissonant character attributed
to the Fourth is apparently due to the Eleventh.
As will be seen hereafter, the minor Tenth,
the Eleventh, and both Thirteenths ought to
be avoided or else treated as dissonances. —
Translator.]
Digitized by. VjOOQIC
192 DEGREE OP HARMONIOUSNESS OF CONSONANCES, part ii.
that number in every case. The intonation is supposed to be that of the scale of
C major with just intervals, but £{> represents the subminor Seventh of c (4 : 7).*
m.
'^m
=(=:
:[::
8i:9i
m=
15, : 16, 8, : 15,
=t=-
Wz
IP
1
6. : 7, S, : 7,
Eg" I ^
I
42:7i
Ss : Sa 57 : 6^
44=53
33 : 53
33 -4.2 2, : 33
The prime tones of the notes in this example are all partials of (7^, which
makes 33 vibrations in a second, and hence their own pitch numbers and those
% of their upper partials are multiples of 33 ; consequently the difference of theso
pitch numbers, which gives the number of beats, must always be 33, 66, or some
higher multiple of 33.
In the low positions here assigned the beats arising from the dissonant xtpper
partials are as effective as their intensity will allow, and in tliis case the Sixths,
Thirds, and even the Fourth are considerably rough. But the major Sixth and
major Third shew their superiority over the minor Third and minor Sixth, by
descending lower down in the scale, and yet sounding somewhat milder than the
others. It is also a well-known practical rule among musicians to avoid these
close mtervals in low positions, when soft chords are required, though there was
no justification for this rule in any previous theory of chords.
My theory of hearing by means of the sympathetic vibration of elastic
appendages to the nerves, would allow of calculating the intensity of the beats
of the different intervals, when the intensity of the upper partials in the corre-
al sponding quality of tone belonging to the instrument used, is known, and the
intervals are so chosen that the number of beats in a second is the same. But
such a calculation would be very different for different qualities of tone, and holds
only for such a particular case as may be assumed.
For intervals constructed on the same lower note a new f&cioT comes into play,
namely, the number of beats which occur in a second ; and the influence of this
feustor on the roughness of the sensation cannot be calculated directly by any fixed
law. But to obtain a general graphical representation of the complicated relations
which co-operate to produce the effect, I have made such a calculation, k-nowing
that diagrams teach more at a glance than the most complicated descriptions, and
have hence constructed figs. 60, A and B (p. 193). In order to construct them
I have been forced to assume a somewhat arbitrary law for the dependence
of roughness upon the number of beats. I chose for this purpose the simplest
mathematical formula which would shew that the roughness vanishes when there
f are no beats, increases to a maximum for 33 beats, and then diminishes as the
number of beats increases. Next I have selected the quality of tone on the violin
in order to calculate the intensity and roughness of the beats due to the upper
partials taken two and two together, and from the final results I have constructed
figs. 60, A and B, opposite. The base lines &&\ d'd" denote those parts of tlie
musical scale which lie between the notes thus named, but the pitch is taken to
increase continuously [as when the finger slides down the violin string], and not by
separate steps [as when the finger stops off definite lengths of the violin string].
It is further assumed that the notes or compound tones belonging to any indi\ddual
part of the scale, are sounded together with the note c\ which forms the constant
lower note of all the intervals. Fig. 60 A, therefore, shews the roughness of all
intervals which are less thaji an Octave, and fig. 60 B of those which are greater
*■ [The ordinal nambers of the partials
which beat 33 times in a second, are here sub-
scribed. Thus 4« : 5, means that the ratio of
the primes is 4:5, and that the beating par-
tials are the 4th of 4, and the 3rd of 5, having
the ratio 16 : 15.- Traii&iaior.l
Digitized by V^OOQIC
CHAP. X. DEGREE OF HARMONIOUSNESS OF CONSONANCES. 193
than one Octave, and less tban two. Above the base line tbere are prominences
marked with the ordinal numbers of the partials. Tbe height of these prominences
at every point of their width is made proportional to the roughness produced by
the two partial tones denoted by the numbers, when a note of corresponding pitch
is sounded at the same time with the note c'. The roughnesses produced by the
different pairs of upper partials are erected one over the other.* It will be seen
that the various roughnesses arising from the different intervals encroach on each
other's regions, and that only a few narrow valleys remain, corresponding to the
position of the best consonances, in which the roughness of the chord is com-
paratively small. The deepest valleys in the first Octave c' c" belong to the
Octave c\ and the Fifth g' ; then comes the Fourth/, the major Sixth a\ and the
major Third «', in the order already found for these intervals. The minor Third
e't>, and the minor Sixth a'^, have * cols ' rather than valleys, the bottoms of their
Fig. 60 A.
IT
Fig. 60 B.
depressions lie so high, corresponding to the greater roughness of these intervals.
They are almost the same as for the intervals involving 7, as 4 : 7, 5 : 7, 6 : y.f ^
In the second Octave as a general rule all those intervals of the first Octave are
improved, in which the smaller of the two numbers expressing the ratio was even ;
thus the Twelfth i : 3 or c'g'\ major Tenth 2 : 5 or c'e", subminor Fourteenth 2 : 7
or c'&"|>— , and subminor Tenth 3 : 7 or c'e"l>— J are smoother than the Fifth 2 : 3
or c'g\ major Third 4 : 5 or c'e', subminor Seventh 4 : 7 or c'h'\)—y and subminor
Third 6: 'jOTc'e'\}—. The other intervals are relatively deteriorated. The Eleventh
or c'f^ or increased Fourth is distinctly worse than the major Tenth or &e" ; the
major Thirteenth or c'a", or increased major Sixth, is similarly worse than the
subminor Fourteenth c'6"|7— . The minor Third or c'e'\}j when increased to a
minor Tenth or c'e^y,t ancl the minor Sixth or c'a'b, when increased to a minor
♦ [The method in which these diagrams
were calculated is shewn in the latter part of
Appendix XV. — Translator.]
t [The interval 4 : 7 is over 6' b — , meaning
'6'b ; the interval 5 : 7 is the • col ' between /
and ^, and the interval 6 : 7 is the next * col '
to the left of e'b.— Translator,]
X [By carrying a line down from e'b in
Digitized by V^OOQIC
194 DEGREE OF HABMONIOUSNESS OF CONSONANCES, part ii.
Thirteenth or c'a"\}, fare still worse, on account of the increased disturbance of
the adjacent intervals. The conclusions here drawn from calculation are easily
confirmed by experiments on justly intoned instruments.* That they are also
attended to in the practice of musical composition, notwithstanding the theoretical
assumption that the nature of a chord is not changed by altering the pitch of any
one of its constituents by whole octaves, we shall see further on, when considering
chords and their inversions.
It has already been mentioned that peculiarities of individual quaUties of tone
may have considerable effect in altering the order of the relative harmoniousness
of the intervals. The quality of tone in the musical instruments now in use has
been of course selected and altered with a view to its employment in harmonic com-
binations. The preceding investigation of the qualities of tone in our principal
musical instruments has shewn that in what are considered good qualities of tone
^ the Octave and Twelfth of the prime, that is the 2nd and 3rd partial, are powerful,
the 4th and 5th partial have only moderate strength, and the higher partiaJs
rapidly diminish in force. Assuming such a quality of tone, the results of this
chapter may be summed up as follows.
I When two musical tones are sounded at the same time, their united sound is
generally disturbed by the beats of the upper partials, so that a greater or less part
of the whole mass of sound is broken up into pulses of tone, and the joint effect is
rough. This relation is called Dissonance.
But there are certain determinate ratios between pitch numbers, for which this
rule suffers an exception, and either no beats at all are formed, or at least only
such as have so little intensity that they produce no unpleasant disturbance of the
united sound. These exceptional cases are called Consonances.
'I. The most perfect consonances are those that have been here called absolute,
in which the prime tone of one of the combined notes coincides with some partial
4 tone of the other. To this group belong the OctavCy Twelfth, and double Octave.
2 . Next follow the Fifth and the i?bi*r^/i,which may be called perfect consonances^
because they may be used in all parts of the scale without any important disturb-
ance of harmoniousness. The Fourth is the less perfect consonance and approaches
those of the next group. It owes its superiority in musical practice simply to its
being the defect of a Fifth from an Octave, a circumstance to which we shall return
in a later chapter.
3. The next group consists of the major Sixth and the major Third, which
may be called medial consonances. The old writers on harmony ponsidered them
as imperfect consonances. In lower parts of the scale the disturbance of the
harmoniousness is very sensible, but in the higher positions it disappears, because
the beats are too rapid to be sensible. But each, in good musical qualities of tone,
is independently characterised, by the fact that any little defect in its intonation
produces sensible beats of the upper partials, and consequently each interval is
II sharply separated from all adjacent intervals.
4. The imperfect consonances, consisting of the minor Third and minor Sixths
are not in general independently characterised, because in good musical qualities of
tone the partials on which their definition depends are often not found for the
minor Third, and are generally absent for the minor Sixth, so that small imper-
fections in the intonation of these intervals do not necessarily produce beats.f
fig. 60 A, it will be seen that e"b belongs to the cVb -and cVb — , the student should take the
little depression to the right of the fraction J same intervals a Fourth lower, as g '6'b and
between c"b- and e". The slight depression g ^bb. All the other notes are on the instm-
for a"b is just under the fraction ^^ to the left ment in all the oct&ves.— Translator.]
of a". The depression for «'b-is just to the f [It must be recollected that in the minor
left of that for e'b,— Translator.] Sixth the 2nd and 3rd partials form the Seml-
* [The student is strongly recommended tone 15 : 16, and the 3rd and 5th form the
to verify all these consonances on the Har- Semitone 24 : 2(; (see note p. 191c), and that
monical, where &b — , that is '&b, is placed on the resulting beats, which in good qualities of
the g\) digital. The Harmonical does not con- tone are never absent, will always be more
tain fib-, that is, 'eb, and hence, in place of powerful than those which arise frcm small
Digitized by V^jOOQlC
CHAP. X. DEGBEE OF HAKMONIOUSNESS OF CONSONANCES. 195
They are all less suited for use in lower parts of the scale than the others, and
they owe their precedence as consonances over many other intervals which lie on
the boundaries of consonance and dissonance, essentially to their being indispens-
able in the formation of chords, because they are defects of the major Sixth and
major Third from the Octave or Fifth. The subminor Seventh 4 : 7 or c'b]}— is
very often more harmonious than the minor Sixtl 5 : 8 or c'a'^, in fact it is
always so when the third partial tone of the note is strong as compared with the
second, because then the Fifth has a more powerfully disturbing effect on the
intervals distant from it by a Semitone, than the Octave on the subminor Seventh,
which is rather more than a whole Tone removed from it.* But this subminor
Seventh when combined with other consonances in chords produces intervals which
are all worse than itself, as 6 : 7, 5 : 7, 7 : 8, &c., and it is consequently not used
as a consonance in modem music.f
5. By increasing the interval by an Octave, the Fifth c'g' and major Third ^
c'e' are improved on becoming the Twelfth c'g'^ and major Tenth cV. But the
errors of intonation, even in qualities of tone * [Reverting to the diagrams before given
in which an 8th partial is well developed. — (p. 19 ic, note), we may compare the effect of
Trofulaton] these intervals thns :
CAb 5i io» *i58 204 t255 t30« 357 40t Il45. Il50i«
Minor Sixth or 5 : 8, cents 814 8, *i6, f24j 1(324 40, |i48«
CjBb--C'£b 4i *8, t"« 164 $20, ^24. 28, 32. II36. 40,,
Subminor Seventh or 4 : 7, cents 969 "^7, ti4j ^21, 284 {I35,
Hence for the minor Sixth the chief beats interval which replaces the 15 : 16 in the
arise from the interval 15 : 16, or the 3rd minor Sixth, being due to those upper partials
partial of the lower and 2nd of the upper which would have coincided for the Fifth,
note, that is, from those tones which would Both CAb and C^B\> can be played on the
coincide for the Fifth, which is what is meant Harmonical, and the effect in the different
in the text by saying that the interval is dis- Octaves should be compared. — Translator,]
turbed by the Fifth. But in the subminor f [In fig. 60 A (p. 193&), the bottom of the ^
Seventh the chief disturbance is from 7 : 8, valley of 4 : 7 above 2/b — , is just a little lower
or the prime of the upper and 2nd partial of than that of 5:7, between / and ^, and than
the lower note, which would coincide for the that of 6:7, which, with that of 7 : 8, lies
Octave. The beats from the interval 12 : 14 between c^ and 0'b. If we take the diagrams
or 6 : 7 are hardly perceptible, but this is the for these intervals we have :
C^b-orG'Bb 6| 12, 18, *244 t30. tS^t 42, §4«e II 54. 6o„
Subminor Third or 6 : 7, cents 267 7, 14, ♦21, f^S* $35, 42^ §49, IJ56,
CGb- or^-Bb Si 10, ♦15. t204 2$^ 30. 357 40^ 45^ 5oi«
Subminor Fifth or 5 : 7, cents 583 7, ♦14, f^ii 28, 35, 42, 49,
CD+oT'BbC 7i 14. 21, 284 *35, t42. 49* S^t 63, 70,0
Snpersecond or 7 : 8, cents 231 8, 16^ 24, ^324 f40, 48^ 56, 64,
The second forms in these examples, O ^Bb, almost the only ones noted in fig. 60 A. In the
E 'Bb, ^BbOj can be played on the Harmoni- Supersecond the continual repetition of the in-
eal. We see, then, that 6 : 7 is disturbed by terval 7 : 8 produces the chief effect, but 32 : 35
a continual repetition of this same intervid from the 4th and 5th partials, and 40 : 42
among its lower partials, and also by the =20 : 21, from the 5th and 6th partials, also f-
intervals 21 : 24 » 7 : 8 from the 3rd and 4th produce much effect, as shewn in the fig. 60 A.
partials, 28 : 30^ 14 : 15 from the 4th and 5th The interval 7 : 9, which is much pleasanter,.
partials, and 35 : 36 from the 5th and 6th has not been considered by Prof. Helmholtz, but
partials. On lookmg at the diagram, fig. 60 is available in all Octaves on the Harmoni-
A (p. 193c), it will be seen that of these four cal. Mr. Poole distinguished 5 : 6, 6 : 7, 7 : 9
the first is chief, but the others are as the minor, minim, and maxim Third, here
active. For the subminor Fifth 5 : 7 the great called minor, subminor, and super-major Third,
disturbance is from 14 : 15, or the 2nd and 3rd There is also the wide (or super) minor Third
partial, but there is also an active one from 14 : 17. I add the anaJysis of the two last,
20 : 21, or the 4th and 3rd partial, and these are both of which are on the Harmonical.
'Bbd 7, 14, 2I3 *284 t35» 424 497 tS^n 63,.
Bnper-major Third or 7 ^ 9» cents 435 9, 18, *27, t36| 45* t54« 63,
'b"b »'d'"b lii 282 42^ ^56; tTol
Super-minor Third or 14 : 17, cents 336 17, 34^ ♦si, 1^84
In the last there are a quantity of beating result is really superior to the Pythagorean
partials, but if "d"'b be kept as here high in minor Third 27 : 32, cents 2g4,— Translator.]
the sciJe, they will not be heard, and the
Digitized by
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196 DEGREE OF HABMONIOUSNESS OP CONSONANCES, past it
Fourth c'f and major Sixth c^d become worse as the Elevenlii df and major
Thirteendi da". The minor Third dd^ and minor Sixth da!>^^ however, become
still worse as the minor Tenth dd'^ and minor Thirteenth dd*^, so that the latter
intervals are &r less harmonious than the subminor Tenth 3:7, dd'^-— [or g '^h*\f\,
and subminor Fourteenth 2:7, c'6"l>— .
• The order of the consonances here proposed is based upon a consideration of
the harmoniousness of each individual interval independently of any connection
with other intervals, and consequently without any regard to key, scale, and
i modulation. Almost all writers on musical theory have proposed similar orders
for the consonances, agreeing in their general features with each other and with
that here deduced from the theory of beats. Thus all put the Unison and Octave
first, as the most perfect of all consonances ; and next in order comes the Fifth,
after which the Fourth is placed by those, who do not include the modulational
% properties of the Fourth, but restrict their observation to the independent har-
moniousness of the interval. There is great diversity, on the other hand, in
the arrangement of the Sixths and Thirds. The Greeks and Romans did not
acknowledge these intervals to be consonances at all, perhaps because in the un-
accented Octave, within which their music, arranged for men*s voices, usually lay,
these intervals really sound badly, and perhaps because their ear was too sensitive
to endure the trifling increase of roughness generated by compound tones when
sounded together in Thirds and Sixths. In the present century, the Archbishop
Chrysanthus of Dyrrhachium declares that modem Greeks have no pleasure in
polyphonic music, and consequently he disdains to enter upon it in his book on
music, and refers those who are curious to know its rules, to the writings of the
West.* Arabs are of the same opinion according to the accounts of all travellers.
This rule remained in force even during the first half of the middle ages, when
the first attempts were made at harmonies for two voices. It was not till towards
f the end of the twelfth centmy that Franco of Cologne included the Thirds among
the consonances. He distinguishes : —
1. Perfect Consonances : Unison and Octave.
2. Medial Consonances : Fifth and Fourth.
3. Imperfect Consonances : Major and minor Thirds.
4. Imperfect Dissonances : Major and minor Sixth.
5. Perfect Dissonances : Minor Second, augmented Fourth, major and minor
Seventh.f
It was not till the thirteenth and fourteenth centuries that musicians began to
include the Sixths among the consonances. Philipp de Vitry and Jean de Muris t
mention as perfect consonances the Unison, Octave, and Fifth ; as imperfect, the
Thirds and Sixths. The Fourth has been cut out. The first author opposes the
major Third and major Sixth, as more perfect, to the minor Third and minor
Sixth. The same order is found in the Dodecachordon of Glareanus, 1557, § who
If merely added the intervals increased by an Octave. The reason why the Fourth
was not admitted as either a perfect or an imperfect consonant, must be looked for
in the rules for the progression of parts. Perfect consonances were not allowed
to follow each other between the same parts, still less dissonances ; but imperfect
consonances, as the Thirds and Sixths, were permitted to do so. But on the other
hand the perfect consonances. Octaves, and Fifths were admitted in chords on
which the music paused, as in the closing chord. Here, however, the Fourth of
the bass could not occur because it does not occur in the triad of the tonic. Again
a succession of Fourths for two voices was not admitted, as the Fourth and Fifth
were too closely related for such a purpose. Hence so far as the progression of
* BtwpTtTiKhr /i4ya rris Mowrucrit vapii Xpvtr- 1 852, p. 49.
dvBov, Tfpy4trni, 1832, cited by Coussemaker, 1 Coussemaker, ibid. p. 66 and p. 68.
Hiaiovre de Vharmoniej p. 5. § [This is the date of the abstract by
f Qerbert, Scrij^Uyfes acclesiastici de Mu- Woneggar of Lithuania, the date of the original
•tea Sacra, Saint-lBlaise, 1784, vol. iii. p. 11, work is 1547, ten years earlier.— rmtwfcitor.] .
— Cousaemaker, Eiitoir^ <fe I'hamwnie, Paris,
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eHAFS. X. XI.
BEATS DUE TO COMBINATIONAL TONES.
197
iMuris was concerned, the Fonrth shared the properties of dissonances, and it was at
once placed among them ; bnt it would have been better to have placed it in an inter-
mediate class between perfect and imperfect consonances. As &r as harmonious-
ness is concerned, there can be no doubt that, for most qualities of tone, the Fourth
is much superior to the major Third and major Sixth, and beyond all doubt better
than the minor Third and minor Sixth. But the Eleventh, or Fourth increased
by an Octave, sounds &r from well when the third partial tone is in any degree
strong.*
The dispute as to the consonance or dissonance of the Fourth has been con-
tinued to the present day. As late as 1840, in Dehn's treatise on harmony we find
it asserted that the Fourth must be treated and resolved as a dissonance ; but Dehn
certainly puts a totally different interpretation on the question in dispute by laying
it down that the Fourth of any bass within its key and independently of the
intervals with which it is combined, has to be treated as a dissonance. Otherwise ^
it has been the constant custom in modem music to allow the reduplication of the
tonic to occur as the Fourth of the dominant in conjunction with the dominant
even in final chords, and it was long so used in these chords, even before Thirds
were allowed in them, and in this way it came to be recognised as one of the superior
consonances.t
CHAPTEB XL
BEATS DUE TO COMBINATIONiVL TONUS.
When two or more compound tones are sounded at the same time beats may arise
£rom the combinational tones as well as from the harmonic upper partials. In
Chapter VII. it was shewn that the loudest combinational tone resulting from two ^
* [See the Eleventh analysed in p. 191c, studied on the Harmonioal,) will shew generally
iootDoie.— Translator,] how they aflfect each other in any combination.
f The following general view of the partials The nnmber of vibrations of each partial of
of the first 16 harmonics of C 66, (which, with each harmonic is given, whence the beats can
the exception of the nth and 13th, can be be immediately foand.
Partials
of C
s
C
3
e
3
9
'c
f'
6
9'
'^t>
8
S"
xo
0"
zx
X3
9"
43.
'y'b
J^'
x6
e"'
I
3
4
S
6
7
3
9
lO'
It
13
'3
14
15
16
17
tS
J9
ao '
44
r as
26
I ^7
^■
1 3'>
33
£6
198
a64
330
396
462
5^8
594
660
736
792
BSB
924
990
1056
iiSS
1^54
1330
i3j86
1453
1518
isa4
1650
17 16
lySa
184S
1980
21 12
13a
396
52S
660
792
934
1056
118S
1320
1453
198
3^
594
7Q2
990
1 188
J386
1SS4
264
528
793
1056
1320
1584
330
660
990
1320
1650
396
792
1 188
1584
463
934
1386
528
1056
31 12
594
1188
1783
660
1320
1980
736
I5I8
792
1584
858
1716
924
1848
990
1980
1056
2IZ3
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J9^
BEATS DUE TQ COMBINATIONAL TONES.
TART ir.
generating tones is that corresponding to tlie difference of iheir pitch numbers, or
the differential tone of the ffrst order. It is this combinational tone, therefore,
which is chiefly -effective in producing beats. Even this loudest combinational tone
is somewhat weak, unless the generators are very loud ; the differential tones of
iiigher orders, and the summational tones, are still weaker. Beats due to such
weak tones as those last mentioned cannot be observed unless all other beats which
would disturb the observer are absent, as, for instance, in sounding two simple
tones, which are entirely free from upper partials. On the other hand the beats of
the flrst differential tones [owing to difference of pitch and quahty] can be heard
very well at the same time as those due to the harmonic upper partials of com-
jpound tones, by an ear accustomed to hear combinational tones.
The differential tones of the first order alone, and independently of the com^
binational tones of higher orders, are capable of causing beats (i) when two
^ compound tones sound together, (2) when three or more simple or compound
tones sound together. On the other hand beats generated by combinational tones
of higher orders have to be considered when two simple tones sound together.
We commence with the differential tones of compound tones. In the same
way that the prime tones in such cases develop combinational tones, any pair of
upper partials of the two compounds will also develop combinational tones, but
such tones will diminish very rapidly in intensity as the upper partials become
weaker. When one or more of these combinational tones nearly coincide with
other combinational tones, or the primes or upper partials of the generators, beats
ensue. Let us take as an example a slightly incorrectly tuned Fifth, having the
pitch numbers 200 and 301, in place of 200 and 300, as in a justly intoned Fifth.
We calculate the vibrational numbers of the upper partials by multiplying those
of the primes by i, 2, 3, and so on. We find the vibrational numbers of the dif
ferential tones of the first order, by subtracting these numbers from each other,
f two and two. The following table contains in the first horizontal line andvertical
column the vibrational numbers of the several partials of the two compound tones,
and in their intersections the differences 6f thbse numbers, which are the pitch
numbers of the differential tones due to th^m.
—
Partials of the Fifth
•
301
602
903
"SS f?^
lOI
402
703
lil 1'
400
99
202
503
)6oo
299
2
303
800
499
'^
103
^ fl^
llDOO
^99
398
97
/
If we arrange these tones by pitch we find the following groups : —
97
198
299
398
499
600
699
99
200
301
400
503
602
703
lOI
202
303
402
103
800 903
1000
The number 2 is too small to correspond to a combinational tone. It only
shews the number of beats due to the two upper partials 600 and 602.* In all the
Other groups, however, tones are found whose vibrational numbers differ by 2, 4, or
6, and hence produce respectively 2, 4, and 6 beats in the same time that the two
first-named partials produce 2 beats. The two strongest combinational tones are
loi and 99, and these also are well distinguished from the rest by their low pitch
We observe in this example that the slowest beats due to the combinational
tones are the same in number as those due to the upper partials [600 and 602].
This is a general rule and appHes to all intervals.f
♦ [The last three, 800, 903, 1000, are
simply non-beating upper partiaJB. — ^Trans-
lator.-]
i [But the beats of the upper partials are
always distinguished by their high pitch.—
Translator.]
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CHAP. XI. BEATS DUE TO COMBINATIONAL TONES. 199
Further it is easy to see that if in our example we replaced 200 and 301, by
the numbers 200 and 300 belonging to the perfect Fifth, all the numbers in our
table would become multiples of 100, and hence all the different combinational and
upper partial tones wliich now beat would become coincident and not generate any
beats. What is here shewn to be the case in this example for the Fifth is also true
for all other harmonic intervals.*
The first differential tones of compounds cannot generate beats, except when the
upper partials of the same compounds generate them, and the rapidity of the beats
is the same in both cases, supposing that the series of upper partials is complete.
Hence the addition of combinational tones makes no essential difference in the
results obtained in the last chapter on investigating the beats due to the upper
partials only. There can be only a slight increase in the strength of the beats.f
But the case is essentially different when two simple tones are sounded together,
so that there are no upper partials to consider. If combinational tones were not f
taken into account, two simple tones, as those of tuning-forks or stopped organ
pipes, could not produce beats unless they were very nearly of the same pitch, and
such beats are strong when their interval is a minor or major Second, but weak for
a Third and then only recognisable in the lower parts of the scale (p. iTid), and
they gradually diminish in distinctness as the interval increases, without shewing
any special differences for the harmonic intervals themselves. For any larger
interval between two simple tones there would be absolutely no beats at all, if
there were no upper partial or combinational tones, and hence the consonant
intervals discovered in the former chapter would be in no respect distinguished
from adjacent intervals ; there would in &ct be no distinction at all between wide
consonant intervals and absolutely dissonant intervals*
Now such wider intervals between simple tones are known to produce beats,
although very much weaker than those hitherto considered, so that even for such
tones there is a difference between consonances and dissonances, although it is ^
very much more imperfect than for compound tones. And these facts depend, as
Scheibler shewed,}: on the combinational tones of higher orders.
It is only for the Octave that the first differential tone suffices. If the lower
note makes 100 vibrations in a second, while the imperfect Octave makes 201, the
first differential tone makes 201 — 100=101, and hence nearly coincides with the
lower note of 100 vibrations, producing one beat for each 100 vibrations. There
is no difficulty in hearing these beats, and hence it is easily possible to distinguish
imperfect Octaves from perfect ones, even for simple tones, by the beats produced
by the former.§
For the Fifth, the first order of differential tones no longer suffices. Take an
imperfect Fifth with the ratio 200 : 301 ; then the differential tone of the first
order is loi, which is too far from either primary to generate beats. But it forms
an imperfect Octave with the tone 200, and, as just seen, in such a case beats ensue.
Here they are produced by the differential tone 99 arising from the tone 10 1 and IT
* This is proved mathematically in Ap- phyaikalische rmd mtuikaUsche Tonmesaer,
pendiz XVI. (fee.) — Essen, G. D. Badeker, 1834, pp. viiL 80,
t [The great difference in the pitch of the 5 lithographed Tables (called 3 on title-page)*
two sets of beats, which are not necessarily and an, engraving of tuning-forks and waves,
even Octaves of each other, keeps them well A most remarkable pamphlet, bat unfortu-
apart. The beating partials, in this case 600, nately very obscurely written, as the author says
602, and the beating differentials, here loi and in his preface, * to write clearly and briefly on a
99, are entirely removed from each other. — scientific subject is a skill (Fertigkeit) I do not
Translator,] possess, and have never attempted.' See also
X [* The physical and musical Tonometer, App. XX. sect. B. No. 7. I do not find any-
-which makes evident to the eye, by means of where that Scheibler attempted to shew that
the pendulum, the absolute vibrations of the combinational tones existed, especially inter-
tones, and of the principal kinds of combina^ mediate ones ; he merely assumed them and
tional tones, as well as the most precise exact- found the beats. — Translator,]
ness of equally tempered and mathematical § [See App. XX. sect. L. art. 3, latter part
chords, invented and executed by Heinrich oid. — Translator.]
^>cheibler, silk manufacturer in Crefeld.' {Der
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aoo BEATS DUE TO COMBINATIONAL TONES. pabt a
the tone 200, and this tone 99 makes two beats in a second with the tone 101.
These beats then serve to distinguish the imperfect from the justly intoned Fifih
even in the case of two simple tones. The number of these beats is also exactly
as many as if they were beats due to the upper partial tones.* But to observe
these beats the two primary tones must be loud, and the ear must not be distracted
by any extraneous noise. Under favourable circumstances, however, they are not
difficult to hear.f
For an imperfect Fourth, having, say, the vibrational numbers 300 and 401 »
the first differential tone is loi ; this with the tone 300 produces the differential
tone 199 of the second order, and this again with the tone 401 the differential tone
202 of the third order, and this makes 3 beats with the differential tone 199 of the
second order, that is, precisely as many beats as would have been generated by the
upper partial tones 1200 and 1203, if they had existed. These beats of the Fourth
f are very weak even when the primary tones are powerful. Perfect quiet and great
attention are necessary for observing them.j: And after all there may be a doubt
whether by strong excitement of the primary tones, weak partials may not have
arisen, as we already considered on p. 1592^, c, §
The beats of an imperfect Tnajor Third are scarcely recognisable, even under the
most favourable conditions. If we take as the vibrational numbers of the primary
tones 400 and 501, we have : —
501—400=101, the differential tone of the first order
400— ioi:=299, ,, ,, ,, second „
501—299=202, „ „ „ third „
400—202=198, ,, ,, fourth ,,
The tones 202 and 198 produce 4 oeats. Scheibler succeeded in counting these
beats of the imperfect major Third.** I have myself believed that I heard them
f under favourable circumstances. But in any case they are so difficult to perceive
that they are not of any importance in distinguishing consonance from dissonance.
Hence it foUows that two simple tones making various intervals adjacent to the
major Third and sounded together wiU produce a uniform uninterrupted mass of
sound, without any break in their harmoniousness, provided that they do not
approach a Second too closely on the one hand or a Fourth on the other. My own
experiments with stopped organ pipes justify me in asserting that however much
this conclusion is opposed to musical dogmas, it is borne out by the fact, provided
that really simple tones are used for the purpose.ff It is the same with intervals
near to the major Sixth ; these also shew no difference as long as they remain
sufficently far from the Fifth and Octave. Hence although it is not difficult to
tune perfect major and minor Thirds on the harmonium or reed pipes or on the
violin, by sounding the two tones together and trying to get rid of the beats, it is
perfectly impossible to do so on stopped organ pipes or tuning-forks without the
% aid of other intervals. It will appear hereafter that the use of more than two
tones will allow these interval3 to be perfectly tuned even for simple tones.
Intermediate between the compound tones possessing many powerful upper
partials, such as those of reed pipes and violins, and the entirely simple tones of
tuning-forks and stopped organ pipes, lie those compound tones in which only the
* [Bat, as before, the pitch is veiy difife* much lower in pitch and so inharmonic to the
rent. — Translator,] others that there is no danger of oonfusing
t [Scheibler, ibid. p. 21. I myself sac- them.— Trafwtotor.]
oeeded in hearing and counting them.— 2Van«- ** [Scheibler, ibid, p. 25, says only * as beats
lator,'] of this kind are too indistinct/ he uses another
X [Scheibler says, p. 24, they are heard as method for tuning the major Third. See
well as for the Fifth. I have not found it so. note *, p. 203^2. He also calculates the int^r-
^-Translator,'] mediate tones differently. But neither he nor
§ [Supposing the pitch numbers of the any one seems to have tried to verify their
mistuned Fourth are 300 and 401, then the existence, which is doubtful. — Translator,^
beating upper partials would be 1200 and ff [Or at any rate tones without the 4th
1203, a very high pitch; but the beating partial, which those of stopped organ pipes do
differentials are 202 and 199, which are so not possess. — Translator.]
Digitized by V^jOOQlC
CHAP.. XI, BEATS DUE TO COMBINATIONAL TONES. aoi
lowest of the upper partials are audible, such as the tones of wide open organ pipes
or the human voice when singing some of the obscurer vowels, as oo in too. For
these the partials would not suffice to distinguish all the consonant intervals, but
the addition of the first differential tones renders it possible.
A. Compound Tones consisting of the prime and its Octave, These cannot
delimit Fifths and Fourths by beats of the partials, but are able to do so by those
of the first differential tones.
a. Fifth. Let the vibrational numbers of the prime tones be 200 and 301,
which are accompanied by their Octaves 400 and 602 ; aU four tones are then too
for apart to beat. But the differential tones
301 — 200=101
400—301= 99
Difference 2 ^
give two beats. The number of these beats again is precisely the same as if they
had been produced by the two next upper partials.* Namely
2 X 301—3x200=2
b. Fourth, Let the vibrational numbers of the primes be 300 and 401, and of
the first upper partials 600 and 802 ; these cannot produce any beats. But the
first differential tones give 3 beats, thus f : —
600—401 = 199
802—600=202
Difference 3
For Thirds it would be necessary to take differential tones of the second ordei*
into account.
B. Compound Tones consisting of the prime and Twelfth. Such tones are
produced by the narrow stopped pipes on the organ {Quintaten, p. 33^^, note). These ^
are related in the same way as those which have only the Octave.
a. Fifth, Primes 200 and 301, upper partials 600 and 903. First differential
tone /
903—600=303
Fifth=3oi
Number of beats 2
b. Fourth, Primes 300 and 401, upper partials 900 and 1203. First dif-
ferential tone *
1203-900=303
Lower prime=3oo
Number of beats 3
Even in this case the beats of the Third cannot be perceived without the help
of the weak second differential tones. 11
C. Compound Tones having both Octave and Twelfth as audible partials. Such
tones are produced by the wide (wooden) open pipes of the organ (Principal^
p, 93^^', note). The beats of the upper partials here suffice to delimit the Fifths,
bat not the Fourths. The Thirds can now be distinguished by means of the first
differential tones.
a. Major Third. Primes 400 and 501, with the Octaves 800 and 1002, and
Twelfths 1200 and 1503. First differential tones J
1002—800^202
1200—1002^198
Number of beats 4
* [The same in number, but observe that existed would beat at pitch 1200,— Translator,]
the first set of beats are at pitch 100, and the t [These are the same two beating tones
second at pitch 600.— Translator.] as calculated on p. 2006, but they are quite dif-
t [At pitch 200, whereas the partials if they ferently deiiyed,—TranslatorJ]
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202 BEATS DUE TO COMBINATIONAL TONES. part n.
b. Minor Third. Primes 500 and 601, Octaves 1202, Twelfths 1500 and 1803.
Differential tones *
1500—1202=298
1803 — 1500=303
Number of beats 5
c. Major Sixth, Primes 300 and 501, Octaves 600 and 1002, Twelfths 900 and
1503. Differential tones
600—501= 99
1002 — 900=102
Number of beats 3
In fact not only the beats of imperfect Fifths and Fourths, but abo those of
^ imperfect major and minor Thirds are easily heard on open organ pipes, and can be
immediately used for the purposes of tuning.
Thus, where upper partials, owing to the quality of tone, do not suffice, the
combinational tones step in to make every imperfection in the consonant intervals
of the Octave, Fifth, Fourth, major Sixth, major and minor Third immediately
sensible by means of beats and roughness in the combined sound, and thus to dis«
tinguish these intervals from all those adjacent to them. It is only perfectly simple
tones that so fsur make default in determining the Thirds ; and for them also the
beats which disturb the harmoniousness of imperfect Fifths and Fourths, are
relatively too weak to affect the ear sensibly, because they depend on differential
tones of higher orders. In reality, as I have already mentioned, two stopped pipes,
giving tones which lie between a major and a minor Third apart, will give just as
good a consonance as if the interval were exactly either a major or a minor Third.
This does not mean that a practised musical ear would not find such an interval
V strange and unusual, and hence would perhaps call it false, but the immediate im«
pression on the ear, the simple perception of harmoniousness, considered indepen-
dently of any musical habits, is in no respect worse than for one of the perfect
intervals.!
Matters are very different when more than two simple tones are sounded
together. We have seen that Octaves are precisely limited even for simple tones
by the beats of the first differential tone with the lower primary. Now suppose
that an Octave has been tuned perfectly, and that then a third tone is interposed
to act as a FiftL Then if the Fifth is not perfect, beats will ensue from the first
differential tone.
Let the tones forming the perfect Octave have the pitch numbers 200 and 400,
and let that of the imperfect Fifth be 301. The differential tones are
400-301= 99
301 — 200=101
H
Number of beats 2
These beats of the Fifth which lies between two Octaves are much more
audible than those of the Fifth alone without its Octave. The latter depend on
the weak differential tones of the second order, the former on those of the first
order. Hence Scheibler some time ago laid down the rule for tuning tuning-forks,
first to tune two of them as a perfect Octave, and then to sound them both at
once with the Fifth, in order to tune the latter .4: If Fifth and Octave are both
perfect, they also give together the perfect Fourth.
The case is similar, when two simple tones have been tuned to be a perfect
* [This was not given for simple tones be- for oases where neither partial nor combina-
fore, but Scheibler calculates the result in that tional tones are present, App. XX. sect. L.
case, p. 26, and says he could use it still less art. 7. — Translator,]
than for the major Third.—Translator.] % (l have been unable to find the passage
t [See Prof. Preyer's theory of consonance referred to.- Translator.'
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CHAP. «. BEATS DUE TO COMBINATIONAL TONES. 203
Fifth, and we interpoee a new tone between them to act as a major Third. Let the
perfect Fifth have the pitch numbers 400 and 600. On intercalating the impure
major Third with the pitch number 501 in lieu of 500, the differential tones are
600—501= 99
501—400=101
Number of beats 2*
The major Sixth is determined by combining it with the Fourth. Let 300 and
^00 be the vibrational numbers of a perfect Fourth, and 501 that of an imperfect
major Sixth. The differential tones are
501—400=101
400—300=100
— H
Number of beats i
If we tried to intercalate an interval between the tones forming a perfect
Fourth, and having the vibrational numbers 300 and 400, it could onlyl)e the sub-
minor Third with the vibrational number 350. Taking it imperfect and =351, we
have the differential tones
400-351=49
351-300=51
Number of beats 2
These intervals 8 : 7 and 7 : 6 are, however, too close to be consonances, and
hence they can only be used in weak discords (chord of the dominant Seventh), f
The above considerations are also applicable to any single compound tone con-
sisting of several partials. Any two partials of sufficient force will also produce ^
differential tones in the ear. If, then, the partials correspond exactly to the series
of harmonic partials, as assigned by the series of smaUer whole numbers, all these
differentials resulting &om partials coincide exactly with the partials themselves,
and give no beats. Thus if the prime makes n vibrations ^in a second, the upper
partials make 2n, 371, 4n, &c., vibrations, and the differences of these numbers are
again n, or 271, or 371, &c. The pitch numbers of the summational tones fall also
into this series.
On the other hand, if the pitch numbers of the upper partials are ever so
slightly different from, those giving these ratios, then the combinational tones will
differ from one another and from the upper partials, and the result will be beats.
The tone therefore ceases to make that uniform and quiet impression which a
compound tone with harmonic upper partials always makes on the ear. How con-
siderable this influence is, we may hear from any firmly attached harmonious
string after we have fastened a small piece of wax on any part of its length. This^ f
as theory and experiment alike shew, produces an inharmonic relation of the
upper partials. If the piece of wax is very small, then the alteration of tone is
also very small. But the slightest mistuning suffices to do considerable harm
to the tunefulness of the sound, and renders the tone dull and rough, like a tin
kettle.
* [On this was fonnded 8cheLbler*8 method asing the perfect Fifth, A 220, Ct 277-1824,
of taniiig the perfect major Third (alluded to -£^330. Then, 277'i824-220« 57-1824, 330—
in p. 20od\ note) and also the tempered major 277*1824 = 52*81 76 and 57* 1824 » 52-81 76 =:
G?hird. 4*3648, and hence the tuning of tiie inter-
First tone a perfect Fifth, and then an mediate fork must be altered till these beats
auxiliary Fifth, 2 vib. sharper. Then if the are heard. These are Scheibler's own ex-
major Third is perfect we have A 220, CZ 275, amples, p. 26, reduced to ordinary double
E 332 and 275-220-55, 332-275*57, and vibrations.— Tnan^Za^.]
^7—55 = 2. Hence the tuning of C% must f [In actual practice, for the chord of the
be altered till the differential tones beat 2 in dominant Seventh the interval is 4 : 7| the in-
a second. terval of the just subminor Seventh 4 : 7 not be-
For the tempered major Third we have, ing used, even in just intonation.— jTrafuZa^or.j
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i
ao4 BEATS DUE TQ COMBINATIONAL TONEa »bt n.
Herein we find the reason why tones with harmonic upper partials play such a
leading part in the sensation of the ear. They are the only sounds which, even
when very intense, can produce sensations that continue in undisturhed repose,
without beats, corresponding to the purely periodic motion of the air, which is the
objective foundation of these tones. I have already stated as a result of the
summary which I gave of the composition of musical tones in Chapter V., No. 2,
p. 1 19a, that besides tones with harmonic upper partials, the only others used (and
that also generally in a very subordinate manner) are either such as have a section
of the series of harmonic upper partials, (like those of well tuned bells), or such
as have secondary tones (as those in bars) so very weak and so fax distant from
their primes, that their differentials have but little force and at any rate do not
produce any distinct beats.
Collecting the results of our investigations upon beats, we find that when two
^ or more simple tones are sounded at the same time, they cannot go on sounding
without mutual disturbance, unless they form with each other certain perfectly
definite intervals. Such an undisturbed flow of simultaneous tones is called a
consonance. When these intervals do not exist, beats arise, that is, the whole
compound tones, or individual partial and combinational tones contained in them
or resulting from them, alternately reinforce and enfeeble each other. The tones
then do not coexist undisturbed in the ear. They mutually check each other's
uniform flow. This process is called dissonance,*
Combinational ^/^rma t^n^ fti^ lyn^af por^nffri nonc^f, pf t^^atfi, They are the sole
cause of beats for simple tones which lie as much as, or more than, a minor Third
apart.f For two simple tones they suffice to delimit the Fifth, perhaps the
Fourth, but certainly not the Thirds and Sixths. These, however, will be strictly
delimited when the major Third is added to the Fifth to form the common major
chord, and when the Sixth is united with the Fourth to form the chord of the
1 Sixth and Fourth, ^.
4
Thirds, however, are strictly delimited, by means of the beats of imperfect
intervals, in a chord of two compound tones, each consisting of a prime and the
two next partial tones. The beats of such intervals increase in strength and dis-
tinctness, with the increase in number and strength of the upper partial tones
in the compounds. By this means the difference between dissonance and conso-
nance, and of perfectly from imperfectly tuned intervals, becomes continually more
marked and distinct, increasing the certainty with which the hearer distinguishes
the correct intervals, and adding much to the powerful and artistic effect of succes-
sions of chords. Finally when the high upper partials are relatively too strong (in
piercing and braying qualities of tone) each separate tone will by itself generate
intermittent sensations of tone, and any combination of two or more compounds of
this description produces a sensible increase of this harshness, while at the same
f time the large number of partial and combinational tones renders it difficult for the
hearer to follow a complicated arrangement of parts in a musical composition.
These relations are of the utmost importance for the use of different instru-
ments in the different kinds of musical composition. The considerations which
determine the selection of the proper instrument for an entire composition or for
individual phrases in movements written for an orchestra are very multi&rious.
First in rank stands mobility and power of tone in the different instruments. On
this there is no need to dwell. The bowed instruments and pianoforte surpass all
others in mobility, and then follow the flutes and oboes. To these are opposed the
trumpets and trombones, which conunence sluggishly, but surpass all instruments
in power. Another essential consideration is expressiveness, which in general
depends on the power of producing with certainty any degree of rapid alterations
in loudness at the pleasure of the player. In this respect also bowed instruments,
♦ [See Prof. Preyer'g addendum to this f [But see App. XX. sect. L. art. 3.— Traiw-
theoryinApp.XX.sect.Ij. art. 7. — rrawiotor.] lator.]
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.fjHAP.xi/ DISSONANCE FOR DIFFERENT QUALITIES OF TONE. 20J
and the human voice, are pre-eminent. Artificial reed instruments, both of wood
and brass, cannot materiallj diminish their power without stopping the action of
the reeds. Flutes and organ pipes cannot greatly alter the force of their tone
without at the same time altering their pitch. On the pianoforte the strength with
which a tone commences is determined by. the player, but not its duration ; so that
the rhythm can be marked delicately, but real melodic expression is wanting. All
these points in the use of the above instruments are easy to observe and have long
been known and allowed for. The influence of quality proper was more difficult
to define. Our investigations, however, on the composition of musical tones have
given us a means of taking into account the principal differences in the effect of
the simultaneous action of different instruments and of shewing how the problem
is to be solved, although there is still a large field left for a searching investigation
in detail.
Let us begin with the simple tones of v)ide stopped organ pipes. In themselves ^
they are very soft and mild, dull in the low notes, and very tuneful in the upper.
They are quite unsuited, however, for combinations of harmony according to
modem musical theory. We have already explained that simple tones of this kind
discriminate only the very small interval of a Second by strong beats. Imperfect
Octaves, and the dissonant intervals in the neighbourhood of the Octave, (the 'p
Sevenths and Ninths,) beat with the combinational tones, but these beats are
weak in comparison with those due to upper partials. The beats of imperfect
Fifths and Fourths are entirely inaudible except under the most favourable condi-
tions. Hence in general the impression made on the ear by any dissonant interval,
except the Second, differs very little from that made by consonances, and as a
consequence the harmony loses its character and the hearer has no certainty in his
perception of the difference of intervals.* If polyphonic compositions containing
the harshest and most venturesome dissonances are played upon wide stopped
organ pipes, the whole is uniformly soft and harmonious, and for that very reason ^
also indefinite, wearisome and weak, without character or energy. Every reader
that has an opporttmity is requested to try this experiment. There is no better
proof of the important part which upper partial tones play in music, than the im-
pression produced by music composed of simple tones, such as we have just
described. Hence the wide stopped pipes of the organ are used only to give
prominence to the extreme softness and tunefcdness of certain phrases in contra-
distinction to the harsher effect of other stops, or else, in connection with other
stops, to strengthen their prime tones. Next to the wide stopped organ pipes as
regards quality of tone stand flutes and the flxie pipes on organs (open pipes, blown
gently). These have the Octave plainly in addition to the prime, and when blown
more strongly even produce the Twelfth. In this case the Octaves and Fifths are
more distinctly delimited by upper partial tones ; but the definition of Thirds and
Sixths has to depend upon combinational tones, and hence is much weaker. The
musical character of these pipes is therefore not much unlike that of the wide ^
stopped pipes already described. This is well expressed by the old joke that nothing
is more dreadful to a musical ear than a flute-concerto, except a concerto for two
flutes.f But in combination with other instruments which give effect to the con-
nection of the harmony, the flute, from the perfect softness of its tone and its
great mobility, is extraordinarily pleasant and attractive, and cannot be replaced
by any other instrument. In ancient music the flute played a much more im-
portant part than at present, and this seems to accord with the whole ideal of
classical art, which aimed at keeping every thing unpleasant from its productions,
eonfining itself to pure beauty, whereas modem art requires more abundant means
* [Bnt see Prof. Preyer in App. XX. sect. L. a concerto or peculiar piece of music for one
art. 7. — Travslator,] instrument, and secondly as a concert^ or piece
f [In the original, * dass einem musikali- of music for several instruments, cannot be
0chen Ohre nichts sohrecklicher sei als ein properly rendered in the translation. — Tram-
Fldtenconcert, ausgenommen ein Ck)ncert von lator,'\
z-wei Floten.' The pun on ' Concert,' first as
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2o6 DISSONANCE FOR DIFFERENT QUALITIES OF TONE, pabt n.
of expression, and consequently to a certain extent admits into its circle what in
itself would be contrary to the gratification of the senses. However this be, the
earnest friends of music, even in classical times, contended for the harsher tones
of stringed instruments in opposition to the effeminate flute.
The open organ pipes afford a favourable means of meeting the harmonic
requirements of polyphonic music, and consequently form the principal stops.*
They make the lower partials distinctly audible, the wide pipes up to the third, the
narrow ones (geigen principal t) up to the sixth partial tone. The wider pipes
have more power of tone than the narrower ; to give them more brightness the
8-foot stops, which contain the ' principal work,' are connected with the 4-foot
stops, which add the Octave to each note, or the principal is connected with the
geigen principal, so that the first gives power and the second brightness. By this
means qualities of tone are produced which contain the first six partial tones in
% moderate force, decreasing as the pitch ascends. These give a very distinct feeling
for the purity of the consonant intervals, enabling us to distinguish clearly between
consonance and dissonance, and preventing the unavoidable but weak dissonances
that result from the higher upper partials in the imperfect consonances, from be-
coming too marked, but at the same time not allowing the hearer's appreciation of
the progression of the parts to be disturbed by a multitude of loud accessory tones.
In this respect the organ has an advantage over all other instruments, as the
player is able to mix and alter the quahties of tone at pleasure, and make them
suitable to the character of the piece he has to perform.
The narrow stopped pipes (Quintaten),t for which the prime tone is ac-
companied by the Twelfth, the reed-flute (Bohrfiote) § where the third and fifth
partial are both present, the conical open pipes, as the goat-horn (Gemshom),**
which reinforce certain higher partials ft more than the lower, and so forth, serve
only to give distinctive qualities of tone for particular parts, and thus to separate
f them from the rest. They are not weU adapted for forming the chief mass of the
harmony.
Very piercing quahties of tone are produced by the reed pipes and compound
stops tt on the organ. The latter, as already explained, are artificial imitations of
the natural composition of all musical tones, each key bringing a series of pipes
into action, which correspond to the first three or first six partial tones of the
corresponding note. They can be used only to accompany congregational singing.
When employed alone they produce insupportable noise and horrible confusion.
But when the singing of the congregation gives overpowering force to the prime
tones in the notes of the melody, the proper relation of quahty of tone is restored,
and the result is a powerful, well-proportioned mass of sound. Without the
assistance of these compound stops it would be impossible to control a vast body
of sound produced by unpractised voices, such as we hear in [German] churches.
The human voice is on the whole not unhke the organ in quahty, so for as
^ harmony is concerned. The brighter vowels, of course, generate isolated high
partial tones, but these are so unconnected with the rest that they can have no
universal and essential effect on the sound of the chords. For this we must look
to the lower partials, which are tolerably uniform for all vowels. But of course in
particular consonances the characteristic tone of the vowels may play an important
part. If, for example, two human voices sing the major Third ^ <)' on the vowel
a in father, the fourth partial of hj;} (or b']}), and the third partial of d' (or a*% £eJ1
among the tones characteristically reinforced by A, and consequently the imperfec-
tion of the consonance of a major Third will come out harshly by the dissonance
a" b']}, between these upper partials ; whereas if the vowel be changed to 0 in no,
the dissonance disappears. On the other hand the Fourth ^ e'[> soimds perfectly
♦ [See p. 141^', note ^.—Translator.] *♦ [See p. 94^, note.— Translator.]
t [See p. 141(2, note.— TranaZator.] ff [Generally the 4th, 6th, and 7th.—
t [See p. 33d, note.— Translat/>r.] Translator.]
§ [See p. 94<?', note.— TransZafor.] tt [See p» 57a', note.-— Tniwsiator.]
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OHAP. XI. DISSONANCE FOR DIFFERENT QUALITIES OF TONE. 207
weU on the vowel a in father, because the higher note e'\} has the same upper par-
tial b"\} as the deeper b\}. But if a in father being incHned towards a in fally or a in
fat, the upper partials /' and e"\} or else d'" and e'"\} might interrupt the con-
sonance. This serves to shew, among other things, that the translation of the
words of a song from one language into another is not by any means a matter
of indifference for its musical effect.*
Disregarding at present these reinforcements of partials due to the characteristic
resonance of each vowel, the musical tones of the human voice are on the whole
accompanied by the lower partials in moderate strength, and hence are well adapted
for combinations of chords, precisely as the tones of the principal stops of the
organ. Besides this the human voice has a peculiar advantage over the organ and
all other musical instruments in the execution of polyphonic music. The words
which are sung connect the notes belonging to each part, and form a clue which
readily guides the hearer to discover and pursue the related parts of the whole body ^
of sound. Hence polyphonic music and the whole modem system of harmony
were first developed on the human voice. Indeed, nothing can exceed the musical
effect of well harmonised part music perfectly executed in just intonation by prac-
tised voices. For the complete harmoniousness of such music it is indispensably
necessary that the several musical intervals should be justly intoned, and our pre-
sent singersf unfortunately seldom learn to take just intervals, because they are
accustomed from the first to sing to the accompaniment of instruments which are
tuned in equal temperament, and hence with imperfect consonances. It is only
such singers as have a delicate musical feeling of their own who find out the
correct result, which is no longer taught them.
Richer in upper partials, and consequently brighter in tone than the human
voice and the principal stops on the organ, are the bowed instruments, which con-
sequently fill such an important place in music. Their extraordinary mobihty and
expressiveness give them the first place in instrumental music, and the moderate ^
acuteness of their quality of tone assigns them an intermediate position between
the softer flutes and the braying brass instruments. There is a slight difference
between the different instruments of this class ; the tenor and double-bass have a
somewhat acuter and thinner quality than the violin and violoncello, that is, they
have relatively stronger upper partials. The audible partials reach to the sixth or
eighth, according as the bow is brought nearer the finger-board for piano, or nearer
the bridge for forte, and they decrease regularly in force as they ascend in pitch.
Hence on bowed instruments the difference between consonance and dissonance is
* [Also, it shews how the musical effect of and tempered intonation in the singing of the
difTerent stanzas in a ballad, though sung to the same choir. It was a choir of about 60 mixed
same written notes, will constantly vary, quite voices, which had gained the prize at the In-
independently of difference of expression. This ternational Exhibition at Paris in 1867, and
is often remarkable on the closing cadence of had been kept well together ever since. After
the stanza. As the vowel changes from a in singing some pieces without accompaniment. ^
father, to a in mat ; e in viet, or i in sit, or and hence in the just intonation to which the
again to o in not, u in hut, and u in put, the singers had been trained, and with the most
musical result is totally different, thou,r^h the delightful effect of harmony, they sang a piece
pitch remains unaltered. To shew the effect with a pianoforte accompaniment. Of course
of the different vowels throughout a piece of the pianoforte itself was inaudible among the
music, I asked a set of about 8 voices to sing, mass of sound produced by sixty voices. But
before about 200 others, the first half of See it had the effect of perverting their intonation,
the conquering hero comes, first to lah, then to and the whole charm of the singing was at
lee, and then to loo. The difference of effect once destroyed. There was nothing left but
was almost ludicrous. Much has to be studied the everyday singing of an ordinary choir,
in the relation of the qualities of vowels to the The disillusion was complete and the effect
effect of the music. In this respect, too, the most unsatisfactory as a conclusion. If the
pitch chosen for the tonic will be found of great same piece of music or succession of chords in
importance. — Translator.] C major or C minor, without any modulation,
t I^This refers to Germany, not to the Eng- be played first on the Harmonical and then be
lish Tonic Soltaists, nor to the English ma- contrasted with an ordinary tempered har-
drigal singers. On Dec. 27, 1869, at a meeting monium, the same kind of difference will be
of the Tonic Solfa College I had an unusual felt, but not so strongly. — Tran^Aitor.J
opportunity of contrasting the effect of just
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2o8 DISSONANCE FOR DIFFERENT QUALITIES OF TONE, part ii.
clearly and distinctly marked, and the feeling for the justness of the intervals very
certain ; indeed it is notorious that practised viohn and violoncello players have a
very delicate ear for distinguishing differences of pitch. On the other hand the
piercing character of the tones is so marked, that soft song-like melodies are not
well suited for bowed instruments, and are better given to flutes and clarinets in
the orchestra. Full chords are also relatively too rough, since those upper partials
fs which form dissonant intervals in every consonance, are sufficiently strong to make
\ the dissonance obtrusive, especially for Thirds and Sixths. Moreover, the im-
\ perfect Thirds and Sixths of the tempered musical scale are on bowed instruments
very perceptibly different in effect from the justly intoned Thirds and Sixths when
. the player does not know how to substitute the pure intervals for them, as the ear
requires. Hence in compositions for bowed instruments, slow and flowing progres-
sions of chords are introduced by way of exception only, because they are not
% sufficiently harmonious ; quick movements and figures, and arpeggio chords are
preferred, for which these instruments are extremely well adapted, and in which
the acute and piercing character of their combined sounds cannot be so distinctly
perceived.
The beats have a pecuHar character in the case of bowed instruments. Regular,
slow, numerable beats seldom occur. This is owing to the minute irregularities in
the action of the bow on the string, abeady described, to which is due the well-
known scraping effect so often heard. Observations on the vibrational figure
shew that every httle scrape of the bow causes the vibrational curve to jump sud-
denly backwards or forwards, or in physical terms, causes a sudden alteration in the
phase of vibration. Now since it depends solely on the difference of phase whether
two tones which are sounded at the same time mutually reinforce or enfeeble each
other, every minutest catching or scraping of the bow will also affect the flow of the
beats, and when two tones of the same pitch are played, every jump in the phase will
^ suffice to produce a change in the loudness, just as if irregular beats were occurring
at unexpected moments. Hence the best instruments and the best players are
necessary to produce slow beats or a uniform flow of sustained consonant chords.
Probably this is one of the reasons why quartetts for bowed instruments, when
executed by players who can play solo pieces pleasantly enough, sometimes sound
so intolerably rough and harsh that the effect bears no proper ratio to the slight
roughness wliich each individual player produces on his own instrument.* When
I was making observations on vibrational figures, I found it difficult to avoid the
occurrence of one or two jumps in the figure every second. Now in solo-playing
the tone of the string is thus interrupted for almost inappreciably minute instants,
which the hearer scarcely perceives, but in a quartett when a chord is played for
which all the notes have a common upper partial tone, there would be from four
to eight sudden and irregular alterations of loudness in this common tone every
second, and this could not pass unobserved. Hence for good combined performance,
f a much greater evenness of tone is required than for solo-playing.f
The pianoforte takes the first place among stringed instruments for which the
strings are struck. The previous analysis of its quahty of tone shews that its
deeper octaves are rich, but its higher octaves relatively poor, in upper partial tones.
In the lower octaves, the second or third partial tone is often as loud as the prime,
nay, the second partial is often louder than the prime. The consequence is that
♦ [To myself, one of the principal reasons rally known. If the music notes could be
for the painful effect here alluded to, which is previously marked by duodenals, in the way
unfortunately so extremely well known, is the suggested in App. XX. sect. E. art. 26, much
fact that the players not having been taught the of this difficulty might be avoided from the
nature of just intonation, do not accommodate first. But the marking would require a study
the pitches of the notes properly. When not yet commenced by the greater number of
quartett players are used to one another they rausici&ns.— Translator,]
overcome this difficulty. But when they learn f [On violins combinational tones are
thus, it is a mere accommodation of the different strong. I have been told that violinists watch
intervals by ear to the playing of (say) the for the Octave differential tone, in tuning their
leader. (See App. XX. sect. G. art. 7.) The real Fitihs.—Trmislator.]
relations of the just tones are in fact not gene-
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CHAP. XI. DISSOKANCE FOB DIFFERENT QUALITIES OF TONE. 209
the dissonances near the Octave (the Sevenths and Ninths) are ahnost as harsh as
the Seconds, and that diminished and augmented Twelfths and Fifths are rather
rongh. The 4th, 5th, and 6th partial tones, on the other hand, on which the Thirds
depend, decrease rapidly in force, so that the Thirds are relatively much less dis •
tinctly delimited than the Octaves, Fifths, and Fourths. This last circumstance is
important, because it makes the sharp Thirds of the equal temperament much more
endurable upon the piano than upon other instruments with a more piercing quahty
of tone, whereas the Octaves, Fifths, and Fourths are delimited with great distinct-
ness and certainty. Notwithstanding the relatively large number of upper partial
tones on the pianoforte, the impression produced by dissonances is fiar from being so
penetrating as on instruments of long-sustained tones. On the piano the note is
powerful only at the moment when it is struck, and rapidly decreases in strength,
so that the beats which characterise the dissonances have not time to become
sensible during the strong conmiencement of the tone ; they do not even begin IT
until the tone is greatly diminished in intensity. Hence in the modem music
written for the pianoforte, since the time that Beethoven shewed how the cha-
racteristic peculiarities of the instrument were to be utilised in compositions, we
find an accumtdation and reduplication of dissonant intervals which would be per-
fectly insupportable on other instruments. The great difference becomes very evi-
dent when an attempt is made to. play recent compositions for the piano on the
harmonium or organ.
That instrument-makers, led solely by practised ears, and not by any theory,
should have found it most advantageous to arrange the striking place of the
hammer so that the 7th partial tone entirely disappears, and the 6th is weak
although actually present,* is manifestly connected with the structure of our system
of musical tones. The 5th and 6th partial tones serve to delimit the minor
Third, and in this way almost all the intervals treated as consonances in modem
music are determined on the piano by coincident upper partials ; the Octave, Fifth, IT
and Fourth by relatively loud tones ; the major Sixth and major Third by weak
ones; and the minor Third by the weakest of all. If the 7th partial tone
were also present, the subminorf Seventh 4 : 7, as cf^h]), would injure the har-
moniousness of the minor Sixth ; the Subminor Fifth 5 : 7 , as c'^fc'[>, that of the Fifth
and Fourth ; and the subminor Third 6 : 7, as g'''h'\}, that of the minor Third ; with-
out any gain in the more accurate determination of new intervals suitable for
musical purposes.
Mention has already been made of a further peculiarity in the* selection of
quahty of tone on the pianoforte, namely that its upper notes have fewer and weaker
upper partial tones than the lower. This difference is much more marked on the
piano than on any other instrument, and the musical reason is easily assigned. The
high notes are usuaUy played in combination with much lower notes, and the
relation between the two groups of notes is given by the high upper partials of the
deeper tones. When the interval between the bass and treble amounts to two or H
three Octaves, the second Octave, higher Third and Fifth of the bass note, are in
the close neighbourhood of the treble, and form direct consonances and dissonances
with it, without any necessity for using the upper partials of the treble note.
Hence the only effect of upper partials on the highest notes of the pianoforte
would be to give them shrillness, without any gain in respect to musical definition.
In actual practice the constmction of the hammers on good instruments causes
the notes of the highest Octaves to be only gently accompanied by their second
partials. This makes them mild and pleasant, with a fiute-like tone. Some
instrument-makers, however, prefer to make these notes shrill and piercing, like
the piccolo flute, by transferring the striking place to the very end of the highest
strings. This contrivance succeeds in increasing the force of the upper partial
* [But Bee Mr. Hipkins' obsenrations on The 7th partial was very distinct on the pianos
PP- 17 y 78, noie.— Translator.'] Mr, Hipkins examined. See also App. XX.
t [For these terms see ttie table on p. 187. Sect. '^,— Translator,^
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2IO DISSONANCE FOE DIFFERENT QUALITIES OF TONE, paet ii.
tones, but gives a quality of tone to these strings which does not suit the
ciiaracter of the others, and hence certainly detracts from their charm.
In many other instruments, where their construction does not admit of such
absolute control over the quality of tone as on the pianoforte, attempts have been
made to produce similar varieties of quality in the high notes, by other means.
In the bowed instruments this purpose is served by the resonance box, the proper
tones of which lie within the deepest Octaves of the scale of the instrument. Since
the partial tones of the sounding strings are reinforced in proportion to their
proximity to the partial tones of the resonance box, this resonance will assist the
prime tones of the higher notes, as contrasted with their upper partials, much
more than it will do so for the deep notes. On the contrary, the deepest notes of
the violin will have not only their prime tones, but also their Octaves and Fifths
favoured by the resonance ; for the deeper proper tone of the resonance box
% hes between the prime and 2nd partial, and its higher proper tone between
the 2nd and 3rd partials. A similar effect is attained in the compound stops of
the organ, by making the series of upper partial tones, which are represented by
distinct pipes, less extensive for the higher than for the lower notes in the stop.
Thus each digital opens six pipes for the lower octaves, answering to the first six
partial tones of its note ; but in the two upper octaves, the digital opens only three
or even two pipes, which give the Octave and Twelfth, or merely the Octave, in
addition to the prime.
There is also a somewhat similar relation in the human voice, although it
varies much for the different vowels. On comparing the higher and lower notes
which are sung to the same vowel, it will be found that the resonance of the cavity
of the mouth generally reinforces relatively high upper partials of the deep notes
of the bass, whereas for the soprano, where the note sung comes near to the charac-
teristic pitch of the vowel, or even exceeds it, all the upper partials become much
% weaker. Hence in general, at least for the open vowels, the audible upper partials
of the bass are much more numerous than those of the soprano.
We have still to consider the artificial reed instruments, that is the wind in-
struments of wood and brass. Among the former the clarinet, among the latter
the horn are distinguished for the softness of their tones, whereas the bassoon and
liautbois in the first class, and the trombone and trumpet in the second represent
the most penetrating quahties of tone used in music.
Notwithstanding that the keyed horns used for so-called concerted music have
a far less braying quality of tone than trumpets proper, which have no side holes,
yet the number and the force of their upper partial tones are far too great for the
harmonious effect of the less perfect consonances, and the chords on these instru-
ments are very noisy and harsh, so that they are only endurable in the open air.
In artistic orchestral music, therefore, trumpets and trombones, which on account
of their penetrative power cannot be dispensed with, are seldom employed for
^ harmonies, except for a few and if possible perfect consonances.
The clarinet is distinguished from all other orchestral wind instruments by
having no evenly numbered partial tones.* To this circumstance must be due many
remarkable deviations in the effect of its chords from those of other instruments.
When two clarinets are playing together all of the consonant intervals will be
delimited by combinational tones alone, except the major Sixth 3:5, and the
Twelfth I : 3. But the differential tones of the first order, which are the strongest
among all combinational tones, will always suffice to produce the beats of imperfect
consonances. Hence it follows that in general the consonances of two clarinets
have but little definition, and must be proportionately agreeable. This is really the
case, except for the minor Sixth and minor Seventh, which are too near the major
Sixth, and for the Eleventh and minor Thirteenth, which are too near the Twelfth.
On the other hand, when a clarinet is played in combination with a violin or oboe,
the majority of consonances will have a perceptibly different effect according as
* [But see Mr. Blaiklej's observations, supra, p. 99&, note.— Trarulafor.]
Digitized by V^jOOQlC
CHAPff* XI. XII. THE CONSONANT CHORDS. 211
the clarinet takes the upper or the lower note of the chord. Thus the major
Third d' f% will sound better when the clarinet takes d' and the oboe f^, so that
the 5th partial of the clarinet coincides with the 4th of the oboe. The 3rd and
4th and the 5th and 6th partials, which are so disturbing in the major Third,*
cannot here be heard, because the 4th and 6th partials do not exist on the clarinet.
But if the oboe takes d' and the clarinet f% the coincident 4th partial will be
absent, abd the disturbing 3rd and 5th present. For the same reason it follows
that the Fourth and minor Third will sound better when the clarinet takes the
upper tone. I have made experiments of this kind with the clarinet and a bright
stop of the harmonium, which possessed the evenly numbered partial tones, and
was timed in just intonation f and not in equal temperament. When ^ was played
on the clarinet, and e'|>, d'y d'\}y in succession on the harmonium, the major
Third i[> d' sounded better than the Fourth &[> e'\}, and much better than the
minor llurd h{} dy. If, retaining llj[} on the clarinet, I played /, ^, ^ in succession f
on the harmonium, the major Third ^ &[> was rougher, not merely than the Fourth
/ 6t>, but even than the minor Third g ^.
This example, to which I was led by purely theoretical considerations that
were immediately confirmed by experiment, will serve to shew how the use of
exceptional quaHties of tone will affect the order of agreeableness of the conso-
nances which was settled for those Usually heard.
Enough has been said to shew the readiness with which we can now account
for numerous peculiarities in the effects of playing different musical instruments
in combination. Further details are rendered impossible by the want of sufficient
preliminary investigations, especially into the exact differences of various qualities
of tone. But in any case it would lead us too far &om our main purpose to pursue
a subject which has rather a technical than a general interest.
CHAPTER XII.
GHOBDS.
Wb have hitherto examined the effect of sounding together only two tones which
form a determinate interval. It is now easy to discover what will happen when
more than two tones are combined. The simultaneous production of more than
two separate compound tones is called a chord. We will first examine the har-
moniousness of chords in the same sense as we examined the harmoniousness of
any two tones sounded together. That is, we shall in this section deal exclusively
with the isolated effect of the chord in question, quite independently of any musical
connection, mode, key, modulation, and so on. The first problem is to determine
under what conditions chords are consonant, in which case they are termed concords.
It is quite clear that the first condition of a concord is that each tone of it should H
form a consonance with each of the other tones ; for if any two tones formed a
dissonance, beats would arise destroying the tunefulness of the chord. Concords
of three tones are readily found by taking two consonant intervals to any one
fundamental tone as c, and then seeing whether the new third interval between
the two new tones, which is thus produced, is also consonant. If this is the case
each one of the three tones forms a consonant interval with each one of the other
two, and the chord is consonant, or is a concord.^
Let us confine ourselves in the first place to intervals which are less than an
Octave. The consonant intervals within these limits, we have found to be : i) the
Fifth cg,i; 2) the Fourth c/, | ; 3) the major Sixth ca,^; 4) the major Third
ce,i; s) the minor Third c 4>, | ; 6) the minor Sixth c at>, f ; to which we may
• [See table on p. 191, note,— Translator.] third are dissonant with each other, I call the
t [Try the Harmonical and clarinet.— result a ' con-dissonant triad.* See App. XX.
Translator.] eect. E. art. 5.— Transiator.!
X [If two tones each consonant with a t
Digitized by VjOQjSiC
212
THE CONSONANT CHOEDS.
PABT n.
add 7) the snbminor or natural Seventh c'h^^, J, which approaches to the minor
Sixth in harmoniousness. The following table gives a general view of the chords
contained within an Octave, The chord is supposed to consist of the funda-
mental tone C, some one tone in the first horizontal line, and some one tone of
the first vertical column. Where the line and column corresponding to these
two selected tones intersect, is the name of the interval which these two latter
tones form with each other. This name is printed in italics when the interval ia
consonant, and in Boman letters when dissonant, so that the eye sees at a glance
what concords are thus produced. [Under the name, the equivalent interval in
cents has been inserted by the Translator.]
c
0
Of
702
U
884
51
Ebi
316
814
6|
702
'J
major
Second
I
204
u
major
Second
major
Third
il
minor
Third
minor
Second
if
112
Fourth
4s
316
major
Third
major
Second
1
Buperflnons
Fourth
minor
Second
1!
70
814
minor
Second
112
minor
Third
minor
Second
If
70
diminished
Fourth
&
Fourth
4.
'Bbi
969
sabminor
Third
sab
Fourth
!i
471
BIlDZIllIlOr
Second
8
snbminor
Fifth
sub
Fifth
&
Bnbmajor
Second
15s
H
From this it follows that the only consonant triads or chords of three notes,
that can possibly exist within the compass of an Octave are the following : —
i) 0 E a 2) C E\}G
3) C F A 4) C F AJ^
5) GE\}A\^ 6) C E A.*
The two first of these triads are considered in musical theory as the funda-
mental triads from which all others are deduced. They may each be regarded as
composed of two Thirds, one major and the other minor, superimposed in different
orders. The chord C E Qyin which the major Third is below, and the minor
above, is a major triad. It is distinguished firom all other major triads by having
its tones in the closest position, that is, forming the smallest intervals with each
other. It is hence considered as tiae fundamental chord or basis of all other major
chords. The triad GE^O, which has the minor Third below, and the major above,
is the fundamental chord of all minor triads,
* [The reader ought to hear the whole set nium, organ, and piano does not permit this,
of triads that could be formed from the table, But they can all (inoludye of those formed by
at least all exclusive of those formed by the the last line) be played on the Harmonioal. —
last line. The ordinary tuning of the harmo- Tramlator.]
Digitized by V^OOQIC
OHAP. xn. DIFFERENCE BETWEEN MAJOE AND MINOB CHORDS. 213
The next two ohords, 0 FA and 0 F A\}, are termed, from their composition,
chords of the Sixth and Fomth, written [OioF being a Fourth, and 0 to il
a major, but (7 to il(> a minor sixth]. If we take O, instead of 0 for the funda-
mental or bass tone, these chords of the Fourth and Sixth become G, G E and
O, C Ej}. Hence we may conceive them as having been formed from the funda-
mental major and minor triads CEO and C JEr[> (?, by transposing the Fifth G an
Octave lower, when it becomes G;.
The two last chords, C E\}A\^ and 0 E A,eae termed chords of the Sixth and
Third, or simply chords of the Sixth, written [OioE being a major Third, and G
to Ej} aminor Third; and G to A ek major Sixth, and C to ii[> a minor Sixth]. If
we take JSr as the bass note of the first, and Ej} as that of the second, they become
E G Ct E\}Gc, respectively. Hence they may be considered as the transpositions H
or in/oersioTts of a fundamental major and a fundamental minor chord, G E G,
C Ej^Gfin which the bass note G is transposed an Octave higher and becomes c.
Collecting these Inversions, the six consonant triads wiU assume the following
form [the numbers shewing their correspondence with the forms on p. 21 2d] : —
1) G E G 2) G E\} G
5) E Gc 6) Ej;} Gc
3) Gee 4) Gce^
We must observe that although the natural or subminor seventh ^J3[> forms a
good consonance with the bass note 0, a consonance which is indeed rather superior
than inferior to the minor Sixth G A\}, yet it never forms part of any triad, because
it would make worse consonances with all the other intervals consonant to 0 than
it does with 0 itself. The best triads which it can produce are G E''JB\} = 4 : 5 : 7>
and 0 G '^j? = 4:6:7. In the first of these occurs the interval E ''B\} = 5 : 7, H
(between a Fourth and Fifth,) in the latter the subminor Third G ^-B^ = 6:7.*
On the other hand the minor Sixth makes a perfect Fourth with the minor Third,
so that this minor Sixth remains the worst interval in the chords of the Sixth and
Third, and of the Sixth and Fourth, for which reason these triads can still be con-
fddered as consonant. This is the reason why the natural or subminor Seventh is
never used as a consonance in harmony, whereas the minor Sixth can be employed,
although, considered independently, it is not more harmonious than the subminor
Seventh.
The triad G E A\}, to which we shall return, [Chap. XVII. Dissonant Triads,
No. 4] is very instructive for the theory of music. It must be considered as a
dissonance, because it contains the diminished Fourth E A\}, having the interval
ratio ff • Now this diminished Fourth E A^ is so nearly the same as a major
Third E G%, that on our keyed instruments, the organ and pianoforte, the two
intervals are not distinguished. We have in fiust ^
^^l> = t* = |.|if
or, approximatively (S A^ = {E G% . Jf t
On the pianoforte it would seem as if this triad, which for practical purposes may
be written either G E A\} or G E Gj|l, must be consonant, since each one of its
tones forms with each of the others an interval which is considered as consonant
on the piano, and yet this chord is one of the harshest dissonances, as all musicians
are agreed, and as any one can convince himself immediately. On a justly intoned
instrument [as the Harmonica!] the interval E A\} is immediately recognised as
dissonant. This chord is well adapted for shewing that the original meaning of
the intervals asserts itself even with the imperfect tuning of the piano, and deter-
mines the judgment of the ear.$
* [Add the consonance G^B}dD = 6 I 7 : 9, cents, difference 42 cents, the great di&is.
—Translator.] See App. XX. sect. T>,— Translator,]
i [E Ab has 428 cents, and E Oti has 386 X [Inserting the values of the iotenrals in
Digitized by V^jOOQ IC
C E
Q
E a
c
Q C
E
C E^
G
E^Q
0
a c
Eb
214 DIFFERENCE BETWEEN MAJOR AND MINOR CHORDS, pabt n--
The hannonious effect of the varioas inversions of triads already found depends
in the first place upon the greater or less perfection of the consonance of the several
intervals they contain. We have found that the Fourth is less agreeable than the
Fifth, and that minor are less agreeable than major Thirds and Sixths. Now the
triad
O has a Fifth, a major Third, and a minor Third
a Fourth, a minor Third, and a minor Sixth
a Fourth, a major Third, and a major Sixth
a Fifth, a minor Third, and a major Third
a Fourth, a major Third, and a major Sixth
a Fourth, a minor Third, and a minor Sixth
For just intervals the Thirds and Sixths decidedly disturb the general tunefbl-
^ ness more than the Fourths, and hence the major chords of the Sixth and Fourth
are more harmonious than those in the fundamental position, and these again
than the chords of the Sixth and Third. On the other hand the minor chords of
the Sixth and Third are more agreeable than those in the fundamental position,
and these again are better than the minor chords of the Sixth and Fourth. This
conclusion will be found perfectly correct for the middle parts of the scale, pro-
vided the intervals are all justly intoned. The chords must be struck separately,
and not connected by any modulation. As soon as modulational connections
are allowed, as for example in a concluding cadence, the tonic feeling, which finds
repose in the tonic chord, disturbs the power of observation, which is here the
point of importance. In the lower parts of the scale either major or minor Thirds
are more disagreeable than Sixths.
Judging merely from the intervals we should expect that the minor triad
C B^Q would sound as well as the major C E Gj e^s each has a Fifth, a major
1 and a minor Third. This is, however, far from being the case. The minor triad
is very decidedly less harmonious than the major triad, in consequence of the
combinational tones, which must consequently be here taken into consideration.
In treating of the relative harmoniousness of the consonant intervals we have seen
that combinational tones may produce beats when two intervals are compounded,
even when each interval separately produced no beats at all, or at least none
distinctly audible (pp. 2oo2»-204&).
Hence we must determine the combinational tones of the major and minor
triads. We shall confine ourselves to the combinational tones of the first order
produced by the primes and the first upper partial tones. In the following
examples the primes are marked as minims, the combinational tones resulting
from these primes are represented by crotchets, those from the primes and first
upper partials by quavers and semiquavers. A downwards sloping line, when
placed before a note, shews that it represents a tone slightly deeper than that
H of the note in the scale which it precedes.
I.) Major Triads with their Combinational Tones :*
cents, the two ohords, A^b 386 C 386 JEr, and triads does not apply to tempered chords, in
C 386 ^1 386 O^ are seen to be identical, none of which are any of the intervals purely
but when the first is inverted to C 386 Ei consonant. — Translator,^
428^ 'bit becomes different from the other. * [As all the differentials most be harmonics
Both, however, remain harshly dissonant. On of C 66, if we represent this note by i, the
tempered instruments of course they become harmonics and hence differentials will 1^ be
identical C 400 E 400 OU , C 400 E 400 A b, and contained in the series
are very harsh. The definition of consonant
I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
C e g d d ^ ''Vb d* d' e" »/' f "a" '6"b ft" C"
First CAonZ.— The notes wiU then be 4, 5, 6, Second CAonl.— Notes 5, 6, 8 ; Octaves 10,
represented by minims, and their Octaves 8, 12, 16.
JO, 12, which are not given in notes. i) Crotchets, 6-5 = 1,8-6=2, 8-5« 3.
3-8=12-1
i) Crotchets, 5—4 =i 6-5 « I, 6— 4 = 2. 2) Quavers, 10-8=12-10=2, 12-8=4.
3) Quavers, 12— ID a 2, 8-5 » 3. 3) Semiquavers, 12-5 = 7, i6-6=io,(but
audible partial,) 16—5
Digitized by V^jOOQlC
3 -
4) Semiquavers, 12— 5 »= 7, 12-4 » 8. this is also an audible partial,) 16-5^11,
CHAT. zii. DIFFEBENGE BETWEEN MAJOB AND MINOB GHOBOS. 215
^P
^^ b^W— ^ '>mm\rjr
s
-J.
I
f
■'i-*
^
2.) Minor Triads with their Combinational Tones :*
$
bb J^ u"W-
^ , ,, ^^i/k^-
m
m
-mm.
^
^
J--t
^i^
In the major triads the combinational tones of the first order, and even the
deeper combinational tones of the second order (written as crotchets and quavers)
are merely doubles of the tones of the triad in deeper Octaves. The higher
combinational tones of the second order (written as semiquavers) are extremely
weak, because, other conditions being the same, the intensity of combinational
tones decreases as the interval between the generating tones increases, with which
again the high position of these combinational tones is connected. I have always f
whioh being more than half an equal Semi-
tone (51 cents) above equally tempered /' is
represented on the staff as a flattened /'8 •
Third C^ord.— Notes 6, 8, lo; Octaves
12, l6, 20.
i) Crotchets, io-8»8-6»2, io-6»4.
2) Qoavers, 12— io»2, 12—8-4.
3) Semiquavers, 20— 6 » 14.
How far these higher notes marked by
semiquavers are effective, except possibly
when they beat with each other, or with some
partials of the original notes, remains to be
ia34S6789
A„b A,b Eb Ab c eb ^gb ab bb
18 20 21 22 24 25 26 27 28
fb c" 'd"b "d^b e"b ^' >«/' f Yb
The omitted harmonics are not ased in this
investigation, though differentials of higher
orders occur up to the 48th harmonic.
First CAori.— Notes 10, 12, 15; Octaves
20, 24, 30.
i) Crotchets, 12— 10-2, 15— 1203, 15—
10 « 5.
2) Quavers, 20-15-5, 20-12 » 8, 24-15
3) Semiquavers, 24-10-14, 30-12-18,
30-10=20.
Second Chord, — Notes 12, 15, 20 ; Octaves
24, 30, 40.
1) Crotchets, 15-12 = 3, 20-15 = 5, 20-
12=^8.
2) Quavers, 24-20 = 4, 24-15 = 9, 30-20
-10.
3) Semiquavers, 30-12=18, 40-15 = 25,
40-12 = 28.
proved.— IZVarwZator.]
* [In minor chords the case is different.
On referring to the list of harmonics in the
last note, it will be seen that the only minor
chord is 10, 12, 15 or 0" &' 6", and this is the
chord upon the major Third above the third
Octave of the fundfunental. Hence in the
example where the chord taken is c' e'b gr'
and its inversions, the harmonics must be
formed on A„b which is the same interval
below &. The list of harmonics in these
examples is therefore
10
II 12
"d'b e'b
13 14 15 16
"/ Vb ff a!b
30 32 33 39 40
f a"b "a"b »V" d"
Third CAori.—Note8 15, 20, 24 ; Octaves ^
30, 40, 48.
1) Crotchets, 24-20-4, 20*15-5, 24—
1S-9.
2) Quavers, 30^24-6, 30-20=10, 40-
24=16.
3) Semiquavers, 40-1^-25, 48-20 = 28,
48 — i^ = 33. This I have nere represented as
"a"b because it is the Twelfth above "d'b, but
in the text it is called a flattened a" because it
is almost the one-sixth of C- 5 28. In fact on
the Harmonical, | x 528 = 880, and A„b would
^}'Cf/='5*33-'26-3, sothat 33 x 26-3 - 867*9
vibrations. The interval 880 : 867*9 ^^ ^
cents, and hence a" is more than a comma too
sharp. The same observation applies as in
the last footnote regarding the audible effect
of the high notes, when not beating with each
other, or with audible partiale.- 2Van32ator.]
Digitized by^OOQlC
2i6 DIFFEEENCE BETWEEN MAJOE AND MINOE CHOEDS. pam n.
been able to hear the deeper combinational tones of the second order, written as
quavers, when the tones have been played on an harmonium, and the ear was
assisted by the proper resonators :* but I have not been able to hear those written
with semiquavers. They have been added merely to make the theory complete.
Perhaps they might be occasionally heard from very loud musical tones having
powerful upper partials. But they may be certainly neglected in all ordinary
For the minor triads, on the other hand, the combinational tones of the first
order, which are easily audible, begin to disturb the harmonious effect. They are
not near enough indeed to beat, but they do not belong to the harmony. For the
fundamental triad, and that of the Sixth and Third [the two first chords], these
combinational tones, written as crotchets, form the major triad A\} C ^, and for
the triad of the Sixth and Fourth [the third chord], we find entirely new tones,
iT A\}9 Bj[}f which have no relations with the originaJ triad.t The combinational
tones of the second order, however (written as quavers), are sometimes partly
above and generally partly below the prime tones of the triad, but so near to them,
that beats must arise; whereas in the corresponding major triads the tones of
this order fit perfectly into the original chord. Thus for the fundamental minor
triad in the example, & e]} g\ the deeper combinational tones of the second order
give the dissonances a|7 ^ (/, and similarly for the triad of the Sixth and Third,
e]} g' c". And for the triad of the Sixth and Fourth g^ c" e"[> we find the disso-
nances B^ & and g' a]}. This disturbing action of the combinational tones on the
harmoniousness of minor triads is certainly too slight to give them the character
of dissonances, but they produce a sensible increase of roughness, in comparison
with the effect of major chords, for all cases where just intonation is employed,
that is, where the mathematical ratios of the intervals are preserved. In the
ordinary tempered intonation of our keyed instruments, the roughness due to the
^ combinational tones is proportionably less marked, because of the much greater
roughness due to the imperfection of the consonances. Practically I attribute
more importance to the influence of the more powerful deep combinational tones
of the first order, which, without increasing the roughness of the chord, introduce
tones entirely foreign to it, such as those of the A\} and ^ major triads in the case
of the C minor triads. The foreign element thus introduced into the minor chord
is not sufficiently distinct to destroy the harmony, but it is enough to give a
mysterious, obscure effect to the musical character and meaning of these chords,
an effect for which the hearer is unable to account, because the weak combinational
tones on which it depends are concealed by other and louder tones, and are audible
only to a practised ear.j: Hence minor chords are especially adapted to express
mysterious obscurity or harshness.f F. T. Vischer, in his Esthetics (vol. iii.
§ 772), has carefully examined this character of the minor mode, and shewn how
it suits many degrees of joyful and painful excitement, and that all shades of
^ feeling which it expresses agree in being to some extent ' veiled ' and obscure.
Every minor Third and every Sixth when associated with its principal com-
♦ [See note f on p. i^jd,— Translator.'] were chosen because the first Third in the
f [From the list of h&rmonics on p. 215c fundamental position is major in the first case
it will be seen that these tones occur as lower and minor in the second. In German the
harmonics of the tone whence the minor chords terms are dur and moZZ, that is, hard and
are derived. — Translator.] soft.] It is well known that the names dur
X [The Author is of course always speaking and moll are not connected with the hard or
of chords in just intonation. When tempered, soft character of the pieces of music written
as on the harmonium, even the major chords in these modes, but are historically derived
are accompanied by unrelated combinational from the angular form of Q and the rounded
tones, sufficiently dose to beat and sufficiently form of b i which were the B durum and B
loud for Scheibler to have laid down a rule molle of the medieval musical notation. [The
for counting the beats in order to verify the probable origin of the forms b B ^ 5 is ^ven
correctness of the tempered tuning (seep. 203(?). from observations on the plates in Gaf onus's
But still the different effects of the two chords Theoricum Opus Ilarmonicae DtscipUnae,
are very marked.— Translator.] 1480, the earliest printed book on music, in a
§ [The English names major and minor footnote, infri p. 31 2(Z. Translator,]
Digitized by V^OOQIC
4CHAP. XII. INVEBSIONS OP CHORDS. aiy
binational tone, becomes at onoe a major chord. C is the combinational tone of
the minor Third ef g^ ; o of the major Sixth g e\ and g of the minor Sixth.e' c".*
Since, then, these dyads naturally produce consonant triads, if any new tone is
Bdded which does not suit the triads thus formed, the contradiction is necessarily
sensible.
Modem harmonists are unwilling to acknowledge that the minor triad is less
consonant than the major. They have probably made all their experiments with
tempered instruments, on which, indeed, this distinction may perhaps be allowed
to be a Httle doubtful. But on justly intoned instruments f and with a moderately
piercing quality of tone, the difference is very striking and cannot be denied. The
old musicians, too, who composed exclusively for the voice, and were consequently
not driven to enfeeble consonances by temperament, shew a most decided feeling
for that difference. To this feeling I attribute the chief reason for their avoidance
of a minor chord at the close. The medieval composers dowd to Sebastian Bach ^
used for their closing chords either exclusively major chords, or doubtful chords
without the Third ; and even Handel and Mozart occasionally conclude a minor
piece of music with a major chord. Of course other considerations, besides the
degree of consonance, have great weight in determining the final chord, such as
the desire to mark the prevailing tonic or key-note with distinctness, for which
purpose the major chord is decidedly superior. More upon this in Chapter XY.
After having examined the consonant triads which lie within the compass of an
Octave, we proceed to those with wider intervals. We have found in general that
consonant intervals remain consonant when one of their tones is transposed an
Octave or two higher or lower at pleasure, although such transposition has some
effect on its degree of harmoniousness. It follows, then, that in all the consonant
chords which we have hitherto found, any one of the tones may be transposed
some Octaves higher or lower at pleasure. If the three intervals of the triad were
consonant before, they will remain so after transposition. We have already seen ^
how the chords of the Sixth and Third, and of the Sixth and Fourth, were thus
obtained from the fundamental form. It follows further that when larger inter-
vals are admitted, no consonant triads can exist which are not generated by the
transposition of the major and minor triads. Of course if such other chords could
exist, we should be able by transposition of their tones to bring them within the
compass of an Octave, and we should thus obtain a new consonant triad within
this compass, whereas our method of discovering consonant triads enabled us to
determine every one that could lie within that compass. It is certainly true that
slightly dissonant chords which lie within the compass of an Octave are sometimes
rendered smoother by transposing one of their tones. Thus the chord i : ^ : |^, or
C, ^^, ^-B[>, J is slightly dissonant in consequence of the interval i : J ; the
interval i : |^, or subminor Seventh, does not sound worse than the minor Sixth ;
the interval ^ : I is & perfect Fifth. Now transposing the tone ^-Et), an Octave
higher to ^4>> ^^^ ^^^ transforming the chord into i : |^ : ^, we obtain i : ^ in ^
place of I : ^, and this is much smoother, indeed it is better than the minor Tenth
of our minor scale i .' V)§ and a chord thus composed, which I have had carefully
tuned on the harmonium, although its unusual intervals produced a strange effect,
is not rougher in sound than the worst minor chord, that of the Sixth and Fourth.
This chord, C, ^^, ^e|7, is also much injured by the unsuitable combinational tones
Gf and F.** Of course it would not be worth while to introduce such strange
* (Tor if : ^^5 : 6, diif. 6— 5=» i or C; t [See these intervals examined in p. 195,
g : 6' = 3 : S, difl. S-3«-2 or c; e' : c"=S ; 8, note *.— Translator.]
diff. 8-5 = 3 o' g.— Translator,] § [The intervals 6 : 7 = y' : '^b, 3 : 7 = g
t See Copter XVI. for remarks upon {ast : '&'b, and $ : I2 = e: ff' can be tried and
and tempered intonation, and for a jastly in- compared on the HarmonicaL — TranslatorJ]
toned instrument suitable for such ezperi- ** [The ratios are 12 ; 21 : 28, and 21 — 12
ments. [The Harmonioal can also be used. ^9, but 9 : I2»3 : 4, hence if 12 is C, 9 is O^
See App. XX. sect. F. for this and other in- Again 28-12-16, 12 : 16 = 3 : 4 and hence
struments.] 16 is F.—Translator.]
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2l8
INVERSIONS OP CHORDS.
PABTII«
tones as ^J9[7, ^4>> ^^ ^^ ^^^ ^^^ ^^ ^^^ ^^ & chord which in itself is not
superior to the worst of our present consonant chords, and for which the tones
could not be transposed without greatly deteriorating its effect.*
The transposition of some tones in a consonant triad, for the purpose of widen-
ing their intervals, affects their harmoniousness in the first place by changing the
intervals. Major Tenths, as we found in Chapter X. p. 1956, sound better than
major Thirds, but minor Tenths worse than minor Thirds, the major and minor
Thirteenth worse than the minor Sixth (p. 196a). The following rule embraces ail
the cases : — Those intervals in which the smaller of the two nmnbers expressing the
ratios of the pitch mtmbers is even, are impbovbd by hamng one of their tones
transposed by an Octave, because the numbers expressing the ratio are thus
diminished.
The Fifth . .2:3 beoomea the Twa/th . . 2 : 6 » i : 3
I The major Third .4:5 „ mcyor Tenth . 4 : 10 = 2 : 5
The stibminor Third 6:7 „ subminor Tenth 6 : 14 » 3 : 7.
Those intervals in which the smaller of the two numbers expressing the ratio of
the vibrational numbers is odd, are made wobse by having one of their tones
transposed by an Octave, as the Fourth 3 : 4 [which becomes the Eleventh 3 : 8],
the minor Third 5 : 6 [which becomes the minor Tenth 5:12], and the Sixths
[major] 3:5, and [minor] 5 : 8 [which become the Thirteenths, major 3 : 10 and
minor 5 : 16].
Besides this the principal combinational tones are of essential importance.
An example of the first combinational tones of the consonant intervals within
the compass of an Octave is given below, the primary tones being represented
by minims and the combinational tones by crotchets, as before.f
Iktbbyal. Octave. Donbl. Oct Fifth. Twelfth. Fourth. Eleventh. Maj.Third. Magor Tenth.
Ratio. 4:8 2:8 ^ :6 4: 12 3:4 3:8 4:5 4: 10
DiFnBBNGS. 4628 15 16
Intebval. Min. Third. Minor Tenth. Mcj. Sixth. Maj. Thirteenth. Min. Sixth. Min. Thirteenth.
Ratio. 5:6 S : 12 3:5 3 '• lo 5:8 S : 16
DiVFSBBNCB. 1727 3 II
The upwards sloping line prefixed to /' denotes a degree of sharpening of about
a quarter of a Tone [53 cents] ; and the downwards sloping line prefixed to 61>
flattens it [by 27 cents] to the subminor Seventh of c. Below the notes are added
* [They are, however, insisted on by Poole,
see App. XX. sect. F. No. e.-^TranslatorJ]
t [Some of the bars and numbers have been
changed to make all agree with the footnote to
p. 21 4d. All these notes and their eombina-
tional notes can by this means be played on
the Harmonical.— Tran32a/or.J
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CHAP. xn.
INVERSIONS OF CHORDS.
919
the names of the intervals, the numbers of the ratios, and the differences of
those numbers, giving the pitch numbers of the several combinational tones.
We find in the first place that the combinational tones of the Octave, Fifth,
Twelfth, Fourth, and major Third are merely transpositions of one of the primary
tones by one or more Octaves, and therefore introduce no foreign tone. Hence these
five intervals can be used in all kinds of consonant triads, without disturbing the
effect by the combinational tones which they introduce. In this respect the major
Third is really superior to the major Sixth and the Tenth in the construction of
chords, although its independent harmoniousness is inferior to that of either.
The double Octave introduces the Fifth as a combinational tone. Hence if the
fundamental tone of a chord is doubled by means of the double Octave, the chord
is not injured. But injury would ensue if the Third or Fifth of the chord were
doubled in the double Octave.
Then we have a series of intervals which are made into complete major triads f
by means of their combinational tones, and hence produce no disturbance in
major chords, but are injurious to minor chords. These are the Eleventh, minor
Third, major Tenth, major Sixth, and minor Sixth,
But the minor Tenth, and the major and minor Thirteenth cannot form part of
a chord without injuring its consonance by their combinational tones.
We proceed to apply these considerations to the construction of triads.
I. Majob Tbiads.
Major triads can be so arranged that the combinational tones remain parts of
the chord. This gives the most perfectly harmonious positions of these chords.
To find them, remember that no minor Tenths and no [major or minor] Thirteenths
are admissible, so that the minor Thirds and [both major and minor] Sixths must
be in their closest position. By taking as the uppermost tone first the Third, then %
the Fifth, and lastly the fundamental tone, we find the following positions of these
chords, within a compass of two Octaves, in which the combinational tones (here
written as crotchets as usual) do not disturb the harmony.
The most Perfect Positiona of Major Triads.*
1234 56
When the Third lies uppermost, the Fifth must not be more than a major
Sixth below, as otherwise a [major] Thirteenth would be generated. But the fun-
damental tone can be transposed. Hence when the Third is uppermost the only
two positions which are undisturbed are Nos. i and 2. When the Fifth lies
uppermost, the Third must be inmiediately under it, or otherwise a minor Tenth
* [CalcQlatioii according to list of har*
monies p. 2i4d, footnote.
i) Chord 4, 6, 10. DifferentialB 6- 4 a a,
10— 6^4, 10-4^6, which is also one of the
tones.
2) Chord 6, 8, 10. Differentials 8-6 »
10— 8b2, 10-6=4.
3) Chord 4, 10, 12. Differentials 12-10
■•2, io~4 = 6, i2-4=-8.
4) Chord 8, 10, 12. Differentials I0-8
eI2-IO»2, 12 — 8-4.
5) Chord 3, 5, 8. Differentials 5-3-2,
8 — 5 = 3, (which is also one of the tones,) 8—3
a 5, (which is also one of the tones).
6) Chord 5, 6, 8. Differentials 6-5 = 1,
8-6 = 2,8-5-3.
These chords should be studied on the
Harmonical, and the combinational tones lis-
tened for, and afterwards the tones played as
substantive notes.— 2Van^2a^.J
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220
INVEESIONS OP CHOEDS-
PAET n.
would be produced ; but the fundamental tone may be transposed. Finallj, when
the fundamental tone is uppermost, the major Third can lie only in the position of
a minor Sixth below it, but the Fifth maybe placed at pleasure. Hence it follows
that the only possible positions of the major chord which will be entirely free firom
disturbance by combinational tones, are the six here presented, among which we
find the three chse positions Nos. 2, 4, 6 ahready mentioned [p. 215a], and three
new ones Nos. i, 3, 5. Of these new positions two (Nos. i, 3) have the funda-
mental tone in the bass, just as in the primary form, and are considered as open
positions of that form, while the third (No. 5) has the Fifth in the bass, just as in the
chord of the Sixth and Fourth [of which it is also considered as an open position].
The chord of the Sixth [and Third] (No. 6), on the other hand, admits of no opener
position [if it is to remain perfectly free from combinational disturbance].
The order of these chords in respect to harmoniousness of the intervals is»
^ perhaps, the same as that presented above. The three intervals of No. i (the
Fifth, major Tenth, and major Sixth) are the best, and those of No. 6 (the Foorth,
minor Third, and minor Sixth) are relatively the most unfiavouiable of the
intervals that occur in these chords.
The remaining positions of the major triads present individual unsuitable com-
binational tones, and on justly intoned instruments are unmistakably rougher than
those previously considered, but this does not make them dissonant, it merely puts
them in the same category as minor chords. We obtain all of them which lie
within the compass of two Octaves, by making the transpositions forbidden in the
last cases. They are as follows, in tiie same order as before, No. 7 being made
from No. i, and so on : —
The less Perfect Positions of Major Triads,"*
89 10 II
Musicians will immediately perceive that these positions of the major triad are
much less in use. The combinational tone ^61>, gives the positions 7 to 10 some-
* [Calculation in oontinaation of the last
note.
7) Chord 3, 4, 10. DiflerentialB 4-3*1,
io-4«6, io-3a-7.
% 8) Chord 3, 8, 10. Differentials 10-8-2,
8-3«5, 10—3 = 7, which gives the interval
7 : 8 with the tone 8.
9) Chord 4, 5, 12. Differentials 5— 4^1,
12— 4^8, 12—5 = 7, the two last differential
tones being 7 : 8.
10) Chord 5, 8, 1 2. Differentials 8-5 = 3,
12—804, 12 — 5 = 7, which gives the interval
7 : 8 with the tone 8.
11) Chord 5, 6, 16. Differentials 6-5=1,
16-6=10, 16-5=11, which two lastform the
dissonant trumpet interval 1 1 : lo of 165 cents
or about three-quarters of an equal tone.
12) Chord 5, 12, 16. Differentials 16-12
»4, 12—5*7, 16—5 = 11, which forms the
same dissonant trumpet interval 11 : 10, but
this time with one of the tones, and therefore
more harshly.
AU these 12 chords should be well studied
on the Harmonical, and for the first 10, the
differential tones can be played also as sub-
stantive notes (remembering that ^Bbison the
Ob digital), which will enable the student to
acquire a better idea of the roughness. The
tones 1 1 and 13 could not be introduced among
the first 4 Octaves on the Harmonical with-
out incurring the important losses of /' and
a". But if we take the chords an Oetave
higher we can play "/" and > V.
The chords should also be played in lower
and higher positions, not only as Octaves of
those given, but from the other major chords
on the Harmonical as FA^Ct OB^D, A^bGJE^b^
E^b OB^b. Particular attention should be
paid to the contrasting of the positions i and 7,
2 and 8, 3 and 9, 4 and 10, 5 and 11, 6 and
12. Unless the ear acquires the habit of
attending to these differences it wiU not pn>.
perly form the requisite conceptions of major
chords. For future purposes the results should
also be contrasted with those obtained by play*
ing the same chords on a tempered instrument,
—if possible of the same pitch, A 44a — Trans-
latoTi]
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OHAP. xn.
INVEESIONS OF CHOBDS.
221
thing of the character of the chord of the dominant Seventh in the key of F
major, ceg}^. The two last, 1 1 and 1 2, are much the least pleasing ; indeed they
are decidedly rougher than the better positions of the minor chord.
2. MiNOB Tbiadb.
No minor chord can be obtained perfectly free from fiedse combinational tones,
because its Third can never be so placed relatively to the fundamental tone, as not
to produce a combinational tone unsuitable to the minor chord. If only one such
tone is admitted, the Third and Fifth of the minor chord must lie close together
and form a major Third, because in any other position they would produce a second
unsuitable combinational tone. The fundamental tone and the Fifth must never
be so placed as to form an Eleventh, because in that case the resulting combina-
tional tone would make them into a major triad. These conditions can be fulfilled 1[
by only three positions of the minor chord, as follows : —
The most Perfect Positions of Minor Triads.*
123
1^
4=4
^
^S
t:
The remaining positions which do not sound so well are : —
The less Perfect Positions of Minor Triads.f
5 678 9 lo
--1^^!t^._ ./^ |.J __isi.
* [Calculation according to the list of har-
monies on p. 215c, footnote.
i) Chord 24, 30, 40. Dijfferentials 30-24
» 6, 40— 30 s 10, 40— 24 B 16.
2) Chord 20, 24,30. Differentials 24-20
-4» 30-248-6, 30-20^10.
3) Chord 10, 24, 30. Differentials 30-24
-6, 24-io»i4, 30-10-20.
These can also be stndied on the Har-
monical, and the differentials to Nos. i and 2
can be played as substantive tones. Not so
No. 3, but the effect may be felt by playing the
chord a major Third higher as eg'^, being the
10, 24, 30 harmonics of C,. and giving the dif-
ferentials O, ^bb, ef which can be played as
sabstantive tones, bnt being so low will make
the effect very rough. — Translator,']
t [Calculation in continuation of the last
note.
4) Chord 12, 15, 40. Differentials 15-12
«=3» 40- 15 = 25, 40- 12 = 28.
5) Chord 12, 30, 40. Differentials 40-30
» 10, 40-12=18, 40-12=28. m
6) Chord 15, 20, 24. Differentials 24-20 ''
-4,20-15 = 5,24-15 = 9.
7) Chord 12, 20, 30. Differentials 20-12
»8, 30— 20» 10, 30— 12= 18, where 18 forms
the dissonance 20 : 18= 10 : 9 with the tone
20.
8) Chord 10, 15,24. Differentials 15-10
= 5, 24— 15 = 9, 24-10=14, which forms the
dissonant interval 15 : 14 with one of the tones
15-
9) Chord 10, 12, 30. Differentials 12—10
B 2, 30— 12 = 18, 30— 10= 20, the two last form
together the dissonance 20 : 18= 10 : 9.
10) Chord 15, 20, 48 referred to A„,b . In-
terpret by taking the Octaves below tne num-
bers in p. 215c, note. Differentials 20—15 =
5 = C; 48-20 = 28=yb. 48- 15 = 33 = "«'*»,
see p. 2i$d\ note, towards the end of the ob-
servations on the Third Chord. ^
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222 INVEESIONS OP CHOEDS. paetil
The positions Nos. 4 to lo each produce two unsuitable combinational tones,
one of which necessarily results from the fundamental tone and its [minor] Third ;
the other results in No. 4 from the Eleventh O C, and in the rest horn the trans-
posed major Third E^ G. The two last positions, Nos. 11 and 12, are the worst
of all, because they give rise to three unsuitable combinational tones [two of which
beat with original tones].
The influence of the combinational tones may be recognised by comparing the
different positions. Thus the position No. 3, with a minor Tenth c' e"[> and major
Third e'^ g^', sounds unmistakably better than the position No. 7, with major
Tenth e^ g" and major Sixth e^ c", although the two latter intervals when struck
separately sound better than the two first. The inferior effect of chord No. 7 is
consequently solely due to the second unsuitable combinational tone, b]}.
This influence of bad combinational tones is also apparent from a comparison
f with the major chords. On comparing the minor chords Nos. i to 3, each of which
has only one bad combinational tone, with the major chords Nos. 11 and 12, each
of which has two such tones, those minor chords will be found really pleasanter
and smoother than the major. Hence in these two classes of chords it is not the
major and minor Third, nor the musical mode, which decides the degree of harmo-
niousness, it is wholly and solely the combinational tones.
Four Part Chords or Tetrads.
It is easily seen that all consonant tetrads must be either major or minor triads
to which the Octave of one of the tones has been added.* For every consonant
tetrad must admit of being changed into a consonant triad by removing one of its
tones. Now this can be done in four ways, so that, for example, the tetrad G E O c
gives the four following triads : —
IF C E G, CEc, E Gc, C G c.
Any such triad, if it is not merely a dyad, or interval of two tones, with the
Octave of one added, must be either a major or a minor triad, because there are
no other consonant triads. But the only way of adding a fourth tone to a major
or minor triad, on condition that the result should be consonant, is to add the
Octave of one of its tones. For every such triad contains two tones, say G and G,
which form either a direct or inverted Fifth. Now the only tones which can be
combined with G and (? so as to form a consonance are E and E^; there are no
others at all. But E and E^ cannot be both present in the same consonant chord,
11) Chord 15, 40, 48 referred to ^^^b ae in 440 : 433*95 or 24 cents, rather more than a
last chord. comma.
Differentials 48 — 40 » 8 = il b, 40 — 1 5 - 25 The student should try all the minor chords
se\ 48- i5 = 33-"a'b as in last chord, which not only in different positions in Octaves, but
Bee. with all the other minor chords on tibe Har-
U 12) Chord 15, 24, 40 referred to A,„b> monical, namely, FA^bC, OB^bD, D^FA^
Differentials 24— 15- 9 = £b, 40— 24»i6»ab, (which contrast with the dissonance DFA^
40— 15 = 25 = 6' where the differentUls 16, 25 for future purposes), AfiE^t JS7,GB„ also in
form the dissonant intervals 16 : 15, 25 : 24 different Octaves, till the ear learns to distin-
with the two tones 15 and 24 respectively. All guish these 12 different forms,
these chords can be studied on the Harmonical, Finally the 12 forms of the major should
and their differentials can be played as sub- be contrasted with the corresponding 12 forma
stantive tones in Nos. 6, 7, and 12. No. 8 can of the minor triad, for the three possible cases
be taken a major Third higher as in chord FA^C and FA^bC; CE^Q and CE^bG ;
No. 3 of the last note, that is as ef b' g" giving GB^D and GB'b-D. To merely read over these
the differentials e, d, 'bb which can be played. pages by eye instead of studying them by ear
Also No. 9 may be played as e' g' 6" giving dif- is useless, and ordinary tempered instruments
ferentials c, d", e". Nos. 4 and 5 do not admit only impede instead of assisting the investi-
of such treatment because e"'b is not on the gator,— Translator,]
instrument. Nos. 10 and 1 1 cannot be so played * [That is, if we exclude the harmonic
because "a'b is not on the instrument. In Seventh from consideration, as on p. 195^,
fact it is the 33rd harmonic of il,,;b" 13*1 5t and those who admit it (as Mr. Poole, App. X.X.
this (see footnote p. 215^', remarks on Third sect. F. No. 6) consider CE^Q'Bb to be a per-
CJujrdf) -33 X 1 3* 15 « 433*95 vib. ; whereas a= fectly consonant tetrad.— Trans totor.]
440, and hence is too sharp by the interval
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OHAP. XII.
INVERSIONS OP CHORDS-
225
and hence every consonant chord of four or more parts, which contains C and G,
must either contain E and some of the Octaves of (7» E, G, or else Ej[} and some
of the Octaves of C, ^, O.
Every consonant chord of three or more parts will therefore he either a major
or a minor chord, and maj be formed firom the fimdamental position of the major
and minor triad, by transposing or adding the Octaves above or below some or all
of its three tones.
To obtain the perfectly harmonious positions of major tetrads, we have again
to be careful that no minor Tenths and no [major or minor] Thirteenths occur.
Hence the Fifth may not stand more than a minor Third above, or a Sixth below
the Third of the chord ; and the fundamental tone must not be more than a
Sixth above the Third. When these rules are carried out, the avoidance of the
minor Thirteenths is effected by not taking the double Octave of the Third and
Fifth. These rules may be briefly enunciated as follows : Those major chords are %
most harmonious in which the fundamental tone or the Fifth does not Ue more than
a Sixth above the Third, or the Fifth does not lie more than a Sixth above or
below it. The fundamental tone, on the other hand, may be as far below the Third
as we please.
The corresponding positions of the major tetrads are found by combining any
two of the more perfect positions of the major triads which have two tones in
common, as follows, where the lower figures refer to the positions of the major
triads ahready given.
The most Perfect Positions of Major Tetrads within the Compass of
Two Octaves,*
a
^
IS
s
m-^
Sf
m
&
?2=
S2I
1+2 1+3 1+4 1 + 5 2 + 4 2 + 5 2 + 6 3 + 4 3 + 6 4 + 6 5 + 6
We see that chords of the Sixth and Third must lie quite close, as No. 7 ; t
and that chords of the Sixth and Fourth t must not have a compass of more than
an Eleventh, but may occur in all the three positions (Nos. 5, 6, 11) in which it
can be constructed within this compass. Chords which have the fundamental tone
in the bass can be handled most freely.
It will not be necessary to enumerate the less perfect positions of major tetrads, ^r
They cannot have more than two unsuitable combinational tones, as in the 1 2th
position of the major triads, p. 220c. The major triads of C can only have the
false combinational tones marked ^6(^ and ^y, [that is, with pitch numbers bearing
to that of C the ratios 7 : i, or 11 : i].
Minor tetrads, like the corresponding triads, must at least have one fedse com-
binational tone. There is only one single position of the minor tetrad which has
only one such tone. It is No. i in the following example, and is compounded of
the positions Nos. i and 2 of the minor triads on p. 2216. But there may be as
* [These major tetrads can all be played on
the Harmonioal, and should be tried in every
-position of Octaves and for all the major chords
on the instrument, namely FA^C, CE^Q,
OBxD.A^bCE^b, ^'bGB*b, tiU the ear is
perfectly familiar with the different forms and
the student can tell them at once and desig-
nate them by their number in this list on hear-
ing another person play them. — Translator.]
t [This chord has the Third both lowest and
highest and is marked , but is more com-
monly marked 6. — Translator.]
X [These chords have the Fifth lowest and
are marked ^. — Translator,]
4
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224
INVEB8I0N8 OP CHORDS.
PABT H;)
many as 4 false combinational tones, as, for example, on eombining positicnis
Nos. 10 and 11 of the minor triads, p. 221c.
Here follows a list of the minor tetrads which have not more than two false
combinational tones, and which lie within the compass of two Octaves. The Mse
combinational tones only are noted in crotchets, and those which suit the chord-
are omitted.
Best Positions of Minor Tetrads.*
3 4 567"
1 + 2
1 + 3
1 + 7
2+3 2+6 2+7
2 + 9
3+S 1+6
The chord of the Sixth and Fourth [marked n occurs only in its closest posi-
tion, No. 5 ; but that of the Sixth and Third [marked ] is found in three positions
(Nos. 3, 6, and 9), namely, in all positions where the compass of the chord does
not exceed a Tenth ; the fundamental chord occurs three times with the Octave of
the fundamental note added (Nos. i, 2, 4), and twice with the Octave of the Fifth
added (Nos. 7, 8).
In musical theory, as hitherto expounded, very little has been said of the
IT influence of the transposition of chords on harmonious efifect. It is usual to give
as a rule that close intervals must not be used in the bass, and that the intervals
should be tolerably evenly distributed between the extreme tones. And even these
rules do not appear as consequences of the theoretical views and laws usually given,
according to which a consonant interval remains consonant in whatever part of the
scale it is taken, and however it may be inverted or combined with others. They
rather appear as practical exceptions from general rules. It was left to the
musician himself to obtain some insight into the various effects of the various
positions of chords by mere use and experience. No rule could be given to guide
him.
The subject has been treated here at such length in order to shew that a right
view of the cause of consonance and dissonance leads to rules for relations which
* previous theories of harmony could not contain. The propositions we have ennn-
^ * [Calcnlation of the combinational tones,
by the list of harmonics in p. 215c.
i) Chord 20, 24, 30, 40. Differentials
24-2o = 4 = ilb, 40--24« i6=sa'b.
2) Chord 10, 24, 30, 40. Differentials
24-10* 14 = Vb» 40-24= i6=a'b.
3) Chord 12, 15, 20, 30. Differentials
20- i2 = 8«ab, 30~i2 = i8s=6'b.
4) Chord 10, 20, 24, 30. Differentials
24-20 = 4 =ilb, 24-10= I4«yb.
5) Chord 15, 20, 24, 30. Differentials
24-20 = 4 = i4b, 24-i5«9=6b.
6) Chord 12, 20, 24, 30. Differentials
24-20=4a^b, 20-i2^8=ab, 30-12=18
= 6'b.
7) Chord 10, 12, 15, 30. Differentials
i2-io=2-il,b, 30— I2 = i8-6'b.
8) Chord 10, 15, 24, 30. Differentials
24-i5-9 = bb,24-io«i4«yb.
9) Chord 12, 15, 20, 24. Differentials
24-20=4=ilb, 20-i2"8=ab, 24-15 = 9
-6b.
These chords should all be studied on the
Harmonical. With the exception of Nos. 2, 4,
7, 8 the differentials can also be played on it
as sabstantive tones. Bnt they can be trans-
posed. Thus No. 2 may be played as « (t' 2/ e"
giving the differentials ^&b, </. No. 4 will be-
come e^ ef' g" h" giving the differential 'b'b,
which can be playeid. No. 7 becomes e' ^ b' b"
giving the differentials C and d". No. 8 be-
comes ef b' ^' b" giving the differentials d* and
'&'b . These chords ^onld also be studied in
aU the minor forms on the Harmonical, not
only in different Octaves, but on all the minor
chords on that instrument, viz. D, jP ^1 „ ^ , C JEr „
JB, G B„ F 4> bC, C iP b (?. (? B» bD. tiU the ear
recognises the form, and the student can name
the number of the position to another person's
playing. — TnwMZafor.J
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CHAP. XII. INVERSIONS OF CHOEDS. 225
<siated agree, however, with the practice of the best composers, of those, I mean,
who studied vocal music principally, before the great development of instrumental
music necessitated the general introduction of tempered intonation, as any one
may easily convince himself by examining those compositions which aimed at
producing an impression of perfect harmoniousness. Mozart is certainly the com-
poser who had the surest instinct for the dehcacies of his art. Among his vocal
compositions the Ave verum corpus is particularly celebrated for its wonder-
fully pure and smooth harmonies. On examining this little piece as one of the
most suitable examples for our purpose, we find in its first clause, which has
an extremely soft and sweet effect, none but major chords, and chords of the
dominant Seventh. All these major chords belong to those which we have noted
as being the more perfectly harmonious. Position 2 occurs most frequently, and
then 8, 10, i, and 9 [of p. 223c]. It is not till we come to the final modulation of
this first clause that we meet with two minor chords, and a major chord in an f
tmfjEivourable position. It is very striking, by way of comparison, to find that the
second clause of the same piece, which is more veiled, longing, and mystical, and
laboriously modulates through bolder transitions and harsher dissonances, has
many more minor chords, which, as well as the major chords scattered among
them, are for the most part brought into unfavourable positions, until the final
chord again restores perfect harmony.
Precisely similar observations may be made on those choral pieces of Palestrina,
and of his contemporaries and successors, which have a simple harmonic construc-
tion without any involved polyphony. In transforming the Roman church music,
which was Palestrina's task, the principal weight was laid on harmonious effect in
contrast to the harsh and unintelhgible polyphony of the older Netherland * system,
and Palestrina and his school have really solved the problem in the most pei^ect
manner. Here also we find an almost uninterrupted fiow of consonant chords, with
some dominant Sevenths, or dissonant passing notes, charily interspersed. Here f
Also the consonant chords wholly, or almost wholly, consist of those major and
minor chords which we have noted as being in the more perfect positions. Only
in the final cadence of a few clauses, on the contrary, in the midst of more powerful
And more frequent dissonances, we find a predominance of the unfavourable posi-
tions of the major and minor chords. Thus that expression which modem music
endeavours to attain by various discords and an abundant introduction of dominant
Sevenths, was obtained in the school of Palestrina by the much more dehcate
shading of various inversions and positions of consonant. chords. This explains the
harmoniousness of these compositions, which are nevertheless full of deep and
tender expression, and sound Uke the songs of angels with hearts affected but
undarkened by human grief in their heavenly joy. Of course such pieces of music
require fine ears both in singer and hearer, to let the delicate gradation of expres-
sion receive its due, now that modem music has accustomed us to modes of
expression so much more violent and drastic. ^
The great majority of major tetrads in Palestrina's Stabat mater are in the
positions i, 10, 8, 5, 3, 2, 4, 9 [of p. 223c], and of minor tetrads in the positions
9, 2, 4, 3, 5, I [of p. 224a]. For the major chords one might almost think that
some theoretical rule led him to avoid the bad intervals of the minor Tenth and
the [major or minor] Thirteenth. But this rule would have been entirely useless
for minor chords. Since the existence of combinational tones was not then
known, we can only conclude that his fine ear led him to this practice, and that
the judgment of that ear exactly agreed with the mles deduced from our theory.
These authorities may serve to lead musicians to allow the correctness of my
arrangement of consonant chords in the order of their harmoniousness. But
any one can convince himself of their correctness on any justly intoned instrument
* [Including both the modern kingdom of 1532, was born in Hainault in the present
the Netherland(«.or Holland, and the still more Belgium.— Translator.]
modem kingdom of Belgium. Josquin, 1450-
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2 26 KETEOSPECT. part n.
(as the Hannonical]. The present system of tempered intonation certainly oblite-
rates somewhat of the more delicate distinctions, without, however, entirely
destroying them.
Having thus concluded that part of our investigations which rests upon purely
BcientiQc principles, it will be advisable to look back upon the road we have travelled
in order to review our gains, and examine the relation of our results to the views of
older theoreticians. We started &om the acoustical phenomena of upper partial tones,
combinational tones and beats. These phenomena were long well known both to
musicians and acousticians, and the laws of their occurrence were, at least in their
essential features, correctly recognised and enunciated. We had only to pursue
these phenomena into further detail than had hitherto been done. We succeeded
in finding methods for observing upper partial tones^ which rendered comp^^tively
f easy an observation previously very difficult to make. And with the help of this
method we endeavoured to shew that, with few exceptions, the tones of all musical
instruments were compounded of partial tones, and that, in especial, those qualities
of tone which are more particularly f&vourable for musical purposes, possess at
least a series of the lower partial tones in tolerable force, while the simple tones,
like those of stopped organ pipes, have a very unsatisfactory musical effect,
although even these tones when loudly sounded are accompanied in the ear itself
by some weak harmonic upper partials. On the other hand we found that, for the
better musical qualities of tone, the higher partial tones, from the Seventh onwards,
must be weak, as otherwise the quahty, and every combination of tones would be
too piercing. In reference to the beats, we had to discover what became of them
when they grew quicker and quicker. We found that they then fell into that
roughness which is the peculiar character of dissonance. The transition can be
effected very gradually, and observed in all its stages, and hence it is apparent to
% the simplest natural observation that the essence of dissonance consists merely* in
very rapid beats. The nerves of hearing feel these rapid beats as roiigh and
unpleasant, because every intermittent excitement of any nervous apparatus affects
us more powerfully than one that lasts unaltered. With this there is possibly
associated a psychological cause. The individual pulses of tone in a dissonant
combination give us certainly the same impression of separate pulses as slow beats,
although we are unable to recognise them separately and count them ; hence they
form a tangled mass of tone, which cannot be analysed into its constituents. The
cause of the unpleasantness of dissonance we attribute to this roughness and
entanglement. The meaning of this distinction may be thus briefly stated : Con-
sonance is a contintious, dissonance an intermittent sensation of tone. Two con-
sonant tones flow on quietly side by side in an undisturbed stream ; dissonant
tones cut one another up into separate pulses of tone. This description of the
distinction at which we have arrived agrees precisely with Euclid's old definition,
f ' Consonance is the blending of a higher with a lower tone. Dissonance is
incapacity to mix, when two tones cannot blend, but appear rough to the ear.' f
After this principle had been once established there was nothing further to do
but to inquire under what circumstances, and with what degree of strength, beats
* [Bat Bee also Prof. Preyer, in App. XX. Fifth, and Fourth) he felt that the tooBs
sect. L. art. 7, infra. — Tra/nslator,] bUndecL But the Sio^wWa (which he applies
t EucUdeSt ed. Meibomius, p. 8 : lS<m 8^ to all other intervals, for he used Pythagorean
ffvfiipc»pta fikp Kpatris S6o ^6yy»Pf h^vrdpov Kot major and minor Thirds, which are really dis>
fiafivT4pov. Atai^yla Sh rohvarriov S^fo ^e6r/ymv sonant) he found to consist in their not eTea
hfu^ia, m4 oimv re icpa^reu, kKXh. rpaxwBfiycu mMng, not even forming a mechanical, mach
rhy &Ko^v. [In translating this passage in the less a chemical unit, so that he goes on to ex-
text, I have endeavoured to make the distinc- plain that this non-mixing of the two tones
tion of fi^is and tcpavu ; the former is taken to consisted in inability to blend, and resulted in
be of the nature of a mechanical, and the producing a roughneast as contradistinguished
latter a ehamical mixture. Mixing and blenui' from a bUnding in the ear. The tones are
ing seem to convey the notion. In mtfi^via ^6yyoi, properly tones sung, but used even for
(which Euclid admitted only for the Octave, tones of the lyre,— Translator.]
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CHAP. XII. RETROSPECT. 227'
would arise in the various combinations of tones through either the partial or tlie
combinational tones. This investigation had hitherto been completely worked out
by Scheibler for the combinational tones of two simple tones only. The law of
beats being known, it became easy to extend it to compound tones. Every
theoretical conclusion on this field can be immediately checked by a proper
observation, when the analysis of a mass of tone is facilitated by the use ofi
resonators. All these beats of partial and combinational tones, of which so much )
has been said in the last chapter, are not inventions of empty theoretical specula-
tion, but rather facts of observation, and can be reaUy heard without difficulty by I
any practised observer who performs his experiments correctly. The knowledge
of the acoustic law facilitates our discovery of the phenomena in question. But
all the assertions on which we depend for establishing a theory of consonance and
dissonance, such as was given in the last chapters, are founded wholly and solely
on a careful analysis of the sensations of hearing, an analysis which a practised f
ear could have executed without any theoretical assistance, although of course
the task was immensely fe^^ilitated by the guidance of theory and the assistance of
appropriate instruments of observation.
For these reasons the reader is particularly requested to observe that my hypo-
thesis concerning the sympathetic vibration of Corti's organs inside the ear has no
inmiediate connection whatever with the explanation of consonance and dissonance.
That explanation depends solely upon observed facts, on the beats of partial tones
«nd the beats of combinational tones. Yet I thought it right not to suppress my
hypothesis (which must of course be regarded solely as an hypothesis), because it
gathers all the various acoustical phenomena with which we are concerned into
one sheaf, and gives a clear, intelligible, and evident explanation of the whole
phenomena and their connection.
The last chapters have shewn, that a correct and careful analysis of a mass of
sound under the guidance of the principles cited, leads to precisely the same dis- ^
tinctions between consonant and dissonant intervals and chords, as have been
established under the old theory of harmony. We have even shewn that these
investigations give more particular information concerning individual intervals
and chords than was possible with the general rules of the former theory, and
that the correctness of these rules is corroborated both by observation on justly
intoned instruments and the practice of the best composers.
Hence I do not hesitate to assert that the preceding investigations, founded
upon a more exact analysis of the sensations of tone, and upon purely scientific,
as distinct firom esthetic principles, exhibit the true and sufficient cause of conso«
nance and dissonance in music.
One circumstance may, perhaps, cause the musician to pause in accepting
this assertion. We have found that firom the most perfect consonance to the
most decided dissonance there is a continuous series of degrees, of combinations of
sound, which continually increase in roughness, so that there cannot be any sharp ^
line drawn between consonance and dissonance, and the distinction would therefore
seem to be merely arbitrary. Musicians, on the contrary, have been in the habit
of drawing a sharp line between consonances and dissonances, allowing of no
intermediate hnks, and Hauptmann advances this as a principal reason against
any attempt at deducing the theory of consonance from the relations of rational
numbers.*
As a matter of fEMst we have already remarked that the chords of the natural
* Harmomk und Meirik, p. 4. [At the the sustained tones of the voice for example,
same time, by accepting equal tonperament grossly dissonant. It is difficult for any ear
they accept as consonant a series of tones brougnt op among these dissonances, to under-
which really form only one consonant interval stand the real distinction between consonance
(the Octave) and only two others even approzi- and dissonance. Hence the absolute necessity
matively consonant (the Fifth and Fourth), of testing all the above assertions by a justly
while the commonest intervals on which har- intoned instrument such as the Harmonical. —
mony rests, the Thirds, with their inversions Translator.]
the Sixths, are not merely dissonant but, on
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228 RETROSPECT. part lu
or subminor Seventh 4 : 7 [c' to ''b]} on the Harmonical], and of the subminor
Tenth 3:7 [g to ^6'[>] in many qualities of tone sound at least as well as the
minor Sixth 5 : 8 [e' to c'^]^ and that the subminor Tenth really sounds better
than the ordinary minor Tenth 5:12 [«' to g^']. But we have already noticed a
cLrcumstauce of great importance for musical practice which gives the minor Sixth
an advantage over the intervals formed with the number 7. The inversion of the
minor Sixth gives a better interval, the major Third [e^ c" inverted gives c' e'], and
its importance as a consonance in modem music is especially due to this very
relation to the major Third ; it isT essentially necessary^ and justified, just because
it is the inversion of the major Third. On the other hand the inversion or trans-
position of an interval formed with the number 7 leads to intervals worse than
itself. Hence, as it is necessary, for the purposes of harmony, to have the power
of transposing the parts at pleasure, we have a sufficient reason for drawing the
9 line between the minor Sixth on the one hand, and the intervals characterised
by 7 on the other. It is not, however, till we come to construct scales, which we
shall have to consider in the next chapter, that we find decisive reasons for making
this the boundary. The scales of modem music cannot possibly accept tones
determined by the number 7.* But in musical harmony we can only deal with
chords formed of notes in the scale. Intervals characterised by 5, as the Thirds
and Sixths, occur in the scale, as well as others characterised by 9, as the major
Second 8 : 9, but there are none characterised by 7, which should form the tran-
sition between them. Here, then, there is a real gap in the series of chords arranged
according to the degree of their harmonious effect, and this gap serves to determine
the boundary between consonance and dissonance.
The decision does not depend, then, on the nature of the intervals themselves
but on the construction of the whole tonal system. This is corroborated by the
fact that the boundary between consonant and dissonant intervals has not been
f always the same. It has been already mentioned that the Greeks always repre*
sented Thirds as dissonant, and although the original Pythagorean Third 64 : 81,
determined by a series of Fifths, was not a consonance, yet even when the natural
major Third 4 : 5 was afterwards included in the so-called syntono-diatonic mode
of Didymus and Ptolemaeus, it was not recognised as a consonance. It has
already been mentioned that in the middle ages, first the Thirds and then tlie
Sixths were acknowledged as imperfect consonances, that the Thirds were long
omitted from the final chord, and that it was not till later that the major, and
quite recently the minor Third was admitted in this position. It is quite a mis-
take to suppose, with modem musical theorists, that this was merely whimsical
and unnatural, or that the older composers allowed themselves to be fettered by
blind faith in Greek authority. The last was certainly partly tme for writers on
musical theory down to the sixteenth century. But we must distinguish carefully
between composers and theoreticians. Neither the Greeks, nor the great musical
m composers of the sixteenth and seventeenth centuries, were people to be blinded by
a theory which their ears could upset. The reason for these deviations is to be
looked for rather in the difference between the tonal systems in early and recent
times, with which we shall become acquainted in the next part. It will there be
seen that our modem system gained the form under which we know it through the
influence of a general use of harmonic chords. It was only in this system that a
complete regard was paid to all the requisitions of interwoven harmonies. Owiiig
to its strict consistency, we were not only able to allow many licences in the use
of the more imperfect consonances and of dissonances, which older systems had to
avoid, but we were often required to insert the Thirds in final chords, as a mode
of distinguishing with certainty between the major and. minor mode, in cases
where tliis distinction was formerly evaded.
* [Poole's scale / 9 a, '5b c' d' e\ /, and monio, which is the only aconstical jastiftcation
Bosanquet's and White's tempered imitation for the greatly harsher dominant Seventh. —
of 'fc'b, properly 969 cents, as 974 cents, shew Translator.'\
the fdeluig that exists for using the 7th har-
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CHAP. XII. RETROSPECT, 329
But if the bonndary between consonance and dissonance has really changed
with a change of tonal system, it is manifest that the reason for assigning this
boundary does not depend on the intervals and their individual musical effect, but
on the whole construction of the tonal system.
The enigma which, about 2500 years ago, Pythagoras proposed to science,
which investigates the reasons of things, ' Why is consonance determined by the
ratios of small whole numbers ? ' has been solved by the discovery that the ear
resolves all complex sounds into pendular oscillations, according to the laws of
sympathetic vibration, and that it regards as harmonious only such excitements of
the nerves as continue without disturbance. The resolution into partial tones,
mathematically expressed, is effected by Fourier's law, which shews how any
periodically variable magnitude, whatever be its nature, can be expressed by a
sum of the simplest periodic magnitudes.* The length of the periods of the
simply periodic terms of this sum must be exactly such, that either one or two ^
or three or four, and so on, of their periods are equal to the period of the
given magnitude. This, reduced to tones, means that the pitch numbers of the
partial tones must be exactly once, twice, three times, four times, and so on,
respectively, as great as that of the prime tone. These are the whole numbers
which determine the ratios of the consonances. For, as we have seen, the con-
dition of consonance is that two of the lower partial tones of the notes combined
shall be of exactly the same pitch ; when they are not, disturbance arises from
l^eats. Ultimately, then, the reason of the rational numerical relations of Pytha-
goras is to be found in the theorem of Fourier, and in one sense this theorem may
be considered as the prime source of the theory of harmony.f
The relation of whole numbers to consonance became in ancient times, in
the middle ages, and especially among Oriental nations, the foundation of extrava-
gant and fanciful speculation. ' Everything is Number and Harmony,' was the
characteristic principle of the Pythagorean doctrine. The same numerical ratios ^
which exist between the seven tones of the diatonic scale, were thought to be found
again in the distances of the celestial bodies from the central fire. Hence the
harmony of the spheres, which was heard by Pythagoras alone among mortal men,
as his disciples asserted. The numerical speculations of the Chinese in primitive
times reach as far. In the book of Tso-kiu-ming, a friend of Confucius (b.c. 500),
the five tones of the old Chinese scale were compared with the five elements of
their natural philosophy — ^water, fire, wood, metal, and earth. The whole numbers
I, 2, 3 and 4 were described as the source of all perfection. At a later time the
12 Semitones of the Octave were connected with the 12 months in the year, and so
on. Similar references of musical tones to the elements, the temperaments, and
the constellations are found abundantly scattered among the musical writings of
the Arabs. The harmony of the spheres plays a great part throughout the middle
ages. According to Athanasius Kircher, not only the macrocosm, but the micro-
cosm is musical. Even Keppler, a man of the deepest scientific spirit, could not ^
keep himself free from imaginations of this kind. Nay, even in the most recent
times, theorising friends of music may be found who will rather feast on arith-
metical mysticism than endeavour to hear upper partial tones.
The celebrated mathematician Leonard EuleriJ: tried, in a more serious and
more scientific manner, to found the relations of consonances to whole numbers
upon psychological considerations, and his theory may certainly be regarded as the
one which found most favour with scientific investigators during the last century,
although it perhaps did not entirely satisfy them. Euler § begins by explaining
that we are pleased with everything in which we can detect a certain amount of
* Namely magnitades which vary as sines tance by Prof. Preyer. See infra, App. XX.
and cosines. sect L. art. 7. — TrafislatorJ]
t [The coincidences or non-coincidences of % Tentamen ttovae thcoriae Mnaicae^ Petro-
combinational tones, which arc independent of poli, 1739.
Fourier's law, are also considered of impor- § hoc, ciL chap. ii. § 7.
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230 RETROSPECT. pakt ii,
perfection. Now the perfection of anything is determined by the co-operation of
all its parts towards the attainment of its end. Hence it follows that wherever
perfection is to be found there must be order ; for order consists in the arrange-
ment of all parts by a certain law from which we can discover why each part lies
where it is, rather than in any other place. Now in any perfect object such a law
of arrangement is determined by the end to be attained which governs all the
parts. For this reason order pleases us more than disorder. Now order can be
perceived in two ways : either we know the law whence the arrangement is de-
duced, and compare the deductions from this law with the arrangements observed ;
or, we observe these arrangements and endeavour to determine the law from them.
The latter is the case in music. A combination of tones will please us when we
can discover the law of their arrangement. Hence it may well happen that one
hearer finds it and that another does not, and that their judgments consequently
^ differ.
The more easily we perceive the order which characterises the objects contem*
plated, the more simple and more perfect will they appear, and the more easily and
joyfully shall we acknowledge them. But an order which costs trouble to discover,
though it will indeed also please us, will associate with that pleasure a certain
degree of weariness and sadness (tristitia).
Now in tones there are two things in which order is displayed, pitch and
duration. Pitch is ordered by intervals, duration by rhythm. Force of tone might
also be ordered, had we a measure for it. Now in rhythm two or three or four
equally long notes of one part may correspond with one or two or three of another,
in which the regularity of the arrangement is easily observed, especially when fre-
quently repeated, and gives considerable pleasure. Similarly in intervals we should
derive more pleasure from observing that two, three, or four vibrations of one tone
coincided with one, two, or three of another, than we could possibly experience if
f the ratios of the time of vibration were incommensurable with one another, or at
least could not be expressed except by very high numbers. Hence it follows that
the combination of two tones pleases us the more, the smaller the two numbers
by which the ratios of their periods of vibration can be expressed. Euler also
remarked that we could better endure more complicated ratios of the periods of
vibration, and consequently less perfect consonances, for higher than for deeper
tones, because for the former the groups of vibrations which were arranged to
occur in equal times, were repeated more frequently than in the latter, and we
were consequently better able to recognise the regularity of even a more involved
arrangement.
Hereupon Euler develops an arithmetical rule for calculating the degree of
harmoniousness of an interval or a chord from the ratios of the periods of the
vibrations which characterise the intervals. The Unison belongs to the first
degree, the Octave to the second, the Twelfth and Double Octave to the third, the
5[ Fifth to the fourth, the Fourth to the fifth, the major Tenth and Eleventh to the
sixth, the major Sixth and major Third to the seventh, the minor Sixth and minor
Thvrd to the eighth, the subminor Seventh 4 : 7 to the ninth, and so on. To the
ninth degree belongs also the major triad, both in its closest position and in the
position of the Sixth and Fourth. The major chord of the Sixth and Third
belongs, however, to the tenth degree. The mn^nor triad, both in its closest and
in its position of the Sixth and Third, also belongs to the ninth degree, but its
position of the Sixth and Fourth to the tenth degree. In this arrangement the
consequences of Euler 's system agree tolerably well with our own results, except
that in determining the relation of the major to the minor triad, the infiuence of
combinational tones was not taken into a^M^ount, but only the kinds of interval.
Hence both triads in their close position appear to be equally harmonious, although
again both the m>ajor chord of the Sixth and Third, and the minor chord of the
Sixth and Fourth are inferior with him as with us.*
* The principle on "which Euler calculated the degrees of harmoniousneBB for interrala
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CH.iP. XII. RETROSPECT. 231
Enler has not confined these speculations to smgle consonances and chords, bat
has extended them to their results, to the construction of scales, and to modula-
tions, and brought out many surprising specialities correctly. But without taking
into account that Euler's system gives no explanation of the reason why a conso-
nance when slightly out of tune sounds almost as well as one justly tuned, and much
better than one greatly out of tune, although the numerical ratio for the former iq
generally much more complicated, it is very evident that the principal difficulty |
in Euler's theory is that it says nothing at all of the mode in which the mind con- I
trives to perceive the numerical ratios of two combined tones. We must not forget I
that a man left; to himself is scarcely aware that a tone depends upon vibrations. ^
Moreover, immediate and conscious perception by the senses has no means of
discovering that the numbers of vibrations performed in the same time are different,
greater for high than for low tones, and that determinate intervals have deter-
minate ratios of these numbers. There are certainly many perceptions of the f
senses in which a person is not precisely able to account for the way in which he
has attained to his knowledge, as when from the resonance of a space he judges of
its size and form, or when he reads the character of a man in his features. But
in such cases a person has generally had a large experience in such relations, which
helps him to form a judgment in analogous circumstances, without having the
previous circumstances on which his judgment depends clearly present to his mind.
But it is quite different with pitch numbers. A man that has never made physical
experiments has never in the whole course of his life had the slightest opportunity
of knowing anything about pitch numbers or their ratios. And almost every one
who delights in music remains in this state of ignorance from birth to death.
Hence it would certainly be necessary to shew how the ratios of pitch numbers
can be perceived by the senses. It has been my endeavour to do this, and hence
the results of my investigation may be said, in one sense, to fill up the gap which
Euler's left. But the physiological processes which make the difference sensible ^
between consonance and dissonance, or, in Euler's language, orderly and disorderly
relations of tone, ultimately bring to light an essential difference between our
method of explanation and Euler's. According to the latter, the human mind
perceives commensurable ratios of pitch numbers as such ; according to our
method, it perceives only the physical effect of these ratios, namely the continuous
or intermittent sensation of the auditory nerves.* The physicist knows, indeed,
that the reason why the sensation of a consonance is continuous is that the ratios
of its pitch numbers are commensurable, but when a man who is unacquainted
with physics, hears a piece of music, nothing of the sort occurs to him,t nor does
the physicist find a chord in any respect more harmonious because he is better
acquainted with the cause of its harmoniousness.^ It is quite different with the
order of rhythm. That exactly two crotchets, or three in a triplet, or four quavers
and chords, is here annexed, because its con- becanse 60 is the least common multiple of m
Bequences are very correct, if combinational 4. 5i 6, that is, the least number which all of ^
tones are disregarded. When p is a prime them will divide without a remainder,
number, the degree is =p. All other numbers * [With possibly Prof. Preyer's addition,
are products of prime numbers. The number see App. XX. sect. L. art. 7. — Translator.']
of the degree for a product of two factors a and f [Li point of fact, as he always hears tem-
6, for which separately the numbers of degree pered tones, he never hears the exact com-
are a and jS respectively »a+i8-i. To find mensurable ratios. Indeed, on account of the
the number of the degree of a chord, which can impossibility of tuning with perfect exactness,
be expressed by p : g : r : «, <&c., in smallest the exact ratios are probably never heard,
whole numbers, Euler finds the least common except from the double siren and wave-siren. —
multiple of p^ 9, r, s, (fee, and the number of Translator,']
its degree is that of the chord. Thus, for :|: [Does a man breathe more easily and
example : aerate his blood better becanse he knows the
The number of the degree of 2 is 2, and of 3 is 3, J^^^*^*"*^?^ °? ^^^^y'^T^^T ""^^ '*^ ''^1**5''''
of 4=- 2x 2 it is 2 + 2-1 = 1 to his carbonised blood? Does a man feel a
of 12= 4 X 1* it is -? + 1 - I « ?* weight greater or less, because he knows the
of 60 = 12 X ^* it is ^ + ^ - 1 = Q ^*^^ °' gravitation ? These are quite similar
^' ^ -^ ^' questions.— rra»«?a/n»-
That of the major triad 4 : 5 : 6 is that of 60,
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9^3 EETROSPECT. - pabt n.
go to one minim is perceived by any attentive listener without the least instmotion.
But while the orderly relation (or commensurable ratio) of the vibrations of two
combined tones, on the other hand, undoubtedly affects the ear in a certain way
which distinguishes it from any disorderly relation (incommensurable ratio), this
difference of consonance and dissonance depends on physical, not psychological
grounds.
The considerations advanced by Rameau* and d'Alembert f on the one side, and
Tartini t on the other, concerning the cause of consonance agree better with our
theory. The last founded his theory on the existence of combinational tones,
the two first on that of upper partial tones. As we see, they had found the
proper points of attack, but the acoustical knowledge of last century did not allow
of their drawing sufficient consequences from them. According to d'Alembert,
Tartini's book was so darkly and obscurely written that he, as well as other well-
^ instructed people, were unable to form a judgment upon it. D'Alembert's book,
on the other hand, is an extremely clear and masterly performance, such as was
to be expected from a sharp and exact thinker, who was at the same time one of
the greatest physicists and mathematicians of his time. Bameau and d'Alembert
lay down two facts as the foundation of their system. The first is that every
resonant body audibly produces at the same time as the prime (gMUrateur) its
Twelfth and next higher Third, as upper partials (harmoniqiies). The second is
that the resemblance between any tone and its Octave is generally apparent. The
first fact is used to shew that the major chord is the most natural of all chords,
and the second to establish the possibility of lowering the Fifth and the Third by
one or two Octaves without altering the nature of the chord, and hence to obtain
the major triad in all its different inversions and positions. The minor triad is
then found by the condition that all three tones should have the same upper partial
or harmonic, namely the Fifth of tlie chord (in fact G, J5t>, and G have all the same
f upper partial g')> Hence although the minor chord is not so perfect and natural
as the major, it is nevertheless prescribed by nature.
In the middle of the eighteenth century, when much suffering arose from an
artificial social condition, it may have been enough to shew that a thing was
natural, in order at the same time to prove that it must also be beautiful and
desirable. Of course no one who considers the great perfection and suitability of
all organic arrangements in the human body, would, even at the present day, deny
that when the existence of such natural relations have been proved as Eameau
discovered between the tones of the major triad, they ought to be most carefully
considered, at least as starting-points for further research. And Bameau had
indeed quite correctly conjectured, as we can now perceive, that this fact was the
(proper basis of a theory of harmony. But that is by no means everything. For
in nature we find not only beauty but ugliness, not only help but hurt. Hence the
mere proof that anything is natural does not suffice to justify it esthetically.
in Moreover if Bameau had listened to the effects of striking rods, bells, and mem-
branes, or blowing over hollow chambers, he might have heard many a perfectly
dissonant chord. And yet such chords cannot but be considered equally natural.
That all musical instruments exhibit harmonic upper partials depends upon the
selection of qualities of tone which man has made to satisfy the requirements of
his ear.
Again the resemblance of the Octave to its fundamental tone, which was one
of Bameau 's initial facts, is a musical phenomenon quite as much in need of
explanation as consonance itself.
No one knew better than d'Alembert himself the gaps in this system. Hence
♦ [Trait4 de Vliarmonic r^duite a de^prirt' 1762.
Hpes iiaiurels, 1721.- Translator.] J [Trattato di Musica secondo la vera
t &Ui)ients de Miisiqiie, suivant les prin- scieiiza delV anttotiia. Padova, 175 1. — IZVoiw-
cipes de M. Rameau, par M. d'Alembert. Lyon, Uitor.]
Digitized by V^OOQIC
CHAP. XII. EETEOSPECT. 233
in the preface to bis book be especially guards bimself against the expression :
* Demonstration of the Principle of Harmony/ wbich Eameau had used. He
declares that so far as be bimself is concerned, be meant only to give a well-
connected and consistent account of all the laws of the theory of harmony, by
deriving them from a single fundamental fact, the existence of upper partial tones
or harmonics, which he assumes as given, without further inquiry respecting its
source. He consequently limits himself to proving the naturalness of the major
and minor triads. In bis book there is no mention of beats, and hence of the
real source of distinction between consonance and dissonance. Of the laws of beats
very little indeed was known at that time, and combinational tones had only been just
brought under the notice of French savants, by Tartini (1751) and Romieu (1753).
They had been discovered a few years previously in Germany by Sorge (1745), but
the fEhct was probably little known. Hence the materials were wanting for build-
ing up a more perfect theory. V
Nevertheless this attempt of Bameau and d'Alembert is historically of great im-
portance, in so fetr as the theory of consonance was thus for the first time shifted from
metaphysical to physical ground. It is astonisliing what these two thinkers effected
with the scanty materials at their command, and what a clear, precise, comprehensive
system the old vague and lumbering theory of music became under their hands.
The important progress which Bameau made in the specially musical portion of
the theory of harmony will be seen hereafter.
If, then, I have been myself able to present something more complete, I owe it
merely to the circumstance that I had at command a large mass of preliminary
physical results, which had accumulated in the century that has since elapsed.
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Google
PART III.
THE EELAHONSHIP OF MUSICAL TONES.
SCALES, AND TONALITY.
CHAPTER Xm.
OfiNEBAL VIEW OF THB DIFFERENT PBINCIPIiES OF MUSICAL STYIiE IN THB
DEVEIiOPMENT OF MUSIC.
Up to this point our investigation has been purely physical. We have analysed
the sensations of hearing, and investigated the physical and physiological caussB
for the phenomena discovered, — partial tones, combinational tones, and beats. Li
% the whole of this research we have dealt solely with natural phenomena, which
present themselves mechanically, without any choice, to all living beings whose
ears are constructed on the same anatomical plan as our own. In such a field,
where necessity is paramount and nothing is arbitrary, science is rightfully called
upon to estabhsh constant laws of phenomena, and to demonstrate strictly a strict
connection between cause and effect. As there is nothing arbitrary in the pheno-
mena embraced by the theory, so also nothing arbitrary can be admitted into the
laws which regulate the phenomena, or into the explanations given for their occur-
rence. As long as anything arbitrary remains in these laws and explanations, it is
the duty of science (a duty which it is generally able to discharge) to exclude it, by
continuing the investigations.
But in this third part of our inquiry into the theory of music we have to famish
a satisfactory foundation for the elementary rules of musical composition, and here
we tread on new ground, which is no longer subject to physical laws alone, although
% the knowledge which we have gained of the nature of hearing, will still find
numerous applications. We pass on to a problem which by its very nature belongs
to the domain of esthetics. When we spoke previously, in the theory of conso-
nance, of agreeable and disagreeable, we referred solely to the immediate impression
made on the senses when an isolated combination of sounds strikes the ear, and
paid no attention at all to artistic contrasts and means of expression ; we thought
only of sensuous pleasure, not of esthetic beauty. The two must be kept strictly
apart, although the first is an important means for attaining the second.
The altered nature of the matters now to be treated betrays itself by a purely
external characteristic. At every step we encounter historical and national dif-
ferences of taste. Whether one combination is rougher or smoother than another,
dopends solely on the anatomical structure of the ear, and has nothing to do with
psychological motives. But what degree of roughness a hearer is inclined to
endure as a means of musical expression depends on taste and habit ; hence tlie
boundary between consonances and dissonances has been frequently changed. Simi-
Digitized by V^jOOQIC
CHAP. XIII. PHYSICAL AND ESTHETICAL PRINCIPLES COMPARED. 235
larly Scales, Modes, and their Modulations have undergone multifarious alterations,
not merely among uncultivated or savage people, but even in those periods of the
world's history and among those nations where the noblest flowers of human
culture have expanded.
Hence it follows, — and the proposition is not even now sufficiently present to
the minds of our musical theoreticians and historians — that the system of Scales,
Modes, and Harmonic Tissues does not rest solely upon inalterable natural laws,
hut is also, at least partly j the result of esthetical principles, which have already
changed, and will still further change, with the progressive development of humanity.
But it does not follow irom this that the choice of those elements of musical art
was perfectly arbitrary, and that they do not allow of being derived from some
more general law. On the contrary the rules of any style of art form a well-
connected system whenever that style has attained a full and perfect development.
These rules of art were certainly never developed into a system by the artists them- ^
selves with conscious intention and consistency. They are rather the result of ten-
tative exploration or the play of imagination, as the artists think out or execute
their plans, and by trial gradually discover what kind or maimer of performance
best pleases them. Yet science can endeavour to discover the motors, whether
psychological or technical, which have been at work in this artistic process. Scien-
tific esthetics have to deal with the psychological motor ; scientific physics with
the technical. When the artist's aim in the style he has adopted, and its prin-
cipal direction, have once been rightly conceived, it can be more or less correctly
determined why he was forced to follow this or that rule, or employ this or that
technical means. In musical theory, namely where the peculiar physiological
functions of the ear, while not immediately present to conscious self-examination,
play an important part, a large and rich field is thrown open for scientific investi-
gation to shew the necessary character of the technical rules for each individual
direction in the development of our art. ^
It does not rest with natural science to characterise the chief problem worked
out by each school of art, and the elementary principle of its style. This must be
gathered from the results of historical and esthetical inquiry.
The relation we liave to treat may be illustrated by a comparison with archi-
tecture, which, like music, has pursued essentially different directions at different
times. The Greeks, in their stone temples, imitated the original wooden construc-
tions ; that was the principle of their architectural style. The whole division and
arrangement of their decorations clearly shew that it was their intention to imitate
wooden constructions. The verticality of the supporting columns, the general
horizontality of the supported beam, forced them to divide all the subordinate parts
for the great majority of cases into vertical and horizontal lines. The purposes of
Grecian worship, which performed its principal functions in the open air, were
satisfied by erections, of this kind, in which the internal spaces were necessarily
narrowly limited by the length of the stone or wooden beams which could be em- ^
ployed. The old Italians (Etruscans), on the other hand, discovered the principle
of the arch, composed of wedge-shaped stones. This discovery rendered it pos-
sible to cover in much more extensive buildings with arched roofs, than the Greeks
could do with their wooden beams. Among these arched buildings the halls of
justice (basil'icae) became important, as is well known, for the subsequent develop-
ment of architecture. The arched roof made the circular arch the chief principle
in division and decoration for Boman {Byzantine) art. The columns, pressed by
heavy weights, were transformed into pillars, on which, after the style was fully
developed, columns merely appeared in diminished forms, half sunk in the mass of
the pillar, as simply decorative articulations, and as the downward continuation of
the ribs of the arches which radiated towards the ceiling from the upper end of the
pillar.
In the arch the wedge-shaped stones press against each other, but as they all
uniformly press inwards, each one prevents the other from falling. The most
Digitized by V^OOQIC
236 PHYSICAL AND ESTHETICAL PRINCIPLES COMPARED, pabt in.
powerful and most dangerous degree of pressure is exerted by the stones in the
horizontal parts of the arch, where they have either no support or no obliquely
placed support, and are prevented from falling solely by the greater thickness of
their upper extremities. In very large arches the horizontal middle portion is con-
sequently the most dangerous, and would be precipitated by the slightest yielding
of the materials. As, then, medieval ecclesiastical structures assumed continu-
ally larger dimensions, the idea occurred of leaving out the middle horizontal part
of the arch altogether, and of making the sides ascend with moderate obliquity
until they met in a pointed arch. From thenceforward the pointed arch became
the dominant principle. The building was divided into sections externally by the
projecting buttresses. These, and the omnipresent pointed arch, made the outlines
hard, and the churches became enormously high. But both characters suited the
vigorous minds of the northern nations, and perhaps the very hardness of the forms,
" thoroughly subdued by that marvellous consistency which runs through the varied
magnificence of form in a gothic cathedral, served to heighten the impression of
immensity and power. -
We see here, then, how the technical discoveries which were associated with
the problems as they rose successively created three entirely distinct principles of
style — the horizontal line, the circular arch, the pointed arch — and how at each
new change in the main plan of construction, all the subordinate individualities^
down to the smallest decorations, were altered accordingly ; and hence how the
individual rules of construction can only be comprehended from the general prin-
ciple of construction. Although the gothic style has developed the richest, the
most consistent, the mightiest and most imposing of architectural forms, just as
modem music among other musical styles, no one would certainly for a moment
think of asserting that the pointed arch is nature's original form of all architectural
beauty, and must consequently be introduced everywhere. And at the present day
^ it is well known that it is an artistic absurdity to put gothic windows in a Greek
building. Conversely any one can unfortunately convince himself on visiting most
of our gothic cathedrals how detestably unsuitable to the whole effect are those
numerous httle chapels of the renaissance period built in the Greek or Roman style.
Just as little as the gothic pointed arch, should our diatonic major scale be regarded
as a natural product. At least such an expression is quite inapplicable, except in
80 far as both are necessary and natural consequences of the principle of style
selected. And just as little as we should use gothic ornamentation in a Greek
temple, should we venture upon improving compositions written in ecclesiastical
modes, by providing their notes with marks of sharps and flats in accordance with
the scheme of our major and minor harmonies. The feeling for historical artistic
conception has certainly made little progress as yet among our musicians, even
among those who are at the same time musical historians. They judge old music
by the rules of modem harmony, and are inclined to consider every deviation from
H it as mere unskilfulness in the old composer, or even as barbarous want of
taste.*
Hence before we proceed to the construction of scales and rules for a tissue of
harmony, we must endeavour to characterise the principles of style, at least for
the chief phases of the development of musical art. For present purposes we may
divide these into three principal periods :-t-
1. The Homophonic or Unison Music of the ancients, to which also belongs
the existing music of Oriental and Asiatic nations.
2. The Polyphonic Music of the middle ages, with several parts, but without
regard to any independent musical significance of the harmonies, extending firom
the tenth to the seventeenth century, when it passes into
3. Harmonic or Modern Music, characterised by the independent significance
♦ Thus in R. G. Kiesewetter's historico- dently an exaggerated zeal to deny everything
musical writings, which are otherwise so rich which will not fit into the modern major and
in facts industriously collected, there is evi- minor modes.
Digitized by VjOOQlC
CHAP. XIII.
PERIOD OP HOMOPHONIC MUSIC.
237
attributed to the harmonies as such. Its sources date back from the sixteenth
century.
I. HoMOFHONic Music.
One part music is the original form of music with all people. It still exists
among the Chinese, Indians, Arabs, Turks, and modem Greeks, notwithstanding
the greatly developed systems of music possessed by some of these nations.* That
music in the time of highest Grecian culture, neglecting perhaps individual instru-
mental ornamentation, cadences, and interludes, was written in one part, or that
the voices at most sang in Octaves, can now be considered as established. In the
problems of Aristotle we find the question : * Why is the consonance of the Octave
alone sung ? For this and no other consonance is played on the mag&dis.' This
was a harp-shaped instrument [with a bridge dividing the strings at one-third
their length]. In another place he remarks that the voices of boys and men fom f
an Octave in singing.f
One part music, considered independently and unaccompanied by words, is too
poor in forms and changes, to develop any of the greater and richer forms of art.
Hence purely instrumental music at this stage is necessarily limited to short
dances or marches. We really find no more among nations that have no harmonic
music4 Performers on the flute § have certainly repeatedly gained the prize in
the Pythian games, but it is possible to perform feats of execution in instrumental
music in concise forms of composition, as, for example, in the variations of a
short melody. That the principle of varying (ficra^oXi;) a melody with reference
to dramatical expression (/u/ti;<ris), was known to tlie Greeks, follows also &om
Aristotle. He describes the matter very plainly, and remarks that choruses must
simply repeat the melodies in the antistrophes, because it is easier for one than
for several to introduce variations. But public competitors (dyoivurrat) and actors
(vroKpiTOL) are able to grapple with these difficulties.** ^
* [See App. XX. sect. K. for some of these the effect of consonances and did not like it,
scales. — Translator.']
f Aifl tI 71 8iik wcuT&¥ <rv/i<p<ovta f Srrai fiSm/i ;
fiayai't(ov<ri yhp rairriyj ttWriy 9k oh9efilay,
Prob. xix. 18. flranslated in the text.] Ai^
rh fuv ikrrt^yoy 4rvfupvy6y iori 8i2^ tnurAy ; iK
wtuBvy ykp yiw kc^ h,vZp&y ylyerai rh iani^yoy
•1 Stcoraffi rois r6yotSf &s v^i? trphs r^f irrdrfjy,
Prob. xix. 39. [* Why is a consonant union of
▼oices pleasanter than a single voice ? Is the
singing of voice against voice, a consonant
anion of voices in Octaves ? This singing of
voice against voice occurs when young boys and
men sing together, and their tones differ as the
highest from the lowest of the scale.'] Towards
the end of the songs the instrumental ac-
companiment seems to have separated itself
from the voice. Probably this is what is meant
by the krouais in the passage rcAcvr»<ratf
8' «tf TotfT^ir, ly Kcd Kotyhy rh tpyov avfu^aivu yiy^
c9aiyKaBdw9prots &whrify«p9riy Kpoiovvi' Kot
yhp oZrot rit &XAa ob irpo<ravXovrrct, 4ay 9ls rav-
rhy Koeraorpi^wriy, Mhippaiyovat fuiWoy r^ r4\9i
l| Kwowri reus wph rov r4\ovs Sio^opcu;, r^ rh
4k ^ta^6pwy rh KOtyhy, liiurroy in rov 9ik wcurAy
yiytireai. Arist Prob. xix. 39. [*But when
they end in the same, the matter is precisely
similar to what occurs when they play an ac-
companiment to a song. For the acoompany-
ists do not follow the rest, but when the singers
return to the same, they please more in the end,
than they displease in the differences before
the end, by which means the common part in
what is generally different, pleases more than
anything but the Octave.*] See also Plutarch,
De Muskar i^x. xxviii. That the Greeks knew
-^V'^
appears by the following passage from Aristotle,
De A^idibilibus, ed. Beldcer, p. 801 : * For this
reason we understand a single speaker better
than many who are saying the same things at
the same time. And so with strings. And
much less, when flute and lyre are played at
the same time, because the voices are confused
by the others. And this is very plain for the
consonances. For both tones are concealed,
one by the other.* Ai^ koI uaKXoy Ms Ajco^ktcs
trvyiefxty^ ^ iroXAwy &fAa rairrii \fy6yruy ' K<»0ir§p
Kad hr\ rAy x^p^^^ ' *f«i wo\h ^rroy, 8ray wpo<r-
av\p ris ifM kclL Ki9api(ih 9tk rh trvyx*ifr9at rks
^cty^ vvb rAy kripnv. Ovx ffxiora Zk rovro iiri
r&y <rvfjup»yi»y i>av9p6y itrriy. *Afi<l>or4povs 7^
iiroKp^wrttrBai cufAfiedytt robs ffx^M vv* iXXiiKuy.
X [In Java long pieces of music in non- m
harmonic scales occur to accompany actions
and develop the feeling of a plot. Many instru-
ments play together, but there is no harmony.
— Translator.]
§ The airXoi were perhaps more like our oboes.
** [At& rl ol /tky y6fi4n ovk iy iufTurrp6^iS
iitoiovyro * ed 8i (UXot ^ctl al xopMoi ; *H Uri ol
fiky y6fA0t kymvivrwv Ijiray, iy Hitri fiifuurdm
Svyofity^y Ked ^tar^iyttrOcu, ii ^8j^ iylyero poKpk
ical iroAu«i3^5; xeiBdirtp ody koI rh, fii\ii rp
lufi-ficfti iiKoKoMu ktl trtpa yiy6fJLtyet, MeiWoy
yhp r^ fi4K€i ivdyicri fjufAtlfrScUf ^ rots ^fuuri.
At h Kcd oi 9tB6p€^ifiotf ivtt^^ fjufirirucol iyiyoyro^
OVK frt ix^^^^^ kyrtarp^ovs, wpirtpoy 84 cf^ov.
hirtov 8c, irt rh ira\euhy oi iKtvSfpot 4x^9*^^^
oJbroi. TloKKovi oiy kywyiartKm 4^*ty^ x"^*^^*^
^y. "SUrr* 4yap/jLuyta fiaWoy /ucAi) 4y^^y. McTOr-
fidWtty yap iroWhs ^crajSuAar t^^ iyl paoy^ l|
rots voWots, fcal ry aytoyiar^^ ^ roiS rh i^vs
Digitized by V^OOQIC
238
PERIOD OF HOMOPHONIC MUSIC.
PART III.
Extensive works of art, in homophonic music, are only possible in connection
with poetry, and this was also the way in which music was applied in classical
antiquity. Not only were songs (odes) and religious hymns sung, but even
tragedies and long epic poems were performed in some musical manner, and
accompanied by the lyre. We are scarcely in a condition to form a conception of
how this was done, because modem taste points in precisely the opposite direction,
, and demands from a great declaimer or public reader that he should produce a
dramatic effect true to nature by the speaking voice alone, rating all approach to
singing as one of the greatest of faults. Perhaps we have some echoes of the
ancient spoken song in the singing tone of Italian declaimers, and the liturgical
recitations (intoning) of the Eoman Catholic priests. Indeed, attentive observa-
tions on ordinary conversation shews us that regular musical intervals involun-
tarily recur, although the singing tone of the voice is concealed under the noises
f which characterise the individual letters, and the pitch is not held firmly, but is
frequently allowed to glide up and down. When simple sentences are spoken
without being affected by feeling, a certain middle pitch is maintained, and it is
only the emphatic words and the conclusions of sentences and clauses which are
indicated by change of pitch.* The end of an aflSrmative sentence followed by a
pause, is usually marked by the voice falling a Fourth from the middle pitch. An
interrogative ending rises, often as much as a Fifth above the middle pitch. For
example a bass voice would say :
m
^
3^
Ich Hn spa - tzie - rew ge - gan - ^en.
I have been walk - ing this mom - ing.
1
^|c==-t
Bist du spa - tzie - ren ge - gam, - genf
Have you been walk - ing this mom - mg?
Emphasised words are also rendered prominent by their being spoken about a Tone
higher than the rest,t and so on. In solemn declamation the alterations of pitch
are more numerous and complicated. Modem recitative has arisen from attempt-
ing to imitate these alterations of pitch by musical notes. Its inventor, Giaconio
Peri, in the preface to his opera of Eurydice, published in i6oo, distinctly says as
much. An attempt was then made to restore the declamation of ancient tragedies
by means of recitative. Ancient recitative certainly differed somewhat from
modem recitative, by preserving the metre of the poems more exactly, and by
^vXdrrov^i. At* h iarXo6<rrfpn iwoiovvro aibrois
m rdfi«Aiy. 'H hk hyriarfo^Sy kirXovv. *Api$fihs ydp
iarif Koi M fierpwai, Tb V ahrh odfrioy kcu 9i6ti
tA flip i.ir6 'njs fficfipvis obx kmlffrpo^ r& (^ rov
X^pov kyrUrrpo^ *0 fiiw yiip ^OKpiriis iyotyurr^s
[ica2 fiifiriT'^s''] 6 9k x^P^^t ^frrov /ufAurai. Arist.
Prob, xix. 15. * Why are themes (nomoi) not
used in antistrophic singing, while all other
choral singing is employed? Is it because
themes belong to public performers who are
already able to imitate and extend, and hence
would make their song long and very figu-
rate? For melodies, like words, follow imi-
tation and change. It is more necessary for
melody to imitate, than for words. Where-
fore dithyrambic poets also when they became
mimetic, disused their previous antistrophic
singing. The reason is that formerly gentle-
men (eleutheroi) used to sing the choruses
themselves. It was difficult for many to sing
like public performers. So they rather intoned
suitable melodies. For it is easier for one to
make numerons variations than for many to
do so, and for a public performer than for
those who retained old usage. Hence the
melodies were made simpler for them. Kow
antistrophic singing is simple, for it depends
on number, and is measured by a unit. The
same reason shews why the parts of the actors
are not antistrophic, but those of the chorus
are so. For the actor is a pnblic performer
[and a mime], but the chorus does not imitate
so well.' — Translator.]
* [Prof. Helmholtz's observations on speak-
ing must be read in reference to North Qerman
habits only. — Translator.]
f [By no means nniformly, even in North
Germany. The habits of different nations
here vary greatly. In Norway and in Sweden
the voice is regularly raised on unemphatie
syllables. In Scotland the emphasis is often
marked by lowering the ^ich,— Translator.]
Digitized by V^jOOQlC
CHAP. XIII.
PERIOD OF HOMOPHONIC MUSIC.
«39
having no accompanying harmonies. Nevertheless our recitative, when well per-
formed, will give us a better conception of the degree in which the expression of
the words can be enhanced by musical recitation, than we can obtain from the
monotonous repetition of the Roman liturgy, although the latter perhaps is more
nearly related in kind to ancient recitation than the former. The settlement of
the Roman liturgy by Pope Gregory the Great (a.d. 590 to 604) reaches back to a
time in which reminiscences of the ancient art, although faded and deformed,
might have been in some degree handed down by tradition, especially if, as we are
probably entitled to assume, Gregory really did little more than finally establish
the Roman school of singing which had existed from the time of Pope Sylvester
(a.d. 314 to 335).* The majority of these formulae for lessons, collects, &c.,
evidently imitate the cadence of ordinary speech. They proceed at an equal
height ; particular, emphatic, or non-Latin words are somewhat altered in pitch ;
&nd for the punctuation certain concluding forms are prescribed, as the following f
for lessons, according to the customs of Miinster.f
if
Sie can-ta com-maj
Thua sing the com -ma,
sic du - o punc'ta:
and thus the co - Ion :
nc ve • ro punc-tum,
and thus the full stop.
i
1^^
s
8ic sig - wu/m in - Ur - ro - ga - ti - o - nis?
Thus sing the mark of in - ter - ro - ga - tion?
These and similar final formulae were varied according to the solemnity of the
feast, the subject treated, the rank of the priest that sang and that answered, and %
Bo on.f It is easy to see that they strove to imitate the natural cadences of
ordinary speech, and to give them solemnity by eliminating their individual irregu-
larities. Of course in such fixed formula no regard can be paid to the grammatical
sense of the clauses, which suffers much in various ways from the intoning.
Similarly we may suppose that the ancient tragic poets prescribed the cadences of
speech to their actors, and preserved them by a musical accompaniment. And
since ancient tragedy kept much further aloof from immediate external realism than
our modem drama, as is shewn by the artificial rhythms, the unusual rolling words,
the immovable strange masks, it could admit of a more singing tone for declamation
than would, perhaps, please our modem ears. Then we must remember that by
emphasising or increasing the loudness of certain words, and by rapidity or slow-
ness of speech, or pantomimic action, much life can be thrown into delivery of this
kind, which would certainly be insufferably monotonous if not thus enlivened.
But in any case homophonic music, even when in olden time it had to ac- ^
company extensive poems of the highest character, necessarily played an utterly
subordinate part. The musical turns must have entirely depended on the changing
sense of the words, and could have had no independent artistic value or connection
without them. A peculiar melody for singing hexameters throughout an epic, or
iambic trimeters throughout a tragedy, would have been insupportable.t Those
* [These are the dates of his reign.
HuHah says the school was fonnded in a.d.
350.— Translator.]
t Antony, * Lehrbuch des Gregorianischen
Kirchengesanges ' [Maniuil of Gregorian
Church-music], Miinster, 1829. According to
the inionnation collected by F^tis (in his
JJistoire gin&rale de Mtisique, Paris, 1869,
vol. i. chap, vi.), it has become donbtfal
whether this system of declamation with pre-
scribed cadences, is not rather to be deduced
from the Jewish ritual chants. In the oldest
manuscripts of the Old Testament, there are
25 different signs employed to denote cadences
and melodic phrases of this kind. The fact
that the corresponding signs of the Greek
Church are Egyptian demotic characters, hints
at a still older Egyptian origin for this nota>
tion.
X [We must remember that the Greek and
Latin so-called accents consisted solely in
alterations of pitch, and hence to a certain
Digitized by V^OOQIC
240 PEEIOD OP HOMOPHONIC MUSIC. part m.
melodies (vofu>i) which were allotted to odes and tragic ohorases, were certainly
freer and more independent. For odes there were also well-known melodies (the
names of some of them are preserved) to which fresh poems were continually
composed.
In the great artistic works just mentioned, then, music must have been entirely
subordinate ; independently, it could only have formed sliort pieces. Now this is
closely connected with the development of homophonio music as a musical system.
Among the nations who possess such music we always find certain degrees of pitch
selected for the melodies to move in. These scales are very various in kind, partly,
it would seem, very arbitrary, so that many appear to us quite strange and incom-
prehensible, and yet the best gifted among those nations which possess them, as
the Greeks, Arabs, and Indians, have developed them in an extremely subtile and
varied manner. [See App. XX. sect. K.]
f When speaking of these systems of tones, it becomes a question of essential
importance for our present purpose, to inquire whether they are based upon any
determinate reference of aU the tones in the scale to one single principal and
fundamental tone, the tonic or key-note. Modem music effects a purely musical
internal connection among all the tones in a composition, by making their rela-
tionship to one tone as perceptible as possible to the ear. Tliis predominance
of the tonic, as the link which connects all the tones of a piece, we may, with
F^tis, term the principle of tonality. This learned musician has properly drawn
attention to the fact that tonality is developed in very different degrees and
manners in the melodies of different nations. Thus in the songs of the modem
Greeks, and chants of the Greek Church, and the Gregorian tones of the Koman
Church, they are not developed in a manner which is easy to harmonise, whereas,
according to F^tis,* it is on the whole easy to add accompanying harmonies
to the old melodies of the northern nations of German, Celtic, and Sclavonic
^ origin.
It is indeed remarkable that though the musical writings of the Greeks often
treat subtile points at great length, and give the most exact information about all
other peculiarities of the scale, they say nothing intelligible about a relation which
in our modem system stands first of all, and always makes itself most disthietly
sensible. The only hints to be found conceming the existence of the tonic are
not in especial musical writings, but as before in the works of Aristotle, who
asks : —
* Why is it that if any one alters the tone on the middle string (fiiayi) after the
others have been tuned, and plays, every thing sounds amiss, not merely when he
comes to this middle tone, but throughout the whole melody ? but if he alters the
tone played by the forefinger t or any other, the difference is only perceived when
that string is struck ? Is there a good reason for this ? All good melodies often
employ the tone of the middle string, and good composers often come upon it,
% and if they leave it recur to it again ; but this is not the case with any other
tone.' Then he compares the tone of the middle string with conjunctions in
language, such as ' and ' [and ' then '], without which language could not exist, and
proceeds to say : * In this way the tone of the middle string is a link between
tones, especially of the best tones, because its tone most frequently recurs.' t
extent determined a melody. See Dionysias joined with a constant quantity or rhythm. —
of HalicamassuSt ircpi trwddatws dvofidruvt Translator.]
chap, xi., where we also find that in his day * Fetis, Biographie univerBelle des Muti-
(first century before Christ) the musical com- dens^ vol. i. p. 126.
posers transgressed at pleasure the rules of f [The forefinger is t Kixaofis^ the note
both accent and quantity. But if the written played by it is ^ Kix^ofoi, accent and gender
accents in Greek, and the accents as deter- both differing. --Tmn«2ator.]
mined by the rules of grammarians in Latin, 1 Ai^ rl, Am fi4y r» rV m^^*' «rir^ 4fU»r.
are carefully examined, it will be found that apfiitras [82] r&f &\Xas x^P^^i K^XPVai r^
every line in a Greek or Latin poem had its hpydvt^^ ov fi6trov Urop Karh rhr r^f fi^aris yitn^
own diRtinct melody, the art of the poet being rcu ^^yyoy^ Ai/irc?, «ral ^ytrat Mt^fMvrop,
shewn by the great variability of pitch con- oAAa kuI xarii rhy iXXtfy fitKq^iay - imp £* riiw
Digitized by V^jOOQlC
CHAP. xin.
PERIOD OF HOMOPHONIC MUSIC.
241
And in another place we find the same question with a slightly different answer,
Why do the other tones sound badly when the tone of the middle string is
altered ? bat if the tone of the middle string remains, and one of the others is
altered, the altered one alone is spoiled ? Is it because that all are tuned and have
a certain relation to the tone of the middle string, and the order of each is deter-
mined by that ? The reason of the tuning and connection being removed, then,
things no longer appear the same.'* In these sentences the esthetic significance of
the tonic, under the name of * the tone of the middle string,' is very accurately
described. To this we may add that the Pythagoreans compared the tone of the
middle string with the sun, and the other tones in the scale with the planets.f It
appears as if it had been usual to begin with the tone of the middle string above
mentioned, for we read in the 33rd problem of Aristotle : * Why is it more agree-
able to proceed from high pitch to low pitch, than from low pitch to high pitch ? Can
it be that we thus begin at the beginning ? for the tone of the middle string is also ^
the leader of the tetrachord and highest in pitch. The second way would be to begin
at the end instead of at the beginning. Or can it be that tones of lower pitch
sound nobler and more euphonious after tones of high pitch ? ' j: This seems also to
shew that it was not the custom to end with the tone of the middle string, which
commenced, but with the tone of lowest pitch [produced by the uppermost string
or Hypate], of which last tone Aristotle, in his 4th problem, says that, as opposed
to its neighbour, the tone of lowest pitch but one, [due to the string of highest
position but one, or Parhypatit] it is sung with complete relaxation of all the effort
that is felt in the other.§ These words of Aristotle may certainly be applied to
the national Doric scale of the Greeks, which, increased by Pythagoras to eight
tones, was as follows : —
Huupiptuf lUvoVj Urcof K&Kcdqi r\s XP^'"'^ : *H
€v\Ayns rovro avfjificdyu ; wdvra ykp rk xf>^<rr&
iyoBol tronrraL, wvicriL xphs riiv fA^trnv kwamStvi *
jc^ harMwfft tox^ hrav^pxovrcu • wphs 8i &AAt}K
O0TW9 ovZffilay. KaBdvep iK rwv \6yutr itrlour
i^atptBirrwv awh^iryLfav^ ovk tfrriv 6 \Syos *Z?<Afiv-
ik6s ' (oTov rh ri, iced rh rol) Koi Ivioi 8i ovOkv
Xuwovci * 8(jt rh rots fi^v, kyayKcuov ttvvu xp^^M
voAA<£iccf , ^ OVK (ffrai \6yos *E?i\rjvuc6s ' rols 8i,
ft^* o5r« KoL r&y ^B6rfyiav ^ inivrt^ &<nr€p ff^l^ir-
Ia6$ 4art, koI fidXtirra rwv KaXAv, htk rh wXturriKis
iyvwdpx*iy "rhy ^^yyoy a^ijs. Arist. Prob. zix.
20. This passage has also been partly quoted
by Ambrosch. [The names of Greek tones
were those of the strings on the lyre by which
they were played, jast as if in English we were
to call the tones g, d\ a', e'\ the tones of the
fourth, third, second, and first strings resi>eo-
tively, because they are produced as the open
notes of these strings on the violin, and con-
tracted them to fourth^ thirds &c., only omit-
ting the word string. As the violin when held
sideways in playing, throws the g string upper-
most, and the e^' string the lowermost, we
might in the same way call g the * up}>ermost
note,' virch-i), although lowest in pit(di, and e"
the * lowermost note,* rfrr% although highest
in pitch. Then d might be called the middle^
fidafi, being really the key-note of the violin
and one of the two middle strings. This illus-
trates the Greek names very closely, for the
lyre was held with the string sounding the
lowest note, uppermost. See tiie scale on the
next page. — Translator,]
* Aik rl, iiiy fi^v ii fiitnj KurnOri, Koi al &AAai
X«pSci2 ^x^'*^^* ^€yy6/i€vai' (one of my col-
leagues. Prof. Stark, conjectures that in place
of ^€yy6fitycu, which makes nonsense, we
should read ^fip6/iweu *) ii» 82 aZ 4i fih^ uAvjf,
r&y 8* dKXmy rls KuniO^, Konfiuva fi6y7i <l>64y- «r
yerai ; (for which Prof. Stark again proposes
^B^lpercu ;) *H 8ti t^ lipfiScBcu i<rrly iiiriffais, [rh
82 Ix**" *■»* ^P^f '^y /jJiniy kwdtreus], koL ^ rd^ts
if iKoffrriSf ff8i| 8t' iKtlyriy ; itpBiyros olyrov ahlov
rod 7ipfi6<rBat koL rod (rvytxoyros, oIk (ri 6fU}lo05
^yercu irdpx^uf- Arist. Prob. xix. 36.
f Nicomachns, Harmonidt lib. i. p. 6,
ed. Meibomii. [The following is Nioomachus'a
arrangement of the comparison, with his
reasons: .
Saturn hypaU, as being highest in position,
traroy ybip rh kytSrraeroy.
Jupiter ]^rhypatli as next highest to
Saturn.
Mars lichanos or hypermesi^ as between
Jupiter and the Sun.
The Sun mesBt as lying in the middle, the
fourth from either end, middlemost string and
planet. «r
Mercury paramese^ as lying between the ^'
Sun and Venus.
Venus paraneatBt as lying just above the
moon.
The Moon neate, as being lowest of all in
position and next the earth, iced yitp yiwov^ rh
Kardararoy, — Translator,]
\ Lib. ri €vapfAOffr6rtpoy iwh rov &^4ot M rh
fiapv, fj iiTh rov fiapdos M rh o^v; Tl6r€pov St<
rh emh rijs if>x^* yiyerai Apx^^Bcu; ^ ykp fjiiari
K<d riy€fiiiy o^vrdrri rov rtrpax^pZov. Th 82 ovk
iir* &pxv^t ^^' A*"^ rtXtvrrjs. *H Jrt r6 fiapb
iLw6 rov 6^4o9 y^yyai&r^poy, koJl th^y^^poy ;
Arist. Prob, xix. 33.
§ Lih rl 82 ra{miy [r^y irapvwdniy] xa^frrcis
[fSovcri], riiy 82 lirrdrriy p^Ziws ' KcUroi SUfftf
4Kar4pas ; ^ Zri fi^r* dy4atus 4i irdrri, ica2 &fxa
fierk r^y avffroffiy i\a^p6y rh &yv fidWfiv ;
Arist. Prob. xix. 4.
Digitized by V^jOQQIC
242 PEEIOD OF HOMOPHONIC MUSIC. part m.
Tetrachord of
lowest pitch
Tetrachord of
highest pitch
'E HypSte [^drii nppermost string]
F Parhypate .... [irapvwdrri next to nppermost string]
G Lichfinos .... [A^x^''^'' forefinger string]
.A Mese (tone of middle string) . {/i4(ni middle string]
B ParamSse .... [wapafU<rri next to middle string]
C Trite [rplrn third string]
D Paranete [waparfrrv next to lowermost string]
E Nete [1^17 lowermost string]
In modem phraseology the last description cited from Aristotle implies that the
Parhypate was a kind of descending ' leading note ' to the Hjpatd. In the leading
tone there is perceptible effort, which ceases on its falling into the fundamental
tone.
If, then, the tone of the middle string answers to the tonic, the Hypate, which
^ is its Fifth, will answer to the dominant. For our modem feeling it is far more
necessary to close with the tonic than to begin with it, and hence we usually take
the final tone of a piece to be its tonic without farther inquiry. Modem music,
however, usually introduces the tonic also in the first beat of the opening bar.
The whole mass of tone is developed firom the tonic and returns into it. Modem
musicians cannot obtain complete repose at the end unless the series of tones con-
verges into its connecting centre.
Ancient Greek music seems, then, to have deviated firom ours by ending on the
dominant instead of the tonic. And this is in full agreement with the intonation
of speech. We have seen that the end of an affirmative sentence is Ukewise
formed on the Fifth next below the principal tone.* This peculiarity has also
been generally preserved in modem recitative, in which the singer usuaUy ends on
the dominant; the accompanying instruments then make this tone part of the
chord of the dominant Seventh, leading to the tonic chord, and thus make a close
^ on the tonic in accordance with our present musical feeling. Now since Greek
music was cultivated by the recitation of epic hexameters and iambic trimeters,
we should not be surprised if the above-mentioned peculiarities of chanting were
so predominant in the melodies of odes that Aristotle could regard them as the
rule.t
From the facts just adduced it follows (and this is what we are chiefly con-
cerned with) that the Greeks, among whom our diatonic scale first arose, were not
without a certain esthetic feeling for tonality, but that they had not developed it so
decisively as in modem music. Indeed, it does not appear to have even entered
into the technical rules for constructing melodies. Hence Aristotle, who treated
music esthetically, is the only known writer who mentions it; musical writers
proper do not speak of it at all. And unfortunately the indications famished by
Aristotle are so meagre, that doubt enough still exists. For example, he says
nothing about the differences of the various musical modes in reference to their
m principal tone, so that the most important point of all from which we should wish
to regard the construction of the musical scale, is almost entirely obscured.
The reference to a tonic is more distinctly made out in the scales of the old
Christian ecclesiastical music. Originally the four so-called authentic scales were
distinguished, as they had been laid down by Ambrose of Milan (elected Bishop
A.D. 374, died A.D. 398). Not one of these agrees with any one of our scales. The
four plagal scales afterwards added by Gregory, are no scales at all in our sense of
the word. The four authentic scales of Ambrose t are :
♦ [This wonld be entirely crossed by the ment of the Homeric Ode to Demeter, which
ancient Greek system of pitch-accents, just as has been published by B. Marcello, shews the
it now is by a similar system in Norwegian, above-mentioned peculiarity very distinctly,
where the pitch may rise for 60^^ affirmative X C^^* Bockstro in his article ' Ambro-
and interrogative sentences. See p. 239^'-, sian Chant,' in Grove's Dictionary of Music^
note %,— Translator.'] ^ ^ states that this attribution of four authentic
f Among the presumed ancient melodies scales to St. Ambrose has not been proved. —
which have been handed down to us, the frag- Translator.]
Digitized by V^jOOQlC
CHAP. XIII. PERIOD OF HOMOPHONIC MUSIC. 243
i)DEFGABcd
2) E F G A B c d e
l) F G A B c d e f
/^) G A B c d e f g
Perhaps, however, the change of B into B^ was allowed from the first, and
this would make the first scale agree with our descending scale of D minor, and the
third scale would become our scale of F major. The old rule was that songs in
the first scale should end in 2), those in the second in E, those in the third in F,
and those in the fourth in G, This marked out these tones as tonics in our sense
of the word. But the rule was not strictly observed. The conclusion might fall
on other tones of the scale, the so-called confinal tones, and at last the confusion
became so great that no one was able to say exactly how the scale was to be
recognised, all kinds of insufficient rules were formulated, and at last musicians
clung to the mechanical expedient of fixing upon certain initial and concluding ^
phrases, called tropes^ as characterising the scale.
Hence although the rule of tonahty had been already remarked in these
medieval ecclesiastical scales, the rule was so unsettled and admitted so many
exceptions, that the feeling of tonahty must have been much less developed than
in modem music.
The Indians also hit upon the conception of a tonic, although their music is
likewise unisonal. They called the tonic AnsaJ* Indian melodies as transcribed
by Enghsh travellers, seem to be very like modem European melodies.f F6tis
and Goussemaker j: have made the same remark respecting the few known remains
of old German and Geltic melodies.
Although, therefore, homophonic music was [possibly] not entirely without a refe-
rence to some tonic, or predominant tone, such a tone was beyond cJl dispute much
more weakly developed than in modem music, where a few consecutive chords
suffice to establish the scale in which that portion of the piece is written. The f
cause of this seems to me traceable to the undeveloped condition and subordinate
part which characterises homophonic music. Melodies which move up and down
in a few tones which are easily comprehended, and are connected, not by some
musical contrivance, but by the words of a poem, do not require the consistent
application of any contrivance, to combine them. Even in modem recitative
tonality is much less firmly established than in other forms of composition. The
necessity for a steady connection of masses of tone by purely musical relations,
does not dawn distinctly on our feehng, until we have to form into one artistic
whole large masses of tone, which have their own independent significance without
the cement of poetry.
♦ Jones, On the Music 0/ the IndianSt " By the word vddi," aays the commentator,
translated by Dalberg, pp. 36, 37. [Sir W. ** he means the note, which announces and
Joneses tract, with many others, is reprinted ascertains the B&ga [tane], and which may be
in Sonrindro Mohnn T^gore's Hindu Music considered as the parent and origin of the
from various Authors, This is what he says graha and nydsa," This clearly shews, I think IT
of the ans'a, p. 149 of Tagore : * Since it [says Sir W. Jones], that the ans'a must be
appears from the N&^yan [a Sanscrit treatise the tonic ; and we shall find that the two
on music], that 36 modes are in general use, other notes are generally its third and fifth,
and the rest very rarely applied to practice, I or the mediant and dominant. In the poem
shall exhibit only the scales of the 6 B&gas entitled Mdgha there is a musical simile,
[tunes] and 30 B^iginis [female personification which may illustrate and confirm our idea.
of tunes in Hindu music] according to S6ma. [I give the translation only.] " From the
. . . Three distinguished sounds in each mode greatness, from the transcendent quaUties of
[as Sir W.Jones translates r^] are called ffra/ia, that hero, eager for conquest, other kings
nydsa, ans'a, and the writer of the Nar&yan march in subordination to him, as other notes
defines them in the two following couplets. are subordinate to the ans'a. '" — Translator J]
[I give the translation only.] ** The note called f [The construction and time are very
graha is placed at the beginning, and that different. The scales are extremely variable.
named nydsa at the end, of a song ; that note. The results are very imi>erfectly represented
which displays the peculiar melody, and to by our present musical notation. — Translator.]
which all the others are subordinate, that, ^ Histoire de VHarmonie au moyen Age,
which is always of the greatest use, is like a Pans, 1852, pp. 5-7.
sovereign, though a mere aTw'a or portion."
Digitized by
'-^QQgle
244 PERIOD OF POLYPHONIC MUSIC. part m.
2. Polyphonic Music.
The second stage of musical development is the polyphonic music of the middle
ages. It is usual to cite as the first invented part-music, the so-called organum or
diaphony, as originally described by the Flemish monk Hucbald at the beginning
of the tenth century. In this, two voices are said to have proceeded in Fifths or
Fourths, with occasional doublings of one or both in Octaves. This would pro-
duce intolerable music for modem ears. But according to 0. Paul* the meaning
is not that the two voices sang at the same time, but that there was a respon-
sive repetition of a melody in a transposed condition, in which case Hucbald would
have been the inventor of a principle which subsequently became so important in
the fugue and sonata.
The first undoubted form of part-music intentionally for several voices, was the
^ so-called discanPus, which became known at the end of the eleventh century in
France and Flanders. The oldest specimens of this kind of music which have
been preserved are of the following description. Two entirely different melodies
— and to all appearance the more different the better — were adapted to one another
by shght changes in rhythm or pitch, until they formed a tolerably consonant whole.
At first, indeed, there seems to have been an inclination for coupling a liturgical
formula with a rather * slippery ' song. The first of such examples could scarcely
have been intended for more than musical tricks to amuse social meetings. It was
a new and amusing discovery that two totally independent melodies might be sung
together and yet sound well.
The principle of discant was fertile, and its nature was suitable for develop-
ment at that period. Polyphonic music proper was its issue. Different voices,
each proceeding independently and singing its own melody, had to be united in
such a way as to produce either no dissonances, or merely transient ones which
f were readily resolved. Consonance was not the object in view, but its opposite,
dissonance, was to be avoided. All interest was concentrated on the motion of the
voices. To keep the various parts together, time had to be strictly observed, and
hence the influence of discant developed a system of musical rhythm, which again
contributed to infase greater power and importance into melodic progression.
There was no division of time in the Gregorian Canttis firmus. The rhythm of
dance music was probably extremely simple. Moreover, melodic movement in-
creased in richness and interest as the parts were multiphed. But the establish-
ment of an artistic connection between the different voices, which, as we have seen,
were at first perfectly free, required a new invention, and this, though it cropped
up at first in a very humble form, has ended by obtaining predominant importance
in the whole art of modem musical composition. This invention consisted in
causing a musical phrase which had been sung by one voice to be repeated by
another. Thus arose canonic imitation, which may be met with sporadically as
If early as in the twelfth century .f This subsequently developed into a higlily
artistic system, especially among Netherland composers, who, it must be owned,
ended by often shewing more calculation than taste in their compositions.
But by this kind of polyphonic music— the repetition of the same melodic
phrases in succession by different voices — it first became possible to compose
musical pieces on an extensive plan, owing their connection not to any union with
another fine art — poetry, but to purely musical contrivances. This kind of music
also was especially suited to ecclesiastical songs, in which the chorus had to express
the feelings of a whole congregation of worshippers, each with his own peculiar
disposition. It was, however, not confined to ecclesiastical compositions, but was
also applied to secular songs (madrigals). The sole form of harmonic music yet
known, which could be adapted to artistic cultivation, was that founded on canonic
* Oeschichte des Claviers [History of the nos, PI. zxvii. No. iy., translated \n p. xxviL
Pianoforte], Leipzig, i868, p. 49. No. xxix.
f Coussemaker, loc. cii- .Discant : Custodi
Digitized by VjOOQIC
CHAP. XIII. PEBIOD OP POLYPHONIC MUSIC. 245
repetitions. If this had been rejected, nothing but homophonio music remained.
Hence we find a number of songs set ad strict canons or with canonical repetitions,
although they were entirely unsuited for such a heavy form of composition. Even the
oldest examples of instrumental compositions in several parts, the dance music of
1529,* are written in the form of madrigals and motets, a character of composition
which, more freely treated, lasted down to the suites of S. Bach and Handel's
times. Even in the first attempts at musical dramas in the sixteenth century,
there was no other way of making the personages express their feelings musically,
than by causing a chorus behind or upon the stage to sing over some madrigals in
the fugue style. It is scarcely possible for us, from our present point of view, to
conceive the condition of an luH; which was able to build up the most complicated
constructions of voice parts in chorus, and was yet incapable of adding a simple
accompaniment to the melody of a song or a duet, for the purpose of filling up
the harmony. And yet when we read how Giacomo Peri's invention of recitative f
with a simple accompaniment of chorus was applauded and admired and what
contentions arose as to the renown of the invention ; what attention Viadana
excited when he invented the addition of a Basso continuo for songs in one or
two parts, as a dependent part serving only to fill up the harmony f ; it is impos-
sible to doubt that this art of accompanying a melody by chords (as any amateur
can now do in the simplest maimer possible) was completely unknown to musicians
up to the end of the sixteenth century. It was not till the sixteenth century
that composers became aware of the meaning possessed by chords as forming an
harmonic tissue independently of the progression of parts.
To this condition of the art corresponded the condition of the tonal system.
The old ecclesiastical scales were retained in their essentials, the first from D io d,
the second from E to 6, the third from Fiof, and the fourth from Gix)g. Of these
the scale from F \of was useless for harmonic purposes, because it contained tlie
Tritone F—B, in place of the Fourth F—B^. Again, there was no reason for ^
excluding the scales from 0 to c and AU> a. And thus the ecclesiastical scales
altered under the infiuence of polyphonic music. But as the old unsuitable names
were retained notwithstanding the (Ganges, there arose a terrible confusion in the
meaning attached to modes. It was not till nearly the end of this period that
a learned theoretician, Glarean, undertook in his Dodecachordon (Basle 1547) to
put some order into the theory of modes. He distinguished twelve of them, six
authentic and six plagal, and assigned them Qreek names, which were, however,
incorrectly transferred. However, his nomenclature for ecclesiastical modes has
been generally followed ever since. The following are Glarean*s six authentic eccle-
siastical modes, keys, or scales, with the incorrect Greek names he assigned to
them.
Ionic . . . CDEFGABc
Doric .
Phrygian
Lydian
Mixolydian
EoUc •
DEFOABcd
EFOABcde
F G A B c d e f
GABodefg
A B c d e f g a
Ionic answers to our major. Folic to our minor system. Lydian was scarcely
ever used in polyphonic music owing to the false Fourth F^B, and when it was
employed it was altered in many different ways.
Inability to judge of the musical significance of a connected tissue of harmonies
again appears in the theory of the keys, by the rule, that the key of a polyphonic
composition was determined by considering the separate voices independently.
Glarean in certain compositions attributes different keys to the tenor and bass, the
soprano and alto« Zarlino assumes the tenor as the chief part for determining the
key.
* Winterfeld, Johannes Oabrieli und sein Zeitalter, vol. ii. p. 41.
t Winterfeld, ibid,t vol. ii. p. 19 and p. 59.
Digitized by VjOOQIC
246 PERIOD OF HARMONIC MUSIC. pabt m.
The practical consequences of this neglect of harmony are conspicuous in
various ways in musical compositions. The composers confined themselves on ihe
whole to the diatonic scale ; ' accidentals/ or signs of alterations of tone, were seldom
used. The Greeks had introduced the depression of the tone Bio JB\}inek peculiar
tetrachord, that of the synimmenoit and this was retained. Besides this/jf, c jf
and g ^ were used, to introduce leading tones in the cadences. Hence modulation,
as we understand it, from the key of one tonic to that of another with a different
signature was almost entirely absent. Moreover, the chords used by preference
down to the end of the fifteenth century, were formed of the Octave and Fifth
without the Third, and such chords now sound poor and are avoided as much as
possible. To medieval composers who only felt the want of the most perfect con-
sonances, these chords appeared the most agreeable, and none others might be used
at the close of a piece. The dissonances which occur are universally those which
^ arise from suspended and passing tones ; chords of the dominant Seventh, which,
in modem harmony, play such an important part in marking the key, and in con-
necting and facihtating progressions, were quite unknown.
Great, then, as was the artistic advance in rhythm and the progression of parts,
during this period, it did Httle more for harmony and the tonal system than to
accumulate an imdigested mass of experiments. Since the involved progression of
the parts gave rise to chords in extremely varied transpositions and sequences, the
musicians of this period could not but hear these chords and become acquainted
with their effects, however httle skill they shewed in making use of them. At any
rate, the experience of this period prepared the way for harmonic music proper,
and made it possible for musicians to produce it, when external circumstances forced
on the discovery.
3. Habmonig Music.
IF Modem harmonic music is characterised by the independent significance of its
harmonies, for the expression and the artistic connection of a musical composition.
The external inducements for this transformation of music were of various kinds.
First there was the Protestant ecclesiastical chorus. It was a principle of Protes-
tantism that the congregation itself should undertake the singing. But a congre-
gation could not be expected to execute the artistic rhythmical labyrinths of
Netherland polyphony. On the other hand, the founders of the new confession,
with Luther at their head, were far too penetrated with the power and significance
of music, to reduce it at once to an unadorned unison. Hence the composers of
Protestant ecclesiastical music had to solve the problem of producing simply
harmonised chorales, in which all the voices progressed at the same time. This
excluded those canonic repetitions of the same melodic phrases in different parts,
which had hitherto formed the chief unity of the whole piece. A new connecting
principle had to be looked for in the sound of the tones themselves, and this was
If found in a stricter reference of all to one predominant tonic. The success of this
problem was facilitated by the fact that the Protestant hymns were chiefly adapted
to existing popular melodies, and the popular songs of the Germanic and Celtic
races, as already remarked, betrayed a stricter feeling for tonality in the modem
sense, than those of southern nations. Thus as early as in the sixteenth century,
the system of the harmony of the ecclesiastical Ionic mode (our present major)
developed itself with tolerable correctness, so that these chorales do not strike
modem ears as strange, although they were still without many of our later contri>
vances for marking the key, as, for example, the chord of the dominant Seventh.
On the other hand, it was much longer before the other ecclesiastical modes, in har-
monising which much uncertainty still prevailed, were fused into the modem minor
mode. The Protestant ecclesiastical hymns of that time produced great effects
on the feelings of contemporaries — a fact emphasised on all sides in the liveliest
language, so that no doubt can exist that the impression made by such music, was
something as new as it was peculiarly powerful.
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CHAP. xm.
PERIOD OP HARMONIC MUSIC.
247
In the Roman Church also a desire arose for altering their music. The divisions
of polyphonic music scattered the sense of the words, and made them unintelligible
to the unpractised public, and occasioned even a learned and cultivated hearer great
difficulties in endeavouring to disentangle the knot of voices. In consequence of
the proceedings of the Council of Trent, and by an order of Pope Pius IV. (a.d.
'559-'565)» Palestrina (a.d. 1524-1594) carried out this simplification and embel-
lishment of ecclesiasticiJ music, and the simple beauty of his compositions is said to
have prevented the complete banishment of part music from the Roman liturgy.
Palestrina, who wrote for choruses of singers practised in their art, did not entirely
drop the more complicated progression of parts found in polyphonic music, but by
appropriate sections and divisions he separated and connected both the mass of
tones and the mass of voices, and generally distributed the latter into several dis-
tinct choirs. The voices also are more or less frequently heard together in such
progressions as were used in chorales, and in this case consonant chords greatly ^
predominated. By this means he made his pieces more comprehensible and
intelligible, and in general extremely agreeable to the ear. But the deviation of
ecolesiastical modes from the new modes invented in modern times for the treat-
ment of harmonies, is nowhere so remarkable as in the compositions of Palestrina
and those of contemporary Italian composers of ecclesiastical music, among whom
Giovanni Gabrieli, a Venetian, should be particularly named. Palestrina was a
pupil of Claude Ooudimel (a Huguenot, slain at Lyons in the massacre of St.
Bartholomew), who had harmonised French psalms in a way which, when the scale
was major, was but very slightly different from modem habits. These psalm
melodies had been borrowed, or at least imitated from popular songs. Hence Pales-
trina was certainly acquainted with this mode of treatment, through his teacher,
but he had to deal with themes from the Gregorian Cantus firrrms that moved in
ecclesiastical tones, which he was forced to maintain strictly even in pieces where
he himself invented or adapted the melodies. Now these modes necessitated a f
totally different harmonic treatment, which sounds very strange to modems. As
a specimen I will only cite the commencement of his eight-part Stabat mater.
Here, at the commencement of a piece, just where we should require a steady
characterisation of the key, we find a series of chords in the most varied keys, ^
from A major to F major, apparently thrown together at haphazard, contrary to all
our rules of modulation. What person that was ignorant of ecclesiastical modes
could guess the tonic of the piece from this commencement ? As such we find D
at the end of the first strophe, and the sharpening of C to (^ in the first chord
also points to D. The principal melody too, which is given to the tenor, shews
from the commencement that D is the tonic. But we do not get a minor chord of D
till the eighth bar, whereas a modem composer would have been forced to introduce
it in the first good place he could find in the first bar.
We see from these characters how greatly the nature of the whole system of
ecclesiastical modes differed from our modem keys. We cannot but assume that
masters Uke Palestrina founded their method of harmonisation upon a correct feel-
ing for the peculiar character of those modes, and that, as they could not foil to be
acquainted with the contemporary advances in Protestant ecclesiastical music,
their work was neither arbitrary nor unskilful.
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248 PEEIOD OF HARMONIC MUSIC. part in.
WLat we miss in such examples as the one just adduced, is first, that the tonio
chord does not play the same prominent part at the very commencement that is
assigned to it in modem music. In the latter, the tonic chord has the same
prominent and connecting significance among chords as the tonic or key-note
among the tones of the scale. Next we miss altogether that feeling for the con-
nection of consecutive chords which in modem times has led to the very general
custom of giving them a common tone. This is evidently related to the fact that,
as we shall see hereafter, it was not possible in the old ecclesiastical modes to
produce chains of chords so closely connected with each other and with the tonic
chord, as in the modem major and minor modes.
Hence, although we recognise in Palestnna and Oabrieli a delicate artistic
sensitiveness for the esthetic effect of separate chords of various kinds, and in so
^ far a certain independent significance in their harmonies, yet we see that the means
of establishing an internal connection in the tissue of chords had still to be dis-
covered. This problem, however, required a reduction and transformation of the
previous scales, to our major and minor. On the other hand, this reduction
sacrificed the great variety of expression which depended on diversity of scale.
The old scales partly form transitions between major and minor, and partly enhance
the character of the minor, as in the ecclesiastical Phrygian mode [p. 245^^. This
diversity being lost, it had to be replaced by new contrivances, such as the trans-
position of the scales for different tonics, and the modulational passage from one
key to another.
This transformation was completed during the seventeenth century. But the
most active cause for ihe development of harmonic music is due to the commence-
ment of opera. Tnis had been occasioned by a revival of acquaintance with
^ classical antiquity, and its avowed object was to rehabilitate ancient tragedy, which
was known to have been recited musically. Here arose immediately the problem
of allowing one or two voices to execute solos ; but these again had to be harmo*
nised so as to fit in between the chomses, which were treated in the polyphonic
manner, the object being to make the solo parts stand prominently forward and
keep the accompanying voices well under. These conditions first gave rise to
Eecitative, invented by Giacomo Peri and Gaccini in 1600, and solo songs with
airs, invented by Claudio Monteverde and Viadana. The new view taken of
harmony shews itself in written music by the appearance of figured basses in the
works of these composers. Every figured bass note represented a chord, so that
the chords themselves were settled, but the progression of the parts of which they
were constituted was left to the taste of the player. And thus what was merely
secondary in polyphonic music, became principal, and conversely.
Opera also necessitated the discovery of more powerful means of expression
f than were admissible in ecclesiastical music. Monteverde, who was extremely
prolific in inventions, is the first composer who used chords of the dominant
Seventh without preparation, for which he was severely blamed by his contempo-
rary Artusi. Generally we find a bolder use of dissonances, which were employed
independently, to express sharp contrasts of expression, and not, as before, as
accidental results of the progression of parts.
Under these influences, even as early as in Monteverde's time, the Doric,
Eolic, and Phrygian ecclesiastical modes [p. 254c, d] began to be transformed and
fused into our modem minor mode. This was completed in the seventeenth cen
tury, and these modes were thus made more suitable for giving prominence to the
tonic of the harmony, as will be more fully shewn hereafter.
We have already given an outline of the nature of the influence which these
changes exerted on the constitution of the tonal system. The mode of connecting
musical phrases hitherto in vogue— canonic repetitions of similar melodic figures
— had necessarily to be abandoned as soon as a simple harmonic accompaniment
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CHAP.xm. PERIOD OF HARMONIC MUSIC. 249
had to be subordinated to a melody. Hence some new means of artistic connection
had to be discovered in the somid of the chords themselves. This was effected,
first by making the harmonies refer their tones much more definitely to one pre-
dominant tonic than before, and secondly by giving fresh strength to the rela-
tions between the chords themselves and between all other chords and the tonic
chord. In the course of our investigations we shall see that the distinctive pecu-
liarities of the modem system of tones can be deduced from this principle, and
that the principle itself is very strictly carried out in our present music. In
reality the mode in which the materials of music are now worked up for artistic
use, is in itself a wondrous work of art, at which the experience, ingenuity, and
esthetic feeling of European nations has laboured for between two and three thou-
sand years, since the days of Terpander and Pythagoras. But the complete for-
mation of the essential features as we now see it, is scarcely two hundred years
old in the practice of musical composers, and theoretical expression was not given ^
to the new principle till the time of Rameau at the beginning of last century. In
the historical point of view, therefore, it is wholly the product of modem times,
limited nationally to the German, Roman, Celtic, and Sclavonic races.
With this tonal system, which admits great wealth of form with strictly defined
artistic consistency, it has become possible to construct works of art, of much
greater extent, and much richer in forms and parts, much more energetic in
expression, than any producible in past ages ; and hence we are by no means
inclined to quarrel with modem musicians for esteeming it the best of all, and
devoting their attention to it exclusively. But scientifically, when we proceed to
explain its contitraction and display its consistency we must not forget that our
modem system was not developed from a natural necessity, but from a freely
chosen principle of style ; that beside it, and before it, other tonal systems have
been developed from other principles, and that in each such system the highest
pitch of artistic beauty has been reached, by the successful solution of more %
limited problems.
This reference to the history of music was necessitated by our inability in this
case to appeal to observation and experiment for establishing our explanations,
because, educated in a modem system of music, we cannot thoroughly throw our-
selves back into the condition of our ancestors, who knew nothing about what we
have been familiar with from childhood, and who had to find it all out for them-
selves. The only observations and experiments, therefore, to which we can appeal,
are those which mankind themselves have undertaken in the development of music.
If our theory of the modem tonal system is correct it must also suffice to furnish
the requisite explanation of the former less perfect stages of development.
As the fundamental principle for the development of the European tonal
system, we shall assume that the whole mass of tones and the connection of har-
monies must stand in a close and always distinctly perceptible relationship to soms
arbitrarily selected tonic, and that the inass of tone which forms the whole compo- %
sition, must be developed from this tome, and mv^t finally return to it. The
ancient world developed this principle in homophonic music, the modem world in
harmonic music. But it is evident that this is merely an esthetical principle, not
A natural law.
The correctness of this principle cannot be established d priori. It must
be tested by its results. The origin of such esthetical principles should not be »
ascribed to a natural necessity. They are the inventions of genius, as we previously
endeavoured to illustrate by a reference to the principles of architectural style.
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CHAPTER XTV.
THE TONALITY OP HOMOPHONIC MUSIC.
Musio was forced first to select artistically, and then to shape for itself, the material
on which it works. Painting and sculpture find the fundamental character of
their materials, form and colour, in nature itself, which they strive to imitate.
Poetry finds its material ready formed in the words of language. Architecture
has, indeed, also to create its own forms ; but they are partly forced upon it by
technical and not by purely artistic considerations. Music alone finds an infi-
nitely rich but totally shapeless plastic material in the tones of the human voice and
artificial musical instruments, which must be shaped on purely artistic principles,
unfettered by any reference to utility as in architecture, or to the imitation of
^ nature as in the fine arts, or to the existing symbolical meaning of sounds as in
poetry. . There is a greater and more absolute freedom in the use of the material
for music than for any other of the arts. But certainly it is more difficult to make
a proper use of absolute fi:eedom,than to advance where external irremovable land-
marks limit the width of the path which the artist has to traverse. Hence also
the cultivation of the tonal material of music has, as we have seen, proceeded much
more slowly than the development of the other arts.
It is now our business to investigate this cultivation.
The first feust that we meet with in the music of all nations, so £Eur as is yet
known, is that alterations of pitch in melodies take place by intervals^ and not by
contimums transitions. The psychological reason of this fact would seem to be
the same as that which led to rhythmic subdivision periodically repeated. All
melodies are motions within extremes of pitch. The incorporeal material of tones
is much more adapted for following the musician's intention in the most delicate
^ and phant maimer for every species of motion, than any corporeal material how-
ever light. Graceful rapidity, grave procession, quiet advance, wild leaping, all
these different characters of motion and a thousand others in the most varied
combinations and degrees, can be represented by successions of tones. And as
music expresses these motions, it gives an expression also to those mental con-
ditions which naturally evoke similar motions, whether of the body and the voice, or
of the thinking and feeling principle itself. Every motion is an expression of the
power which produces it, and we instinctively measure the motive force by the
amount of motion which it produces. This holds equally and perhaps more for the
motions due to the exertion of power by the human will and human impulses,
than for the mechanical motions of external nature. In this way melodic pro-
gression can become the expression of the most diverse conditions of human dis-
position, not precisely of hximei^a feelings * but at least of that state of sensitiven>ess
which is produced by feelings. In English the phrase out of tune, unstrung j and
^ in German the word stimmung, literally tuning, are transferred from music to
mental states. The words are meant to denote those peculiarities of mental con-
dition which are capable of musical representation. I think we might appro-
priately define gemilthsstimmung, or mentaZ tune, as representing that general cha-
racter temporarily shewn by the motion of our conceptions, and correspondingly
impressed on the motions of our body and voice. Our thoughts may move fa*st or
slowly, may wander about restlessly and aimlessly in anxious excitement, or may
keep a determinate aim distinctly and energetically in view; they may lounge
about without care or effort in pleasant fancies, or, driven back by some sad
memories, may return slowly and heavily from the spot with short weak steps.
All this may be imitated and expressed by the melodic motion of the tones, and
the listener may thus receive a more perfect and impressive image of the ' tune ' of
* Hanslick seems to me to have the advan- means of clearly oharacterising the object ol
tage over other esthetic writers in this point, feeling,
because music, unassisted by poetry, has no
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CHAP. XIV. PROGEESSION BY INTERVALS. 251
another person's mind, than by any other means, except perhaps by a very perfect
dramatic representation of the way in which such a person really spoke and acted.
Aristotle also formed a similar conception of the effect of music. In his 29th
problem he says : * Why do rhythms and melodies, which are composed of sound,
resemble the feelings ; while this is not the case for tastes, colours, or smells ? Can
it be because they are motions, as actions are also motions ? Energy itself belongs
to feeling and creates feeling. But tastes and colours do not act in the same way.' *
And at the end of the 27th problem he says : * These motions, i.e. rhythms and
melodies, are active, and action is the sign of feeling.' f
Not merely music but even other kinds of motions may produce similar effects.
Water in motion, as in cascades or sea waves, has an effect in some respects
similar to music. How long and how often can we sit and look at the waves
rolling in to shore I Their rhythmic motion, perpetually varied in detail, produces
a peculiar feeling of pleasant repose or weariness, and l^e impression of a mighty V
orderly life, finely linked together. When the sea is quiet and smooth we can
enjoy its colouring for a while, but this gives no such lasting pleasure as the rolling
waves. Small undulations, on the other hand, on small surfaces of water, follow
one another too rapidly, and disturb rather than please.
But the motion of tone surpasses all motion of corporeal masses in the delicacy
and ease with which ;t can receive and imitate the most varied descriptions of
expression. Hence it arrogates to itself by right the representation of states of
mind, which the other arts can only indirectly touch by shewing the situations
which caused the emotion, or by giving the resulting words, acts, or outward
appearance of the body. The union of music to words is most important, because
words can represent the cause of the frame of mind, the object to which it refers,
and the feeling which lies at its root, while music expresses the kind of mental
transition which is due to the feeling. When different hearers endeavour to de-
scribe the impression of instrumental music, they often adduce entirely different %
situations or feelings which they suppose to have been symbolised by the music.
One who knows nothing of the matter is then very apt to ridicule such enthusiasts,
and yet they may have been all more or less right, because music does not represent
feelings and situations, but only frames of mind, which the hearer is unable to
describe except by adducing such outward circumstances as he has himself noticed
when experiencing the corresponding mental states. Now different feelings may
occur under different circumstances and produce the same states of mind in dif-
ferent individuals, while the same feelings may give rise to different states of mind.
Love is a feeling. But music cannot represent it directly as such. The mental
states of a lover may, as we know, shew the extremest variety of change. Now
music may perhaps express the dreamy longing for transcendent bhss which love
* Aid ri ol pvBfxoi Kcd rh, fA4\ri ^mvif qIvo, the only sensation which excites the feelings ?
'f^ctf'u^ loucer • •/ Hk x^f^^ ^^* ^^^' ^^^^ '''^ Even melody without words has feeling. But
Xp^fwra Kcd €d6(rfiai;*H6riKurfi(r€is€Ur\y/&ffir€p this is not the case for colour, or smell, or ^
Kcd al trpd^tis ; ffiri Z\ ^ fiky Mpytta ii9uchy, koL taste. Is it because they have none of the
woui liBos' ol 9h X"f">^ *^^ "^^ XP^M^'''^ ^^ iroiovffiy motion which sound excites in us ? For the
Afioiws, Arist. Prob, xix. 29. others excite motion ; thus colour moves the
t [The above words conclude the problem, eye. But we feel the motion which follows
which it seems best to cite in full. Aiii r( rh sound. And this is alike, in rhythm, and
iucovarhv lUvov ^dos Ixci r&v ataBrrruy ; kc^ 7^ alteration in pitch, but not in united sounds.
ihf f &ycv kj6yov fA4\os, ZfjL»t %x^i ^Bor &AX* ob Sounding notes together does not excite feel-
T^ Xpif/Mf oitik ^ ifffiii, ov9h 6 xvfi^f ^X<t« *H ing. This is not the case for other sensations.
8ti Klr/iffiu ^x^i iiovovovx^i hv ^ ^6<bos ii/xas Kivti; Kow these motions stimulate action, and this
Totwdrri fjL^y yikp Kcd rois JixkoLs iiripxfi, fctyti ykp action is the sign of feeling.' Aristotle seems
«a2 T^ Xf^MA [f^oX] r^y i^iy ■ tiXAA rris ivofitimis to have required motion to excite feeling, and
T^ TQio^^ ^^^ tdada»6tA§$a Kty4ia9ms, ASrti in sounding two notes together, there was no
9i tx^i SfioUrnrOf Hy re rots fve/iois koX 4y rp motion of one towards the other. It is evi-
T&y ^$6yywy rd^ti r&y 6^4»y xai fiapimy, o^k iy dent that he had not the slightest inkling of a
T$ /tf|ci. *AAA' ii ifvfA^yia ovk Ix^' ^^'- '^'^ progression of /larmomes, and this utter blank
Si Toif &AAOU al(r07irois rovro oIk forty. Ai 8i in his mind is one of the strongest proofs that
KUfiiTtts o^rai, itptutrtKai flaiy. At 9^ irpd^tis, ijBovs the Greeks had never tried harmony. *Apfioyia
irriiutMrta i<rrl. Arist. Prob, xix. 27. Which we had the modem meaning of melody ; fi€\<f9ia
may perhaps translate thus : ' Why is sound was words set to mvLOC.— Translator,]
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252 PROGRESSION BY INTERVALS. pabt in.
may excite. But precisely the same state of mind might arise from religious
enthusiasm. Hence when a piece of music expresses this mental state it is not
a contradiction for one hearer to find in it the longing of love, and another the
longing of enthusiastic piety. In this sense Vischer's rather paradoxical state-
ment that the mechanics of mental emotion are perhaps best studied in their
musical expression, may be not altogether incorrect. We really possess no other
means of expressing them so exactly and deUcately.
As we have seen, then, melody has to express a motion, in such a manner that
the hearer may easily, clearly, and certainly appreciate the character of that motion
by immediate perception. This is only possible when the steps of this motion,
their rapidity and their amount, are also exactly measurable by iromediate sensible
perception. Melodic motion is change of pitch in time. To measure it perfectly,
the length of time elapsed, and the distance between the pitches, must be measur-
^ able. This is possible for iromediate audition only on condition that the altera-
tions both in time and pitch should proceed by regular and determinate degrees.
This is immediately clear for time, for even the scientific, as well as all other
measurement of time, depends on the rhythmical recurrence of similar events, the
revolution of the earth or moon, or the swings of a pendulum. Thus also the
regular alternation of accentuated and unaccentuated sounds in music and poetry
gives the measure of time for the composition. But whereas in poetry the con-
struction of the verse serves only to reduce the external accidents of linguistic
expression to artistic order ; in music, rhythm, as the measure of time, belongs to
the inmost nature of expression. Hence also a much more delicate and elaborate
development of rhythm was required in music than in verse.
It was also necessary that the alteration of pitch should proceed by intervals,
because motion is not measurable by immediate perception unless the amount of
space to be measured is divided off into degrees. Even in scientific investigations
^ we are unable to measure the velocity of continuous motion except by comparing
the space described with the standard measure, as we compare time with the seconds
pendulum.
It may be objected that architecture in its arabesques, which have been
justly compared in many respects with musical figures, and which also shew
a certain orderly arrangement, constantly employs curved lines and not lines
broken into determinate lengths. But in the first place the art of arabesques really
began with the Greek meander, which is composed of straight lines set at right
angles to each other, following at exactly equal lengths, and cutting one another
off in degrees. In the second place, the eye which contemplates arabesques can
take in and compare all parts of the curved lines at once, and can glance to and
fro, and return to its first contemplation. Hence, notwithstanding the continuous
curvature of the lines, their paths are perfectly comprehensible, and it became
possible to renounce the strict regularity of the Grecian arabesques in favour of
^ the curvilinear freedom. But whilst freer forms are thus admitted for individual
small decorations in architecture, the division of any great whole, whether it be a
series of arabesques or a row of windows or colunms, &c., throughout a building, is
st'U tied down to the simple arithmetical law of repetition of similar parts at equal
intervals.
The individual parts of a melody reach the ear in succession. We cannot per-
ceive them all at once. We cannot observe backwards and forwards at pleasure.
Hence for a clear and sure measurement of the change of pitch, no means was
left but progression by determinate degrees. This series of degrees is laid down
in the musical scale. When the wind howls and its pitch rises or falls in insensible
gradations without any break, we have nothing to measure the variations of pitch,
nothing by which we can compare the later with the earlier sounds, and compre-
hend the extent of the change. The whole phenomenon produces a confused, un-
pleasant impression. The musical scale is as it were the divided rod, by which we
measure progression in pitch, as rhythm measures progression in time. Hence
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CHAP. XIV. RATIONAL CONSTRUCTION OF DIATONIC SCALES. 253,
the analogy between the scale of tones and rhythm naturally occurred to musical
theoreticians of ancient as well as modem times.
We consequently find the most complete agreement among all nations that use
music at all, from the earliest to the latest times, as to the separation of certain
determinate degrees of tone from the possible mass of continuous gradations
of sound, all of which are audible, and these degrees form the scale in which
the melody moves. But in selecting the particular degrees of pitch, deviations
of national taste become inunediately apparent. The number of scales used by
different nations and at different times is by no means small.
Let us inquire, then, what motive there can be for selecting one tone rather
than another in its neighbourhood for the step succeeding any given tone. We
remember that in sounding two tones together such a relation was observed. We
found that under such circumstances certain particular intervals, namely the con-
sonances, were distinguished from all other intervals which were nearly the same, ^
by the absence of beats. Now some of these intervals, the Octave, Fifth, and
Fourth, are found in all the musical scales known.* Recent theoreticians that
have been bom and bred in the system of harmonic music, have consequently
supposed that they could explain the origin of the scales, by the assumption that
all melodies arise from thinking of a harmony to them, and that the scale itself,
considered as the melody of the key, arose from resolving the fundamental chords
of the key into their separate tones. This view is certainly correct for modem
scales ; at least these have been modified to suit the requirements of the harmony.
But scales existed long before there was any knowledge or experience of harmony at
all. And when we see historically what a long period of time musicians required
to learn how to accompany a melody by harmonies, and how awkward their first
attempts were, we cannot feel a doubt that ancient composers had no feeling at all
for harmonic accompaniment, just as even at the present day many of the more
gifted Orientals are opposed to our own harmonic music. We must also not forget ^
that many popular melodies, of older times or foreign origin, scarcely admit of any
harmonic accompaniment at all, without injury to their character.
The same remark applies to Rameau's assumption of an * understood ' funda-
mental bass in the constraction of melodies or scales for a single voice. A modern
composer would certainly imagine to himself at once tlie fundamental bass to the
melody he invents. But how could that be the case with musicians who had never
heard any harmonic music, and had no idea how to compose any ? Granted that
an artist's genius often unconsciously * feels out * many relations, we should be
imputing too much to it if we asserted that the artist could observe relations of
tones which he had never or very rarely heard, and which were destined not to be
discovered and employed till many centuries after his time.
It is clear that in the period of homophonic music, the scale could not have
been constructed so as to suit the requirements of chordal connections uncon-
sciously supplied. Yet a meaning may be assigned, in a somewhat altered form, ^
to the views and hypotheses of musicians above mentioned, by supposing that the
same physical and physiological relations of the tones, which become sensible
when they are sounded together and determine the magnitude of the consonlint
intervals, might also have had an effect in the construction of the scale, although
under somewhat different circumstances.
Let us begin with the Octave, in which the relationship to the fundamental tone
is most remarkable. Let any melody be executed on any instrument which has a
good musical quahty of tone, such as a human voice ; the hearer must have heard
not only the primes of the compound tones, but also their upper Octaves, and, less
strongly, the remaining upper partials. When, then, a higher voice afterwards
executes the same melody an Octave higher, we hear again a part of what we
heard before, namely the evenly numbered partial tones (p. 49^^) of the former
* [It will be seen in App. XX. sect. E. that the Fourth and Fifth are often materially inexact
or designedly eMered, -Traiislator,]
Digitized by V^jOOQlC
254 RELATIONSHIP OF COMPOUND TONES. part in.
compound tones, and at the same time we hear nothing that we had not previously
heard. Hence the repetition of a melody in the higher Octave is a real repetition
of what has been previously heard, not of all of it, but of a part.* If we allow a
low voice to be accompanied by a higher in the Octave above it, the only part
music which the Greeks employed, we add nothing new, we merely reinforce the
evenly numbered partials. In this sense, then, the compound tones of an Octave
above are really repetitions of the tones of the lower Octaves, or at least of part of
their constituents. Hence the first and chief division of our musical scale is that into
a series of Octaves. In reference to both melody and harmony, we assume tones
of different Octaves which bear the same name, to have the same value, and, in the
sense intended, and up to a certain point, this assumption is correct. An accom-
paniment of Octaves gives perfect consonance, but it gives nothing additional ; it
merely reinforces tones already present. Hence it is musically applicable for in-
% creasing the power of a melody which has to be brought out strongly, but it has
none of the variety of polyphonic music, and therefore is felt to be monotonous,
and it is consequently forbidden in polyphonic music.
What is true of the Octave is true in a less degree for the Twelfth. 11 a melody
is repeated in the Twelfth we again hear only what we had already heard, but the
repeated part of what we heard is much weaker, because only the third, sixth,
ninth, &c., partial tone is repeated, whereas for repetition in the Octave, instead of
the third partial, the much stronger second and weaker fourth partial is heard, and
in place of the ninth, the eighth and tenth occur, &c. Hence repetition of a
melody in the Twelfth is less complete than repetition in the Octave, because only
a smaller part of what had been already heard is repeated. In place of this
repetition in the Twelfth, we may substitute one an Octave lower, namely in the
Fifth. Bepetition in the Fifth is not a pure repetition, as that in the Twelfth is.
Taking 2 for the pitch number of the prime tone, the partials are (197c, d)
H for the fundamental compound .2 4 6 S 1012-
for the Twelfth .... 6 12
for the Fifth 3 69 12
When we strike the Twelfth we repeat the simple tones 6 and 12, which already
existed in the fundamental compound tone. When we strike the Fifth, we continue
to repeat the same simple tones, but we also add two others, 3 and 9. Hence for
the repetition in the Fifth, only a part of the new sound is identical with a part of
what had been heard, but it is nevertheless the most perfect repetition which can be
executed at a smaller interval than an Octave. This is clearly the reason why
unpractised singers, when they wish to join in the chorus to a song that does not
suit the compass of their voice, often take a Fifth to it. This is also a very evident
proof that the uncultivated ear regards repetition in the Fifth as natur^J. Such
an accompaniment in the Fifth and Fourth is said to have been systematically
developed in the early part of the middle ages. Even in modem music, repetition
II in the Fifth plays a prominent part next to repetition in the Octave. In normal
fugues the theme, as is well known, is first repeated in the Fifth ; in the normal
form of instrumental pieces, that of the Sonata, the theme in the first movement
is transposed to the Fifth, returning in the second part to the fundamental tone.
This kind of imperfect repetition of the impression in the Fifth induced the Greeks
also to divide the interval of the Octave into two equivalent sections, namely two
Tctrachords. Our major scale on being divided in this manner would be : —
d e f g a b c' d' e' f
II. m.
* [Some considerations have been omitted, powerful as in the higher tone. The upper
probably by design. The quality of tone of partials of the higher tone, which are still
the Yoice which sings the Octave above is quite effective, would be inaudible in the lower
materially different. The evenly numbered tone. — Translator.']
partials of the lower tone are by no means so
Digitized by V^jOOQlC
CHAP. XIV. RELATIONSHIP OF COMPOUND TONES. 255
The suocession of tones in the second tetrachord is a repetition of that in the
first, transposed a Fifth.* To pass into the Octave division, the successive tetra-
chords must be alternately separate and connected. They are said to be connected,
or conjunct, when, as in II. and III., the last tone c of the lower becomes the first
of the higher tetrachord ; and separate, or disjunct, when, as in I. and 11., the last
tone of the lower is different from the upper. In the second tetrachord ^ to 0, every
ascending series of tones necessarily leads to d as the final tone, and this c' is also
the Octave of the fundamental tone of the first tetrachord. Now this & is the Fourth
of g, the fundamental tone of the second tetrachord. To make the succession of
tones the same in both tetrachords, the lower tetrachord had to be increased by the
tone / which answers to c'. The Fourth /, however, would have suggested itself
in the same way as the Fifth, independently of this analogy of the tetrachords.
The Fifth is a compound tone in which the second partial is the third partial of the
fundamental compound tone ; the Fourth is a compound tone in which the third ^
partial is the same as the second of the Octave. Hence the limits of the two
analogous divisions of the Octave are settled, namely : —
c— /, g — c,
but the mode of filling up these gaps remains arbitrary, and different plans for
doing so were adopted by the Greeks themselves at different periods, and others
again by other nations. But the division of the scale into octaves, and the octave
into two analogous tetrachords, occurs everywhere, almost without exception.
Boethius (De Musica, lib. i. cap. 20) informs us that according to Nicomachus
the most ancient method of tuning the lyre down to the time of Orpheus, con-
sisted of open tetrachords,
Q—f—g — d,
with which certainly it was scarcely possible to construct a melody. But as it ^
contained the chief degrees of the pitch of ordinary speech, a lyre of this kind
might possibly have served to accompany declamation.
The relationship of the Fifth, and its inversion the Fourth, to the fundamental
tone, is so close that it has been acknowledged in all known systems of music.f
On the other hand, many variations occur in the choice of the intermediate tones
which have to be inserted between the terminal tones of the tetrachord. The
interval of a Third is by no means so clearly defined by easily appreciable partial
tones, as to have forced itself from the first on the ear of unpractised musicians.
We must remember that even if the fifth partial tone existed in the compound
tones of the musical instruments employed, it would have had to contend with the
much louder prime tone, and would also have been covered by the three adjacent
and lower partials. As a matter of fact, the history of musical systems shews that
there was much and long hesitation as to the tuning of the Thirds. And the doubt
is even yet felt when Thirds are used in pure melody, unconnected with any har- fl
monies. I must own that on observing isolated intervals of this kind, I cannot
come to perfectly certain results, but I do so when I hear them in a well-constructed
melody with distinct tonahty. The natural major Thirds of 4 : 5 thus seem to me
calmer and quieter than the sharper major Thirds of our equally tempered modern
instruments, or with the still sharper major Thirds which result from the Pytha-
gorean tuning with perfect Fifths. Both of the latter intervals have a strained
effect. Most of our modem musicians, accustomed to the major Thirds of the
equal temperament, prefer them to the perfect major Thirds, when melody alone is
concerned. But I have convinced myself that artists of the first rank, like Joachim,
use the Thirds of 4 : 5 even in melody. For harmony there is no doubt at all.
* [This applies to the Pythagorean scale whereas ^ to a is a minor Tone and a to 5 a
and hence to Greek music, and also to all major Tone. These distinctions were of coarse
tempered music. But in just intonation c to (2 purposely omitted in the text. — Translator.]
is a major Tone, and d to e & minor Tone, f C^^t see App. XX. sect. K,— Translator,]
Digitized by V^jOOQlC
256 RELATIONSHIP OP COMPOUND TONES. part iii.
Every one chooses the natural major Thirds. In Chapter XVI. I shall describe on
instrument which will enable any one to perform experiments of this kind.*
Under these circumstances another principle for determining the small intervals
of the scale was resorted to during the in&ncy of music, and seems to be still
employed among the less civilised nations. This principle, which has subsequently
had to yield to that of tonal relationship, consists in an endeavour to distinguish
equal intervals by ear, and thus make the differences of pitch perceptibly uni-
form.
This attempt has never prevailed over the feeling of tonal relationship for
the division of the Fourth, at least in artistically developed music. But in the
division of smaller intervals we shall find it applied as an auxiliary in many of the
less usual divisions of the Greek tetrachord and in the scales of Oriental nations.
But arbitrary divisions which are independent of tonal relationship, disappeared
^ everywhere in exact proportion to the higher development of the musical art.
We will now inquire what kind of a scale we should obtain by pursuing to its
consequences the natural relationship of the tones. We shall consider musical
tones to be related in the first degree which have two identical partial Umes ; and
related in the second degree, when they are both related in the first degree to some
third musical tone. The louder the coincident in proportion to the non-coincident
partials of compound tones related in the first degree, the closer is their relation-
ship and the more easily will both singers and hearers feel the common character
of both the tones. Hence it follows that the feeling for tonal relationship ought to
differ with the qualities of tone : and I believe that this states a &ct in nature,
because flutes and the soft stops of organs, on which chords are somewhat colour-
less owing to an absence of upper partials and a consequent incomplete definition
of dissonances, retain much of the same colourless character in melodies. This,
I think, depends upon the fact, that, for such qualities of tone, the recognition of
IT the natural intervals of the Thirds and Sixths, and perhaps even of the Fourths
and Fifths, does not result from the immediate sensation of the hearer, but at most
from his recollection. When he knows that on other instruments and in singing
he has been able by immediate sensation to recognise the Thirds and Sixths as
naturally related tones, he acknowledges them as well-known intervals even when
executed by a flute or on the soft stops of an organ. But the mere recollection of
an impression cannot possibly have the same freshness and power as the immediate
sensation itself.
Since the closeness of relationship depends on the loudness of the coincident
upper partial tones, and those having a higher ordinal number are usually weaker
than those having a lower one, the relationship of two tones is generally weaker,
the greater the ordinal number of the coincident partials. These ordinal numbers,
as the reader will recollect from the theory of consonant intervals, also give the
ratio of the vibrational numbers of the corresponding notes.
^ In the following table, the first horizontal line contains the ordinal numbers of
the partial tones of the tonic o, and the first vertical column those of the corre-
sponding tone in the scale. Where the corresponding vertical columns and hori-
zontal lines intersect, the name of the tone of the scale is given for which this
coincidence holds. Only such notes are admitted as are distant from the tonic by
less than an Octave. Below each degree of the scale are placed the two ordinal
numbers of the coincident partials, which will serve as a scale for measuring the
closeness of the relationship.
* [Other experimental InstrameDts will be not so harsh but quite near enough to shew its
described in App. XX. sect. F. The Harmoni- character. For the intervals used by violinists,
cal gives only the just major Third 4 : 5. Its see also App. XX. sect. G. arts. 6 and 7. —
nearest approach to the Pythagorean 64 : 81, Translator. '\
or 408 cents, is 'Bb : X>, = 63 :*8o, or 413 cents,
Digitized by
Google
CHAP. XIV.
PENTATONIC SCALES.
257
Partial Tones of the Tonio
X
9
3
4
5
6
c
I : I
I : 2
C
2 : I
2 : 2
9
2:3
2:4
F
3:2
c
3:3
/
3:4
3:5
3:6
C
4:2
0
4:3
c
4:4
4:5
4?6
5:3
ilb
5:4
c
5:5
6b
s:6
C
6:3
F
6:4
A
6:5
c
6:6
IF
In this systematic comparison we find the following series of notes lying in
the octave above the fandamental note c, and related to the tonic 0 in the first
degree, arranged in the order of their relationship :
c c^ g f a c eb
1:1 1:2 2:3 3:4 3:5 4:5
and the following series in the descending octave :
C F O E^}
5:6
0
I : I
4: 3
5 :3
5:4
A
6:5
The series is discontinued when the resultant intervals become very close.
Intervals adapted for practical use must not be too close to be easily taken and ^
distinguished. What is the smallest interval admissible in a scale is a question
which different nations have answered differently according to the different
direction of their taste, and perhaps also according to the different dehcacy of
their ear.
It seems that in the first stages of the development of music many nations
avoided the use of intervals of less than a Tone, and hence formed scales, which
alternated in intervals from a Tone to a Tone and a half. According to examples
collected by M. F^tis,* a scale of this kind is found not only among the Chinese
but also among the other branches of the Mongol race, among the Malays of Java
and Sumatra, the inhabitants of Hudson's Bay, the Papuas of New Guinea, the
inhabitants of New Caledonia, and the Fullah negroes.f The five-stringed lyre
(Kissar) of the inhabitants of North Africa and Abyssinia, which is represented in
the bas-r^liefis of the Assyrian palaces as an instrument play ed on by captives, was
also, according to Villoteai},^ tuned by the scale of five degrees : ^
^ — a — 6 — d' — e'§
Traces of an ancient scale of this kind are clearly furnished by the five-stringed
lyre or lute {KiBdpa) of the Greeks. At least Terpander (circa B.C. 700-650), who
played a conspicuous part in the development of ancient Greek music, and who
added a seventh string to the former Cithara of six strings, used a scale composed
of a tetrachord and a trichord, having the compass of an Octave and tuned
thus : —
e^f^g^a-'b^ — d' — e'**
* Hiataire GinSrale de la Musique^ Paris,
1869, vol. i.
t [See App. XX. sect. E. for pentatonio
scales in Java, China, and Japan. — Trans-
latar.]
X DescripHons des Instrum&nts de Mtisique
des Orieniaux ; chap. xiii. in the Description
de VEffypte. 6tat Modeme.
§ [This is probably only a rude approxi-
mation or a guess. See App. XX. sect. K. for
observations on existing pentatonic scales actu-
ally heard.— Translator.]
** Nicomachus makes Philolaus say (edit.
Meibomii, p. 17), * From the Hypate Jte) to '
Digitized by VJ
8^'§k
258 PENTATONIC SCALES. part ni.
in which there is no c', and the upper tetrachord has no interval of a Semitone,
although there is an interval of this kind in the lower.*
Olympos (circa B.C. 660-620), who introduced Asiatic flute music into Greece
aiid ieidiBbpted it to Greek tastes, transformed the Greek Doric scale into one of five
tones, the old enharmonic scale
b^^c e^^f at
This seems to indicate that he brought a scale of five tones with him &om Asia,
and merely borrowed the use of the intervals of a Semitone firom the Greeks.
Among the more cultivated nations, the Chinese and the Celts of Scotland and
Irelsiiid still retain the scale of five notes without Semitones, although both have
also become acquainted with the complete scale of seven notes.
Among the Chinese, a certain prince Tsay-yu is said to have introduced the
IF scale of seven notes amid great opposition firbm conservative musicians. The
division of the Octave into twelve Semitones, and the transposition of scales have
also been discovered by this intelligent and skilful nation. But the melodies tran-
scribed by travellers mostly belong to the scale of fi^e notes. The Gaels and Erse
have likewise become acquainted with the diatonic scale of seven tones by means
of psalmody, and in the present form of their popular melodies the missing tones
are sometimes just touched as appoggiature or passing notes. These are, however,
in many cases merely modern improvements, as may be seen on comparing the
older forms of the melodies, and it is usually possible to omit the notes which do
not belong to the scale of five tones without impairing the melody. This is not
only true of the older melodies, but of more modem popular airs which were com-
posed during the last two centuries, whether by learned or unlearned musicians.
Hence the Gaels as well as the Chinese, notwithstanding their acquaintance with
the modem tonal system, hold fast by the old.t And it cannot be denied that by
if avoiding the Semitones of the diatonic scale, Scotch airs receive a peculiarly bright
and mobile character, although we cannot say as much for the Chinese. Both
Gaels and Chinese make up for the small number of tones within the Octave by
great compass of voice. §
The scale of five tones admits of a certain variety in its construction. Assume
c as the tonic and add to it the nearest related notes in the ascending Octave, till
you come to a Semitone. This gives
c — c' — g — / — a.
The next note e would form a Semitone with/. In the descending Octave we find
in the same way
c—-C — F—-G — Ei).
The great gaps in the scales between c and/ in the first, and between G and c
H in the second are filled up by tones related in the second degree. Since the tones
related to the Octave can only be repetitions of those directly related to the tonic,
Mese (a) wag a Fourth, from the Mese (a) to bition in London, 1884, gives many varieties
the Ncte (e') a Fifth, from the Nete {e') to the of this scale, see App. XX. sect. E. Japan. —
Trite (6) a Fourth, from the Trite (6) to the Translator,]
Hypate (e) a Fifth.* This shews that c, not 6, % Chinese Melodies, in Ambrosch's Oe-
was the missing note. schichte der Muaikt vol. i. pp. 30, 34, 35. Of
* [The upper tetrachord was thus reduced Scotch melodies there is a fine collection
to a trichord, while the lower remained a per- with reference to the authorities and the older
feet tetrachord. If we take Pythagorean into- forms in G. F. Graham's Songs of Scotland,
nation the cents are e 90 / 204 g 204 a 204 3 vols. Edinburgh, 1859. The modem piano-
b 294 d' 204 e'. — TranslatorJ] forte accompaniment which has been added, is
f [Taking Pythagorean intonation, the often ill enough suited to the character of the
cents in the intervals are 6 90 c 408 egof 408 airs.
a 204 b. The account of the popular tuning § [Exclusive of the two drones there are
of the Eo-to, the national Japanese instru- only 9 tones on the bagpipe. For the whole
ment, furnished by the Japanese, but in Euro- of these observations see App. XX. sect. K. —
pean notes, at the International Health Ezhi- Translator.]
Digitized by V^jOOQlC
CHAP. XIV.
PENTATONIC SCALES.
259
the next tones to be considered are those related to the upper Fifth g, and lower
Fifth F, and these are d (the Fifth above the upper Fifth g) and B^ (the Fifth
•below the lower iFiith F). We thus obtain the scales*
i) Ascending
c — d — ^-f — g — a — ^c'
If $ t *
2) Descending
C — ^E^—F—O — ^B0—c
* J t V 2
But in place of the tones more distantly related to the tonic in the first degree,
both systems of tones related in the second degree might be used, and this would
give a scale resulting from a simple progression by Fifths, as
3)
c — d — -*-/ — g — -^ftt> — cf
Then there are also some more irregular forms of this scale of five tones, in
which the major Third e replaces the Fourth /, which is more nearly related to
the tonic c. This transformation is probably due to the modem preference for the
major mode, and it has made its appearance in very many Scotch melodies. The
scale is then
4) c — d — e — ^-^g — a — ^^c'
f
The examples of a similar exchange of the Fifth g for the minor Sixth at> are
doubtful. This would give the scale
s) 0— ^JSb— i?"—- '4)— jBi>— c
The scale 6) c — — 4> — / — 9 — « — ~^c
f * i 4 » ^
in which all the notes are related in the first degree, but for which the nearest
notes to the tonic, either way, are a Tone and a Semitone distant from it, has not
jet been discovered in actual use.
The above five forms of the scale of five tones can all be so transposed that
they can be played on the black notes of a piano without touching the white ones.f
•This is the well-known simple rule for composing Scotch melodies.J Any one of
4) c 204 d 182 e, 316 g 182 a, 316 c'.
5) C3i6^'bi82ii^3i6-4»b 204B»bi82c.
6) c 316 g'b 182/ 204 g 182 a, 316 c—
Translator.]
■f [In the following way— the numbers re-
ferring to the BohemeB in this page, and the
oorresponding cents, of course, belonging to
equal temperament :
ZOO eft
300 eft 200 d't
300 c't 200 d't 300 ft 200 g't
300 eft 200 d't 300/15
Anderson's College, Glasgow, in The Thistle,
* a miscellany of Scottish song, with notes
critical and historical ; the melodies arranged
in their natural modes ; with an introduction,
explaining the construction and characteristics
of Scottish music, the Principles, Laws, and
Origin of Melody * (Glasgow, Bee. 1883), says,
p. viii. : * The pentatonic form of the scale is
used in Scotland, but not to a greater extent
than in the national music of some other
countries. A general idea seems to prevail
that Scottish music can be played upon the
five black digitals of the pianoforte , re|j|e^ej^ing
* [In the following investigation the
Author all along assumes harmonic forms of
•the intervals, which are certainly modem.
The cents in the five forms cited, as deter-
mined from the ratios given, are :
i) c 204 d 294 / 204 g 204 a 316 c'.
2) C3i6^*b 182^204 0294 Bb 204c.
3) c 204 (Z 294/ 204 gf 294 6 b 204 c'.
1) cj 200 d» 300/5 200 g» 200 aJJ
2) dt 300 ft 300 gt 200 at
3) gt 200 at
4) ^ 200 ^« 200 at
5) At 300 ct 200 dt 300/1 200 gt 200 at
And this shews that all five are formed by a
Bimi^e succession of tempered Fifths, for the
five black notes arranged in order of Fifths are
ft 700 et 700 gt 700 dt 700 at . The piano
being tuned in equal temperament gives very
nearly perfect Fifths, and hence very well
imitates a succession of five notes thus tuned.
If, however, the Fifths are perfect, then every
200 and 300 cents in the above scheme becomes
204 and 294, differences which few ears will
perceive in melody.
% [Mr. Colin Brown, Euing Lecturer on
the Science, Theory, and History of Music,
26o
PENTATONIC SCALES.
PART nr.
the five black notes may then be used as a tonic, bat the ^ or Ajj^ having no Fifih
(F or Eijf) among the black notes has a very doubtful effect as a tonic.
The following are examples of the use of these various scales of five tones :—
The First Scale without Third or Seventh. Chinese, after John Barrow.*
I.
»
-#-^-
^^m
^m
;st=s^
±r-
i
m
wt:
df ij-
^
at^
■mzM
4--
m
i
^^^f=^
it:=it
:#tz«b;i*
"CT-
2. To the Second Scale, without Second or Sixth, belong most Scotch airs
which have a minor character. In the modem forms of these airs one or other of
the missing tones is often transiently touched. Here follows an older form of the
air called Cockle Shells :t
^^^
3^^^
-wj-
W 0 ^ '- '— ^ #
1!
what is popularly known as the Caledonian
scale ; but any one who will take the trouble
to examine Scottish music will find that not
more than a twentieth part of our old melodies
are pentatonic, or constructed upon this form
of the scale. In Dauney's work, where the
Skene MSS. (the oldest collection extant) are
noted, this statement is fully verified/ I have
examined the first 36 airs as printed in The
Thistle^ and I found only one which was
strictly pentatonic, p. 5i» No. 8, Lament for
Huaridh Mar, Macleod of Macleod— Dun-
vegan 1626. But in nearly a quarter of the airs
the Semitones were introduced by an unac-
cented note which looked to be modem, as in
Jioxfs Wife, p. 10, and the Banks and Braes
o' Bonnie Boon, p. 48, on the last of which
Mr. Brown observes, p. 49 = * With pentatonic
theorists Ye Banks and Braes is a favourite
example of this assumed peculiarity of Scottish
music. But it can only be brought into the
pentatonic scale by being played in an incom-
plete form-.' The only places in which the
Seventh gt occurs are the cadence e'/5 9^ «'
(which occurs twice, and is evidently out of
character, and should be e' fl a' o'), and the
flourished ad libitum cadence /'« e" d" c"5 h"
containing the Fourth d, (which should clearly
be f'% e" c"t b'). And many of the others
can be probably ' restored ' in a similar fashion.
Thus of Bxry's Wife Mr. Brown himself aays,
p. 1 1, * played as a dance tune it is pentatonic,'
and gives the substitutes for his version, which
are clearly the more ancient forms. Mr. Brown
gives as the marks of Scotch music (pp. ix., x.)
I. its modal character, being constructed on
the ancient seven modes ; 2. its modtdatton or
change of mode, which is constant ; 3. almost
absence of transition or change of key; 4.
preponderance of minor forms of the scale ;
5. almost absence of sharp Sevenths in the
minors \ 6. cadences on to every note of the
scale, and double cadences closing on an unac-
cented note, which are simple (repeating the
cadential tone) or compound (the unaccented
tone differing from the preceding). — 2Vaiu-
lator.]
* [Scale, tempered d 200 e 300 g 200 a 200
b 300, d\ That is, no /B and no c» . All
these scales are merely the best representa-
tives in European notation of the sensations
produced by the scales on European listeners.
They cannot be received as correct represen-
tations of the notes actually played. — Trans-
lator,]
t Playford's Dancing Master, ed. 1721,
The first edition appeared in 1657. — Songs of
Scotland, vol. iii. p. 170. [Scale, d 300 / 200
^ 200 a 300 c' 200 d', without e or b\>.— Trans-
lator.]
Digitized by V^jOOQlC
CHAP. XIV.
PENTATONIC SCALES.
261
3. F<yr the Third Scale, without Third and Sixth,
bagpipe tune.*
Gaelic. Probably an old
^
±
^
^^
g^^
:5tEr*t
P
Blythe,blytheand mer-ry are we,
Gan-ty days we've of - ten seen ; A
Blythe are we,
night like this we
one and
ne - ver
^
^feg
rbe
«=»c
^
*=^^
¥
The gloam-ing saw us all sit down, And mei-kle mirth has been our fa'.
P
Then
D.C.
b^JLj^zfe
^S
^
let the toast and sang go round. Till chan - ti - oleer be - gins to craw :
4. To the Fourth Scale, without Fourth or Seventh, belong most Scotch airs
which have the character of a major mode. Since dozens of Scotch tunes of this
kind are to be found in every collection, and are perfectly well known, T give here
a Chinese temple hymn, after 6it8churin,t as an example :
H
T
^
I
b I r?
?2I
=t
S^
=^2=
e^^g=f
2^
zcz:
a^
22=
S:
rt
g
-p2:
z^
5. For the Fifth Scale, without Second and Fifth, I have found no perfectly
pure examples. But there are melodies with either only the Fifth or else with a IT
mere transient use of both Second and Fifth. In the latter case the minor Second
is used, giving it the character of the ecclesiastical Phrygian tone, for example in
the very beautiful air, Auld Bohin Gray. I give an example with the tonic J%^
in which the Second (^ or g) is altogether absent, and the fifth c^ is only once
transiently touched, so that it might just as well have been omitted.
JM^^s^^
l—^^\ ^ J^
Will
Let
ye
UB
go»
las - sie.
go
To the braes
Bal
i
s^
w
- quhid - der, Where
the
blae - her -
ries grow, Mang the
^
^
5:
=F
bon - nie bloom - in hea - ther ; Where the deer and the
Sport the lang aim - mer
^^
-j^ J 1 J
^
rae,
day,
Light
On
the
bound
braes
inff
o
the ' ge - ther,
Bal - quhid -der?
* There is a Chinese tone of the same kind
in Ambrosch, Zoc. cU, vol. i. p. 34, second piece.
Another, with a single occnrrence of the Sixth,
My Peggie is a young thing, may be seen in
Songs of Scotland^ vol. iii. p. 10. [Scale,
€ 200 /g 300 a 200 6 300 d' 200 e\ without g
or c. On the bagpipe, see App. XX. sect. £.
Probably the scale of the bagpipe has been
unaltered since its importation from the East,
and it probably never could have played such
a scale as it is here supposed capable of per-
forming.—TnansZator.]
t Ambrosch, loc. cit. vol. i. p. 30. To the
same class belongs the first piece on p. 35 after
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262 TETEACHOBDS. pabt iix-
We might also in this example assume b as the tonic» and regard the conclusions
as formed upon the dominant and subdominant in the old-fashioned way.* In
these scales of five tones the determination of the tonic is much more doubtful
than in the scales of seven tones.
The rule usually given for the G-aeHc and Chinese scales, to omit the Fourth
and Seventh, applies therefore only to the fourth of the above scales, which cor-
responds to our major scale. True this scale is often used in the usual Scotcli
airs of the present day, and is probably due to the reaction of our modem tonal
system. But the examples here adduced shew that every possible position may be
assumed by the tonic in the scale of five tones, if indeed we allow these scales to
have a tonic at all. In Scotch melodies the omissions in both major and minor
scales are so contrived as to avoid the intervals of a Semitone, and substitute fior
them intervals of a Tone and a half. Among the Chinese airs, however, I have
% found one which belongs rather to the old Greek enharmonic system, to be con-
sidered presently, and it will be explained at the same time (p. 265c).
We now proceed to the construction of scales with seven degrees. The first
form was developed in Greece under the influence of the tetrachordal divisions.
The ancient Greek melodies had a small compass and few degrees, a peculiarity
especially emphasised even by later authors, as Plutarch, but it is also found among
most nations in the early stages of their musical cultivation. Hence the scale was
at first formed within a less compass than an Octave, namely within the tetrachord.
On looking within this compass for the tones nearest related to the limiting tonic
(/AcoTTy), we find only the Thirds. Thus if we assume e (the last tone in the tetra-
chord, b — e) as a tonic, its next related tone within the compass of that tetrachord
is c, the major Third below e. This gives : —
1. The afunent enharmonic tetrachord of Olympos —
b^^ c e
^ i i ^
Archytas was the first to settle that the tuning of c : e must be 4 : 5 in the
enharmonic mode. The next most closely related tone to e would be the minor
Third below it. Adding this we obtain :
2. The older chromatic tetracJiord of the Greeks —
5 w c ^' cjf — ^-^a
I f J
The method of tuning the intervals here assigned agrees with the data of
Eratosthenes (in the third century before Christ). The interval between c and cjf
in this case corresponds to the small ratio ^ [= 70 cents], which is less than the
Semitone |f [= 112 cents]. Next to it comes the much wider interval, cjf — e,
corresponding to a minor Third. We should obtain a more even distribution of
% intervals, by measuring the minor Third upwards from the lowest tone of the
tetrachord. This gives rise to
3. The diatonic tetrachord —
few c — d — e
This is the tuning assigned by Ptolemy for the diatonic tetrachord. Here we
Bftrrow and Amiot. fSoale,/ 200 g 200 a 300 to tonic, dominant and subdominant, implies
c' 200 i' 3C» /, without 6 b or e.— Translator,] harmonic aoales, which pentatonio scales could
* [Taking /5 as the Tonic, the scale would not have been originally. Mr. C. Brown gives
be No. 5, without Second and Fifth, thus : this air (Thistle, p. 198) as here printed, but
/5 300 a 200 6 300 d 200 « 200 /» ?g;8 it varies between hifl modes of the ^
. , * . .,. X . XI- 1 1^ k« (Greek Done, Ecclesiastical Phrygian) and 5th
but taking b as the tonic the scale would be J^ ^^^ ^^^^ ^^^ ^^^^^ E^l Mixolydian).
No. 2, without Second and Sixth, as fj^^ spelling of the words has been corrected
6 300 d 200 « 200/5 300 a 200 6 by his edition.— Trntwiator.]
which is altogether different, -^ny referenoe
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CHAP. XIV. TETEACHORDS. 263
must observe that if e contmued to be regarded as the tonic, d would hi^ve only a
distant relation with it in the second degree through the auxilia2;y tone b, J£ two
tetrachords had been connected, as was very oarly done, thus :
b e a
a closer connection in the second degree between d and e might have been obtained
by tuning (2 as a fifth below a. Taking g as i, a will be f , and the Fifth below it
is d = |. We thus obtain the tetrachord
b -^ c — d — (
^ i
which agrees with the tuning assigned by Didymus (in the first century before
Christ). %
According to the old theory of Pythagoras, which will be examined presently,
all the intervals of the diatonic scale should be tuned by means of intervals of a
Fifth, giving :
5. b ^- c — d — e
* -Si « I
9
The tetrachord thus obtained is the Greek Doric, which is considered as normal,
and made the basis of all considerations on other scales. Accordingly those tones
which formed the lower notes of the semitonic intervals of the scale, were, at least
theoretically, considered as the immovable limiting tones of the tetrachord while
the intermediate tones might change their position. Practically the intonation of
even these fixed tones was a Uttle changed, as Plutarch tells us, which may mean
that in the Lydian, and Phrygian modes, &c., the tonic is not selected from one of IF
these so-called fixed tones of the tetrachords. Thus we shall see further on, that
when d is the tonic, the b in the natural intonation of such a scale does not form
a perfect Fifth with e.
The tetrachord could, however, be differently completed by inserting tones which
formed major or minor Thirds with either of the extreme tones.
Two minor Thirds give the Phrygian tetrachord —
6. d — e ^^ f — g
If a major Third were taken upwards from the lower extreme tone, and a minor
Third downwards from the upper extreme, we should obtain the Lydian tetrachord —
_d — e — /
3
i n I
IT
8. Two major Thirds, as in b ^- c — cqf— ' e, would form a variety of the
chromatic scale, which does not seem to have been used, or at any rate not to
have been distinguished from the chromatic form.*
* [Adopting the notation explained later on may be seen, and the oorreotness of the trans-
in this chapter, these tetrachords may be ac- positions verified,
curately written as follows ; Nos. i, 3, 4 and 7 i. Olympos . . 6, 112 c' 3866,'
may be played as they stand on the Harmonioal, 2. Old Chromatic . 6, 1 1 2 c' 70 c/t 3160/
and Nos. 2, 6, 8 by transposition as shewn (play . a iiza^iyoai^iOcf)
below, but No. 5 requires the six notes forming 3. Diatonic . . S, 1 12 c' 204 d^ 182 e/
5 perfect Fifths, and these do not occur on the 4. Didymus . . 6, 1 12 (/ 182 (2/ 2046/
Uarmonioal, but can be played sufficiently well 5. Doric . . 6 90 (/ 204 d' 204 e'
on any tempered harmonium. Between the (not playable on the Harmonical)
names of the notes are inserted the number of 6. Phrygian . . d 182 e, 134/* 182 ^
cents in the interval between them. By re- (play . ^ 182 a, 134 6* b 182 c')
f erring to the table called the Duodenarium, 7. Lydian . . c 1S2 di 204 e, 112/
App. XX. sect. E. art. 18, which employs the 8. Unused . . 6, 1 12 c' 274 ^,'9 112 e/
same notation, the exact position of the notes (play . ^ 1 12 a'b 274 6, 112 c')
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264 TETRACHOEDS. pabt m.
These are all the normal snbdivisioiiB of the tetraohord that have been used*
But other subdivisions occur which the Greeks themselves termed irrational (SXoya) ,*
and we do not know with certainty -how for they were practically used. One of
them, the soft diatonic mode, makes use of the interval 6:7, which is at any rate
very near to a natural consonance, being that between the Fifth and the subminor
Seventh of the fundamental note, an interval occasionally used in harmonic musio
when unaccompanied singers take the minor Seventh of the chord of the dominant
Seventh. The intervals j are :
9- 3JJ
V r
6:7
By lowering the Lichanos the Parhypate is also flattened. However, the small
f interval || is very nearly the Pythagorean Semitone, which expressed approxi-
mately is f ^.
The equal diatonic mode of Ptolemy, which was divided thus : t
10. 3:4
contained a perfect minor Third divided as evenly as possibla
There is a similar succession of tones, in an inverse order, in the modem
Arabic scale as measured by the Syrian, Michael Meshaqah.§ In this case the
Octave is divided into twenty-four Quartertones ** ; the tetrachord 10 has ten of
them, its lowest interval four, and each of the upper intervals three. Under these
circumstances the two upper intervals together form very nearly a minor Third,
^ which, as in the equal diatonic scale of the Greeks, is divided into two equal
intervals, without paying regard to any sensible relationship of the intermediate
tone thus produced.
The closer the interval, the more easy and certain is its division into two
intervals, by the mere feeling for difference of pitch. This is, in particular, poe-
Bible for intervals which approach to the limits at which dififerences of pitch are
If the minor Thirds df and e g were taken as monioal, as d'" c/" "/" 9^"» downwards as g**'
Pythagorean « 294 cents, tetrachord 6 would ^^f"e^"'d"\ The division of the minor Hiird
become d 204 e 90 / 204 g^ which is more in- g : e| = 3i6 cents into 151 and 165 cents is of
telligible. course only approximative. But it is a purely
On referring to App. XX. sect. D. the natural tetrachord of which ^204/ 112 e, 182 d(
ratios corresponding to each of these numbers is a deformation. — Translator.']
of cents will be found. — Translator.'] § Journal of the American Oriental Sodeiy^
* [That is, strictly, having a ratio not ex- vol. i. p. 173, 1847.
pressible by whole numbers.— Tronstotor.] *♦ [If the Octave is divided equally into 24
f [The notes which would form tetrachord 9 quarters, each of which is half an equal Semi-
m might be written in the Translator's notation, tone or 50 cents, we can write it by using the
descending from left to right, additional sign q (a turned b, standing for q
'db 85 a, 182 g 231 '/. ^6 initial of quarter) to represent an added
The three first notes could be played on the Q^arteriione, 5 being two Quartertones. and
Harmonical. The interval 23ilBents could be «*» ^^^ Quartertones thus ascendmg c c<K
played on it downwards as C 231 '6b, but the ^« <; «*» ^ ^^ descending^ dM d\>c^c, usmg db
whole tetrachord cannot be played on it. Here »« the equiviUent of c« . Then the principal
85 cents represent 21 : 20, while the Pytha- »<^»^« <>* Meshaqah (see App. XX. sect. K.) is
gorean Semitone 256 .* 243 is 90 cents. The a 2006 150 c'q 150 (2' 200 e' 150/q 150 ^ 200 a'.
di^rence Ib smaU but perceptible- rmn.- ^^^^ ^^^ tetrachord « : d', which represent.
t [Using the Dotation ■■/ for the i ith har- ^°^ *«*«»*. ^ °"« ^^^".'^ «' «» f°*« »'
monic of C.I0 that " is equivalent to 33 : 3* or 4 Quirtertones. and two of 150 cents or 3
53 cents, tetrachord loTay be written down- Q^rtertones. This mterval of 3 Quartertonea
wards- representsthe trumpetmtervals*'/ I gf* II : 12
a lu 'Y i6«; e 182 d "" '5i cents, and c : »'/- 10 : 11 « 165 cents,
y -> J ^ ^ ' and was introduced into Arabia by the luti&t
This is simply, in order, the 12, 11, 10, and 9th Zalzal, who died about 1000 years ago, and ia
harmonic of c, and can be played on the horn much used in the East. — Translator^
or trumpet, and on the 5th octave of the Har-
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CHAP. XIV. TETRACHORDS, 265
distinguishable by the ear. The distinctness with which the yet sensible difference
can be felt then famishes a measure of its magnitude. In this sense we have
probably to explain the possibility of the later enharmonic mode of the Greeks,
which, however, had already feillen into disuse in the time of Aristoxenus, and was
perhaps hunted up again by later writers as an antiquarian curiosity. In this
mode the Semitone of the ancient enharmonic mode already mentioned (No. i ,
p. 2626) was again subdivided into two Quartertones, so that a tetrachord was
produced like the chromatic one, but with closer intervals between the adjacent
tones. The division of this enharmonic tetrachord * was
II. 3:4
n U i
This Quartertone can only be considered as a transition in the melodic move-
ment towards the lowest extreme of the tetrachord. A similar interval occurs in H
this way in existing Oriental music. A distinguished musician whom I requested
to pay attention to it on a visit to Cairo, wrote to me as follows : ' This evening
I have been listening attentively to the song on the minarets, to try to appreciate
the Quartertones, which I had not supposed to exist, as I had thought that the
Arabs sang mit of tune. But to-day as I was with the dervishes I became certain
that such Quartertones existed, and for the following reasons. Many passages
in litanies of this kind end with a tone which was at first the Quartertone and
then ended in the pure tone.f As the passage was frequently repeated, I was able
to observe this every time, and I found the intonation invariable.* The Greek
writers on music themselves say that it is difficult to distinguish the enharmonic
Quartertones.^
The later interpreters of Greek musical theory have mostly advanced the
opinion that the above-mentioned differences, which the Greeks called colourings
(xpoat), were merely speculative and never came into practical use.§ They con- ^
sider that these distinctions were too delicate to produce any esthetic effect except
on an incredibly well cultivated ear. But it seems to me that this opinion could
never have been entertained or advanced by modem theorists, if any of them had
practically attempted to form these various tonal modes and to compare them by
ear. On an harmonium which will shortly be described I am able to compare
* [It is not to be supposed that these nnmber of vibrations for the sharpened and
two Quartertones, differing only by two cents normal note, which gave the interval as 48
(32 : 31=55 cents, 31 : 30 = 57 cents), were cents. The effect was very peculiar, but can
exactly produced. The lutist or lyrist would of course be easily imitated on the violin. On
tune his Fourth c : /by ear (tolerably correctly), the classical Indutn instrument, the Vina, the
then a major Third below f or d^\> also by ear frets are very high, sometimes about an inch,
(and probably very incorrectly on account of Hence by pressing down the string behind the
the great difficulty of tuning a major Third), fret, the tension could be greatly increased, and
and then would by feeling divide the remaining as much as a Semitone could be easily added,
interval in halves as well as he could. Using so that the scale could be indefinitely altered
c : c<\ for the approximate Quartertone he without changing the frets, which were fixed ^
would have a&ou^c 56 cq 56 <2'b 386/, or some- with wax. On the Arabic Babab and the
thing sufficiently like it. Meshaqah's c 50 curious Chinese fiddles, which have no frets or
cq 50 cS 400 /would doubtless have been near finger-board, a note could be instantaneously
enough. Probably no two lyrists tuned alike. sharpened in a similar manner by pressing more
My experience of tuning by ear is quite against strongly. — Translator.l
any approach to the accuracy which the figures % [And yet a quarter of a Tone is between
in the text would imply. — Translator.] 2 and 3 commas, and all the difficulties of tuning
f [Probably the effect was like that which in just and tempered intonation arise from in-
I heard produced by Baja R&m P41 Singh on tervals of a single comma or less. — Translator,']
his Sit4r. Here the tone of the note, played § Even Bellerman is of this opinion
by pressing the string against a fret, was shar- (Tonleiter der Qriechen, p. 27). Westphal,
pened a quarter of a Tone by sliding the finger in his Fragmenten der Griechischen Rhyth-
along the fret (thus deflecting the string miker^ p. 209, has collected passages from
and increasing the tension), and then it was Greek writers proving the real practical use of
allowed to glide on to the proper note by these intervals. According to Plutarch (De
straightening the string without repluoking Mtisica^ pp. 38 and 39), the later Greeks had
it. I determined the amount of sharpening even a preference for these surviving archaic
by observing the distance of deflection, and intervals,
then, at leisure, measuring by my forks the
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266 VAEIOUS DIATONIC SCALES. part iu-
natural intonation with Pythagorean, and to play the diatonic mode at one time
after the method of Didymus and at another after that of Ptolemy, and also to
make other deviations. It is not at all difficult to distingoish the difference of a
comma |^ in the intonation of the different degrees of the scale, when well-laiown
melodies are performed in different ' colourings,' and every musician with whom I
have made the experiment has immediately heard the difference. Melodic passages
with Pythagorean Thirds have a strained and restless effect, while the natural
Thirds make them quiet and soft, although our ears are hahituated to the Thirds
of the equal temperament, which are nearer to the Pythagorean than to the natural
intervals. Of course where delicacy in any artistic ohservations made with the
senses, comes into consideration, modems must look upon the Greeks in general
as unsurpassed masters. And in this particular case they had very good reason
and abundance of opportunity for cultivating their ear better than ours. From
^ youth upwards we are accustomed to accommodate our ears to the inaccuracies of
equal temperament, and the whole of the former variety of tonal modes, with their
different expression, has reduced itself to such an easily apprehended difference aa
that between major and minor. But the varied gradations of expressions which
modems attain by harmony and modulation, had to be effected by the Greeks and
other nations that use homophonic music, by a more deUcate and varied gradation of
the tonal modes. Can we be surprised, then, if their ear became much more finely
cultivated for differences of this kind than it is possible for ours to be ?
The Greek scale was soon extended to an octave. Pythagoras is said to have
been the first to estabUsh the eight complete degrees of the diatonic scale. At first
two tetrachords were connected in such a way as to have a common tone, the fjJtnj :
.f^g — a^ l\} — c — d
^ which produced a scale of seven degrees. Then this scale was changed into the
following form :
e — / — g — a — b ^^ — d — e
and thus made to consist of a tetrachord and a trichord, of which mention has
already been made (p. 2576?). Finally Lichaon of Samos, (according to Boethius,)
or Pythagoras, (according to Nicomachus,) completed the trichord into a tetrachord,
and thus established a scale consisting of two disjunct tetrachords.
The diatonic scale thus obtained could be continued either way at pleasure by
adding higher and lower octaves, and it then produced a regularly alternating
series of Tones and Semitones. But for each piece of music a portion only of this
unlimited diatonic scale was employed, and the tonal systems were distinguished
by the character of the portions selected.
These sectional scales might be produced in very different ways. The first
% practical object which necessarily forces itself on attention, as soon as an instm-
ment with a limited number of strings, like the Greek lyre, is used for executing a
piece of music, is, of course, that there should be a string for every musical tone
required. This prescribes a certain series of tones which must be provided and
tmied on the instrument. Now as a rule when a certain series of tones is thus
prescribed as a scale for the tuning of a lyre, no question is raised as to whether a
tonic is to be distinguished or not, or if so which it should be. A tolerable number
of melodies may be found in which the lowest tone is the tonic : others in which
an interval below the tonic is touched ; and others, again, in which the Fifth or
Fourth above the Octave below the tonic is used. This is the kind of difference
between the authentic and plagal scales of the middle ages. In the authentic
scales the deepest tone of the scale, in the plagal its Fifth below or Fourth above,
was the tonic ; thus : * —
* [See Mr. Rockstro's article, ' Modes Eocle- on * Gregorian Modes,' vol. i. p. 625, in Grove's
siastioal,' vol. ii. p. 340, and Bev. T. Helmore's Dictionary of Music, What Prof. Helmholts
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CHAP. XIV. VARIOUS DIATONIC SCALES. 267
First Authentic Eccleslastical SoAiiE, tonic d.
d — e — / — g — a — b — c — d
V. ..p. /
FouBTH Plagal Scale, tonic g.
d — e — / — g — a — b — c — d
y , ^
The scales were looked upon as composed of a Fifth and a Fourth, as the
braces shew. In the authentic tone the Fifth lay below ; in plagal, above.
Now if we have nothing else before us but a scale of this kind, which marks out
the accidental compass of a series of melodies, we can collect but Httle respect-
ing the key. Such scales themselves may be fittingly termed accidental. They
comprise, among others, the medieval plagal scales. On the other hand, those f
scales which, like the modem, are bounded at each extremity by the tonic, may be
termed essential. Now practical needs clearly lead in the first place to accidental
scales alone. When a lyre had to be tuned to accompany the human voice in
unison, it was indispensably necessary that all the tones required should be present.
There was no immediate practical need for marking the tonic of a song sung in
miison, or even to become fully aware that it had a tonic at all. In modem
music, where the structure of the harmony essentially depends on the tonic, the
case is entirely different. Theoretical considerations on the structure of melody
could alone lead to distinguishing one tone as tonic. It has been already mentioned
in the preceding chapter, that Aristotle, as a writer on esthetics, has left a few
notices indicating such a conception, but that the authors who have specially
written on music say nothing about it.
In the best times of Greece, song was usually accompanied by an eight- stringed
lyre, tuned so as to embrace an Octave of tones selected from the diatonic scale. ^
These were the following :
1. Lydian c — d — e — / — g — a — b — c
2, Phrygian d — e — / — g — a — b — c — d
3. Doric e — / — g — a — b — c — d — e
4. Hypolydian .... / — g — a — b — c — d — e — /
5. Hypophrygian (Ionic) . . g — a — 6 — c — d — e — / — g
6. Hypodoric (Eolic or Locrian) . a — b — c — d — e — / — g — a
7. Mixolydian .... b — c — d — e — / — g — a — b — (c)
Hence any one of the tones in the diatonic scale could be used as the initial
or final extremity of such a tonal mode. The Lydian and Hypolydian scales
contain Lydian, the Phrygian and Hypophrygian contain Phrygian, and the Doric
and Hypodoric contain Doric tetrachords. In the Mixolydian two Lydian tetra-
chords seem to have been assumed, one of which was divided, as shewn by the braces
in the above examples.*
calls the ionic was termed the final. What on the piano and organ in equally tempered
was the exact intonation of this music it id intonation, as their ancestors played them
perhaps impossible to say. Perhaps we may in meantone intonation. Bat either of the
assume it to have been Pythagorean, as latter admit of being harmonised ; not so the
d2O4e90f2O4a2Oda2OAbQ0C2OAd. former, so that there is an essential difference.
■^ "^ ^ ^ "» ^ ^Translator.]
Of course, modern musicians play them * [By a reference to p. 263<2, note, it will be
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268
VARIOUS DIATONIC SCALES.
PART III.
The scales or tropes of the best Greek period have hitherto been considered aB
essential, that is, the lowest tone or hypate has been considered as the tonic. Bat
I cannot find any definite ground for this assumption. What Aristotle says, as we
have seen, makes the middle tone or mese, function as the tonic, but yet it cannot
be denied that other attributes of our tonic belong to the hypate.* Whatever may
have been the real state of the case, whether the mese or hypate be regarded as the
tonic, whether the scales be considered as all authentic or all plagal, it is extremely
probable that the Greeks, among whom we first find the complete diatonic scale,
took the Hberty of using every tone of this scale as a tonic, just as we have seen
that every one of the five tones forming the scales of the Chinese and Gaels
occasionally functions as a tonic. The same scales are also found, probably
handed down immediately by ancient tradition, in the ancient Christian ecclesi-
astical music.
Hence if we disregard the chromatic and enharmonic scales, and the apparently
arbitrary scales of the Asiatics, none of which have shewn themselves capable of
«r further development,t homophonic vocal music developed seven diatonic scales,
which differ from one another in about the same way as our major and minor
scales. These differences wiU be better appreciated by making them all begin
with the same tonic c.t
seen that this paragraph materially alters the
intonation from what would result from a
mere beginning of each mode with a different
note of the Pythagorean or diatonic scale. I
therefore repeat the scales as defined by this
paragraph in the notation explained on pp. 276a
to 277a and note •, and write between each pair
of notes the number of cents in the interval
between each pair of notes, which will be found
useful in future comparisons. These scales
^ should be traced out on the Duodenarium,
App. XX. sect. £. art. 18. They cannot be
played on the Harmonical.
1. Lydian, c 182 di 204 ei 1 12 / 204 g 182 Aj 204
5, 112 c
2. Phrygian, d 182 ej 134 /* 182 g 204 a 182
6, 134 c* 182 d
3. Doric, ego f 204 g 204 a 204 & 90 c 204
d 204 e
4. Hypolydian, / 204 g 182 ai 204 &i 112 c 182
^, 2046, 112/
5. Hypophrygian (Ionic), g 204 a 182 61 112
c 204 d 182 e, 134/' 182 g
6. Hypodoric (Eolic or Locrian), a 204 b 90
c 204 d 204 6 90 / 204 g 204 a
7. Mixolydian, 5i 112 c 182 di 204 0, 112/204
g 182 a^ 204 6,. — Translator.]
* B. Westphal, in his Oeschichte der alien
gr UTid miltelalterlichen Musik^ Breslau, 1864,
>' which is unfortunately still incomplete, uses the
previous citations from Aristotle, to frame an
hypothesis on the tonic and final cadence of the
above scales. But he applies the remarks of
Aristotle only to the Doric, Phrygian, Lydian,
Mixolydian and Locrian scales, and not to the
Eolic and Ionic, which were also known at
that time, although the ground for their ex-
clusion is not apparent. In the first four of
these he takes the mese as tonic and the
hypate as the terminal tone. In those scales
distinguished by the prefix Hypo-t the hypate
was both tonic and terminal; but in those
having the prefix Syntono-t the hypate was
both the terminal and the Third of the tonic,
and the same was the case perhaps for the
Boeotian scale, which is only mentioned once.
Hence it follows that the minor scale of A
occurs as Doric with the terminal e, as Hypo-
doric with the teiTninal a, as Boeotian with
the terminal c. Moreover the Ifixolydian
would be a minor scale of JE7, with a minor
Second, and a terminal in 5; the Locrian a
minor scale of D with a major Sixth, and a
terminal in a ; the Phrygian, Hypophrygian or
lastic, and the Syntonoiastic, major scales of
Of with a minor Seventh, the terminals being
dy g, and b respectively. Finally the Lydian,
Hypolydian and Syntonolydian would be
major scales of F, with superfluous Fourth,
and with the terminals c, /, and a respectively.
But according to Westphal the normal major
scale was entirely absent. If the Ionic were
interpreted according to the words of Aristotle,
it would yield a correct major scale. The
tonic F with B (instead of JBb) as its Fourth,
has a totally impossible appearance to modem
musical feeling.
f [In India there is a highly developed
system with a vast variety of scales. — Trans-
lator.]
X [Continuing to use the notation of p. 2686,
note, these transposed scales may be written
as follows. As the order is different from that
in p. 267c, the numbers there used are added
in n. The number of cents in each interval
will complete the identification. I give only
the ancient Greek names, and the names pro-
posed by Prof. Helmholtz.
1. Lydian— Mode of the First (Major) (i),
c 182 di 204 6| 112 / 204 g 182 ai 204
61 112 c.
2. Ionic or Hypophrygian — Mode of the
Fourth (5), c 204 d 182 e^ 112 / 204 g 182
«! 134 &*b 182 c
3. Phrygian — Mode of the minor Seventh (2),
c 182 d, 134 e^b 182 /204 g 182 a. 134
6'bi82c
4. Eolic— Mode of the minor Third (Minor)
(6), c 204 d 90 eb 204 / 204 g 90 ab 204
6b 204 c
5. Doric— Mode of the minor Sixth (3), c 90
db 204 eb 204/ 204 9 90 ab 204 6b 204 e
6. Mixolydian — Mode of the minor Second (7),
c ii2<2^b I82 6b204/ii2 9*b204a*bi82
6b 204 c
7. Syntonolydian— Mode of the Fifth, not in
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CHAP. XIV.
VARIOUS DIATONIC SCALES.
269
▲nolent Qreek Names
Scales beginning with <?
Olarcan'B
Ecclesiastical
Names
Proposed new
Name8«
1. Lydian ....
2. Ionic or Hypophrygian .
3. Phrygian
4. EoUc ....
5. Doric ....
6. r Mixolydian .
7. t Syntonolydian
c-d -e -/ -g -a -6 -</
c— <J — e -/ —g —a — 6b— c'
c-d -«b-/ -g -a — ftb-c/
c-d -eb~/ -jT -ab-6b-c'
c— rfb— «b-/ — ^ — ab — 6b— c'
c— db— eb— / — grb— ob~6b— c'
c-d — e -ft -g -a -6 -c'
Ionic
Mixolydian
Doric
Eolic
Phrygian
Lydian i
Mode of the :—
First {major)
Fourth
minor Seventh
minor Third
(minor)
minor Sixth
minor Second
Fifth
To assist the reader I have added the names assigned to the ecclesiastical
modes by Olarean, which were wrongly distributed among the scales owing to his
confdsing the older tonal modes with the later (transposed) minor Greek scales, ^
but which are more known among musicians than the proper Greek names. But
I shall not use Glarean's names without expressly mentioning that they refer to an
ecclesiastical mode. It would be really better to forget them altogether. The old
numerical notation of Ambrose was much more suitable, but as his figures have
been altered again and do not suffice for all modes, I have ventured to propose a
new nomenclature in the above table, which will save the reader the trouble of
memorising the systems of Greek names, of which Glarean's are certainly wrong,
and the others are also perhaps not quite correctly applied. The principle of the
new nomenclature is this. By ' the mode of Fourth of C,' is meant a mode of
which C is the tonic, but which has the same signature (or additional $ and \}
signs) as the major scale formed on the Fourth of the diatonic scale beginning
with C ; that is on F. The minor Seventh, minor Third, minor Sixth, and minor
Second must always be understood as the intervals intended in this case.f If the
major intervals were selected the tonic would not occur in their scales. Thus^
' the mode of the minor Third of C ' is the scale with the tonic C, having the
signature of JSf) major (that is B{}, E\}, A\}), because E[} is the minor Third of C ;
this is therefore C minor, at least as it is played in the descending scale. I hope
the reader will have no difficulty in understanding what is meant by this notation.^
This was tlie tonal system in the best times of Greek art, up to the Macedonian
empire. Airs were at first Hmited to a tetrachord, as is still often the case in the
Boman Catholic Uturgy. They were afterwards extended to an Octave. Longer
scales were not necessary for singing, as the Greeks refused to employ the straining
upper notes, and unmetallic deep notes of the human voice. Modem Greek songs,
of which Weitzmann has made a collection,§ have also a surprisingly small com-
pass. If Phrynis (victor in the Panathenaic competitions, B.C. 457) added a ninth
string to his cithara, the chief advantage of the arrangement was to allow of
passing from one kind of scale to another.
The later Greek scale, which first occurs in Euclid's works of the third century %
B.C., embraces two Octaves, thus arranged :
the former table under this name, but
really the Hypolydian (4), c 204 d 182 6, 204
/iS 112 g 182 a| 204 6, 112 c
Befer to the Duodenarium, App. XX.
sect. £. art iS.—Translator.]
* (If we subtract each of the nombers in
the names of the modes here proposed, from
9, (reckoning i as 8 its Octave,) we obtain the
nambers on p. 267c, which shew the number of
the note in the major scale determined by the
signature, on which the special scale begins.
Thus as 9 less 7 is 2, the mode of the minor
Seventh is that numbered 2 on pp. 267c, 268c.
If we call the major scale, when reduced to a
harmonisable form, i. do, 2. r«, 3. tni, 4. /a,
5. so, 6. 2a, 7 tiy then these transformed modes
may be called with the Tonic Sol-faists the do,
re, mi, Ac, modes respectively. — Translator,]
f [The qualification minor will therefore
be always used in this translation, and has
been inserted in the above table. — Translator.]
X [In App. XX. sect. E. No. 10, I have
endeavoured to deduce scales for harmonic use,
from a general theory of harmony which de-
termines the precise value of each tone as
part of a chord, and I have given precise
names for them, there exemplified. This har-
monic deduction of scales is quite independent
of the historical melodic deduction in itie text.
— Translator.]
§ Oeschichte der Oriechischen Musik^
Berlin, 1855.
Digitized by V^OOQIC
270
LATER GREEK SCALES.
PART? UU
. lowest Tetrachord
A added Tone
B\
c
d
f middle Tetrachord
9
a
Proslambanom'enos
Tetra. hyp'aton
Tetra. mes'on
&
d'
fe'
f
9'
disjunct Tetra.
T. diezeug'mendn
superfluous Tetr.
T, hyperbolai'oTL
a \
&
d'
conjunct Tetr.
T, synem'menon.
This scheme gives first the Hypodoric [Eolic, or Locrian] scale* for two
Octaves, and then an added tetrachord which introduces a,Ij\}ia addition to the b,
and thus, in modem language, allows of modulation from the principal scale into
that of the subdominant.t
This scale, essentially of a minor character, was transposed, and thus a new
series of scales were generated that correspond with the (descending) minor scales
of modem music. To these were applied the old names of the tonal modes, by
giving originally to each minor mode the name belonging to that tonal mode
which was formed by the section of the minor scale which lay between the
extreme tones of the Hypodoric t scale. According to the Greek method of repre-
senting the notes, these extreme tones would have to be written /../. Their
% actual pitch was probably a Third lower. Thus the minor scale of D was called
Lydian, because in this scale —
d—e— \f—g—a-'l^-'C'-d''e'-f\g—a-'l^^c—d
the section of the scale lying between the extreme tones /and/ belonged to the
Lydian tonal mode. Li this way the old names of the tonal modes altered their
meaning into those of tonal keys. The following table shews the correspondence
of these names : —
i) Hypo-doric
= F minor
8) Phrygian «
9) Eolio =
C minor
2) Hypo-ionio
« Ft minor
CI minor
(deeper Hypo-phrygian)
(deeper Lydian)
3) Hypo-phrygian
■a Q minor
10) Lydian «
D minor
4) Hypo-edic
s GS minor
II) Hyper-dorio =
Eb minor
(deeper Hypo-lydian)
(Mixo-lydian)
5) Hypo-lydian
« A minor
12) Hyper-ionic «
(higher Mixo-lydian)
E minor
6) Doric
= B\> minor
13) Hyper-phrygian
/ minor]
CB
(Hyper-mizo-lydian)
7) lonio
« B minor
14) Hyper-eolic «
/8 minor
3^
(deeper Phrygian)
15) Hyper-lydian =
g minor ^
'S
Within each of these scales each of the previously mentioned tonal modes
might be formed, by using the corresponding part of the scale. Besides this it
was possible to pass into the conjunct tetrachord and thus modulate into the tonal
key of the subdominant.
The experiments on transposition which formed the basis of these scales
♦ [See No. 6, of p. 267c, text, asBoming
Pythagorean intonation. — Translator.]
t Singularly enough this species of musical
scale has been preserved in the Zillerthalin
Tyrol, for the wood-harmonioon. This scale
has two rows of bars. One forms a regular
diatonic scale with the disjunct tetrachord.
The other, which lies deep, has the conjunct
tetrachord in its upper part.
X [This seems to be an error for Hypo-
lydian, No. 4 of p. 267c, of which the extreme
tones are / and /. — Translator,]
Digitized by V^jOOQlC
CHAP. XIV. ECCLESIASTICAL SCALES. 271
shewed that the Octave might be considered as composed approximativelj of twelve
Semitones. Even Aristoxenus knew that by taking a series of twelve Fifths we
reached a tone that was at least very near 'to a higher Octave of the initial tone.
Thus in the series
he identified e/jj^ with /, and by thus closing the series of tones he obtained a cycle
of Fifths. Mathematicians denied the fact, and with reason, because if the Fifths
are taken perfectly true, ^ is a little sharper than /. For practical purposes,
however, the error was quite insensible, and might be justly neglected in homo-
phonic music in particular.*
This closes the development of the Greek tonal system. Complete as is our
acquaintance with its outward form, we know but little of its real nature, because
the examples of melodies which we possess are not only few in number, but very f
doubtful in origin.
Whatever may have been the nature of tonahty in Greek scales, and however
numerous may be the questions about it that are still unresolved, yet so far as the
theory of the general historical development of tonal modes is concerned we leam
all we want from the laws of the earhest Christian ecclesiastical music, wbich at
its commencement touched upon the ancient construction as it died out. In the
fourth century of our era. Bishop Ambrose, of Milan, established four scales for
ecclesiastical song, which in the untransposed diatonic scale were :
First mode : d — e — / — g — a — b — c — d mode of the minor Seventh.
Second mode : e — / — g — a — b — c — d — e mode of the minor Sixth.
Third mode : / — g — a — b — c — d — e — / mode of the Fifth (unmelodic).
Fourth mode : g — a — b — c — d — e — / — g mode of the Fourtb.
The variable character of the tone b, which was transmutable into l^ in the ^
later Greek scales, remained, and produced the following scales : —
First: d — e — f—g — a — 1\} — c — d mode of the minor Third.
a^^^A. >.^««Au^^^ ( mode of the minor Second
Second: e — / — g — a — en — c — a — e \. i j- x
•^ ^ ^ . I (unmelodic).
Third : / — g — a— 6^ — c — d — e — / mode of the First (major).
Fourth: g — a — ^ — c — d — e — / — g mode of the minor Seventh.
There can be no doubt that these Ambrosian scales are to be regarded as
essential (see p. 2676), for the old rule is that melodies in the first are to end in d,
those in the second in e, those in the third in /, and those in the fourth in gr, and
this marks the initial tones of the scale as tonics. We may certainly assume that
this arrangement was made by Ambrose for his choristers as a practical simplifica-
tion of the old musical theory, which was overloaded with an inconsistent nomen-
clature. And this leads us to conclude that we were right in conjecturing that the ^
similar older Greek scales could have been really used as different essential scales.
Pope Gregory the Great inserted between the Ambrosian essential scales the
same number of accidental scales (p. 267a), called plagal, proceeding from the
Fifth to the Fifth of the tonic. The Ambrosian scales were, then, called authentic
* It is by no means an nnimportant fact, Bepresentations of sach Antes are found in
for oar appreciation of the Greek scale, that a the very oldest Egyptian monuments. They
flute was foand in the royal tombs at Thebes are very long, the holes are all near the end,
in Egypt (now in the Florentine Museum, and hence the arms must have been greatly
No. ^88), which, according to M. F^tis, who stretched, giving the player a characteristic
examined it, gave an almost perfect scale of position. The Greeks can scarcely have been
Semitones for about an Octave and a half ; ignorant of this scale of Semitones. That it
namely, was not introduced into their theory till after
Series of primes, abb b & c'id' the time of Alexander, clearly shows the pre-
First upper partial tones, a' b'b b' c" c"U d" ferenoe they gave to the diatonic scale. [M.
Second upper partial tones, t!' f ft ^' g"t a" F6tis*s deductions must be treated with much
Third upper partial tones, a" 6"b b" d" c'"5 d" caution.— Tmiwtotor.]
Digitized by V^OOQIC
272 KATIONAL CONSTEUCTION OF DIATONIC SCALES, part i.
for distinction. The existence of these plagal ecclesiastical scales helped to increase
the confusion which broke over the ecclesiastical scales towards the end of the
middle ages, as composers began to neglect the rules which fixed the terminal
tones, and this confusion assisted in favouring a freer> development of the tonal
system. This confusion also shewed, as we remarked in the last chapter (p. 2436),
that no feeling for the thorough predominance of the tonic was much developed
in the middle ages. But a step, at least, was made in advance of the Greeks, by
recognising as a rule that the piece should close on the tonic, although this rule
was not always observed.
Glarean endeavoured in 1547 to reduce the theory of the scales to order again,
in his Dodecachordon. He shewed by an examination of the musical compositions
of his contemporaries, that six, and not four, authentic scales should be distinguished,
and adorned them with the Greek names in the table on p. 269a. Then he assumed
^ six plagal scales, and hence on the whole distinguished twelve modes, whence the
name of his book. Hence down to the sixteenth century essential and accidental
scales were reckoned as parts of one series. Among Glarean*s scales one is
unmelodic, namely the mode of the Fifth, which he calls the Lydian. There are
no examples of these to be found, as we know from a careful examination of
medieval compositions made by Winterfeld,* and this confirms Plato's opinion of
the MixolydiEkn and Hypolydian modes.
Hence there remain the following five melodic tonal modes applicable strictly
for homophonic and polyphonic vocal music, namely :
In oar Nomenclature
Ajiclent Greek
Olarean's Names
Scale
I
2
3
4
5
Major Mode ....
Mode of the Fourth
Mode of the minor Seventh .
Mode of the minor Third
Mode of the minor Sixth
Lydian
Ionic
Phrygian
EoUc
Doric
Ionic
Mixolydian
Doric
Eolic
Phrygian
H
The rational construction of these scales when extended to the Octave or beyond
the Octave results from the principle of tonal relationship already explained.f The
limits of the extent to which tones related in the first degree should be used, are
determined by the necessity of avoiding intervals too close to be distinguished with
certainty. The larger gaps thus left have to be filled with the tones most nearly
related in the second degree.
The Chinese and Gaels made the whole Tone V [=182 cents] the smallest
interval.^: The Orientals, as we have seen, still retain Quartertones. The Greeks
experimented with them, but soon gave them up and kept to the Semitone {f
[=112 cents] as the smallest.
European nations have followed Greek habits, and retained the Semitone ff as
^ the limit. The interval between ^ (f ) [=316 cents] and E (|) [=386 cents], and
between A\} (f) [=814 cents] and A (f) [=884 cents], in the natural scale is
smaller, being || [=70 cents], and we consequently avoid using both ^ and E,
or both A\} and A in tlie same scale. We thus obtain the following two series of
intervals between the most nearly related tones for ascending and descending
scales :
Ascending : c e — / — g — a c'
* if I V *
Descending : c A\} — O — F — ^ C
n I
10
* von Winterfeld'B Johannes Gabrieli und
sein Zeitalter, Berlin, 1834, vol. i. pp. 73 to
iu8.
t [The following is not an attempt to
restore the Greek originals, which have already
been treated, but to form harmonic scales on the
same, and these are obtained by another pro-
cess in App. XX. sect. E. art. ^.—TranslaUn'.]
X [I have found much smaller intervals in
Chinese instruments. See App. XX. sect. K.
— Translator,]
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CHAP. XIV. RATIONAL CONSTRUCTION OF DIATONIC SCALES. 273
The numbers below the series shew the intervals between the two tones between
which they are placed.*
It is at once seen that the intervals from and to the tonic are too large, and
might be further divided. But as we have come to the limits of relationship in
the first degree, we have to fill these gaps by tones related in the second degree.
The closest relationship in the second degree is necessarily famished by the
tones most nearly related to the tonic. Among these the Octave stands first. The
tones related to the Octave of the tonic are of course the same as those related to
the tonic itself; but by passing to the Octave of the tonic we obtain the descending
in place of the ascending scale, and conversely.
Thus, ascending from c we found the following degrees of our major scale —
c e— / — g — a &
But taking the tones related to c', we obtain —
0 eb — /— 9 — c^ c'
Hence the second degree of relationship to the tonic gives an ascending minor
scale. In this scale e[> is given as the major Sixth below c'. But it has also the
weak relationship to c marked by 5 : 6. Now we found that the sixth partial of a
compound tone was clearly audible in many qualities of tone for which the seventh
or eighth could not be heard ; for example, on the pianpforte, the narrower organ
pipes, and the mixture stops of the organ. Hence the relationship expressed by
5 : 6 may often become evident as a natural relationship in the first degree. This,
however, could scarcely be the case for the relationship e — of) or 5 : 8. Hence it
is more natural to change e into ^ than a into at> in the ascending scale. The
latter, 0(7, can only be related to the tonic la the second degree. The three
ascending scales la order of intelligibility are, therefore — f ^
e e — / — g — a c'
c ^—f—g — a 0'
c eb — /— g — (^ c'
These distinctions based on a relationship in the second degree, through the
medium of the Octave, are certainly very slight, but they make themselves felt
in the well-known transformation of the ascending minor scale, to which these
distinctions clearly refer.
Descending from 0, instead of the relations in the first degree, given in
c A\^—G — F — E\} C
we may assume relations in the second degree, that is of the deeper 0, and
obtain „
c A — G — F — E G ^
Tn the latter, A is connected with the initial tone by the distant relationship in the
first degree, 5 : 6, and E only by a relationship in the second degree. Hence the
third descencUng scale
c A — G — F — E\} C
which we also found as an ascending scale. For descending scales we have there-
fore the following series.^
* [With the subsequent notation and inter- c 386 e^ 112/ 204 g 182 a, 316 c'
yals expressed in cents : c 316 e* bi82 / 204 9 182 A) 316 (/
C386., 112/204^ 182a, 316C c3i6e'bi82/204(7ii2a'b386C
c 386 i'b 112 G 2b4 F 182 £»b3i6 C Translator.]
Translator,'] + [These are the same three scales as in
f [With the subsequent notation and inter- the last note, read backwards.— 2Vans^^.J
vals in cents ;
Digitized by
^.jo^gle
274 EATIONAL CONSTEUCTION OF DIATONIC SCALES, part in.
c ^.A\}'-G — F — E^ G
c A —G^F — E\) G
c A —G — F — E G
Generally, since all Octaves of the tonic, distant or near, higher or lower, are
80 closely related that they can be almost identified with it, all higher and lower
Octaves of the individual degrees of the scale are almost as closely related to the
tonic, as those of the next adjacent tonic of the same name.
Next to the relations of the Octave & of c, follow those of g^ the Fifth above,
and F the Fifth below e. We must therefore proceed to study their effect in the
construction of the scale. Let us begin with the relations of g, the Fifth above
the tonic*
ABCENDINa SCALES.
Belated to c : c e — / — g — a c'
H Belated to g : c d fi^ — — g b — c'
Uniting the two, we have —
i) The Major Scale (Lydian mode of the ancient Greeks) :
c — d — e — / — g — a — h — cf
I I f J f f V 2
The change of e into e[> is here facilitated by its second relationship to g. This
gives —
2) The Ascending Minor Scale :
c — d — 4> — / — 9 — ^ — * — ^
I I * * * * V ^
DESOENDING SCALES.
^ Belated toe: c il^— G^ — i^ — E^ G
Belated to flf: cJBb G E^ — D—G
giving:—
if The Descending Minor Scale (Hypodoric or Eolic mode of the ancient
Greeks — our mode of the minor Third) :
c — S^ — A)^ — G — F^E(^ — D — G
2 * f I $ t I I
or in the mixed scale, changing A\^ into A,
* [In the complete notation, and mfh This is different from the Greek Phrygian,
intervals in eents, these scales are : p. 26Sd\ note {, No. 3, in the two last intervals.
Ascending Scales. It is i C ma.mi.mi. {ibid, VH.).
Related to c : c 386 c, 1 12 / 204 ^ 182 a, 316 c' Ascending Scales.
Belated to g: c 20^diJ2e^b 386 (7386 &i 112 c' Related to c : c 386 e, 112 / 204 ^ 182 a, 316 c'
1) Major Scale: Belated to F: c 182 d, 316 / 386 a, 112
^ c 204 d 182 0, 112/204 g 182 a, 204 6, 112 c 6b 204 c^
" This is not quite the Greek Lydian, see p. 2684', 5) Mode of the Fourth :
note %, No. i. It is i C7 ma.TDa.nia. of App. XX. c 182 (2, 204 d, 1 12 / 204 ^ 182 a, 1126b 204 t/
sect. E. art. 9, 1. l^is is not quite Greek Ionic or Hypophrygian,
2) The Ascending Minor Scale : p. 268^', note :{:, No. 2. It is 5 ^ majnaona.
c 204 (2 112 e'b 182 / 204 (7 182 a, 204 61 112 c' {ibid I.).
This is I C majni.ma. {ibid, HI.), 6) New form of mode of the minor Seventh :
Descending Scales. c 182 d i^ .ibi82/2CH^ 182 a, 112 6b204c'
A/&Bi.j9rii^xx«u wvai^o. rpjj^g IS 5 ^ ma.majm. {ibid, V.).
Belated to c: c 386 -4>b 112 G 204 F 182 ^ ^ ^ ^ '
^'b 316 C Descending Scales.
Belated to g: c 182 B^b 316 O 386 E^b Ii2 Belated to c: C386 A^b 112 0 204 F 182
D204G E^bsiSC
3) The Descending Minor Scale: Belated to F: 0204 Bb 112 ^1386 F386
c 182 B'b204 A^b 112 G204 F 182 Jfi?»b 112 D»b 112 C
D 204 C 7) Mode of the minor Sixth:
This is not quite the Greek Eolic, see p. 26^', c 204 Bb 182 A^b 112 G 204 JP 182 ^b 204
note t, No. 4. It is i C mi.mi.mi. {ibid. VIII.). D' b 1 12 C
4) Mode of the minor Seventh : This is not quite the Greek Doric, p. 26Sd\
c 182 B>b 134 A^ 182 G 204 F 182 E^b 112 note J, No. 5. It is 5 F mi.ini jni. (i6i(i. VIII-).
D 204 C — Translator.]
Digitized by V^jOOQlC
CHAP. XIV. RATIONAL CONSTBUCTION OF DIATONIC SCALES. 275
4) Mode of the minor Seventh (ancient Greek Phrygian) :
c — Bi} — A — G — F — Ei} — D—G
2 » f I * * I I
On examining the relations of F, the Fifth below the tonio c, the following
scales result :
ABCENDma SCALES.
Belated to c : c e — / — g — a c'
Related to -P: c—d / a — Jt> — (/
This gives —
5) The mode of the Fourth (ancient Greek Hypophrygian or Ionic) :
c — <Z — e — / — g — a — l\} — &
I V f f t * V 2
By changing e into e\}, we again obtain — ^
6) TTie mode of the mdnor Seventh, but with a different determination of the
intercalary tones d and b\}, from those in No. 4 :
c — d — e\} — / — g — a — b\} — (/
I V * I I * V 2
DESCENDINO SCALES.
Belated toe: c A\}—a — F—Ej;} C
Belated ioF: c — Bl;} — A F JD\}—C
giving :—
7) The mode of the minor Sixth (ancient Greek Doric) :
c — Bi;} — J\} — G — F — E\;} — JD\}—C
2 V i i i i n ^ H
In this way the melodic tonal modes of the ancient Greeks and Christian
Church have all been rediscovered by a consistent method of derivation. As long
as homophonic vocal music is alone considered, all these tonal modes are equally
justified in their construction.
The scales have been given above in the order in which they are most naturally
deduced. But, as we have seen, each of the three scales
b e — / — g — a c'
c i\}— f—g — a &
c ^— f—g — c^ c'
can be played either upwards or downwards, although the first is best suited to
ascending and the last to descending progression, and hence the gaps of any one
of them may be filled up with either the relations of F or the relations of g, or
even one gap with those of F and the other with those of g. IT
The pitch numbers of the tones directly related to the tonic are of course fixed *
and unchangeable, because they are given by the condition that the tones should
form consonances with the tonic, and are thus more strictly determined than by any
more distant connection. On the other hand, the intercidary tones related in the
second degree are by no means so precisely fixed.
Taking c=i, we have for the Second —
i) the d derived from g =f , [=204 cents]
2) the d derived from / = V =ff x |, [== 1 82 cents]
3) the e2t> derived from/=|J, [=112 cents]
* Thus I eannoi agree with Hanptmann, damental bass d. But this would indicate a
in allowing a Pythagorean a, the Fifth above (2, distinct modulation into O major, which is not
in the ascending minor scale of c. d'Alembert required when the nataral relations of the
introdnoes the same tone even in the major tones to the tonic are preserved. See Haupt-
scale, by passing from ^ to 6 through the fun- mann, Hamumik und Metrik^ p. 60.
Digitized by VjCJC3QIC
276 RATIONAL CONSTRUCTION OP DIATONIC SCALES, pabt m.
and for the Seventh—^
i) the 6 derived from gr = Y, [=1088 cents]
2) the 6l> derived from g =§, [=1018 cents]
3) the ^b derived from / = V =|f x f , [=996 cents]
Hence while b and d\} are given with certainty, l^ and d are uncertain. Either of
them may he distant from the tonic hy the major Tone f [=204 cents] or the
minor Tone y, [=182 cents].
In order henceforth to mark this difference of intonation with certainty and
without amhiguity, we will introduce a method of distinguishing the tones deter-
mined by a progression of Fifths, from those given by the relationship of a Third
to the tonic. We have already seen that these two methods of determining the
tones lead to somewhat different pitches, and hence in accurate theoretical re-
f searches both kinds of tones must be kept distinct, although in modem music they
are practically confused. 4
The idea of this notation belongs to Hauptmann, but as the capital and small
letter which he uses, and which I also, in consequence, employed in the first edition
of this book, have a different meaning in our method of writing tones, I now
introduce a slight modification of his notation.
Let C be the initial tone, and write * its Fifth (?, the Fifth of this Fifth jD, and
so on. In the same way let the Fourth of G be F, the Fourth of this Fourth £\}
and so on. In this way we have a series of Tones, here written with simple
capitals, all distant from each other by a perfect Fifth or a perfect Fourth : f
Bj;}±F±C±G±D±A±E,&o.
The pitch of every tone in the whole series is, therefore, known when that of any
one is known.
If The major Third of (7, on the other hand, will be expressed by Ei, that of F
by ill, ^^^^ ^ 0^* Hence the series of tones
is a series of alternate major and minor Thirds. It is therefore clear that the
Tones
Di±Ai±Ei±By± Fjf^, &c.,
also form a series of perfect Fifths.
We have already found that the tone D, , that is the minor Third below or major
Sixth above jP, is lower in pitch than the tone D, which would be reached by a
series of Fifths from F, and that the difference of pitch is that known as a comma,
the numerical value of which is |^, or musically about the tenth part of a whole
Tone.t Since, then, D-±A and I>i±ili are both perfect Fifths, A must be also a
% comma higher than A | , and so also every letter with an inferior number, as i , 2, 3, &c.,
attached to it, will represent a tone which is i, 2, 3, &c., commas lower in pitch than
♦ Die Naiur der Harmonik und Meirik^ quently, and change (— ) into ( + ) for the
Leipzig, 1853, pp. 26 and following. I cannot major Third of 386 cents. In the case of
but join wi& C. E. Naamann in expressing my Fifths which consist of a major and a minor
regret that so many delicate musical appercep- Third 702 = 386 + 316 cents, the symbol ia
tions as this work contains, should have been properly ±. which I here also take the liberty
needlessly buried under the abstruse termi- to use. For other intervals I shall use (...) for
nology of Hegelian dialectics, and hence have (— ), and generally give the precise interval in
been rendered inaccessible to any large circle cents elsewhere. I trust that this change will
of readers. be found suggestive as well as convenient, and
f [Prof. Helmholtz uses (—) between the may therefore not be considered presamptuons.
letters in all such cases. I have taken the — Translator,']
liberty from this place onwards, whenever a % [The comma 81 : 80 is just over 2i4
line or combination of Thirds occurs to leave cents, for which I use 22 cents, see App. X]^
( — ) only in the just minor Thirds of 316 cents, sect. A. art. 4, and sect. D. Hence a major tone
to use ( I ) in the Pythagorean minor Thirds of 204 cents contains about 9^^ commas. —
of 294 cents, as Prof. Helmholtz does subse- Translator,']
Digitized by V^jOOQlC
CHAP. XIV. INTEODUOTION OF MOKE PEECISE NOTATION.
277
that represented by the same letter with no inferior number attached, as is easily
seen by carrying on the series.
A major triad will therefore be written thus :
and a minor triad
Ai-C+Ei or Ci-^+<?i
Now if we lay it down as a rule that as every inferior figure, i, 2, 3, &o., depresses
its tone by the i, 2, 3, &c., conmia, every superior figure, i, 2, 3, &c., shall raise its
tone by the same i, 2, 3, &c., commas, we may write the major triad as
c+ei-g or c^ + e-g^
and the minor triad as
or even
c-e^l^+g or Ci-t^-^gi,
c^-e^-g^ or Ca-eib-gfj.*
The three series of Tones directly related to C are consequently to be written
thus :
C El ^F-G-Ai c
and the intercalary tones are —
Between the tonic and Third, jD, Dj, or D*t>.
Between the Sixth and Octave, Bi and £\} or B^\}.
Consequently the melodic tonal modes of the ancient Greeks and old Christian
Church are,t
* In the ist [German] edition of this
book, as in Hanptmann's, the small letters
were supposed to be a comma lower than the
capital letters, and a stroke above or below the
letters was only occasionally used for raising
or depressing the pitch by two commas. Hence
a major triad was written C — e— Ootc — E
— J ; a minor triad a — C — c, or 4 — c — -B,
&c. The notation used here [in fhe 3rd and
4th German and the ist English editions]
and also in the French translation is due to
Herr A. v. Oettingen, and is much more readily
comprehended. [Herr v, Oettingen's notation
of lines above and below, which was at Prof.
Helmholtz's request retained in the ist Eng-
lish edition of this translation, was found
extremely inconvenient for the printer, and
actually delayed the work three months in
pa<»8ing through the press. I have now for
some years employed the very easy substitute
here introduced. By referring to the table
called the Duodenarium, in App. XX. sect. E.
art. 18, where this new notation is systemati-
cally carried out for 117 notes, the whole bear-
ing of it will be better appreciated. Another
notation which I had used formerly, and into
which I translated Herr v. Oettingen 's in the
footnotes to the ist edition of this translation,
and employed in Table IV., there correspond-
ing to my present Duodenarium, is conse-
quently abandoned, and is now only mentioned
to account for the difference in notation be-
tween the two editions of this translation.
The spirit of Herr v. Oettingen's notation is
therefore retained, while its use has been
rendered typographically convenient. — Trans-
lator.]
t [This variation of the intercalary tones
really amounts to a change of mode, so that
the names used in the text become ambiguous. ^
This difficulty is overcome by the trichordal
notation proposed in App. XX. sect. E. art. 9.
i) The major mode of C with D, has the 3
major chords 2?^+ .4, -C, C tE.-O, Q-k-B^-
D and is i C ma.ma.ma. But with D^ in
place of D, it has the 3 minor chords D^—F-{-
ii„ il, - O + J5|, £, - G + jB, (of which the two
last belong also to the first form), and is there-
fore 3 A^ mi.mi.mi. This is a related, but very
different, mode.
2) The mode of the Fourth, as it stands in
the first line, is not trichordal, but by using
D and jB'b it has the 3 chords JP-»-4,-C,
G-k-E^-Q, 0''B^\}+D, and is hence i 0
ma.ma.mi. If we take I), and Bb it has the
3 chords Bb + Di-F, F+A^-C, C + J5,-
G, and is hence 5 F ma.ma.ma. With both
D, and B'b it is again not trichordal.
3) The mode of the minor Seventh. If we ^
take the upper line as it stands, this is also not
trichordal. But if we use D and B* b, it has the
3 chords F+A^-C, C-E'b + Q, G-B'b + A
and is hence I C ma.mi.mi. If we take D,
and Bb , the 3 chords are Bb +Di-F, F+A^
-C, C-^'b + G, and the scale is 5 JP
ma.majni. With D^ and £' b the scale is again
not trichordal.
4) Mode of the minor Third. The first
line as it stands is not tri^ordal. Taking
D and S'b the 3 chords are F-A^b-i- C,C-
£'b + G, G-B'b + D, and the scale is i C
mi.mi.mi. Taking i>, and Bb the 3 chords are
Bb-^Di^F.F-A^b + CrC-E^b +G,andthe
scale is 5 JP ma.mijni. With Dj and B'b
again the scale is not trichordal.
5) Mode of the minor Sixth. The first
line as it stands gives the 3 chordsBb— i)'b +
^, ^-^'b+C, C--E'b + G, and the scale is
Digitized by V^jOOQlC
278 INTBODUCTION OF MOBE PRECISE NOTATION, pabt m. |
i) Majob mode,
C...D.,.Ei...F.,.G.,.Ai,..Bi,..c
Di
2) Mode op the Foubth,
3) Mode of the minob Seventh,
4) Mode of the mikob Thibd,
^ 5) Mode of the minob Sixth,
J3»b
By this notation, then, the intonation is always exactly expressed, and the kind
of consonance which each tone makes with the tonic, or the tones related to it is
clearly shewn.
In the ancient Greek Pythagorean intonation these scales would have to be
written:
Major mode-- C...D...E,.,F.,.a...A..,B...C.
and the others in a similar manner, all with letters of the same kind, belonging
to the same series of Fifths.*
In the formulae here given for the diatonic tonal modes, the intonation of the
Second and Seventh is partly undetermined. In these cases I have given D the
^ preference over D^, and ^ the preference over B^\}, because the relationship of
the Fifth is closer than that of the Third ; but ^ and D stand in the relation of
the Fifth respectively to F, (?, the tones nearest related to the tonic, while Dj and
B^\} are only in the relation of the minor Third to F and G. But this reason is
certainly not sufficient entirely to exclude the two last tones in homophonic vocal
music. For if in a melodic phrase, the Second of the tonic came into close con-
nection with tones related to jP— for instance, if it fell between F and il|, or
followed them — an accurate singer would certainly find it more natural to use the
Di, which is directly related to F and Ai, than |the D which is related to them
only in the third degree. The slightly closer relationship of the latter to the
tonic could scarcely give the decision in its favour in such a case.
This ambiguity in the intercalary tones cannot, I think, be considered as a
fault in the tonal system, since in our modem minor mode, the Sixth and Seventh
of the tonic are often altered, not merely by a comma, but by a whole Semitone,
f according to the direction of the melodic progression. We shall find, however,
more decisive reasons for the use of D in place of Di in the next chapter, when we
pass from homophonic music to the influence of harmonic music on the scales.
The account here given of the rational construction of scales and the corre-
sponding intonation of intervals, deviates essentially from that given to the Greeks
by Pythagoras, which has thence descended to the latest musical theories, and
even now serves as the basis of our system of musical notation. Pythagoras
constructed the whole diatonic scale from the following series of Fifths : —
F±C±G±D±A±E±B,
5 F mijnijni. If we use B^b in place of distinct, though purposely confused in the
£b, the 3 chords are i)'b + JP—>i*b, il'b + C— nomenclature of the text, apparently as aa ^
E^bi E^b +0~B^b, and the scale is 3 A^b accommodation to the usual tempered nota-
nujnijni. Hon.— TrmislatorJ]
The modes formed by taking one inter- * [In this case the intonation becomes
calary tone or the other are therefore quite altogether different. - Translator.}
Digitized by V^jOOQlC
CHAP. XIV. INTRODUCTION OP MORE PRECISE NOTATION.
279
and oalcnlated the intervals from it as they have been given above. In his diatonic
scale there are but two kinds of small intervals, the whole Tone f , [s 204 cents]
and the Limma f4§> [= 9<> cents].*
In this series if C be taken as the tonic, A would be related to the tonic in the
Third degree, E in the Fourth, and B in the Fifth. Such a relationship would be
absolutely insensible to any ear that has no guide but direct sensation.
A series of Fifths may certainly be tuned on any instrument, and continued as
far as we please ; but neither singer nor hearer could possibly discover in passing
from c to e that the latter is the fourth from the former in the series of Fifths. Even
in a relation of the second degree through Fifths, as of c to d, it is doubtful whether
a hearer can discover the relation of the tones. But in this case when we pass
from one tone to the other we can imagine the insertion of a silent g,* so to speak,
forming the Fourth below c, and the Fifth below d, and thus establish a comiection,
for the mind's ear at least, if not for the body's. This is probably the meaning to ^
be attached to Rameau*s and d'Alembert's explanation that a singer effects the
passage from ciodhy means of the fundamental bass G. If the singer does not
hear the bass note G at the same time as d, he cannot possibly bring his d into
consonance with that G ; but the melodic progression may certainly be facilitated
by conceiving the existence of such a tone. This is a well-known means for
striking the more difficult intervals, and is often applied with advantage. But of
course it completely fieuls when the transition has to be made between tones widely
separated in the series of Fifths.f
* [The fact that the Greek scale was
derived from the tetrachord, or divisions of
the Fourth, and not the Fifth, leads me to
suppose that the tuning was founded on the
Fourth and not the Fifth. On proceeding up-
wards from C by Fourths, we get C F Bb Eb
Ab Db Ob Cb Fb Bbb Ebb Abb Dbb, and
on proceeding dou^wards we get C G D A E.
Now these notes after Gf b in the first series, are
precisely those of Abdulqadir, written as ^j5, ^E^
M, *Di G, *C, on p. 2826, according to the no-
tation explained on p. 2816, note *. Of course
the Arabic lute, tuned in Fourths, naturally led
to this. It is most convenient for modem habits
of thought to consider the series as one of
Fifths. But I wish to draw attention to the
fact that in all probability it was historically a
series of Fourths. — Translator,]
t [One of the practical results of the Tonic
Solfa system of teaching to sing the diatonic
major scale as marked on p. 2746, No. i, in just
intonation, (see App. XVIII.,) has been the
discovery that it is not so easy to learn to
strike the proper tone by a knowledge of the
interval between two adjacent tones in a
melodic passage, as by a knowledge of the
mental effect produced by each tone of the
scale in relation to the tonic. These mental
effects are perhaps not veiy clearly character-
ised by the mere names given to them in the
Tonic Solfa books, but the teacher soon makes
his class understand them, and then finds
them the most valuable instrument which he
possesses for inspiring a feeling for just intona-
tion. On these characters of each tone in the
(just) diatonic scale, a system of manual signs
has been formed, by which classes are con-
stantly led. Particulars are given in * The
Standard Course of Lessons and Exercises in
the Tonic Solfa Method of Teaching Music,
with additional exercises, by John Curwen,
new edition, re-written, a.d. 1872.' But it
may be convenient to mention in this place
the characters and manual signs there given
{ib, p. iv.).
I. First step.
Do J Tonic, * the strong or firm tone,* fist
closed, horizontal, thumb down.
So, Fifth, ' the okand or bright tone,* the
fingers extended and horizontal, hand with
little finger below and thumb above, so that
the palm of the hand is vertical. ar
Mi, Major Third, * the stsadt or calm tone/
fingers extended and horizontal, palm of hand
horizontal and undermost.
II. Second step.
Re, Second, ' the bousino or hopefid tone,*
fingers extended, hand forming half a right
angle with ground pointing upwards, palm
downwards.
Ti, Seventh, *the pdsbcino or sensitive
tone,* only the forefinger extended and pointing
up, the other fingers and thumb closed, hand
forming half a right angle with ground, back
of hand downwards.
III. Third step.
Fa, Fourth, * the desolate or awe-inspiring
tone,* only the forefinger extended and point-
ing down, at half a right angle with the ground,
the back of hand upwards.
La, major Sixth, * the bad or weeping tone,* m
fingers fully extended, whole hand pointing
down with a weak fall, back of hand upwards.
It is thus seen that the order of teaching
takes the tonic chord first, then the dominant,
and lastly the subdominant. The doubtful
Second thus comes early on. * The teacher
first sings the exercise to [the names of] con-
secutive figures, telling his pupils that he is
about to introduce a new tone (that is one not
DO, n, or so), and asking them to tell him on
which figure it falls. When they have distin-
guished the new tone, he sings the exercise
again — laa-ing it [this is calling each note la]
— and asks them to tell him how that tone
'* makes them feel.** Those who can describe
the feeling hold up their hands, and the
teacher asks one for a description. But others,
who are not satisfied with words, may also
perceive and feci. The teacher can tell by
Digitized by V^jOOQlC
28o ARABIC AND PERSIAN MUSICAL SYSTEM. pabt m.
Finally there is no perceptible reason in the series of Fifths why they should
not be carried further, after the gaps in the diatonic scale have been supplied.
Why do we not go on till we reach the chromatic scale of Semitones ? To what
purpose do we conclude our diatonic scale with the following singularly unequal
arrangement of intervals —
1, I, i, I, I, I, i
The new tones introduced by continuing the series of Fifths would lead to no closer
intervals than those which already exist. The old scale of five tones appears to
have avoided Semitones as being too close. But when two such intervals already
appear in the scale, why not introduce more ?
The Arabic and Persian musical system^ so far as its nature is shewn in the
writings of the older theorists, also knew no method of tuning but by Fifths. But
^ this system, which seems to have developed its peculiarities in the Persian dynasty
of the Sassanides (a.d. 226-651) before the Arabian conquest, shews an essential
advance on the Pythagorean system of Fifths.
In order to judge of this system of music, which has been hitherto completely
misunderstood, the following relation has to be known. By tuning four Fifths
upwards from G
C±G±D±A±E
we come to a tone, E, which is f ^ or a comma higher than the natural major
Third of O, which we write Ei. The former E forms the major Third in the
Pythagorean scale. But if we tune eight Fifths downwards from C, thus —
C±F±JB\}±E\}±A\}±JD\)±G\}±Gj;}±F\}
we come to a tone, F\}, which is almost exactly the same as the natural Ei* The
m interval of C to F\} is expressed by
liH=f x^ilMi or nearly Jx|ff:f [=384 cents].
their' eyes whether they have done so. He ^1, B|. We can partially judge of them by
multiplies examples until all the class have the effects of equal temperament, which melo-
their attention fully awakened to the effect of dically cannot differ much, although they
the new tone. This done he tells his pupils certainly differ sensibly, from those of Pytha-
the Sol-fa name and the manual sign for the gorean intonation. And it must be remem-
new tone, and guides them by the signs to bered that singers actually learn to sing in
Sol-fa the exercise and tiiemselves produce the equal temperament, in which all major Thirds
proper effect. The signs are better in this are 14 cents too sharp, and then find just
case than the notation, because with them major Thirds intolerably flat ! To this I would
the teacher can best command the attention add the following anecdote quoted from F^tis
of every eye and ear and voice, and at {Hist. G&rUrale de la Musique, vol. ii. p. 27) by
the first introduction of a tone, attention Prof. Land (Oamme Arabe, p. 19 footnote),
should be acute ' (ibid, p. 15). This passage, containing ' a fact,' as he says, * which could
the result of practice with hundreds of thou- not be believed, if it were not attested by the
sands of chilchren, shews that a totally new person whom it concerns. The celebrated
^ principle of understanding the relation of the organist M. Lemmens, who was bom in a
tones in a scale to the tonic has not only been village of Gampine [or Eempenland, a district
introduced, but worked out on a large scale in the Belgian province of Limbourg, 5i°i5' N.
practically,and,a8lmyBelf know, successfully, lat., 5^20^ £. long.], studied music in early
See Prof. Helmholtz's own impression of the youth upon a clavecin (harpsichord), which
success, as long ago as 1864, in App. XVIII. had been long dreadfully out of tune, because
Since that time great experience has been no tuner existed in the district. Fortunately*
gained and many methods improved. But the an organ-builder was summoned to repair the
object of introducing this notice here is to organ at the abbey of Everbode near that
shew that proper training (such as the ancient village. By chance he called upon the young
Greeks certainly had) could produce the corre- musician's father, and heard the boy play on
Bjpondjngfeeling for the effect of any tone in any his miserable instrument. Shocked at the
scale anyhow divided, independently of the rela- multitude of false notes which struck his ear,
tion of consonances , and that this consideration he immediately determined to tune the clavecin,
may help to explain the persistence of many When he had done so, M. Lemmens experi-
scales which are harmonically inexplicable, enced the most disagreeable sensations, and it
No doubt Pythagorean singers hit the degrees was some time before he could habituate his
of their scale quite correctly, and no doubt the ear to the correct intervals, having been bo
' mental effects ' of their A^ E^ B, were very long misled by different relations.' Uenoe,
different from those of the harmonisable ^p false intervals may seem natural. — TransUUorJ]
Digitized by V^jOOQlC
CHAP. XIV. AKABIC AND PEESIAN MUSICAL SYSTEM.
281
Hence the tone F\} is lower than the natural major Third ^1 [=386 cents] by
the extremely small interval ffj [=2 cents], which is about the eleventh part of a
comma [=22 cents]. This interval between F\} and ^1 is practically scarcely per-
ceptible, or at most only perceptible by the extremely slow beats produced by the
chord C...F\}..,G [=C 384 jP^ 318 G] upon an instrument most exactly tuned.
Practically, then, we may without hesitation assume that the two tones Fi^ and E^
are identical, and of course that their Fifths are also identical, or
F\} = ^1, q? = -Bi, Gb = FS, &c.*
Now in the Arabic and Persian scale the Octave is divided into 16 intervals, but
in our equal temperament it is divided into 6 whole Tones. Modern [European]
interpreters of the Arabic and Persian system of music have hence been misled
into the conclusion that each of the 17 degrees of the scale corresponded to about
the third of a Tone in our music. In that case the intonation of the degrees in IT
the Arabic and Persian scale would not be executable on our instruments. But
in Eiesewetter's work on the Music of the Arabs,t which was written with the
* [On this substitation, which amounts to
a temperament with perfect Fifths, and major
Thirds too flat by a skhisma, or nearly the
eleventh of a comma, and which I therefore
call skhismic temperament, see Appendix XX.
section A. art. 1 7. It is convenient to use a grave
accent prefixed thas, ^^p to show flattening by
a skhisma, and to read it as skhismic, thus,
* skhismic E one.' The above equations can
therefore be made precise by writing ^b»^JS7i,
Cb«'B„ Gb = '^,» , &o.— Translator.]
t R. G. Kiesewetter, Die Musik der Araher
nach Originalquellen dargestellt, mit einem
Vorworte von dem Freiherm von Hammer-
PurgstalU Leipzig, 1842, pp. 32, 33. The
directions given in an anonymous manuscript
of the 666th year of the Hegira, a.d. 1267, in
the possession of Prof. Salisbury [of Yale Coll.],
are essentially the same. See Journal of the
American Oriental Society, vol. i. p. 1 74. [Since
the publication of the 4th German edition
of this work in 1877, the whole history of the
Arabic scale has been reinvestigated from the
original Arabic sources by Herr J. P. N. Land,
D.B., Professor of Mental Philosophy at
Leyden, an Oriental scholar and a musician,
and the results were published first in Dutch
as a paper in the Transactions of the Dutch
Academy of Sciences, division Literature, 2nd
series, vol. ix., and separately under the title
of Over de Toonladders der Arabische Mvaiek
(On the Scales of Arabic Music) in 1880, and
secondly in French as a paper conmiunicated
to the International Congress of Orientalists at
Leyden in 1882 and pubUshed in vol. ii. of their
* Transactions,' and also separately in 1884 as
liecherches sur Vhistoire de la Garrnne Arabe,
This paper supersedes in many respects the
work of Kiesewetter and von Hammer-Furgstall,
of whom the first was a musician but not an
Orientalist, and the second an Orientalist but not
a musician. Alf arables scale was produced by a
succession of Fifths [or rather Fourths, see
p. 42, note], but a century and a half previously
Zalzal had introduced a new interval 22 : 27 »
355 cents, which Prof. Land terms a neutral
Third. It is actually if x § or 151 + 204 cents,
that is, three quarters of a Tone sharper than
a major tone, whereas the major Third is 182
cents or a minor Tone sharper, and the minor
Third was only a diatonic Semitone 112 cents
sharper. The interval 12 : 11 » 151 cents is the
well-known trumpet interval between the shar-
pened Fourth and Fifth, the nth and 12th har-
monics, as may be heard in the fifth Octave of
the Harmonical *'/" : g"'. This on the Arabic
lute was necessarily accompanied by a similar
interval on the next string, 498 -(-3 55^853
cents. These two notes eventually superseded
the old Pythagorean minor Third of 294 cents
and the Fourth above it of 792 cents; and
seem entirely out of the reach of a succession
of Fifths or Fourths. But it was the object of
Abdulqadir and others to form a succession of
Fifths (or rather Fourths) which would include
these two intervals, at least approximately, f
This they accomplished within less than 30 cents
by their 384 and 882 cents. It does not appear
to have been Abdulqadir's object to approxi-
mate to the just major Third 386, and just
major Sixth 884, but to get by means of Fifths
or Fourths certain tones which would pass as
Zalzal 's. The list in the text (p. 2826) gives
the seventeen tones thus produced with the in-
tervals that they form with each other, and Prof.
Helmholtz's names of the notes, completed by
a grave accent. Here I re-arrange them in
order of Fifths down or Fourths up, the ap-
proximate Thirds being added immediately to
the right, and the numbers showing the interval
in cents from C :
E 408
A 906, D\> ='Ci5 90
D 204, Ob «-'^,« 588
Q 702, Cb «'^i 1086 ^
C o, F\> -i'^i 384
F 498, Bbb^'il, 882
jBb996, i;bb«='A 'So
£b294, ilbb='Oi 678
Ab192, Dbb = 'Ci 1176
Observe that the real major Third was the
Pythagorean 408 cents, as the minor Third
was the Pythagorean 294 cents. Also that
180 cents was within two cents of the minor
tone 182 cents. But these approximations
were probably not contemplated.
An English concertina, which has fourteen
notes to the Octave, was tuned with thirteen
consecutive Fifths from Gb to CS, so that I
was able to try the chords ADbE DObA,
that is, A^CJ^E, D'F^ZA, where the major
Thirds are two cents too flat, and compare
them with the Pythagorean chords ACtEt
Digitized by V^OOQIC
282 ABABIC AND PEESIAN MUSICAL SYSTEM. pabt ru.
assistance of the celebrated Orientalist von Hammer-Purgstall, there is given a
translation of the directions for the division of the monoohord laid down by Abdul
Kadir, a celebrated Persian theorist of the fourteenth century of oar era, that lived
at the courts of Timur and Bajazet. These directions enable us to calculate the
intonation of the Oriental scale with perfect certainty. These directions also agree
in essentials both with those of the much older Farabi,* (who died in a.d. 950), and
of his own contemporary, Mahmud Shirazi,t (who died in 1315), for dividing the
fingerboard of lutes. According to the directions of Abdul Eadir all the tonal
degrees of the Arabic scale are obtained by a series of 16 Fifths, and if we call the
lowest degree C, and arrange them in order of pitch within the compass of an
Octave, they will be the following, as expressed in our notation [with the addition
of the grave accent explained in p. 281&, note *].
I) C - 2) Db- 3) 'D,^4) D^ s)^>- 6) 'E,^
IT 7) ^- 8) i^ - 9) Gb-io)^Gi--ii) G -12) ^b-
13) ^ili— 14) A -is) B[} -16) ^^1-17) "ci ^iS) c
where the line— between two tones indicates the interval of a Pythagorean Linmia
fi7 (which is nearly f^ [=90 cents]), and the sign -^ a Pythagorean comma
[=531441 : 524288=24 cents]. The Limma is about i and the Pythagorean
comma a little more than ^ and less than f of the natural Semitone \^ [=112
cents].
Abdul Eadir assigns the following intonation to the three first of the 12 prin-
cipal tonal modes or Makamiat : —
Arabic Ancient Qreek
1. Uschak C...D ,..E ..,F...O ...4 ...J5l>...c Hypophrygian or Ionic.
2. Newa C..D ...Ei^.„F.,.G ...-4b...J5b...c Hypodorian or Eolio.
-r 3. Buselik C...I)t>...j&b-- -^•••G^l>--^b--^-c Mixolydian, [allonp. 269a.]
These three are therefore completely identical with the ancient Greek scales in
Pythagorean intonation.^: Since the Arabic theoreticians divide these scales into
the Fourth G...F and the Fifth F±,c, and since O, F and B\} are considered to be
invariable tones, and the others to be variable, it is probable that F must be
regarded as the tonic. In this case
1. Uschak would be = i^ major.
2. Newa would be = the mode of the minor Seventh of i^.§
3. Buselik would be = the mode of the minor Sixth of F.
all three in Pythagorean intonation. The Persian school also considers the scales
to be related.
DFt A, The latter were offensive, the former their dftssical lute, to which alone the above
^ indistiiiguishable from just. It seems re- refers. — Translator.]
markable, therefore, how with suoh a collection * J. G. L. Eosegarten, Alii IspaJutnensiB
of notes the Arabs escaped harmonic music. Zdber CaiUUenartun^ pp. 76-86.
But it will be seen on examining the scales f Eiesewetter, Die Mtuik der Amber nach
formed from them (see espeoia^y p. 284^, note), Originalquellen darg., p. 33.
that they were perfectly unadapted for har- \ [Not therefore according to the forms on
mony, which would have occasioned a perfect p. 26&i', note, but on the more recent Pytha-
revolution in their musical systems. gorean imitation of those forms. They are
There was certainly no attempt to divide respectively the representatives of scales 2, 4,
the scale as Villoteau supposed into seventeen and 6 of that note. — Translator,']
equal parts each of about 70*6 cents, for no § [In the German text, Quartengsscklecht^
such intervals occur, still less any third parts or the mode of the Fourth of F. The tones
of a (tempered) tone of 66f cents, which was a in the mode of the Fourth of ^ are those in
mere hallucination of Yilloteau's. the Pythagorean scale of Bb , or, in order of
This system of Abdulqadir prevailed from Fifths, E\f ±B\>±F±C±Q±B±At and the
the thirteenth to the fifteenth century. The tones of the mode of the minor Seventh of F
modem division into twenty-four Quartertones are those in the Pythagorean scale otE\>% or, in
is noticed on p. 2646 and note **. order of Fifths, A\) ±E\> ± B>> ±F ± C ±0 ± D.
The Arabs, however, had also entirely The correction is therefore evident. — Trans-
dilterent scales for other inbtruments than lalor,]
Digitized by VjOOQIC
CHAP. XIV. ARABIC AND PERSIAN MUSICAL SYSTEM. 283
The next group consists of five tonal modes having just or natural intonation,
namely :
4. Rast 0 ... ^D, ... 'E, ... F ... G ... 'A^ ... JB\}
5. Husseini. . . . C ... 'Di „. Ej[} ... F ...'G^... A\} ... B\}
6. Hidschaf. . . . C ...'Dy ... E\} ... F ...'G^...'!^ ... J^
7. Rahewi • . . . C ... 'D^ ...'F^ ... F ...'Gi... A\) ... B\}
8. Sengule . . . . G ... D ...'E^ ... F ...'G^... 'A^ ... B\}
.. c
.. c
.. G
.. C
.. c
Bast may be regarded as the mode of the Fourth of C ; Hidschaf as the mode
of the Fourth of F ; Husseini as the mode of the Fourth oi B^\ as such they
would have perfectly natural intonation. In Bahetvi, if we refer it to the tonic F,
the minor third A\} is in Pythagorean, not natural, intonation. It might be re-
garded as the mode of the minor Seventh of i^ in which the major Seventh E^ is m
used as the leading note in place of the minor Seventh, as in our own minor mode.
The natural intonation of such a tonal mode cannot, indeed, be properly repre-
sented by the existing 17 tonal degrees. It becomes necessary to take either
Pythagorean minor Thirds and natural major Thirds or conversely. Husseini may
be regarded as the same tonal mode with Bahetvi, having the same false minor
Third, but a minor Seventh. Finally Sengule may be regarded as F major with a
Pythagorean Sixth. Bast may be conceived in the same way ; they are merely dis-
tinguished by the different values of the Seconds G oi^Gi.
The four last Makamat have each 8 tones, new intercalary tones being em-
ployed. Two of them resemble the modes Bast and Sengule, and between B\}
and C there is an intercalary tone ^Cy introduced ; named
9. Irak . . . C...'D,...'Ey...F...G ...'A,...B\}...'c,..x
10. Iszfahan . . G...D ...'E^... F..:Gy..:Ai... B\}..:cy...c ^
The last transposed a Fourth gives
11. Biisiirg . • C... D ...'Ey... F...'Gi...G ... A ...'By...c
The last tonal mode is
12. Zirefkend. . C...'Dy...E\}...F..:Gy...A\}...''Ay...'By...c
which certainly, if rightly reported, is a very singular creation. It might be
looked upon as a minor scale with a major Seventh, and both a major and minor
Sixth, but then the Fifth ^Gx is wrong. On the other hand, if F is taken as the
tonic, it has no Fourth, for which certainly there is some analogy in the Mixo-
lydian and Hypolydian scales. The instructions for scales of eight notes are very
contradictory, to judge by the different authorities cited by Kiesewetter.
The following four are distinguished as the principal modes of the Makamat : — ^
1. Uschak = Pythagorean F major.
2. Rast = Natural mode of the Fourth of (7, or natural F major with acute
Sixth.
3. Husseini = Natural mode of the minor Seventh of F.
4. Hidschaf = Natural mode of the Fourth of F.
We find, then, a decided predominance of scales with a perfectly correct natural in-
tonation, which has been attained by a skilful use of a continued series of Fifths.
This makes the Arabic and Persian tonal system very noteworthy in the history of
the development of music. Moreover, in some of these scales we find ascending
leading notes, which are perfectly foreign to the Greek scales. Thus in Rahewi,
E\ is the leading note to F, although the minor Third -4|> stands above F, while no
Greek scale could have allowed this without at the same time changing ^^ into E\},
Digitized by V^OOQIC
284
ARABIC AND PERSIAN MUSICAL SYSTEM.
PABT lU.
Similarly in Zirefkend the Bi is used as a leading note to C, although the minor
Third E\} is used above 0.*
♦ [Prof. Land (Gamme Arabe^ p. 38, note 3)
says * some of the descriptions of Prof. Helm-
holtz, borrowed from Kiesewetter, do not quite
correspond with the original data.' It will be
interesting therefore to give these scales as Prof.
Land describes them with his (more exact)
French orthography of the Arabic names and
in his order. The notation is the Translator's,
M| being 24 cents flatter than A.
1. 'OcJiaq, Our F major commencing (as
shewn by [) with the dominant, FOABb
[CDEF. * This commencement is the inevitable
consequence of the progression by conjunct
tetrachords which belongs to the lute. 'Ochaq
^ is as it were the type of all these maqdmdty the
others of which differ at one time like the
tropes or modes of the Greeks and of the
middle ages, by the displacement of both the
Bemitones at once, and at other times like the
Greek genera, by exchanges of intervals with-
out disturbing the scheme of two conjunct
tetrachords followed by a tone, with the ex-
ception of Nos. 7 and 8, which are more distinct
from the model maqd?na.*
2. Nawd, ' We may say that the scale is
that -of E\> major, beginning with the Sixth.'
EbFQABb[CDE\>,
3. B<msiUk or Ahou-sillk, 'The scale of
D\> major beginning at the Seventh, DbE\>
FGb AbBb [CDb.' The Pythagorean into-
nation of the three first scales renders them
non-harmonic.
% 4. Bast, ' The same as 'Ochaq except that
the Third A and the Seventh E are depressed
by a Pyth. comma, FQ'A^Bb [CD'E^F, which
makes them just rather than Pythagorean.'
The subdominant BbDF is non-harmonic.
5. ^Iraq. * Like Bast, but with the second
and the sixth above diminished by a Pyth.
comma, which makes the second nearly the
minor Second 10 : 9, and with grave supple-
mentary Fifth.' F'O, 'A^Bb ^C, [ C 'D, 'E, F.
Tuis has the proper subdominant Bb^D^F, but
the double Fifth is quite non-harmonic.
6. I^ahdn. ^Bast enriched with a grave
supplementary Fifth.' FG'A^Bb'C^[CD'E^F.
Here both the subdominant BbDF and double
Fifth render the scale non-harmonic.
7. Zirafkend, C 'D, EbF'G^Ab M, ^J5, C.
* An artificial scale composed of fragments of
those of Eb (eb/ Vi<*t> c *d, fib, Third and
^ Seventh almost just) and of C (c ^d^f^a^ ^5, c,
Second minor and Sixth nearly just) varied
also with Pythagorean A ot D and ^^p' Of
course entirely non -harmonic.
8. Bousourk, * C major with the Second,
Third, and Seventh diminished by a Pyth.
comma, and with a grave supplementary Fifth.'
C D;E^FG^GA'Bfi. Both subdominant and
dominant are non -harmonic.
9. Zetikouleh. * Differs from Bast only in
having the Second minor.' F^'Gi^AiBbiC
D'E^F, Subdominant non-harmonic.
10. Bdhawi. *F minor commencing with
the Fifth, but with the Sixth and Seventh
each increased by a Limma = 90 cents, and the
Second diminished by a Pythagorean comma,
very nearly our just ascending scale of F
minor.' F'GiAbBblC'D^E.F. The Pyth.
scale of F minor is FGAbBb[C DbEbF.
Here Db=^90 cents; 90 + 90=180^204-24
cents -^Dp 24 cents being the Pyth. comma.
Similarly J?b « 294 cents ; 294 + 90 = 384 « 408
— 24 cents = ^E^.■ Entirely non-harmonio.
11. HhosaKni, *Like Nawa, but with the
Third and Seventh diminished by a Pyth.
comma.' EbF^G^ABb [C'D^ Eb. Entirely
non-harmonic.
12. Hhidjdzi, *Bb major, beginning with
the Second and with the Third, Sixt£, and
Seventh diminished and therefore nearly just.'
Bb[C'D^EbIf''G^'A^Bb. This is the only one
of tiiese scales which is practically harmonic.
If we restore the proper names of the notes
in the series of Fifths or Fourths, (as in p. 28 id')
calculate the cents between each pair of notes
and from the first to each note, and begin with
the note indicated, we shall have a better idea
of the real nature of these scales, thus :
1. 'Ochaq, C204 D 20^ E ^ F 21^ G204
o sn4 408 498 7«
il 90 JBb 204 C
906 996 xaoo
2. Nawd. C 204 D 90 £b 204 F 204 O 204
o 904 394 498 70a
A^ Bb 204 C
906 99^ 1200
3. BousilXk. C 90 Db 204 Eb 204 F 90
o 90 394 49B
Gb 204 Ab 204 Bb 204 C
4. Bast, C 204 D 180 Fb 114 F 20^ Q i8q
O 904 384 498 TQB
Bbb 114 Bb204 C
88b 996 X900
5. 'Irdq, C 180 ^bb 204 Fb 114 F 180
o i8b 384 498
^bb 204 Bbb 114 Bb 180 Dbb 24 C.
678 88b 996 XX76 laoo
This double initial Z>bb,C maybe compared to
our double second in just major scales, and pos-
sibly has to be explained in the same way as a
real modulation.
6. I^ahdn. O 180 Ebb 20^ Fb iiA ^ 204
o 180 3B4 49«
G 180 Bbb 114 Bb 180 Dbb 24 G
70a 88b 996 X176 xaoo
7. Zirafkend. C 180 £bbii4 £b204 F 180
o 180 994 49B
Abb 114 Ab 90 Bbb 204 Cb 114 C
678 79a SSa 1086 xaoo
8. Bomourk, C 180 Ebb 204 ^b 114 F 180
o x8o 3B4 498
Abb 24 G 204 A iSo Cb 114 O
678 70a 906 1086 xaoo
9. Zenkouleh. C 204 B 180 ^b 114 F 180
o ao4 384 498
Abb 204 Bbb 114 Bb 204 C
678 88a 996 xaoo
io» Bdhawi. C 180 Ebb 204 ^b 114 F 180
0x80 384 49B
Abb 114 Ab 204 Bb 204 C
678 79a 996 laoo
II, IDvosaXni. C iSo Ebb 114 i?b204 J^ 180
Abb 228 il 90 Bb 204 G
678 906 996 xaoo
12. Hhidjazi, C 180 Ebb 114 J5b 204 F 180
o 180 994 498
^bb 204 Bbb 114 Bb 204 C
678 883 996 xaoo
Of these I have been able to play i, 2, 3 direct,
and 4, 5, 10, 12 by transposition upon my Pytha-
gorean concertina (p. 28 id'). When 12 beKHis
with Bb, or is played by transposition abd'\>
d'cfj'ba'b a't it is indistinguishable from the
Digitized by^OOQlC
CHAP. XIV. MEANING OP THE LEADING NOTE. 285
At a little later period a new musical system was developed in Persia with
12 Semitones to the Octave, analogous to the modem European system. Kiese-
wetter here hazards the very unlikely hypothesis that this scale was introduced
into Persia hy Christian missionaries. But it is clear that the system of 17 tonal
degrees which had been previously in popular use, merely required the feehng for the
finer distinctions to grow dull so that intervals which differed only by a [Pythagorean]
comma should be confused, in order to generate the system of 12 Semitones.* No
foreign influence was necessary here. Moreover, the Greek system of music had
long been taught to the Arabs and Persians by Alf&rabl. Again, the European
theory of music had not made any essential advance in the fourteenth and fifteenth
centuries, if we except the study of harmony, which never found favour with the
Orientals. Hence the Europeans of those days could teach the Orientals nothing
that they did not already know better themselves, except some imperfect rudiments
of harmony which they did not want. There is much more reason, I think, for ^
asking whether the imperfect fragments of the natural system which we find
among the Alexandrine Greeks, do not depend on Persian traditions, and also,
whether the Europeans in the time of the Crusades did not learn much music from
the Orientals. It is very probable that they brought tlie lute-shaped instruments
with fingerboards and the bowed instruments from the East. In the construction
of tonal modes we might especially instance the use of the leading note, which
we have here found existing in the East, and which at that time also began to figure
in the Western music.
The use of the major Seventh of the scale as a leading note to the tonic marks
a new conception, which admitted of being used for the further development of
the tonal degrees of a scale, even within the domain of purely homophonic music.
The tone Bi in the major scale of C has the most distant relationship of all the
tones to the tonic 0, because as the major Third of the dominant G, it has a less
close connection with it than its Fifth X>. We may perhaps assume this to be the IT
reason why, when a sixth tone was introduced into some Gaehc airs, the Seventh
was usually omitted. But, on the other hand, the major Seventh Bi developed a
peculiar relation to the tonic, which in modem music is indicated by caUing it
the leading 7U>te. The major Seventh B^ differs from the Octave c of the tonic by
the smallest interval in the scale, namely a Semitone, and this proximity to the
tonic allows the Seventh to be struck easily and pretty surely, even when starting
from tones in the scale which are not at all related to jBj. The leap F...B1
[=45 ' 32=590 cents], for example, is difficult, because there is no relationship at
all between the tones. But when a singer has to perform the passage F...Bi...c,
he conceives the interval F...C, which he can easily execute, but does not force his
voice up sufficiently high to reach c at first, and thus strikes Bi on the way. Thus
Bi assumes the appearance of a preparation for c, and this view alone justifies it
to the ears of a hstener, by whom the transition into c is, therefore, expected.
Hence it has been said that Bi leads to c ; or that Bi is the leading note to c. In f
jast «oale abCi%d'e'f/tg/Za', The three melodies the Orientals, as we do, selected
chords d^g'ba't ad'\>e\ ef^bh are perfectly from them several series of 7 [occasionally 8]
good, and the passage d'b 0V, d^b a', e'tt'ii^'b, tones, very slightly different from our dia-
e^a'b b't d! b tla! perfectly good, much better than tonic scales.* But so materially different that
on the piano. Yet it never occurred to Arabs any attempt to play harmonies upon them
to play in harmony. would result in frightful dissonance. — Traws-
* In face of these historical scales,' observes latorJ]
Prof. Land (t&u2.p.38), * it is difiScult to conceive * [If we suppose the pairs of notes in ( )
how Eiesewetter could say that the 17 degrees of to have been confused into one by neglecting
the complete scale were not treated like sharps the Pythagorean comma, then the series of
and flats, but that each one had tiie same notes on p. 2826 becomes C Db CA D) Eb {'E^
importance. On the contrary, the 17 degrees E) F Gb fO^O) Ab ^A^A) Bb 'B, i^CiC),
were like our 12 Semitones to the Octave, or, wnence the equally tempered scale C Db D
still better, like the 17 intervals of the so- Eb E F Ob G Ab A Bb B c immediately
called enharmonic scale, which distinguishes follows. In Meshaqah's scale of 24 Quarter-
sharps and flats, without dividing the Semi- tones, p. 264c, that of 12 Semitones is also
tones E to Ff and B to C To compose their implicitly contained.— Traru^a^.]
Digitized by^OOQlC
286 MEANING OF THE LEADING NOTE; part in.
this sense it also becomes easy to sharpen Bi somewhat, making it B, for example,
to bring it near to c, and mark its reference to that tone more distinctly.
According to my own feeling, the leading effect of the tone ^i is much more
marked in such passages as i^...jBi...c otF-\-Ax..,Bi,.,c, in which ^i is not related
to the preceding tones, than in such a passage as 0+Bi.,.c where it is. But as I
have found nothing on this point in musical writings, I do not know whether
musicians are likely to agree with me in this opinion. For the other Semitone of
the scale, Ei.,.F, the Ei does not seem to lead to F, if the tonality of the melody
is well preserved, because in this case Ei has its own independent relation to the
tonic, and hence is musically quite determinate. The hearer, then, has no
occasion to regard Ei as a mere preparation for F. Similarly for the interval
G..,A^\} [=112 cents] in the minor mode. The O is more nearly related to the
tonic C than A^\} is. On the other hand, Hauptmann is probably right in
.f considering the interval D...E^\} [=112 cents] in the minor mode, as one in which
D leads to E^\}, because D has only a relationship of the second degree to the tonic
0, although its relationship is certainly closer than that of Bi to 0.
But the relation of D^\} in descending passages of the mode of the minor
Sixth of C (the old Greek Doric) is perfectly similar to the effect of ^1 in the
ascending scale of C major. It really forms a kind of descending leading note,
and since in the best period of Greek music descending passages were felt as
nobler and more harmonious than ascending ones,* this peculiarity of the Dorio
mode may have been of special importance and have been a reason for the
preference given to this scale. The cadence with the chord of the extreme sharp
Sixth [ratio 128 : 225, cents 976] —
C — JE;»[> + (?... c ^
f is almosi the only remnant of the ancient tonal modes. It is quite isolated and
misunderstood. This is a (Greek) Doric cadence, in which D^\} and Bi are both
used at the same time as leading notes to O.f
The relation of the second or pa/rhypate of the Greek Doric scale, to the lowest
tone or hypate, seems also to have been perfectly well felt by the Greeks them<
selves, to judge by Ari8totle*s remarks in the 3rd and 4th of his problems on
harmony. I cannot abstain from adducing them here because they admirably and
V delicately characterise the relation. Aristotle inquires why the singer feels his
voice more taxed in taking the parhypate than in taking the hypatS, although they
are separated by so small an interval. The hypate is sung, he says, with a remission
of effort. And then he adds that in order to reach an aim easily it is necessary that
in addition to the motive which determines the will, the kind of volitional effort
should be quite feuniliar and easy to the mind.t The effort felt in singing the lead-
* Aristotle, Problems zix< 33. [The pas- robots 9^ ir6vos' xovovvra 84 fxaXXop Sto^cfperoi*
^ sage has already been cited at full, p. 241(2', Aid rl 84 ravrvp x'^^^h I'V 84 iirw^v pifBtmr
note I. — Translator,] icdrot ZUais Ijcor^pas; *H 8rt fur* kifiauos 1^
f [This cadence is a union of the ancient ^^i}, koX dfia /xtrii. r^v tHtvraffuf 4ka^¥ rh
Doric, beginning with c, rendered harmonisable ivot fidXXttv ; Ath. rainh 84 Ifoocc irp6s fiiay ^^ccr-
as the mode of the minor Sixth, (c ... c2' b ... 0' b Bcurii itpbs rairrfiv ^ mpatr^imv j^fuwa' 8€< 7^
. . . / . . .pr . . . a* b . . . b' b . . . c', p. 27&2, note) with the fi^rit trwpoias icol Kwraffrda-€»s ohc^undrris r^
modem minor, beginning with c, (c. . .d. . .«• b . . ./ ^^i irphs riiv fiofiKriffUf. Arist. ProbL xix. 3, 4.
...</... a' b ... bi .. . c',) and will be more partica- [The whole passage may perhaps be translated
larly considered in the next chapter, pp. 2o6d- thus : * Why do those who sing the parhypaU
308c. The intervals expressed in cents are break down not less than those who sing the
i)'b 386 F 204 O 386 B, and C 316 J7'b 386 G nJete and higher tones, though with a greater
498 c— Translator.] disagreement (hiiartura) ? Is it because they
% This periphrasis seems to me to render sing this with the greatest difficulty, even when
correctly the last clause in the following cita- this is the beginning ? Does not difficulty
tion : Ath ri r^v trofnnrdrriv $8orrcs tAdhiora arise from straining [and forcing] the voice ?
itiro^pfirYruvrtUy oitx i^Toy ^ r^v jrfirrip KpX rh. 6»t»^ This occasions effort, and things done with
fitrii 84 9uurrdtrws wKtiovos; *H 8ri xo^cin^crra effort are most apt to fail. But why do they
rairjfy fSovo'i, icol cStti i| kpxh ; t^ 84 xo^-c^o^ sing the parhypaU with difficulty, and yet
9tk riiv iwh-wrw [irat wUcw] riis ^»tnis ; iv take the hypate easily, although there is on^
Digitized by V^jOOQlC
OTAP.xiv. MEANING OF THE LEADING NOTE. 287
ing note does not lie in the larynx, but in the difficulty we feel in fixing the voice
upon it by mere volition while another tone is already in our mind, to which we
desire to pass, and which by its proximity conducted us to the leading note. It is
not till we reach the final tone that we feel ourselves at home and at rest, and this
final tone is sung without any strain on the will.
Proximity in the scale then gives a new point of connection between two tones,
which is not merely active in the case of the leading note, just considered, but also,
as already mentioned, in interpolating tones between two others in the chromatic
and enharmonic modes. Intervals of pitch are in this respect analogous to mea-
surements of distance. When we have the means of determining one point (the
tonic) with great exactness and certainty, we are able by its means to determine
other points with certainty, when they are at a known small distance from it (the
interval of a Semitone), although perhaps we could not have determined them with
so much certainty independently. Thus the astronomer employs his fundamental f
fixed stars, of which the positions have been determined with the greatest possible
accuracy, for accurately determining the positions of* other stars in their neigh-
bourhood.
We may also remark that the interval of a Semitone plays a peculiar part as the
introduction {appoggiatura) to another note. As an appoggiatura in a melody any
tone can be used, even when not in its scale, provided it makes the interval of a
Semitone with the note in the scale which it introduces ; but a foreign tone which
makes the interval of a whole Tone with that note in the scale, cannot be so used.
The only justification of this use of the Semitone is certainly its existence as a
well-known interval in the diatonic scale, which the voice can sing correctly and
the ear can readily appreciate, even when the relations on which its magnitude
ddipends are not clearly sensible in the passage where it is used. Hence also no
arbitrarily chosen small interval can be thus employed. Although slight changes
in the interval of the leading note may be introduced by practical musician^Ho give ^
a stronger expression to its tendency towards the tonic, they must never go so fia.r
as to make those changes clearly felt.*
Hence the major Seventh in its character of leading note to the tonic acquires
a new and closer relationship to it, unattainable by the minor Seventh. And in
this way the note which is most distantly related to the tonic beeomes peculiarly
valuable in the scale. This circumstance has continually grown in importance in
modem music, which aims at referring every tone to the tonic in the clearest pos-
sible manner ; and hence, in ascending passages going to the tonic, a preference
has been given to the major Seventh in all modem keys, even in those to which it did
not properly belong. This transformation appears to have begun in Europe during
the period of polyphonic music, but not in part songs only, for we find it also in the
bomophonio Gantus firrmiB of the Eoman Catholic Church. It was blamed in an
edict of Pope John XXII., in 1322, and in consequence the sharpening of the lead-
ing note was omitted in writing, but was supplied by the singers, a practice which ^
Winterfeld believes to have been followed by Protestant musical composers even
down to the sixteenth and seventeenth centuries, because it had once come into
use. And this makes it impossible to determine exactly what were the steps by
which this change in the old tonal modes was effected.f
Even to the present day, according to A. v. Oettingen's report,^ the Esthonians
a diesis (Semitone or Qaartertone) between clearly a connection in the writer's mind be-
them ? Is it because the hypatS is snng with tween Sicbroo-is, HffrturiSy and Kardaraffis^
a remission of eiffort, and at the same time it which influenced his reasoning, but evaporates
is easy to go upwards after getting oneself in translation. — Translator.]
together for the effort {avcrouris) ? For the ♦ [See App. XX. sect. G. art.' 6.—TranS'
same reason it is easy to sing what leads up latorJ]
to any note, or the paraneU. For the will f Def evangeUsche Kirchengesang, Leip-
requires not only conscious thought {Hwota) zig, 1843, vol. i. introduction.
but an inclination {Kardtrrcuris) which is per- J Das Harmoniesystem in dualer Ent-
fectly familiar to the habit of mind {^Bos).* wickelung, Dorpat und Leipzig, 1866, p. 113,
The passage is very difficult, and there was
Digitized by VjOOQlC
288 MEANING OP THE LEADING NOTE. part m.
struggle against singing the leading note in minor scales, although it may be clearly
struck on the organ.
Among the ancient tonal modes, the Greek Lydian (major mode) and the un-
melodic Hypolydian (mode of the Fifth, p. 269a, No. 7) had the major Seventh as
the leading note to the tonic, and hence the first was developed into the principal
tonal mode of modem music, the major mode. The Greek Ionic (mode of the
Fourth) differed from it only in having a minor Seventh. On simply altering this
into a major Seventh, this mode also became major. On giving a major Seventh
to each of the other three, they gradually converged to our present minor mode
during the seventeenth century. From the Greek Phrygian (mode of the minor
Seventh) by changing B\} into By we obtain
THE ASCENDING MINOR SCALE.
' G„.D...E'\}...F.„G...Ai..,By...c
as we had already found from a simple consideration of the relationship of tones
[p. 2746, No. 2]. The Greek Hypodoric or EoKc (mode of the minor Third), which
answers to our descending minor scale, gives on changing B^\} into ^1,
THE INSTRUMENTAL MINOR SCALE.
C ... D ... E^\} ... F ... G ... A'\} ... 5, ... c
which is difficult for singers to execute, on account of the interval A^\} ... 5,
[= ratio 64 : 75, cents 274J, but frequently occurs in modem music both ascending
and descending.
The Greek Doric (mode of the minor Sixth) with a major instead of a minor
Seventh, is still discoverable in the final cadence mentioned on p. 286^.
The general introduction of the leading tone represents, therefore, a continually
increasing consistency in the development of a feeling for the predominance of the
«r tonic in a scale. By this change, not only is the variety of character in the ancient
tonal modes seriously injured, and the wealth of previous means of expression
essentially diminished, but even the links of the chain of tones in the scale were
disrupted or disturbed. We have seen that the most ancient theory made tonal
systems consist of series of Fifths, and that each system had at first four and after-
wards six intervals of a Fifth. The predominance of a tonic as the single focus
of the system was not yet indicated, at least externally ; it became apparent at
most by a limitation of the number of Fifths to contain those tones only which
occurred in the natural scale. All Greek tonal modes may be formed from the
tones in the series of Fifths —
F±C±G±D±A±E±B.
Directly we proceed to the natural intonation of Thirds, the series of Fifths ia
interrupted by an imperfect Fifth, as in
m F ± C ± G ± D ... Ay ± El ± Bi
where the Fifth D...A1 [= 680 cents] is imperfect. And when finally the sharp
leading note is introduced, as by the use of G^if^ for G in Ay minor, the series
is entirely interrupted [0 : (?2 J = 16 : 25 = 772 cents].
In the gradual development of the diatonic system, therefore, the various links
of the chain which bound the tones together were sacrificed successively to the
desire of connecting all the tones in a scale with one central tone, the tonic. And
in exact proportion to the degree with which this was carried out, the conception of
tonality consciously developed itself in the minds of musicians.
The further development of the European tonal system is due to the cultivation
of harmony, which will occupy us in the next chapter.
But before leaving our present subject, some doubtful points have still to be
considered. In the preceding chapter I have shewn that the melodic relationship
of tones can be made to depend upon their upper partials, precisely in the same
Digitized by VjOOQlC
CHAP. XIV. MELODY IN SIMPLE TONES. 289
way as their consonance was shewn to be determined in Chapter X. Now this
method of explanation may in a certain sense be considered to agree with the
favourite assertion that * melody is resolved harmony,* on which mtisicians do
not hesitate to form musical systems without staying to inquire how harmonies
could have been resolved into melodies at times and places where harmonies had
either never been heard, or were, after hearing, repudiated. According to our
explanation, at least, the same physical peculiarities in the composition of musical
tones, which determined consonances for tones struck simultaneously, would also
determine melodic relations for tones struck in succession. The former then would
not be the reason for the latter, as the above phrase suggests, but both would have
a common cause in the natural formation of musical tones.
Again, in consonance we found other peculiar relations, due to combinational
tones, which become effective even when simple tones, or tones with few and faint
upper partials, are struck simultaneously. I have already shewn that combina- ^
tional tones very imperfectly replace the effect of upper partial tones in a con-
sonance, and that consequently a chord formed of simple tones is wanting in
brightness and character, the distinctions between consonance and dissonance
being only very imperfectly developed.
In melodic passages, however, combinational tones do not occur, and hence the
question arises as to how far a melodic effect could be produced by a succession of
simple tones. There is no doubt that we can recognise melodies which we have
already heard, when they are executed on the stopped pipes of an organ, or are
whistled with the mouth, or merely struck on a glass or wood or steel harmonicon,
as a musical box, or are played on bells. But there is also no doubt that all these
instruments, which generate simple tones, either alone or accompanied by weak
and remote inharmonic secondary tones, are incapable of producing any effective
melodic impression without an accompaniment of musical instruments proper.
They may be often extremely effective for performing single parts when accom- ^
panied by the organ, or the orchestra, or a pianoforte, but by themselves they produce
very poor music indeed, which degenerates into absolute unpleasantness when the
inharmonic secondary tones are somewhat too loud.
We are bound, however, to give some reason why any impression of melody at
all can be produced by such instnmients.
Now we must first remember that, as shewn at the end of Chapter VII., the
actual construction of the ear favours the generation of weak harmonic upper par-
tials within the ear itself, when powerful but objectively simple tones are sounded.
Hence it is at most very weak objectively simple tones which can be regarded as
also subjectively simple.
Next, there is an effect of memory to be brought into account. Supposing
that I have been used to hear Fifths taken at all possible pitches, and have recog-
nised them by aural sensation as having a very close melodic relationship, I should
know the magnitude of this interval by experience for every tone in the scale, ^
and should retain the knowledge thus acquired by the action of a man's memory
of sensations, even of those for which he has no verbal expression.
When, then, I hear such an interval executed on tuning-forks, I am able to
recognise it as an interval I have often heard, although its tones have either none,
or only some faint remnants of those upper partials which formerly gave it a right
to be considered as a favourite interval of close melodic relationship. And just in
the same way I shall be able to recognise, as previously known, other melodic pas-
sages or whole melodies which are executed in simple tones, and even if I hear a
melody for the first time in this way, whistled with the mouth or chimed by a clock,
or struck on a glass harmonicon, I should be able to complete it by imagining how
it would soimd if executed on a real musical instrument, as the voice or a violin.
A practised musician is able to form a conception of a melody by merely reading
the notes. If we give the prime tones of these notes on a glass harmonicon, we give
a firmer basis to the conception by really exciting a large portion of the impression
Digitized by V^ODQIC
290 MELODY IN SIMPLE TONES. pabt iu.
on the senses which the melody would have produced if sung. Simple tones, how-
ever, merely exhibit an outline of the melody. All that gives the melody its charm
is absent. We know, indeed, the individual intervals which it contains, but we have
no inamediate impression on our senses which serves to distinguish those which are
distantly from those which are closely related, or the related from the totally un-
related. Observe the great difference between merely whistling a melody or play-
ing it on a violin ; between striking it on a glass harmonicon or on the piano I The
difference is somewhat of the same kind as that between viewing a single photograph
of a landscape, and seeing two corresponding photographs of it through a stereo-
scope. The first enables me, by means of my memory, to form a conception of the
relative distances of its parts, and this conception may be often very satisfactory.
But the stereoscopic fusion of the two figures gives me the real impression on the
senses which the relative distances of the parts of the landscape would have them-
[ selves produced, and which I am obhged in the case of a single image to supply by
experience and memory. Hence the stereoscopic picture is more lively than the
simple perspective view, exactly in the same way as immediate impressions on our
senses are more Hvely than our recollections.
The case seems to be the same for melodies executed in simple tones. We
recognise the melodies when we have heard them otherwise performed ; we can
even, if we have sufi&cient musical imagination, picture to ourselves how they
would sound if executed by other instruments, but they are decidedly without the
immediate impression on the senses which gives music its charm.
CHAPTER XV.
THE CONSONANT CHORDS OF THE TONAL MODES.
^ Polyphony was the form in which music for several voices first attained a certain
degree of artistic perfection. The peculiar characteristic of this style of music
was that several voices were singing each its own independent melody at the same
time, which might be a repetition of the melodies already sung by the other voices,
or else quite a different one. Under these circumstances each voice had to obey
the general law of tonaHty common to the construction of all melodies, and, more-
over, every tone of a polyphonic passage had to be referred to the same tonic.
Hence each voice had to commence separately on the tonic or some tone closely
related to it, and to close in the same way. In practice each part of a polyphonic
piece was made to begin with the tonic or its Octave. This fulfilled the law of
tonality, but necessitated the closing of a polyphonic piece with a unison.
The reason why higher Octaves might accompany the tonic at the close, lies, as
we saw in the last chapter, in the fa^i that higher Octaves are merely repetitions of
portions of the fundamental tone. Hence by adding its Octave to the tonic at the
H close, we merely reinforce part of its compound tone ; no new compound or simple
tone is added, and the union of all the tones contains only the constituents of the
tonic itself.
The same is true for all the other partial tones which are contained in the tonic.
The next step in the development of the final chord was to add the Twelfth of the
tonic. Now the chord c...c'±^' contains no element which is not also a con-
stituent of the compound tone c when sounded alone, and consequently, being a
mere representative of the single musical tone o, it is suitable for the termination
of a piece of music having the tonic c.
Nay even the chord &±g\..c" might be so used, for when it is struck we hear,
weakly indeed, but still sensibly, the combinational tone c, so that the whole mass
of tone again contains nothing more than the constituents of the tone c. It must
be owned, however, that this combination would answer to a rather unusual quality
of tone, with a proportionably weak prime partial.
On the other hand, it was not possible to use the chords c.c'.../' or c'.../'dic"
Digitized by VjOOQlC
CHAP. XV.
MEANING OF THE TONIC CHORD.
291
to end a piece having the tonic c, although these chords are consonances as well as
the preceding, because / is not an element of tlie compound tone c, and hence the
closing chord would contain something which was not the tonic at all. It is here
probably that we have to look for the reason why some medieval theoreticians
wished to reckon the Fourth among the dissonances. But perfect consonance was
not sufficient to make an interval available for the final chord. There was a second
condition which the theoreticians did not clearly understand. The tones of the
final chord had to be constituents of the compound tone of the tonic. This was
the only case in which those tones could be employed.
The Sixth of the tonic is as ill suited as the Fourth for use in the final chord.
But the major Third can be used, because it occurs as the fifth partial tone of the
tonic. Since the qualities of tone which are fit for music generally allow the fifth
and sixth partial tone to be audible, but make the higher partials either entirely
inaudible or at least very funt, and since, moreover, the seventh partial is dissonant ^
i¥ith the fifth, sixth, and eighth, and is not used in the scale, the series of tones
available for the closing chord terminates with the Third. Thus we actually find
down to the beginning of the eighteenth century, that the final chord has either
no Third, or only a major Third, even in tonal modes which contain only the minor
and not the major Third of the tonic. To attain fulness, it was preferred to do
violence to the scale by using the major Third in the closing chord. The minor
Third of the tonic can never stand for a constituent of its compound tone. Hence
it was originally as much forbidden as the Fourth and Sixth of the tonic. Be-
fore a minor chord could be used to close a piece of music the feeling for harmony
bad to be cultivated in a new direction.
The ear is the more satisfied with a closing major chord, the more closely the
order of the tones used imitates the arrangement of the partial tones in a compound.
8ince in modem music the upper voice is most conspicuous, and hence has the
principal melody, this voice must usually finish with the tonic. Bearing this in f
mind, we can use any of the following chords for the close (combinational tones
are added as crotchets) : —
12345
5
rz2i
=g=
IZ2I
izar
122=
m
IC2I
r?2=
r
r
In the chords i and 2 all the notes coincide with partials of 0, and they
therefore most closely resemble the compound tone C itself. And then closer
positions of the chord can be substituted, provided they resemble the first by ^
having C for the fundamental tone as in 3, 4, 5. They still retain sufficient resem-
blance to the compound tone of the low G to be used in its place. Moreover, the
combinational tones, written as crotchets in 3, 4 and 5, assist in the effect of
making the deeper partials of the compound C, at least faintly, audible. But the
first two positions always give the most satisfactory close. The tendency towards
a deep final tone in harmonic music is very characteristic, and I beheve that the
above is its proper explanation. There is nothing of the kind in the construction
of homophonic melodies. It is peculiar to the bass of part music.
Precisely in the same way that the tonic, when used as the bass of its major
chord at the close, gives it a resemblance to its own compound tone, and is hence
felt as the essential tone of the chord, all major chords sound best when the
lowest tone of their closest triad position (No. 4, p. 219c) is made the bass. The
other major chords in the scale are those on its Fourth and Fifth, and hence for
the scale of C major, are F+Ai — C and G+^i— i>. Hence if we make the
Digitized by V^JiQQQlC
292 REPLACEMENT OF COMPOUND TONES BY CHORDS, pabt m.
harmony of a piece of masic to consist of these major chords only, each having
its fundamental tone in the bass, the effect is almost that of a compound tonic in
different qualities of tone passing into its two nearest related compound tones, the
Fourth and Fifth. This makes the harmonisation transparent and definite, but it
would be too uniform for long pieces. Modem popular tunes, songs and dances,
are however, as is well known, constructed in this manner. The people, and
generally persons of small musical cultivation, can be pleased only by extremely
simple and intelligible musical relations. Now the relations of the tones are
generally much easier to feel with distinctness in harmonised than in homophonic
music. In the latter the feeling of relationship of tone depends solely on the
sameness of pitch of two partials in two Consecutive musical tones. But when we
hear the second compound tone we can at most remember the first, and hence we
are driven to complete the comparison by an act of memory. The consonance, on
^ the other hand, gives the relation by an immediate act of sensation ; we are no
longer driven to have recourse to memory ; we hear beats, or there is a roughness
in the combined sound, when the proper relations are not preserved. Again, when
two chords having a common note occur in succession, our recognition of their
relationship does not depend upon weak upper partials, but upon the comparison of
two independent notes, having the same force as the other notes of the corre-
sponding chord.
When, for example, I ascend from C to its Sixth Ai,l recognise their mutual
relationship in an unaccompanied melody, by the fact that «',, the fifth partial of
C, which is already very weak, is identical with the third partial of -4i. But if I
accompany the -4, with the chord jP+^Ii — c, I hear the former c sound on power-
fully in the chord, and know by immediate sensation that Ai and C are consonant,
and that both of them are constituents of the compound tone F,
When I pass melodically from C to i9i or X), I am obliged to imagine a kind of
% mute G between them, in order to recognise their relationship, which is of the
second degree. But if I audibly sustain the note G while the others are sounded,
their common relationship becomes really sensible to my ear.
Habituation to the tonal relations so evidently displayed in harmonic music,
has had an indisputable influence on modem musical taste. Unaccompanied
songs no longer please us ; they seem poor and incomplete. But if merely the
twanging of a guitar adds the fundamental chords of the key, and indicates the
harmonic relations of the tones, we are satisfied. Again, we cannot fail to see
that the clearer perception of tonal relationship in harmonic music has greatly
increased the practicable variety in the relations of tones, by allowing those which
are less marked to be freely used, and has also rendered possible the construction
of long musical pieces which require powerful links to connect their parts into one
whole.
The closest and simplest relation of the tones is reached in the major mode,
^ when all the tones of a melody are treated as constituents of the compound tone
of the tonic, or of the Fifth above or the Fifth below it. By this means all the
relations of tones are reduced to the simplest and closest relation existing in
any musical system — that of the Fifth.
The relation of the chord of the dominant G to that of the tonic C, is some-
what different from that of the chord of the subdominant F to the tonic chord.
When we pass from C-^-Ei—G to G-^Bi—d we use a compound tone, G, which
is already contained in the first chord, and is consequently properly prepared, while
at the same time such a step leads us to those degrees of the scale which are most
distant firom the tonic, and have only an indirect relationship with it. Hence this
passage forms a distinct progression in the harmony, which is at once well assured
and properly based. It is quite different with the passage from C+J5J|— G to
F-^Ai^c, The compound tone F is not prepared in the first chord, and it has
therefore to be discovered and struck. Hence the justification of this passage as
correct and closely related, is not complete until the step is actually made and it is
Digitized by V^jOOQlC
CHAP. XV,
HARMONY OF THE MAJOR MODE.
293
felt that tlie chord of F contains no tones which are not directly related to the tonic
(7. In the passage from the chord of 0 to that of F, therefore, we miss that distinct
and well-assured progression which marked the passage from the chord of C to that
of G, Bat as a compensation, the progression from the chord of C to that of F
bas a softer and calmer kind of heaaty, due, perhaps, to its keeping within tones
directly related to the tonic C, Popular music, however, favours the other passage
from the tonic to the Fifth ahove (hence called the dominant of the key), and many
of the simpler popular songs and dances consist merely of an interchange of tonic
and dominant chords. Hence also the common harmonicon (accordion, German
concertina), which is arranged for them, gives the tonic chord on opening the
bellows, and the dominant chord on closing them. The Fifth below the tonic is
called the sz^idominant of the key. Its chord is seldom introduced at all into usual
popular melodies, except, perhaps, once near the close, to restore the equilibrium
of the harmony, which had chiefly inclined towards the dominant.
When a section of a piece of music terminates with the passage of the dominant
into the tonic chord, musicians call the close a complete cadence. We thus return from
tbe tones most distantly related to the tonic, to the tonic itself, and, as befits a close,
make a distinct passage from the remotest parts of the scale to the centre of the
system itself. If, on the other hand, we close by passing from the subdominant to the
tonic chord, the result is called an imperfect or plagal cadence. The tones of the
subdominant triad are all directly related to the tonic, so that we are already close
upon the tonic before we pass over to it. Hence the imperfect cadence corresponds
to a much quieter return of the music to tbe tonic chord, and the progression is
much less distinct than before.
In tbe complete cadence tbe chord of tbe tonic follows that of the dominant,
but to preserve the equilibrium of the system in relation to tbe subdominant, its
chord is made to precede that of the dominant as in i or 2.
IF
IS
-4-
^
-<sf-
"Ts — :
ESE
321
-^ r:>-
IZ2I
A^^
"7S>"
This succession really forms tbe complete close, by bringing all the tones of the
whole scale together again, and thus in conclusion collecting and fixing every part
of tbe key.
Tbe major mode, as we bavei seen, permits tbe requisitions of tonality to be
most easily and completely united with harmonic completeness. Every tone of its
scale can be employed as a constituent of tbe musical tone of tbe tonic, the domi IT
nant, or the subdominant, because these fundamental tones of tbe mode are also
fundamental tones of major chords. This is not equally tbe case in tbe other
ancient tonal modes.
I. Majob mode *
Mode of the
FOUBTH *
major
^"~ • * "<k
f + ai — c +«! " g +&1— ^
• , ^ .
major major
major
/ + ai - c + ei - g " b^\} + d
major minor
* [Of course when the modes are thas
reduced to harmonic combinations, the tones
ol the old modes, as given in footnote to p. 268c,
are all altered, and become those in footnote
to p. 274c. See also App. XX. sect. £, arts. 9
and 10,— Translator.]
Digitized by V^OOQIC
294
AMBIGUITY OF THE MINOR CHORD.
PART m.
3. Mode op the
MiNOB Seventh *
4. Mode op the
MINOB Thibd*
{minor mode)
5. Mode op the
MINOB Sixth*
minor
/ 4- ai - c - e^\}-h g - b^\}+ d
major minor
minor
/ - a^\>+ c - e^\>+ g - 2>'t>+ ^
minor
minor
minor
5t?- ^*t>+ / - a»t?+ c - e^\}+ g
minor minor
In the minor chords, the Third does not belong to the compomid tone of its fdnda-
% mental note, and hence cannot appear as a constituent of its quality ; so that the
relation of all the parts of a minor chord to the fundamental note is not ao im-
mediate as that for the major chord, and this is a source of difficulty in the final
chord. For this reason we find almost all popular dance and song music written
m the major mode f ; indeed, the minor mode forms a rare exception. The people
must have the clearest and simplest intelligibility in their music, and this can only
be furnished by the major mode. But there was nothing like this predominance of
the major key in homophonic music. For the same reason the harmonic accom-
paniment of chorales in major keys was developed with tolerable completeness as
early as the sixteenth century, so that many of them correspond with the cultivated
musical taste of the present day; but the harmonic treatment of the minor and
the other ecclesiastical modes was still in a very unsettled condition, and strikes
modem ears as very strange.
In a major chord c+^i —g, we may regard both g and e, as constituents of the
f compound tone of c, but neither c nor g as constituents of the compound tone of e^,
and neither c nor Ci as constituents of the compound tone of g.t . Hence the major
chord c+Ci^g is completely unambiguous, and can be compared only with the
compound tone of c, and consequently c is the predominant tone in the chord, its
rooty or, in Rameau's language, its fundamental bass ; and neither of the other two
tones in the chord has the sHghtest claim to be so considered.
In the minor chord c—e^\}+gj the ^ is a constituent of the compound tones of
both c and e^\}. Neither e^^ nor c occurs in either of the other two compound
tones c, g. Hence it is clear that g at least is a dependent tone. But, on the other
hand, this minor chord can be regarded either as a compound tone of c with an
added e^\} or as a compound tone of e^\} with an added c. Both views are enter-
tained at different times, but the first usually prevails. If we regard the chord as
the compound tone of c, we find g for its third partial, while the foreign tone 6^t>
only occupies the place of the weak fifth partial e|. But if we regarded the chord
f as a compound tone of e^\}, although the weak fifth partial g would be properly repre-
sented, the stronger third partial, which ought to be b^\}, is replaced by the foreign
tone c. Hence in modem music we usually find the minor chord c—e*b+^ treated
as if its root or fundamental bass were c, so that the chord appears as a some-
what altered and obscured compound tone of c. But the chord also occurs in the
position e^\}-^g.,,c (or better still as e^\}-^g...c^) even in the key of B^\} major, as a
substitute for the chord of the subdominant e'j^. Rameau then calls it the chord of
the great Sixth [in Enghsh ' added Sixth '], and, more correctly than most modem
theoreticians, regards e^\} as its fundamental bass.§
♦ [See p. 293, note.] t [T^is remark does not apply to old English moBio,— Translator.^
X [Taking only six partials, we have for —
Oompoimd ToneB
Simple Partial Tones
12345 6
C
C c g & e\ g'
Ex e, 6, e\ g'^l b\
^1
G,
0 g d' g' b\ d"
£'b
£"bc'b6'bc"b<7' 6"b.
— Translator. 1
§ [The scale of B^ b major has the chords
«^b + gr— 6'b + d-/* + a— c*; hence, regarding
the chord as made up of the notes of this
scale, it would be c^ \ e'b + (7, which is not a
minor chord at all, like c-e'b + f7, because it
has a Pythagorean in place of a just minor
Third. It was only tempered intonation which
confused the iwc?;c^|e^^|^f£^ wUl be
CHAP. XV, AMBIGUITY OF THE MINOR CHORD. 295
When it is important to guide the ear in selecting one or other of these two
meanings of the minor chord, the root intended may be emphasised by giving it a
low position or by throwing several voices upon it. The low position of the root
allows such other tones as could be fitted into its compound tone, to be considered
directly as its partials, whereas the low compound tone itself cannot be considered
as the partial of another much higher tone. In the first half of last century, when
the minor chord was first used as a close, composers endeavoured to give pro-
minence to the tonic by increasing the loudness of the tonic note in comparison
with its minor Third. Thus in Handel's oratorios, when he concludes with a
minor chord, most of the conspicuous vocal and instrumental parts are concen-
trated on the tonic, while the minor Third is either touched by one voice alone, or
merely by the accompanying pianoforte or organ. The cases are much rarer where
in minor keys he gives only two voices to the tonic in the closing chord, and one
to its Fifth and another to its Third, which is his rule in major modes. IT
When the minor chord appears in its second Subordinate signification, as
e^\}+g..x with the root e^\}, this (aoi is shewn partly by the position of e^\} in the
bass, and partly by its close relationship to the tonic &^t>. Modem music even
makes this interpretation of the chord still clearer by adding Z»^t> as the Fifth of e^t>>
so that the chord becomes dissonant in the form e^l}—g-\-b^\}..,c^.*
The disinclination of older composers to close with a minor chord, may be
explained partly by the obscuration of its consonance from false combinational tones,
and partly because, as already mentioned, it does not give a mere quality of the
tonic tone, but mixes foreign constituents with it. But in addition to the minor
Third, which does not fit into the compound tone of the tonic, the combinational
tones of a minor chord are equally foreign to it. As long as the feeling of tonaHty
required a definite single compound tone for the connecting centre of the key, it
was impossible to form a satisfeu^tory close except by a reproduction of the pure
compound tone of the tonic with no foreign admixture. It was not till a farther ^
development of musical feeling had given the chords of the mode an independent
significance, that the minor chord, notwithstanding its possession of constituents
foreign to the compound tone of the tonic, could be justified in its use as a close.
Hauptmann f gives a different reason for avoiding the minor chord at the close.
He asserts that before the chord of the dominant Seventh came into use, there was
no voice-part suitable for fedling into the minor Third of the tonic. Thus if the final
cadence consisted of the chords G+Bi— -D, C— ^'t> + G^, the D of the first chord
was the only one which could pass melodiously in E^\}, but this would have ap-
peared like the passage of the leading note D in the key of E^\} major into its tonic
E^\}, and hence have called up the feeling of E^\} major in Heu of C minor. We
may admit that this relation of the leading note would have drawn the hearer's
special attention to the two tones in question, and to a certain extent disturbed his
recognition of the key, but yet it is clear that even without the help of this chord
of the dominant Seventh, there were several ways for the voices to pass through ^
dissonances into the minor Third of the closing chord, if composers had felt any
wish to do so. Thus in the plagal cadence
c -e>|>-^ g ...&
0 - e^\}+ g ... cf
which is so often used on other occasions, the Fourth / could be made to descend
to the minor Third e^\} without any inconvenience. Indeed, we find that when
hereafter drawn to this important distinction, Observe that it is c* which is now introduced
see p. 299a. — Translator,] in the text, in place of c. If c is retained,
* [Transposing the c* the chord becomes thus c— e*b + <;— 6'b, the chord is one of those
c* I e'b + i;— 6' b, BO that we have a major chord chords of the Seventh considered in Chapter
with the Sixth of its root added, that is, the X\l.- -Translator,]
snbdominant of the key of JB'b rendered dis- f Harmonik und Mclrik, Leipzig, 1853,
sonant by introducing c\ the Second of the p. 216.
key, or the Sixth above the Bubdominant c' b. V^OOQIC
2 96 DEGREES OF RELATIONSHIP OF CHORDS. part hi.
the chord of the dominant Seventh had actually come into use, and the Saventh F
of the chord (?+5i — Z) | F ought by every right to have descended into the minor
Third E^\} of the closing chord, musical pieces of the fifteenth century* avoid this
progression, and make this Seventh F either ascend to the Fifth G, or descend to
the major Third ^i of the final chord, instead of to E^\}y its minor Third. This
custom prevailed down to Bach's time.
In Chapter XUI. (p. 249a) we characterised modem harmonic music, as con-
trasted with medieval polyphony, by its development of a feeling for the independent
significance of chords. In Palestrina, Gabrieli, and still more in Monteverde and the
first composers of operas, we find the various degrees of harmoniousness in chords
carefully used for the purposes of expression. But these masters are almost entirely
without any feeling for the mutual relation of consecutive chords. These chords
often follow one another by entirely unconnected leaps, and the only bond of union
. II is the scale, to which all their notes belong.
The transformation which took place from the sixteenth to the beginning of the
eighteenth century, may, I think, be characterised by the development of a feeling
for the independent relationship of chords one to the other, and by the establish-
ment of a central core, the tonic chordy round which were grouped the whole of
the consonant chords that could be formed out of the notes of the scale. For these
chords there was a repetition of the same effort which was formerly shewn in the
construction of the scale, where interrelations of the tones were first grounded on
a chain of intervals, and afterwards on a reference of each note to a central com-
pound tone, the tonic.
Two chords which have one or more tones in common will here be termed
directly related.
Chords which are directly related to the same chord will be here said to be
related to each other in the second degree,
H Thus c+Ci —g and g-\'hx—d are directly related, and so are c+ei—g and aj —
cH-^i ; but g + bi—d and ai—c+Ci are related only in the second degree.
When two chords have two tones in common they are more closely related than
when they have only one tone in common. Thus c+ej — ^ and Ui — c+ei are more
closely related than c -f- e 1 — gf and gf -h 6 , — d.
The tonic chord of any tonal mode can of course only be one which more or
less perfectly represents the compound tone of the tonic, that is, that major or
minor chord of which the tonic is the root. The tonic note, as the connecting core
of all the tones in a regularly constructed melody, must be heard on the first
accented part of a bar, and also at the close, so that the melody starts from it and
returns to it ; the same is true for the tonic chord in a succession of chords. In
both of these positions in the scale we require to hear the tonic note, accompanied
not by any arbitrary chord, but only by the tonic chord, having the tonic note
itself as its root. This was not the case even as late as the sixteenth century, as
1[ is seen by the example on p. 247c taken from Palestrina.
When the tonic chord is major, the domination of all the tones by the tonic
rote is readily reconciled with the domination of all the chords by the tonic chord*
for as the piece begins and ends with the tonic chord, it also begins and ends at the
same time with the pure unmixed compound tone of the tonic note. But when
the tonic chord is minor, all these conditions cannot be so perfectly satisfied. We
are obliged to sacrifice somewhat of the strictness of the tonality in order to admit
the minor Third into the tonic chord at the beginning and end. At the com-
mencement of the eighteenth century we find Sebastian Bach using minor chords
at the end of his preludes, because these were merely introductory pieces, but not
at the end of fugues and chorales, and at other complete closes. In Handel and even
in the ecclesiastical pieces of Mozart, the close in a minor chord is used alternately
• See an example in Anton Brnmel, in will be found, iMd, p. 550, where the voices
Forkers Oeschichte der Musikf vol. ii. p. 647. might have easily been led to the minor
Another, with a plagal cadence by Josquin, Third.
Digitized by VjOOQlC
CHAP. XV. MINOR CHORD IN THE CLOSE. 29?
with the close in a chord without any Third, or with the major Third. And the
last composer cannot be accused of external imitation of old habits, for we find
that in these usages they always observe the expression of the piece. When at the
close of a composition in the minor mode, a major chord is introduced,, it has the
effect of a sudden and unexpected brightening up of the sadness of the minor key,
producing a cheering, enlightening, and reconciling effect after the sorrow, grief,
or restlessness of the minor. Thus a close in the major suits the prayer for the
peace of the departed in the words, * et lux perpetua luceat eis,' or the conclusion
of the Confutatis Tnaledictis, which runs thus : —
Oro Bupplex et acclinis,
Cor contritum quasi ciuis ;
Gere coram mei finis.
But such a closing major chord is certainly somewhat startling for our present f
musical feeling, even though its introduction may, at one time, add wondrous
beauty and solemnity, or, at another, dart like a beam of hope into the gloom
of deepest despair. If the restlessness remains to the last, as in the Dies irae of
Mozart's Bequiem, the minor chord, in which an unresolved disturbance exists,
forms a fitting close. Mozart was wont to terminate ecclesiastical pieces of a less
decided character with a chord that had no Third. There are many similar
examples in Handel. Hence although both masters stood on the very same plat-
form as modem musical feeling, and themselves gave, as it were, the finishing
touch to the construction of the modem tonal system, they were not altogether
strangers to the feeling which had prevented older musicians from using the minor
Third of the tonic in the final chord. They followed no strict rule, however, but
acted according to the expression and characteir of the piece and the sense of the
words with which they had to close.
Those tonal modes which furnish the greatest number of consonant chords «r
related to one another or to the common chord, are best adapted for artistically
connected harmonies. Since all consonant chords, when reduced to their closest
position and simplest form, are triads consisting of a major and a minor Third, all
the consonant chords of any key can be found by simply arranging them in order
of Thirds, as in the following tables. The braces above and below connect the
chords together. The ordinary round braces, which are placed above, point out
minor chords ; the square braces below indicate major chorda. The tonic chord
is printed in capitals.
i) Majob mode
di - f + ai — C ± El - G + bi -- d
I 1 1 ! I !
2) Mode of the Foubth
6l>+ dt -f + at - C + Et- G - i'|?+ d
3) Mode of the hinob Seventh
b\>+ di-f + at- C - £'t>+ a - 6'1>+ d
I II
J I I
4) Mode of the minob Thibd
I I ! ^1 ^1
5) Mode of the uinob Sixth
-E'\}+ G-
1 1 _
Digitized by VjOOQlC
Jl>-i't) + /-o'|?+ C - E'\}+ G- i'l)
I ! I II I
298 INTONATION OP THE INTEBCALABY TONES. pabt m.
In this ajrangement I have introduced the different intonations of the Second
and Seventh of the key, which we found in the construction of the scales for
homophonic music* But we observe that the chords directly related to the tonio
chord contain every tone in the scale, excepting in the mode of the minor Sixth.
The Second and Seventh of the tonic occur first in the chord of G, which is directly
related to the tonic chord, and next in chords containing jP, which are, however, not
directly connected with the tonic chord. The supplementary tones of the scale
which are related to the dominant thus acquire in harmonic music an important
preponderance over those related to the subdominant. We must necessarily prefer
direct to indirect relations for determining scalar degrees. Hence by confining
ourselves to the chords which are directly related to the tonic chord, we obtain the
following arrangement of the tonal modes :t —
f i) Major mode
f+ai-C+Ei-G+bi-d
I II II I
2) Mode of the Foubth
I II I
3) Mode op the minob Seventh
/+tt,-C-B»t> + (?-6't>+i
4) Mode of the hinob Thibo
V I 11 I
5) Mods of the minob Sixth
d'\>+f^a'\>+C-E'\}-hG-b%
A glance at this table shews that the major mode and mode of the minor Third
(minor mode) possess the most complete and connected series of chords, so that
these two are decidedly superior to the rest for harmonic purposes. This is also
the reason which led to the preference given to them in modem music.
And in this way we obtain a final settlement of the proper intonation of the
supplementary tones of the scale, at least for the first four modes. Hauptmann,
with whom I agree, considers the tone D alone to be the essential constituent of
both the major and minor modes of G. This D forms an imperfect (Pythagorean)
H minor Third with F^ so that the chord D \ F-k-A^ must be considered as dissonant.^
This chord thus intoned is in reaHty most decidedly dissonant to the ear. On the
other hand, Hauptmann admits a major mode which reaches over to the sub-
dominant, and uses Dx in place of D. I consider this conception to be a very
* [These soales differ from those tran- the double modality alluded to in the last
scribed in pp. 293^ and 294a, only in the addi- note, and fixes the modes in the meanings
tion of the secondary forms of intercalary tones, of App. XX. sect. E. art. 9, as
dp bbt or 6^b, which, in fact, imply modula- (i) i C ma.ma.ma.
tions into adjacent modes, or else give (2) i C ma.ma.mL
dp bbt or 6^b, which, in fact, imply modula- (i) i C ma.ma.ma.
tions into adjacent modes, or else give (2) i C ma.ma.mL
a double and ambiguous character to each (3) i C majm.mi.
mode, as shewn on p. 277, footnote f, and by (4) i C mi.mi.mi.
referring to the Duodenarium, App. XX. sect. (5) 3 ^'bma.ma.ma.
£. art. 18, it will be seen that there is a real In the last scale it is more usual, however, to
change of duodene, which always must happen take 6 b in place of 6'b» which makes the scale
when changes of a comma occur. — Trans- 6b-d*b+/— o'b + C— ^'b +0«5Fmijnijiii.
latorJ] But temperament obscures all these dififer-
t [The first four are the same as in pp. zg^d ences.—TraTtslator.]
and 294^. The settlement in the text avoids \ [See p. 295(i, note *. — Traiislaior.]
Digitized by V^jOOQlC
CHAP. XV. HARMONIC DIFFERENCE BETWEEN MAJOR AND MINOR. 299
happy expression of the real state of things. When the consonant chord
Di—F+Ai occurs in any composition it is impossible to return immediately,
without any transitional tone, to the tonic chord C-^E^ — G, The result would
be felt as an harmonic leap without adequate notice. Hence it is a correct
expression of the state of affairs to look upon the use of this chord as the
beginning of a modulation beyond the boundaries of the key of G major, that is,
beyond the limits of direct relationship to its tonic chord. In the minor mode
this would correspond to a modulation into the chord of D^\}+F—A^\}. Of course
in the modem tempered intonation the consonant chord Di—F+Ai is not dis-
tinguished from the dissonant D | F+Ai, and hence the feeling of musicians has
not been sufficiently cultivated to make them appreciate this difference on which
Hauptmann insists.*
As regards the other supplementary tone b^\}, which may occur in the chords
e'b+gf— ^*[> SiJidg — b^\}-^d\ 1 have already shewn in the last chapter that even in ^
homophonic music it is almost always replaced by &i. Harmonic considerations
likewise favour the use of b^ independently of melodic progression. It has been
already shewn that when the two tones of the scale which are but distantly
related to the tonic, make their appearance as constituents of the dominant, they
enter into close relation to the tonic. Now this can only be the case with the
compound tones of the major chord g+bi—d, and not with those of the minor
chord g^b^\}+d» Considered independently, the tones b^\} and d are quite as
closely related to c as the tones b^ and d. But by regarding the two latter as
constituents of the compound tone g, we connect them with c by the same
closeness of relationship that g is itself connected with c. Hence, in all modem
music, wherever b^\} might occur as a constituent of the dominant chord of the key
of 0 minor, or of some dissonant chord replacing the dominant chord, it is usual to
change it into bi and otherwise to use either b^\}OT bi according to the melodic
progression, but more frequently the latter, as I have already remarked when ^
treating of the construction of minor scales. It is this systematic use of the major
Seventh bi in place of the minor Seventh 5^t> of the key which now distinguishes
the modem minor mode from the ancient Hypodoric,t or the mode of the minor
Third. Here again some part of the consistency of the scale is sacrificed in
order to bind the harmony closer together.
The chain of consonant chords in the mode of the minor Third is certainly
impaired when that mode is transformed into our minor by the introduction oibi.
In place of the chain
f^a'\>+C'-E'\>+G-b^M
our minor furnishes only
I I J I
which has one triad less. But the composer is at liberty to alternate the two
tones b^\} and bi.
The introduction of the leading note 5i into the key of c minor generated a new
difficulty in the complete closing cadence of this key. When the chord g + bi—d
is followed by the chord c—e*b+gr, the first being a perfectly harmonious major chord,
and the latter an obscurely harmonious minor chord, the defect in the harmonious-
* [This was referred to in p. 2g4d, note §. be traced on the Duodenarinm. — Translator.']
See App. XX. sect. £. art. 26, example of the f [Hyi)odoric, also caUed Eolio, p. 26Sd, foot-
uae of duodenals. It is a real, though tern- note No. 6, but here the harmonic alteration
porary modulation into a new duodene, one of that mode is meant as in p. 274, footnote
Fifth lower. But for D | F+il, we might use No. 3. This confusion is here regular and in-
D—F^ + A, which is again a modulation into ienUoB&l.^TransUUor,]
a new duodene, one Fifth higher. This should
Digitized by
Google
300 HARMONIC DIFFERENCE BETWEEN MAJOR AND MINOR, pabtiik
ness of the latter is made much more e\ddent by the contrast. But it is precisely
in the final chord that perfect consonance is essential to satisfy the feeling of the
hearer. Hence this close could not become satisfactory until the chord of the
dominant Seventh had been invented, which changed the dominant consonance
into a dissonance.
The preceding explanation shews that when we try to institute a close con-
nection among all the chords peculiar to a mode similar to the close connection
among the tones of the scale, (that is, when we require all the consonant triads
in the harmonic tissue to be related to one of tlieir number, the tonic triad, in a
manner analogous to that in which the notes of the scale are related to one of
their number, the tonic tone), there are only two tonal modes, the major and
wiTior, which properly satisfy such conditions of related tones and related chords.
The major mode fulfils the two conditions of chordal relationship and tonal
II relationship in the most perfect manner. It has four triads which are immediately
related to tlie tonic chord
L II II I
Its harmonisation can be so conducted, (indeed, in popular pieces which must be
readily inteUigible, it is so conducted,) that all tones appear as constituents of the
three major chords of the system, those of the tonic, dominant, and subdominant.
These major chords, when their roots lie lo^, appear to the ear as reinforcements
of the compound tones of the tonic, dominant, and subdominant, which tones are
themselves connected by the closest possible relationship of Fifths. Hence in
this mode everything can be reduced to the closest musical relationship in existence.
And since the tonic chord in this case represents the compound tone of the tonic
% immediately and completely, the two conditions —predominance of the tonic tone
and of the tonic chord — go hand in hand, without the possibility of any contra-
diction, or the necessity of making any changes in the scale.
The major mode has, therefore, the character of possessing the most complete
melodic and harmonic consistency, combined with the greatest simplicity and
clearness in all its relations. Moreover, its predominant chords being major, are
distinguished by full unobscured harmoniousness, when such positions are selected
for them as do not introduce inappropriate combinational tones.
The major scale is purely diatonic, and possesses the ascending leading note of
the major Seventh, whence it results that the tone most distantly related to the
tonic is brought into closest melodic connection with it.
The three predominant major chords furnish tones sufficient to produce two
minor chords, which are closely related to them, and can be employed to diversify
the succession of major chords.
i) The minor mode is in many respects inferior to the major. The chain of chords
for its modem form is —
/-a»b+C-^^»b+ G+h.-d
Elinor chords do not represent the compound tone of their root as well as the
major chords ; their Third, indeed, does not form any part of this compound tone.
The dominant chord alone* is major, and it contains the two supplementary tones of
the scale. Hence when these appear as constituents of the dominant triad, and
therefore of the compound tone of the dominant, they are connected with the tonic
by the close relationship of Fifths. On the other hand, the tonic and subdominant
triads do not simply represent the compound tones of the tonic and subdominant
notes, but are accompanied by Thirds which cannot be reduced to the close relation-
* [That is, among the characteristic chords, textf contain the tones of one major chord,
The two minor chords, as is 8he>»'n in the a'b +c-c'b.— Transiator.]
Digitized by V^OOQIC
CHAP. XV. HARMONIC DIFFERENCE BETWEEN MAJOR AND MINOR. 301
ship of Fifths. The tones of the minor scale can therefore not be harmonised in
such a way as to link them with the tonic note by so close a relationship as in the
major mode.
The conditions of tonality cannot be so simply reconciled with the predominance
of the tonic chord as in the major mode. When a piece concludes with a minor
chord, we hear, in addition to the compound tone of the tonic note, a second
compound tone which is not a constituent of the first. This accounts for the long
hesitation of musical composers respecting the admissibility of a minor chord in
the close.
The predominant minor chords have not the clearness and unobscured har-
moniousness of the major chords, because they are accompanied by combinational
tones which do not fit into the chord.
The minor scale contains an interval a^b...6i, which exceeds a whole Tone in
the diatonic scale,* and answers to the numerical ratio 75 : 64 [=274 cents]. ^
To make the minor scale melodic it must have a different form in descending from
what it has in ascending, as mentioned in the last chapter.
The minor mode, therefore, has no such simple, clear, intelligible consistency as
the major mode ; it has arisen, as it were, from a compromise between the different
conditions of the laws of tonality and the interlinking of harmonies. Hencfe it is
also much more variable, much more inclined to modulations into other modes.
This assertion that the minor system is much less consistent than the major,
win be combated by many modem musicians, just as they have contested the
assertion already made by me, and by other physicists before me, that minor triads
are generally inferior in harmoniousness to major triads. There are many eager
assurances of the contrary in recent books on the theory of harmony.f But the
history of music, the extremely slow and careful development of the minor system
in the sixteenth and seventeenth centuries, the guarded use of the minor close by
Handel, the partial avoidance of a minor close even by Mozart, — all these seem to ^
leave no doubt that the artistic feeling of the great composers agreed with our
conclusions.^ To this must be added the varied use of the major and minor
Seventh, and the major and minor Sixth of the scale, the modulations rapidly
introduced and rapidly changing, and finally, but very decisively, popular custom.
Popular melodies can contain none but clear transparent relations. Look through
collections of songs now preferred by those classes among the Western nations which
have often an opportunity of hearing harmonic music, as students, soldiers,
artisans. There are scarcely one or two per cent, in minor keys, and those are
mostly old popular songs which have descended from the times of homophonic
music. It is also characteristic that, as I have been assured by an experienced
♦ [The interval ie so strange, when nnaccom- falling upon the same note with which they
panied, that if it had to be taken merely as an began, will take e\^ the major Third of c'.
interval, a* b 2745,, a singer would probably fail. Hence the difficulty is not avoided but in-
But the a> b is taken as the minor Third of / creased by introducing the ambiguity of the ^
with ease, and the 6, is taken as the leading major key, into which this is a real modulation
note to c', with 6qual ease, so that the per- from q onwards. — Translator.]
f ectly unmelodic and inharmonic interval a' b f [Can this be due to temperament ? The
274&, never comes into consideration at all. sharp equally tempered major Third of 400
To get rid of it, the subdominant is often taken cents is worse of its kind than the flat equally
major, producing the chords of i C ma.mi.ma., tempered minor Third of 300 cents, which
App. XX. sect. E. art. 10, III., which makes the approaches close to 16:19-298 cents, an
Bcalec204dii2e'bi82/204gi82a,2046,ii2c', interval which many like, and which maybe
and this differs from lie major only by having tried as ef" I -V'b on the Harmonical. — Trans-
e^b in place of e^ In many pianoforte in- lator.]
stmction books this is given as the only form { [These composers played in meantone
of tl e ascending minor. Mr. Curwen (Standard temperament (App. XX. sect. A. art. 16), in
Course, p. 86) says that this major Sixth which the minor Third of 310 cents was much
* ascending is very difficult to sing,* and * has rougher than the equally tempered one of 300
a hard and by no means pleasant effect,* and cents, having much slower beats. Possibly
points out that it leads singers to forget the this difference in the modes of tempering the
key, and in such a phrase as g a, 6, c' d' c" b, the minor Third, may have led to the difference of
pupils will sing e\ instead of e'*b; and even in opinion mentioned in the text.— Tf aiwtotor.l
singing such a passage as ^ a, 6, c' g, instead of
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302
HAKMONISATION OF THE TONAL MODES.
PABT in.
teacher of singing, pupils of only moderate musical talent have much more difficulty
in hitting the minor than the major Third.
But I am by no means of opinion that this character depreciates the minor
system. The major mode is well suited for all frames of mind which are completely
formed and clearly understood, for strong resolve, and for soft and gentle or even
for sorrowing feelings, when the sorrow has passed into the condition of dreamy
and yielding regret. But it is quite unsuited for indistinct, obscure, unformed
frames of mind, or for the expression of the dismal, the dreary, the enigmatic, the
mysterious, the rude, and whatever offends against artistic beauty ; — and it is pre-
cisely for these that we require the minor mode, with its veiled harmoniousness,
its changeable scale, its ready modulation, and less intelligible basis of construc-
tion. The major mode would be an unsuitable form for such purposes, and
hence the minor mode has its own proper artistic justification as a separate
f system.
The harmonic peculiarities of the modem keys are best seen by comparing
them with the harmonisation of the other ancient tonal modes.
Major Modb.
Among the melodic tonal modes the Lydian of the Greeks (the ecclesiastical
Ionic [p. 274, note, No. i] ), in agreement with our major, is the only one which has
an ascending leading note in the form of a major Seventh. The four others had
originally and naturally only minor Sevenths, which even in the later periods of
the middle ages began to give place to major Sevenths, in order that the Seventh
of the scale, which was in itself so loosely connected with the tonic, might be more
closely united to it by becoming the leading note to the tonic at the close.
Mode op the Fourth.
^ The inode of the Fourth (the Greek Ionic, and ecclesiastical Mixolydian) is
principally distinguished from the major mode by its minor Seventh. By merely
changing this into the major we obliterate the difference between them. Taking
C as the tonic the chain of chords in the unaltered mode are as on p. 2986, No. 2,
I II I
If we attempt to form a complete cadence in this mode, as in the following
examples i and 2, they will sound dull from want of the leading note, even when
the dominant chord is extended to a chord of the Seventh g—b^\}+d\f, as in 2.
'I
i[C]*
2[C]
3[C]
fM=j=ii
\fS fJ-:$^;=si
E^^
U
^EE
^
* I
I
^
ite
::tsi
The second example, in which the leading note li^^ lies uppermost, is even duller
than the first example, in which that note Z^^[>ismore concealed. The i^^in these
examples has a very uncertain sound. It is not closely enough related to the
tonic, it is not part of the compound tone of the dominant note g, it is not suffi-
ciently close in pitch to serve as a leading note to the tonic, and it has no tendency
* [The [0] ifl the duodenal of App. XX.
sect. E. art. 26, shewing the exact pitch of all
the notea. These examples have been trans-
posed to admit of their being played on the
Harmonioal, — Translator,]
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CHAP. XV. HARMONISATION OP THE TONAL MODES- 303
to push on to the tonic. Hence when the older composers wished to distinguish
pieces written in the mode of the Fourth from those in the major mode, by their
close, they employed the imperfect or plagal cadence, as in example 3. And as
such a cadence wants the decisive progression required for a close, the sluggishness
previously caused by the absence of a leading tone ceases to be striking.*
In the course of a piece written in this tonal mode, the leading note bi may of
course be used in ascending passages, provided the minor Seventh b^\} is employed
often enough in descending passages. But the effect of the mode is destroyed
when an essential tone of the scale is changed at the close. Hence pieces in the
mode of the Fourth sound like pieces in a major mode which have a decided incli-
nation to modulate into the major mode of the subdominant.f For reasons
> already given, transition to the subdominant appears to be less active than transi-
tion to the dominant. This tonal mode has also no decided progression at the
close, whereas major chords, of which the tonic is one, predominate in it owing to ^
their greater harmoniousness. The mode of the Fourth is consequently as soft
and harmonious as the major mode, but it wants the powerful forward impetus of
major movement. This agrees with the character assigned to it by Winterfeld.t
He describes the ecclesiastical Ionic (major) mode, as a scale which, ' strictly self-
contained and founded on the clear and bright major triad— a naturally harmonious
and satisfactory fusion of different tones, — also bears the stamp of bright and
cheerful satisfsustion.' On the other hand, the ecclesiastical Mixolydian (mode of
the Fourth) is a scale ' in which every part by sound and movement hastens to the
source of its fundamental tone ' (that is, to the major mode of its subdominant),
' and this gives it a yearning character in addition to the former cheerful satisfaction,
not unlike to the Christian yearning for spiritual regeneration and redemption, and
return of primitive innocency, though softened by the bliss of love and faith'.'
Mode of the minob Seventh.
The mode of the minor Seventh (Greek Phrygian [p. 274, note, No. 4] eccle-
siastical Doric) has a minor chord on c as the tonic, and originally another on g
as the dominant, while it has a major chord on its subdominant /, and this last
chord distinguishes the mode from the w/>de of the minor Third (Eolic [p. 294(2,
note, No. 3] ) ; thus
I ! I I
Both of these modes of the minor Seventh and minor Third may, without
destroying their character, change the minor Seventh h^\^ into a leading note 6,,
and our minor mode is a fusion of both. The ascending minor scale belongs
to the mode of the minor Seventh, in which the leading note is used, and the ^
descending to the mode of the minor Third. But when the mode of the minor
Seventh admits the leading note, its chain of chords reduces to the three essential
chords of the scale
This tonal mode has all the character of a minor, but the transition to the chord
of the subdominant has a brighter effect than in the normal minor, where the sub-
* [These can be played on the Harmonical. with its subdominant fga^hbcfd\6\f. —
—Translator.] Translator,]
t [This inolination seems to arise from the X Johannes GabrieK und ssin Zeitalter,
tempered confusion of h^\>d with &b (2, so that vol. i. p. 87.
the scale c(ie|/(^aj d'bc' becomes confused ^-^ ^
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3^4
HAEMONISATION OF THE TONAL MODES.
PABT III.
dominant chord is also minor. On forming the complete cadence both dominant
and subdominant chords are major, while the tonic remains minor. This has of
course an mipleasant effect in the close, because it makes the final chord obscurer
than either of the other two principal chords. Hence it is necessary to introduce
strong dissonances into these two chords, to restore the balance. But if we follow
the old composers and make the final chord major, we give the closing cadence of
this mode an unmistakably major character. As in ecclesiastical modes it is
always allowed to change Ai into A^\}, which would change the subdominant
chord of the mode of the minor Seventh into a minor chord,* we can protect the
mode of the minor Seventh from confusion with the major mode in its final
cadence, but then again it will entirely coincide with the old minor cadence.
Sebastian Bach introduces the major Sixth of the tonic, which is peculiar to
this tonal mode, into other chords for the closing cadence, and thus avoids the
f major triad on the subdominant. He very usually employs the major Sixth as the
Fifth of the chord of the Seventh on the Second of the scale,t as in the following
examples. No. i is the conclusion of the chorale : Was mein Gott loill, das
gescheh' allzeit, in the St. Matthew Passion-Music. No. 2 is the conclusion of the
hymn Veni redemptor gentium, at the end of the cantata : Schwingt frevdig Euch
empor zu den erhabenen Sternen, In both the tonic is bi, the major Sixth g^-t
* [In the original the scale was g + b^ — D
— F^ + A + C|Jl -e in order that it might run
from D to d; and hence the statement was
that it is allowable to change B into B'b.
But in order to keep to the same notes as
were used previously, and to allow of the
scale being played on the Harmonical, I have
transposed it, and hence have had to make
the same change here. The result is precisely
the same, merely meaning that the Seventh
might be taken minor. — Translator.]
t [In the scale / + a,-c-e*b + Sf + 6, -<^,
a, is the major Sixth of the tonic c and d the
Second. The chord of the Seventh on the
^ Second of the scale is therefore d +/ilf — a | c.
hence if S. Bach makes this Fifth a agree with
the major Sixth of the scale a„ he is thinking
in tempered music. When just intonation is
restored, this occasions a restless modulation
as shewn by the duodenals which I hav« in-
troduced over the following examples. — Trotis-
lator.]
t [The notes in the stafif notation are the
usual tempered scale, but the inserted duodenals
convert them into just notes, on the principle
of App. XX. sect. £. art. 26. The tonic is
taken as B, in order to be within the duo^ne
of C, and hence the subdominanr is j^| and -
the dominant i^,K , giving the three duodenes :
In Ex. I the [Bj] indicates that the first two
Subdominant J?|
Tonic By
Dominant F,ff
D F,n A,n
A C,« E^
E G,t B,5
0 B, DS
D F,l A.,l
A C,« E^
C Fs, 0,ff
0 JB, D^t
D F,t Aji
F A, C,ff
C E, G,n ■
0 B, Dji
chords are in the duodene of B,. Then [E{\ shews
that the Dext two chords are in the duodene
of Ey. The difference relates to the chords
with A in the tirst case and A^ in the second.
But the next pair of chords return to the
duodene of B,, which remains till the last bar,
when the notes are in the duodene of F^Xi .
This is rendered necessary by the chord of the
Seventh on C,S the second of the scale, the
Fifth of which is G,ff and not Gf-J , which is the
Sixth of the scale of /?,. That is, it is C,ff + E.,l
— G,5 I J5,. This is, however, only a tem-
porary modulation, and the piece ends in the
d u odene of B, . In E x. 2 the modulations are only
J5,. F,5 and B,, that into F^t being necessitated
by the same chord as before. If these modu-
lations were not taken, but the duodene of B,
were persisted in throughout, frightful dis-
sonances (much worse than the old * wolves ')
would ensue from the imperfect Fifths E^A
and C,« GjU .-Translator,]
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OHAF. XT.
HABMONISATION OF THE TONAL MODES.
305
2. [Bi-] I'M
[Bil
i
Ei&
i
^ U 1^^"^="^^
V, iTJ -^
.-^ ^ ^
^E^
y
*
g
There are many similar examples. He evidently evades a regular olose.
Minob-Majob Mode.
Modem composers, when they wish to insert a tonal mode which lies between ^
Major and Minor, to be used for a few phrases or cadences, have generally preferred
giving the minor chord of the mode to the subdominant and not to the tonic.
Hauptmann calls this the minor-major mode {Moll'Durtonart).* Its chain of
chords ]
I ^^1 I
^his gives a leading note in the dominant chord, and a complete final cadence in
the major chord of the tonic, while the minor relation of the subdominant chord
remains undisturbed. This minor-major mode is at all events much more suitable
for harmonisation than the old mode of the minor Seventh. But it is unsuitable
for homophonic singing, unless in the ascending scale a^[> is changed into ai,
because the voice would otherwise have to make the complicated step a*[>. . . &i [= 2 74
cents, see p. 301a, d]. The old modes were derived from homophonic singing, for ^
which the mode of the minor Seventh is perfectly well fitted, as we know from its
being still used as our ascending minor 8cale.t
Mode of the Mikob Sixth.
While the mode of the minor Seventh oscillates indeterminately between major
and minor without admitting of any consistent treatment, the m^de of the minor
Sixth (Greek Doric, [p. 274^', note No. 7] ecclesiastical Phrygian), with its minor
Second, has a much more peculiar character, which distinguishes it altogether from
all other modes. This minor Second stands in the same melodic connection to the
tonic as a leading note would do, but it requires a descending progression. Hence
for descending passages this mode possesses the same melodic advantages as the
major mode does for ascending passages. The minor Second has the more distant
relationship with the tonic, due entirely to the subdominant. The mode cannot If
form a dominant chord without exceeding its limits. If we keep c as the tonic,
the chain of chords is
In this case the chords 6(>— d^|>-|-/and ^^[>+/— a'^) are not directly related to the
tonic. The tone d^\} cannot enter into any consonant chord which is directly
related to the tonic. But since d^\} is the characteristic minor Second of the mode,
such chords cannot well be avoided, not even in the cadence. Although, then.
* [It is I C mijiia.ma. of App. XX. sect. E.
art. g,— Translator,]
f [After the introduction of the leading note
to form a major dominant chord. — Translator.]
^ [The notes have been transposed in
order to keep the same tonic chord C—E^b + O.
Obserre that both Sevenths hb and b^b are in-
troduced. If &'b be omitted, the system of
chords is that of 5 ^ mi.mi.mi. On the Har-
monica!, on account of the absence of 6 b» it is
necessary to use the system of chords e2, — /+
a, - c + JK, - G + JBi - d.'-TranslatorJ}
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3o6 EEMNANTS OF OLD TONAL MODES. pabtih.
there is a close relationship between the consecntive links of the chain of chords,
some of its indispensable terms are only distantly related to the tonic. Moreover,
in the course of a piece in this mode, it will always be necessary to form the
dominant chord g-i-bi^d* although it contains two tones foreign to the original
mode, as otherwise we could not prevent the prevalence of the impression that/
is the tonic and/— a*[>+c the tonic chord. It follows, therefore, that the mode of
the minor Sixth must be still less consistent in its harmonisation and still more
loosely connected than the minor mode, although it admits of very consistent
melodic treatment. It contains three essential minor chords, namely the tonic
c— e^b + ^r, the subdominant /—a'b+c, and the chord which contains the two tones
slightly related to the tonic H^—d^\}+f, It is exactly the reverse of the major
mode, for whereas that mode proceeds towards the dominant, this mode proceeds
towards the subdominant.
f Major: f+a^ -C+JS?i ^G+b^-d
Mode of I I I
minor Sixth : 6t>-d>b+/-a'l>+0-JS7>b+C?
For harmonisation the difference of the two cases is, first, that the related tones
introduced into the sgale by the subdominant /, namely ^ and d^\}j are not
partials of the compound tone of the subdominant, whereas tones hi and d, which
are introduced by the dominant, are some of the partials of the tonic ; and,
secondly, that the tonic chord always lies on the dominant side of the tonic tone.
Hence in the harmonic connection, the tones l^BXid d^\) cannot be so closely united
with either the tonic tone or the tonic chord, as is the case with the supplementary
tones introduced by the dominant. This gives a kind of exaggerated minor cha-
racter to the mode of the minor Sixth, when harmonised. Its tones and chords are
certainly connected, but much less clearly and intelligibly than those of the minor
IT system. The chords which can be brought together in this key, without obscuring
reference to c as the tonic, are ^ minor and d^\} major on the one hand and g
major on the other, chords which in the major system could not be brought together
without extraordinary modulational appUances.f The esthetical character of the
mode of the minor Sixth corresponds with this fact. It is well suited for the
expression of dark mystery, or of deepest depression, and an utter lapse into
melancholy, in which it is impossible to collect one's thoughts. On the other
hand, as its descending leading note gives it a certain amount of energy in descent,
it is able to express earnest and majestic solemnity, to which the concurrence of
those major chords which are so strangely connected gives a kind of peculiar
magnificence and wondrous richness.
Notwithstanding that the mode of the minor Sixth has been rejected from
modem musical theory, much more distinct traces of its existence have been left
in musical practice than of any other ancient mode ; for the mode of the Fourth
IT has been fiised into the major, and the mode of the minor Seventh into the minor.
Certainly a mode like that we have described is not suitable for frequent use ; it is
not closely enough connected for long pieces, but its peculiar power of expression
cannot be replaced by that of any other mode. Its occurrence is generally marked
by its peculiar final cadence which starts from the minor Second in the root. In
Handel the natural cadence of this system is used with great effect. Thus in the
* [The introdaction of this chord shews 3 il'b majna.ma. of App.XX. sect. E. art. 9. —
that the composer is writing in the key of c, but Translator.]
has a prevailing tendency to modulate into the t [This, in fact, lengthens the original chain
subdominant, from which 6 b , <i' b are chosen, of chords into bb-d}b +f—a^b +c—e^b -h g +
When &'b is used for 6b, or b^ for &*b, the 5, -<2, and leads to the treatment of the mode
modulation into the subdominant does not as merely C minor, with a tendency to modu-
take place. The major chord e*b + ^— 6'b is late into F minor. The C minor is, however,
entirely adventitious. If it is used in ascend- the modem minor C mi.mi.ma., and the F
ing, thus, c 112 d'b 204 ^'b 182/204^ 112 minor is F mi.mi.mi., whidi is much more
a*b 204 &*b 182 c\ the result is the scale of gloomy, ^Translator.]
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CHAP. XV.
EEMNANTS OF OLD TONAL MODES.
307
Messiah, the magnificent fagae And with his stripes we are healed, which has the
signature of F minor, but by its frequent use of the harmony of the dominant
Seventh on G, shews that C is the real tonic, introduces the pure [ecclesiastical
Phrygian] Doric cadence as follows : •
[^0
^
^
^
521
lar
132=
122:
=22=
^
^
i
J J.
m
zjan
-7zr
Similarly in /SarrMon,t the chorus, flie^r, Jacob's God, which, written in the ^
Doric mode of E, finely characterises the earnest prayer of the anxious Israelites
as contrasted with the noisy sacrificial songs of the Philistines in G major, which
immediately follow. The cadence here also is purely Doric.it
The chorus of Israelites which introduces the third pwrt : In Thwnder come, 0 God, ^
from heaven I and is chiefly in A minor, has likely an intermediate Doric section.
Sebastian Bach also, in the chorales which he has harmonised, has left them
in the mode of the minor Sixth, to which they melodically belonged, whenever the
text requires a deeply sorrowful expression, as in the De Proftmdis or the AtLS
tiefer Noth schreV ich zu dir, and again in Paul Gerhardt's song, Wenn ich einmal
soil scheiden, so scheide nicht von mvr. But he has harmonised the same melody
arranged for other texts, as Befiehl Du deine Wege, and 0 Haupt von Blut und
Wunden, &c., as major or minor, in which case the melody ends on the Third or
Fifth of the key, instead of on the Doric tonic.
Fortlage § had already observed that Mozart had applied the Doric mode in
* [The cadence is produced by passing
from the minor subdominani Bb^D^b + F to
the major dominant, C + E^ — 0,m the key of
F minor. This is the concluding cadence of
the whole fugne, and for this reason appa-
rently, the signature in Novello's edition is
that of C minor, not of F minor, and the d^ b
18 n&arked as an accidental throughout. That
is, Kovello takes the key to be C minor with a
constant tendency to modulate into the key of
the subdominant, from which it borrows the
chord Bb—D^b + F, But the fugue begins
with F...f in the bass, and the opening sub-
ject, in the treble,is c", a"b, d»"b, «/, /, g", a»'b,
6'b, c", which is clearly in the scale of F minor,
with the chordal system bb—d^b +f-a^ b + c + e,
— ^, of which it contains every note. In the
text ihe [FJ] is the duodenal and refers to
the duodene of F, which contains all the tones
in the passage. The whole fugue oscillates
between the duodenes C and F.— Translator.]
f [Mr. H. Keatley Moore informs me that
this chorus was taken by Handel from
PloratefiMae Israel in Carisaimi's Jephthah,
— Translator.]
X [The duodene is that of A^. The sncces- m
sion of chords, each reduced to the simplest
form, as referred to by the bracketed figures
below the notes, is i. e,— ^ + 6„ 2. a^ — c + e^,
3. e^-g + b^, 4. Z+Oi-c, S-<^i-/+«ii 6. e,+
i7«5 — 6, 7- «!— c + e„ 8. e^+g^—bi. Hence,
assuming the scale to have the chordal system
d,-/+a,-c + e,-5f + 6„ with e,-gf + 6, as the
tonic chord, taken major as e^+ g^ -^biin the
close, we have the * Boric cadence' between
chords 5 and 6, which is then lengthened by
introducing the remaining tones of the key in
7, the whole closing as in 8. It would be most
probably received as in A^ minor, closing in
the dominant. — Translator.]
§ Examples from instrumental music are
mentioned by Ekert in his Habilitationsschrift,
Die Principien der Modulation und musikal-
ischen Idee. Heidelberg, i860, p. 12. ^
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3o8 REMNANTS OF OLD TONAL MODES. i»abt m.
Famina*B air in the second act Of II Fla/uto Magioo [No. 19]. One of the finest
examples for the contrast between this and the major mode occurs in the same
composer*s Don Giovanni in the Sestette of the second act [No. 21], where Ottavio
and Donna Anna enter. Ottavio sings the comforting words—
Tergi il oiglio, 0 vita mia,
£ d& calma al tuo dolore
in D major, which, however, is pectdiarly coloured by a preponderating, although
not uninterrupted, inclination to the subdominant, as in the mode of the Fourth.
Then Anna, who is plunged in grief, begins in perfectly similar melodical phrases,
and vnth a similar accompaniment, and after a short modulation through D minor,
establishes herself in the mode of the minor Sixth for C, vnth the words —
Sol la morte, 0 mio tesoroi
11 n mio pianto pa6 finir.
The contrast between gentle emotion and crushing grief is here represented with a
most wonderfully beautiful effect, principally by the change of mode. The dying
Commandant also, in the introduction to Don Giovanni, ends with a Doric cadence.*
Similarly the Agni^ Dei of Mozart's Bequiem-^aliiioxxgh, of course, we (ure not
quite certain how much of this was vmtten by himself.
Among Beethoven's compositions we may notice the first movement of the
Sonata, No. 90, in E minor, for the pianoforte, as an example of peculiar de-
pression caused by repeated Doric cadences, whence the second (major) movement
acquires a still softer expression.
Modem composers form a cadence which belongs to the mode of the minor
Sixth, by means of the minor Second and the major Seventh, the so-called chord
of the extreme sharp Sixth,t /'+a...e2ijjl) where both /> and dijj^ have to move
f half a tone to reach the tonic e [p. 2S6b], This chord cannot be deduced fix>m the
major and minor modes, and hence appears very enigmatical and inexplicable to
many modem theoreticians. But it is easily explained as a remnant of the old
mode of the minor Sixth, in which the major Seventh dijj^, which belongs to the
dominant chord h+d^jj^-fj/^, is combined vdth the tones f^+a, which are taken
from the subdominant side.it
These examples may suffice to shew that there are still remnants of the mode
of the minor Sixth in modem music. It would be easy to adduce more examples
if they were looked for. The harmonic connection of the chords in this mode is
not sufficiently firm and intelligible for the construction of long pieces. But in
short pieces, chorales, or intermediate sections, and melodic phrases in larger
musical works, it is so effective in its expression, that it should not be forgotten,
especially as Handel, Bach, and Mozart have used it in such conspicuous places
in their works. §
f * [No. I of the opera. Bepresenting major Leipzig, 1866), has carried oat» in a moat in-
chords by capitals and minor by small letters, teresting manner, the complete analogy be-
the final chords of the vocal music are /, D* bi tween the mode of the minor Sixth and the
0*b, /, C, ft so that all the tones will lie in major mode, of which it is the direct conTer-
the scheme g*b + 6b -d> b + / -a» b + c + «, - (7, sion ; and has shewn how this conversion l^da
or c- e* b + ^. The tonic is F. — Translator.] to a peculiarly characteristic harmonisation of
f [Callcott {Mtuical Qrammar, 1809, art. the mode of the minor Sixth. In this respect
441) calls it *the chord of the extreme sharp I wish emphatically to recommend this book
Sixth,* and says that * this harmony when to the attention of musicians. On the other
accompanied simply by the Third, has been hand, it seems to me that it is necessary to
termed the Italian Sixth,* Of course he has shew by musical practice, that the new prin-
no theory for it ; the tone is * accidentally ciple, which is made the basis of that writer's
sharpened.' — Translator.] theory of the mode of the minor Sixth, con-
i [That is the chords of the scale are sidered by birn as the theoretically normal
taken as (2 -/* + a - c* + e - g* + 6 + (iff — ^ , of minor mode, really suffices for the construction
which the two notes last are modem additions.' of great musical pieces. The author, namely.
See p. 2S6d, note f.— Translator.] considers the minor triad c— e'b + g as repre-
§ Herr A. von Oettingcn, in his Harmonie- senting the tone g" which is common to the
system in dualer Entwickelujtg (Dorpat and three compound tones of which it is composed
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CHAP. XV.
TEICHORDAL TONAL MODES-
309
Similar relations exist for the mode of the Fourth and of the minor Seventh,
although these are less specifically different from the major and minor modes re-
spectively. They are, however, capable of giving a peculiar expression to certain
musical periods, although dif^culties would arise in consistently carrying out these
peculiarities through long pieces of music. The harmonic phrases which belong to
these two last-named modes can of course also be executed within the usual major
and minor systems. But perhaps it would fEM^ilitate the theoretical comprehen-
sion of certain modulations, if the conception of these modes and of their system
of harmonisation were definitely laid down.
The only point, then, as historical development and physiological theory alike
testify, for which modem music is superior to the ancient, is harmonisation. The
development of modem music has been evoked by its theoretical principle, that the
tonic chord should predominate among the series of chords by the same laws of
relationship as the tonic note predominates among the notes of the scale. This If
principle did not become practically efiective till the commencement of last century,
when it was felt necessary to preserve the minor chord in the final cadence.
The physiological phenomenon which this esthetical principle brought into
action, is the compound character of musical tones which are of themselves chords
composed of partials, and consequently, conversely, the possibility under certain
circumstances of replacing compound tones by chords. Hence in every chord the
principal tone is that of which the whole chord may be considered to express its
compound form. Practically this principle was acknowledged from the time that
pieces of music were allowed to end in chords of several parts. Then it was im-
mediately felt that the concluding tone of the bass ought be accompanied by a
higher Octave, Fifth, and, finally, major Third, but not by a Fourth, or minor
Sixth, and for a long time also the minor Third was rejected ; and we know that
the first three intervals (the Octave, Fifth, and major Third) occur among the
partials of the compound tone which lies in the bass, and that the others do not. f
The various values of the tones of a chord were first theoretically recognised
by Bameau in his theory of the fundamental bass, although Bameau was not ac-
quainted with the cause here assigned for these different values. That compound
tone which represents a chord according to our view, constitutes its Fundamental
Bas8^ Radical Tone or Boot^ as distinguished from its hass^ that is, the tone which
belongs to the lowest part. The major triad has the same TOot whatever be its
inversion or position. In the chords c+ 6]—^, orgr...c+ei,therootisstillc. The
minor chord d^P-\-a has also as a rule only d as its root in all its inversions, but
in the chord of ttie great [or added] Sixth/* + a... (i^ we may also consider/* as the
root, and it is in this sense that it occurs in the cadence of c* major. Bameau's
successors have partly given up this last distinction ; but it is one in which Bameau's
fine artistic feeling fully corresponded with the facts in nature. The minor chord
really admits of this double interpretation, as we have already shewn (p. 294^).
The essential difference between the old and new tonal modes is this : the old ^
have their minor chords on the dominant, the new on the subdominant side.
The reasons for the following construction have been already investigated.*
In the
The chord of the
Subdominant
l8
Tonio
iB
■ • ■■->
Dominant
is
( Mode of the minor Third .
Old-i Mode of the minor Seventh
(Mode of the Fourth .
Major Mode . . . .
v^« i Minor-Major Mode .
^^^iMinorMode . . . .
Minor
Major
Major
Major
Minor
Minor
Minor
Minor
Major
Major
Major
Minor
Minor
Minov
Minor
Majpr
Major
Major
(being a higher Octave of g, of the Fifth of c,
and of the major Third of 6'b), and hence
caUe it * the phonic g tone,* whereas he con-
siders c + e,— <7 in the same way as we do, as
the * tonic c tone.'
* [It will be seen that this arrangement
Digitized by V^OOQ IC
310 DIFFERENT CHAEACTERS OP KEYS. paet m*
CHAPTER XVL
THE SYSTEM OF EETS.
Thebb is nothing in the nature of music itself to determine the pitch of the ionic
of any composition. If different melodies and musical pieces have to be execated
by musical instruments or singing voices of definite compass, the tonic must be
chosen of a suitable pitch, differing when the melody rises fax above the tonic and
when it sinks much below it. In short, the pitch of the tonic must be chosen so
as to bring the compass of the tones of the piece within the compass of the execu-
tants, vocal or instrumental. This inevitable practical necessity entails the con-
dition of being able to give any required pitch to the tonic.
f Moreover, in the longer pieces of music it is necessary to be able to make a
temporary change of tonic, that is, to modulate^ in order to avoid uniformity and
to utilise the musical effects resulting &om changing and then returning to the
original key. Just as consonances are made more prominent and effective by means
of dissonances, the feeling for the predominant tonality and the satisfaction which
arises from it, is heightened by previous deviations into adjacent keys. The variety
in musical turns produced by modulational connection has become all the more
necessary for modem music, because we have been obliged entirely to renounce,
or at any rate materially to circumscribe, the old principle of altering expression
by means of the various tonal modes. The Greeks had a free choice among seven
different tonal modes, the middle ages among five or six, but we can choose between
two only, major and minor. Those old tonal modes presented a series of different
degrees of tonal character, out of which two only remain suitable for harmonic
music. But the clearer and firmer construction of an harmonic piece gives modem
f composers greater freedom in modulational deviations from the original key, and
places at their command new sources of musical wealth, which were scarcely
accessible to the ancients.
Finally I must just touch on the question, so much discussed, whether each
different key has a peculiar character of its own.
It is quite clear that, within the course of a single piece of music, modulational
deviations iato the more or less distantly related keys on the dominant or sub-
dominant side produce very different effects. This, however, arises simply from
the contrast they offer to the original principal key, and would be merely a rela-
tive character. But the question here mooted is, whether individual keys have an
absolute character of their own, independently of their relation to any other key.
This is often asserted, but it is dif&cult to determine how much tmth the
assertion contains, or even what it precisely means, because probably a variety of
different things are included under the term character^ and perhaps the amount of
H effect due to the particular instrument employed has not been allowed for. If an
instrument of fixed tones is completely and uniformly tuned according to the equal
temperament, so that all Semitones throughout the scale have precisely the same
magnitude, and if also the musical quality of all the tones is precisely the same,
there seems to be no ground for understanding how each different key should have
a different character. Musicians folly capable of forming a judgment have also
admitted to me, that no difference in the character of the keys can be observed on
the organ, for example. And Hauptmann,* I think, is right when he makes the
same assertion for singing voices with or without an organ accompaniment. A
great change in the pitch of the tonic can at most cause all the higher notes to be
strained or all the lower ones obscured.
On the other hand, there is a decidedly different character in different keys on
does not inolnde the mode of the minor Sixth, art. 9, and thence to the general theory of
It was this tabulation which led me to the duodenes in that section. — TVansZafor.]
richordal theory developed in App. XX. sect. E. * Harmonik und Metrik, p. 188.
Digitized by V^jOOQlC
CHAP. XVI. DIFFERENT CHARACTEEB OF KEYS. 311
pianofortes and bowed instmments. 0 major and the adjacent D\> major have
different effects. That this difference is not caused by difference of absolute pitch,
can be readily determined by comparing two different instruments tuned to differ-
ent pitches. The D\} of the one instrument may be as high as the G of the other,
and yet on both the G major retains its brighter and stronger character, and the
X>t> its soft and veiled harmonious effect. It is scarcely possible to think of any
oth^ reason than that the method of striking the short narrow black digitals of
the piano must produce a somewhat different quality of tone, and that difference of
character arises &om the different distribution of the stronger and gentler quality
of tone among the different degrees of the scale.* The difference made in the
timing of those Fifths which the tuner keeps to the last, and on which are crowded
the whole of the errors in tuning the other Fifths in the circle of Fifths, may
possibly be regular, and may contribute to this effect, but of this I have no person^
experience. [See App. XX. sect. G. art. 17.] f
In bowed instruments the more powerful quality of tone in the open strings is
conspicuous, and there are also probably differences in the quality of tone of strings
which are stopped at short and long lengths, and these may alter the character of
the key according to the degree of the scale on which they fall. This assumption
is confirmed by the inquiries I have made of musicians respecting the mode in
which they recognise keys under certain conditions. The inequaHty of intonation
wiQ add to this effect. The Fifths of the open strings are perfect Fifths. But it
is impossible that all the other Fifths should be perfect if in playing in different
keys e%ch note has the same sound throughout, as appears at least to be the inten-
tion of elementary instruction on the violin. In this way the scales of the various
keys will differ in intonation, and this will necessarily have a much more important
influence on the character of the melody. [See App. XX. sect. O. arts. 6 and 7.]
The differences in quality of tone of different notes on wind instruments are
still more striking. ^
If this view is correct, the character of the keys would be very different on differ-
ent instruments, and I believe this to be the case. But it is a matter to be decided
by a musician with delicate ears, who directs his attention to the points here raised.
It is, however, not impossible that by a peculiarity of the human ear, abeady
touched upon in p. 11 6a, certain common features may enter into the character of
keys, independent of the difference of musical instruments, and dependent solely
on the absolute pitch of the tonic. Since ^'"' is a proper tone of the human ear,
it sounds peculiarly shrill under ordinary circumstances, and somewhat of this
shrillness is common to /'"J and a""l>. To a somewhat less extent those musical
tones of which ^"" is an upper partial, as gr"', 0"', and gf", have a brighter and more
piercing tone than their neighbours. It is possible, then, that it is not indifferent
for pieces in G major to have its high Fifth g'^ and high tonic c'" thus distinguished
in brightness from other tones, but these differences must in all cases be very slight,
and for the present I must leave it undecided whether they have any weight at all. ^
All or some of these reasons, then, made it necessary for musicians to have free
command over the pitch of the tonic, and hence even the later Greeks transposed
their scales on to all degrees of the chromatic scale. For singers these trans-
positions offer no difficulties. They can begin with any required pitch, and find
in their vocal instrument all such of the corresponding degrees as lie within the
extreme limits of their voice. But the matter becomes much more difficult for
musical instruments, especially for such as only possess tones of certain definite
degrees of pitch. The difficulty is not entirely removed even on bowed instruments.
It is true that these can produce every required degree of pitch ; but players are
unable to hit the pitch, as correctly as the ear desires, without acquiring a certain
* [Mr. H. Eeatley Moore, Mas. B., thinks they gain this by a quicker motion, each arm
the difference is due to the different leverage of the lever being shorter, and short keys
of the black digitals. Although in well con- differing altogether from long ones in the feel-
structed digitals the black have as much ing produced in the hand. See also App. XX.
action at the further end as the white ones, sect. N. No. 6. — Translator.]
Digitized by V^jOOQ IC
3X2 TEMPEBED INTONATION. .pabt m.
mechanical use of their fingers, which can only result from an immense amoant
of practice.
The Greek system was not accompanied with great difficulties, even for injstra-
ments, so long as no deviations into remote keys were permitted, and hence but
few marks of sharps and flats had to be used. Up to the beginning of the seven-
teenth century musicians were content with two signs of depression for the notes
B\} and E\}, and with the sign $ for ^, df^^ O^, in order to have the leading
tones for the tonics G, D, and A. They took care to avoid the enharmonicallj
equivalent tones il J for B^}, Djj^ for E\}, 0\} for i^, D\> for C#, and A\} for G^.
By help of B\} for B* every tonal mode could be tremsposed to the key of its sab-
dominant, and no other transposition was made.
In the Pythagorean system, which maintained its predominance over theory
to the time of Zarlino in the sixteenth century, tuning proceeded by ascending
IF Fifths, thus—
CaDAEBFJj^C^GiDjj^Ajj^EJiBH^
Now if we tune two Fifths upwards and an Octave downwards, we make a step
having the ratio f x f x ^=f , which is a major Second. This gives for the pitch of
every second tone in the last list —
G D E 1% a^ A^ B^
I I (I)' (I)' (ir («)* (f)'
Now if we proceed c2(m^wards by Fifijis from C we obtain the series —
G F B\> E\} A\> D\> G\> C\> iT> B\}\> E\>\> A\}\} 2)t>t>.
If we descend two Fifths and rise an Octave, we may obtain the tones —
0 B\} A\} G\} F\} E\}\} D\}\}
I I (1)^ m' HY {ir (i)«
If Now the interval If )«=Mmt=i x UUH
or, approximately (|)^=i x ^
Hence the tone Bjj^ is higher than the Octave of G by the small interval
^ [=24 cents] , and the tone D[}\} is lower than the Octave below G by the same
interval. If we ascend by perfect Fifths from G and i>t>t>, we shall find the same
constant difference between
G a D A E B 1% G% Gi D% Ai Ejf^ Bjtfiad
D\}\} A\}\> E\^\} B\}\> F^ G\} G\} D\} A\} E\> B\^ F G.
The tones in the upper line are all higher than those in the lower by the small
interval || [=24 cents]. Our staff nptation had its principles settled before the
development of the modem musical system, and has consequently preserved these
differences of pitch. But for practice on instruments with fixed tones the distinc-
5) tion between degrees of tone which lie so near to each other, was inconvenient, and
attempts were made to fiise them together. This led to many imperfect attempts,
in which individual intervals were more or less altered in order to keep the rest
* [In the oldest printed book on mnsic, almost the cursive ^written form f} of Ii in
{Franchini Oafori Laudenaia Muaici pro- Germany. Qn the other hand it was often .
fessorU theoricum opus artnonice discipline, made with two strokes || afterwards crossed,
Neapolis M.CCCC.LXXX., for a sight of which like B , and then it degenerated into || » which
I am indebted to Mr. Quaritch, who bought it is apparently the precursor of our 8 . In tiiis
at the sale of the SysUm Library) 6 b is in the case both q and S and also h would have
printed text represented by a small Boman arisen from the same square-bottomed b» the
b, and &l| by a capital Boman B. But in a French bicarrey and PraBtorius Q quadratum^
plate attached are given eight varieties of the which, however, he identified witii h, H in
written form of &Q , by which it would seem subsequent writing. The Italian names for
to have been intended for b with a square h,b\>9ktQsiminoretS%maggiore, Whether these
instead of a round bottom, like 0 , which is refer to the musical intervals a 5bi a 6l| , which
almost indistinguishable from a mutilated Gafori printed a b, a B, or to these printed
Boman b. As it was clearly made in two forms, it is difficult to say with certainty,
parts t 1 , the second was often long, and then The Germans accepted the forms b b , as b h,
the resemblance to Q was great, and this was calling the latter ha. The meaning that
Digitized by V^OOQIC
CHAP. ZVI.
TEMPEEED INTONATION.
313
true, producing the so-oalled unequal temperaments, and finally to the system
of equal temperament, in which the Octave was divided into 12 precisely equal
degrees of tone.* We have seen that we can ascend from C by 12 perfect Fifths to
B$, which differs firom c by about j^ of a Semitone, namely by the interval |}. In
the same way we can descend by 12 perfect Fifths to D\)>\}, which is as much lower
than C, as ^ is higher. If, then, we put G^BJj^=iD\}\}, and distribute this httle
deviation of J4 equally among all the 12 Fifths of the circle, each Fifth will be
erroneous by about |^ of a Semitone [or ^ of a comma or 2 cents], which is
certainly a very small interval. By this means all varieties of tonal degrees within
an Octave are reduced to 12, as on our modem keyed instruments.
The Fifth in the system of equal temperament, is, when expressed approximately
in the smallest possible numbers,ss f x |f f . It is very seldom that any difficulty
could result from its use in place of the perfect Fifth. The root struck with its
tempered Fifth makes one beat in the time that the Fifth makes 442^ complete If
vibrations. Mow since a' makes 440 vibrations in a second, it follows that the
tempered Fifth d'dta' will produce exactly one beat in a second. In long-sustained
tones this would, indeed, be perceptible, but by no means disturbing, and for
quick passages it would have no time to occur. The beats are still less disturb-
ing in lower positions, where they decrease in rapidity with the pitch numbers of
the tones. In higher positions they certainly become more striking ; d'"±^o}'*
gives four, and a'"±.e"' six beats in a second ; but chords very seldom occur with
such high notes in slow passages. The Fourths of the equal temperament are
$ X iff [=49^ + 2 cents]. There is one beat for every 22i| vibrations of the lower
tone of the Fourth. Hence the Fourth a . d' makes one beat in a second, the
same as the Fifth d'±:a'. The pure consonances retained in the Pythagorean
system are therefore not injured to any extent worth notice by equal temperament.
In melodic progression of tones the interval ||| borders on the very limits of dis-
tinguishable differences of pitch, according to Preyer's experiments (see p. 147&). H
In the doubly accented Octave it would be easily distinguished. In the unac-
cented or lower Octaves it would not be felt at all.
The Thirds and Sixths of the equal temperament are nearer the perfect
intervals than are the Pythagorean.f
Intervals
Perfect
Equally Tempered
Major Third •
ntioa cents
1 386
ratioe cents
fxta 400
xatios cents
Sxij 408
Minor Sixth •
! 814
i^m 800
fxK 79»
Biinor Third .
1 3>6
JxiH 300
fxH a94
Major Sixth .
i 884
fxifi 900
IxfJ 906
Semitone
if '8a
BorJfHi 100
IHorHxJf 90
The dissonances occasioned by the upper partial tones are consequently some- IF
what milder than those due to Pythagorean intervals, but the combinational tones
Oafori attached to b B (which in one plate he
also gives in the same sense in black letter),
is shewn by the following quotation which he
makes from Guidons hexachord, and this also
shews that he used Pythagorean intonation,
meaning in our notation:
/
Ffaut
tonus
G sol re ut
tonus
a la mi re
semitonium
bfa
apothome
B mi Trittonus
semitonium
0 sol fa ut
204 cents s Tone
204 cents a Tone
a
90 cents » Semitone
6b
1 14 cents » Apotome
h Tritone 612 cents
90 cents a Semitone
c' Fifth 702 cents
The Germans generally speak of b and S
as Be, Ereuz (cross). I do not remember ever
having heard the I) named, but I find in Flii-
gers Dictionary Beqiuidra/twn (square b) and
Wiederhersiellungazeichen (sign of restitution,
for which Q was not used till the seventeenth
century). Germans never have occasion to use
the word, because, instead of ^d flat, d natural,
d sharp,' they say * des, de, dis,' while &b, 6 Q
are termed ' be, ha.* On older organ pipes b 4
are constantly used for &b 6 K , and some organ*
builders still use them. — Translator,']
♦ [The general relations on which the
schemes of temperai;^ent depend will be found
in App. XX. sect, k.— Translator.]
f [The cents in the Table were of course
inserted by me. — Translator.]
Digitized by VjOOQIC
3M
TEMPEBED INTONATION.
PABTXn.
are much more disagreeable. For the Pythagorean Thirds V+e' and d—(f the
combinational tones are nearly C)|; and B,, both differing by a Semitone &om the
combinational tone C» which woiiLd result from the perfect intervals in both cases.
For the Pythagorean minor chord e'—g'-^-h' the combinational tones are B, and
very nearly (rj. The first, J5y, is very suitable, better even than the combina-
tional tone G which resxdts from perfect intonation. But the second, (?$, belongs
to the major and not to the minor chord of E. However, as in perfect intonation
one of the two combinational tones C and Q is fedse, the Pythagorean minor chord
can hardly be considered inferior in this respect. But the combinational tones of
the equally tempered Thirds lie between those of the perfect and Pythagorean
Thirds, and are less than a Semitone different horn those of just intonation. Hence
they correspond to no possible modulation, no tone of the chromatic scale, no dis-
sonance that could possibly be introduced by the progression of the melody ; they
IT simply sound out of tune and wrong.*
These bad combinational tones have always been to me the most annoying
part of equally tempered harmonies. When moderately slow passages in Thirds
at rather a high pitch are played, they form a horrible bass to them, which is all
the more disagreeable for coming tolerably near to the correct bass, and henea
sounding as if they were played on some other instrument which was dreadiiilly
out of tune. They are heard most distinctly on the harmonium and violin. Here
every professional and even every amateur musician observes them immediately,
when their attention is properly directed. And when the ear has once become
accustomed to note them, it can even discover them on the piano. In the
Pythagorean intonation the combinational tones sound rather as if some one were
intentionally playing dissonances. Which of these two evils is the worse, I will
not venture to decide. In lower positions where the very low combinati<mal tones
can be scarcely, if at all, heard, the equally tempered Thirds have the advantage
% over the Greek, because they are not so rough, and produce fewer beats. In.
higher positions the latter advantage is perhaps destroyed by their combinational
tones. However, the equally tempered system is capable of effecting everything
that can be done by the Pythagorean, and with less expenditure of means.
C. E. Naumann,t who has lately defended the Pythagorean as opposed to the
equally tempered system, grounds his reasons chiefly on the fact that the Semi-
tones which separate the ascending leading tone from the tonic, and the descend-
ing minor Seventh from the Third of the chord on which it has to be resolved, are
smaller in the Pythagorean (where they are about ^ ; as appears in the Table on
P* 3^3^) than in the equally tempered, where they are about ||; while they are
greatest of all in just intonation, viz. \i. Now in the equally tempered scale there
is only one tone between / and g, which is accepted at one time as /jf to be a
leading note to ^, and at another as g\^ to act as a Seventh resolving upon /; but
in the Pythagorean there are two tones, /{); and g\}, of which the latter is the flatter.
% * [This may be seen more clearly by oalcalating the pitch numbers, assmning (/ to be
264 as on p. 17. Then —
Notes
Just, Difference
Pythagorean, Difference
Tempered, Difference
c'
264
66«C
330
66«C
396
99«Qf
495
264
7012
334*12
6r88 = J5,
396
105-19
501-19
264
68-61
332*61
62-94
395*55
I02-8I
49836
The * differences' give the pitch nmnbers of
the combinational tones. Now we have by
p. 17a, C = 66, G « 99, B^«6r88, but the others
correspond to no precise tones. The nearest
equally tempered intervals are B 62-3, Off
« 69-93, ^^^ ^t = 104-76. — TranslatoT.\
t Veher die verschiedenen Bestimmungen
der Tonverhdltnisse, [On the various deter-
minations of the ratios of tones.] Leipzig,
1858.
Digitized by V^OOQIC
CHAP. XVI. TEMPERED INTONATICttl. 315
Hence the Semitone always approaches the tone on to which it would fall in regular
resolution, and the height of the pitch determines the direction of resolution. But
although the leading tone plays an important part in modulations, it is perfectly
clear that we are not justified in changing its pitch at will in order to bring it
nearer to the note on which it has to be resolved. There would otherwise be no
limit to our making it come nearer and nearer to that tone, as in the ancient Greek
enharmonic mode.* Suppose we replace the Pythagorean Semitone, which is
about f of the natural Semitone, by another still smaller, about f of the natural
one, say 44 x |^ x |^ ; the result would be perfectly unnatural as a leading note.f
We have already seen that the character of the leading note essentially depends
upon its being that tone in the scale which is most distantly related to the tonic,
and hence most uncertain and alterable [melodically]. Hence we are perfectly
unjustified in deducing from such a tone the principle of construction for the whole
scale. ^
The principal fault of our present tempered intonation, therefore, does not lie
in the Fifths ; for their imperfection is really not worth speaking of, and is
scarcely perceptible in chords. The fault rather lies in the Thirds, and this error
is not due to forming the Thirds by means of a series of imperfect Fifths, but it is
the old Pythagorean error of forming the Thirds by means of an ascending series of
four Fifths. Perfect Fifths in this case give even a worse result than flat Fifths.
The natural relation of the major Third to the tonic, both melodically and harmo-
nically, depends on the ratio f of the pitch numbers. Any other Third is only a
more or less unsatisfactory substitute for the natural major Third. The only
correct system of tones is that in which, as Hauptmann proposed, the system of
tones generated by Fifths should be separated from those generated by major
Thirds. Now as it is important for the solution of many theoretical questions to
be able to make experiments on tones which really form with each other the
natural intervals required by theory, to prevent the ear from being deceived by IT
the imperfections of the equal temperament, I have endeavoured to have an in-
strument constructed by which I could modulate by perfect intervals into all keys.
If we were really obliged to produce in all its completeness the system of tones
distinguished by Hauptmann, in order to obtain perfect intervals in all keys, it
would certainly be scarcely possible to overcome the diflBculties of the problem.
Fortunately it is possible to introduce a great and essential simplification by means
of the artifice originally invented by the Arabic and Persian musicians, and pre-
viously mentioned on p. 281a.
We have afready seen that the tones of Hauptmann's system which are generated
by Fifths, and are marked by letters without any subscribed or superscribed lines, as
c±:gf±^±a±, Ac, are one comma or f| [=22 cents] higher than the notes which
bear the same names, when generated by major Thirds, and which are here dis-
tinguished by an inferior figure as Cj ±gf 1 ±(Zi ±ai ±, &c. We have further seen that
if we descend from 6 by a series of 12 Fifths down to ct>, the last tone, reduced to IT
the proper Octave, is lower than b by about fj [=24 cents]. Hence we have —
b : 5,=8i : 80
b : ct>=74 : 73
Now these two intervals are very nearly alike ; bi is rather higher than c\}, but
only in the proportion —
c\} : 51=32768 : 32805 [=2 cents]
* [However justifiable BDch alterations may place of 112 cents). No diminution of the
be in unaccompanied melody, they are de- just Semitone can be made without injury to
Btructive of harmony, and hence do not belong the major Thirds. — Translator.]
to harmonic music proper. Of all the older f [This would have 112 — 2x22^68 cents,
temperaments, the meantone is most har- which approaches very closely to the small
monious, but this makes the leading tones Semitone 25 : 24970 cents, so that the effect
still further from the tone on which they are can be judged from playing &'b...&i on the
resolved, than even in just intonation (117 in Harmonical.— 2Va»wZator.]
Digitized by
\^oogle
3i6
HABMONIUM IN JUST INTONATION.
PABT UI.
or, using the approximation obtained by continued fractiouB —
c|> : 61=885 : 886.
The interval between ct> and bi is consequently about the same as that between
a perfect and an equally tempered Fifth.*
Now bi is the true major Third of g, and if we descend 8 Fifths from g we
arrive at c\} thus :
g±c±f±b\}±e\}±a\}±d\}±g\}±c\}
Now, as c\} is flatter than &i, if we diminished f all the Fifths by i of the small
interval fff we should arrive at &i instead of c\}.
Now, since the interval f|f is itself on the limits of sensible difference of pitch,
the eighth part of this interval cannot be taken into account at all, and we may
1 consequently identify the following tones of Hauptmann's system, by proceeding in
a series of Fifths from c[>=&i, that is, the upper line with the lower, or —
f\}±c\}±g\} ±d\} ±a\} ±e\} ±b\>
:=e,±b, ±/J±c,#±^,#±d,#±aj#
Among musical instruments, the harmonium, on account of its uniformly sus-
tained sound, the piercing character of its quality of tone, and its tolerably distinct
combinational tones, is particularly sensitive to inaccuracies of intonation. And as
its vibrators also admit of a delicate and durable tuning, it appeared to me pecu-
liarly suitable for experiments on a more perfect system of tones. I therefore
selected an harmonium of the larger kind, j: with two manuals, and a set of vibrators
for each, and had it so tuned that by using the tones of the two manuals I could
play all the major chords from JP]> major to Fjj^ major. The tones are thus dia--
f tributed :
/b + a,b-cb + e,b-gb + 6,b— <ib+/,-
On Lower Manual
eb + flfi - 6b + i| - /+ a,-|-c
„•; II fj D
On Upper Manual
I II II ^^1
On Lower Manual
-e+g^U -6 + d,« -Jt +a,J -cj +
On Upper Manual
eiS
This instrument therefore furnishes 15 major chords and as many minor
chords, with perfectly pure Thirds, but with Fifths too flat by i of the interval by
which an equally tempered Fifth is too flat.§ On the Lower Manual we have the
* [On aooount of the approximate character
ol the calculation, the extreme closeness of
result is not well shewn. Taking the accurate
f numbers, the ratio
^+cb«S?i{. giving cents r95372i.
Perfect Fifth + tempered Fifth ^f+'i'/,
giving cents 701 •955001 — 700 « I -955001. Dif-
ference *ooi28o cents. Human ears, however
much assisted by human contrivances, could
never hear the difference. — Translator.]
f [Accidentally misprinted * increased,* that
is, * zu gross,' and ' too sharp,* that is, * za
hoch,* instead of * zu klein,* and * zu tief * in
all the four Oerman editions. This error
evidently arose merely from forgetting for the
moment that the Fifths were taken down and
not ttjp. Now 8 perfect Fifths down ■■ — 8 x 702
cents » — 5616 cents, which on adding 5 octaves
b6ooo cents, gives 384 cents, and this is less
than the major Thirds of 386 cents by 2 cents.
Hence if we diminish each Fifth by i cent,
8 diminished Fifths down « — 8 x 7oi| = — 5614
cents, which, on adding 5 octaves or 0000 cents
gives 386 cents, and this is the correct major
Third. But to tune Fifths of this kind, if pos-
sible, would be a work of immense labour even
with tuning-forks, the most permanent of exist-
ing conveyors of pitch, and the most perfeot
apparatus known. Thus such a Fifth reckoned
from cf 264 vib. gives ^ 395*944 vib., while the
perfect g' is 396*000 vib., difference '056 vib.,
which it is hopeless to tune exactly. Henoe
these Fifths can only be regarded as products
of calculation which could not be realised. In
App. XX. sect A. art. 18 I term the result-
ing temperament Helmholtzian, although, as
will be seen in the following note §, Prof.
Helmholtz himself did not attempt to realise
it. — TranslatorJ]
X Made by Messrs. J. & P* Schiedmayer,
in Stuttgart.
§ The tuning of this instrument was easily
managed. Herr Schiedmayer succeeded at the
first attempt by the following direction. Start-
ing from a on the lower manual, tune the
Fifths d±a,g±dtC±g perfectly just, and thus
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OHAP. XYI.
HABMONIUM IN JUST INTONATION.
317
complete scales of C\> major and O major and in the upper the complete scales of
II\} major and B major inclusive complete. All the major scales exist from C\}
major to B major, and they can all be played with perfect exactness in the natural
intonation. But to modulate beyond B major on the one side and C\} major on the
other, it is necessary to make a really enharmonic interchange between B^ and Ct>,
which perceptibly alters the pitch (by a comma |^).* The minor modes on the
lower manual are Bi minor or G\} minor complete, on the upper manual DiJj^ minor
or E\} minor.
For the minor keys this series of tones is not quite so satisfactory as for the
major keys. The dominant of a minor key is the Fifth of a minor triad and the
root of a major triad. But as the minor chord has to be written as ai — c + ^i,
and the major chord as / 1> + ^i [> — c[>, the corresponding dominant must be written
in the first chord with an inferior number, and in the second with a letter without
any number attached ; that is, they must be tones of the kind which we have ^
identified by means of the assumptions here made, as in the present case where Ci
obtain c, g, d. Then tone the major chords
c + e,— <7, g + b^—d.d + fji -a, giving the Ihred
tones ei&i/iS 1 and finally the Fifth, fit ± CjS ,
to obtain CiS, Then patting e^^fb^ 2>i«cb,
/18 "I7b» c,S adbi tune the major chords
/b+a,b-cb, cb + eib-S'b, gb-^-h^b-db with
pnre Thirds giving no beats, thas obtaining
a, b, ei b, &i b , and finally the Fifth 5| b ±/i > giving
/,. This completes the tuning of the notes on
the lower manual. For the upper manual first
tune e as the perfect Fifth of the a in the
lower manual, and then the three major chords
e + Jf,! -6,6 + d,» -/5,/5 +a,5 -c»,andthe
Fifth a,» ±e.J,giving6,y5,clf ,andthengr,5,
d|8, OiS and also e^^. Then put g^% -ab,
d|S -eb,a|8 »6b, e^t »/, and tune the Thirds
in the major chords ab + c,~eb» eb + gi — &b,
6b+di-/, and the Fifth d^jto^. This gives
^1* fif ^n <^^ ^1* <^^ completes the whole
tuning, which is much easier than for a series
of equally tempered tones.
[The theoretical flattening of all the Fifths
by ) of a skhisma is here neglected, as it
woiJd be impossible by ear only, and in all
probability many other errors in tuning were
eommitted, which could not be detected. The
result is that the two manuals were tuned to
the following tones, using capital letters to
represent the large or white digitals, and the
small letters the small or black digitals. The
Boman letters below shew the secondary mean-
ing attached to the letters above them for the
tuning of the notes marked with a * above
them.
Upper ManuaL
61 c« A d,t E E,t ft G, g,t J, a,« B
eb f ab bb
Lower Manual.
0 Cj« D «I> E, F, /,5 G Jjb A 6*b B,
db Fb gb Cb
To make it more dear how the 24 notes of ^
this instmment represent 48 by neglecting the
skhisma, I have below arranged the scale on
the duodenary system (major Thirds in lines.
Fifths in columns, App. XX. sect. E. art. 18),
and given the proper number of cents for each
note, using capitals for the notes actually
tuned, and smiUl letters for those obtained by
substitution. The notes above the horizontal
line were in the upper manual, those below it
on the lower.
It is thus seen i) that the notes in cols. L, II.,
L
II.
in.
IV.
V.
VL
Ct 114
Ft 612
B mo
E 408
E,t 500
4.« 998
D,J 296
Oit 794
gjit
886
184
682
1 180
d'b 112
0*b 610
c'b iio8
fb 406
/ 498
6b 996
eb 294
ab 792
B, 182
G, 680
C, I 178
A 906
D 204
G 702
C 0
J^it 590
B, 1088
E, 386
gji
478
976
274
772
6»bb 904
e»bb 202
a»bb 700
d'bb 1198
db 90
gb 588
cb 1086
/b 384
F, 476
B,b974
E^b 272
A,b 770
III.are exactly 2 cents sharper than those in cols.
IV., v., VI. 2) that only cols. I., H., VI. were
tuned, and that IV., V., and HE. without being
tuned were assumed to be identical with them
respectively. 3) That cols. I., II., III. form a
series of Fifths down or Fourths up, of which
only two, namely C to ^,S and E^ to 9^ S ,
are defective, being both 700 cents down or
500 up, in place of 702 and 498 as all the
others. 4) That the simplest way of tuning
would be to take A to pitch, and then A + C,S,
and Cit + ejS as perfect major Thirds, and then
from A, Cit , e^t to tune the rest of the notes
in their columns by perfect Fifths and Fourths,
naming the notes in col. IIL for convenience as
those in col. VI. Afterwards the identity of the
first three with the last three columns would
be assumed. All the properties and defects
of this system of tuning can be immediately
deduced from the above diagram.— TraYuZa-
tor.]
* [For instead of the keys of J^ and JPb,
the absence of Gt^ and 6b b, <2,b obliges us
to use the keys of F,8 and F^b, which are
respectively a comma lower and higher. —
Translator,]
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3i8 HARMONIUM IN JUST INTONATION. part irf.
is identified with /[>• Hence the inBtniment famishes the following eight perfectly
just minor scales [where the letters in brackets indicate those which are not written
in the account of the manuals in the text of p. 316c] : —
i) ai or b\}\} minor : di — f + ai — c -h ei + \gjj^ -- ^i ]
f\} + a,b - c\}
2) Bi or/t) minor: ay ^ c -h «! -^ + 6, + [djj^ -/i#]
3) 6, orct> minor: Bi -g -f ^1 -d +/i#+ [^ajf - CiiO
g\} + 6,> - d\}
4)/i#orgrt> minor: ii - d +/,# -a + c,# + [«2# - 9\9\
d\> + fi -a\>
5)Ci#ord|> minor :/,#-. a + cj - « + ^J + [^a# - ^i#]
IT at>' + Ci — el>
6) ^,#or a^ minor : Ci#- 6 + grj - 6 + dj + [/jfttf- «i#]
«> + ^1 - ^t>
7) di# or eb minor : gf, J- 6 + d,# -/J + a,# + [cJJ - e,#]
^b + d, -/
8) aij; or 6t> minor : dij-yj + a,# - cjf + c,# + [Sfafttf- *iftl
/ + a, - c
Of these, the six last tonics from C|> to B\} are also provided with major scales.
Hence there are complete minor scales on all degrees of the scales of Bi major and
^1 major ; and complete minor and major scales on all degrees of the scale of Bi
major, with the exception of Ei.
After previous experiments on another harmonium, where I had at command
only the two sets of tones of one octave common to two stops with one manual, I
f had expected that it would be scarcely observed if either the other minor keys had
a somewhat too sharp Pythagorean Seventh, or if minor chords which are them-
selves rather obscurely harmonious, were executed in Pythagorean intonation.
When isolated minor chords are struck the difference is, indeed, not much observed.
But when long series of justly-intoned chords have been employed, and the ear has
grown accustomed to their effect, it becomes so sensitive to any intermixture of
chords in imperfect intonation, that the disturbance is very appreciable.*
The least disturbance is caused by taking the Pythagorean Seventh, because
this leading tone is in modem compositions scarcely ever used but in the chord of
the dominant Seventh, or other dissonances. In a pure major triad its effect is
certainly very harsh. But in a discord it has a less disturbing effect, becau^ by
its sharpness it brings out the character of the leading note more distinctly. On
the other hand, I have found minor chords with Pythagorean Thirds absolutely
intolerable when coming between justly-intoned major and nunor chords.* By
^ allowing, then, a Pythagorean Seventh in the scale, or a Pythagorean major Third
in the chord of the dominant Seventh, we may form the following minor scales : t —
9) di minor : gr, — fc[> + dj — / + aj ... cS I «i
10) gi minor : Ci — eb + ^1 — 6i> + dj .../iff I ai
11) Ci minor : /j — at> + Ci — et> + fiTi ... 61 I di
12) /, minor : 6,l> — dt> + /i — at> + Cj ... e, I ^,
13) 6,t> minor : a,t> — {/b + *il> — ^t> + /i ••• ^1 I ^1
14) e,b minor : a^]} — c^ + Cjb — g\} + ^\\> ••• ^1 1 /i
* [My own experience is that the minor ments enable me to compare these effects
chords even more than the major shew the readily, and both arise from similar, though
vast superiority of the just intonation over the not the same causes.— TranaZafor.]
equal temperament; and that the occasional f [In which (...) represents the Pythago-
introduction of Pythagorean among justly- rean major Third of 408 cents and ( | ) the
intoned chords, major or minor, is comparable Pythagorean minor Third of 294 cents.—
only to the • wolves * on the * bad keys,* as Eb, Translator,]
or E, of the old organ tuning. My instru-
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CHAP. XVI. HAEMONIUM IN JUST INTONATION. 319
In the former series, Nos. 8 and 7, we had already b\} minor and e\^ minor,
which are a comma sharper than Nos. 13 and 14. Hence the series of minor
keys is also completed by the fusion of their extremities through enharmonic inter^
change.
In most cases it is possible to transpose the music to be played on such instru-
ments, so as to avoid the necessity of making these enharmonic interchanges, pro-
vided the modulations do not extend too far into different keys. But if it is not
possible to avoid enharmonic interchanges, they must be introduced where two
unrelated * chords follow each other. This is best done between dissonant chords.
Naturally this enharmonic change is always necessary when a piece of music
modulates through the whole circle of Fifths — from C major to Bjj^ major, for
example. But Hauptmann is certainly right when he characterises such circular
modulation as unnatural artificiality, which could only be rendered possible by the
imperfections of our modem system of temperament. Such a process must cer- IT
tainly destroy the hearer's feeling for the unity of the tonic. For although Bjj^
has very nearly the same pitch as 0, or can be even improperly identified with it,
the hearer can only restore his feeling for the former tonic by going back on the
same path that he advanced. He cannot possibly retain his recollection of the
absolute pitch of the first tonic C after his long modulations up to ^, with such
a degree of exactness as to be able to recognise that they are identical. For any
fine artistic feeling Bjj^ must remain a tonic far removed from G on the dominant
side ; or, more probably, after such distant modulations, the hearer's whole feeling
for tonality will have become confased, and it will then be perfectly indifferent to
him in what key the piece ends. Generally speaking, an immoderate use of strik-
ing modulations is a suitable and easy instrument in the hands of modem com-
posers, to make their pieces piquant and highly coloured. But a man cannot live
upon spice, and the consequence of restless modulation is almost always the
obliteration of artistic connection. It must not be forgotten that modulations f
should be only a means of giving prominence to the tonic by contrasting it with
another and then returning into it, or of attaining isolated and peculiar effects of
expression.
Since harmoniums with two manuals have usually two sets of vibrators for
each manual of which the above system of tuning only uses one, I have had the
two others (an 8-foot and a 16-foot stop) tuned in the usual equal temperament,
which renders it very easy to compare the effect of this tuning with just intona-
tion, as I have merely to pull out or push in a stop to make the difference.f
As regards musical effect, the difference between the just and the equally-
tempered, or the just and the Pythagorean intonations, is very remarkable. The
justly-intoned chords, in &vourable positions, notwithstanding the rather piercing
quality of the tone of the vibrators, possess a fall and as it were saturated har-
moniousness ; they flow on, with a full stream, calm and smooth, without tremor
or beat. Equally- tempered or Pythagorean chords sound beside them rough, dull, f
trembling, restless. The difference is so marked that every one, whether he is
musically cultivated or not, observes it at once. Chords of the dominant Seventh
in just intonation have nearly the same degree of roughness as a common major
chord of the same pitch in tempered intonation. The difference between natural
and tempered intonation is greatest and most unpleasant in the higher Octaves of
the scale, because here the false combinational tones of the tempered intonation
are more observable, and the number of beats for equal differences of pitch becomes
larger, and hence the roughness greater.
A second circumstance of essential importance is, that the differences of effect
between major and minor chords, between different inversions and positions of
* [That is, chords not haying a common and at the same time greatly facilitating
tone. — Translator.] ^ fingering by the use of a single manual, will
f Proposals for making the series of tones be found in Appendix XVII.
in this system of intonation more complete
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32Q HARMONIUM IN JUST INTONATION. pabtiii.
chords of the same kind, and between consonances and dissonances are much
more decided and conspicuous, than in the equal temperament. Hence modu-
lations become much more expressive. Many fine distinctions are sensible, which
otherwise almost disappear, as, for instance, those which depend on the different
inversions and positions of chords, while, on the Other hand, the intensity of the
harsher dissonances is much increased by their contrast with perfect chords. The
chord of the diminished Seventh, for example, which is so much used in modem
music, borders upon the insupportable, when the other chords are tuned justly.*
Modem musicians who, with rare exceptions, have never heard any mnsio
executed except in equal temperament, mostly make light of the inexactness of
tempered intonation. The errors of the Fifths are very small. There is no doubt
of that. And it is usual to say that the Thirds are much less perfect consonances
than the Fifths, and consequently also less sensitive to errors of intonation. Tlie
f last assertion is also correct, so long as homophonic music is considered, in which
the Thirds occur only as melodic intervals and not in harmonic combinations. In
a consonant triad every tone is equally sensitive to false intonation, as theory and
experience alike testify, and the bad effect of the tempered triads depends especially
on the imperfect Thirds.f
There can be no question that the simplicity of tempered intonation is ex*
tremely advantageous for instrumental music, that any other intonation requires
an extraordinarily greater complication in the mechanism of the instrument, and
would materially increase the difficulties of manipulation, and that consequently
the high development of modem instrumental music would not have been possible
without tempered intonation. But it must not be imagined that the difference
between tempered and just intonation is a mere mathematical subtilty without any
practical value. That this difference is really very striking even to unmuaieal ears,
is shewn immediately by actual experiments with properly tuned instruments^
f And that the early musicians, who were still accustomed to the perfect intervals of
vocal music, which was then most carefully practised, felt the same, is immediately
seen by a glance at the musical writings of the latter half of the seventeenth and
the earlier part of the eighteenth centuries, at which time there was much dis-
cussion about the introduction of different kinds of temperament, and one new
method after another was invented and rejected for escaping the difficulties, and
the most ingenious forms of instrument were designed for practically executing
the enharmonic differences of the tones. Praetoriust mentions a universal
cymbalum, which he saw at the house of the court-organist of the Emperor
Budolph n. in Prague, and which had 77 digitals in 4 octaves, or 19 to the
octave, the black digitals being doubled, and others inserted between those for s
and/, and between those for b and c.§ In the older directions for tuning, several
tones are usually tuned by Fifths which beat slightly, and then others as perfect
major Thirds. The intervals on which the errors accumulated were called wolves*.
^ * [This should be tried on the Hannonioal where the Fifth is only one-eleventh of a
as h^'-d\f-a}\>. Although it has two per- comma or 2 cents too flat, and tke major
feet minor Thirds and only one Pythagorean, Third is seven-elevenths of a comma or 14
it is a mere piece of noise, of a much worse cents too sharp, and hence the minor Third is
kind than the noise of the equally-tempered eight-elevenths of a comma or 16 cents too
imitation of the same chord in the same flat. The effect is much more strongly felt
quality of tone. In just intonation the chord in playing passages than in playing isolated
of the diminished Seventh can therefore be chords.— 2Va?wZa<or.]
used only with mild qualities of tone. But % Syntagma musicum, II., Chap. XL, p.
the real intonation of this chord is to : 12 : 14 63.
: 17, which can also be played on the Har- § [This was to make the meantone acala
monical as 0/'3i6 ^267 'd^'b 336 "d'^'b. — more complete, the scale being C cff db -D dt
Tramlator.-] e\> E el F /« g\> Q gt a)> A a« 6b B 6j C,
f [A triad in which the major Third is where the capitals represent the white, and
perfect, but the Fifth and minor Third both small letters the black digitals, all in mean-
too small by a quarter of a comma or 5} cents tone temperament, the effect of which would
(as in the meantone temperament, in which I have been very good on the organ. For the
have a concertina tuned), has a very much intonation of these notes see App. XX. sect. A.
better effect than the equally -tempered triad, art. 16, —TransUUor.]
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CHAP. XVI. DISADVANTAGES OF TEMPERED INTONATION. 321
PraBtorius says : * It is best for the wolf to remain iii tlie wood with its abominable
howling, and n6t disturb our hannonicas concordantias,' Hameau, too, who at a
later period contributed greatly to the introduction of equal temperament, in 1726 *
still defended a different style of tuning, in which the Thirds of the more usual
keys were kept perfect at the expense of the Fifths and of the unusual keys.
Thus he tuned up from G, in Fifths so much diminished, that the fourth Fifths
instead of being E, became the perfect Third of C, namely ^i=i^t>. Then again
four Fifths more to i4i|7, the perfect Third of F\}t instead of to A\}. But then the
four Fifths between this Ai\} and G had necessarily f to be made too large, because
it is not Ai\} but A\} which is four perfect Fifths distant from C This plan of
tuning gives the perfect major Thirds, C+-E,, G-l-5,, D-fi^iJ, -Kj + Gajj!, but
when we proceed further from E on the dominant side, or from G on the sub-
dominant side, we find Thirds which become worse and worse. The error in the
Fifths is about three times that in the equal temperament. Even in 1762, this ^
system could be characterised by d'Alembert as that commonly used in France, in
opposition to the equal temperament which Bameau subsequently proposed. Mar-
purg t has collected a long series of other systems of tuning. Since players found
themselves compelled by the use of only 12 digitals to the octave, to put up with a
series of false intervals, and to let their ears become accustomed to them, it was
certainly better to make up their minds to give up their few perfect major Thirds
Btill remaimng in the scale, and to make all the major Thirds equally erroneous. It
necessarily produces more disturbance to hear very falsely tuned Thirds amidst
correct intervals, than to hear intervals which are all equally out of tune and are
not contrasted with others in perfect intonation. Hence as long as it is necessary
practically to hmit the number of separate tones witliin the octave to 12, there caif
be no question at all as to the superiority of the equal temperament with its 12 equal
Semitones, over all others, and, as a natural consequence, this has become the sole
acknowledged method of tuning. It is only bowed instruments, with [including ^
the Tenor] their four perfect Fifths G±G±D±A±E, which still deviate from it.
The equal temperament came into use in Germany before it was introduced into
France. In the second volume of Matheson's Gritica Mtisicdy which appeared in
1752, he mentions Neidhard and Werckmeister as the inventors of this tempera-
ment.§ Sebastian Bach had already used it for the clavichord (clavier), as we
must conclude from Marpurg's report of Kimbeirger's assertion, that when he was
a pupil of the elder Bach he had been made to tune all the major Thirds too sharp.'
Sebastian's son, Emanuel, who was a celebrated pianist, and published in 1753 a
* Nouveau Systime de Muaiquet Chap. D, O on the other, fience oame the wolves.
XXIV. And a system of tuning was blamed for not
t [That is, if only twelve digitals might doing what it never professed to do. As long
be used, so that the temperament became as twelve digitals only are insisted on, the
unequal. But this style of tuning, which at equal temperament, by dividing the Octavo
first was the meantone temperament, where into twelve equal Semitones, is a necessity.
the Fifths are made a quarter of a comma too But with Mr. Bosanquet*s fingerboard (App. ^
flat, should be carried out through twenty-six XX. sect. F. No. 8) there is no longer any need to
Fifths, requiring twenty-seven tones (namely, limit organs to 12 notes to the Octave. When,
7 natural, 7 sharp, 7 flat, 3 double sharp, and however, he played on such an organ before
3 double flat) to be really effective, and if any the Musical Association, great objection was
fewer are employed no attempt should be made taken to the flatness of the leading note,
to modulate into keys not provided with proper which was 5^ cents flatter than just, as musi-
tones. It is a temperament with which I am cians are accustomed to on6 which is 12 cents
practically familiar. It is harmonically far too sharp.— Tratw^a^.]
superior to the equally tempered, and is even { Versuch Uber die musikalische Tempera-
endurable on the concertina, which used to be tur^ Breslau, 1776.
always so tuned, but having fourteen digitals, § Op, cit. p. 162. The following works of
extends from ^ b to Dt . The twelve digitals these two authors are cited by Forkel : Ncid-
could play then only in Bb, F, C, G, D, and hard (Royal Prussian Band-conductor), Die
A major, and in G, D, and A minor, which of teste und leickteste Tempcratur des Mono-
course failed to satisfy the requirements of cJtordi (the best and easiest temperament of the
modulation. Hence players sought to identify monochord), Jena, 1706 ; Sectio carwnis har-
Dff , ^5 , J??jr , FJJ , C« « , «a J with E\), B\>, monici, KSnigsberg, 1724. Werck7n€ister (or-
F. G, />, A on the one hand, and I>b, Gb, Cb, ganist at Quedlinburg, bom 1645), Miisikalische
Ftfj Bbby Kb by Abb with CJJ, FU , D, JC, A, Tempcratur, Frankfurt and Li'ipziq, i69i.
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322 DISADVANTAGES OF TEMPERED INTONATION, part m.
work of great autliority iii its day * on the true art of playing the clavier,' requires
this instrument to be always tuned in the equal temperament.*
The old attempts to introduce more than 12 degrees into the scale have led
to nothing practical, because they did not start from any right principle. They
always attached themselves to the Greek system of Pythagoras, and imagined only
(hat it was necessary to make a difference between dfi and d\}f or between /$ and
g\}y and so on. But that is not by any means sufficient, and is not even always
correct. According to our system of notation we may identify c,Jjl with d\}, but
we must distinguish the cfjj^ found from the relation of Fifths, from the c,j|! found
from the relation of Thirds.f Hence the attempts to construct instruments with
complex arrangements of manuals and digitals, have led to no result, which was
at all commensurate with the trouble bestowed upon them, and the increased
difficulties of fingering which they occasioned. The only instrument of the kind
f which is still used is the pedal harp a double mouvement, on which tlie intonation
can be changed by the foot.
Not only habitual use, and the absence of any power to compare its effects
with those of just intonation, but some other circumstances are &vourable to equal
temperament.
First, it should be observed that the disturbances due to beats in the tempered
scale, are the less observable the swifter the motion and the shorter the duration of
the single notes. When the note is so short that but very few beats can possibly
occur while it lasts, the ear has no time to remark their presence. The beats pro-
duced by a tempered triad are the following :
I. Beats of the tempered Fifth. Suppose we take the number of vibrations of
o' to be 264, the tempered Fifth c''±.g' would produce 9 beats in 10 seconds, partly
by the upper partials, and partly by the combinational tones. These beats are
always quite audible.
^ 2. Beats of the two first combinational tones of c'+6' and ^— ^' in tempered
intonation ; 5^ in the second. These are plainly audible in all qualities of tones,
if the tones themselves are not too weak.
3. Beats of the major Third c'+^' alone, 10^ in the second, which, however,
are not plainly audible unless the quahties of tone employed have high upper
partials.
4. Beats of the minor Third e'— ^', 18 in the second, mostly much weaker than
those of the major Third, and also heard only in qualities of tone having high
upper partials.
All these beats occur twice as fast when the chord lies an Octave higher, and
half as fast when it occurs an Octave lower.
Of these beats, the first, arising from the tempered Fifths, have the least
injurious influence on the harmoniousness of the chord. They are so slow that
they can be heard only for very slow notes in the middle parts of the scale, and
^ then they produce a slow undulation of the chord which may occasionally have a
good effect. Beats of the second kind are most striking for the softer quality of
tone. In an Allegro, of four crotchets in a bar, two bars occupy about three
seconds. If, then, the tempered chord c'+e'— gr' is played on a crotchet in this
bar, 2^ of these beats will be heard, so that if the tone begins soft, it will swell,
decrease, swell again, decrease and then finish. It would be certainly worse if
this chord were played an Octave or two higher, so that ^\ or 8^ beats could be
heard, because these could not fail to strike the ear as a marked rouglmess.
For the same reason the beats of the third and fourth kinds, arising from the
Thirds, which are clearly audible on harsher qualities of tone (as on the har-
jnonium), are also decidedly disturbing in the middle positions, even in quick time»
* [Equal temperament was not commer- perament, pp. 27^32. — Translator.]
cially established in England till i84i>i846. f V^^^^ is* ct found from c±g±d^a±e
See App. XX. sect. N. No. 5. With regard to ±b±fZ ±cff, from c,9 found from first the
both Sebastian and Emanuel Bach's relation to major Third c + e^ and then the Fifths ^| ±
it, see Bosanquel's Musical Intervals aitd Tern- 6, ±/,5 ±c,J .- Translator.]
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CHAP. XVI. DISADVANTAGES OF TEMPERED INTONATION. 323
and essentially injure the calmness of the triad, because they are twice and thrice
as fast as the others. It is only in soft qualities of tone that they are but little
observed, or, when observed, are so covered by stronger unbroken tones as to be
very slightly marked.
Hence in rapid passages, with a soft quality and moderate intensity of tone, the
evils of tempered intonation are but little apparent. Now, almost all instrumental
music is designed for rapid movement, and this forms its essential advantage over
vocal music. We might, indeed, raise the question whether instrumental music
had not rather been forced into rapidity of movement by this very tempered into-
nation, which did not allow us to feel the full harmoniousness of slow chords to the
8ame extent as is possible from well-trained singers, and that instruments had
consequently been forced to renounce this branch of music.
Tempered intonation was first and especially developed on the pianoforte, and
hence gradually transferred to other instruments. Now, on the pianoforte circum- f
stances really f&vour the concealment of the imperfections due to the temperament.
The tones of a pianoforte are very loud only at the moment of striking, and their
loudness rapidly diminishes. This, as I have already had occasion to mention,
causes their combinational tones to be audible at the first moment only, and hence
makes them very difficult to hear. Beats from that source must therefore be left
out of consideration. The beats which depend on the upper partials have been
eliminated in modern pianofortes (especially in the higher Octaves, where they
would have done most harm), owing to the mode in which upper partials are
greatly weakened and the quality of tone much softened by regulating the striking
place, as I have explained in Chap. V. (p. 776). Hence on a pianoforte the defi-
ciencies of the intonation are less marked than on any mstrument with sustained
tones, and yet are not quite absent. When I go from my justly-intoned harT-
monium to a grand pianoforte, every note of the latter sounds false and disturbing,
especially when I strike isolated successions of chords. In rapid melodic figures f
and passages, and in arpeggio chords, the effect is less disagreeable. Hence older
musicians especially recommended the equal temperament for the pianoforte alone.
Matheson, in doing so, acknowledges that for organs Silberman*s unequal tempera-
ment, in which the usual keys were kept pure,* is more advantageous. Emanuel
Bach says that a correctly tuned pianoforte has the most perfect intonation of all
instruments^ which in the above sense is correct. The great diffusion and conve-
nience of pianofortes made it subsequently the chief instrument for the study of
music and its intonation the pattern for that of all other instruments.
On the other hand, for the harsher stops on the organ, as the mixture and reed
stops, the deficiencies of equal temperament are extremely striking. It is con-
sidered inevitable that when the mixture stops are played with full chords an aw ful
din (hollenldrm) must ensue, and organists have submitted to their fate. Now this
is mainly due to equal temperament, because if the Fifths and Thirds in the pipe^
for each digital of the mixture stops were not tuned justly, every single note would ^
produce beats by itself. But when the Fifths and Thirds between the notes
belonging to the different digitals are tuned in equal temperament, every chord
famishes at once tempered and just Fifths and Thirds, and the result is a restless
blurred confusion of sounds. And yet it is precisely on the organ it would be so
easy by a few stops to regulate the action for each key so as to produce harmonious
chords.t
Whoever has heard the difference between justly-intoned and tempered chords,
can feel no doubt that it would be the greatest possible gain for a large organ to
omit half its stops, which are mostly mere toys, and double the number of tones
* [Probably this was the xneantone tern- 1850 there is a description of an organ by
penunent explained on p. 32 1, note t.— Trans- Poole, which is tuned justly for all keys
lalor.] by means of stops. fSee App. XVIII. second
t From Zamminer'B book (p. 140), I see that paragraph, and App. XX. sect. F. No. 7, where
in Silliman's American Journal of Science for Poole's new keyboard without stops is figured.]
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324 DISADVANTAGES OF TEMPERED INTONATION. part iil
in the Octave, so as to be able, by means of suitable stops, to play correctly in all
keys.*
The case is the same for the harmonium as for these stops on the organ. Its
powerful false combinational tones and its gritty trembling chords, both due U>
tempered intonation, are certainly the reason why many musicians pronounce this
instrument to be out of tune, and dismiss it at once as too trying to the nerves.
Orchestral instruments can generally alter their pitch slightly. Bowed instru-
ments are perfectly unfettered as to intonation ; wind instruments caji be made a
little sharper or flatter by blowing with more or less force. They are, indeed, all
adapted for equal temperament, but good players have the means of indulging tlie
ear to some extent. Hence, passages in Thirds for wind instruments, when
executed by indifferent players, often sound desperately false (verzweifelt falsch),
whereas good performers, with delicate ears, make them sound perfectly well.
^ The bowed instruments are peculiar. From the first they have retained their
tuning in perfect Fifths. The violins themselves have the perfect Fifths, G±D
±i4±jB. The tenor and violoncello give the Fifth C:hG in addition. Now,
every scale has its own peculiar fingering, and hence every pupil could be easOy
practised in playing each scale in its proper intonation, and then, of course, tones
of the same name but in different keys could not be rtopped in the same way, and
even the major Third of tlie major scale of C, when the C of the tenor is taken
as the tonic, must not be played on the E string of the violin, because this gi^-es
E and not ^j. Nevertheless, tlie modem school of violin -playing since the time
of Spohr, aims especially at producmg equally-tempered intonation, although this
caimot be completely attained, owing to the perfect Fifths of the open strings. At
any rate, the acknowledged intention of present violin-players is to produce only
1 2 degrees in the Octave. The sole exception which they allow is for double-stop
passages, in which the notes have to be somewhat differently stopped from what
fjf they are when played alone. But this exception is decisive. In double-stop
passages the individual player feels himself responsible for the harmoniousness <tf
the interval, and it lies completely within his power to make it good or bad. Any
violin-player will easily be able to verify the following &ct. Tune the strings of
the violin in the perfect Fifths G±D±.A±,E, and find where the finger mast be
pressed on the A string to produce the B, which will give a perfect Fourth B...E,
Now, let him, without moving his finger, strike this same B together with the
open D string. The interval D...B would, according to the usual view, be a
major Sixth, but it would be a Pythagorean one [of 906 cents]. In order to
obtain iiie consonant Sixth D...B1 [of 884 cents], the finger would have to be
drawn back for about ij Paris lines (nearly ^% inch), a distance quite appreciable
in stoppings, and sufficient to alter the pitch and the beauty of the consonance
most perceptibly.
But it is clear that if individual players feel themselves obliged to distinguish
^ the different values of the notes in the different consonances, there is no reason
why the bad Thirds of the Pythagorean series of Fifths should be retained in
quartett playing. Chords of several parts, executed by several performers in
a quartett, often sound very ill, even when each single one of these performers
can perform solo pieces very well and pleasantly ; and, on the otlier hand, when
quartetts are played by finely-cultivated artists, it is impossible to detect any false
consonances. To my mind the only assignable reason for these results is that
practised violinists with a delicate sense of harmony, know how to stop the tones
they want to hear, and hence do not submit to the rules of an imperfect school.
* [That is, as correctly as on the Author's more than three-quarters instead of only on€-
justly-intoned harmoniam, but that is far too half of the stops. And then musicians would
deficient in power of modulation into minor have to learn how to use a practically just
keys, to make it worth while to construct it scale, and how to adapt tempered music to it,
on a great orgali. Nothing short of the 53 both of which present considerable difiBiculties.
division of the Octave (p. 328c) would suffice, It is, the refore, safe to say that nothing of the
and this would necessitate the omission of kind will be done. — Translator. ^^
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CHAP. XVI. DISADVANTAGES OF TEMPERED INTONATION. 323
That performers of the highest rank do really play in just intonation, has been
directly proved by the very interesting and exact results of Delezenne.* This
observer determined the individual notes of the major scale, as it was played by
distinguished violinists and violoncellists, by means of an accurately gauged string,
and found that these players produced correctly perfect Thirds and Sixths, and
neither equally tempered nor Pythagorean Thirds or Sixths. I was fortunate
enough to have an opportunity of making similar observations by means of my
liarmonium on Herr Joachim. He tuned his violin exactly with the g±d±a±e
of my instrument. I then requested him to play the scale, and immediately he
had played the Third or Sixth, I gave the corresponding note on the harmonium.
By means of beats it was easy to determine that this distinguished musician used
bi and not b as the major Third to g, and Sy not e as the Sixth.f But if the best
players who are thoroughly acquainted with what they are playing are able
to overcome the defects of their school and of the tempered system, it would ^
certainly wonderfully smooth the path of performers of the second order, in their
attempts to attain a perfect ensemble^ if they had been accustomed from the first
to play the scales by natural intervals. The greater trouble attending the first
attempts would be amply repaid by the result when the ear has once become
accustomed to hear perfect consonances. It is really much easier to apprehend
the differences between notes of the same name in just intonation than people
usually imagine, when the ear has once become accustomed to the effect of just
consonances. A confusion between a^ and a in a consonant chord on my har-
monium strikes me with the same readiness and certainty as a confusion between
A and A\} on b^ pianoforte.it
I am, however, too httle acquainted with the technicalities of violin-playing,
to attempt making any proposals for a definite regulation of the tonal system of
bowed instruments. This must be left to masters of this instrument who at the
same time possess the powers of a composer. Such men will readily convince %
themselves by the testimony of their ears, that the facts here adduced are correct,
and perceive that, far from being useless mathematical speculations, they are
practical questions of very great importance.
The case is precisely similar for our present singers. For singing, intonation
is perfectly free, whereas on bowed instruments, the five tones of the open strings
at least have an unalterable pitch. In singing the pitch can be made most easily
and perfectly to follow the wishes of a fine musical ear. Hence all music began
with singing ;§ and singing will always remain the true and natural school of all
music. The only intervals which singers can strike with certainty and perfection,
are such as they can comprehend with certainty and perfection, and what the
singer easily and naturally sings the hearer will also easily and naturally under-
Btand.**
* Recueil des Travaux de la SodiU des when isolated from the rest of the scale, I
Sciences f de V Agriculture, et des Arts de Lille, find it difficult to distinguish between the just ^
1826 et premier semestre 1827 ; Mimoire sur and the Pythagorean major Third. But when
les Valeurs nuirUrique des Notes de la Oatnme, I play on my harmonium the complete melody
par M. Delezenne. [See especially pp. 55-6.] of some well-known air without harmonies the
For observations on corresponding circum- Pythagorean Third always feels tome strained,
stances in singing, see Appendix XVIII. the perfect Third calm and soft. It is only in
t Messrs. Comu and Mercadier have indeed the leading note, perhaps, that the sharper
published contradictory observations. {Comptes Third is more expressive. [See App. XX.
liendus de VAcad. des Sc, de Paris, 8 et «2 sect. G. arts. 6 and 7, for the results of later
F^vrier, 1869.) They let a musician play the experiments by Messrs. Comu and Mercadier.
Third of a major chord firet in melodic sue- — Translator.]
cession, and then in harmonious consonance. X U^ ^ consonant chord the difference is
In the latter case it was always 4:5. But striking, melodically not so. An eminent
in melodic succession the performer selected teacher of singing could only by great atten-
a somewhat sharper Third. I am bound to tion tell the difference when I alternated a„ a
reply, that in melodic succession the major and d„ c2 in the major scale of C. — Translator.]
Third is not a very characteristically deter- § [It must not be forgotten, however, that
mined interval, and that all living musicians the voice was the only musical instrument at
have been accustomed to sharp Thirds on the first knovfn.— Translator.]
pianoforte. In the simple succession c + e-y, ♦♦ [That this must also apply to non-har*
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326 DISADVANTAGES OF TEMPEEED INTONATION. PAKTin.
Down to the seventeenth century singers were practised by the monocbord, for
which Zarlino in the middle of the sixteentli century reintroduced the correct
natural intonation. Singers were then practised with a degree of care of which
we have at present no conception. We can even now see from the Italian music
of the fifteenth and sixteenth centuries that they were calculated for most periiBct
intonation of the chords, and that their whole effect is destroyed as soon as this
intonation is executed with insufficient precision.
But it is impossible not to acknowledge that at the present day few even of our
opera singers are able to execute a httle piece for several voices, when either
totally unaccompanied, or at most accompanied by occasional chords, (as, for
example, tlie trio for the three masks, Protegga it giusto cielOy from the finale to
the first act of Mozart *s Don Giovanni,) in a manner suited to give the hearer a
full enjoyment of its perfect harmony. The chords almost always sound a little
% sharp or uncertain, so that they disturb a musical hearer. But where are our
singers to learn just intonation and make their ears sensitive for perfect chords ?
They are from the first taught to sing to the equally-tempered pianoforte. If a
major chord is struck as an accompaniment, they may sing a perfect consonance
with its root, its Fifth, or its Third. This gives them about the fifth part of a
Semitone for their voices to choose from without decidedly singing out of harmony,
and even if tliey sing a little sharper than consonance witli the sharp Third
requires, or a little flatter than consonance with the flat Fifth requires, the
harmoniousness of the chord will not be really much more damaged. The singer
who practises to a tempered instrument has no principle at all for exactly and
certainly determining the pitch of his voice.*
On tlie other hand, we often hear four musical amateurs who have practised
much together, singing quartetts in perfectly just intonation. Indeed, my own
experience leads me almost to affirm that quartetts are more frequently heard
5 with just intonation when smig by young men who scarcely sing anything else,
and often and regularly practise them, than when sung by instructed solo singers
who are accustomed to the accompaniment of the pianoforte or the orchestra. But
correct intonation in singing is so far above all others tlie first condition of beauty,
that a song when sung in correct intonation even by a weak and unpractised voice
always sounds agreeable, whereas the richest and most practised voice offends the
hearer when it sings false, or sharpens.
The case is the same as for bowed instruments. The instruction of our present
singers by means of tempered instruments is unsatisfactory, but those who possess
good musical talents are ultimately able by their own practice to strike out the
right path for themselves, and overcome the error of their original instruction.
They even succeed the earlier, perhaps, the sooner they quit school, although, of
course, I do not mean to deny that fluency in singing, and the disuse of all kinds
of bad ways can only be acquired in school.
f It is clearly not necessary to temper the instruments to which the singer
practises. A single key suffices for these exercises, and that can be correctly
tuned. We do not require to use the same piano for the teaching to sing and for
playing sonatas. Of course it would be better to practise the singer to a justly*
intoned organ or harmonium in which by means of two manuals all keys may be
used.t Sustained tones are preferable as an accompaniment because the singer
himself can immediately hear the beats between the instrument and his voice
when he alters the pitch slightly. Draw his attention to these beats, and he will
monic scales is evident from the fact that wo aid not suit. Still the Harmonioal, or, for
music has existed for thousands of years, but modulating purposes, the just harmoniom or
harmonic scales have been in use only a few just concertina, may prove of service. Other-
centuries, and are far from being even yet wise special instruments must be used, as Mr.
univeraal.— Translator.] Colin Brown's Voice-Harmonium. The Tonic
♦ See Appendix XVIII. So^.faists teach without any accompaniment,
t ; Voices differ so much that the bame not even that of the teacher's voic^e, but
pitch for the tonic, that is the same key, rapidly introduce part music— IVaHfiiotor."
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CHA?. XVI. DISADVANTAGES OF TEMPERED INTONATION. 327
then have a means of cbeckiiig his own voice in the most decisive manner. This
is very easy on my justly-intoned harmonium, as I know by experience. It is
only after the singer has learned to hear every slight deviation from correctness
announced by a striking incident, that it becomes possible for him to regulate the
motions of his larynx and the tension of his vocal chords with sufficient delicacy
to produce the tone which his ear demands. When we require a delicate use of
the muscles of any part of the human body, as, in this case, of the larynx, there
must be some sure means of ascertaining whether success has been attained. Now
the presence or absence of beats gives such a means of detecting success or failure
when a voice is accompanied by sustained chords in just intonation. But tempered
chords which produce beats of their own are necessarily quite unsuited for such a
purpose.
Finally, we cannot, I think, fail to recognise the influence of tempered intona-
tion upon the style of composition. The iirst effect of this influence was favourable, f'
It allowed composers as well as players to move freely and easily into all keys, and
thus opened up a new weaAth of modulation. On the other hand, we hkewise cannot
fail to recognise that the alteration of intonation also compelled composers to have
recourse to some such wealth of modulation. For when the intonation of consonant
chords ceased to be perfect, and the differences between their various inversions
and positions were, as a consequence, nearly obhterated, it was necessary to use
more powerful means, to have recourse to a frequent employment of harsh disso-
nances, and to endeavour by less usual modulations to replace the characteristic
expression, which the harmonies proper to the key itself had ceased to possess.
Hence in many modem compositions dissonant chords of the dominant Seventh
form the majority, and consonant chords the minority, yet no one can doubt that
this is the reverse of what ought to be the case ; and continual bold modulational
leaps threaten entirely to destroy the feehng for tonahty. These are unpleasant
symptoms for the further development of art. The mechanism of instruments ^
and attention to their convenience, threaten to lord it over the natural require-
ments of the ear, and to destroy once more the principle upon which modem
musical art is founded, the steady predominance of the tonic tone and tonic chord.
Among our great composers, Mozart and Beethoven were yet at the commence-
ment of the reign of equal temperament. Mozart had still an opportunity of
making extensive studies in the composition of song. He is master of the sweetest
possible harmoniousness, where he desires it, but he is almost the last of such
masters. Beethoven eagerly and boldly seized the wealth offered by instrumental
music, and in his powerful hands it became the appropriate and ready tool for
producing effects which none had hitherto attempted. But he used the human
voice as a mere handmaid, and consequently she has also not lavished on him the
highest magic of her beauty.
And after all, I do not know that it was so necessary to sacrifice correctness of
intonation to the convenience of musical instruments. As soon as violinists have f
resolved to play every scale in just intonation, which can scarcely occasion any
difliculty, the other orchestral instruments will have to suit themselves to the
correcter intonation of the violins. Horns and trumpets have already naturally
just intonation.*
Moreover, we must observe that when just intonation is made the groundwork
of modulations, even comparatively simple modulational excursions will occasion
enharmonic confusions (amounting to a comma) which do not appear as such in
the tempered system.f
To me it seems necessary that the new tonic into which we modulate should
'*' [Beferring to this passage, Mr. Blaikley in practice/ See the whole of this paper,
says (Proceediyigs of the Musical Association^ and the discussion on it, and also see supra,
vol. iv. p. 56) : ' This must be taken as being pp. 97^, 996, notes.— Translator,]
particularly and not generally true, that is, t [See p. 324«{,note '*', and p. 340c, note *•
though the ideal instrument has such charac- —Translator.]
teristics, this ideal is not necessarily attained
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3f8 RULES OF MODULATION. . pawt iii.
be related to the tonic in which we are playing ; the nearer the relationship, the
more striking the transition. Again, it is not advisable to remain long in a key
which is not related to the principal tonic of the piece. With these principles the
rules for modulation usually given coincide. The easiest and most usual tran-
sitions are into the key of the dominant or subdominant, these tones being, as is
well known, the nearest relations of the first tonio> Henoe if the original key is C,
we can pass immediately into G major, and thus change the tones F and Ai of the
scale of C major into F^jf^ and A. Or we can pass into F major by exchanging i?,
and D for B\} and Di. After this step has been made, the music will often pass
into a key with a tonic related to C in the second degree only, as from GtoDor
from F to B\}. By proceeding in this way we should come to keys as A and ^t>,
of which the relation to the original tonic C would be very obscure and in which
it would certainly not be advisable to remain long for fear of too much weakening
f the feeling for the original tonic.
Again, we may also modulate from the principal tonic C to its Thirds and
Sixths, to E^ and A^ or E^\} and A^\}. Li tempered intonation these changes
seem to be the same as from O and D to A and E^ or from F and B\} to E\} and
A\}, But they differ in the pitch, as shewn by the different marks A and ^i, &c.
In the tempered intonation it seems allowable to go by a Sixth from c to the key a,
and then by Fifths back, to d, g, and then c again. But in reality we thus reach
a d fferent c from that with which we began. By such a transition, which is
certainly not quite natural, we should be obliged to make an enharmonic exchange
[alteration of pitch by a comma], and this would be best done while in the key of <f,
since both d and dy are related to c in the second degree. In the comphcated
modulations of modem composers such enharmonic changes will of course have
to be often made. A cultivated taste will have to judge in each individual case
how they are to be introduced, but it will be probably advisable to retain the roles
f already mentioned, and to choose the intonation of the new tonics introduced by
modulation m such a manner as will keep them as closely related to the prin*
cipal tonic as possible. Enharmonic changes are least observed when they are
made immediately before or after strongly dissonant chords, as those of the
diminished Seventh. Such enharmonic changes of pitch are already sometimes
clearly and intentionally made by violinists, and where they are suitable even pro*
duce a very good effect.*
t If we desire to produce a scale in almost precisely just intonation, which will
allow of an indefinite power of modulation without having recourse to enhar-
monic changes,^ we can effect our purpose by the division of the Octave into 53
exactly equal parts, as was long ago proposed by Mercator [to represent Pytha-
gorean intonation]. Mr. E. H. M. Bosanquet§ has recently provided this tem-
perament, as realised on an harmonium, with a symmetrically arranged finger-
board. When the Octave is divided into 53 equal intervals or degrees, 31 such
^ degrees give an almost perfect Fifth, the error of which is only ^\ of the error
of the Fifth of the usual equal temperament, and 17 of these degrees give a major
Third, of which the error in defect is only ^ of the above-named error of the Fifth
in equal temperament.** The error of the Fifth in this system must be considered
* See examples in C. E. Naumann's JSes^im- in Boom Q of the Scientific Collections ai the
mungen der Tonverhdltnisse (Determinations South Kensington Maseam].
ol the Tonal Katios), Leipzig, 1858, pp. 48, sqq. ** On converting the ratio of the extent of
f [From here to the end of the chapter the interval of a Fifth to that of an Octave
is an addition to the 4th German edition. — (that is log. 1-5 .* log. 2) into a continued frac-
Translator.] tion, we get the following approximations :
t [This is unfortunately not the case j^ -- y^ Fifths,
when translating equally tempered music, as nearly = 7 31 179 Octaves
shewn by the last example.— Tratwia/or.] \
§ An Elementary Treatise on Musical And by a similar approximation
Intervals and Temperament, London, Mac- ^ 28 59 major Thirds-
millan, 1875. The instrument described was nearly -i 9 19 Octaves,
exhibited in the Scientific Loan Exhibition at
South Kensington [in May 1876, and is still TAs these approximations give no coneep-
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QHAP. XVI.
THE CYCLE OF FIFTY-THREE.
329
as quite inappreciable, that of the major Third is still more difficult to perceive than
that of the equally tempered Fifth.* In these degrees the major scale will be
C D E, F G Ay By G
degrees o 9 17 22 31 39 48 53
differences 9859895
These differences of 9, 8, 5 correspond to the major, minor, and half Tone of
the just scale. Each separate degree of the scale corresponds nearly with the
interval 77 : 76 [=2 2 '6 cents] and is therefore extremely little greater than the
comma 81 : 80 [=21*5 cents], which in the just scale gives the difference between
a large or diatonic Semitone [16 : i5=:ii2 cents] and a small Semitone or limma
[256 : 243=90 cents] . The ear cannot distinguish this scale from the juBt,t and
in its practical applications it admits of unhmited modulation in what is equal to
exact intonation. The difference between our c, and c, or our c and c' would answer
to sharpening by one degree. Mr. Bosanquet therefore employs the convenien' IT
signs \c for c, and /c for c*, Wc for c^, &c. These signs, \and /, he also
employs before notes on the staff, exactly as we employ jf and \}, The fingerboard is
arranged in a very comprehensible and symmetrical way to make the fingering of
all scales and all chords the same in all keys.t A diagram of the keyboard will
be found in App. XIX. §
Perhaps a justification is needed for our having in this whole theory of keys
and modulations, identified the key of the Octave with that of its root, while
tion of the extreme closeness with which the
53 division, if accurately taned, woald approxi-
mate to just intonation, I annex the following
table :
Note
Ju ^t cents
53 divUion
cents
No. of
degrees
c
0
0
0
c»
21-506
22-642
I
^»
182*404
181-132
8
D
203910
203-774
9
E^b
3'5-64i
316-981
14
E
386-314
384906
17
ip
470781
475-472
21
F
498-045
498-113
22
0
701955
701-887
31
At,
813-687
815-094
36
At
884-359
883019
39
'Bb
968-^26
973-585
43
Bb
996091
996-226
44
J5'b
1017-597
1018-868
45
^»
1088-269
1086-792
48
C
1200000
1200000
53
»'Db
1304-955
1313-208
58
1
Hence for all tertian intervals the approxima-
tion is within 2 cents, often within i cent.
The septimal comma being greater than I
degree, the 'Bb is too sharp by 5 cents.
StiU the 53 div. chord C I E, Gl'Bb ish
great improvement on the just C : E^ : G : Bb.
The 17^1 harmonic is 8 cents too sharp, but
the chord of the diminished Seventh in the
53 div. Ej : O : 'Bb : ^^Bb is much superior
to the just form E^: O: Bb: D'b, though
from its just surroundings inferior in. effect
to the equally tempered E : O : Bb I Db.—
Translator,]
* [Both, however, give rise to beats which
are of great importance to the tuner. See
App. XX. sect. G. art. 20.— Translator.]
t [Melodically ; but harmonically, at least
as the intervals were tuned on Mr. Bosanquet's
instrument, there was a decidedly perceptible
difference to an ear, accustomed as mine was,
to listen to just intonation. Traimlator.]
% [Prof. Helmholtz adds, * after a plan in-
vented by the American, Mr. H. W. Poole.* I fl
have omitted this line because, although Mr.
Poolers remarkable fingerboard (figured in
App. XX. sect. F. No. 6) also allows of playing
with the same fingering in all keys, it was
not intended for the 53 division, and it bears
no resemblance to that of Mr. Bosanquet,
who has the exclusive merit of inventing and
practically carrying out his extraordinary * gene-
ralised keyboard,' which is suitable for all cycles
(except the ordinary one of 12) that resolve the
tones used into a series of tempered Fifths. See
App. XX. sect. A. art. 20 ^({.—Translator,]
§ [In App. XX. sect. F. No. 8, 1 have added
a further account of this invention, and (ibii.
No. 9) a notice of another keyboard for a reed
instrument called the Harmon^ also using the
53 division, invented and executed by Mr. James
Paul White, a tuner, of Springfield, Massachu-
setts, U.S. kmetioik. —Translator.]
Digitized by VjOOQlC
330 DISSONANT INTERVALS. pakt in.
we have distinguished the key of the Twelfth. lii tlie usual school of musical
theory, the meaning of the sound of the Octave is completely identified with that
of its root, and is so treated. For us, on the other hand, the Octave is only the
Tone most nearly and clearly related to the root; but its relationship is the same
in kind as that of the Twelfth, or the next higher major Third (Seventeenth) to
the root.
Now we have shewn in p. 273a that in the particular relation of the forma-
tion of scales, that is of the determination of the key, the higher Octave introduces
the same series of directly related tones as does the lower, although in a somewhat
different order of strength of relationship. Only throughout the formation of the
lower Octave the tones of the major scale are favoured, and in the formation of
the upper Octave those of the minor scale are preferred, but not to the exclusion
of those of other scales.
m When we proceed beyond the limits of the first Octave, the relationships of
tone depending on the six first partials give only the Tenth and Twelfth. The
other steps of the scales have then to be filled up with tones related in the second
degree, and, among these, the relations of the Octave must have the preference,
and next those of the Twelfth. H.nce in the second Octave we have necessarily
a repetition of the scale of the first. By this means, in the formation of scales an
equivalence of Octaves is established, without any necessity for assuming a speci-
fically different relation of similarity between them and the root, as we had to do
for the other consonances. In the formation of consonant intervals, the usual
theory of music also considers the Octaves as equivalent to the roots. This is
within certain hmits correct, because the intervals usually considered as consonant,
remain consonant when one of their tones is transposed by an Octave, or at least
produce intervals which lie on the limits of consonance. But here the usual rule
of the school really gave a very imperfect expression of the facts^ since, as we
^ have shewn in Chapters X., XI., and XII., the degree and sequence of the con-
sonance are really materially altered by these changes, and composers who have
outgrown the rules of the school, have also very clearly had regard to tliese
alterations.
CHAPTER XVII.
OF DISGOBDS.
When voices move foi-ward melodically in part music, the general rule is that they
must form consonances with each other. For it is only as long as they are con-
sonant, that there is an uninterrupted fusion of the corresponding auditory seusa-
tions. As soon as they are dissonant the individual parts mutually disturb each
other, and each is a hindrance to the £ree motion of the other. To this estlietic
reason must be added the purely physical consideration, that consonances cause
an agreeable kind of gentle and uniform excitement to the ear which is distin-
guished by its greater variety from that produced by a single compound tone,
whereas the sensation caused by intermittent dissonances is distressing and ex-
hausting.
However, the rule that the various parts should make consonances with each
other, is not without exception. The esthetic reason for tliis rule is not opposed
to an occasional and temporary dissonance among the various parts, provided the
motion of the parts is so contrived as to make the directions of the different voices
perfectly easy to follow by the ear. Hence, in addition to the general laws of scale
and. key, to which the direction of every part is subject, there are particular rulc^
for the progression of voices through discords. Again, dissonances cannot be en-
tirely excluded because consonances arc physically more agreeable. That which
is physically agreeable is an important adjunct and support to esthetic beauty, but
Digitized by V^OOQIC
CHAP. XVII. DISSONANT INTERVALS. 331
it is certainly not identical with it. On the contrary, in all arts we frequently
employ its opposite, that which is physically disagreeable, partly to bring the
beauty of the first into relief, by contrast, and partly to gain a more powerful
means for the expression of passion. Dissonances are used for similar purposes in
music. They are partly means of contrast, to give prominence to the impression
made by consonances, and partly means of expression, not merely for peculiar and
isolated emotional disturbances, but generally to heighten the impression of musical
progress and impetuosity, because when the ear has been distressed by dissonances
it longs to return to the calm current of pure consonances. It is for this last
reason that dissonances are prominently employed immediately before the conclu-
sion of a piece, where they were regularly introduced even by the old masters of
medieval polyphony. But to effect this object in using them, the motion of the
parts must be so conducted that the hearer can feel throughout that the parts are
pressing forward through the dissonance to a following consonance, and, although T
tliis may be delayed or frustrated, the anticipation of its approach is the only motive
which justifies the existence of the dissonances.
Since any relation of pitch which cannot be expressed in small numbers is
dissonant, and it is only the number of the consonances which is limited, the
number of possible dissonances would be infinite were it not that the individual
parts composing a discord in music must necessarily obey the laws of melodic
motion, that is, must lie within the scale. Consonances have an independent right
to exist. Our modem scales have been formed upon them. But dissonances are
allowable only as transitions between consonances. They have no independent
right of existence, and tlie parts composing them are consequently obhged to
move within the degree of the scales, by the same laws that were established in
favour of the consonances.
On proceeding to a detailed consideration of the separate dissonant intervals, it
should be rememberod that in theoretical music the normal position of discords is H
taken to be that which arranges their tones as a series of Thirds. This, for example,
is the rule for the chord of the dominant Seventh, which consists of the root, its
Third, Fifth, and Seventh. The Fifth forms a Third with the Thurd, and the
Seventh forms a Third with the Fifth. Hence we can consider a Fifth to be com-
posed of two, and a Seventh of three Thirds. By inverting Tliirds we obtain
Sixths, by inverting Fifths we obtain Fourths, and by inverting Sevenths we
obtain Seconds. In this way all the intervals in the scale are reproduced.
Using the present modification of Hauptmann's notation, it is easily seen how
different intervals of the same name must differ from each other in magnitude.
We have only to remember that c^ is a comma higher than 0, and C| two commas
lower than c^ and one comma lower than c, and that the comma is about the fifth
part of a Semitone.
To obtain a general view of both the magnitude and roughness of the dissonant
intervals, I have constructed fig. 6t (p. 333a), in which the curve of roughness is %
copied from fig. 60 A (p. 1936). The base line X Y signifies the interval of an Octave,
upon which the individual consonant and dissonant intervals are set off from X,
according to their magnitude on this scale.* On the lower side of the base are
marked the twelve equal SemitcHies of the equally-tempered scale [each distant
from the other by 100 cents], and on the upper side the consonant and dissonant
intervals which occur in justly -intoned scales. The magnitude of the interval is
always to be measured on the base line from X to the corresponding vertical line.
The vertical lines corresponding to the consonances have been produced to tlie
upper margins of the diagram, and those for the dissonances have been made
shorter. The length of the verticals intercepted between the base and the curve
of roughness shews the comparative degree of roughness probably possessed by tlie
interval when played in a violin quality of tone.
♦ [That is, assuming X Y to represent the X of any line shewing the interyal, givca the
cents in an Octave or 1200, the distance from centb in that interval. -Translator.]
Digitized by V^jOOQlC
332
DISSONANT INTERVALS.
PART III.
* [Tabclab Expbbssiom or thb Duobam, Fio. 6i Opposite.
Intenrala
No.
Helmholtz's
Notation, as in
Diagram
£lll«*s Nototion
of InternUs
reckoned from c
Batio
I : I
I :'^^2
Cents
Rouglincss
Unison
I
c : C
c : C
0
0
11 76
Minor Seconds
2
lie : ct
lie : ct
II 100
3
br.c
c :db
15 : 16
112
70
38
4
die,
c:d,
9 : 10
182
Major Seconds
5
6
Herd
cld
\\c:d
c:d
i:'^
8:9
11 200
204
0 25
32
7
8
6i : d'b
c : e'^bb
225 : 256
224
30
a»b : 6,
c:d,u
64 : 75
274
24
Minor Thirds
9
lO
d:f
Ijcrdg
c : eb
lie : dt
27:32
I : '^8
294
II 300
26
II 24
II
12
♦c:c»b
•cre'b
♦5:6
• 3'6
♦ 20
• 8
*c:e,
♦c : e.
♦4:5
♦ 386
Major Thirds
13
\\c : e '
ic:e
I : »ri6
II 400
II 18
14
b, : e»'b
clf'b
25:32
428
* 498
25
* 2
'5
*c:/
*c:/
♦3:4
Fourths
i6
lie:/
lie:/
1 : 'l'32
« 500
i 3
17
i8
/».:d'
ciP
20: 27
520
568'
27
6'b:«'»
clfji
18:25
32
Sharp Fourths
or Flat Fifths
19
20
|lc:/l
c:/.»
l|c:^
32:45
i:^V64
590
II 600
20
H 18
21
6.:/
e : gf'b
45:64
610
28
22
«, : 6'b
e:g'b
25:36
632
35
23
d : a,
cifl'i
27:40
680
44
. Fifths
24
lo:g
Wcig
I :»Vi28
II 700
11 I
25
*c:g
*c:g
♦2:3
• 702
772
♦ 0
39
26
e'b : 6,
c:g^
16 : 25
Minor Sixths
27
Ic : (75
He : gt
1 : ^^256
1 800
H 22
28
♦c : a'b
*c : a»b
♦5:8
* 814
♦ 20
29
*c:a.
*c : a,
•3:5
♦ 884
♦ 3
Major Sixths
30
31
Ijc : a
fid'
lie: a
e : a
I : i«/5i2
16 : 27
II 900
906
II 22
24
32
6, : a'b
c : 6'bb
75 : 128
926
24
_
33
d'b : 6,
c : a.,t
128 : 225
9 : 16
976
996
15
34
d:c'
c : db
23
Minor Sevenths
35
l|c : 6b
He : db
I : »^I024
II 1000
II 24
30
e,:d'
c : 6'b
5:9
1018
25
Major Sevenths
37
38
c : 6,
lie: 6
e:6
lie: 6
8:15
I : ^^^2048
1088
1 iioo
42
U 48
• Octave
39
♦c : c'
♦c:c'
•i :2
* 1200
» 0
Digitized by VjOOQlC
CIL\P. XVII.
DISSONANT INTERVALS.
333
Pig. 6i. (See iwte * opposite,)
The preceding tabular expression of the
diagram will be often found eonvenient. The
degree of roughness was determined by mea-
suring the lengths of the verticals in the dia-
gram in hundredths of an inch. The names of
the notes are given in the notation of the text,
using superior and inferior figures for the lines
above and below in the diagram. The sign ||
means * equally tempered/ and '*' ' consonance.'
The cross lines group the just intervals repre-
sented by a single tempered interval. The
cents are cyclical, as in the Duodenarium,
App. XX. sect. E. art. i8.
The intervals in the diagram are not noted
as from C to another tone, but as between the
two tones where they usually occur, except in
the equal intonation below. In the Table both
are given. The verticals for the dissonances
Were placed in two rows in re-cutting the
diagram for the ist edition of this translation,
pierely for the purpose of clearness, to prevent
the letters from coming too close to each other,
but without attaching any meaning to the differ-
ence of row ; the other differences described in
the text have been retained. The diagram also
uses the lines above and below the letters em-
ployed in the ist edition, (p. 2770, note *) and
separates the letters by ( — ), (p. 276^, note f)
as it was not considered advisable to re-engrave
it. In the Table, however, the notation of the
text is restored.
Tabic of Roughness.
The following is a comparative arrangement
of these intervals in order of roughness, the
consonances being marked *, and the tempered
intervals ||. The number in a parenthesis is
that of the interval when it is contained in the
preceding Table. The name given to each in-
terval in App. XX. sect. D. is annexed, fol-
lowing by its roughness t marked * ro.'
Rough iicffs
* o— (25) just Fifth.
Roughness
II I — (24) tempered Fifth representing (25)
just Fifth, ro. o, and (23) grave
Fifth, ro. 44.
♦ 2— (15) just Fourth.
g 3— ( 16) tempered Fourth, representing ( 1 5) _
just Fourth, ro. 2, and (17) H
acute Fourth, ro. 27.
♦ 3 -(29) just major Sixth.
♦ 8— (12) just major Third.
15— (33) extreme sharp Sixth.
II 16 — (13) tempered major Third, represent-
ing (12) just major Third, ro. 8,
and (14) diminished Fourth, ro.
25,— and also the Pythagorean
major Third, if required, ro. 19.
II 18— (20) tempered sharp Fourth or flat
Fifth, representing (19) false
Fourth or Tritone, ro. 20, (21)
diminished Fifth, ro. 28, (18)
superfluous Fourth, ro. 32, and
(22) acute diminished Fifth,
ro. 35.
19— Pythagorean major Third c I e "SJ-
408 cents. See p. 334, note J. g|
♦ 20 — (28) just minor Sixth. "
♦ 20— (I I) just minor Third.
2a— (19) false Fourth or Tritone.
II 22— (30) tempered major Sixth, represent-
ing (29) just major Sixth, ro. 3,
(31) PyUiagorean major Sixth,
ro. 24, and (32) diminished
Seventh, ro. 24.
II 22 — (27) tempered minor Sixth, represent-
ing (28) just minor Sixth, ro.
20, and (26) grave superfluous
Fifth, ro. 3Q.
23 — (34) minor Seventh.
II 24- (35) tempered minor Seventh, repre-
senting (33) the extreme sharp
Sixth, ro. 15; (34) the minor
Seventh, ro. 23, and (36) the
acute minor Seventh, ro. 25.
[Continued on next page]
Digitized by V^jOOQlC
334
THIRDS AND SIXTHS.
PART III.
The various Thirds, Fifths, and Sevenths of the scale are foand by arranging
it in Thirds thus: —
A. Tones op the MiUOR Scale.
hx-dlf+ai — c + Bi-g + bi'-dlf-ax
5T T T ¥ f
32 A
2T ¥
B. Tones of the Minor Scale.
^r l i; T^ ¥ ¥ IS
A
%\
For the minor scale I have assumed the usual form with the major Seventh,
^ because scales with the minor Seventh yield the same intervals as the major
scale.*
I. Thirds and Sixths.
The above schemes shew that in the justly-intoned major and minor scales,
three kinds of Thirds occur, and their inversions give three kinds of Sixths. These
are:
i) The jiistly -intoned major Third ^,
inversion the minor Sixth |^,
0
[i2, cents 386, roughness 8],t and its
[28, cents 814, roughness 20], both consonant.
-Therjttstlf intonod mi;iior~ ThrcL {jD [11, cents 316, roughness 20], and its
inversion the viajor Sixth f , [29, cents ^4, roughness 3], also both consonant.
3) The Pythagorean minor Third ^^, [9, cents 294, roughness 26], between
the extreme tones of the key, d and /. If we used dy in place of d, this interval
would occur between by and d^. On comparing this dissonant minor Third d \f
with the consonant minor Third dy^f, we find that the former is a comma
closer than the latter, since d ib a, comma sharper than dy. The Pythagorean
minor Third is somewhat less harmonious than the just minor Third, but tlie
difference between them is not so great as that between the two corresponding
major Thirds.^ The difference of the two cases consists, first, in the major Tliird
being a much more perfect consonance than the minor Third, and consequently
much more liable to injury from defects of intonation ; and secondly in the nature
RouRhneas
1} 24— (10) tempered minor Third, represent-
ing (11) just minor Third, ro.
20 ; (8) acute augmented Tone,
ro. 24, and (9) Pythagorean
minor Third, ro. 26.
24— (31) Pythagorean major Sixth.
24 — Pythagorean minor Sixth c : ab -^ii
= 792 cents.
I 24 — (32) diminished Seventh.
24— (8) augmented Tone.
II 25— (5) tempered major Second or whole
Tone, representing (7) diminished
minor Third, ro. 30, (6) major
Tone, ro. 32, and (4) minor Tone,
ro. 38.
25— (14) diminished Fourth.
25 — (36) acute minor Seventh.
26 — (9) Pythagorean minor Third.
27— (17) acute Fourth.
28— (21) diminished Fifth.
29 — grave major Seventh c : b^ = {;? - 1067.
30 — (7) diminished minor Third. '
32 —(6) major Tone.
I acute diminished Fifth.
32 — (18) superfluous Fourth^
3S-(22)
38— (4) minor Tone.
39— (26) grave superfluous Fifth.
42— (37) just major Seventh.
Roughness
44— (2^) grave Fifth.
ii 48 -{38) tempered major Seventh, repre.
senting (37) just major Seventh,
ro. 42.
56— great Limma c : d*b =f? = 134 cents.
70— (3) just minor Second, just or diatonic
Semitone.
II 76 — (2) tempered Semitone, representing
(3) just Semitone, ro. 70. —
Translator,]
* [The remainder of this chapter should
be followed step by step on the Harmon ical,
wherever it is possible, as is most frequently
the case. — Translator.]
f [For immediate comparison I have, after
each interval as it arises, inserted in square
brackets, the number of the interval, the
number of cents it contains, and its degree of
roughness as given in the Table on p. 332.
— Translator.]
X [The roughness of the just major Third,
c + e, is only 8, while that of the Pythagorean «
l{ (which is not given in the Table on p. 332,
becausie it does not occur in the scale) is
necessarily close to that of the tempered major
Third, 18, and may probably be taken as 19,
as will be seen by the curve in fig. 61, p. ;i^^.
— Translator.]
Digitized by VjOOQlC
CHAP. XVII. FIFTHS AND FOURTHS. 335
of the two combinational tones. The just minor Third d/" — /" has 6[> for its
combinational tone, which completes it into the just major triad of b\}. The
Pythagorean minor Third d"' \ f" has a, for its combinational tone, which com-
pletes it into the chord d\ f •\- a^^ and this is not a perfectly correct minor chord.
But as the incorrect Fifth a^ lies among the deep combinational tones and is very
weak, the difference is scarcely perceptible. Moreover, it is practically almost im-
possible to tune the interval so precisely as to insure the combinational tone a^ in
place of a. But for the Pythagorean major Third c" ., e" the combinational tone
is (4f, which is, of course, much more annoying than the rather imperfect Fifth a^
when added to the chord d \ f.*
The Pythagorean major Third does not occur in scales tuned according to the
conditions of harmonic music. If we used the minor Seventh b\} in place of ^^l>
for the minor scale, b\}...d would be a Pythagorean major Third.t
The inversion of the Third ^ |/is the Pythagorean major Sixth f...d\ f J, [31, c
cents 906, roughness 24] » which is a comma wider than the just major Sixth, and
is greatly inferior to it in harmoniousness, as is clearly seen in fig. 61 (p. 333a).
II. Fifths and Fourths.
The Fifth is simply composed of two Thirds, and the different varieties of
Fifths depend upon the nature of those Thirds.
4) The just Fifth |, [25, cents 702, roughness o], consists of a just major and
a just minor Third, or f = J x ^ [cents 702 = 386 + 316]. Its inversion is the
just J^ourth f , [15, cents 498, roughness 2]. Both are consonant. Examples in
the major scale, /±c', ai ± e/, c±^, ei ±&i, ^± d.
5) The grave or imperfect Fifth d...ai f^, [23, cents 680, roughness 44], a
comma [of 22 cents] less than the just Fifth, consists of a Pythagorean minor and
a just major Third, f ? = ^f x f [cents 680 = 294 + 386]. It sounds like a badly- ^
tuned Fifth, and makes clearly sensible beats. In the Octave C'...&\ the number
of these beats in a second is 11. Its inversion, the acute or imperfect Fourth,
ai...d\ f J, [17, cents 520, roughness 27], is also decidedly dissonant. The Fourth
Ai...d makes as many beats in a second as the Fifth d...aiy the d being the same
in each, [see App. XX. sect. G. art. 16].
6) The false or diminished Fifth, bi.., /, ^, [21, cents 610, roughness 28],
consists of a just and Pythagorean minor Third, f| = J x ^, [cents 610 = 316 +
294] and is, hence, as the composition shews, [92 cents or] about half a Tone closer
than the just Fifth. It is a tolerably rough dissonance, nearly equal in roughness
to a major Second [6, cents 204, roughness 32]. Its inversion is the false Fourth
or Tritone, /... 61, ^, [19, cents 590, roughness 20], consisting of three whole
Tones, major/...gf,minorgf...a,,and major ai...6,, f x V x | = ^, [cents 590 =
204 + 182 -f 204] ; it has very nearly the same degree of roughness as the last [or
false Fifth], and is [20 cents or] about a comma closer. For the false Fifth ^1./^
is nearly the same as c|>.. ./, and if we diminish this interval by a comma we obtain
cl> — /i, which is a false Fourth. Strictly speaking, as c\} is not precisely the same
as 5 1, the difference between the intervals is not precisely a comma, |^, but about
^, [or 1^ of a comma = 20 cents]. On keyed instruments they coincide.
7) The superfliums or extreme sharp Fifth of the minor scale, e*t> -^u t¥> [26,
cents 772, roughness 39], consists of two major Thirds, e^\} + g, and ^ + 6,, ^ =
J X f [cents 772 = 386 -f 386]. It is seen to be [42 cents or] nearly two commas
[44 cents] closer than the minor Sixth, [cents 814] by putting for b- the nearly
identical ol>, so that e^t>...5, is nearly the same as e^|>...c[>, whereas the consonant
minor Sixth is e||;>...c|;>, where «it> is two commas flatter than e^\}. The superfluous
Fifth [26, cents 772, roughness 39] is markedly rougher than the minor Sixth [28,
* [In just intonation, however, the difference f [The Pythagorean major Third of 408
between <2, — / and d \ f i& very marked, as cents does not occur on the Harmonical. The
may be readily observed on the Harmonical. — nearest interval '&b...(i, of 413 cents is supe-
Translator.] rior in eflect.^Tra7tslator.]
Digitized by VjOOQlC
336 SEVENTHS AND SECONDS. p.uiT iii.
cents 814, roughness 20], with wliich it coincides upon keyed instruments. Its in-
version, the diminished Fourth 6,...e*'t>» ^I» [14, cents 428, roughness 25], is [42
cents or] about two commas higher than the just major Third, [12, cents 386,
rouglmess 8], and considerably rougher, although the two intervals coincide on
keyed instruments, [13, cents 400, common rouglmess 18].
Two just or two Pythagorean minor Thirds cannot occur consecutively in the
natural series of Thirds of the just major and minor scales. In the modes of the
minor Seventh and of the Fourth, we may find the intervals a|...e*|;> and ex...b^\}
= JJ, [22, cents 632, roughness 35], composed of two minor Tliirds, ^f = J x f
[cents 632 •— 316 H- 316] ; these are a comma wider than the usual false Fifths
6,../ (or ap. e'\} m the key of b\} major, and <j,...6t> in the key of /major), and
are decidedly rougher than these, [21, cents 610, roughness 28].
III. Sevenths and Seconds.
Any three successive Thirds give a Seventh. Beginning with the smallest we
obtam the following different magnitudes :
8) The diminished Seventh of the minor scale fep..a*'t> [32, cents 926, rough-
ness 24], = (6, — d') + (rf' +/) H- (/ — a*'l>), or two just minor Thirds and one
Pythagorean minor Third. Its numerical ratio is VV = ir ^ 5t ^ J, [cents 926 =
316 + 294 -f- 316,] which is [42 cents or] about two commas greater than tlie major
Sixth [29, cents 884, roughness 3], as is seen by putting 6, ...a*'t> = c[>.. a*'l>. The
interval c\^...a\'\}^ which is two commas flatter than the last, would be a just
major Sixth. Its dissonance is harsh and rough, the same as that of the Pytha-
gorean major Sixth c. a, [31, cents 906], which is [20 cents or] about a conmia
less. But its inversion, the superfiuoiLs Second a^\}...hx [8, cents 274, having tlie
K same roughness 24], is not much rougher than the just minor Third [i i, cents 316,
roughness 20 ; the tempered minor Third 10, cents 300, has exactly the same
roughness 24]. Its numerical ratio 7ff [c®ii*s 274] is very nearly J [cents 267]
(since^ = | x 5.JJ [cents 274= 267 -f 7]). If this Second is extended to a Ninth,
7,^, [Ijaving 1474 cents or] nearly ^, [cents 1467] it becomes tolerably harmonious,
as much so as the minor Tenth V» [cents 15 16] which, however, is a very imper-
fect consonance, [see fig. 60 B, p. 193c].*
9) The closer minor Seventh of the scale g-'-fj bx . a/, or d—c\ V, [34, cents
996, roughness 23], consists of a just major, a just minor, and a Pythagorean
minor Third, ^ - / = ( ^ + &,) + (6, - ci') + (d'' - /), [or V = ^ x f x 5|
cents 996 = 386 -I- 316 + 294.] It is a comparatively mild dissonance, milder than
the diminished Seventh [32, cents 926, roughness 24], and this is of importance
for the effect of the chord of the dominant Seventh, in which the Seventh has
this form. This closer minor Seventh is the interval of a Seventh in the scale
H nearest to the natural Seventh or seventh harmonic, J [= V x JJ, cents. 969 =-996
— 27], although not so close as the extreme sharp Sixth [33, cents 976=996- 20,
roughness 15]. It has been already shewn that the natural Seventh belongs to
harmonious combinations (pp. 195a, 217c). The inversion of this Seventh is the
major Second, c...d, ai,..by, /., g, |, [6, cents 204, roughness 32], a powerful
dissonance.
10) The actcte or wider minor Seventh, e,...rf', ai...g', ^, [36, cents 1018,
roughness 25] , a comma greater than the last, is distinctly harsher than that interval,
because it is nearer to the Octave ; its roughness [25] is nearly the same as that of
tlie diminished Seventh [24]. It consists of a just major and two just minor
Thirds : e^.^.d' = (e, - g) + (g > b^) -f- (b^ - d'), [or J = ^ x ^ x f , cents 1018
= 316 + 386 + 316.] The last-mentioned closer minor Seventh has its root on the
* [Compare on Harmonical a''b...6," with g\..b^"b. The (/'... '6"b will be found much
l/'...'6"b, and a'b...6/ with g^..,^b"\> and with the most hannonious.— rrawsfa/cw.;;
Digitized by V^OOQIC
CHAP. XVII. MAJOR SEVENTH AND EXTBEME SHARP SIXTH, 337
dominant side of the scale, and its Seventh on the subdominant side, because it
contains the Pythagorean minor Third d \ /. The wider minor Seventh, on the
other hand, has its Seventh on the dominant side. Its inversion, the minor Tone,
^f d...ei, g...aif [4, cents 182, roughness 38] is somewhat harsher than the major
Tone [6, cents 204, roughness 32].
11) The major Seventh f...e\, c.hy, */, [37, cents 1088, roughness 42], con-
sists of two just major and one just minor Third : C...5, = (c + ei) + (^i — g) +
{g + bi) [or V = T X I >< T» cents 1088 = 386 -f 316 + 386]. It is a harsh dis-
sonance, about the same as the minor Tone [4, cents 182, roughness 38]. Its
inversion, the minor Second or Semitone ^i-.c', ei.../, |f, [3, cents 112, roughness
70], is the harshest dissonance in the scale.*
In the mode of the Fourth or minor Seventh, we find a somewhat closer
major Seventh, b^\}.,.a\, which is a comma closer than the usual major Seventh,
and hence somewhat milder in effect.t H
Finally we have to mention an interval peculiar to the Doric mode of the
minor Sixth, namely :
12) The sujperfltious or extreme sharp Sixth d^\}...bi, which arises in this mode
from connecting the peculiar minor Second of the mode d^\} with the leading note
bi [see p. 2S6b].
The numerical ratio is f||, [33, cents 976, roughness 15], so that it is [20 cents or]
about a comma less than the closer minor Seventh of the chord of the dominant
Seventh [cents 996], as is seen by putting d2*t>... 6, =c^^t>--c'l>; the interval d\^ .c'\}
would be the closer minor Seventh, and d^\} is a comma higher than d\}. The
superfluous Sixth may be conceived as composed of two just major Thirds and
one just major Tone : {d^\} +f) + (f...g) + (S' + ^'i) = ^*b ... &i, [or|?i = f x f
X f , cents 976 = 386 + 204 + 386]. Its harmoniousness is equal to that of the
minor Sixth, because it is almost exactly the natural Seventh ^^i since i||= 7 x
Hf [or 976 = 969 + 7]- Taken alone it cannot be regarded as a dissonance, but H
it makes no other consonant combinations, and hence is unfit for use in consonant
chords. When it is inverted into the diminished Third |ff [cents 224], or nearly
^ [cents 231], it is, as already observed, considerably damaged [7, cents 224, the
roughness rises to 30], but it is improved by taking the upper tone bi an Octave
higher, in which case it is [cents 2176 or] nearly ^ [= cents 2169]. Its near
agreement with the natural Seventh and its comparative harmoniousness, seem to
have preserved this remarkable interval in certain cadences, although it is quite
foreign to our present tonal system. It is characteristic that musicians forbid its
inversion into the diminished Third (which lessens its harmoniousness), but allow
its extension into the corresponding Thirteenth (which improves its harmonious-
ness). On keyed instruments this interval coincides with the minor Seventh [35,
cents 1000, roughness 24].
Generally, a glance at fig. 61 (p. 333a) will shew to what an extraordinary extent
different intervals are fused on keyed instruments.§ On the lower side of the base ^
line X Y are marked the places of the tones of the equally tempered scale, and the
small braces below the base line shew those different tonal degrees which are
* [That is in the jast major scale; the 1 = 969 cents. As a matter of fact, on my
8emitone of the tempered scale, 2, reaches 76 meantone concertina I find / 966 d% much
degrees of roughness. — Translator.] smoother than / 1007 eb. The chord intro-
t [Its numerical ratio is ^ = V' ^ a?* cents ducing this interval occurs in three forms.
1088 -22 =1066, so that it is the interval c...6g, The 'Italian' D'b 386 F 590 J5„ and the
which by fig. 61 (p. 333a) should have a rough- * German * D»b 386 F 316 A^b 274 J5,, are
ness of about 29, in place of 42, the roughness simply imitations of the true chord of the
of c..,b^.— -Translator.] dominant Seventh D»b 386 F 316 A»b 267
J [The diagram, fig. 61 (p. 333a), gives the 'C*b. The * French ' form, (the only one con-
roughness of the superfluous Sixth as 15, and sidered in the text and on p. 2866,) Z>'b 386
that of the minor Sixth as 20 ; see p. 3330^, d'. F 204 O 386 B^ is the harshest of all. The O
This would make the former more harmonious seems to be merely an anticipation of the note
tiian the latter. This interval does not exist on of the chord C 316 -B'b 386 G 498 c on which
the Harmonical. In meantone intonation, the it resolveB.— Translator.]
extreme sharp Sixth has only 966 cents, and is § [This is shewn in detail on pp. 332-4, note,
therefore still closer to the subminor Seventh -^Translator,]
Digitized by VjO@QlC
338 DISSONANT TKIADS- pabt in.
usually expressed by the corresponding tone of the tempered scale. The interval
bi...a^\} [cents 926] is identified on the pianoforte with the major Sixth c\}...a\}
[cents 884, or 42 cents closer], while the interval d^\}...bi [cents 976, or 50 cents
wider than the first] is made a (tempered) Semitone [cents 100] wider [being
identified with 1000 cents], and yet the last is scarcely more different from the
first, than the first from a major Sixth. The figure 61 shews also very clearly
what an immense difference of harmoniousness ought to exist between the first
and either of the two last of the following intervals c.ai, /...d', and bi.,.a^\},
[29, 31, 32, respective cents 884, 906, 926, respective roughness 3, 24, 24],
which are all expressed by the sufficiently harsh sound of the tempered inter-
val c.,.a [30, cents 900, roughness 22]. The justly-intoned harmonium with two
rows of keys* allows all these intervals to be given accurately, by which the differ-
ence of their sound becomes extremely striking. In this evidently lies one of the
IT greatest imperfections of tempered intonation.
Dissonant Triads.
Dissonant triads with a single dissonant interval are obtained by taking two
tones which are consonant to the root, but dissonant to each other. Thus :
i) Fifth and Fourth : c...f...g, [orfdtc±g],
2) Third and Fourth : c + ej.../ or c — e^\} ./, [or/± c + Bi and/± c— «*W.
3) Fifth and Sixth : c ± gr . a, or c ±:g.,.a^\}, [or a—c ± g, and a^\} -\- c:t.g],
4) Dissimilar Thirds and Sixths: c— c'|:>...ai or c + ei...a*l>, [orai— c— e"l>
anda^b + c + «i].t
In all these c is consonant with each of the other two tones. The first chord
alone plays a great part in the older polyphonic music as a chord of suspension,
% The others we shall meet with again in the chords of the Seventh.
The chords named in the fourth series above t admit of an inversion which
makes them appear as triads with diminished or superfluous Fifths, namely :
ai — c — e^\} and a^\} + c + gj.
The first of these is composed of two just minor Thirds, [so that the Fifth
ax,..e^\}, No. 22, ratio 25 : 36, cents 632, roughness 35, is the acute diminished
Fifth,] and the second of two just major Thirds, [so that the Fifth a^\}...ei. No. 26,
ratio 25 : 16, cents 772, roughness 39, is the grave superfluous Fifth], Both are
dissonant on account of the altered Fifth, although the dissonance of the second
has to be played as the consonance ^2$... e [minor Sixth 814 cents] upon keyed
instruments. The first of these two chords can only appear in the mode of the
minor Third, and the above would be heard in that of F.§ The second, on the
other hand, belongs £0 J^ minor.**
5F If we suppose this series of tones to be continued as
a»l> + c'+ e/... a»'t> + c"+ e/'
i i n i i
* [And, with the exception of the extreme — c— 6*b + g, that is in one of the forms
^rp Sixth d'b...6,,on the Harmonic^ also. 6b + d,-/+a,-c-e»b + ^=i Fma.maanL
The extreme sharp Sixth c...a^ may be. ^^ ftb-d'biz+a^-c-e^b + i^^i FmijnaanL
oiently imitated as c.Jhb.^Translatar.'] /+a,-c-e»b + ^ + 6,-d = i C ma.mi.ma.
t [These triads I propose to term can^^ /+a -c-<i«b + g-6'b + d-i Cma.mi.mL
sonant^ and the two last especially I caU the '' '
minor and major trine. See App. XX. sect. E. Bat not one of these belongs to the mode of
art. 5. — Translator.'] the minor Third, which for F is i F mijnijni.,
X [From p. 338c, beginning with these words unless the second is taken to be such with
to the paragraph ending * as in concords,' on a major tonic. The last, however, is the
p. 339&, is an insertion in the 4th German edi- mode of the minor Seventh of C — TraMS-
tion. —Translator,] latorJ]
§ [It is evident that a,— c— e'b can only ** [In the major dominant fonn 6 b—cl^b-f
occur when the chain of chords contains / + a, /— a' b + e + 61 — g. — Translaior.]
Digitized by V^jOOQlC
CHAP. xvn. DISSONANT TRIADS. 339
an interval glides in of §| = f • f|f = J • || approximatively [cents 428 == 386 + 42] ,
which is slightly (about 2 commas) greater than a just major Third. By small
alterations of pitch other chords are formed which belong to other keys :
A^\} + c + 61 .,.a^\} in i'' minor
Oiij^,.,c + ei + gjj^ in -4, minor
U i i
A^\} + c.../'t>+a^t> ^ -D*t> minor
The roots of these three minor keys
D'\} + F+A,
form a similar chord, of which the roots are a Semitone higher than those of the ^
preceding.* Since ^*|;> is nearly the same as G^jf^, and F^\} nearly the same as
£ji, these transformations alter the pitch of one of the tones in the chord by
about two commas, or, at least in the resolution of the chord, this tone is treated
as a leading note just as if it were thus altered. Hence we obtain modulations
which with a single step lead us to comparatively distant keys, and we can as
easily resolve into the minor as into the major keys of the three roots named.
This means of modulation is often employed by modem composers, (for example
B. Wagner) in place of using the chord of the diminished Seventh, which is much
rougher but was also applied for the same purpose. In just intonation these
chords are not by any means so unpleasant as in the tempered intonation of the
pianoforte. Generally it may be observed that when one is accustomed to play in
just intonation, the ear becomes quite as sensitive to a pitch which is wrong by a
comma in discords as in concords.
For modem music triads with two dissonances, formed by including the ^
extremes of the key, are more important.
In the series of chords in any key, major and minor Thirds follow each other
alternately, and any two adjacent Thirds produce a consonant triad. But the
interval between the extreme tones d and / is a Pythagorean minor Third, and
when these are connected as a chord with one of either of the two adjacent tones
to make a new triad, it will be dissonant.
Major : o + Bi — gr + 61— i|/+a, —c + ei ^g
Minor : c — e*b + gr + 61 — d \ f— a^\} -^ c — e^\} + g
far, 83S6 06 A
T 1 f VT f Tir ^
The major system gives two triads of this kind :
bi-d+f and d I /+ ai %
in Mi
The minor scale also gives two :
bi-d \f and d \ f - a^\>
i u n i
In the two triads bi— d \ /and d \ f — a^|?, which combine a Pythagorean
with a just minor Third, there are also second dissonances, namely the false Fifths
&!.../ and d...a^\}f which make the chord more strongly dissonant than the Pytha-
gorean minor Third ff alone could make them. They are hence called diminished
* [Only in the form c + 6, +^^. From apart, and could not pOBsibly be confounded in
what follows it is evident that the transforma- just intonation. Of course Wagner thought
tion could only take place in tempered intona- only in equal temperament, in which the tones
tion. The tones confounded are all 42 cents are absolutely identical.— Tmn^Zator.]
Digitized by V^(^€>Q1C
340
DISSONANT TRIADS.
FABT nL
triads^ The chord d | / -f ^i, which in the usual musical notation is not distiii*
guished from the minor triad c^i — / + aj, and may hence be called the false miner
triad, is, as Hauptmann has correctly shewn, dissonant, and on justly-intoned
instruments it is very decidedly dissonant. It sounds almost as rough as the chord
6, — (i I /. If in G major, without confounding d with di, we form the cadence
I or 2
r, - ^j ^ -[0] _ [sodded]
lE^^
1"-
the chords a^^...d" \ f and/ -f ai',,.d" \ f are quite as dissonant in their effects
as the following hi' -- d" \f and g' -»- hi — d" \ f". But on account of the in-
correct intonation of our musical instruments we cannot produce the same effect
^ without combining an inverted chord of the Seventh with the subdominant in the
cadence, as /+ ai — d ,..d\ Hauptmann doubts whether in practice the iialse
minor chord of the key of Q major can be distinguished from the minor chord of D.
I find that this is most distinctly and undoubtedly effected on my justly intoned
harmonium, but allow that we cannot expect the correct intonation from singers.
They will involuntarily pass into the minor chord, unless the progression of the
parts which execute D, strongly emphasise its connection with the dominant G.*
These chords, and among them most decisively and distinctly the chord
hi — d Iff have for musical composition the especially important advantage of
combining those limiting tones of the key, which separate it from the nearest
related keys, and are consequently extremely well suited for marking the key in
which the harmony is moving at any given time. If the harmony passed into G
major or G minor, / would have to be replaced by /i J. If it passed into -F major,
d would become di and if into F minor d would become d^\} and Z>, would in the
^ same chords become h^\}. Thus —
in G major :
b, -d +/,#
d +/,#-«
in C major :
b, -d \f
d 1 / + a,
in F major :
b\> +d, -f
di -/ +a,
in G minor :
b'\> + d +/,#
d + /,# - a
in C minor :
b, -d 1/
d 1/ -a't)
in F minor :
b\} -d'\}+/
d'\}+f -a't)
This shews that the chords in the nearest related keys are all distinctly different,
with the exception of d \f + ai and ^i — / + ^i, the distinction between which in
♦ [The chord on the Second of the major
scale is in fact the crux of the translation of
tempered into just intonation. It is easy to
^ play Ex. I and 2, and Ex. 3, here added, as
a' d^' f f a' d," f f a( d,"
6/ d" /'andfl' 6/ d" /'and^ V d"
d' c" e," c' c" c" e," f 6/ d"
e' d' d'
and the effect is not bad. In the first the d('
might be held on to the second chord, as 6' d"
f\ without materially increasing the harshness
of the dissonance, but in the second this would
give ^ 6/ d/' /", where the grave Fifth is very
harsh. In the second case, then, there is
least harshness in playing d!' in both chords.
And in both cases there is most smoothness
in playing them as just written. The effect is
one on which I have repeatedly experimented,
'but I find that the smali interval d^' d" in the
highest or lowest part, produces a strange effect,
which in singing, and perhaps on the violin,
seems to be overcome by a gUde, if the other
voices are strong enough to pull this voice out
of its course, even though the words and parts
are written so as to imply that this note is sus-
tained. WTien the d" is in the principal pari
in the melody, as in the third example, I find
it best on the whole not to play as written, d/'
d'\ but to sustain d'\ In some cases an at-
tempt to avoid the dissonance, which is really
harsh, would lead to such melodic phrases as
d d, d, which would be simply impossible for
an unaccompanied voice. If in the third ex-
ample d" were held throughout, and the ac-
companying voices sang the minor chord, wa
should get the succession P'a'd", g' 6/ d*\ f b/
d'\ e* c" c\ which amounts to a modulatioa
into the minor of the dominant, instead of
into the subdominant. Whether such is pos-
sible depends on the preceding chords. As/*
does not occur on the Harmonical, I played
Ex. 3 on my just concertina in A^ major as d'
fx't W, e/ g^t 6/, d,' g^l 6/, Cj'J a/ a\ and
found that such chords produced the best effect
of all for this isolated phrase. — TraJisl(UoT.\
Digitized by V^jOOQlC
CHAP. XVII. CHORDS OF THE SEVENTH. 341
singing might be doubtful. The rest are much more clearly distinguished from the
chords in the nearest adjacent keys. Nevertheless
b^-^d \ f and d \ f -a'\}
6 33 3 2 8
5 TT irr f
axe easily confused with
&, I d, -/ and d-f I a»t>
U i i ^
I
of which the former belongs to Ai minor, and the latter to E^\} major or minor,
where A^ minor is the minor key nearest related to C major, and E^\} major is the
major key nearest related to C minor.
Finally when we remember that the Pythagorean minor Third ff [cents 294] ^
is nearer the superfluous second J J [cents 274] than to the normal minor Third
[cents 316] (^? = 5 X Sy [cents 294 = 316 — 22] and Jf = J-J x fJ^J [cents 294
= 274 + 20] or nearly = H ^ f 3^» ^^ requires comparatively slight changes in
intonation to convert the chord bi— d \ /into
by — d ...CjiJ and c^}[).,.di—f
6 7.'} 70 n
which belong to i^jjlj! minor and E\) minor. Hence the diminished triad bi — d | /,
by sUght changes of intonation,* never exceeding |^, can be referred to the keys
of
C major, C minor, Ai minor, F^jj^ minor, and E\} minor.
Hence although the use of the diminished triad 61 — ^ | / excludes the keys f
most nearly related to C, it allows of a confusion with more distant keys, and hence
also the characterisation of the key by these triads will not be complete without a
fourth note, converting the triad into a tetrad. This leads us to the chords of the
Seventh proper.
Chords of the Seventh.
A. Formed of two Consonant Triads,
Consonant tetrads, or chords in four parts, as shewn on p. 222^, cannot be
constructed without using the Octave of one of the tones, but dissonant tetrads are
easily constructed. The least dissonant of such chords are those in which only a
single interval is dissonant, and the rest are consonant. These are most teadily
formed by uniting two consonant triads which have two tones in common. In this
case the tones which are not in common to the two chords are dissonant to each ^
other, and the rest are consonant, so that the dissonance is comparatively un-
observed t amid the mass of consonances. Thus the triads
c + ^i -gr
on being fused give the tetrad
c + Bi—g + bi
in which the major Seventh c.&i is a dissonant interval and the other intervals
are consonant, as the annexed scheme shews : —
* [Which are made spontaneously in denariom, App. XX. sect. E. art. 18. — TranS'
equally tempered intonation, where all three latorj]
chords are absolutely identical, bat would other- f [To my sensation the dissonant tones
wise require an entire sacrifice of the feeling utterly destroy the consonance. — Translator.^
of tonality. Follow these chords on the Duo-
Digitized by
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342 CHORDS OP THE SEVENTH. pabt in.
[In cents :
c 702 gr, ei 702 bi
C 386 El 316 G 386 Bi
c 1088 6 J.]
¥
This position of the chord of the Seventh, deduced from the closest positions of
the triads, is regarded as fundamental or primary. The intervals between the
individual tones appear as Thirds, and when chords of the Seventh are formed
from the consonant triads of the scale, these Thirds will be alternately major and
minor, because consonant triads always unite a major with a minor Third. Haupt-
mann calls these chords of the Seventh which occur spontaneously in the natnial
f series of Thirds of a key
the chords of the direct system or simply direct chords. There are two kinds of
these chords. In one a minor Third lies between two major Thirds, as in the
tetrad c+61— gf+bi already cited, and similarly in / + ai— c'+ e/ in G major,
and il*[?H-c— c*b+gr in 0 minor. In the other a major Third lies between two
minor Thirds, as in
% [In cents :
1' ' ^ a 702 e'l, c' 702 g'
aj-c' + £jV-i/' ^1 316 G 386 E, 316 G
V yK.,^^y — ^ ai loiS g\]
i i i
f
^ and similarly in ei — ^ + 6, — dj in C major and /— a*|?+c— «*l> in C minor. In
this second species the dissonance is a minor Seventh, f, [roughness 25, p. 332,
Table, No. 36, cents 1018,] which is much milder than the major Seventh, '^ [ibid.
No. 37, cents 1088, roughness 42].
B. Chords of the Seventh formed of Dissonant Triads,
Other chords of the Seventh may be formed from the dissonant triads of the
key, each united with one consonant triad, and also from the two dissonant triads
themselves. By thus uniting the limiting tones of the series of chords in the key,
major: c+Ci ^g+bi— d \ f+a^ —c
and minor: c— e*b+gf-f 6,— d | /— a*l>+c
we obtain the following Chords of the Seventh in the reverted system^ or iiidirect
f tetrads :
[In cents :
1) 1^ i) g ^02 d\ bi 610 f
J' ' ^ G 386 J5, 316 D 294 F
gJtb,-d' \ f 17 996 /]
f f
33
V-
2) I [2) d 680 a,,/ 702 c'
JO. -_ D 2942^386^1, 316 C
d 996 c']
d I f -^-a^-c'
Digitized by
Google
CHAP. XVII. CHORDS OP THE SEVENTH.
3) f [3) d6joa%f'jo2e'
^.^^~" ^ D 294 F 316 A^\} 386 G
d I f-a^]}+& dgg6&]
7T
343
31 e J
V-
4) ^ [4) 6, 610/, d 680 a,
fl4 .
ty— " ^ Bi 3i6Z)294jP386il,
6 3 3 A
5) n [5) b, 610 f,d 610 a'\} f
B, 3i6D294F3i6il»t>
bi 926 a^W
The Sevenths of these chords all come pretty near to the natural Seventh ^
[cents 969], and are all smaller than the Sevenths in the chords of the Seventh
formed from two consonant triads [cents 1088 and 1018]. The principal disso-
nances in these chords are the false and imperfect Fifths bi...f, d...aif and d,..a^\},
that is, the intervals $| and ^ [p. 332, Table, Nos. 21 and 23, cents 610 and 680,
roughnesses 28 and 44]. Hence the first three of these chords of the Seventh,
g + bi — d \ ff d I / + ^1 — c, and d \ f"-a^\} + c, each of which contains only one
of these imperfect Fifths, are less harshly dissonant than the two last, each off
-which contains two of them. Such of these chords as contain a major triad,
namely —
g + fei — d I / and d | / + ^i — c
I ! I I
are about equal in dissonance to the milder chords of the Seventh in the direct
system, which contains the larger and rougher kind of Sevenths, and, at the same
time, only perfect Fifths, viz. :
ai—c + Bi—g and Bi ^ g + bi — d
The chord of the dominant Seventh g + bi— d^ \ f can be even rendered
much milder by lowering its / to//. The interval g..»fi corresponds to the
ratio VW [cents 974], which is very nearly equal to J [cents 969], being = Jx f
^i% [cents 969 + 5], or approximately i x|^^. Hence the chord gf + 61— ^ | /i
is on the verge of consonance.*
But the chord of the Seventh which contains a fialse Fifth and a minor triad,
namely No. 3 above, or
is about as rough as the tetrads of the direct system containing a major Seventh,
namely —
* [That is, allowing g h^d^f or c e^g'hb to be satisfactory imitation — especially by ears nn-
consonant. In the 53 division the player uses accustomed to the true interval, because it is so
44 degrees » 996 cents for ^.. ./, and 43 degrees much superior to the former of 44 degrees => 996
■1974 cents for g...7, and the latter is found a oent8.-*TnirM2ator.]
Digitized by VjOOQlC
344
CHOBDS OF THE SEVENTH.
PABT m.
It is carious that the first of these three tetrads oontains exactly the same intervals
as the chord of the dominant Seventh itself, g + b^ — d \ f, only in inverse order,
thus —
U
In the first the consonant portion is a minor triad, and this makes it decidedly
harsher than the second where the consonant portion is a major triad.
Here also the difference of harshness depends on the nature of the combina-
tional tones, of which those generated by the closer intervals are most distinctly
heard. These are
for ^' + 6/-(i" I /' and for d" \ /'-a»"t> + c'"
f
G G Ai
Ai d^\} a^\}
Hence one combinational tone in the first chord, and two in the second, are un-
suitable to the chord. "
The harshest chords of the Seventh are those which each contain two false
Fifths,. namely, No. ^orbi—d] Z+a/ and No. 5 or &i-d' | /-a*'t>- But the
first of these can be made much milder by a slight change in its intonation. Thus
bi — d...fi'...a' contains tones which all belong to the compound tone of G,, and
these sound tolerably well together.*
The chords of the reverted system play an important part in modulations, by
serving to mark the key precisely. The most decisive in its action is the chord of
the Seventh on the dominant of the key, that is the chord g + bi — d | / for the
tonic C. We saw (p. 341 5) that the diminished triad bi — d \ f could be adapted
by slight changes in its intonation to the keys of
IT G major, C minor, Ai minor, i^, J minor, and E^\} minor.
Of these only the two first contain the tone G, so that the chord g + bi — d \ f
can belong to no tonic but G.
The imperfect minor triad [or chord of the added Sixth] d \ /+ ^i, which,
when the intonation is correct, belongs only to the key of G major, admits of being
confused [and is in equal temperament always identified] with di — /+ ai, which
* [This is only to be taken as an approxi-
mative statement, grounded on the assumption
that the interval g to // is correctly J, in which
case the primes of the tones 6,, d,/,', o' are
the 5th, 6th, 7th, 9th partials of G. This
chord in its true formation is used on Mr.
Poole*8 double or dichordal scale F G A''Bb
cdCify the two chords being F : A^ : c = 4.l $ :
6, and C : ^, : G : '-Bb : d = 4 : 5 : 6 : 7 : 9.
^ In the text it is, in point of fact, proposed to
use B^ b in the chord C Ei G B^b d^ as an imi-
tation of Mr. Poole's natural chord, which
"^'ould be still closer than C E G A^ d^ with
the extreme sharp Sixth in place of the
natural Seventh. In fact, C : ^Bb = 969 cents,
C : B,b = 974 cents, and C : A^t «976 cents.
To test the effect of septimal intonation I
had an instrument tuned to give the chords —
BbdJ,FA,C, CE.G'Bbd, GB^d,DF'A^
perfectly (of which the second, third, and
fourth occur on the Harmonical). The effect
of the third of these chords far surpasses my
expectations, and it is beyond comparison
better than the usual chord of the Ninth with
Bb d in place of ^Bbd (for which on the Har-
monical gb^dfa^ can be played). The chord
of the Bubminor Seventh and its inversions
CE^Q 'Bb. E^G^Bbc, G "Bb c Cp »Bb ce,g
are all decidedly superior to the chord of the
dominant Seventh, with Bb in place of '£b,
and its inversions (which on the Harmonical
must be tried as ^ &, (2 / and its inversions).
The septimal minor triad G 'Bb <2 is far snpe-
rior to the Pythagorean minor triad D P A
(not on the Harmonical), or the false minor
triad D F A^^ and is not far inferior to the
true minor triad Bj F A^ or D F^ A (on the
Harmonical compare g^bbd' with gb^bd). The
septimal diminished triad ^^ G 'Bb approaches
consonance much more nearly than the nsaal
diminished triad E^ G Bb (play 6, d/ on the
Harmonical). Though Poolers ascending scale
makes too great a gap between 'Bb and c, yet
by using 'Bb B as alternative tones with Bb
Bp ascending with the sharper and descending
with the flatter forms, we obtain the perfectly
melodious scales of
F G A^Bbc d e^f and f e^d^c 'Bb A^QF
'of which the first, being the ordinary scale of
i^ major, does not exist on the Harmonical,
which has no Bb, but the second can be played
upon it). These facts shew the acoustic possi-
bility of a septimal theory of harmony, which
would include the tertian, or ordinary harmony
of just intonation. — Translator>1
^'
Digitized by VjOOQIC
CHAP. XVII. CHOKDS OF THE SEVENTH. 345
belongs to the keys of Ai minor, J^ major, and £J> major. This confusion is not
entirely obviated by adding the tone c, and the consequence is the chord of the Seventh
d I / + a, — c is usually employed only in alternation with the chord of the
dominant Seventh in the cadence, where it distinguishes G major from C minor.*
But the addition of the tone b^ to the triad d | / + ^i [as fej — ei \ f '\- a{] is
characteristic, because this last can at most be confused with 5| | ^1 — / + ai,
which belongs to A^ minor. The chord h] — d \ f -^ a^, however, sounds com-
paratively harsh in every position for which a, is not the highest note, and hence
its application is very limited. It is often united with the chord of the dominant
Seventh as a chord of the Ninth, thus g + b^ — d' | / + a/, in which g and ai'
must remain the extreme tones. More upon this hereafter.
In the key of C minor, the triad d \ f ^ cl^\} would, in just intonation, be
characteristic, but yet it is easily confused with other chords. Thus
H
d \ f — a^\} [in cents d 294/316 a^\}] belongs to G minor
^1 ~ / I a|? [in cents d 316 / 294 a\}] to E\} major and E\} minor
6 3 2
S ST
d — /^..{/ij [in cents d 316/* 274 <7iJ] to A minor
d^.., e,J -gfj [in cents d^ 274 eijf 316 ^ to Fjj^ minor.
75 r.
The addition of the tone G in the first chord of the Seventh above, thus
^ I / - ct*|? 4- c, would decisively exclude the key i^ minor only, and the addition
of the tone ^i (which in tempered intonation is confused with b or c^\yj would also
readily be adapted to all the above keys. Thus altered it becomes the chord 61 — IT
d I f — a^\}, and is called th^ chord of the diminished Seventh, which on keyed
instruments appears as a series of minor Thirds. In reality a Pythagorean minor
Third or else an acute augmented Second separates the normal minor Thirds, thus :
bi-d \ f-^a'\) . b.-d I /-a»t>...^^i
[In cents: 316294 316 274 316 294316 274]
Since the three intervals |, -|f, and J-{ [cents 316, 294, 274] differ but very
slightly [by 20, 22, 42 cents respectively], they are readily confused,t and we obtain
the following, nearly identical, series of tones :
bi— d I / — a*l>...Z>i [in cents 6, 316 ^ 294/316 a*|:> 2746,] in C minor
6 33 6 75
b 1 d" f ... gi$ ^ b \m cents b 294 d 316 f 274 gjj^ 316 b]inA minor
b — d^ ... 61J — (/jf I 5 [in cents b 316 d^ 274 ei# 316 gf# 294 6] in i^ minor
6 7 5 6 3 2
c^l> ... ^1— / I a\} - c*t> [in cents c^\} 274 di 316/294 a\} 316 c^\}] in E\} minor.
75 6 3 2 6
* [This arises entirely from temperament, hence in all written music they are treated as
which identifies the two chords d |/+a, — c, identical. The four following forms of the
and rf, -/+ ai—c. Listen to the difiference on chord (of which only the first can be played on
the Harmonioal.— TmrwZator.] the Harmonical) are struck with absolutely the
t [It is quite impossible to confuse them same digitals on a pianoforte. Trace them on
in the just intonation of any harmonic inter- the Duodenarium, App. XX. sect. E. art. 18. —
▼als, but they are absolutely identified in Translator.}
equally tempered intonation as 300 cents, and
Digitized by
Google
S46 CHOEDS OF THE SEVENTH. part m.
These chords of the diminished Seventh do not form so sharp a contrast with
the consonances in the minor mode, as the corresponding chord does in the major
mode, although if the intonation is just the dissonance is always extremely harsh
and cutting.* When they are followed by the triad of the tonic, the two chords
together contain all the tones of the key, and hence completely characterise it.
The chief use of the chord of the diminished Seventh is due to its variability,
which readily leads the harmony into new keys. By merely subjoining the minor
chords of i^. A, C or ^ the new key will be completely established. It is readily
seen that this series of keys itself forms a chord of the diminished Seventh, the
tones of which lie a Semitone higher than those of the given chord. This gives a
simple means of recollecting them.f
The comprehension of the whole of a key by these chords is of special impor-
tance in the cadence at the end of a composition or of one of its principal sections.
% For this purpose we have also to determine what fundamental primary tones can
be represented by these chords of the Seventh.
It is clear that a single musical tone can never be more than imperfectly
represented by the tones of a dissonant chord. But as a general rule some of
these tones can be accepted as the constituents of a musical tone. This gives rise
to a practically important difference between the different tones of such a chord.
Those tones which can be considered as the elements of a compound tone, form a com-
pact, well-defined mass of tone. Any one or two other tones in the chord, which do
not belong to this mass of tone, have the appearance of unconnected tones, acci-
dentally intruding. The latter are called by musicians the dissonances or the dis-
sonant notes of the chord. Considered independently, of course, either tone in a
dissonant interval is equally dissonant in respect to the other, and if there were only
two tones it would be absurd to call one of them only the dissonant tone. In
the Seventh c.bi, cis dissonant in respect to &i, and bi in respect to c. In the
% chord c + ^1 — gf + 61 the notes c -h 61 — gf form a single mass of tone corresponding
to the compound tone of c, and bi is an unconnected tone sounding at the same
time. Hence the three tones c + ^1 — ^have an independent steadiness and compact-
* [As the ratios 4 : 5 : 6 : 7 are the justifi- four keys into which a slight alteration of the
cation of the chord of ihe dominant Seventh pitches of the notes in the chord of the dimi-
4 : 5 : 6 : 7j, so the ratios 10 : 12 : 14 : 17 are nished Seventh will make it fit, are FJl, A^ C,
the justification of the chord of the diminished E\}. These notes, however, do not form a chord
Seventh 10 : 12 : 14J : 17^ taking the ratios with the same intervals, but Ft 294 A 294 C
of No. 5, p. 343&, and commencing with 10. 294 ^b, that is a succession of Pythagorean minor
That is, e"3i6 ^'267 '6"b 336 *'d'"b, which Thirds, the result of which is simply hideooA.
can be played on the Harmonical, is the just It is only in equally tempered intonation in
chord of the diminished Seventh, for which which the four forms above given of the chord
the form of ordinary just intonation is e''3i6 of the diminished Seventh agree absolutely in
g^'2f)^ &''b3i6 d'b, which must be played as sound, though they di£Fer in writing, because
^^316 2)/294 /'316 a^"\> on the Harmonical, signs originally intended for other tempera-
an intensely harsh chord, for which is played ments (as the Pythagorean, meantone, or other
in equal temperament /300 6^300 /'300 a"b. which distinguished Cjf and Db, but did not
IT Observe that the dimini&ed Seventh 10 : 17 distinguish the conmia) have continued in use,
has 919 cents, the diminished Seventh of with confounded meanings. This is precisely
ordinary just intonation 10 : 17^ has 926 the same as in ordinary English speUing,
cents, 7 too sharp ; while in equally tempered where combinations of letters originally repre-
intonation it is only 900 cents or 19 too flat. senting very different sounds, are now con-
And the tempered major Sixth is repre- fused, as I have demonstrated historically in
sented by the same interval of 900 cents, my Early English Pronunciation. In equally
which is 16 cents too sharp. It is remarkable tempered intonation the roots ft 300 a 300 c
that any sense of interval or tonality survives 300 ^b do also form a chord of the diminished
these confusions. Of course the introduction Seventh. But this does not end the confusion,
of the 17th harmonic into the scale is a sheer for the key oift may betaken as that of grb, of
impossibility. The chord 10 : 12 : 14J : 17^ a as that of &bb, c as that of &S , eb as that of
is simple noise. The chord 10 : 12 : 14 : 17 dt ,and these four roots, ab, &b \>^ht^dt , being
which I have tried on Appunn's tonometer in played with the same digitals represent the
its inversions, is a comparatively smooth dis- same chord, but the four keys are now totally
cord superior to the tempered form. But the unrelated. What then becomes of the feeling
chord is really due to tempered intonation of tonality? and how are we to feel the right
only. For further notes on this chord see amid this mass of wrong, as Sir George Mac-
App. XX. sect. E. art. 23, and sect. F. towards farren says we can, and as I must thereforo
end of No. 7.— SVansIator.] suppose he himself hjM sacceeded in doing ? —
t [It is correctly stated in the text that the Translator,'] Digitized by ^OOQIC
CHAP, XVII. DISSONANT NOTES IN CHOBDS OF THE SEVENTH. 347
ness of their own. But the unsupported solitary Seventh bi has to stand against
the preponderance of the other tones, and it could not do so either when executed
by a singer, or heard by a listener, unless the melodic progression were kept very
simple and readily intelligible. Consequently particular rules have to be observed
for the progression of the part which produces this note, whereas the introduction
of c, which is sufficiently justified by the chord itself, is perfectly free and unfettered.
Musicians indicate this practical difference in the laws of progression of parts by
terming by alone the dissonant note of this chord; and although the expression
is not a very happy one, we can have no hesitation in retaining it, after its real
meaning has been thus explained.
We now proceed to examine each of the previous chords of the Seventh with a
view to determine what compound musical tone they represent, and which are their
dissonant tones.
1. The chard of the dominant Seventh, g + bi—d\f, contains three tones f
belonging to the compound tone of G, namely^, ^j, and (2, and the Seventh / is the
dissonant tone. But we must observe that the minor Seventh ^ . . / [or *^ = |^ x J, or
cents 996 = 969 + 27] approaches so near to the ratio i [cents 969] which would be
almost exactly represented by ^ . . ./, , [cents 974] , that / may in any case pass as the
seventh partial tone of the compound G* Singers probably often exchange the/
of the chord of the dominant Seventh for/i, t partly because it usually passes into e|,
partly because they thus diminish the harshness of the dissonance. This can be
easily done when the pitch of/ is not determined in the preceding chord by some
near relationship. Thus if the consonant chord g-\'bi—d had already been struck
and then / were added, it would readily fall into/i, [^h.Q,t is ^/j because/ is to itself
unrelated to g, b^j or d.t Hence, although the chord of the dominant Seventh
is dissonant, its dissonant tone so nearly corresponds to the corresponding partial
tone in the compound tone of the dominant, that the whole chord may be very well
regarded as a representative of that compound. For this reason, doubtless, the f
Seventh of this chord has been set free from many obHgations in the progression
of parts to which dissonant Sevenths are otherwise subjected. Thus it is allowed
to be introduced freely without preparation, which is not the case for the other
Sevenths. In modem compositions (as E. Wagner's) the chord of the dominant
Seventh not unfrequently occurs as the concluding chord of a subordinate section
of a piece of music.
The chord of the dominant Seventh consequently plays the second most impor-
tant part in modem music, standing next to the tonic. It exactly defines the key»
more exactly than the simple triad g + b^—d, 01 than the diminished triad bi—d | /.
As a dissonant chord it urgently requires to be resolved on to the tonic chord,
which the simple dominant triad does not. And finally its harmoniousness is so
extremely little obscured, that it is the softest of all dissonant chords.§ Hence we
could scarcely do without it in modem music. This chord appears to have been
discovered in the beginning of the seventeenth century by Monteverde. f
2 . The chord of the Seventh upon the Second of a major scale, d \ /+ a, — c, has
three tones,/, a^, c, which belong to the compound tone of F. When the intona-
tion is just, d is dissonant with each of the three tones of this chord, and hence must
♦ [It has, however, a very different effect on such points) that /, is very remote indeed
on the ear. — Translator.'] from g. — Translator.]
t [Here /, must be considered as the repre- § [As we hear it only in tempered music as a
sentative of y. Singers would not naturally rule, with the harsh major Third, which makes
take such a strange artificial approximation as the major triad almost dissonant, the addition
/,, unless led by an instrument. Unaccom- of the dominant Seventh increases the harsh-
panied singers could only choose between / and ness surprisingly little. But in just intonation
y, and singers of unaccompanied melodies are g b^df is markedly harsher than g b^ d % as
said often to choose '/ when descending to e. I have often had occasion to observe in Ap-
What is the custom in unaccompanied choirs, punn's tonometer, where g b^ d can be left
which have not been trained to give /, has, so sounding, and / suddenly transformed to */ and
far as I know, not bepn recorded. — Translator.] conversely. On the Harmonical we must oom-
% [And '/ is, but/i again is not. It will be pare g b^d f with ce^g ^5b, and that in aU
seen by the Duodenarium (App. XX. sect. E. their inversions and positions. — Translator,]
art. 18) (which should be constantly consulted * Digitized by V^OOQIC
348 DISSONANT NOTES IN CHOKDS OP THE SEVENTH, part in.
be regarded as the dissonant note. This would make the fundamental position of
this chord to be that which Eameau assigned, making/the root, thus :/+ ^i — c ...fZ,
which is a position of the Sixth and Fifth, and the chord is called by Eameau the
cJiord of the great Sixth [grande Sixte, in English * added Sixth']. This is the
position in which the chord usually appears in the final cadence of C major. Its
meaning and its relation to the key is more certain than that of the false minor
chord d \ f+a^^ mentioned on p. 340a, which as executed by a singer or heard by
a listener is readily apt to be confused with di —f + ai in the key of Ai minor.
By changing d | /+«! into d^ — / + ai we obtain a minor chord, to which there
will be a great attraction when the relation of ^ to ^ is not made very distinct.
But if we were to change d into di in the chord d \ /H-ai— c, thus producing
di — /+ «! — c, although di would be consonant with /and ai it would not be so
with c ; on the contrary, the dissonance dy-.x' [p. 332, No. 36, cents 1018, rough-
f ness 25] is much harsher than d...c\ [ibid.. No. 34, cents 996, roughness 23,
much the same as the other] , and, after all, it would be only the tone ai which
would enter into the compound tone of d^, so that, notwithstanding this change, /,
which contains three tones of the chord in its own compound tone, would predomi-
nate over d,, which has only two. In accordance with this view, I find the chord
/-h ^i — c...{? when used on the justly-intoned harmonium, as subdominant of C
major, produces a better effect than/+ ai—c.di.
3. The corresponding chord of the Seventh on the Second of the minor scale^
d \f— a^\} -h c, has only one tone, c, which can be regarded as a constituent of the
compound tone of either / or a^\}. But since c is the third partial of / and only
the fifth partial of a*[7, / as a rule predominates, and the chord must be regarded
as a subdominant chord /— a^|? + c with the addition of dissonant d. There is
still less inducement to change d into di in this case than in the last.
4. The chord of the Seventh on the Seventh of the major scale, 61 — d | /+ ^i,
^ contains two tones, bi and d, belonging to the dominant g, and two others, /, and
ai , belonging to the subdominant /. Hence the chord splits into two equally im-
portant halves. But we must observe that the two tones / and Ay approach very
closely to the two next partial tones of the compound tone of (7. The partials of
this compound tone from the fourth onwards may be written —
g-]-bi-d..,fi.,.g...a
4 5 6 7* 8 9
Hence the chord of the Ninth g + b^—d \ f+a^ may represent the compound tone
of the dominant g, provided that the similarity be kept clear by the position of the
tones, g being the lowest and a^ the highest ; it is also best not to let / [standing
for y] fall too low. Since a is the ninth partial tone of the compound g, which is
very weak in all usual quahties of tone, and is often inaudible, and since there
is the interval of a comma between a and ax , and also between /j and / [but y and /
% differ by 27 cents], care must be taken to render the resemblance of the chord of
the Ninth to the compound tone of g, as strong as possible, by adopting the device
of keeping a, uppermost, and then tiie use of /, a,, for /i, a, [meaning y a] wiU
not be very striking. In this case / and a^ must be considered as the cUssonant
notes of the chord of the Ninth g + bi—d \ f+ai, because although they are very
nearly the same, they are not quite the same, as the partial tones of G. No pre-
paration is necessary for the introduction of a^ into the chord, for the same reasons
that / is allowed to be introduced into the chord of the dominant Seventh,
g+bi—d I / without preparation. Lastly, some of the tones of the pentad chord
of the Ninth may be omitted, to reduce it to four parts ; for example, its Fifth, as
in g + bi ..f+ai, or its root, as in bi^d \ /+«!. If only the order of the tones is
preserved as much as possible, and especially the a^ kept uppermost, the chord will
always be recognised as a representative of the compound tone of G.
* (That is, supposing /i to be used for so that the above chord represents gbid^fg a.
y, as abready explained, see p. 347<J, note f, — Trcmslaior.^
Digitized by V^OOQIC
CHAP. XVII. DISSONANT NOTES IN CHOKDS OP THE SEVENTH. 349
This seems to me the simple reason why musicians find it desirable to make ai
the highest tone in the chord bi—d\ /H-ap Hauptmann, indeed, gives this as a
rule without exception, and assigns rather an artificial reason for it. The ambiguity
of the chord will thus be obviated as far as possible, and it receives a clearly in-
telhgible relation to the dominant of the key of C major, whereas in other positions
of the same chord there would be too great a .chance of confusing it with the sub-
dominant of A I minor.* When the intonation is j ust , the chord gr + 6 , — 6? . . ./i . . . a,f
which consists (very nearly indeed) of the partial tones of the compound tone of g,
sounds very soft, and but slightly dissonant ; the chord of the Ninth in the key
of C major, g-^bi-^d' l/'+a/, and the chord of the Seventh in the position
hi—d' IZ+a/, sound somewhat rougher, on account of the Pythagorean Third
d' I/, and the imperfect Fifth ^'...a/, but they are not very harsh. If, however,
a/ is taken in a lower position, they become veiy rough indeed.
The chord of the Seventh hi—d [/+«! and the following triad c-he^—g, as ^
already observed, contain all the tones in the key of C major, and hence this
chordal succession is extremely well adapted for a brief and complete characterisa-
tion of the key.
5. The chord of the diminished Seventh , bi—d\ f—a^\}, and the minor chord
c — e^\}+g, have the last mentioned property for the minor key of C, and for this
reason as well as for its great variability (p. 345^) it is largely, perhaps far too
largely (p. 320^^), employed in modem music, especially for modulations. It con-
tains no note which belongs to the compound tone of any other note in the chord,
but the three tones bi—d\ /may be regarded as belonging to the compound tone
of ^, so that it also presents the appearance of a chord of the Ninth in the form
g+bi—d I/— a'b- It therefore imperfectly represents the compomid tone of the
dominant, with an intruded tone a*b, and/ anda*t> ^^J therefore be regarded as its
dissonant tones. But the connection of the three tones bi-—d | / with the compound
tone of g is not so distinctly marked as to make it necessarjf to subordinate the pro- «r
gression of the tones/ and a^\} to that of bi and d. At least the chord is allowed
to commence without preparation, and it is resolved by the motion of all its tones
to those tones of the scale which make the smallest intervals with them, for its
elements are not sufficiently well connected with one another to allow of wide steps
in its resolution.
6. The chords of the major Seventh in the direct system of the key, as
/-t-ai— c+Ci and c+«i— g^ + ^i in C major, and a'b+c— e^b+S' ^^ ^ minor, as
already remarked, mainly represent a major chord with the major Seventh as dis-
sonant tone. The major Seventh forms rather a rough dissonance, and is decidedly
opposed to the triad below it, into which it will not fit at all.
7. The chords of the minor Seventh in the direct system of the key, as
ai — c + ex—g and ex—g + bi—df give greatest prominence to the compound tone
of their Thirds, to which their bass seems to be subjoined. Thus c-f e| —g...ai is
the compound tone of c with an added a,, and g-{-bx—d...ei is the compound tone m
of g with an added Cj. But since c + Cj— {/ and g-\-bx—d, being the principal
triads of the key, are constantly recurring, this addition of a^ and 6, respectively
gives by contrast great prominence to these tones ; moreover, the a, and e^ in these
chords of the Seventh are not so isolated as the d'uid\ f-^-a^ — c, where d has no true
Fifth in the chord. The a, in aj— c+Cj-gf has the Fifth e,, and even the
Seventh gt which belongs to its compound tone ; and in the same way the b^ and
* [The rootless chord of the Ninth on the pare its effect with that of the next three
dominant of C major is b^—d \ f+a^^ and the chords as given in the text. — Translator,']
snbdominant of A^ minor is 6 | dj— /+a„ J [The tone gj of course represents the
which would not be confused with the former third partial of a,. Does the Author mean
in just intonation, but in equal temperament that the acute minor Seventh g represents the
is identical with it. — Translator,] seventh partial '^| for which it is 49 cents, or
f [This is the form in which the Author about a quarter of a Tone too sharp? The
was obliged to play it on his instrument, which usual minor Seventh ^, has been allowed to do
had /„ see p. 317c, note, but not Y. On the so, although 27 cents too sharp. Perhaps the
Harmonical play c + c,— y...'6l)...o and com- expression * even the Seventh' (alien/alls auch
Digitized by V^jOOQlC
35© PKOGEESSION OP PAETS. pabt hi.
dotei—g+bi—d may be considered to belong to the compound tone of e,. Hence
the tone a^ in the first and Ci in the second are not necessarily subject to the laws
of the resolution of dissonant notes.
Writers on harmony are accustomed to consider the normal position of all these
chords to be that of the chord of the Seventh, and to call the lowest tone its root.
Perhaps it would be more natural to consider c + e, —g...ai as the principal position
of the chord ai— c+^i— gr and c as its root. But such a chord is a compound
tone of c with an inclination to aj , and in modulations this intrusion of the tone
of ai is utilised for proceeding to those chords related to aj which are not related
to the chord c+e, —g, for example to d^ — /+ai. In the same way we can proceed
from ^ + 6,— d...ei to ai— c+«i, which would be a jump from ^-f 6,— (^. For
modulation, therefore, the a^ and ei are essential parts of these chords respectively,
and in this practical light they might be called the fundamental tones of their
I respective chords.
8. The chord of the Seventh on the tonic of the minor hey, c-e^b+f^-f ^i, is
seldom used, because b^ in the minor key belongs essentially to ascending motion,
and a resolved Seventh habitually descends. Hence it would be always better to
form the chord c— e*t)+^ — 6^l>, which is similar to the chords considered in No. 7.
CHAPTER XVm.
LAWS OF THE PBOGRESSION OF PARTS.
Up to this point we have considered only the relations of the tones in a piece of
«- music with its tonic, and of its chords with its tonic chord. On these relations
depends the connection of the parts of a mass of tone into one coherent whole.
But besides this the succession of the tones and chords must be regulated by natural
relations. The mass of sound thus becomes more intimately bound up together,
and, as a general rule, we must aim at producing such a connection, although,
exceptionally, peculiar expression may necessitate the selection of a more violent
and less obvious plan of progression. In the development of the scale we saw that
the connection of all the notes by means of their relation to the tonic, if originally
perceived at all, was at most but very dimly seen, and was apparently replaced by
the chain of Fifths ; at any rate, the latter alone was sufficiently developed to be
recognised in the Pythagorean construction of tonal systems. But by the side of
our strongly developed feeling for the tonic in modem harmonic music, the necessity
for a linked connection of individual tones and chords is still recognised, although
the chain of Fifths, which originally connected the tones of the scale, as
If f±c±g± d ±a±e±b,
has been interrupted by the introduction of perfect major Thirds, and now appears
as
/± o±^g ±d...di±^i±:^i±, *i-
The musical connection between two consecutive notes may be effected :
I. By the relation of their compound tones.
This is either :
a.) direct, when the two consecutive tones form a perfectly consonant interval,
in which case, as we have previously seen, one of the clearly perceptible partial
tones of the first note is identical with one of the second. The pitch of the follow-
die Septime) is intended to shew that this — gf, and the chord of a is ||a-4-cS — e-^. Bat
view is rather too loose. In eqaal tempera- this is mere confosion. — TranslaiorJ]
ment, indeed, the dissonant chord is g a-c + 6
Digitized by
^.joogle
CHAP. XVIII. MUSICAL CONNECTION OP TONES. 351
ing compound tone is then clearly determined for the ear. This is the best and
surest kind of connection. The closest relationship of this kind exists when the
voice jumps a whole Octave ; but this is not usual in melodies, except with the
bass, as the alteration of pitch is felt to be too sudden for the upper part. Next
to this comes the jump of a Fifth or Fourth, both of which are very definite
and clear. After these follow the steps of a major Sixth or major Third, both of
which can be readily taken, but some uncertainty begins to arise in the case of
minor Sixths and minor Thirds. Esthetically it should be remarked, that of all
the melodic steps just mentioned, the major Sixth and major Third have, I might
almost say, the highest degree of thorough beauty. This possibly depends upon
their position at the limit of clearly intelligible intervals. The steps of a Fifth or
Fourth are too clear, and hence are, as it were, drily intelligible; the steps of
minor Thirds, and especially minor Sixths, begin to sound indeterminate. The
major Thirds and major Sixths seem to hold the right balance between darkness ^
and light. The major Sixth and major Third seem also to stand in the same
relation to the other intervals harmonically.
b.) indirect, of the second degree only. This occurs in the regular progression
of the scale, proceeding by Tones or Semitones. For example :
c ... d d ... $1 Bi f
^IT ^"5^ ^c^
The whole major Tone c ... d proceeds from the Fourth to the Fifth of the
auxiliary tone G, which Eameau supposed to be subjoined as the fundamental bass
of the above melodic progression. The minor Tone d...e\ proceeds from the Fifth
to the major Sixth of the auxiHary tone (7, and the Semitone 61 .../ from the major
Third to the Fourth of the auxiliary tone C But in order that these auxiliary
tones may readily occur to both singer and hearer, they must be among the f
principal tones of the key. Thus the step a^.^.h^ in the major scale of G causes
the singers a little trouble, although it is only the interval of a major Tone, and
could be easily referred to the auxiliary tone E^, But the sound of ei is not so firm
and ready in the mind, as the sound of G and its Fifth G and Fourth F. Hence the
Hexachord of Guide of Arezzo, which was the normal scale for singers throughout
the middle ages, ended at the Sixth.* This Hexachord was sung with different
pitches of the first note, but always formed the same melody :
m
Be
Mi
Fa
Sol
La
either
G
A
B
C
D
E
or
C
D
E
F
G
A
or
F
G
A
B\}
C
D
So that the interval Mi.., Fa always marked the Semitone. t
For the same reason Eameau preferred, in the minor scale, to refer tlie steps ^
d...e't) aiid e^[}...f to G and G as auxiliary tones, rather than to B}}, the Seventh of
* For the same reason d'Alembert explains certainly ased for training singers in meantone
the limits of the old Greek heptachord, by temperament. It could not have been used for
means of two connected tetraohords — just intonation, because the melody cdef is
K Q g ^ f Q a assumed to be identical with^ a 6 cin the same
'' "* *' '" scale, whereas in just intonation c 204 d 182 «»
in which the step a ... & is avoided. But this 112/ and ^ i82a| 20461 112c' are different. For
explanation would only suit a key in which c an excellent account of the Hexachord see
was the tonic, and this ¥ras probably not the Mr. Bockstro's article * Hexachord,' in Grove's
case for the ancient Greek scale. Dictionary. To shew, however, how intona-
t [Prof. Helmholtz leaves the intonation tions are mixed up, it may be observed that he
unmarked. Guido d' Arezzo, the presumed in- illustrates the use of the Hexachord in * Beal
ventor of the Hexachord, is said to have intro- Fugue ' by an example of Palestrina, who lived
duced it about 1024 a.]>., that is long before in the sixteenth century, and is often credited
meantone temperament existed. Hence we with just intonation, but who being junior to
must assume Pythagorean intonation (see p. Salinas and Zarlino must have used meantone
3i3fi). Yet in later times the Hexachord was temperament.— Tratwiator.]
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352 MUSICAL CONNECTION OF TONES. part in.
the descending scale,* which had not a sufficiently close relationship to the tonic,
and hence was not well enough impressed on the singer's mind for such a purpose.
Taking g and c, the Octaves of G and C as the auxihary tones, the motion in
d...e^\} is from the Fourth below g to the major Third below it, and in e^\}...f from
the major Sixth below c to the Fifth below it. On the other hand, it is impossible
to reduce the step a^\}...bx [cents 274] in the minor scale to any relationship of the
second degree. [See p. 301c, note *.] It is therefore also decidedly unmelodic
and had to be entirely avoided in the old homophonic music, just as the steps
of the false Fifths and Fourths, as 6 1 . . ./ [cents 6 1 o] , or / . . . 5 / [cents 590] . Hence
the alterations in the ascending and descending miaor scales already mentioned.
In modem harmonic music many of these difficulties have disappeared, or
become less sensible, because correct harmonisation can exhibit the connections
which are absent in the melodic progression of an unaccompanied voice. Hence
% also it is much easier to take a part at sight in a harmony, written in pianoforte
score, which shews its relations, than to sing it from an unconnected part. The
former shews how the tone to be sung is connected with the whole harmony, the
latter gives only its connection with the adjacent tones.f
2. Tones may be connected by their ajp'proxhnation in pitch.
This relation has been considered previously with reference to the leading note.
The same holds good for the intercalated tones in chromatic passages. For
example, if in C major, we replace C...D hj C.CJff ..D, this Cj has no relation
either of the first or second degree with the tonic (7, and also no harmonic or
modulational significance. It is nothing but a step intercalated between two
tones, which has no relation to the scale, and only serves to render its discontinuous
progression more Uke the gliding motion of natural speech, or weeping or howling.
The Greeks carried this subdivision still further than we do at present, by splitting
up a Semitone into two parts in their enharmonic system (p. 265a). Notwithstand-
^ ing the strangeness of the tone to be struck, chromatic progression in Semitones
can be executed with sufficient certainty to allow it to be used in modulational
transitions for the purpose of suddenly reaching very distant keys.
ItaHan melodies are especially rich in such intercalated tones. Investigations
of the laws^ under which they occur will be found in two essays of Sig. A. Basevi.J
The rule is without exception that tones foreign to the scale can be introduced only
when they differ by a Semitone § from the note of the scale on to which they
resolve, while any tones belonging to the scale itself can be freely introduced
although out of harmony with the accompaniment, and even requiring the step of
a whole Tone for their resolution.
* [The Anthor writes £b, and calls it ' the tones, the small 90 and the large 1 14 cents, and
Seventh of the descending scale ' of C minor, the rule was to make the Semitone closest to the
which, however, is B'b, and this answers for note to which it led, thus cii4c](9od, (i 114
the first interval (2 ... e^ b, owing to 6^ b + d^ and db 9od, And this notation was retained even
^ e^b ± 6*b ; but it does not answer to the second in meantone temperament, where the relations
interval e'b .../ as / ...6*b is dissonant, and it were reversed, as c 76 cj 117 d^ d 76 db 117 c;
would not do to use &b, although 6b ±/ is con- but practically this made no difference except
sonant, because e'b ... &b is dissonant. Buta'b . to the singer, as the player had only one Semi-
would do, as we see from a'bie^b, /— a'b. tone at command. Ttus writing is still con-
Hence if the text gives Rameau's notes, he must tinned in equal temperament, although the
have been misled by temperament. — Trans- two Semitones are now equalised as 100 cents,
lator.] thus c 100 cH 100 d, and d 100 db 100 c. Bat
f [Hence any means of shewing the relation in just intonation We have Semitones of varioos
of each tone to the tonic of the moment, as in dimensions, c 114 eft 90 c2, c 112 d^b 92 d, c 92
the Tonic Solfa system, materially facilitates c,]( 112 d, c 90 db 114 d, c 70 CjS 134 d ; which
sight-singing, as perhaps the use of the duode- of these is the player to play, or the singer to
nal (App. XX. sect. E. art. 26) when thoroughly sing (a question of importance when each part
understood might also do. — Translator.'] is sustained by many unaccompanied voices) ?
X Introduction d un nouveau System Practically the player will take the most handy
d*Harmoniet traduit par L. DeUtre ; Florence, interval, and the singers must arrange in re-
1855. Studj sulV Armoniat Firenze, 1865. hearsal, but would possibly take c 92 c^t 1 12 d,
§ [Of course those who laid down the rule d 92 d'b 112 c, as these are the intervals used
thought only of a tempered Semitone. But in in modulation from / to /,S to the dominant,
Pythagorean temperament there were two Semi- and 6, to 6 b to the Bubdominant. — Translator.]
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CHAP. xvm. EESOLUTION OF DISSONANCE. 353
In the same way steps of a whole Tone may be made, provided the notes lie in
the scale, when they serve merely to connect two other tones which belong to
chords. These are the so-called passing or changing notes. Thus if while the
triad of C major is sustained, a voice sings the passage c...d...ei.,f...g, the two
notes d and / do not suit the chord, and have no relation to the harmony, but
are simply justified by the melodic progression of the single voice. It is usual to
place these passing notes on the unaccented parts of the bar, and to give them a
short duration. Thus in the above example c, e,, ^ would fall on accented parts
of the bar. Then d is the passing note between c and ei and /between e^ and g.
It is essential for their intelligibihty that they should make steps of Semitones or
whole Tones. They thus produce a simple melodic progression, which flows on
freely, without giving any prominence to the dissonances produced.
Even in Che essentially dissonant chords the rule is, that dissonant tones which in-
trude isolatedly on the mass of the other tones must proceed in a melodic progression, f
which can be easily understood and easily performed. And since the feeling for
the natural relations of such an isolated tone is almost overpowered by the simul-
taneous sound of the other tones which force themselves much more strongly on
the attention, both singer and hearer are thrown upon the gradual diatonic pro-
gression as the only means of clearly fixing the melodic relations of a dissonant
note of this description. Hence it is generally necessary that a dissonant note
should enter and leave the chord by degrees of the scale.
Chords must be considered essentially dissonant, in which the dissonant notes
do not enter as passing notes over a sustained chord, but are either accompanied
by an especial chord, differing firom the preceding and following chords, or else are
rendered so prominent by their duration or accentuation, that they cannot possibly
escape the attention of the hearer. It has been already remarked that these chords
are not used for their own sakes, but principally as a means of increasing the
feeling of onward progression in the composition. Hence it follows for the motion IT
of the dissonant note, that when it enters and leaves the chord, it will either ascend
on each occasion or descend on each. If we allowed it to reverse its motion in the
second half, and thus return to its original position, there would seem to have been
no motive for the dissonance. It would in that case have been better to leave the
note at rest in its consonant position. A motion which returns to its origin and
creates a dissonance by the way, had better be avoided ; it has no object.
Secondly it may be laid down as a rule, that the motion of the dissonant note
should not be such as to make the chord consonant without any change in the
other notes. For a dissonance which disappears of itself provided we wait for the
next step, gives no impetus to the progress of the harmony. It sounds poor and
unjustified. This is the principal reason why chords of the Seventh which have
to be resolved by the motion of the Seventh, can only permit the Seventh to
descend. For if the Seventh ascended in the scale, it would pass into the octave
of the lowest tone, and the dissonance of the chord would disappear. When Bach, %
Mozart, and others use such progressions for chords of the dominant Seventh, the
Seventh has the effect of a passing note, and must be so treated. In that case it
has no effect on the progression of the harmony.
The pitch of a single dissonant note in a chord of many parts is determined
with greatest certainty, when it has been previously heard as a consonance in the
preceding chord, and is merely sustained while the new chord is introduced. Thus
if we take the following succession of chords :
0...d... g + b^
c +«!— g + bi
the hi in the first chord is determined by its consonance with G. It simply remains
while the tones c and Bi are introduced in place of G and d, and thus becomes a
dissonance in the chord of the Seventh c-^ei— g+b^. In this case the dissonance
is said to be prepared. This was the only way in which dissonances might be
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354 RESOLUTION OF DISSONANCE. pabt ra.
introduced down to the end of the sixteenth century. Prepared dissonances produce
a peculiarly powerful effect : a part of the preceding chord lingers on, and has to
be forced from its position by the following chord. In this way, an effort to advance
against opposing obstacles wliich only slowly yield, is very effectively expressed.
And for the same reason the newly introduced chord (c 4 Ci — 5^ in the last
example) must enter on a strongly accented part of the bar ; as it would otheonrise
not sufficiently express exertion. The resolution of the prepared dissonance, on
the contrary, naturally falls on an unaccented part of the bar. Nothing sonnds
worse than dissonances played or sung in a dragging or uncertain manner. In
that case they appear to be simply out of tune. They are, as a rule, only justified
by expressing energy and vigorous progress.
Such prepared dissonances, termed sttspensions, may occur in many other chords
besides those of the Seventh. For example :
'' Preparation : G .,.c + «i
Suspension : G ...c ...d
Resolution : G -\- Bi— d
The tone c is the prepared dissonance ; in the second chord, which must faXL on
an accented part of the bar, d the Fifth of G is introduced and generates the dis-
sonance c..,d, and then c must give way, and according to our second rule, must
go further from d, which results in the resolution G -{- Bi ^ d. The chords might
also be played in the inverse order, and then d would be a prepared dissonance which
was forced away by c. But this is not so good, because descending motion is better
suited than ascending motion to an extruded note. Heightened pitch always gives
us involuntarily the impression of greater effort, because we have continually to
exert our voice in order to reach high tones. The dissonant note on descending
seems to yield suitably to superior force, but on ascending it as it were rises by its
^ own exertion. But circumstances may render the latter course suitable, and its
occurrence is not unfrequent.
In the other case, especially frequent for chords of the Seventh, when the dis-
sonant note is not prepared but is struck simultaneously with the chord to which it
is dissonant, the significance of the dissonance is different. Since these unprepared
Sevenths must usually enter by the descent of the preceding note, they may be
always considered as descending from the Octave of the root of their chord, by
supposing a consonant major or minor chord having the same root as the chord
of the Seventh to be inserted between that and the preceding chord. In this case
the entrance of the Seventh merely indicates that this consonant chord begins to
break up immediately and that the melodic progression gives a new direction to the
harmony. This new direction, leading to the chord of resolution, must be empha-
sised, and hence the dissonance necessarily falls on the preceding unaccented pari
of a bar.
^ The introduction of an isolated dissonant note into a chord of several parts
cannot generally be used as the expression of exertion, but this character will
attach to the introduction of a chord as against a single note, supposing that this
single note is not too powerful. Hence it lies in the nature of things that the
first kind of introduction takes place on unaccented and the last on accented parts
of a bar.
These rules for the introduction of dissonances may be often neglected for the
chords of the Seventh in the reverted system, in which the Fourth and Second of
the scale oecur, and notes from the subdominant side are mixed up with notes from
the dominant side. These chords may also be introduced to enhance the dynamical
impression of the advancing harmony, for they have the effect of keeping the
extent of the key perpetually before the feeling of the hearer, and this object
justifies their existence.
Of several voices which are leaving the chord of the tonic C, it is quite easy
for some to pass on to notes of the dominant chord g + bi — d, and for others to
Digitized by V^OOQIC
CHAP. xvin. CHORDAL SEQUENCES. 355
proceed to the notes of the subdominant chords/ + «i — c or/ — a't> + c, as each
voice will be able to strike the new note with perfect certainty, on account of the
close relationship between the chords. When, however, the dissonant chord has been
thus formed and sounded, the dissonant notes will have the feeling for their more
distant relations obscured by the strangeness of the other parts of the chord, and
must generally proceed according to the rule of resolution of dissonances. Thus
the singer who sounds / in the chord g-^-bi^d \f, would vainly endeavour to
picture to himself the sound of the ai which is related to / with sufficient clearness
to leap up or down to it with certainty ; but he is easily able to execute the small
step of half a Tone, by which / descends to 6] in the chord ci-ei—g. But the
note g itself, on the other hand, having its own compound tone approximatively
indicated by the chord of the Seventh, has no difficulty in passing by a leap to its
related notes, as c for example, or bi to g.
In the chords hi —d \ f-\-ai and 61 —d | /— a^t), in which neither dominant nor ^
Bubdomiiiant prevails, it would not be advisable to let any note proceed by a leap.
And it would also not be advisable to pass by a leap into the chords of the
reverted system from any other chord but the tonic, because that chord alone is
related to both dominant and subdominant chords at the same time.
It is not possible to pass to chords of the Seventh in the direct system, from
another chord related to both extremities of the chord of the Seventh, and hence
in this case the dissonance must be introduced in accordance with the strict rules.
Musicians are divided in opinion as to the proper treatment of the subdominant
chord with an added Sixth, f-\-ai — c...d in C major. The rule of Eameau is
probably correct (p. 347^), making d the dissonant note, to be resolved by rising
to Bi. This is also decidedly the most harmonious kind of resolution. Modem
theorists, on the other hand, regard this chord as a chord of the Seventh on d^ and
take c as the dissonant note to be resolved by descent ; whereas when c remains, d
is quite free and may therefore even descend. cr
Chordal Sequences.
Just as the older homophonic music required the notes of a melody to be linked
together, modern music endeavours to link together the series of chords occurring
in a tissue of harmony, and it thus obtains much greater freedom in the melodic
succession of individual notes, because the natural relationship of the notes is'
much more decisively and emphatically marked in harmonic music than in homo-
phonic melody. This desire for linking the chords together was but sHghtly
developed in the sixteenth century. The great Italian masters of this period allow
the chords of the key to succeed each other in leaps which are often surprising,
and which we should at present admit only in exceptional cases. But during the
seventeenth century the feeling for this peculiarity of harmony also was developed,
so that we find Eameau laying down distinct rules on the subject in the beginning
of the eighteenth century. In reference to his conception of fundamental bass, f
Bameau worded his rule thus : ' Ths fundamental bass ma/y, as a general rule,
proceed only in perfect Fifths or Thirds, upwards or downwards.' According to
our view the fundamental bass of a chord is that compound tone which is either
exclusively or principally represented by the notes of the chord. In this sense
Bameau's rule coincides with that for the melodic progression of a single note to
its nearest related notes. The compound tone of a chord, like the voice of a
melody, may only proceed to its nearest related notes. It is much more difficult
to assign a meaning to progression by relationship in the second degree for chords
than for separate notes, and similarly for progression in small diatonic degrees
-without relationship. Hence Rameau's rule for the progression of the funda-
mental bass is on the whole stricter than the rules for the melodic progression
of a single voice.
Thus if we take the chord c+^i — gr, which belongs to the compound tone of C,
we may pass by Fifths to ^+61— d, the compound tone of G, or to/-f aj — c the
Digitized by V^jOAQIC
356 CHORDAL SEQUENCES. pabt in.
compound tone of F. Both of these chords are directly related to the first
0+^1— 9» because each has one note in common with it, g and c respectively.
But we can also allow the compound tone to proceed in Thirds, and then we
obtain minor chords, that is, provided we keep to the same scale. The transition
from the compound tone of C to that of ^i is expressed by the sequence of chords
c+e^—g and e,— ^+&if which are related by having two notes, «i, gr, in common.
, The sequence c+^i— ^ and a^— c+^i from the compound tone of C to that of Ai
is of the same kind. The latter is even more natural than the former, because the
chord ai — c+^i represents imperfectly the compound tone of Ax into which that of
C intrudes, so that the compound tone of 0, which was clearly given in the pre-
ceding chord, persists with two of its tones, c, ^i, in the second chord, a relation
which did not exist in the former case.
But if we prefer to leave the key of G major, we can pass to perfect compound
f tones in Thirds, as from c + «i — ^ to fii + g^ — hi or a^ + c^ — 6i, as is very
usual in modulations.
Bameau will not allow a simple diatonic progression of the fundamental baas of
consonant triads, except where major and minor chords alternate, as from g-¥hi^d
to a|— c+<?it that is from the compound tone of O to that oi Ai, but cedls this a
* hcence.' In reality this progression is readily explicable from our point of view,
by considering aj— c + ^i as a compound tone of G with an intrusive ai. The
transition is then one of the usual close relationship, from the compound tone of
G to that of C, and the aj appears as a mere appendage to the latter. Every
minor chord represents two compound tones in an imperfect manner. Bameaa
first formulated this ambiguity {double emploi) for the minor chord with added
Seventh, which, in the form ^i — / + ai— c, may represent the compound tone of
Di, and in the form/+ai— c.d that of Fy or in Bameau's language its funda-
mental bass might be Di or F,* In this chord of the Seventh the ambiguity is
f more marked, because it contains the compound tone of F more completely ; but
the ambiguity belongs in a less marked degree to the simple chord also.
With the false cadence in the major key
g + 6i— d to ai — c + Ci
must be associated the corresponding cadence in the minor key»
g + bi—d to a*t>— c + c^t>
where the chord a* |> — o + e^ t> replaces the normal resolution c^e^\}+g. But here
there is only a single note of the compound tone of G remaining, and the false
cd.dence therefore becomes much more striking. It will be rendered milder by
adding the Seventh /to the G chord, because/ is related to a^\}.
When two chords having only a relationship of the second degree, are placed
in juxtaposition, we usually feel the transition to be very abrupt. But if the chord
^ which connects them is one of the principal chords of the key, and has con-
sequently been frequently heard, the effect is not so striking. Thus in the final
cadence it is not unusual to see the succession/ -h ai— c and y+6i — d, the two
chords being related through the tonic chord c+ei^ g, thus :
/ + ai-c 5r + 6i-d
c — «i — ^
> ,
Generally we must remember that all these rules of progression are subject
to many exceptions, partly because expression may require exceptional abruptness
of transition, and partly because the hearer's recollection of previous chords may
* [Of ooorse Bamean, writing in tempered and f+a^-^c.^d were to him identioaL See
notation, did not distinguiBh d, and <2, so that pp. 340a, 345a, 34&1. — Translator.]
the actual notes in the two chords c{, — /+ a, • e
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CHAP. xvin. CHORDAL SEQUENCES. 357
saffioiently strengthen a naturally weak relationship. It is clearly an entirely false
position which teachers of harmony have assumed, in declaring this or that to be
' forbidden ' in music. In point of fact nothing musical is absolutely forbidden,
and all rules for the progression of parts are actually violated in the most effective
pieces of the greatest composers. It would have been much better to proceed from
the principle that certain transitions, which are disallowed, produce striking and
unusual effects upon the hearer, and consequently are unsuitable except for the
expression of what is unusual. Generally speaking, the object of the rules laid
down by theorists is to keep up a well-connected flow of melody and harmony, and
make its course readily intelligible. If that is what we aim at, we had best observe
their restrictions. But it caimot be denied that a too anxious avoidance of what is
unusual places us in danger of becoming trivial and dull, while, on the other hand,
inconsiderate and frequent infringement of rules makes compositions eccentric and
unconnected. %
When disconnected triads would come together it is frequently advantageous to
transform them into chords of the Seventh, and thus create a bond between them.
In place of the preceding sequence of two triads
/-hai— c to gf + 5i— d
we can use a sequence of chords of the Seventh which r^resent the same 00m-
pound tones
/ + tti— c ...d to g + &i— d I /.
In this case two of the four notes remain unchanged ; in the chord of F, the
d belongs to the compound tone of the dominant, and in the chord of G the / to
that of the subdominant.
Hence chords of the Seventh come to play an important part in modem music
for the purpose of effecting well-connected and yet rapid transitions from chord to f
chord, and urging them forward by the action of dissonances. In this way par-
ticularly, transitions to the compound tone of the subdominant are easily effected.
Thus, for example, beginning with the triad g -{• bi ^ d vfe can not merely
pass to the chord of C, or c -h ei — g, but, letting g remain as a Seventh, to the chord
of the Seventh a^ — c + ei — g, which unites the two chords c -f e, — ^ and
ai — c -t- fii, and then immediately pass to t^i — /+ a,, which is related to the
latter chord, so that two steps bring us to the other extremity of the system of C
major. This transition also gives the best progression for the Seventh {g in the
example), because it has been prepared in the previous chord, and is resolved by
descent (to /) in the succeeding chord. If we tried the same transition backwards,
we should have to obtain the Seventh g by progression from aj in the chord of
di —/ + «!, and then be compelled to introduce the c of the chord of the Seventh
abruptly, because we should have a prohibited succession of Fifths {dx + ai and
c + ^) if we tried to descend from di. We must rather obtain c by a leap from/, ^
because aj in the first triad must furnish both the a^ and g of the chord of the
Seventh. Thus the transition to the dominant is by no means easy, fluent, and
natural ; it is much more embarrassed than the passage to the subdominant. Con-
sequently the regular and usual progression of the chord of the Seventh is for
its Seventh to descend to the triad whose Fifth is the root of the chord of the
Seventh. Supposing we denote the root of the chord of the Seventh by I, its
Third by III, &c., a falling Seventh will lead us to either of these chords :
i_ni-v-vn and i-in-v-vn
I - IV - VI I - m - VI
Of these two transitions, the first, which leads to a chord of which IV is tiie root,
10 the liveliest, because it introduces a chord with two new tones. Tbe.other,
Digitized by V^OOQIC
IT
358 CHORDAL SEQUENCES. pabt m,
which leads to a triad having VI for its root, introduces only one new tone. Henoe
the first is regarded as the principal method of resolving chords of the Seventh.
For example :
g + by — d \ f Ci - g + bi — d
\ \/ / \ \/ /
g ... c + e\ Ci ... ai — c
<5 + «i — 9 + *i bi -- d \ f + ai
I V / i \/ /
c ... / + ai bi ... Ci — g
The descent of the tone VII introduces the tone YI. In the first case this is the
Third of the new triad, and in the second its root.* But it may be its Fifth :
i-m-v-vn
11 -IV - VI
This, however, could only occur naturally in the two chords :
bi—d I f + ai and bi-^d | / — a't>
\ V" /
c + ei — g
because the two chords of the Seventh represent the compound tone of G, and the
tonic chord estabHshes the bond of union between its two sections. In other cases
our scheme gives so-called false cadences :
(/ + 61 — d I / and g + by — d \ f
a, — c + fij a*p + c — e*p
which are justified (the first as most natural) by the fact that either c + «i or c — e'b
belongs to the chord of the normal resolution. Eameau therefore justly observes
that this kind of resolution is only permissible when the IV of the second chord is
the normal Fourth of the I in the chord of the Seventh.
This exhausts the resolutions by the descent of the Seventh. Those in which
it remains unchanged take place according to the schemes :
I-m-V-Vnt and I-IH-V-Vnt
I
vn
ii^iV - VII n — i^ —
In the first the Seventh becomes the root, in the second the Third of the new
chord. If it were the Fifth, the new chord would coincide with part of the chord
% of the Seventh :
I - m - V - VII
I I I I
vn - m - V - vn.§
* [As examples of the second method have triad 6.— <2 | /, itself a dissonanoe ; and in the
been omitted in the text, take third tne imperfect minor d | Z+Oj. — TVymm-
lator.]
g+b,...ei e^-g
-g ... c g + ^-cti/, C + 6,-^ + 0,
^Translator.] X/ I I \/ I ^
a, d f, d ... g + b^
t [Examples : . - -^ -7*1
\Y...) \y...i yX..d v'rVi rtTi'
Here in the first example we obtain the major /•••^'i— a | / 0|...6i— gf + 0,
triad /+a|~c; in the second the diminished — TransUUof.
Digitized by VjOOQIC
CHAP. xvm. CONSECUTIVE FIFTHS AND OCTAVES. 359
Tti these conneGtions the resolution is towards the dominant side. The transition is
most decisive in the first, where the Seventh becomes the root. These resolutions
are on the whole less usual, because we pass more easily and frequently from
chords on the dominant side into chords of the Seventh of the direct system. In
tlae chords of the reverted system these transitions occur more frequently, because
tlieir Sevenths may enter by ascent, and hence we avoid the sequences of Fifths,
Tvliich greatly embarrass the transitions from a triad to a chord of the Seventh on
its dominant side.
As to the transitions from one chord of the Seventh to another, or to a dissonant
triad of the direct system which may be regarded as a mutilated chord of the
Seventh, all these matters are sufficiently developed in the ordinary manuals of
Thorough Bass, and offer no difficulties that would justify us in dwelling upon
tbem here.
On the other hand, we have to say a few words on certain rules respecting the IT
progression of the individual parts in polyphonic compositions. Originally, as we
liave already remarked, all these parts were of equal importance, and had usually
to repeat the same melodic figures in succepsion. The harmony was a secondary
consideration, the melodic progression of the individual voices was the principal
matter. Hence it was necessary to take care that each voice should stand out
clear and distinct from all the others. The relation between the importance of
harmony and melody has certainly altered essentially in modern music ; the former
has attained a much higher independent significance. But, after all, perfection of
harmony must arise from the simultaneous performance of several voices, each of
-which has its own beautiful and clear melodic progression, and each of which
therefore moves in a direction that the hearer has no difficulty in understanding.
On this rests the prohibition of consecutive Fifths and Octaves, The meaning
of this prohibition has given rise to much disputation. The meaning of pro-
hibiting consecutive Octaves has been made clear by musical practice. In poly- %
phonic music two voices which lie one or two Octaves apart, are forbidden to
move forward in such a way that after their next step they should be also one or
two Octaves apart. But precisely in the same way, two voices in a polyphonic
piece are forbidden to go on in unison for several notes, while for complete musical
compositions it is not forbidden that two voices, or even all the voices, should
proceed in Unisons or Octaves, for the purpose of strengthening the melodic pro-
gression. It is clear that the reason of this rule must he in the Umiting the wealth
of the progression of parts by Unisons and Octaves. This is allowable when it is
intentionally introduced for a whole melodic phrase, but it is not suited for a few
notes in the course of a piece, where it can only give the impression of reducing
the richness of the harmony by an unskilful accident. The accompaniment of a
lower part by a voice singing an Octave higher, merely strengthens part of the
compound tone of the lower voice, and hence where variety in the progression of
parts is important, does not essentially differ from a Unison. ^
Now in this respect the nearest to an Octave are the Twelfth, and its lower
octave, the Fifth. Hence, then, consecutive Twelfths and consecutive Fifths par-
take of the same imperfection as consecutive Octaves. But the case is somewhat
worse. It is possible to accompany a whole melody in Octaves when desirable,
without committing any error, but this cannot be done for Fifths and Twelfths
without changing the key. It is impossible to proceed by a single diatonic step
from the tonic as root with an accompaniment of Fifths, without departing from
the key. In G major, we ascend from the Fifth c ± gr to the Fifth (i ± a, but a
does not belong to the scale, which requires the deeper a^ ; we descend to fei±/, J,
and there is no /j J in the scale at all. The other upward steps from d exclusive
to ai can of course be accompanied by perfect Fifths in the scale, as ^i ±61,
/±c', g'i^', ai±Ci'. It is therefore impossible to use the Twelfth consistently
for increasing the richness of the tone. But again, when the intervals of a Twelfth
or Fifth are continued for a few steps in melodic progression, they have simply the
Digitized by V^jOOQlC
36o CONSECUTIVE FIFTHS AND OCTAVES. pabt ra.
effect of strengthening the root. For the Twelfth this arises from its directly
corresponding to one of the upper partial tones of the root. For the Fifth c ± ^,
the c and g are the two first upper partials of the combinational tone G, which
necessarily accompanies the Fifth. Hence an accompaniment in Fifths above,
when it occurs isolatedly in the midst of a polyphonic piece, is not only open to the
charge of monotony, but cannot be consistently carried out. It should therefore
be always avoided.
But that consecutive Fifths merely infringe the laws of artistic composition,
and are not disagreeable to the natural ear, is evident from the simple fact that all
the tones of our voice, and those of most instruments, are accompanied by Twelfths,
and that our whole tonal system reposes upon that fsuat When the Fifths are
introduced as merely mechanical constituents of the compound tone, they are
therefore fully justified. So in the mixture stops of organs. In these stops the
f pipes which give the prime tones of the compound, are always accompanied by
others which give its harmonics, as the Octaves, Twelfths variously repeated, and
even the higher major Thirds. By this means the performer is able to compose a
tone of a much more penetrating, piercing quaUty, than it would be possible to
produce by the simple organ pipes with their relatively weak upper partial tones.
It is only by such means that an organ is able to dominate over the singing of a
large congregation. Almost all musicians have blamed an accompaniment of
Fifths, or even Thirds, but fortunately have not been able to effect anything
against the practice of organ-builders. In fa^t the mixture stops of an organ
merely reproduce the masses of tone which would have been created by bowed
instruments, trombones, and trumpets, if they had executed the same music. It
would be quite different if we collected independent parts, from each of which we
should have to expect an independent melodic progression in the tones of the
scale. Such independent parts cannot possibly move with the precision of a
f machine ; they would soon betray their independence by slight mistakes, and we
should be led to subject them to the laws of the scale, which, as we have seen,
render a consistent accompaniment in Fifths impossible.
The prohibition of Fifths and Octaves extends also, but with less strictness, to
the next adjacent consonant intervals, when two of them are so placed as to form
a connected group of upper partials in a compound tone. Thus transitions like
d ,,, g + bi to c.,,/+(ii,
are rulad by musical theorists to be inferior to transitions like
bi — d* ... g' to a, — c' .../.
For df g, by are the third, fourth, and fifth partial tones of the compound G^, but
61, d\ g' could only be regarded as its fifth, sixth, and eighth. Hence the first
f position of the chord expresses a single compoimd tone much more decidedly than
the second, which is often allowed to be continued through long passages, when of
course the nature of the Thirds and Fourths varies.
The prohibition of consecutive Fifths was perhaps historically a reaction against
the first imperfect attempts at polyphonic music, which were confined to an ac-
companiment in Fourths or Fifths, and then, like all reactions, it was carried
too far, in a barren mechanical period, till absolute purity from consecutive Fifths
became one of the principal characteristics of good musical composition. Modem
harmonists agree in allowing that other beauties in the progression of parts are
not to be rejected because they introduce consecutive Fifths, although it is advisable
to avoid them, when there is no need to make such a sacrifice.
There is also another point in the prohibition of Fifths to which HauptmAnn
has drawn attention. We are not tempted to use consecutive Fifths when we pass
from one consonant triad to another which is nearly related to it, because other
progressions lie nearer at hand. Thus we pass from the triad of G major to the
Digitized by VjOOQlC
CHAP. XVIII. CONSECUTIVE FIFTHS AND OCTAVES. 361
four related triads in the following manner, the fundamental bass proceeding by
Thirds or Fifths :
c -\- Bx — g c + ^i — ^ c + ei-gr and c + «i — ^
to c + 64 ... ai, to c ... / + ai, to B^,,. ex — g^ to Bx ... d ... g.
Bat when the fundamental bass proceeds in Seconds, and hence does not pass
to a directly related chord, the nearest position of the new chord is certainly one
which produces consecutive Fifths. For example :
^ + 61 — d' or (/ + 61 - d'
to a — c' + Cxt to / + ai — c
In such cases, then, we must have recourse to other transitions by larger
intervals, as : "
g -{■ hx — d/ or ^ + &i — d'
to ^1... ax — c', toai— c ... /
which avoid consecutive Fifths.
Hence when the chords are closely connected by near relationship and small
distance in the scale, consecutive Fifths do not present themselves. When they
occur, therefore, they are always signs of abrupt chordal transition, and it is then
better to assimilate the progression of parts to that which spontaneously arises in
the case of related chords.
This consideration respecting consecutive Fifths, which was emphasised by
Hauptmann, appears to give the law greater importance. That it is not the only
motive for the prohibition of consecutive Fifths appears from the fact, that the
forbidden sequence
gr + 61 - d' to / + a^ — </ ^
is allowed, when the chords are in the position
hx^d'„.g to ax— c' .,.f,
although the step in the fundamental bass is the same.
The prohibition of so-called hidden Fifths and Octaves has been added on to
the prohibition of consecutive Fifths and Octaves, at least for the two extreme
voices of a composition in several parts. This prohibition forbids the lowest and
uppermost voice in a piece to proceed by direct motion [that is, both parts ascending
or both parts descending] into the consonance of an Octave or Fifth (including
Twelfth). They should rather come into such a consonance by contrary motion
(one descending and the other ascending). In duets this would also hold for the
unison. The meaning of this law must certainly be, that whenever the extreme
voices unite to form the partial tones of a compound, they ought to have reached ^
a state of relative rest. It must be conceded that the equihbrium will be more
perfect when the extreme parts of the whole mass of tone approach their junction
from opposite sides, than when the centre of gravity, so to speak, of the sonorous
mass is displaced by the parallel motion of the extreme voices, and these voices
catch one another up with different velocities. But where the motion proceeds in
the same direction, and no relative rest is intended, the hidden Fifths are also not
avoided, as in the usual formulae :
m
e=ff
in which the g±.diB reached by passages involving hidden Fifths.
Another rule in the progression of parts, prohibiting false relations ^ must have
had its origin in the requirements of the singer. But what the singer finds a
Digitized by V^jOOQlC
362 RESULTS OP THE INVESTIGATION. pabt m.
difficulty in hitting, must naturally also appear an unusual and forced skip to the
hearer. By fake relations is meant the case when two tones in consecutive
chords, which belong to different voices, form false Octaves or fedse Fifths. For
example, if one voice in the first chord sings bi and another voice in the next
chord sings 5[>, or the first has c and the second cjj^, there are false Octave rela-
tions. False Fifth relations are forbidden for the extreme voices only. Thus in
the first chord the bass has ^p in the second the soprano has /, or conversely,
where bi .,./ ia a false Fifth. The meaning of this rule is, probably, that the
singer would find it difficult to hit the new tone which is not in the scale, if
he had just heard the next nearest tone of the scale given by another singer.
Similarly, when he has to take the false Fifth of a tone which is prominent in
present harmony as lowest or highest. There is therefore a certain sense in the
prohibition, but numerous exceptions have arisen, as the ear of modem musicians,
f singers and hearers, has become accustomed to bolder combinations and livelier
progressions. All these rules were essentially intended for the old ecclesiastieal
music, where a quiet, gentle, well-contrived, and well-adjusted stream of sound
was aimed at, without any intentional effort or disturbance of the smoothest
equihbrium. Where music has to express effort and excitement, these rules
become meaningless. Hidden Fifths and Octaves and even false relations of
Fifths are found in abundance in the chorales of Sebastian Bach, who is other-
wise so strict in his harmonies, but it must be admitted that the motion of his
voices is much more powerfully expressed than in the old Italian ecclesiastical
music.
CHAPTER XIX.
ESTHETIOAL RELATIONS.
Let us review the results of the preceding investigation.
Compound tones of a certain class are preferred for all kinds of music, melodic
or harmonic ; and are almost exclusively employed for the more delicate and
artistic development of music: these are the compound tones which have har-
monic upper partial tones, that is compound tones in which the higher partial tones
have vibrational numbers which are integral multiples of the vibrational number of
the lowest partial tone, or prime. For a good musical effect we require a certain
moderate degree of force in the five or six lowest partial tones, and a low degree
of force in the higher partial tones.
This class of compound tones with harmonic upper partials is objectively dis-
tinguished by including all sonorous motions which are generated by a mechanical
f process that continues to act uniformly, and which consequently produce a uniform
and sustained sensation. In the first rank among them stand the compound tones
of the human voice, man*s first musical instrument in time and value. The com-
pound tones of all wind and bowed instruments belong to this class.
Among the bodies which are made to emit tones by striking, some, as strings,
have also harmonic upper partials, and these can be used for artistic music.
The greater number of the rest, as membranes, rods, plates, &c., have inhar-
monic upper partial tones, and only such of them as have not very strong secondary
tones of this kind can be singly and occasionally employed in connection with
musical instruments proper.
Although sonorous bodies excited by blows may continue to sound for some
time, their tones do not proceed with uniform force, but diminish more or less
slowly and die away. Constant power over the intensity of tone, therefore, which
is indispensable for expressive performance, can only be attained on instruments
of the first kind, which can be maintained in a state of excitement, and which
Digitized by V^jOOQlC
CHAP. XIX. RESULTS OP THE INVESTIGATION. 363
produce only harmonic upper partial tones. On the other hand, bodies excited by
blo^w^s have a peculiar value for clearly defining the rhythm.
A second reason for preferring compound tones with harmonic upper partials
is subjective and conditioned by the construction of our ear. In the ear even
every simple tone, if sufficiently intense, excites feeble sensations of harmonic
npper partials, and each combination of several simple tones generates combina-
tional tones, as I have explained at the end of Chap. VII. (p. 157^1590). A single
compound tone with irrational partials, when sounded with sufficient force, thus
produces the sensation of dissonance, and simple tones acquire in the ear itself
something of the nature of composition out of harmonic upper partial tones.
We are justified in assuming that historically all music was developed from
song. Afterwards the power of producing similar melodic effects was attained by
means of other instruments, which had a quality of tone compoimded in a manner
resembling that of the human voice. The reason why, even when constructive H
art was most advanced, the choice of musical instruments was necessarily limited
to those which produced compound tones with harmonic upper partials, is clear
from the above conditions.
This invariable and peculiar selection of instruments makes us perfectly certain
that harmonic upper partials have from all time played an essential part in musical
constructions, not merely for harmony, as the second part of this book shews, but
also for melody.
Again, we can at any moment convince ourselves of the essential importance
of upper partial tones to melody, by the absence of all expression in melodies
executed with objectively simple tones, as, for example, those of wide-stopped
organ pipes, for which the harmonic upper partials are formed only subjectively
and weakly in the ear.
A necessity was always felt for music of all kinds to proceed by certain definite
degrees of pitch ; but the choice of these degrees was long unsettled. To distin- ^
guish small differences of pitch and intonate them with certainty, requires a greater
amount of technical musical power and cultivation of ear, than when the intervals
are larger. Hence among almost all uncivilised people we find the Semitones
neglected, and only the larger intervals retained. For some of the more cultivated
nations, as the Chinese and Gaels, a scale of this kind has become established.*
It might perhaps have seemed most simple to make all such degrees of pitch
of equal amount, that is, equally well distinguishable by our sensations. Such a
graduation is possible for all our sensations, as Fechner has shewn in his investi-
gations on psychophysical laws. We find such graduations used for the divisions of
musical rhythm, and "the astronomers use them in reference to the intensity of
light in determining stellar magnitudes. Even in the field of musical pitch, the
modem equally tempered chromatic scale presents us with a similar graduation.
But although in certain of the less usual Greek scales and in modem Oriental
music, cases occur where some particular small intervals have been divided on the ^
principle of equal graduations, yet there seems at no time or place to have been a
system of music in which melodies constantly moved in equal degrees of pitch,
but smaller and larger intervals have always been mixed in the musical scales
in a way that must appear entirely arbitrary and irregular until the relationship of
compound tones is taken into consideration.*
On the contrary, in all known musical systems the intervals of Octave and
Fifth have been decisively emphasised. Their difference is the Fourth, and the
difference between this and the Fifth, is the Pythagorean major Tone 8 : 9, by
which (but not by the Fourth or Fifth) the Octave might be approximatively
divided.
The sole renmants that I can find in modem music of the endeavour some-
times made in homophonic music to introduce degrees depending on equality of
interval and not on relationship of tone, are the chromatic intercalated notes, and
* [See, however, App. XX. sect. E.— Translator.]
Digitized by V^OOQ IC
364 RESULTS OP THE INVESTIGATION. pabt in.
the leading note of the key when similaxly ased. But this is always a Semitone
(p. 352c), an interval well known in the series of related tones, which, owing to
its smallness, is easily measured hy the sensation of its difference, even in places
where its tonal relationship is not immediately sensible.
The decisive importance acquired by the Octave and Fifth iu all musical scales
from the earliest times shews, that the construction of scales must have been
originally influenced by another principle, which Anally became the sole regulator
of every artistic form of a complete scale. This is the principle which we have
termed tonal relationship,
Belationship in the first degree between two compound tones consists in their
each having a partial tone of the same pitch.
In singing, the similarity of two musical tones which stand in the relation of
Octave or Fifth to one another, must have been very soon observed. As already
% remarked, this gives also the Fourth, which has itself a sufficiently perceptible
natural relationship to have been remarked independently. To discover the tonal
similarity of the major Third and major Sixth, required a finer cultivation of the
musical ear, and perhaps also peculiar beauty of voice. Even yet we are easily led
by the fomiliar sharp major Thirds of equal temperament, to endure any major
Thirds which are somewhat too sharp, provided they occur melodically and are
not sounded together. On the other hand, we must not forget that the rules of
• Archytas and Abdul Kadir,* both of which were applicable to homophonic musie
only, gave a preference to the natural major Third, although its introduction
obliged both musicians to renounce a musical system so theoretically consistent
and invested with such high authority as that of Pythagoras.
Hence the principle of tonal relationship did not at all times exclusively deter-
mine the construction of the scale, and does not even yet determine it exclusively
among all nations. This principle must, therefore, be regarded to some extent as
% & freely selected principle of style, as I have endeavoured to shew in Chapter XTTT.
But, on the other hand, the art of music in Europe was historically developed
from that principle, and on this fact depends the main proof that it was really as
important as we have assumed it to be. The preference first given to the diatonic
scale, and finally the exclusive use of that scale, introduced the principle of tonal
relationship in all its integrity into the musical scale. Within the diatonic scale
various methods of execution were possible, and these generated the ancient modes,
which had equal claims to attention in homophonic song, and hence stood on a
level.
But the principle of tonal relationship penetrated far deeper in its harmonic
than it did in its melodic form. In melodic sequence the identity of two partial
tones is a matter of memory, but when the notes are sounded together the im-
mediate sensible impression of the beats, or else of the undisturbed flow of sound
forces itself on the hearer's attention. The liveliness of melodic and harmonic
f impressions differs in the same way as a recollected image differs from the actual
impression made by the original. As an immediate consequence arose that £Eur
superior sensibility for the correctness of the intervals which is seen in the har-
monic union of tones, and which admitted of being developed into the finest
physical methods of measurement.
It must also be remembered that relationship in the second degree can in
harmonic music be reduced to audible relationships of the first degree, by a proper
selection of the fundamental bass, and that generally more distant relationships
can easily be made clearly audible. By this means, notwithstanding the variety
of progression, a much clearer connection of all parts with their origin, the tonic,
can be maintained and rendered objectively sensible to the hearer. It cannot be
doubted that these are the essential foundations of the great breadth and wealth of
expression which modem compositions can attain without losing their artistic unity.
• [For Archytas of Tarentnm, about b.o. note ^.^TranskUor.']
400, see p. 262c, and for Abdolqadir, see p. 281,
Digitized by VjOOQIC
CHAP. XIX. RESULTS OF THE INVESTIGATION. 365
We then saw that the requirements of harmonio musio reacted in a peculiar
manner on the construction of scales ; that properly speaking only one of the old
tonal modes (our major mode) could be retained unaltered,* and that the rest after
undergoing peculiar modifications were fused into our minor mode, which, though
most like the ancient mode of the minor Third, can at one time resemble the mode
of the minor Sixth, and at another time that of the minor Seventh, but does not
perfectly correspond with any one of these.
This process of the development of the elements of our modem musical
system lasted down to the middle of the last century. It was not until composers
ventured to put a minor chord at the close of compositions written in the minor
mode, that the musical feeling of European musicians and hearers can be admitted
to have become perfectly and surely habituated to the new system. The minor
chord was allowed to be a real, although obscured, chord of its tonic.
Whether this admission of the minor chord expressed a feeling for another V
mode of unifying its three tones, as A. von Oettingen f has assumed, — relying
on the fwt that the three tones c—e^\}'^g have a common upper partial gr", —
must be left to future experience to decide, should it be found practicable to con-
struct long and well-connected musical compositions in Oettingen's phonic system
(this is the name which he gives to the minor system which he has theoretically
developed, and which is essentially different from the historical minor mode). At
any rate, the minor mode has historically developed itself as a compromise
between dififerent kinds of claims. Thus it is only major triads which can per-
fectly indicate the compound tone of the tonic ; minor chords contain in their
Third sm element which, although nearly related to the tonic and its Fifth, does
not thoroughly fuse with them, and hence in their final cadence they do not so
thoroughly agree with the principle of tonality which had ruled the previous
development of music. I have endeavoured to make it probable that the peculiar
esthetic expression of the minor mode proceeded partly from this cause and partly If
from the heterogenous combinational tones of the minor chord.
In the last part of my book, I have endeavoured to shew that the construction
of scales and of harmonic tissue is a product of artistic invention, and by no
means famished by the natural formation or natural function of our ear, as it has
t)een hitherto most generally asserted. Of course the laws of the natural function
of our ear play a great and influential part in this result ; these laws are, as it
were, the building stones with which the edifice of our musical system has been
* [But see Bupra, p. 274, note ♦, scale i.— in the theory of composition. For the rest
Translator.] this author justifies (p. 54) the assertion I have
f The System of Harmony Dually Deve- made in the text by remarking : ' I am sorry
loped, Dorpat and Leipzig, 1866. Herr y. Get- to say that I am unable to adduce a single
tingen, as already observed, p. 308, note §, example from the whole of our musical litera-
regards the minor chord as representing the ture, of the carrying out of (v. Oettingen's) pure
harmonic undertones of its Fifth, and hence as minor mode harmony even in the simplest
standing in place of a part of its compound tone. manner.* I have not been able to convince^
He caUs it the Aphonic 'chord, as opposed to the myself of the correctness of the fact adduced
* tonic * major chord which stands in place of on p. xiii. and p. 6, that the undertones of a
the upper partials of its root. He proceeds to tone strongly struck on the piano sound when
deduce the formation of the minor system from the corresponding dampers are raised. Perhaps
the relations of the harmonic undertones in a the author has been deceived by the circum-
manner precisely analogous to that by which stance that with very resonant instruments
I have deduced the major system from the (especiaUy older ones) any strong shake, and
relations of the upper partial tones. The therefore probably a violent blow on the digitals,
tonal mode thus constructed is, however, in our will cause some one or several of the deeper
language the mode of the minor Sixth (p. 274, strings to sound its note. [The undertones
note *, scale 7), and the usual minor, a mixed have always each an upper partial tone of the
mode. Latterly Dr. Hugo Biemann has given pitch of the note struck ; the striking of this
in his adhesion to this view, and in his lately note must then sympathetically excite those
published Musical Syniaxis has attempted to upper partials of the undertones, and thus
examine and establish the consequences of reinforce the prime of the note struck, just as
this system by examples from acknowledged striking the undertone sympatheticaUy excites
composers. The application of this critical the higher tone itself. Can this have deceived
method appears to me very commendable, and Dr. Biemann ? — Translator,]
to be the indispensable condition to advancing
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366 ESTHETIC ANALYSIS OF WORKS OP ART. past m.
erected, and the necessity of accurately understanding the nature of these
materials in order to understand the construction of the edifice itself, has been
clearly shewn by the course of our investigations upon this very subject. But
just as people with differently directed tastes can erect extremely different kinds of
buildings with the same stones, so also the history of music shews us that the
same properties of the human ear could serve as the foundation of very different
musical systems. Consequently it seems to me that we cannot doubt, that not
merely the composition of perfect musical works of art, but even the construction
of our system of scales, keys, chords, in short of all that is usually comprehended
in a treatise on Thorough Bass, is the work of artistic invention, and hence most
be subject to the laws of artistic beauty. In point of fact, mankind has been at
work on the diatonic system for more than 2500 years since the days of Terpander
and Pythagoras, and in many cases we are still able to determine that the pro*
«|[ gressive changes made in the tonal system have been due to the most distin-
guished composers themselves, partly through their own independent inventions,
and partly through the sanction which they gave to the inventions of others, by
employing them artistically.
The esthetic analysis of complete musical works of art, and the comprehension
of the reasons of their beauty, encounter apparently invincible obstacles at almost
every point. But in the field of elementary musical art we have now gained so
much insight into its internal connection that we are able to bring the results of
our investigations to bear on the views which have been formed and in modem
times nearly universally accepted respecting the cause and character of artistic
beauty in general. It is, in fact, not difBcult to discover a close connection and
agreement between them ; nay, there are probably fewer examples more suitable
than the theory of musical scales and harmony, to illustrate the darkest and most
dif&cult points of general esthetics. Hence I feel that I should not be justified in
^ passing over these considerations, more especially as they are closely connected
with the theory of sensual perception, and hence with physiology in general.
No doubt is now entertained that beauty is subject to laws and rules dependent
on the nature of human inteUigence. The dif&culty consists in the forct that these
laws and rules, on whose fulfilment beauty depends and by which it must be judged,
are not consciously present to the mind, either of the artist who creates the work,
or the observer who contemplates it. Art works with design, but the work of art
ought to have the appearance of being undesigned, and must be judged on that
ground. Art creates as imagination pictures, regularly without conscious law,
designedly without conscious aim. A work, known and ackno\^ledged as the pro-
duct of mere intelligence, will never be accepted as a work of art, however peifect
be its adaptation to its end. Whenever we see that conscious reflection has acted
in the arrangement of the whole, we find it poor.
Man fiihlt die Absioht, and man wird verstimmt.
IT (We feel the purpose, and it jars upon as.)
And yet we require every work of art to be reasonable, and we shew this by
subjecting it to a critical examination, and by seeking to enhance our enjoyment and
our interest in it by tracing out the suitability, connection, and equilibrium of all its
separate parts. The more we succeed in making the harmony and beauty of all
its peculiarities clear and distinct, the richer we find it, and we even regard as
the principal characteristic of a great work of art that deeper thought, reiterated
observation, and continued reflection shew us more and more clearly the reason-
ableness of all its individual parts. Our endeavour to comprehend the beauty of
such a work by critical examination, in which we partly succeed, shews that we
assume a certain adaptation to reason in works of art, which may possibly rise to
a conscious understanding, although such imderstanding is neither necessary for the
invention nor for the enjoyment of the beautiful. For what is esthetically beau>
tiful is recognised by the immediate judgment of a cultivated taste, which declaims
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CHAP. XIX. ESTHETIC ANALYSIS OF WORKS OF ART. 367
it pleasing or displeasing, without any comparison whatever with law or concep-
tion.
But that we do not accept delight in the beautiful as something individual, but
rather hold it to be in regular accordance with the nature of mind in general,
appears by our expecting and requiring from every other healthy human intellect
the same homage that we ourselves pay to what we call beautiful. At most we
allow that national or individual peculiarities of taste incline to this or that artistic
ideal, and are most easily moved by it, precisely in the same way that a certain
amount of education and practice in the contemplation of fine works of art is
undeniably necessary for penetration into their deeper meaning.
The principal difficulty in pursuing this object, is to understand how regularity
can be apprehended by intuition without being consciously felt to exist. And this
unconsciousness of regularity is not a mere accident in the effect of the beautiful
on our mind, which may indifferently exist or not ; it is, on the contrary, most ^
clearly, prominently, and essentially important. For through apprehending every-
where traces of regularity, connection, and order, without being able to grasp the
law and plan of the whole, there arises in our mind a feeling that the work of art
-which we are contemplating is the product of a design which far exceeds anything
we can conceive at the moment, and which hence partakes of the character of the
illimitable. Remembering the poet's words :
Da gleichst dem Gteist, den da begreifst,
(Thoa*rt like the spirit thoa oonceivest),
we feel that those intellectual powers which were at work in the artist, are far above
our conscious mental action, and that were it even possible at all, infinite time,
meditation, and labour would have been necessary to attain by conscious thought
that degree of order, connection, and equilibrium of all parts and all internal
relations, which the artist has accomplished under the sole guidance of tact and ^
taste, and which we have in turn to appreciate and comprehend by our own tact and
taste, long before we begin a critical analysis of the work.
It is clear that all high appreciation of the artist and his work reposes essen-
tially on this feeling. In the first we honour a genius, a spark of divine creative
fire, which hx transcends the limits of our intelligent and conscious forecast. And
yet the artist is a man as we are, in whom work the same mental powers as in our-
selves, only in their own peculiar direction, purer, brighter, steadier ; and by the
greater or less readiness and completeness with which we grasp the artist's language
we measure our own share of those powers which produced the wonder.
Herein is manifestly the cause of that moral elevation and feeling of ecstatic
satisfaction which is called forth by thorough absorption in genuine and lofty works
of art. We learn from them to feel that even in the obscure depths of a healthy
and harmoniously developed human mind, which are at least for the present
inaccessible to analysis by conscious thought, there slumbers a germ of order that ^
is capable of rich intellectual cultivation, and we learn to recognise and admire in
the work of art, though draughted in unimportant material, the picture of a similar
arrangement of the universe, governed by law and reason in all its parts. The
contemplation of a real work of art awakens our confidence in the originally healthy
nature of the human mind, when uncribbed, unharassed, unobscured, and un-
falsified.
But for all this it is an essential condition that the whole extent of the regularity
and design of a work of art should no^ be apprehended consciously. It is precisely
from that part of its regular subjection to reason, which escapes our conscious
apprehension, that a work of art exalts and delights us, and that the chief effects
of the artistically beautiful proceed, not from the part which we are able fully to
analyse.
If we now apply these considerations to the system of musical tones and har-
mony, we see of course that these are objects belonging to an entirely subordinate
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368 APPLICATION TO MUSIC. pabt nn
and elementary domain, but nevertheless they, too, are slowly matured inventions
of the artistic taste of musicians, and consequently they, too, must be governed by
the general rules of artistic beauty. Precisely because we are here still treading
the lower walks of art, and are not dealing with the expression of deep psycho-
logical problems, we are able to discover a comparatively simple and transparent
solution of that fundamental enigma of esthetics.
The whole of the last part of this book has explained how musicians gradually
discovered the relationships between tones and chords, and how the invention of
harmonic music rendered these relationships closer, and clearer, and richer. We
have been able to deduce the whole system of rules which constitute Thorough
Bass, from an endeavour to introduce a clearly sensible connection into the series
of tones which form a piece of music.
A feeling for the melodic relationship of consecutive tones, was first developed,
^ commencing with Octave and Fifth and advancing to the Third, We have taJ^en
pains to prove that this feeling of relationship was founded on the perception of
identical partial tones in the corresponding compound tones. Now these partial
tones are of course present in the sensations excited in our auditory apparatus, and
yet they are not generally the subject of conscious perception as independent sensa-
tions. The conscious perception of everyday life is limited to the apprehension of
the tone compounded of these partials, as a whole, just as we apprehend the taste
of a very compound dish as a whole, without clearly feeling how much of it is due
to the salt, or the pepper, or other spices and condiments. A critical examination
of our auditory sensations as such was required before we could discover the exist-
ence of upper partial tones. Hence the real reason of the melodic relationship of
two tones (with the exception of a few more or less clearly expressed conjectures,
as, for example, by Bameau and d'Alembert) remained so long undiscovered, or at
least was not in any respect clearly and definitely formulated. I believe that I have
^ been able to furnish the required explanation, and hence clearly to exhibit the
whole connection of the phenomena. The esthetic problem is thus referred to the
common property of all sensual perceptions, namely, the apprehension of compound
aggregates of sensations as sensible symbols of simple external objects, without
analysing them. In our usual observations on external nature our attention is so
thoroughly engaged by external objects that we are entirely unpractised in taking
for the subjects of conscious observation, any properties of our sensations them-
selves, which we do not already know as the sensible expression of some individual
external object or event.
After musicians had long been content with the melodic relationship of tones,
they began in the middle ages to make use of harmonic relationship as shewn in
consonance. The effects of various combinations of tones also depend partly on
the identity or difference of two of their different partial tones, but they likewise
partly depend on their combinational tones. Whereas, however, in melodic
-r relationship the equahty of the upper partial tones can only be perceived by
remembering the precedLag compound tone, in harmonic relationship it is deter-
mined by immediate sensation, by the presence or absence of beats. Hence in
harmonic combinations of tone, tonal relationship is felt with that greater liveli-
ness due to a present sensation as compared with the recollection of a past sensa-
tion. The wealth of clearly perceptible relations grows with the number of tones
combined. Beats are easy to recognise as such when they occur slowly ; but those
which characterise dissonances are, almost without exception, very rapid, and are
partly covered by sustained tones which do not beat, so that a careful comparison
of slower and quicker beats is necessary to gain the conviction that the essence of
dissonance consists precisely in rapid beats. Slow beats do not create the feeling
of dissonance, which does not arise till the rapidity of the beats confuses the ear
and makes it unable to distinguish them. In this case also the ear feels the dif-
ference between the undisturbed combination of sound in the case of two consonant
tones, and the disturbed rough combination resulting from a dissonance. But, as
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CHAP. XIX. UNCONSCIOUS SENSE OF RESEMBLANCE. 369
s, general rule, the hearer is then perfectly unconscious of the cause to which the
disturbance and roughness are due.
The development of harmony gave rise to a much richer opening out of musical -
art than was previously possible, because the far clearer characterisation of related
combinations of tones by means of chords and chordal sequences, allowed of tlie
use of much more distant relationships than were previously available, by modu-
lating into different keys. In this way the means of expression greatly increased
as well as the rapidity of the melodic and harmonic transitions which could now
be introduced without destroying the musical connection.
As the independent significance of chords came to be appreciated in the fifteenth
and sixteenth centuries, a feeling arose for the relationship of chords to one another
and to the tonic chord, in accordance with the same law which had long ago
unconsciously regulated the relationship of compound tones. The relationship of
compound tones depended on the identity of two or more partial tones, that of ^
chords on the identity of two or more notes. For the musician, of course, the law
of the relationship of chords and keys is much more intelligible than that of com-
pound tones. He readily hears the identical tones, or sees them in the notes before
him. But the unprejudiced and uninstructed hearer is as little conscious of the
reason of the connection of a clear and agreeable series of fluent chords, as he is
of the reason of a well-connected melody. He is startled by a false cadence and
feels its unexpectedness, but is not at all necessarily conscious of the reason of its
unexpectedness.
Then, again, we have seen that the reason why a chord in music appears to be
the chord of a determinate root, depends as before upon the analysis of a com-
pound tone into its partial tones, that is, as before upon those elements of a
sensation which cannot readily become subjects of conscious perception. This rela-
tion between chords is of great importance, both in the relation of the tonic chord
to the tonic tone, and in the sequence of chords. %
The recognition of these resemblances between compound tones and between
chords, reminds us of other exactly analogous circumstances which we must have
often experienced. We recognise the resemblance between the faces of two near
relations, without being at all able to say in what the resemblance consists,
especially when age and sex are different, and the coarser outlines of the features
consequently present striking differences. And yet notwithstanding these differ-
ences— notwithstanding that we are unable to fix upon a single point in the
two countenances which is absolutely alike — the resemblance is often so extra-
ordinarily striking and convincing, that we have not a moment's doubt about
it. Precisely the same thing occurs in recognising the relationship between two
compound tones.
Again, we are often able to assert with perfect certainty, that a passage not
previously heard is due to a particular author or composer whose other works we
know. Occasionally, but by no means always, individual mannerisms in verbal or ^
musical phrases determine our judgment, but as a rule we are mostly unable to fix
upon the exact points of resemblance between the new piece and the known works
of the author or composer.
The analogy of these different cases may be even carried farther. When a
father and daughter are strikingly alike in sonie well-marked feature, as the nose
or forehead, we observe it at once, and think no more about it. But if the resem-
blance is so enigmatically concealed that we cannot detect it, we are fekscinated, and
cannot help continuing to compare their countenances. And if a painter drew two
such heads having, say, a somewhat different expression of character combined
-with a predominant and striking, though indefinable, resemblance, we should
undoubtedly value it as one of the principal beauties of his painting. Our ad-
miration would certainly not be due merely to his technical skill ; we should
rather look upon his painting as evidencing an unusually delicate feehng for the
BB
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370 UNCONSCIOUS SENSE OP TONAL RELATIONSHIP, pabt hi.
significance of the human countenance, and find in this the artistic justification
of his work.
Now the case is similar for musical intervals. The resemblance of an Octave to
its root is so great and striking that the dullest ear perceives it ; the Octave seems
to be almost a pure repetition of the root, as it, in fact, merely repeats a part of the
compound tone of its root, without adding anything new. Hence the esthetical
effect of an Octave is that of a perfectly simple, but little attractive interval. The
most attractive of the intervals, melodically and harmonically, are clearly the
Thirds and Sixths, — ^the intervals which he at the very boundary of those that the
ear can grasp. The major Third and the major Sixth cannot be properly appre-
ciated unless the first five partial tones are audible. These are present in good
musical qualities of tone. The minor Third and the minor Sixth are for the most
part justifiable only as inversions of the former intervals. The more compHcated
^ intervals in the scale cease to have any direct or easily intelligible relationship.
They have no longer the charm of the Thirds.
Moreover, it is by no means a merely external indifferent regularity which the
employment of diatonic scales, founded on the relationship of compound tones, has
introduced into the tonal material of music, as, for instance, rhythm introduced
some such external arrangement into the words of poetry. I have shewn, on the
contrary, in Chapter XIY., that this construction of the scale furnished a means of
measuring the intervals of their tones, so that the equaUty of two intervals lying
in different sections of the scale would be recognised by immediate sensation.
Thus the melodic step of a Fifth is always characterised by having the second
partial tone of the second note identical with the third of the first. This produces
a definiteness and certainty in the measurement of intervals for our sensation,
such as might be looked for in vain in the system of colours, otherwise so
similar, or in the estimation of mere differences of intensity in our various sensual
IT perceptions.
Upon this reposes also the characteristic resemblance between the relations of
the musical scale and of space, a resemblance which appears to me of vital impor>
tance for the peculiar effects of music. It is an essential character of space that
at every position vTithin it like bodies can be placed, and like motions can occur.
Everything that is possible to happen in one part of space is equally possible in
every other part of space and is perceived by us in precisely the same way. This
is the case also with the musical scale. Every melodic phrase, every chord, which
can be executed at any pitch, can be also executed at any other pitch in such a way
that we immediately perceive the characteristic marks of their similarity. On the
other hand, also, different voices, executing the same or different melodic phrases,
can move at the same time within the compass of the scale, like two bodies in
space, and, provided they are consonant in the accented parts of bars, without
creating any musical disturbances. Such a close analogy consequently exists in
^ all essential relations between the musical scale and space, that even alteration of
pitch has a readily recognised and unmistakable resemblance to motion in space,
and is often metaphorically termed the ascending or descending motion or progres-
sion of a part. Hence, again, it becomes possible for motion in music to imitate
the peculiar characteristics of motive forces in space, that is, to form an image of
the various impulses and forces which lie at the root of motion. And on this, as I
believe, essentially depends the power of music to picture emotion.
It is not my intention to deny that music in its initial state and simplest forms
may have been originally an artistic imitation of the instinctive modulations of the
voice that correspond to various conditions of the feelings. But I cannot think that
this is opposed to the above explanation ; for a great part of the natural means of
vocal expression may be reduced to such facts as the following : its rhythm and
accentuation are an immediate expression of the rapidity or force of tiie corre*
spending psychical motives — all effort drives the voice up — a desire to make a plea-
sant impression on another mind leads to selecting a softer, pleasanter quality of
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CHAP. XIX. EXPEESSION OF MOTION IN MUSIC. 371
tone — and so forth. An endeavour to imitate the involuntary modulations of the
voice and make its recitation richer and more expressive, may therefore very pos-
sibly have led our ancestors to the discovery of the first means of musical expres-
sion, just as the imitation of weeping, shouting, or sobbing, and other musical
delineations may play a part in even cultivated music, (as in operas), although
such modifications of the voice are not confined to the action of free mental
motives, but embrace really mechanical and even involuntary muscular contrac-
tions. But it is quite clear that every completely developed melody goes far beyond
an imitation of nature, even if we include the cases of the most varied alteration
of voice under the influence of passion. Nay, the very fact that music introduces
progression by fixed degrees both in rhythm and in the scale, renders even an
approximativdy correct representation of nature simply impossible, for most of
the passionate affections of the voice are characterised by a ghding transition in
pitch. The imitation of nature is thus rendered as imperfect as the imitation of IT
a picture by embroidery on a canvas with separate little squares for each shade of '
colour. Music, too, departed still further from nature when it introduced the
greater compass, the mobility, and the strange qualities of tone belonging to musical
instruments, by which the field of attainable musical effects has become so much
wider than it was or could be when the human voice alone was employed.
Hence though it is probably correct to say that mankind, in historical develop*
ment, first learned the means of musical expression from the human voice, it can
hardly be denied that these same means of expressing melodic progression act,
in artistically developed music, without the slightest reference to the application
made of them in the modulations of the human voice, and have a more general
significance than any that can be attributed to innate instinctive cries. That this
is the case appears above all in the modem development of instrumental music,
which possesses an effective power and artistic justification that need not be gain-
said, although we may not yet be able to explain it in all its details. V
Here I close my work. It appears to me that I have carried it as far as the
physiological properties of the sensation of hearing exercise a direct influence on
the construction of a musical system, that is, as &r as the work especially belongs
to natural philosophy. For even if I could not avoid mixing up esthetic problems
with physical, the former were comparatively simple, and the latter much more
complicated. This relation would necessarily become inverted if I attempted to
proceed further into the esthetics of music, and to enter on the theory of rhythm,
forms of composition, and means of musical expression. In all these fields the
properties of sensual perception would of course have an influence at times, but only
in a very subordinate degree. The real difficulty would lie in the development of
the psychical motives which here assert themselves. Certainly this is the point IT
where the more interesting part of musical esthetics begins, the aim being to ex-
plain the wonders of great works of art, and to learn the utterances and actions of
the various affections of the mind. But, however alluring such an aim may be, I
prefer leaving others to carry out such investigations, in which I should feel myself
too much of an amateur, while I myself remain on the safe ground of natural
phdUosopby, in which I am at home*
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APPENDICES-
APPENDIX I.
«r ON AN ELEGTBO-MAON^ITIC DBIYIKG MACHINB FOB THE 6IBEN.
(See p. 13a.)
I HAVE lately had a small electro-magnetic machine constructed with a constant
velocity of rotation, and it has proved of great service in driving the siren. A
rotating electro-magnet, in which the direction of current is changed every semi*
rotation, moves between two fixed magnetic poles. The current in this electro-
magnet is interrupted, as soon as the velocity begins to exceed the desired amount,
by the centrifugal force of a mass of metal febstened to its axis of rotation. Two
spiral springs whose elasticity is opposed to the centrifugal force, maybe tightened
or loosened, and thus made stronger or weaker at pleasure. By this means the
velocity can be maintained at any required rate. A figure and description of this
machine were given by Herr S. Exner, in the ' Proceedings ' {Sitzungsberichte)
of the Vienna Academy : * Math. Naturw. CI.* vol. Iviii. part 2, 8 Oct. 1868.
The machine was improved in 1875 by separating from it the centrifugal
ir regulator, and* letting it only open and close the weaJk current for a relay. It
is the relay which makes or breaks the strong current that drives the electro-
magnetic machine.
The siren is connected with the machine by a thin driving band, and then
it does not require to be blown. Instead of blowing, I placed on the disc a
small turbine constructed of stiff paper, which drove the air through the open-
ings whenever they coincided with those in the chest. This arrangement gave
me extremely constant tones on the siren, rivalling those on the best constructed
organ pipes. Latterly I have given the siren straight holes, so that the strength
of the wind has no longer any influence on its speed, and then I blow through the
box. [See App. XX. sect. B. No. 2.]
APPENDIX H.
ON THE SIZE AND CONSTRUCTION OF BBSONATOBS»
(See pp. 44b and i66d, note ♦.)
Sphebioal Resonators with a short fimnel-shaped neck for insertion into the
ear as shewn in fig. 16 a (p. 4$b), are most ^cient. Their advantage consists
partly in their other proper tones being very distant indeed from their prime tones,
and nence being very slightly reinforced, and partly in the spherical form giving
the most powerful resonance. But the walls of the sphere must be firm and
smooth, to oppose the necessary resistance to the powerful vibrations of air whioh
take place within them, and to impede the motion of the air as litUe as possible
by firiction. At first I employed any spherical glass vessels that came to hand,
as the receivers of retorts, and inserted into one of their openings a glass tube
which had been adapted to my ear. Afterwards Herr R. Koenig, maker of acous-
tical instruments, Paris [now of 27 Quai d'Anjou], constructed a series of these
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APP. II.
SIZE AND CONSTRUCTION OF EESONATORS.
373
glass spheres properly tuned, and afterwards had them made of brass in the form
shewn in fig. i6 a, p. 436. This is the most appropriate form for resonators.
When the openings are relatively very narrow, their pitch can be determined by
the formula which I have developed, namely*
n
where a is the velocity of sound, a the area of the circular opeiung, and S the
volume of the cavity* Or if we assume as its value
a = 332*260 metres,
which corresponds with a temperature of zero centigrade, the above formula gives
~ =56174 4^ (^) ,
Herr Sondhauss [Pogg. vol. Ixxxi. pp. 347-373] had discovered the same formula
experimentally, but used 52400 for the numerical coefficient, which agrees with
the experimental results better when the openings are not very small. When
the diameter of the opening is smaller than one-tenth of the diameter of the
sphere, the formula deduced from theory agrees well with Wertheim's experiments.
For resonators which have the diameter of their opening between a fourth and a
fifth of the diameter of the sphere, I have experimentally determined the coeffi-
cient as 47000. The second opening of the resonator may be regarded as closed
because it is brought firmly against the ear. If the cavity is spherical with radius
jB, while r is that of the opening, the above formula becomes
'-'■' (rfs.)
Here follows a hst of the measurements of my glass resonators.
Fitoh
I) 9
r
5)
6)
il
d'
8) V\>
9) 6"b
10) <r"
%
Diameter of the
Sphere
in millimetres
[and inohes]
131
130
115
79
76
70
[606]
[5-i6]
[5-12]
[4-53]
[3-1 1]
[2-99]
[276]
53*5 [2-11]
46
43
[r8i]
[1-69]
Diameter of the
Oriflce
in millimetrea
[and inches]
35'S [i'4o]
285 [I 1 2]
302 [119]
30 [118]
" [73]
i8-5
22
205
8
IS
IS
[•87]
[■81]
[•31]
[•S9]
[•S9]
Volumes of the
Interior
in cubic centimetres
[and cnbic inches]
1773 [108-19]
1092 [66*64]
1053 [6426]
546 [33-32]
23s [14-341]
214 [1306]
162 [9-89]
74 [452]
49
37
[2-99]
[2-26]
Remarks
Neck
'funnel shaped
Neck oylindrioal
{Keck cylindrical,
month at side
Neck oylindrioal
Smaller spheres did not answer welL In order to tune the resonators, Herr
£oenig has also made them of two short cylinders of which one runs into tiie
other, each having a pierced lid at its external end. One opening serves* for a
connection with the ear or a sympathetic flame, the other is free.f The measure-
ments given in App. IV., p. 377^^, will serve for the manufibcture of such tubes, as
the second opening is of no consequence because it is firmly inserted intOithe ear.
Since metal cubes are troublesome to manufacture and hence proportionately
dear, we can use double cones of tin or pasteboard, the vertices of which have
•been removed. The cone next the ear is more shfurply poizited so that its end
readily fits the ear.
Conical resonators of thin sheet zinc, such as HerrG. Appunn % of Hanau manu-
* * Theory of Vibrations of Air in Tubes
with Open Ends/ in Jourjial fUr reine u, an-
gew. Mathematikf vol. Ivii., equation 30a, and
following.
t Poggendorff's Annals ^ vol. ezlvi. p. 189.
X [Deceased, now Anton Appunn.^
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374 MOTION OF PLUCKED STRINGS. app. ii. in.
factures,* are easily made, and are useful for most purposes. But they reinforce
all the partials of their fundamental tone at the same time. Their length is about
the same as that of open organ pipes of the same pitch.
Eesonators with a very narrow opening generally produce a much more con-
siderable reinforcement of the tone, but then there must be a much more precise
agreement between the pitch of the tone to be heard, and the proper tone of the
resonator. It is just as in microscopes ; the greater the magnifymg power, the
smaller the field of view. Eeducing the size of the orifice also deepens the pitch
of the resonator, and this gives an easy means of tuning it to any required pitch.
But, for the above reason, the opening must not be reduced too much.
I should also mention Herr Koenig*s plan of transferring vibrations of air to
gas flames, and thus making them visible. Flames of this sort act well when
connected with resonators, which are then best made of a spherical form, and
should have two equal openings. To one of the openings the small gas-chaonber
is fixed. This chamber is a small flat box, about big enough to contain two
5[ shilling pieces laid flat on one another. It is cut out of a plate of wood, and
closed on the side next the resonator with a very thin membrane of india-rubber,
which, wkLle it completely separates the air of the resonator from the gas in the
chamber, allows the vibrations of the air to be freely communicated to the gas.
Through the plate of wood two narrow pipes enter the chamber, one introducing
the inflammable gas and the other conducting it away. This last ends in a Teiy
fine point at which the gas is lighted. The vibrations of the air in the re-
sonator being communicated to the gas cause the flame to leap up and down.
These oscillations of the flame are so rapid and regular that, when viewed directly,
the flame appears to be quite steady. Its altered condition, however, betrays itself
by an altered form and colour. Thus to recognise the beats of two tones rein-
forced by the resonator, it is enough to look at the flame and observe how it alter-
nates between its forms of rest and of oscillation. But to see the separate
oscillations the flames should be viewed in a rotating glass, in which the flame at
rest appears to be drawn out into a long uniform ribbon, while the oscillating flame
f appears as a series of separate images of flames. It is thus possible to allow a
large niunber of persons at once to determine whether or not a given tone is
reinforced by the resonator .f
An extremely sensitive means of making the vibrations of the air in a resonator
visible, is a flat film of glycerined soap and water which is placed over its opening.
[Mr. Bosanquet finding that for observations on beats all these resonators im-
perfectly plug one ear and leave the other open, has invented another kind, for
which see App. XX. sect. L. art. 4, b.]
APPENDIX in.
ON THE MOTION OF PLUCKED STBINGS.
(See p. 526.)
Let X be the distance of a point in the string from one of its extremities, and I
the length of the string, so that for one extremity a?=o, and for the other a;=Z.
It is suflicient to investigate the case for which the motion takes place in one
Elane passing through the position of rest. Let y be the distance of the point x
:om its position of rest at the time t And let f( be the weight of the unit
* [There is a set in the Soienoe Collections octaves were sang to the French vowels, were
at South Kensington Museum. — Translator.] also exhibited. See Koenig's paper on the
t [All these instruments and appliances subject, with plates, in Philosophical Maga-
can be obtained of Herr Eoenig, by whom ^tm, 1873, vol, xlv. pp, 1-18, 105-14. — On
they were exhibited in London at the Inter- the principles of the use of revolving mirrors,
national Exhibition of 1872. Large drawings first experimentally used by Sir C. Wheats
of the appearances of the flames just described, stone, see Donjcin's Acoustics, 1870, p. 142.—
as viewed in the rotating mirror while two Translator*]
Digitized by V^OOQIC
APP. in. MOTION OF PLUCKED 8TBINGS. 375
of length, and T the tensiou of the string. The differential equation of motion is
then
'"'w -^'W ^'
Then, since the extremities of the string are assumed to be at rest, we must have
^ = o when rcsro, and also when a; = { (la)
The most general integral of the equation (i) which fulfils the conditions (la)
and corresponds to a periodic motion of the string, is
y = ill . sin ^ . cos ^^n^ + ilj . sin ?^. cos ^ifnt^
+ ils . sin 3^ . cos 6 nnt + &c.
+ Si . sin !!^ , sin 2imt +^2 • sin ^ . sin 4'tmt
+ ^3 . sin 5^ . sin Simt + &c.
(lb)
T
where n* = — =. and Ai, A^, A^, &o., and Bi, B^, B^, &c., are any constant co«
4/1 1*
efficients, which can be determined when the form and velocity of the string aro
known for any determinate time t.
For t ss o, the form of the string will be
y = 4,.flin^+^.55i^+4,.Bin2p + &c (ic)
and the velocity of the string will be
g = a«»(B,.sin!:^ + 2S,.8in?^ + 3B,.8in3p + &c.)... (id) ^
Now suppose the string to have been drawn aside by a sharp point, and that
the point was withdrawn at the time ^ = o, so that the vibration commenced at
that moment, then for t=o there was no velocity, that is -^ was = o for all
^values of ic. This can only be the case when in equation (id), o^Bi ^B^ s=58=&c.
The coefficients A^yA^^A^^ &c., depend on the shape of the string at the time
^ =: o. At the moment the sharp point quitted it the string must have assumed
the position of fig. i8 A (p. 54a), that is, it must have formed two straight lines
proceeding from the sharp point to the fixed extremities of the string. Supposing
the position of the sharp point at that moment to be determined by a; = a and
y = 6, then for the time ^ = o, the value of y^ was
y= — if a>a;>o (2)
and y = 6.7^^if Z>«>a (2a)
*— a
and the values of y in (ic) and Y2), or else (2a) respectively, must be identical.
To find the coefficient A^^ tne well-known metiiod is to multiply both sides of
the equation (ic) by sin ^^ . dx^ and to integrate between the limits 0;= o and
x^L In this case equation (ic) reduces to
|V.sin!^.dx=^..|)in»'?p.«fe (2b)
in which y must be replaced by its values in (2) and (2a). Performing the into-
grations indicated in (2b) we find
^" = mVa(Z-a)-"°-r ^3>
Digitized by
\^oogle
376 MOTION OF PLUCKED STRINGS. app. iii-
Hence A„ will =s o, and consequently the mOi tone of the string will disappear,
when sin — ^- =o, that is, when a = — or= -, or = ^, Ac. Hence if we sup-
pose the string to be divided into m equal parts, and to be plucked in one of these
divisions, the mth tone disappears, and this is the tone whose nodes fail upon
these points.
Every node for an mth tone is also a node for the 2n»th, 3mth, 4mth, &c., tone,
and hence all these tones also disappear.
The integral of equation (i) may also, as is well known, be exhibited in the
following form : —
y^<^(x^at) + ^x + at) (4)
T
where a' = — , and ^, \l/ are arbitrary functions. The function <l>(X'-at) denotes
any form of the string which advances in the direction of positive x with the
^ velocity a, but without any other change ; and the fonotion yl/(x + at) denotes a
similar form proceeding with the same velocity in the direction of negative x.
For any given value of the time t we must suppose both fonctions to be given
from a;=— 00 tQa;= + oo, and then the motion of the string is determined.
The determination of the motion of a plucked string will result in this second
form of solution, from determining the functions ^ and \l^, so that
1) for the values x = o and a; = 2, the value of y for any value of t will be
constantly ss o. This will be the case, if for any value of ^
^( -a^)= ^^+at) (4^)
and ^(Z - a^) = - i/'(i + a^) (4b)
If in the first equation we put at= —v, and in the second 2 + a^ = — t;, we
obtain
iT and 0(2Z + v)= — >/'( — '»)
so that ^(2i + v) = ^(v) (5)
Hence the function ^ is periodic, for its value becomes the same when its
argument is increased by 2 2. The same results for \j/.
2) For t = o, we must have ^= o between the limits a; = o and x^L Henee
at
writing }l/{v) for J^^^J, and putting -^ = o in equation (4), we obtain
av at
And integrating this with respect to x, we have
^(a:) = i/^{x) + C
^1 ISTow since neither y nor -^ are altered by adding the same constant to f and
dt
subtracting it from 4^, the constant G is perfectly arbitrary, and we may conse-
quently assume it to be = q, and hence write
9(^)^^x) (5a)
3) Since finally at the time t^o, and within the limits x^o, x^l^ the
magnitude
y, which is = ^(x) + i(x) = 2<p{x),
must have the value shewn in fig. 18 A (p. 54a), the ordinates of this figure imme-
diately give the value of 2^(x) and of 2}l/(x), by means of equation (5) : —
between x = o and x^zl
„ x=2l „ a; = 3Z
„ a; = 4Z „ x^s^
and 60 forth.
Digitized by VjOOQIC
^Fp. m. IV.
SIMPLE TONES FEOM BESONANCE.
377
But since from (4a, 4b, 5) it follows that ^( — v) = — ^(t?), and ([^(1 — 1?) =
— f{l + v), the value of 2^(0;) is given by the triangle in fig. 18 G (p. 54^)»
between a? =— iandaj=s o
„ a;= — 3Z „ a; = — 2Z
and in the same way between x^ I ,, x^ 2I
and so forth.
By this means the functions ^ and ij/ are completely determined, and on sup-
posing that the two wave-Hnes proceed in opposite directions with the velocity a,
we obtain the forms of the string given in fig. 18, p. 54a, b, which represent the
changes of the string for every twelfth part of the periodic time of its vibration.
[See Donkin's Acotistics, Chaps. V. and VI.]
APPENDIX IV.
ON THE PRODUCTION OF SIBiPLE TONES BY BESONANCE.
(See pp. 55a and 69c.)
I HAVE given the theory of tubes and hollow spaces filled with air, so fax as it can
be at present mathematically expressed, in my paper, entiiled * The Theory of
Aerial Vibrations in Tubes with Open Ends* (±heorie der Luftschwingungen in
Bohren mit offenen Etiden), in Crelle*s Journal filr Mathematik, vol. Ivii. A
comparison of the upper partial tones of tuning-forks and the corresponding reso-
nance tubes, will be found in my paper, * On Combinational Tones ' ( Ueber Com-
hinationstone), in PoggendorflTs Annalen^ vol. xcix. pp. 509 and 510.*
I add here the dimensions of the resonance tubes mentioned on p. 54a, which
were made for me by Herr Fessel, in Cologne, in connection with the tuning-forks m
kept in motion by electricity as described in Appendix VIII. These were cylindrical
tubes of pasteboard, with terminal surfaces of zinc plate, one entirely closed, the
other provided with a circular opening. These tubes therefore had only one
opening, not two like the resonators which were intended for insertion in the ear.
A resonance tube of this kind can have its tone flattened by diminishing its opening.
To sharpen the tone, when necessary, I threw in a little wax, and placed the closed
end of the tube on a warm stove or hob, until the wax was melted, and imifbrmly
distributed over the surface. It was then allowed to cool in the same position.
To try whether the tone of a tube is a httle sharper or flatter than that of the
fork, cover its opening sHghtly while the excited fork is held before it. If the
covering strengthens the resonance the tube was too sharp. But if the resonance
begins to decrease decidedly as soon as any part of the opening is covered, the
tube was too flat. The dimensions in millimetres [and inches] are as follows : —
No. Pitch
Length of Tube
Diameter of Tube
Diameter of Opening
I Bb
425 [1673]
138 [543]
82 [323]
315 [1-24]
2 bb
210 [827]
23-5 [-93]
3 /'
117 [4-^iJ
65 2-56J
16 63]
4 h'b
88 [3-46]
55 2-17]
14-3 -56]
5 d-
58 [2-28]
55 217]
14 -55.
6 r
53 [2-09]
44 173]
12-5 -49
7 a"b'
50 [I-97J
39 1-54]
"•2 -44
8 V'b
40 [1-57]
39 i:i-54]
"•5 -45
9 d"'
35 [138]
30-5 [I-20]
10-3 41
10 /'"
26 [I -02]
26 [I -02]
8-5 [-34]
The theory of the sympathetic resonance of strings is best developed by means
* The hannonic uppQr partials of the air
▼ibrating in the neighbourhood of a tuning-
fork, there mentioned, have also been observed
with an interference apparatus by Herr Stefan
{Proceedings of the Vienna Academy, vol. Izi.
part 2, pp. 491-8) and by Herr Quincke
(Poggcndorfif's Annals, vol. xxviii.).
Digitized by
\^oogle
378 SIMPLE TONES FEOM RESONANCE. app. iv.
of the experiments mentioned on p. 55c. Retain the notations of Appendix III.
and assume that the end of the string for which a; = o, is connected with the stem
of the tuning-fork, and must move in the same way, and that its motion is given
hy the equation
y^^A . sinm^, for ic = o (6)
Suppose the other end of the string to rest on the bridge which stands on the
sounding board. The following forces act upon the bridge : —
i) The pressure of the string, which will increase or diminish according to the
angle under which the extremity of the string is directed against the bridge. The
tangent of this angle between the variable direction of the string and its position
of rest is ^, and hence we can put the variable pressure of the string = — T. -J',
for a; = Z, supposing the bridge to lie on the side of negative y,
2) The elastic force of the sounding board, which acts to bring the bridge back
IT into its position of rest, may be put = — /V*
3 ) The sounding board, which moves with the bridge, is resisted by the air, to
whicn it imparts some of its motion. The resistance of the air may be considered
to be approximatively proportional to the velocity of its motion, and hence be
Then putting M for the mass of the bridge, we obtain the following equation
for the motion of the bridge, and hence for that of the extremity of the string
which rests upon it :
^•S=-^-l-^^-^'4?' ^'-' (^*)
For the motion of the other points in the string, we have, as in Appendix III.,
the condition
'•§=^-g <•>
Since part of every motion of the string must be constantly given off to the
air in the resonance chamber, the motion would gradually die away if it were not
kept up by some continuous cause. Hence we may neglect the variable initial
conditions of the motion, and proceed at once to determine the periodic motion,
which finally remains constant under the influence of the periodic agitation of
the tuning-fork. It is manifest that the period of the motion of the string must
be the same as the period of the motion of the fork. Hence the required integral
of (i) must be of the form
y = Z>. cosjpaj. sin m^ + jB7. cos ^2;. cos mt\ / x
-{-F , sin^aj . sin mt-^- G . sin^a;. cos mtj
And to satisfy equation (i) we must then have
f ^Trf^Tf (7a)
From the equation (7) we have, when a; =30,
y = D . sin mi + jE? . cos mi,
and on comparing this with equation (6) we find
I> = i4, and £ = 0 (8)
The two other coefficients of the equation (7), namely F and G, must be deter-
mined by means of equation (6a). On substituting in (6a) the values of y from
(7), the equation (6a) splits into two, as we must put the sum of the terms mul-
tiplied by sin mi separately = o, and also the sum of those multiplied by cos vni
separately = o. These two equations are :
F . \{p-Mm^). sin^^+2)!r. cos ^q - Gmgr* . sin |?r
=—-4 . [(/*— Jtfm') . cos2?^-p!r. sin^pZ] /g^)
-P77i^«.sinj5Z+a.[(/»-ilfm2).sinpi+2)r.cosi?q ^ '
=— .4f/^wi. cos pZ
^ Digitized by ^vjOOgie
App. rv.
SIMPLE TONES FROM RESONANCE.
379
Assume for abbreviation
f*-Mm>
(/»-Jifm«)«+l>»T«=C«
Then the values of F and O -will be as follows :
p__il C* . sin 2 (p? + ft) + g*m* . sin 2pt
1' (P. axx^ {pl + k) + g*m* . sin^yi
G = -A
Cmg* . sin k
(8b)
(8c)
C* . sin" (i?^ + A;) + g*m^ . sin' jji
Putting the amplitude of the vibration of the extremity of the string which
rests upon the bridge = F, equation (7) becomes
V^^[F.Bmpl + A, GOB pJ]^ + CP. sin« pi, <
and on putting in this equation the values of F and G from (8c) we find
AC . sin k
F =
V [0* .Bia^(pL + lc)+ g^w? . sin ^pj;\
(9)
The numerator in this expression is independent of the length of the string.
Any alteration of its length therefore affects the denominator only. Under the
radical sign is the sum of two squares, which can never = o, because m, g, p, T,
and hence k, can never = o. The coefficient of the resistance of air, g, must cer-
tainly be considered as infinitesimal. Hence the denominator is a maximum, and
F is a minimum, when sin (pi + A;) = o, or when
pl = vTr—k , ,
where v is any whole number. The maximum value of F is
AC
(9a)
f
F^ = -
g'm
Hence, other circumstances beiag the same, this maximum value increases as
Qy the coefficient of the resistance of the air, decreases, and as C increases. To
ascertain the circumstances on which the magnitude of C depends, put for p^ in
the second of the equations (8b), which defines the meaning of C, its value from
(7a), and also put n* =^r>; this gives
C« = Jlf » . (n«-m8)» + TfAmK
Now n is the number of vibrations which the bridge would perform in 2t
seconds, under the influence of the elastic sounding board alone, without the
string and the resistance of the air ; and m is the same number of vibrations for
the tuning-fork. Hence the maximum value of F can now be written f
^'-V["'-(-5)'*^'-]
in which everything is reduced to the weights M, T, fi and the magnitude of the
interval i — J? •
m
JSm>n, which is usually the case, it is advantageous to make the weight of
the bridge if, rather large. Hence I have had it constructed of a plate of copper.
If Jlf is very large, k will be very small in consequence of (8b), and then the equa-
tion (9a) shews that the varidus tones of greatest resonance approach all the more
nearly to those which correspond with the series of simple whole numbers. The
heavier the bridge the sharper the boundaries of the tones of the string.
Observe that the rules here given for the influence of the bridge hold only
for the assumption that the string is excited by a tuning-fork, and not, so far as
this investigation extends, for other cases.
Digitized by VjOOQIC
38o VIBBATION OF PIANOFORTE STRINGS. app. ▼.
APPENDIX V.
ON THB VIBRATIONAIi FOBMS OF PIANOFORTE STRINGS.
(See pp. 74c to Sob.)
When a stretched string is struck by a perfectly hard and narrow metal point,
which is immediately withdrawn, the blow conveys a certain velocity to the point
struck, while the rest of the string receives no motion. Let the moment of im-
pact correspond to ^ = o ; the motion of the string can then be determined on the
condition that at that moment the string as a whole was in its position of equilibrium,
and that it was only the point struck that had any velocity. Hence in equations
-. (ic) and (id) of Appendix III. (p. 3756) put both y = o and -^^ = o for ^ = o, at all
n (It
points except that which is struck, for which suppose the co-ordinate to be a.
Hence it follows that in those equations
o = i4i = ila = il3 = &c.,
and the values of E are determined by an integration similar to that in (2b),
P- 375^1 giving
J) I jodt I
and vnmlB^ = c . sin
Ttiva
where c is the product of the velocity imparted to the struck portion of the string
and of its infinitesimal length. Consequently
vnl ' \
sm y- . sm -y- . Bin 2init
+ -. sin?I- * sin - . sin 4-^nt + i. sin ^. sin 5-^. sin 6wnt + &c^
^ «-=ss-"°? <""
The mth partial tone of the string, therefore, disappears in this case also when
it is struck in a node of this string. Also the upper partial tones are stronger in
comparison with the prime tone, than when the string is plucked, because the
value of A^ in equation (3), p. %i^d, has m^ as a divisor, whereas the value of B^ io
(10) has only m as a divisor. This is immediately confirmed by experiment, on
striking the strings with the sharp edge of a metal ruler.
For a pianoforte, the discontinuity in the motion of the string is diminished by
covering the hammer with an elastic pad. This sensibly diminishes the force df
f the higher upper partials, because the motion is no longer conveyed to a single
point, but is imparted to a sensible length of string, and this too, not in an indivi-
sible moment of time, as would be the case for a blow with a perfectly hard body.
On the contrary, the elastic pad yields to the blow at first, and then recovers itself,
so that while the hammer is in contact with the string, the motion is capable
of extending over a considerable length. An exact ansJysis of the motion of a
string excited by the hammer of a pianoforte would be rather compUcated. But
observing that Uie string moves but very slightly from its position of rest, and
that the elastic pad of the hammer is very yielding and admits of much com-
pression, we may simplify the mathematical theory, by assuming the pressure
exerted by the hammer during the blow which it gives to the string to be as great
as it would be if the string were a perfectly fixed and perfectly unyielding body.
AVe are then able to assume the pressure of the hammer to be
P = i4 sin mU
for such moments of time that o<^<-. This last magnitude ^ is the length of
771 m
Digitized by V^jOOQiC
APP.v. VIBRATION OF PIANOFORTE STRINGS. 381
time during which the hammer iff in contact with the string. The magnitude of
m increases therefore as the elastic power of the hammer increases and its weight
decreases.
We have first to determine the motion of the string during the interval of time
that the hanmier is in contact with it, that is, from ^ = 0 to ^= ~. During this
time, the hammer divides the string into two sections, and the motion of each
section has to be separately determined. At the place of impact let x be written
Xq. When xKx^, distinguish the values of y by writing them y,, and when x>Xq,
by writing them y^. At the point struck the pressure of the string against the
hammer must be equal to the pressure P, which the hammer exerts against the
string. The pressure of the string is to be calculated as in Appendix IV., equation
(6a) (p. 3786), and we consequenUy obtain the equation
P=^.dn^=r. (*-!') (.., t
Waves proceed towards both ends of the string from the place struck. Hence
yi must have the form
for values of tj for which o < ^ < ^, and fl;o> x > Xq— at, and y^ must have the
m
form
for the same values of ^ and for values of x for which tCo < ^ < ^0 + ^^« Using f
for the differential coefficient of the function ^, equation (11) gives
P=:A.saimt=^2T.<i^' (at) (iia)
Integrating with respect to ^ we find %
C- — . cosmi=-^. A (at),
m a
and then, determining the constant C, so that yi = o when js = Xo + at, and y ^ = o
when x=^Xq — at, we have
2mT
< I — cos I -(a; —Xo) + mt\ \ ,
I i-cosf— (aJo-«) + mM j.
This determines the motion of the string for the time t, when o < ^ < _ , and
m
on the supposition that the two waves proceeding firom the place of impact have
not reached one of the ends of the string. If the latter had been the case, tibey ^
would have been reflected there.
When at has become greater than ^ , the pressure P will be ss o, and hence it
follows from equation (iia) that from thenceforward
<&' (at)^o, and hence ^=s constant, when at >^.
m
aA
Hence both y^ and y^ remain = — = for all those parts of the string over which
the waves have already advanced, until portions of the waves reflected from the
extremities reach those parts of the string on their return.
To introduce the influence of the extremities of the string properly into calcu-
lation, suppose the string to be infinitely long, and that at all points distant from
Xq by multiples of 2I, similar blows are given to it, so that from all these places
Digitized by VjOOQ IC
382 VIBRATION OF PIANOFORTE STRINGS. app. v.
waves proceed similar to those which proceed from Xq. Moreover suppose that in
all those places for which x = — Xq^ 2yl, a blow be applied equal to that given to
Xq and at the same time, but in the opposite direction, so that from all these latter
points waves will proceed of an identical form, but with a negative height. Those
points of the infinite string which correspond with the extremities of the finite
string will then be agitated by positive and negative waves of equal magnitude at
the same time, and will hence be completely at rest, and consequently all the
conditions of the real finite string will be fulfilled by the state of this section of
the infinite string.
From the moment that the hammer quits the string, the motion of the string
may be regarded as two systems of waves, one advancing (or in the direction of
positive x), and the other retreating (or in the direction of negative x). Of these
systems of waves we have as yet found only certain isolated portions, namely
those which correspond to the sections of the string lying nearest the point struck.
We have now to complete these waves properly and obtain a connected advancing
IF and retreating system.
Advancing in the direction of the positive x on the string, we have y = o untfl
we come to a positive retreating wave, and then it rises to ~-^, which is its value
in the positive striking points. If we proceed beyond the striking point, and over
the wave thence proceeding, we again find values of y which = o, and sink to
——- as soon as the first negative retreating wave has been passed over. This is
the value of y in the first negative striking point. To connect the positive and
negative retreating waves properly with each other, we must suppose the values of
yi to be increased between every positive striking point and the next foUowing
negative striking point, by the magnitude + ^, so that the height of the wave
retains this value, which it had at Xq^ until the corresponding negative wave begins.
■r Here then the height of the wave becomes — rn'^Vi ^^^ sinks to zero. Similarly,
suppose that =is added to the height of the wave in advancing waves between
ml
any negative striking point and the next following positive striking point. The
retreating waves will thus be everywhere positive, and the advancing waves every-
where negative, and the waves at the same time are so constituted that their con-
tinued motion will generate that kind of motion which we have found to exist in
the string after the hammer quits it.
We have now to express this system of waves as the sum of simple waves.
The length of the wave is 2Z, because the points of simultaneous impact lie at
intervals of 2h Let us take the positive retreating waves at the time tss-
then
i)y, =0, from a? = o to a? = a:o— —
from a; = a?o— — to x^x^
m
3)2/1=— 7Ti» from X = Xo to iC=2/— Xo— —
ml VI
from a; = 2Z-rro-— to ar = 2Z-a-o
m
5) ^1 = o» from X = 2^—3^0 to rr = 2/.
Digitized by
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App. V. VIBRATION OF PIANOFORTE STRINGS. 383
Hence if we assume
y , = ilo + ill . cos J (a; + c) + ^a . cos ^J (a? + c) + ilg . cos ^(x + c) -\- &c.
+ Bi . siny (x + c)+B^. sin y (a; + c) + J?, . sin 35 (x -^ c) -{- &o. (12)
we shall have
*' .nw,
I yi . cos ^ (« + c) . da; = A J,
yi . sin !^ (a? + c) . da; = JB„Z.
^
If we put c = — , every B becomes = o, because yi has the same values for
— + ( and — — (, and the limits of the integration may be selected at pleasure «r
provided only that their difference is 2Z. But on the other hand
1 2aA7nP '^ fnv _ N ^^^ fnv av\ , .
K^—in r-n-a am- sm ( ^ .a;© ) . cos ( -^ . ^ ) (12a)
This equation gives the amplitude A^ of the several partial tones of the com-
pound tone of the string which has been struck. When the point of impact is a
node of the nth partial, the factor sin [ ^ . x^ will = o, and hence all those par-
tial tones disappear which have a node at the point of impact. The table on
p. 79c was calculated from this equation.*
To determine the complete motion of the string we must further substitute
a; -(- a^ for a; in the equation (12) for y^ The corresponding expression for y^ then
becomes
y*=— ilo""-^! -coSy (a; + ai—c)— -4s. cos y (a;— a^— c)— Ac. ^
and finally
y = yi + y> = 2A1 . COS J a? . COS ^ {at +c) + ^A^ . cos — a; . cos ^ {at + c) + &c-
which completely solves the problem.
If m be infinite, that is, if the hammer be perfectly hard, the expression for
A^ in (12a) becomes identical with that of B^ in equation (10), p. 380c. It must be
remembered that m in (10) is identical with n in (12a) (and that a in (10) is then
identical with Xq in (i2aj, while a in (12a) has a different meaning).
If m is not infinite, as n increases the coefficients il« decrease as ^, but if m
be infinite they decrease as i- ; for plucked strings they decrease as -^. This corrc'
spends to the theorems proved by Stokes {Cambridge Transactions, vol. viii.
PP- 533 to 5S4) concerning the effect of the discontinuity of a function, when f
developed in Fourier's series, upon the magnitude of the terms with high ordinal
numbers. Thus, if y is the fdnction to be developed in a series
y = ilo + -4i . sin {ma + Ci) + A^ . sin {2fnz + c^ + &c.
the coefficient of A^ when n is very great,
I ) is of the order L when y itself suddenly alters ;
n
* [It is shewn in the notes on p. 76(2' and foroe of the corresponding partial is materially
p. 77c, that when the blow is made with an weakened, it is not absolutely extinguished,
ordinary pianoforte hammer, the partial tone, It may therefore be necessaiy to re-open the
eorresponding to the node struck, does not mathematical investigation without having re-
wholly vanish. The subject is resumed in course to the facilitation due to the funda-
App. XX. sect. N. No. 2, where the result of mental (but certainly only approximative) as-
later experiments is given. In the meantime sumption on p. 'fiod, giving P = ^ sin nU. —
it must be borne in mind that though the TranslcUor,"]
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384 MOTION OF VIOLIN STRINGS. app. v. vi,
2) is of the order -^ when the first differential coefficient S suddenly alters ;
3) is of the order J^ when the second differential coefficient -^ suddenly
alters ;
4) is at most of the order e-" when the function itself and all its differential
coefficients are continuous. [See note, p. 35^2.]
Hence follow the laws of musical tones so often mentioned in the text, that
their upper partial tones generally increase in power, with the greater discontinuity
of the corresponding motion of the resonant body.
[See Donldn's Acoicstics, pp. 1 19-126, where, on p. 124, equation (14) corre*
^ spends to equation (12a) above.]
APPENDIX VI.
ANALYSIS OP THE MOTION OP VIOLIN STRINGS.
(See p. 83a.)
AssuBfE the lens of the vibration microscope to make horizontal vibrations, then
vibrational curves wiU be observed like those represented in fig. 23, p. 826, c. Call
the vertical ordinate y and the horizontal x ; then y is directly proportional to Uie
displacement of the vibrating point, and x to that of the vibrating lens. Tlie
latter performs a simple pendular vibration. If the number of its vibrations be n
^ and the time t, we have generally
re =s 4 . sin {iimt + c)
where A and c are constant.
Now if y also makes n vibrations, x and y are both periodic and have the
same periodic time. Hence, at the end of each period, x and y have the same
values as before, and the observed point is at exactly the same place as at the
beginning of the period. This holds for every point in the curve and for every
fresh repetition of the vibratory motion, so that the curve appears stationary.
Suppose a vibrational form of the kind depicted in figs. 5, 6, 7, 8, 9, pp. 20
and 21, in which the horizontal abscissa is directly proportional to the time, to
be wrapped round a cylinder^ of which the circumference is equal to a single
period of those curves, so that the time t is now to be measured along the cir-
cumference of the cylinder. Gall x the distance of a point from a plane drawn
through the axis of the cylinder. Then in this case also
^ a; = -4 . sin (2imt + c),
where ii . sin c is the value of x for ^ = o» and A is the radius of the cylinder.
Hence, if the curve drawn upon the cylinder be viewed by an eye at an infinite
distance in the line x =s o, ^ = o, the curve has exactly tide same appearance as in
the vibration microscope.
If X and y have not exactly the same period ; if, for example, y makes n vibra-
tions while X makes n + A n, where A n is a very small number, Uie expression for
X may be written
a? = il . sin [2imt + (c + 2irtAn)].
In this case, then, c which was formerly constant, increases slowly. But c r«>
presents the angle between the plane x =0 and the point in the drawing for which
^ = o. In this case, then, the imaginary cylinder round which the drawing is
supposed to be wrapped, revolves slowly.
Since a magnitude which is periodic after the period ^, may be also considered
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APP. VI. MOTION OF VIOLIN STRINGS. 385
as periodic after the periods 2ir, or 3ir, or I'lr, where r is any whole number, these
remarks apply also for the case when the period of y is an aliquot part of the
period of x, or conversely, or both are aliquot parts of the same third period, that
is, for the case when the tones of the tuning-fork and of the observed body stand in
any consonant ratio ; the only limitation is that the common period of vibration
must not exceed the time required for a luminous impression to become extinct
in the eye. [See Donkin's Acoustics, pp. 36-44.]
Prom the observed curves, fig. 23 B, C, p. 826, and fig. 24 A, B, p. 836, it
follows that all points of the string ascend and descend alternately, that the ascent
is made with a constant velocity, and also the descent with a constant velocity,
which is however different from the velocity of ascent. When the bow is drawn
across a node of one of the upper partials of the string, the motion takes place in
all nodes of the same tone precisely in the manner described. For other points
of the string, little crumples are perceptible in the vibrational figure, but they do
not prevent us from clearly recognising the principal motion, IT
Pia. 63.
If in fig. 62 we reckon the time from the abscissa of the point a, so that for
a, ^ = o, and further for the point fi put ^ = r, and for the point y put ^ = T, so
that the last represents the length of a whole period ; then *
yss/^ + &, from ^ = 0 to ^asr; 1 ,^
y =zg {T -t)-^h, from t=^T to t^T, J ^''
whence for ^ = r, it results that
fr=g{T-T). 1
Now suppose y to be developed in one of Foiurier's series :
y = il, . sin ?^ + -4a . sin ^ + ^8 • sin -^ &o.
+ Bi.C08^+JB,.C0S^* + B,,C0S^&C.
then it results from integration that
and this gives the following values for A^ and B^ :
and y may then be written in the form
In equation (2), y denotes merely the distance of any determinate point of the
string from its position of rest. U x denotes the distance of this point from the
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386 MOTION OF VIOLIN STRINGS. app. vt.
beginning of the string, and L the length of the string, then the general form of
y, as in equation (ib) of App. III., p. 5756, is
+2::r {^-"-^-^('-i)} 13)
By comparing equations (2) and (3) we find immediately that all D*8 Yanish,
or
2)^ = o, and
n .sin^^=^.-^..8in5!^ (3a)
^ Here g +/ and r are independent of x^ but not of n. On taking the equations
for n= I and n^2, and then, dividing one by the other, there residts
^«.C08 2=-.008 5,.
T m
From which it follows that for « = -, t is also= -, as observation shews.
2 2
But if a; = o, then according to observations r is also = o. Hence
C, = i.CH andj^l, (3b)
so that ^ +/ i& independent of x. Let v be the amplitude of the vibration of the
point X in the string, then
IF /r = 5r(T-T) = 2t;,
„ . ^ 2V , 2V 2VT 2VL*
T T-T r(T^T) Txijj-x)
And since ^ +/ is independent of x, we must have
where Fis the amplitude in the middle of the string. From equation (3b) it follows
that the sections a/3 and fiy of the vibrational figure, fig. 62, p. 3856, must be
proportional to the corresponding parts of the string on both sides of the observed
point. Hence we have finally
for the complete expression of the motion of the string.
If we put t— - =0, y will s=o for all values of x, and hence all parts of the
string pass through their position of rest simultaneously. From that time the
velocity/ of the point x is
,_2v_SV(L'-x)
^^V LT
TV
But this velocity lasts only during the time r, where r=s - . Hence after the
time t, and
Tx SV
as long as ^< -r-» ^® ^*^® y='f^^f-m- (i — «)^ (4)
SV
and hence y < ^=^,x(L''x),
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APP. VI. MOTION OP VIOLIN STRINGS. 387
From that point y retumB with the velocity
2V 8 7x
9 — i
■-T ' JjT
And hence after the time <s=r 4- ti.
And since L — a: = — >y,- • L
we find y=^.{r-(r + «.)}
=^. (r-«) (4a) f
The deflection on one part of the string is therefore given bv the equation (4),
and on the other part by the equation ^4a). Both equations show that the form
of the string is a straight line, which m (4) passes through x^L, and in (4a)
through a; = o. These are the two extremities of the string. The point where
these straight hnes intersect is given by the condition
Whence (L-x)t^x(T-i)
and Lt^xT.
Hence the abscissa x of the point of intersection increases in proportion to the
time. This point of intersection, which is at the same time the point of the string
most remote from the position of rest, passes, therefore, with a constant velocity
from one end of the string to the other, and during this passage describes a ^
parabolic arc, for which
Hence the motion of the string may be briefly thus described. In fig. 63 the
foot d of the ordinate of its highest point moves backwards and forwards with a
constant velocity on the horizontal line ab,
Fio. 63.
while the highest point of the string describes in succession the two parabolic arcs
acib and bcsa, and the string itself is always stretched in the two lines aC| and
bo, or aca and bcj. [See Donkin's Acoustics, pp. 131-138.]
The small crumples of the vibrational figures which are so frequently observed,
fif?' 25, p. 846, probably arise from the damping and disappearance of those tones
which have nodes at the point bowed or in its immediate neighbourhood, and
are consequently either unexcited or but slightly excited by the bow. When the
bow is drawn across the string in a node of the mth partial tone situate near to
the bridge, the vibrations of this mth, and further of the 2mih, 3mth, &c., tone
have no influence on the motion of the point in the string touched by the bow,
and they may consequently disappear, wi&out changing the effect of the bow upon
the string, and this really explains the crumples observed in the vibrational figure.
[See also App. XX. sect. N. No. 5, on Prof. Mayer's Harmonic Curves.] I have
not been able to determine by observation what happens when the bow is drawn
across the string between two nodal points.
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38S THEORY OF PIPES. app. \^I.
APPENDIX Vn.
ON THE THBOBY OP PIPES*
A. Influence of Resonance in Reed Pipes.
(See p. 1026.)
• The laws of resonance for cylindrical tubes have been developed maihematicallj
in my paper on the * Theory of Aerial Vibrations in Tubes with Open Ends '
(Theorie der Luftschwingungen in Rdhren mit offenen Enden, * Journal fiir reine
und angewandte Mathematik/ vol. Ivii.). The example treated in § 7 of that paper
is applicable to reed pipes, where the motion at the bottom of the pipe is assumed
% to be given. Let Vdt be the volume of air which enters the reed pipe in the in-
finitesimal time dtj then as this magnitude is periodical we can express Fin one of
Fourier's series, thus : —
F= Co + Ci . cos {2irnt + Tx) + Cj . cos (41m* + ^2) + &c (i)
The resonance must be determined separately for each term, because the vibra-
tions corresponding to each partial tone are superposed without modification.
If we assume Z to be 4ihe length of the tube, S its section, Z + a the reduced
length of the tube (where in cylindrical tubes the difference a is equal to the
radius multiplied by ^ [but see p. 9ieZ, note t]), fe the magnitude ^ (where X is the
4 ^
length of the wave), and put the potential of the wave in free space, for the tone
having the vibrational number i^n.
%
= .J! , cos [vkr^ irmt + c),
where r is the distance from the middle of the opening; then the equations (15)
and (12b) of the paper referred to, give
V (4fl-« cos« rk (Z r a) 4- y*k*S^ sin^ vkl)
Since the magnitude k^S must always be considered as very small to make our
theory applicable, this equation, for cases in which Z + a is not an uneven mul-
tiple of the length of a quarter of a wave, becomes approximatively,
3f^— 9i .
2T.C0Sr^(Z + a)*
Hence the resonance is weakest when the reduced length of the tube is an
even multiple of the length of a quarter of a wave, and becomes stronger as it
>[ approaches an uneven multiple of that length. When it absolutely reaches such
a multiple the complete formula gives
Hence the maximum of resonance increases as the length of the wave of the
tone in question increases and the transverse section decreases. The smaller the
transverse section, the more sharply defined is the limit of the pitch of the tone
which is strongly reinforced by resonance ; while when the transverse section is
large, the reinforcement of resonance extends over a much greater length of the
scale.
For hollow bodies of other shapes, with naiTow mouths, similar equations
may be obtained by means of the propositions given in § 10 of the paper cited.
Since the condition of powerful resonance is that cos lA; (Z -H o) = o, cylin-
drical tubes (clarinet) reinforce only the prime and other unevenly numbered partial
tones [but see note p. 99c].
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APP. vn. THEORY OF PIPES. 389
In the interior of conical tubes we may assume the potential of the motion of
the air for the tube n to be
F= - . sin (ftr + c) cos 2nnL
T
where r is the distance from the vertex of the cone. If a vibrator is introduced at
a distance a from the vertex, and I be the length of the tube, so that for the open
end r = Z + a, we may assume as an approximatively correct Umiting condition for
the free end, that the pressure there vanishes. This is the case when
dV A
-3- = — 2?m . . . sin [k{l 4- a) + c] . sin 2nnt = o, and hence
at I + a
Hence we may assume
sin [k{l + a) + c] = ©•
A
and F= - , sin fc(r— i— a) . cos 2imL
T
The most powerful resonance, then, in this case, as well as in cylindrical tubes,
belongs to those tones which have a minimum velocity at the place where the
vibrator is placed. For as during the development of velocity in the mouthpiece in
equation (i) the coefficients C^ have a determinate value which depends only on
the motion of the vibrator and the pulses of air which it occasions, the coefficient
A of the last equation must increase, as the velocity produced by the correspond-
ing train of waves in the mouthpiece of the tube decreases. The velocity in the
other parts of the tube will then increase. Now the velocity of a particle of air is
dV A
= _- . cos 2rmt • [kr • cos fc(r— i— a)— sin fe(r— i— a)].
Hence for maximum resonance we have the condition that for r s a ^
either hr = tan A;(r— Z— a)
or ia = — tanfcZ.
If, then, the magnitude a, that is, the distance of the vibrator from the vertex
of the cone, be very small, ka and also tan kl will be very small, and kl—w must
also be very small, if i' is any whole number. Hence we may develop the tangents
according to powers of their arc, and retaining only the first term of this develop-
ment we have
ka = mr— kl, and k{a + Z) = ^t
or, putting fc = ~ , we have
A
a + l=iy.-.
f
This shews that conical tubes reinforce all ihose tones for which the whole length
of the cone, reckoned up to its imaginary vertex, is a multiple of half the length
of the wave, on the assumption that the distance of the vibrator from this ima-
ginary vertex is infinitesimal in comparison with the length of the wave» Hence
if the prime of the compound tone is reinforced by the tube, all the partial tones,
both the evenly and unevenly numbered^ will be reinforced up to a pitch where
the wave-lengths of the higher partial tones cease to be very large in comparison
with the distance a.*
CcBteris paribus, the number and magnitude of the terms of the series (i),
which represents the exciting aerial motion will be the greater, the more perfectly
the entering stream of air is interrupted. Free reeds must therefore tit their
* [The remainder of this Appendix VII. to principally from the ist English edition.-—
p. 3966, is an addition to the 4th German Translator.]
edition. The additions on pp. 396-7 are
Digitized by VjOOQIC
390 THEOBY OF PIPES. app. vii.
frames very exactly, in order to produce a powerful tone. Strikmg reeds, which
effect a more perfect stoppage, are superior in this respect. According to the
information obtained by Mr. A. J. Ellis [see p. 95^^', note t], organ-builders have
really been more inclined in recent times to use striking reeds. But the vibrating
laminsB are very sUghtly curved, so that they do not strike the frame all at once,
but roll themselves gradually out upon it.
B. Theory of the Blotuing of Pipes.
When longitudinal waves have once been excited in the mass of air in a tube,
they may be reflected backwards and forwards many times between the ends of
the tube, and form constant, periodically returning vibrations, before they die
away. At the closed end of a stopped pipe, the reflexion of ever^ train of waves
is tolerably complete, but at the open ends a perceptible fraction of the wave
always passes into the open air, and hence the reflected wave has not the same
f intensity as the incident wave possessed. Indeed the intensity of the waves re-
flected backwards and forwards in the tube continually diminishes, and finally
dies off, if the lost work is not replaced at every backwards and forwards reflexion
by some other kind of action. What has to be replaced, however, is usually only
a small part of the whole vis viva of the modulatory motion, that is, just as much
as is lost by reflexion at the open ends. If the inner radius at the open end of a
cylindrical tube be i?, the fraction of the amplitude which is lost at the open end
for a tone having the wave-length A, is, according to theory,
where B is small in comparison with X. In the pipes examined by Zamminer,
the wave-length X varied between S4B and i^'6B. In the first case the loss
would be T^T^th, and in the latter about ^th of the amplitude.
Now, this loss of vis viva can be replaced in various ways. Supposing that
% the small volume dV, which was under the pressure ^0* ^^re forced over into a
space filled with air under the pressure p, the required work would be (p -'PQ)dV.
Hence if during the vibrations of sound, at those places and times where the air
is condensed, either a small quantity of air is regularly forced in, or the pressiue
of the compressed air is increased by heating, this mass of air generates by its
expansion a greater quantity of vis viva than was lost by its resistance to the
condensation at the time the loss occurred. The first of the two pauses is effective
in reed pipes, the second in the tubes of the Pyrophone [see App. XX. sect. N.
No. 4] , In which, together with the air which streams back into the tube, an
increased quantity of gas is poured in firom the gas tube, and this on burning
increases the pressure during the time of re^expansion.
The conditions which must be fulfilled to make reed pipes speak were given by
me in the ' Transactions of the Association for Natural History and Medicine '
(Verhandlungen des naturhistorischr^medicinischen Vereins) at Heidelberg (26 July
i86j), and I take the liberty of reprinting this short explanation here with a few
IT improvementn.
I. The Blowing of Bsed Pipes.
By a reed pipe I mean any kind of wind instrument in which the path of the
stream of wind is alternately opened and closed by means of a vibrating elastic
body, The first work which made the mechanics of reed pipes accessible waa
that of W, Weber. But he experimented chiefly with metal reeds, which on
account of their great mass and elasticity, could not be powerfully moved by the
air unless the tone given by the pipe was not materiallv different from the proper
tone of the reed independently of the pipe. Hence pipes with metal reeds are
usually capable of producing only a single tone, namely that one among those
theoretically possible which is closest to the proper tone of the reed.
The case is different for reeds of light material which offers but little resistance,
such as the cane reeds of the clarinet, oboe, bassoon, and the muBcular reeds of
the human lips in trumpets, trombones, and horns, Beeds of vulcanised India*
rubber, placed similarly to the vocal chords in the larynx, are also well adapted for
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APP. vu. THEORY OF PIPES. 391
experiments ; but, to make them speak easily and well, care must be taken to
place them obliquely to the current of air (p. 976).
The action of reeds differs essentially according as the passage which they close
is opened when the reed moves against the wind towards the windchest, or moves
with the wind towards the pipe. I shall say that the first strike inwardsy and the
second strike (mtwards. The reeds of the clarinet, oboe, bassoon, and organ all
strike inwards. The human lips in brass instruments, on the other hand, are
reeds striking outwards. The india-rubber reeds that I employ may be arranged
to strike either inwards or outwards.
The laws for the pitch of reed pipes are completely found, when we have deter-
mined the motion of the reed as influenced by the alternating pressure of the air
in the pipe and air chamber [see fig. 29, B, p p, on p. g6b] ; remembering that the
effluent air cannot attain its maximum velocity until the passage closed by the
reed has been ojpened as widely as possible.
i) Beeds with cyUndrical pipe without air chamber. The reed is regarded as ^
a body which returns to its position of equilibrium by elastic forces, and is again
brought out of that position by the pressure of the aur in the pipe, which changes
periodically with the sine of the time. The equations of motion* show that the
instant of greatest pressure within the pipe must fall between a maximum dis-
placement of the reed outwards, which precedes it, and a maximum displacement
of the same inwards, which follows it. If the vibrational period be divided into
360®, like the circumference of a circle, the angle c, by which the maximum pres-
sure precedes the passage of the reed through its position of equilibrium, is given
by the equation
tanc = ^'-A\
where L is the length of the wave of the reed in the air without the pipe, X the
wave-length of the tone which is actually produced, and /S^ a constant which is
greater for reeds of light material and greater friction than for heavy and perfectly
elastic materials. The angle c must be taken between — 90^ and + 90"^. %
In the same way we must determine the time at which the greatest pressure
within the pipe separates from the greatest velocity. The latter must coincide
with the position of the reed for which the opening is greatest. The calculation
of this magnitude results from my investigations on ^e motion of air within an
open cylindrical tube {Jotimal filr reine uvd angewamdte Mathematik, Ivii.). The
maximum of the velocity in the direction of the opening precedes the maximum
of pressure by an angle 8 (considering the vibrational period aa the cireumfsrence
of a circle) which is given oy the equation
where S is the transverse section, I the length of the tnbe» and a a constant de-
pending on the form of the opening, being 45** [but see note f and t V' 9}\ ^^^
cylindrical tubes, of which the section has the radius p. The angle 8 in this case
is again to be taken between — 90® and -h 90^.
Now as air can only enter the pipe when the reed leaves the passage open, it f
follows that for reeds which strike inwa/rds the maximum velocity of the air dnrected
outwards must coincide with tiie maximum displacement of the teed inwards.
Hence we must have
2
and both 8 and c must be negative.
For reeds which strike outwards^ on the other hand, the maximiam effluence of
the air must coincide with the maximum di^lacement of the reed outwards, and
we must have
_ir = 8 + €,
2
and both 8 and c must be positive.
* To be treated as in Seebeck*8 theory of the following App. IX. Bat the c there is the
sympathetic resonance, Bepertorium der Phy- complement of &e c here, and wave-lengths are
ti/p, vol. vUi. pp. 60-64. Also see equation 4c in here used instead of pitch numbers, as there.
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392 THEORY OF PIPES. app. vu.
Both cases are included in the equation
tan CSS cot S
or
8inlli^ = ^S^.^ (,j
in which the reeds must strike inwards or outwards respectively, according as the
quantities on each side of equation (i) are positive or negative.
Since S and ^ are very small quantities, sin {4^(2 -|- a) -i-X} cannot have any
sensible value unless \*—L^ is very small, that is, unless the pitch of the mpe
nearly coincides with that of the reed sounded separately from the pipe. This
is generally the case with metal reeds. The viJue of X is determined from
equation (i).
On the contrary when the difference of the two tones X— 2/ is great, then
f sin {4ir(Z + a) -f- X} must be very small, and hence approzimatively
4
where v is any whole number.
The alteration of pressure within the tube is proportional to sin {zw{l + a)-i- X} ,
and hence is a minimum when
and a maximum when
4
In the first case the force of the pressure of the air does not suffice to move the
m reed. In the second case it suffices when the reeds are not too heavy and have
not too great a power of resistance. Hence the tones speak well for which
approzimatively
Z + a = (2i^ + i).^,
4
that is, for which the column of air in the pipe vibrates as in a stopped pipe. At
the same time we see that these tones are almost independent of the proper tone
of the reed.
Of this kind are the tones of the clarinet. Membranous reeds of india-rubber
which strike inwards, attached to glass tubes 16 feet long, also speak easily and
allow various upper partials to be produced which agr^ well with equation (i).
The reeds which strike outwards must be tuned very low in order to give the
pure tones of the tube. Hence the human lips are well adapted for this purpose,
as the bundles of elastic fibres of which they are composed, are loaded with a
IF large quantity of watery inelastic tissue [see footnote^ p. 97^. Cylindrical glass
tubes may easily be blown as trumpets, and give the tones of a stopped pipe.
Of these the upper tones, for which the difference Zr^— X^ is large, can be pro-
duced with firmness and correct intonation, but the lower tones, on the other
hand, being not quite independent of the v^ue of L, that is, of the tension and
density of the lips, are uncertain and variable.
2) Beeds with conical pipes without air chamber. There is a remarkable
difference between cylindrical and conical pipes. The motion of the air in the
interior of the latter may be determined on the same principles that I have used
for cylindrical pipes.
Put the potential of the motion of the air, inside the pipe, equal to
^ { Bin p^ (jB-r + a)l cos 2imt j + ?^^? . cos ?^ {B-r) . sin 2imt,
where r is the distance from the vertex of the cone, B the value of r at the open-
ing [or base] of the cone, S its section, a the difference between the true and
Digitized by VjOOQlC
JLPP, vn.
.THEORY OF PIPES.
393
reduced length, n the pitch number. Considering a -f- X to be very small, and
putting JS— r = i, this gives
. « —X* . 2ir(l + a)
tan h = — - . sm ^ — '^J
• r«
27r(Z + a)
2Tr(l 4- O^
, - , COS _ii^_i:i + _:l. sin ""\' "\
2irb XL X 27rr A J
in which I is to be referred to the place of the reed.
Here also we have to put
cot ^ s= tan c.
We are at present chiefly interested in those tones of pipe which differ much
from the tone of the reed, for which therefore L^—P is great, tan € is also con-
sequently very great, and tan 3 very small. For these then we must either have
approximatively
sin [2ir{l + a) -4- X] = o,
which gives no tone at all, because the alteration of pressure in the interior of the ^
pipe is too weak, or else
tan [2ir(i + a)-i-X] = — 2irr-HX (2)
This is the equation for the powerful upper tones of the pipes.
Below is the series of tones calculated from equation (2) for a conical pipe of
zinc of the following dimensions :
Length I = 1297 centimetres [ = 5i'o6 inch].
Diameter of the openings 5*5 and 0*7 ctm. [ = 2*17 and '28 inch].
Reduced length Z + a, calculated I24'77 ctm. [ = 49*12 inch].
Approximate
Tone
WaTe-Iengths
calcuUiLed
open j stopped
pipes
I. B-
centimetres
28361 -
centimetres
f X 141-80
centimetres
« Jx 70-90
2. 6-
139-83 =
1x139-84
= J X 104-88
3./'»
91*81 -»
|x 137-71
- }x 114-76
4.6' +
6794 =
ix 135-88
= fx 118-89
5. d'% +
5376-
1 X 134-39
« 1 X I20-95
6. ^'-
44.40=1
Jx 133-21
= ^X I22-II
7. ft^b-
37-79-
f X 132-26
»^x 122-82
8. c'"-
3287 =
|x 131-50
= ^x 123-28
9. d'"-
29-22 »
|x 131-47
-^x 124-17
The tones firom the 2nd to the 9th could be observed, and were found to agree
perfectly with the calculation. It appears from the last two columns that the
higher tones were almost exactly those of a stopped pipe, the length of which is ^
equal to the reduced length of the pipe 124-77 ctm., and that the deeper tones
approach nearer to those of an open pipe, the length of which was that from the
vertex to the foot of the cone. The reduced length of this would be JR + a= 142:6
ctm. [ = 56' 1 5 inch]. The tones of brass instruments are usually assumed to be
the same as those of an open pipe, but the higher tones -of these instruments are
relatively too sharp* for the lower ones, in the present case by more than half
* [The text has * flat/ but this is against
the figures. As it will appear that the notes
assigned to the pitches in the first eolnmn
are only roughly approximative, it is best to
calculate out the intervals in cents, and as-
suming that the pitch varies inversely as the
wave-length, we hawe in cents —
For tones .
I
0
0
0
2
3
4
5
6
7
8
9
Reed pipe .
Harmonics
1221
1200
1953
1902
2474
2400
2879
2786
3210
3102
3489
3369
3731
3600
3935
3804
Difference .
21
51
74
93
108
120
Jii
131
Digitized by V^OOQ IC
394
THEORY OP PIPES.
App. vn.
a Tone. In trampets and horns this error is perhaps io some extent compensated
by the caps of the mouthpiece. In trombones the slides assist.*
Whereas trampets, trombones, and horns belong to the reed pipes of this
class with conical pipes, and deep reeds which strike outwards, oboes and bas-
soons have high reeds which strike inwards. When strongly blown they also
give the higher Octave and then the Twelfth, like an open pipe. The calculation
from equation (2) for the oboe agrees very well with Zamminer's measurements.
[Zam. ibid, p. 306.]
n. The Bloiving of Flue Pipes.
In my memoir on * The Discontinuous Motions of Fluids * (Monthly Proceed-
* ings of the Academy of Sciences at Berlin, April 23, 1868}, I nave described the
mechanical peculiarities of such motions, and deduced from the theory how ihey
are brought about by means of the blade-shaped current of air at the mouth of an
m organ pipe which is blown, as described on p. 92a to p. 93a. The bounding surfaces
, of this current which cuts through and across the mass of air that runs into and
out of the mouth of the pipe, are to be considered as vortical surfaces, that is,
surfaces which are faced with a continuous stratum of vortical filaments or thread-
like eddies. Such surfe^es have a very unstable equilibrium. An infinitely ex-
tended plane surfEM^ uniformly covered with parallel straight vortical filaments
might indeed continue stable ; but where the least flexure occurs at any time,
the surface curls itself round in ever narrowing spiral coils, which continually
involve more and more distant parts of the surface in their vortex.
shewing that the tones of the reed pipe are
always too sharp, and not too flat, as the
German text states, the sharpness being a
comma for tone 2, a Quartertone for tone 3,
^ Tone for 4, nearly a Semitone for 5, more
than a Semitone for the rest, the last two
being equally too sharp by aboat 1} Semitone.
% The misprint of {2'"8 tor d'" in the German text
For tones
made the last tone appear to be a whole l\ne
too sharp. In determining the notes in column
I, the Author has probably assumed the velo-
city, of sound at 342 metres (which gives 11 22
feet, or the velocity at about 60' F., see note
p. god), and divided it by the wave-lengths
reduced to metres. This would give —
I
2
3
4
5
6
7
8
1 206
1246
244-6
249-2
372-5
373-5
5034
498-4
6362
628-0
770*3
781-6
905-0
9400
1040-5
10560
B
b
ft
b'
d'%
9"'
6"b
&"
the pitch nos.
While pitch nos. .
belong to equally \^
tempered notes J
Whence it appears that g" and b"\> are much,
and cf'\ d'" somewhat, too sharp, so that the
d"% of the German text is a manifest error.
This rough mode of comparing by vibrational
1170-4.
1 184-2
d'"
numbers, does not however convey a proper
conception. If we calculate the cents from
C66, we find—
For tones
I
2
3
4
5
6
7
8
1 '
the cents
But cents
belong to equally 1
m tempered notes J
1044
IIOO
B
2268
2300
b
2996
3000
n
3517
3500
V
3923
3900
d"i
4254
4300
4600
6"b
4774
4800
4978.
5000
d"
Differences
-56
-32
-4
+ 17
+ 23
-46
-67
-26
-22.
And this shews how very rough are the ap-
proximations to the pitch which are made in
the text by means of equally tempered notes. —
TramlaUyr^
♦ [* The conical tube examined by Prof.
Helmholtz,* says Mr. Blaikley (see note^
p. 97<2), *wa8 not a perfect but a truncated
cone, and any such would have its series of
intervals intermediate between i, 2, 3, 4, 5,
&c., and I, 3, 5, 7, 9, <&c. ; that is, a perfect
cone, or one truncated to an infinitely small
extent, would have the first, and an infinitely
long cone ( » a stopped cylindrical tube)
would have the second. Such a cone as
Helmholtz describes is wit a representative of
the brass instrument family, for if cylindrical
tubing were added at the small end, the series
with this added tube would not even be so
near the theoretical i, 2, 3, 4, 5, <J^o., as on the
original cone. There are brass instruments in
which the series, so far from getting sharp on
the higher tones, gets flat, i, - 2, — 3, — 4, >- 5,^fco.
Technically such instruments are said to be
** sharp at the bottom.** In short, trumpets
and trombones, <&c., are not conical in the
ordinary sense of the word, but have in most
cases a cylindrical tube expanding into a bell
by a line of increasing curvature, so that the
boundaries are approximately hyperbolic*
MS. communication. See also note f* P- 99^
— TraiiilaU>r,'\ .
Digitized by VjOOQIC
APP. VII. THEORY OP PIPES. 395
This inclination of the dividing surfaces of masses of air when moved discon-
tinuously, to resolve themselves into vortices, can also be clearly seen on cylin-
drical streams of air, driven from cylindrical pipes and mixed with a little smoke
to make them visible. In perfectly quiet air and under favourable conditions, they
may reach a length of one foot to three feet. The least noise however makes
them shrink up, as the vortices then commence close to their origin. Professor
TyndaU has also observed and described a great number of similar phenomena of
this kind, in burning gas jets.*
This resolution into vortices takes place in the blade of air at the mouth of
the pipe, where it strikes against the lip. From this place on it is resolved into
vortices, and thus mixes with the surrounding oscillating air of the pipe, and
accordingly as it streams inwards or outwards, it reinforces its inward or outward
velocity, and hence acts as an accelerating force with a periodically alternating
direction, which turns from one side to the other with great rapidity. Such a
blade of air follows the transversal oscillations of the surrounding mass of air ^
without sensible resistance. During the phase of entrance of air, the blade is
also directed inwards, and thus on its part reinforces the vis viva of the inward
currents. Conversely, for the outward current.f
If we suppose the accelerating force of the current of air to be represented by
one of Fourier's series, the amplitude of any term of the order m will in general
diminish as i -nm (see p. 35^2). In fact we require only to use the expreRsion
given in App. HI. p. 375, equations (ib) and (3) for the displacement ^ of a
plucked string, in order to find the value of -^, for the time ^ = o. We thus find the
ax
series which expresses the periodical alternation between a greater and smaller
value of y, as shewn ia fig. 19, p. 54c.
From my memoir * On the Vibrations of Air in Tubes with Open Ends,'
already cited (p. 388a), it follows that throughout the tube a positive component of
pressure coincides with the maximum velocity in the direction of the opening, and
when multiplied by such velocity this component has the value %
aA^k^S
where
a is the velocity of sound,
A the maximum velocity at the end of the tube,
S the transverse section of the cylindrical portion of the tube,
k = 27r -i- \, X being the length of the wave.
If, then, two trains of waves start from any transverse section in the directions of
the two ends, and have the same velocity at that section, the above component
of pressure must be directed in opposite ways in the two trains of waves. This
holds for the place of blowing even when it is quite close to the end of the tube, so
that one train of waves is infinitesimally short. Under these circumstances the
acceleration produced by the air blown in, must correspond to twice that difference
of pressure. Since A is the velocity at the opening, twice this difference of pres- H
sure for the mtii tone, wiQ be
a A^ 2ir Sm^
This would be the only difference of pressure if the tone blown exactly corre-
sponded to the proper tone of the tube. But it may be shewn that this cannot be
made to agree with the mechanism of blowing, and there is always a length fi
which must be intercalated between the two trains of waves in order to reduce
* J. TyndaU On Sounds Lect. VI. , also, in Journal January, 1867 ; Nature^ vol. viii. pp.
Philosophical Magassine, series iv. vol. xxxiii. 25, 45, 383, vol. x. pp. 161, 481, vol. xi. p. 325
pp. 92-99, and 375-391. [24 Jane 1875, P- »45 '» 27 April 1876, p.
f The formation of this blade of air has Sn])* Herr Sohneebeli also gave a mechani-
been described by Messrs. Schneebeli (Pogg., cal explanation of the principal features of
Ann. cliii. p. 301), Sonreck (ib. clviii. p. 129), the process. [See the Translator's addition
and Hermann Smith (English Mechanics^ at the end of this Appendix, p. 3966.]
Digitized by V^jOOQlC
396
THEORY OF PIPES.
APP. VTL
them to an accordant series of constant vibrations. lu this case there is another
additional difference of pressure equal to
—a 4^ sin
2'frmP
For the smaller numbers expressing the order, the sine may be replaced by
the arc, and this latter term considered as the greater. Consequently the lower
partials of the musical tone produced, allow the coefficient m A^io increase as
i-7-m, that is A^ as i -f- m^, and the higher partials allow A^ to increase as
I ^ rn^. The velocities of the partial vibrations in more distant parts of the
external free air contain the factor k once more than the velocities in the tubes
(see equations i2g and i2h in my memoir). These will consequently increase as
I -t- m for lower values of m, which is also the case for the velocities of violin
strings, but for higher values of m they decrease as i h- m*. The greater S is, the
f sooner will this more considerable decrease of the higher partials occur. It is
for this reason especiaUy that organ-builders compare the tones of narrow metal
organ pipes with those of the violin and violoncello.
The circumstances which affect the blowing of pipes and the value of the mag-
nitude P, require a more extended investigation, which I hope to be soon aUe to
give elsewhere.
Additions bt Tbanslator.
[It may be conTenient for those not con-
versant with mathematics to reproduce the
aoconnt of the phenomena, which was given by
me in pp. 708-711 of the ist English edition.
Uerr Sohneebeli had an experimental pipe
constructed in the usual way ; with glass back
and a movable lip and slit or windway, through
which was driven air impregnated with smoke,
as is frequently done to make it visible. When
^ he so placed the lip and slit that the stream
of air passed entirely otUside of the pipe, no
sound occurred; but if he blew gently upon
this sheet of air, at right angles, the pipe
sounded, and the tone continued until he blew
through the other end of the pipe ; neverthe-
less under these circumstances it was very
rare indeed to find that any smoke penetrated
into the pipe. If the sheet of smoked air
passed entirely inside of the pipe, there was
also no sound ; but then on blowing through
the open end, so as to force some of it out,
sound was produced, and it was stopped by
blowing against the slit. This case was there-
fore the converse of the last. He concludes :
*That the stream of air which issues from
the slit forms a species of air-reed (Luft-
Lamelle, aerial lamina), and that this plays
in the generation of vibrations in the mass of
H air, a part analogous to that of reeds in reed
pipes.* He states that the vibrational nature
of motion of the air between the slit and the
lip can be shewn by attaching a piece of silk-
paper to the edge of the lip or the split, and
pressing a point against it. He further pro-
poses a theory founded on Helmholtz's hydro-
dynamical investigations in the Berichte der
Berliner Akademie^ 1868, 23 April, Crelle 60,
and states it to this effect : When the split is
in the normal position the air-reed strikes the
lip, a portion of the stream enters, and pro-
duces a compression as in reed pipes; the
reaction of this compression affects the air-
reed and bends it outwards ; on the pressure
ceasing the air-reed returns to its original
position and tiie process begins afresh.
Mr. Hermann Smith states that the air
from the bellows is iwt directed ' against the
edge of the lip,* and that, if it were so directed,
the pipe would not speak. He also states that
the sharpness of the lip is immaterial to mere
speaking, and that a pipe that speaks well
may have the edge of its lip half an inch
thick. (Compare supril, p. 6qc, where the
wind which excites the sound in the bottle
is blown across its mouth and the edges of
the opening are rounded, not sharps)
The source of tone, according to both Mr.
Hermann Smith and Herr Schneebeli, is the
formation of what the former calls an ^ aero-
plastic reed,* and also simply an '^air-reed,*
and the latter a * Luft-Lamelle * (Prof. Helm-
holtz's * Luft-Blatt,* air-blade, oi; blattfdrmige
Schicht, ' blade-shaped stratum or sheet,* as
used BupriL, p. 394a), which is produced out-
side of the pipe, and bends partly within
it. For the formation of this reed both agree
that it is' essential for the exciting air to pass
the lip, certainly not to enter the pipe. The
existence of the reed is shewn by Mr. Her-
mann Smith by interposing a thin lamina, a
shaving, or crisp tissue paper, which is caught
by the air and vibrates as a reed*, and by Herr
Schneebeli by the smoke mixed with air which
enables the experimenter to see its motion
directly, and also by a piece of silk-paper.
Herr Schneebeli supposes this air -reed to
act by producing condensation, but Mr. Her-
mann Smith's theory of its origin seems to
be as follows. The air driven rapidly and
closely from the slit past the mouth of the
pipe, in a flat stream, just and only just
avoiding the edge of the lip, creates a vacuum,
precisely as in the tubes for ether spray or
perfume spray in common use, or in ordinary
chimney-pots. The air in the pipe under
the action of the atmospheric pressure at
• Press a piece of crisp, bat very light, thin paper
firmly Against the oatalde of the windway by moans oC
a card or piece of ynxA. Let the paper project upwards
till it nearly oorers the mouth, but is quite dear of
both Up and ears. The paper then resembles a free
reed. On now blowing in the u^nal way through the
slit, the pajier wi'.l commence its Tibrations which Mr.
Hermann Smith ponsidcfi to correspond to thoae of his
air-rocd. The eilect is caciiy observed.
Digitized by V^jOOQlC
.APP. VII.
THEORY OF PIPES.
397
the upper open end immediately descends to
supply this vacumn, and by so doing not
<only bends the flat exciting stream of air
outwards, but also of course produces a rare-
faction in the tube, which by extending from
the mouth upwards necessarily weakens the
force of the outward rush of wind. The ex-
ternal (not the exciting) air, taking advantage
of this relaxation of force, enters the tube at
the lip, causing a condensation in the lower
part of the pipe, and the resulting wave of
condensation before it has proceeded half-way
meets the former wave of rarefaction, which
continued to proceed from the further end of
the tube, and thus forms a node. The node
is consequently always nearer the mouth than
the end of the pipe. The ratio of the length
of the segment further from the mouth to
that nearer to it varies from 4 : 3 to 7 : 6 ac-
cording to the diameter or scale of the pipe
and the strength of wind (see p. 91 6, and
footnotes t, %). In the meantime the exciting
stream of air rights itself, passes over the
vertical, bends inwards, and a small portion
of it enters the pipe with the external air, to
be cast out again by the returning wave of
rarefaction, and by this time the exciting
stream of air has been converted into a vibra-
ting air-reed. That the first (but momentary)
effect of the upward rush of the exciting
stream of air is to abstract air from the pipe
Mr. Hermann Smith considers to be demon-
strated by inserting on the 'languid,' just
within the mouth, some filaments of cotton,
fluff, or down, which, in larger pipes, are shot
out with energy. He supposes the air-reed
afterwards to abstract and admit air in con-
stant succession, thus producing the necessaiy
stimulu» for the sound heard, which would on
this theory depend, among other conditions,
upon the force of blast, its inclination to the
mouth, the size and form of the mouth and
ears, the interposition of obstructions between
the reed and outer air (as the shading bar of
the Oamba), the capacity and length of the
pipe, and whether a node has to be formed in
the pipe in the mode explained for open pipes,
or in the mode used for stopped pipes, which
are acknowledged to speak more readily than
the former. The external air of course passes
continually, but intermittently, tiirough tiie
mouth of the pipe.
The law of vibration of the air-reed as
stated by Mr. Hermann Smith, but unob*
served and possibly unobtained by Herr Schnee-
beli, is : * As its arcs of vibration are less,
its speed is preater,* 6r * the times {of vibra-
tion) vary with the amplittidet* being different
from the usual law of a vibrating reed in
which the time is independent of the ampli-
tude of vibration. If by extraneous influence
the pitch of the pipe is flattened, as by partly
shading the mouth from the external air, Mr.
Hermann Smith states that the path of the
air-reed is lengthened, and conversely. When
an organ pipe speaks the tones of the air-
reed and pipe are distinct and may be sepa-
rated or combined; and when a pipe is said
* to fly off to its Octave,* hd says that the air-
reed leaps back to its Octave speed, compel-
ling the pipe to follow, and that this can be
made visible. The natural pitch of the air-
reed is also, he states, far higher than the
pitch of the tones of the pipe.
Herr Sonreck's account, although tho-
roughly independent, agrees with Mr. Her-
mann Smith's so closely as regards the origin
of the motion of air in the pipe that it seems
unnecessary to cite it. He calls the blade of
air merely the Anblasestrom, i.e. * blow-cur-
rent ' or ' blast.'
But so far as I have hitherto found, the
first person who drew attention to the mode in
which flue pipes were made to speak, was M.
Aristide Cavaill6-Coll, the celebrated French
organ-builder, who in a paper presented to the
French Academy of Sciences on February 24,
1840 (which was never printed, owing appa- IT
rently to the death of M. Savart, one of the
referees), * demonstrated,' as he states in his
paper printed in the Comptes Rendus for i860
(even then anterior to all the other writers),
vol. 1. p. 176, 'more dearly than had hitherto
been done, the real function of the originator
of the sound in the mouth of flue pipes, which
originator he assimilated to a free aerial reed
(anche Ubre airienney He also investigated
the mode of blowing the flute, by the mouth
or a mouthpiece. And lastly he treated of
the analogy between 'the transversal vibra-
tions of vibrating lamine of air, and solid
vibrating laminiB,' which he supposed to be
governed by the same laws, and from this
examination he deduced ' positive data for
determining the height of mouths of flue pipes
in relation to their intonation and the eUstic f
force of air which excites them.' He informs
me that he intends to publish this paper.
In conclusion, it diould be observed that
this blade or flat current of air acts like a
metal reed only so far as it oscillates back-
wards and forwards, and not in other respects.
It does not shut off and open out a stream of
air alternately. It is moved inwards by the
outward air, and outwards by the inclosed air,
whereas the metal reed is moved only one way
by the current of air, and the other way by
its own elasticity, l^ere is therefore simply
an analogy and not a substantive similarity,
so that tiie terms aero-plastic reed, air-reed,
free aerial reed, suggesting a different opera-
tion to what actually ensues, might be disused
with advantage. The action seems really to
be one of alternate rarefaction and condensa-
tion. But there are numerous litUe points IT
which require veiy careful study — ^the shape
of the upper lip; straight (as usual) or arched
(as in Benatus Harris's flue pipes), the height
of the opening of the mouth, the exact direc-
tion of the blade of air in relation to that of the
upper lip, the presence and shape of the ears,
and the general arts of the * voicer ' whereby
he makes a pipe ' speak ' satisfactorily. All of
these matters influence both pitch and quality
of tone, and though they are daily practised,
their theory is as good as unknown.
For observations on the action of reeds see
App. XX. sect. N. No. 8.]
Digitized by
Google
398 EXPERBIENTS ON COMPOSITION OP VOWELS, app. viil
APPENDIX Vm.
PBACTICAIi DIBECTIONS FOB PBBPOBMINQ THE EXPEBIMBNTS ON THE
COMPOSITION OF VOWELS.
(See p. 1 2 2d.)
To make the forks vibrate powerfully, it is necessary that the ratios of their piteh
numbers should agree with the simple arithmetics^ ratios to the utmost mcety.
Afber the forks have been toned by the maker by ear and to the piano as accu-
rately as is possible in this way, the necessary greater exactness is obtained by the
electrical current itself. First the interrupting fork, fig. 33, p. 122c, is connected
with the fork that gives the prime tone, and the movable clamp on the former is
^ arranged so as to make the unison perfect. This gives a maxunum intensity to
the prime tone, easily recognised by both epre and ear. The vibrations of this
lowest fork are so powerful that the excursions of the extremities of the prongs
under favourable circumstances amount to 2 or 3 millimetres (from '08 to *i2
inch). It should also be observed that if the unison has not been perfectly at-
tained, a few beats of the fork are heard when the electric current is first brought
to bear upon it, although these disappear when the apparatus is in full action.
[This is accounted for in Appendix IX.]
After perfect unison has oeen accomplished between the interrupting fork and
that of the prime tone, the other forks are successively brought into electrical con-
nection, wiui their resonance chambers wide open, and they are tuned until they
reach a maximum intensity when excited by the current. The tuning is first per-
formed wiUi the file. The forks are sharpened, as is well known, bv taking off
some metal from their extremities, and flattened by reducing the thickness of the
root of the prongs. Both must be done with the greatest possible uniformity to
each prong. To discover whether the fork is too sharp or too flat, stick a httle
f piece of wax at the ends of its prongs (which flattens the fork) and observe whether
the tone becomes louder or weaker. If louder, the fork was too sharp ; if weaker,
it was too flat. Since alterations of temperature, and, perhaps, other causes exert
a slight influence on the pitch of the forks, I have preferred to make the higher
forks a little too sharp by filing, and to bring them into exact tune by attaching
small quantities of wax to the extremities of the prongs. The quantity of wax is
easily altered at pleasure, and by this means shght accidental variations of pitch
can be readily corrected.
It is not necessary to tune the resonance tubes so accurately ; if, when blown
across, they give the same pitch as the forks, the;^ are sufiiciently well tuned. If
they are too flat, some melted wax may be poured in to sharpen them. If they are
too sharp, the opening must be reduced.
It cost me some trouble to get rid of the noise of the spark at the point of in-
terruption. At first I inserted a large condenser of tin foil such as is used in
induction machines. But this merely reduced the spark to a certain size. No
% good effect followed from increasing the size of the condenser. The lasers of the
condenser are separated by thin varnished paper ; one is connected with the in-
terrupting fork, and the other with the cup of quicksilver into which its end dips.
After many vain attempts, I at last found that, by inserting a very long and thin
wire between the two extremities of the conduction at the point of interruption^
the noise of the spark was almost entirely destroyed, without injuring the action
of the current on the forks. The wire thus inserted must have an amount of
resistance far greater than that occasioned by the coils in all the electro-magnets
taken together. When this is the case no sensible portion of the current wUl go
through tliis wire. It is not till the conduction is broken and the thin wire forms
the ouly connection for the extra current of the electro-magnets, that the current
discharges itself through the wire. But to prevent the thin wire itself from
generating any secondary current, it must not be coiled round a cylinder, but must
be stretched up and down on a board in such a way that two adjacent pieces of the
wire should be traversed by two currents, proceeding in opposite directions. For
this purpose I screwed two hard india-rubber combs at the two ends of a board
Digitized by V^jOOQlC
APP. VIII. EXPERIMENTS ON COMPOSITION OF VOWELS.
399
(one foot long), and passed a thin plated copper wire, such as is used for spinning
over with silken threads, backwards and forwards (90 times) between the teeth of
^ese combs. By this means a great length (90 feet) of this wire was brought
i^ell insulated into a comparatively small space, in such a way as not to produce
any sensible secondary current. For when on breaking the primary current a
secondary current would be formed in the wire, this would have a direction in the
circuit formed by the electro-magnets and the thin wire, opposite to the secondary
current in the electro-magnets, and the latter would be entirely or partly pre-
vented from discharging itself through the thin wire.
For moving the forks I used two or three cells of a Orove's battery. The
electro-magnets were placed in two rows beside one another. The whole arrange-
ment is shewn in a diagram in fig. 64, below. The figures i to 8 shew the
resonant chambers of the tuning-forks ; the dotted lines which lead to mi and mg
are the threads which remove the cover from the opening of the resonance .
chambers ; ai to ag are the electro-magnets which set in motion the tuning-forks f
Fig. 64.
tar^
m SH^
- — my
^^^
If
between their legs ; h is the interrupting fork, and f its own electro-magnet. The %
relative position of the two last has been somewhat changed in order to make the
connection of the direction of the currents more intelligible. The cells of the
galvanic battery are marked Ci and e, ; the great resistance-wire dd ; the condenser
c, but its plates which are rolled in a spiral are seen only in section.
The electric current passes from e^, through all the electro-magnets in order,
up to the handle of the interrupting fork g. It is sometimes more advantageous
to arrange this part of the conduction so that it should be separated into two
parallel branches, and that the three highest forks, which are the most difficult to
set in motion, should be inserted into one branch, and the five lower forks into the
other, thus allowing a stronger stream to pass through the former than the latter.
The remainder of the conduction from g to the second pole of the battery Ci
contains the interrupting apparatus, which is here so arranged that each vibration
of ttie fork makes the current twice ; once when the upper prong dips into the
cup of mercury h, and once when the lower prong dips into the cup i. When
the conduction is closed at h, it passes from g through the fork to h, and then
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400 PHASES OF WAVES CAUSED BY RESONANCE, app. vm. re.
through the electro-magnet f to k and e^ Between h and k it is generally neces-
sary to insert a lateral branch h 1 k, having moderate resisting power, to weaken
the current in the eleotro-magnet f , sufficiently to prevent the fork b from maJ^g
violent vibrations. The zigzag at 1 shews this branch.
When the prongs of the fork move apart, the conduction will be broken at h,
and after a short interval again completed at i, so that the current now passes
from g through the lower prong of the fork to i, and thence by k to the battery at
C). But at the moment the conduction is broken either at h or at i, poweifdl
secondary currents are formed by induction in the 8 electro-magnets of the tuning-
forks, which would emit luminous and noisy sparks at the points of interruption,
if the rush of electricity were not partly stored for the moment in the condenser c,
and partly discharged through the very great resistance dd.
This resistance dd, as is seen by the figure, forms a perpetual connection be-
tween g and the battery, but it conducts so badly that no sensible part of the cnr-
.f rent can pass through it, except at the moment when, on breaking the conduction*
the great electro-motive force of the secondary currents is generated.
The arrangement just described is preferable when the fork in front of the
resonance chamber i is the Octave above the fork b. But if the fork opposite i
makes the same number of vibrations as the fork b, the wire i k must be removed,
and both the other wires ending in i must be connected with h.
To exclude particular forks from the circuit, short secondary connections of the
coils of wire of their electro-magnets are introduced. The arrangement is shewn
in fig. 32, p. J 2 lb. The metal knobs h h are connected condnctively with the
clamping screw g in which the wire of the electro-magnet terminates. If the
lever i is moved down, it presses with some friction on the nearer knob h, and
forms so good a secondary conducting connection for the wire of the electro-
magnets, tiiat the greater part of the electric current passes by h h, and only
an infinitesimally small part travels round by the much longer path of Uie electro-
magnets.
As regards the theory of the motion of the forks, it is immediately seen that
IT the force of the current in the electro-magnets must be a function of the time.
The length of the period is equal to the period of a vibration of the interrupting
fork b. Let the number of interruptions in a second be n. Then the strength of
the current in the electro-magnets, and consequently the magnitude of the force
exerted by the electro-magnets on the forks, will be of the form :
Aq-^Ai. cos (2'jmt + Ci) -I- ila . cos {4imt + c^)
4- ilj . cos (67m^ -f c,) + &c.
The general term of this series A^ . cos {iirmnt •{- c^) is adapted for setting in
motion a fork making mn vibrations in a second, but would have little effect on
forks otherwise tuned.
\ APPENDIX IX.
ON THE PHASES OP WAVES CAUSED BY RESONANCE.
(Seep. 1246.)
Let a tuning-fork be brought near the opening of a resonance chamber, and sup-
pose the ear of the Ustener to be at a great distance off in comparison with the
dimensions of the opening. In the Journal filr reine und arwewandte Mathe*
matiky vol. Ivii. pp. 1-72, in my paper on the * Theory of Aerial Vibrations in Tubes
with Open Ends,' I have shewn that if a sonorous point exists at jB in a space partly
bounded by firm walls, and partly unbounded, the motion of sound at another
point Ay in the same space, will be identical in intensity and phase with that
which would have existed at B if A had been the sonorous point. Let B be the
position of the tuning-fork (or more properly of the end of one of its prongs), and
A that of the ear. The motion of the air which begins when the tuning-fork ia
placed near the opening of the resonance chamber, is not easily determined, but
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APP.tt. PHASES OP WAVES CAUSED BY EESONANCE. 401
I have been able (in pp. 47 and 48 of the paper quoted above) to determine the
motion when the tuning-fork is at a great distance. Let us suppose, then, that the
fork is removed to the position of the ear A, and we shall then have to determine the
motion of the sound at the point B near the opening. This motion is composed
of two parts ; the first, having its potential denoted by ^ in the paper cited, corre-
sponds with the motion which would also exist if the opening to the resonance
chamber were closed, and in'the above case is too small to be sensible ; the second,
there marked ^, has, in open space and at some distance from the opening, the
following value, using the notations explained in the above paper (p. 28, equation
12 h),
ir = -^.cos(A;^-27mi) (i)
2irp
where Q is the sectional area of the resonance tube, p the distance from the middle
point of the opening, n the pitch number, ^ the length of the wave. The motion IF
at an infinitesimal distance r from the sonorous point A is given by the equation.
^^^ cos(2irn^-c) .
r
and if r^ be the distance of the imaginary sonorous point A from the middle
of the op^iing of the resonance tube, we find from equations (i6c) and (13a) of
the paper cited :
. /, , N . A;* . Q . sin W . cos Ara , •
-tan (A;ri+c) = tanr2 = ^ j--t- — -- (2a)
^ 27r . cos k\^L + a) ^ '
(Z length of tube, and a a constant depending on tlie form of its opening), and
finally by the same equations, the magnitude there called I is :
r TT 2A: . sin A;Z . ^ X-. Zf^ . sin ^Z
Tx 2T . sm ra H
whence it follows that ^ = ±H.4!;ii^5Ji (3)
The si^ ± is to be so determined that the consonants A and B. have the same
sign, and m that case r^ must lie between o and ir.
Iq this case the strength of the resonance A is expressed in terms of the in-
tensity of the sonorous point H, the section of the resonance tube Q, the distance
Ti of the sonorous point from the opening of the tube, and the magnitude r,. The
difference of phase between the points A and B is shewn by equations (i), (2), and
(2a) to be
TT — Zf/i + c = TT — A;p — fcri — r^.
But the magnitude Up at all such distances of the point B firom the middle of
the opening as we can use, may be regarded as iofinitesimally small, so that when
we weaken the tone by withdrawing the fork further from the opening of the tube, f
we do not sensibly change the phase of the aerial vibration. But if we change
the pitch of the tube, the expression for the phase will be altered only through a
change in r^, which by equation (2a) depends on hi, and to this change there
always corresponds a change in the strength of the resonance, since sin r^
appears as a fEkctor in the expression for that resonance in equation (3). The
resonance is strongest when sin r, = i, or 73 = \v. Calling this maximum reson-
ance il], we have
and for other pitches of the tube, supposing its sectional area Q to remain un-
changed,
A
sm rj = -J- .
Whether rj is to be taken smaller or greater than a right angle, depends upon
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402 PHASES OF WAVES CAUSED BY RESONANCE. app.i^.
whether the value of tan r, from equation (2a) is positive or negative. Bat
since k, Q and cos ka are always positive, the value of tan r^ depends on the
factor — ?J5L The maximum resonance corresponds to cos Ar(Z + a) = o : the
cos k(l'\- a) ^ V / »
minimum to sin A;Z = o. Hence r^K^ when bv lengthening the tube the reson-
ance is brought towards its minimum ; but T2>ffr when the resonance is brougfat
towards its maximum. In actual application the tube is always near its posi^on
of maximum resonance, and hence r2<^ir when the tube is too flat in pitch, and
r2>^7r when the tube is too sharp in pitch.
If we put the tube out of tune to such an extent that A* = ^.A^^,iihe phase of
vibration alters by ^. Hence we are always able to estimate the amount of altera-
tion of phase by the alteration of strength of resonance.
x = -
A similar law holds for the phases of the vibrating forks as compared with
^ those of the exciting current. To simplify the treatment, I will consider the case
of a single vibrating heavy point, which is constantly restored to its position of
rest by an elastic force. When the heavy point is moved to the distance x from
its position of rest, let— a^^ be the elastic force. Suppose moreover that there
act, first a periodic force, similar to that generated by the electrical currents in our
experiments, which may be represented by il . sin nt, and secondly a force which
damps the vibrations and is proportional to the velocity, so that we may write it
— ft* -7>. A force of the latter kind arises in our experiments partly from friction
at
and resistance of the air, and also partly from the currents induced by the tuning-
forks set in motion, and this latter part has most effect in damping the vibrations.
If m is the mass of the heavy vibrating point, we have, therefore,
m.f^ = -a»ir-6«^ + i4.sinn^ (4)
dt^ at
^ The complete integral of this equation is
^.J55J.sin(n^-f) + ^e"^.sin ( -.>/(a«m-J6^) +c j ... (4a)
where tan£= + -H = (4b)
a^ — mri'
The term having the coefficient B in (4a) is sensible only at the beginning of
the motion ; on account of the factor e >» it decreases with the increase of the
time tf and ultimately vanishes. But its existence at the beginning of the motion
occasions those transient beats mentioned in App. VIII., p. 3986, when n is slightly
different from
I >/ (a«m-i6«).
m
f The term with the coefficient A in equation (4a), on the other hand, corresponds
to the sustained vibration of the heavy point. The vis viva t* of this motion is
equal to the maximum value of ^ m . [ ~ ] , or to
., m^«.8in»e .
2b* ^^'
When the pitch of the exciting tone, that is, n, can be altered, i^ will reach its
maximum (which we will call P), when
8in*€=i, dr tan € = ±00,
givmg ^'^Ib^'
Hence we may also write
••-. (5«)
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APt^.ix. PHASES OF WAVES CAUSED BY BESONANCE. 403
The same magnitude e therefore determines in equation (4a) the difference
of the phases between the periodically changing displacements x of the heavy point
and the changing values of the force, and, in equation (5a), the strength of the
resonance.
The condition tan £ = ± 00 is by (4b) falfilled when a* « m n*.
Hence if ^ be the value of n which answers to the maximum of the sym-
pathetic vibration, we shall have
3
J^ = ^ (Sb)
This tone of strongest resonance is the same as the tone which the heavy
point would occasion, if it were set in vibration solely by the influence of the
elastic force, without friction and without external excitement. Somewhat, differ-
ent from this is the proper tone of the body, which it produces under the influence
of friction and resistance of the air. The pitch of this proper tone y is given in %
the second term of the equation (4a)
Not until 6 = 0, that is, not until the friction and resistance of the air vanish,
will r^z^^^IP.
m
Now in all practical cases where we have to observe the phenomenon of
sympathetic vibration, b is infinitesimal, so that the difference between the tone
of greatest resonance and the proper tone of the vibrating body may be disregarded,
as in the text. Introducing the magnitude N the equation (4b) becomes
*-'=^K^«) '^) ^
On account of the question raised on p. 150& as to the behaviour of the basilar
membrane of the ear for noises, we are interested farther in the integral of an equa-
tion in which il sin n^ of equation (4) (p. 4026) is replaced by an arbi^ary function of
Uie time i/^^. Of course, if this function vanishes for very great positive and negative
values of the time, it could be transformed, by means of Fourier's integral, into a
Bum (integral) of terms such as il sin {nt -f* c), and then for each one of these
terms, the solution just found might be applied, and finally the sum of all these
solutions might be taken. But this form of solution becomes mcomprehensible,
because it eiddbits a continuous series of tones each of which exists from ^ss— 00
to ^= + 00 . Hence we must proceed differently.
The differential equation to be integrated is
.-^+^'l+'»'-='^ (s) ,
in which x is the required, and \p the given function of the time, 1/^ being assumed
to be finite for all values of t.
Assume
y + sA^A r tJ/,.e-^'-^.ds (6)
where k represents one of the roots of the equation
mic« + &»ic + a» = o (6a)
that IB
«=_^±^f^-i*.) (6b)
whieh we will represent by
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404 .PHASES OP WAVES CAUSED BY EESONANCE. app.h.
That is, X is shewn to be a sum of superposed expiring oscillations, of which the
initial time is 8, and the initial amplitude -^^ and every moment preceding the
point of time t contributes to the result. But this contribution vanishes for those
parts of the motion which were excited for a long time before the moment con-
^ sidered, that is/for those for which the exponent a {t—s) is a large number, and
the motion therefore depends at every instant only on those forces i^ which have
acted a short time previously.
If the action of the force \l/ takes place only during a limited time from to to
ti, then X of the equation (6d) will not be=so except up to the time ^o, lifter
which it will differ from nothing, and after ti the motion will be that of simply
expiring vibrations. Also the magnitude of x will depend upon how often large
positive values of 4/ occur at the same time as large positive values of sin fit, and
negative with negative. The value of x will be comparatively the greatest when
i// and sin fit change their sign nearly at the same time.
If ;//( has had a constant value p from ^ = ^0 to ^3=^,, then
on putting
assuming the coefficient of damping to be smaU enough for the root, which we '
represent by fi, to be possible.
Hence, if 1// is a continuous function, j
|^(y + rci)=A-P J...e'('-\& + ^^, (6c)
£^{y^xi)^AK^Vj..e<*-'\ds + AKx(^,+A^^ (6d)
Then, multiplying (6) by a*, (6c) by b^, (6d) by m, and adding, and taking account
of (6a), we obtain the following equation between the imaginary parts of the le-
spective expressions.
Then assuming A = -—
pm
equation (6) gives a value for x which satisfies the di£Eerential equation in 8, and
is finite for all values of the time, namely
X^He -^'-^) Bm{fi{t-ti)-he-ri]
k cos ri = — a, fcsini/=)8
-^ ^^ ' "" ^ ^' "^ '^'''^^ ''''' A^i -^0)]
If tire suppose k to have the positive value of %/(♦«* + j8*), )? wiU be an obtuse
angle. If we give H the sign of the pressure p, then the angle c, which hes
between + ^^ and — '^, will have the same sign as sin fi{ti -~ <o)* ^ this case the
expression for x represents expiring vibrations, of which the initial amplitude
(patting r s the length of the action ss t| — ^0) has the value
H^^JSL^s/ (1-2 c-'cos/Jr + e -»•').
This is a maximum for different values of r, when cos (/3r + ij) =5: cos ly . c^*, or,
for small values of a and r, when fir approximately contains an uneven number
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A^. IX. X. SYMPATHETIC RESONANCE. 405
of half periods of vibration of the proper tone ; and on the other hand f is a
minimmn for an even number of &uch vibrations.
After long continued action of the force jp, however, the exponential functions
vanish, and H receives the constant value
H ^ P
On the other hand, for very small values of r the initial maxima for /3r = w, may
attain the value
If the pressure p changes its sign whenever cos j3r does so, the ampUtude H after
n such changes of sign will be ^
or
H= JL^ . i±£Z . (I ^ e-(»+i)-M.
fimk I — e — ' ^ ^
This expression shews the reinforcement, increasing with every change of sign,
which ensues upon the coincidence of the period of change of pressure with the
period of the proper tone. The denominator (i — e ~") gives the amount of damp-
ing during half the periodic time of vibration. Finally, when this is very smtJl
H will be very large, and at last, afker an infinite number of repetitions.
I— e"
APPENDIX X.
BSLATION BETWEEN THE STRENGTH OF SYMPATHETIC BEBONANCE AND THE
LENGTH OF TIME BEQITIBED FOB THE TONE TO DIE AWAY.
(See pp. 113a and 142^2.)
Retain the notation of App. IX., for the motion of a heavy point, reduced to
its position of rest by an elaistic force. When such a point is agitated by an ex-
ternal periodic force, its motion is given by equation (4a), p. 402c. If we assume
Af the intensity of this force, to vanish, equation (4a) reduces to f
x^B.e a»» •. sin (fi + c)
where y=^ • V (a^w— i 6^).
On account of the factor which contains ^ in the exponent, the value of x
continually diminishes. As in the text, measure t by the number of vibrations of
the tone of strongest resonance, and for this purpose put
2n
/J IT 6* /IT n\ , f^.
/Vjoogle
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4o6 VIBRATION OF THE BASILAR MEMBRANE. app. x. n.
Let L be the vis viva of the vibrations at the time tso, and I at the time
i^U then
.c-*^^
BO that l = e-»^^
Li
and T = iz.lognat~ (6a)
In the table on p. 143a, it is assumed that L^iol, and the value of T is cal-
culated on this assumption, as follows, after finding the value of p. In equation
(6), sin* e is put sss^, corresponding with the condition that the strength of the
% tone of the sympathetically resonant body should be ^ of the maximum strength
it can attain ; and the ratio ^ : n is calculated from the numerical ratios corre-
sponding to the intervals mentioned in the first column of that table.
Equation (4b) in App. IX., p. 402c, may be whttei^
V p
tan £ = =
„ fN n\ fN n\
In this equation N, giving the pitch of strongest resonance ; 6', the strength
of the friction; and m, the mass may be different for various fibres of Gorti.
Hence in applications to the ear, we must consider b^ and m to be functions of N.
Now since the degree of roughness of the closer dissonant intervals remains tolerably
constant for constant intervals throughout the scale, the magnitude represented
by tan t muEt assume approximatively the same values for equal values of --, and
[ **
hence the magnitude — ^= niust be tolerably independent of the values of N.
No very exact result can be obtained. Hence in the calculations which will
follow hereafter p is assumed to be independent of N.
APPENDIX XI.
VIBBATION OF THE MEMBRANA BASXLABIB IN THE OOOHLEA.
(See p. 1466.)
f The mechanical problem here attempted is to examine whether a connected mem*
brane with properties similar to those of the membrana basilaris in the cochlte,
could vibrate as Herr Hensen has supposed this particular membrane to do; that
is, in such a way that every bundle of nerves in the membrane could vibrate sym-
pathetically with a tone corresponding to its length and tension, without being
sensibly set in motion b^ the adjacent fibres. For this investigation we may dis-
regard the spiral expansion of the basilar membrane, and assume it to be stretched
between the legs of an angle, of the magnitude 217. Let the axis of x bisect this
angle, and the axis of ^ 1^ drawn at right angles to it through the vertex of the
angle. Let the tension of the membrane parallel to the axis of x be = P, and
that parallel to the axis of ^ be = Q, both measured by the forces which when
exerted on the sides of a unit square, parallel to x and y respectively, would balance
the tension of the membrane. Let /j. be the mass of this unit square, t the time,
and z the displacement of a point in the membrane from its position of equilibrium.
Moreover let Z be an external force, acting on the membrane in the direction of
positive z, and setting it in vibration. The equation of the motion of the mem-
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Ai»r. XI. VIBRATION OF THE BASILAR MEMBRANE. 407
brane, deduced without material difficulty from Hamilton's principle by EirchhofTs
process, is then
^+^-®+«-i'='-4' • <'^
The limiting conditions are
1 ) that 2r = o along the legs of the angle, that is, jsr = o, when y = ± a? . tan >/.
2) that £r = o, when a; = y = o, that is, at the vertex of the angle, and finally
3) that z is finite, when x is infinite.
The farther development of the problem will shew how these fcwo last limiting
equations, which suffice for our purpose, may be replaced by certain determinate
curves acting as fixed boundaries between the legs of the angle (p. 41 ic, d).
By putting x^i , s/P and y =1; . n/ Q, the equation (i) may be reduced to the ^
better known form
which is the equation of motion for a membrane stretched uniformly in all direc-
tions, I and V being the rectangular co-ordinates on its surface.
For this notation the limiting conditions become
i) ;?=:o for i;=:±£ . n/— . tan j;,
2) ;j=o for f = v = o,
3) z finite, for 5 = 00 .
The transformed problem consequently differs from the original merely in
baying a uniformly stretched membrane, and a different amount of angle, which
we will represent by 2e. H
Since in the applications which we have to make of the result, P will be very
small in comparison with Q, the angle c for the transformed membrane will also
be very small, and upon this circumstance mainly depend the analytical difficulties
of the problem.
After these preliminary remarks, we proceed to the analytical treatment of the
equations (i) and (la) by introducing polar co-ordinates, assuming
icssf . s/Pssr . n/P . cos taf 1 i y^
y = v . s/q^r . s/Q . sin «/ ^'^^
The equations (i) and (la) then take the form
d^z . 1 dz , 1 d^z d^z „ ,-.
dP+f jr+p-5;?=''-5F-'^ ^"^
The limiting conditions are now, that ^
* p
i) esso, when utss + c, and hence tan £=:^/-.. tan 17,
2) 2r = o for r=o,
3) z is finite, when r is infinite.
As regards the nature of the force Z, we shall assume that it consists of two
dz
parts ; the first depending on the friction, which we may put = — v . -,- , where y
is a positive real constant ; the second, depending on a periodically variable pres-
sure exerted by the surrounding medium on the membrane, uniformly over its
whole surface. Consequently we put
at
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4o8 VIBRATION OF THE BASILAB MEMBRANE. app. xi.
and obtain as the equation of motion
£'+?-l+^-&='-»+-|-^-"- <»>
Of the various motions which the membrane could execute under these circam-
stances, we are interested solely in those which are maintained by the continaons
periodical action of the force, and which must themselves have itxe same period.
Let us consequently assume
j? = f .e'"*, where i=-v/(-i) («»)
and determine C by the equation
t ^K-fr*^-^^^---'"'-'—^ «"■'
In this case the real part of the value of z will satisfy equation (2) and corre-
spond to a uniformly sustained oscillation of the membrane.
Having thus eliminated the variable t from the differential equation, we proceed
to do the same for w by means of the first limiting equation, after transforming
both C and the constant A into a series of cosines of uneven multiples of the
angle — = ^. It is well known that between the limits ^ s=s + ^ and — ^
2e
il=:-^ . /"cOS^— i.C08 3^ + i.COSS;i« + ...} (3)
IT V 3 5 ^
If in the same way we put
4:=si . cos ^« — i .S3, cos 3^+ ' •»6-oo8 s^+ (3ft)
«' 35
then for each coefficient 8^ we must have
And since the first of our limiting conditions is satisfied by the equation (3a).
whenever the series converges, there remain only the conditions that
i) s,. = oforr=o,
2) s^ finite, for r = 00 .
It is easily seen that every s^ is perfectly determined by these conditions. For
if there were two different functions which satisfied the equation (3b) and the two
limiting conditions, then their difference, which we will call S, would satisfy the
conditions
5^ + r-5-r-^(''~-^^^-^-^^^ (^"^
and hence be a Bessel's function, and at the same time we should have
i) 3s=o for r = o,
2) 2 finite for r = oo.
But these two conditions cannot be satisfied at the same time by a Bessel's func-
tion, when V has a value differing ever so little from o. It is only when i^so,
that is, when there is no friction, that the determination is insufficient. In that
case oscillations once induced may continue for ever, even when there is no force
to give fresh impulses.
Particular integrals of the equation (3b) may be easily developed in the form
of series, resembling the series for the related BessePs functions which satisfy
equation (3c). One of these series proceeds according to integral powers of r and
is always convergent. But when the angle c is very small, the number of terms
Digitized by V^jOOQlC
APP.xi. VIBRATION OP THE BASILAR MEMBRANE. 409
in this series which are necessary to determine s is very large, and hence the series
ctmnot be used for determining the progress of the function. — A second series
which proceeds according to negative powers of r and gives a second particular
integral is semi-convergent, and will not become an algebraical function, unless h
is an uneven number. But in the latter case the first mentioned series will be
infinite in its separate terms.
It is therefore preferable for our present purpose to obtain the expression for
8 in the form of definite integrals.
Let ^ and }p denote the following pair of integrals : —
^=1 u.e \ uj
du
(4)
where ic= v'lftn*— i»v) (4a) %
and the sign of the root is so chosen that the possible part of i ic is positive*
Then
3^=:^. (mA.i/' + m/t.^,cos Jw&ir— i) (4b)
which is the required expression for 8^,
To shew that the expression in (4b) really satisfies the equation (3b), substitute
this value for s^ in that equation, and in differentiating under the integral signs of
1/^ and ^, use partial integration to eliminate the ia^tora cos t and [1^—- ] which
appear under the integral signs.
For r = o we find
f du ^ I
mh
and hence 9,^ = 0.
For r=oo , we have ^s=\/.s=:o, and hence
8 — ^
Hence the function 5,,, also satisfies the two limiting conditions, which have
been already shewn to be sufficient to determine it.
The equation (4b) may be used to determine the value of 8^ when P, the ten-
sion of the membrane in the direction x, is infinitesimal. Li this case, as (ib)
shews, r must be the iofinite ; as also h, of which the value is -r
ft- ^>VQ
2 . v P . tan ly*
Hence putting
rzszhp
p will be the finite, namely
2X . tan 71
It is easily seen that under these circumstances mhf will = o. For we may
write
mki,^r mh.e-^f^'^ogu-il-i.)M.(u^iy^ ^^^
u
where I have put
iics=i-iA^
Digitized by
Google
4IO VIBRATION OF THE BASILAR MEMBRANE. app. xl
and I according to the above supposition will be positive. Since within the whole
extent of the integration t^ > i and hence log u>o, the possible part of the ex<
ponent will be negative throughout the same extent, and will contain the infinite
factor h. Consequently every part of the integral vanishes, and hence also the
whole value h <f>.
On the other hand the integral rp or
^-i: e
— (Z— iX) . ^.sin ^
. sin mht . dt
will have the possible part of the exponent negative and infinite for all those parts
of the integral for which t is not infinitesimal, so that these will all = o. Bat
this is not the case for those parts of the integral for which t vanishes.
Hence for an infinite h we may replace the above equations for ^ by the
f following :
^ Jo . emmht .dt
In this last form the integration may be effected and gives
''"^ ^.[(Z-iA)«.f)« + m«] J ^^*^
and S^ = , a a a^
or, by (4a), ^
,^ = _^^i£.^.,-^j (5b)
Or if, in order to get rid of the auxiliary magnitude p, we represent by ^fi the
^ value of y on the limits of membrane, we have
ifi^x tan 7/,
and hence 0 = — '- — --,
so that [using S^ for the modulus of ^J,*
S««=-
4A
!?^!![i^-».i»V4..«.i1 (5d)
^[(^-A^n^)» + n».«]
This value is quite independent of the magnitude of the angle through which
the membrane is stretched. In place of the distance p or a; from the vertex, we
have only /3 the breadth of the membrane at the point in question. Hence this
expression will still hold when the angle is = o, and the membrane vibrates like a
string between two parallel lines, thus forming m vibrating segments which are
^ separated by lines of nodes parallel to the edges.
The same expression also results for a string, if j? is regarded from the first in
equation (i^ as only a function of ^ in a line, and supposed to be independent of
X, but the limiting condition is retained that when ^=: ±/3, then 2r = o. Hence
the motion of the membrane is the same as that of a series of juxtaposed but
unconnected strings.
The value of — . 5^ in (sd) gives us the amplitude of the corresponding fonn
m
of vibration having the pitch number ^ , and having m vibrating transverse divi-
sions of the membrane. The maximum of S^ will occur when
m*ir«0-/3V»* = o (6)
* [In the 3rd German edition S^ is used <« is henceforth nsed for S^ ; consequentlj
without the explanation here inserted; in the the reading of the 3rd edition has been re-
4 th German edition by an error of the press tained. — Translator,]
Digitized by V^OOQIC
APP. XI. VI. THEORY OF COMBINATIONAL TONES. 411
The valae of this mftTimTiTn, which we call 2„ is
any
The smaller the coefficient of friction v, the larger ^nll be this maximum at the
point in question.
If we cskH h the value of /3 which satisfies the equation (6), we may write the
equation (sd) thus
/r mU^Q^ f\ I \n
When V is infinitesimal, and the condition of the maximum is not fulfilled in
equation (6), the denominator of this expression becomes infinite, and hence S^
infinitesimal. The amplitude of the vibrations 1- . £>,„, will become finite, for If
those values of /3 only which are so nearly = &, that & — /? is of the same order
as V, Under these circumstances, therefore, each simple tone sets in vibration
only some narrow strips of the membrane in the direction of x, of which the first
has one, the second three, the third four, &c., vibrating segments, and in which ^,
that is the length of the vibrating segments, has always the same value.
The greater the coefficient of friction r, the greater in general will be the
extent of the vibrations of every tone over the membrane.
The present mathematical analysis shews that every superinduced tone must
also excite all those transverse fibres of the membrane on which it can exist as a
proper tone with the formation of nodes. Hence it would follow, that if the
membrane of the labyrinth were of completely uniform structure, as the membrane
here assumed, every excitement of a bundle of transverse fibres by the respective
fundamental tone must be accompanied by weaker excitements of the unevenly ^
numbered harmonic undertones, the intensity of which would, however, be mul-
tiplied by the &ctors i, i- , and generally -L. Although this hypothesis has been
9 25 w*
advanced by Dr. Hugo Biemann in his Musikalische Logik, there is nothing of the
kind observable in the ear. I think, however, that this cannot necessarily be urged
as an objection against the present theory, because the appendages of the basilar
membrane probably greatly impede the formation of tones with nodes.
The solution can also be extended without difficulty to the case where the
membrane in the field of £, v is bounded by two circular arcs, with their centre at
the vertex of the angle. To this case correspond as boundaries in the real case,
that is, in the field of x, y, two elliptic limiting arcs, which when P vanishes be-
come straight lines. It is only necessary to add to the value of 5,^ in (4b), a com-
plete integral of the equation (3c), which can be expressed by BessePs functions
with two arbitrary constants, and to determine these constants in such a manner ^
as to make «,» = o on the limiting curves selected. When y is small this change
in the limits has no essential efi'ect on the motion of the membrane, except when
the maximum of vibration itself falls in the neighbourhood of the limiting curves.
APPENDIX Xn.
THEOBY OF COMBIKATIONAIi TONES.
(See pp. 152, note f, and 158, note *.)
It is well known that the princ^)le of the undisturbed superposition of oscillatory
motions, holds only on the supposition that the motions are small, — so small,
indeed, that the moving forces excited by the mutual displacements of the par-
ticles of the oscillating medium should be sensibly proportional to these displace-
Digitized by^OOQlC
412 THEORY OP COMBINATIONAL TONES. apf.mt.
ments. Now it may be shewn that combinational tones must arise whenever the
vibrations are so large that the square of the disjplacements has a sensible in^
fiuence on the m>otions. It will suffice for the present to select, as the simplest
example, the motion of a single heavj point onder the influence of a system of
waves, and develop the corresponding result. The motions of the air and other
elastic media may be treated in a pei^ectly similar manner.
Suppose that a heavy point having the mass m is able to oscillate in the
direction of the axis of x. And let tiie force which restores it to its position of
equilibrium be
isKoaj + ia?*-
Suppose two systems of sonorous waves to act upon it, with the respective
forces
/ . sin ptf and g .em (qt + c)
% then its equation of motion is
cPx
—m. -^ ss ax -^ bx^ -hf . Bin pt + g . Bm {qt -^ c)»
This equation may be integrated by a series, putting
and then equating the terms multiplied by like powers of e, separately to zero.
This gives
1) axi+m.-j^ = -fi.anpt-gi .em(qt + c),
af
3) ax^+ m . -7^ = — 2bxi ^8» ^^^ so on.
From the first equation we obtain
Xi^A . sin^i . N/^-+fc j + t* . sin^i + v . Bin (g^ + c)
where tt=s — /l — and v = — ^ — .
mpr^a mq^—a
This is the well-known result for infinitesimal vibrations, shewing that the
body which vibrates sympathetically produces only its proper tone >/^, together
% with those communicated to it, p and q. Since the proper tone in this case
rapidly disappears, we may put Jl a=o. And then equation (2) gives
2a '
2(4mp'
^ .COS2pt -, ^ 1 . COS 2(qt + c)
^^ . cos [(i, - q)t^c] -._--^^^^_ . COS [(;, + q)t +c].
The second term of the series for x [involving x^, contains, then, a constant,
and also the tones 2p, 2q, (^—9)9 and (j> + ^)» If the proper tone n/^ of the
body which vibrates sympathetically is deeper than (2? — g)» aa mav be certainly
assumed in most cases lor the drumskin of the ear in connection with the auditory
Digitized by V^jOOQlC
App.xn. xm. MECHANISM OF THE POLYPHONIC SIREN. 413
ossicles, and if the intensities u and v are nearly the same, the tone {p — q) -will
have the greatest intensity of all the tones in the terms of x^ ; it corresponds with
the weU-biown deep combinational tone. The tone (p + q) will be much weaker,
and the tones 2p and 2q will be heard with difficulty as weak harmonic upper
partial tones of me generating tones.
The third term x^ [of the series for x] contains the tones ^p, $q, 2^+?, 2^—9,
p-h2q,p^2q^p and q. Of these 2p^q or 2q^p is a combinational tone of the
second order according to Hallstroem's nomenclature (p. 154^). Similarly the
fourth term x^ [of the series for x] gives combinational tones of the third order ;
and so on.
If, then, we assume that in the vibrations of the tympanic membrane and its
appendages, the sq^uare of the displacements has an effect on the vibrations, the
preceding mechamcal developments give a complete explanation of the origin of
combinational tones. Thus the present new theory explains the origin of the
tones (^ -fg), as well as of the tones {p^q)t and shews us, why when the intensities
t^ and V of the generating tones increase, the intensity of the combinational tones, «-
which is proporaonal to uv, increases in a more rapid ratio.
The previous assumption respecting the magnitude of the force called into
action, namely
k=:ax + boc^
implies that when x changes its sign, k changes not merely its sign, but also its
absolute value. Hence this assumption can hold only for an elastic body which is
nnsymmetrically related to positive and negative displacements. It is oiUy in such
that tiiQ square of the displacement can affect the motion, and combinational
tones of the first order arise. Now among the vibrating parts of the human ear,
the drumsldn is especially distinguished by its want of symmetry, because it is
iorcibly bent inwards to a considerable extent by the handle of the hammer, and I
venture therefore to conjecture that this peculiar form of the tympanic membrane
conditions the generation of combinational tones.
[See especially App. XX. sect. L. art. 5.] ^
APPENDIX Xm.
DBSOSIPTIOK OF THE IfEOHANIBM EMPtiOTED FOB OPENING THE SEVERAL SERIES
OF HOLES IN THE POLYPHONIC SIREN.
{See p. 162, note *.)
Fia. 65 (p. 414a, b) shews the vertical section of the upper box of the double siren, in
order to display its internal construction. E is the wind pipe which is prolonged
into the interior of the box, and firmly fixed in the cross beam AA of the sup-
port of the apparatus. The prolongation of the wind pipe into the box B has
conical surfaces at its upper and lower ends, on which slide corresponding hollow tr
surfaces in the bottom and top surfaces of the box, so that this box can revolve
freely about the wind pipe as an axis. At a may be seen a section of the toothed
wheel fastened to the cover of the box. At /3 is the driving wheel which is turned
by the handle y ; and 2 is a pointer which is directed to the graduation on the
edge of the disc te.
D is the upper extremity of the axis of the movable discs, of which only the
npper one CC is here shewn. The axis turns on fine points in conical cups. The
upper cup is introduced into the lower end of the screw 17, which can be more or
less tightened by the milled screw head introduced above, so that any required
degree of ease and steadiness in the motion of the axis may be attained.
Inside the box are seen the sections of four pierced rings k\, X/i, fiv, and vo,
which fit on to one another with oblique, tile-shaped edges, and thus mutually hold
each other steady. Each of these rings lies beneath a series of holes in the cover,
and contains precisely the same number of holes as the corresponding series of the
cover and of the rotating disc. By means of studs, shewn at ii in fig. 56 (p. 162),
Digitized by V^OOQIC
414
VARIATION OP PITCH IN BEATS.
App. xin. xrv.
these four rings can be slightly displaced, so as either to make the holes of the
ring coincide with the holes of the box, and thus give &ee passage to the air and
produce the corresponding tone ; or else to close the holes of the cover by the
interspaces between the holes of the ring, and thus cut off their corresponding
tone idtogether.
In this way it is possible to sound the various tones of this siren in succession
or simultaneously, and hence obtain separate or combined tones at pleasure.
APPENDIX XIV.
VABUTION IN THE PITCH OF SIMPLE TONUS THAT BEAT TOGETHEB.
(See p. 1656 and note *.)
Let t; be the velocity of a particle, which vibrates under the influence of two tones,
so that
V =A . Bm mt + B . an (nt + c)
where m differs very slightly from w, and A>B. We may then put
n^ + c = m^ — [[m ^n)t^c\, and
^ f)= [A +3 . cos [(m — n) t^c]] . Bmmt—B . sin [(w— n) * — c] . eosm^
Assume
A-\
B . sm I
and
80 that
A+B ,QOB [im — n) ^ — c] =s: C . cos €,
sm £,
t? = C . sin {mt — c),
in which C and e are functions of the time f, which will alter slowly, if, as we have
assumed, m — n is small in comparison with m.
The intensity (? of this osciUation is determined by
C^^A^+2 AB .QOB [(m-w) *-c] + 5«,
and it will be a maximum,
C= {A + J5)*, when cos [(m - n) * - c] = + x,
Digitized by V^OOQlC
App. XIV. XV. INTENSITY OP BEATS OF DIPFEBENT INTERVALS. 415
and a minimum,
C^=(il— B)*, whencos [(m — w) ^ — c] = — I,
The variable phase e of the motion is determined by
tanc- ^.Bin[(m-n)^-c]
A'\-B .COS [{m — n) t — c]
A.B A>B, this tangent never becomes infinite, and hence e remains included
between the limits +^ and — ^ir, to which it alternately approaches. As long
as e increases, mt — e mcreases more slowly than mt ; as long as c diminishes,
mt — € increases faster than nU ; hence in the first case the tone flattens and in
the second it sharpens.
The pitch number of the variable tone, multiplied by 2 tt, is under these cir-
cumstances equal to 1
_ Jg ^ m i4'+ {m-^-n) , AB . COB [(m — w) t — c] + n jg'
dt il«+ 2 .ili?. cos [(m-n)^-c] + B^
The limits for the pitch number therefore correspond
to cos [(m — n)t — c] becoming + i or — i,
and hence also to a maximum or minimum strength of tone.
i) When the strength of tone is a maximum, the pitch number varies as
mA +nB __^_{m — n) B __^ ^(m'-n) A
A+B "^ TTB ""^ A+B '
3) When the strength of tone is a minimuTn, the pitch number varies as
mA —nB_^. (m— n) B ^ .im — n) A f
-Zirfi-='*+ ^-iri- =»+ -JITS-
Hence in the first case, [or during the maximum strength,] the pitch of the
variable tone Ues between the pitches of the two separate tones. But during the
minimum strength, if the stronger tone is also the sharper, the pitch of the
variable tone is sharper than that of either of the single tones; and if the stronger
tone is the flatter, the pitch of the variable tone is flatter than that of either of
the single tones.
These differences are well heard with two stopped pipes ; and also with two
tuning-forks when first the higher and then the lower is placed nearer to the
resonance chamber.
[See Mr. Sedley Taylor's paper on this subject, Philosophical Maaazine^ July
1872, pp. 56-64, where he gives several figures illustrating the variabiUty of the
pitch, and deduces the above results (i) and (2) firom the figures only.]
APPENDIX XV.
CALCULATION OF THE INTENSITY OF THE BEATS OF DIFFEBENT INTEBVALS.
(See pp. xSya and note* and 193, note *.)
We shall again employ the formulae for the strength of the sympathetic vibration
developed in Appendix IX., equations (4a) and (4b), p. 402c, and (5) and (5a),
p. 402S. For the tone of strongest resonance in one of Corti's elementary organs,
let n be its number of vibrations in 2ir seconds, n^ and n^ the corresponding num-
bers of vibrations in 2ir seconds for two tones heard, and B, B" the greatest
velocities of the vibrations which they superinduce in those Corti's organs which
have the same pitch, and B^^ B^ the greatest velocities which both attain in their
Digitized by V^OOQIC
4i6 INTENSITY OP BEATS OP DIFFEBENT INTEBVALS. app.xv.
representation of the number of vibrations n. Then by equation (5a) of Appen-
dix IX., p. 402d, we have
Bi =B' sin ei, and B^^B'^ . sin e,
where «• . tan «, = — — — , and «• . tan t^ = — 1- — ,
ni n n^ n
and /3 is a magnitude which may be regarded as independent of n. Hence the
intensity of the vibrations of the organ for the number of vibrations n, when both
tones Uy and n^ affect it simultaneously, fluctuates between the values
{B^+B^f and (B^^B^)\
The difference of these two magnitudes, which measures the strength of the
f beats, is
4B, -82 = 4 B' B" . sin e, . sin «, (7)
Hence for equal differences in the amount of pitch, the strength of the beats is
dependent on the product B' B". For the mth partial tone of the compound tone
of a violin, we may, by Appendix VI., p. 597c, put 5'* = — ^ , and hence \1 the With
rn,
and m^th partial tone of two compound tones of a violin, beat, we may put the
intensity of their beats for equal differences of interval = .
m\ . m^
This is the expression &om which the numbers in the last column of the table
on p. 187& have been calculated. [They are therefore 100 times the reciprocak of
the products of the two numbers which give the ratio of the pitch numbers in the
corresponding line of its third column.]
For tiie calculation of the degree of roughness of the various' intervals, men-
tioned in pp. 193, 332, and 333, the following abbreviations of notation are in
1[ troduced :
w, + n, = 2 J»f.
n, =JV(i+5).
n^ =iV (i - 1).
n =^(i + >')-
So that
V . tan ei= — ; — - — —5 8^d V . tan £5=-
I -^ S 1 -j- y I — oi-f-r
Since powerful sympathetic resonance ensues only when v and S are very small,
we may assume that, approximatively,
tan c,= — -1- ., and tan €•= 1 .
27r(»'— d) 23r(i'-f-<))
U Putting these values for cj and f2> ^ equation (7) we have
4 5. B,= 4 B' B" . ^^^ ^^, ^ ^^ _ ^^^ '^ ^_^ ^^, ^^ _^ ^^,^ ... (7a)
If then we consider y, that is, the pitch of the Gorti's organ which vibrates
sympathetically, to be variable, 4 B^ B^ will reach its maximum when v==^, and
hence n=iVs=^ [n^ + n^), and if we call the value of this maximum s we have
'-'^'^'-w^ ('»')
In calculating the degree of roughness arising from sounding two tones together
which differ &om each other by the interval 2I, I have thought it sufficient to
consider this maximum value, which exists in those Corti's organs which are most
favourably situated. Undoubtedly other beats of a weaker kind will be excited in
Digitized by V^jOOQlC
APP. XV. INTENSITY OF BEATS OF DIFFERENT INTERVALS. 417
the neighbouring fibres, but their intensity rapidly diminishes. It might therefore
appear to be a more exact process to integrate the value of 4 5, J?, in equation
(7a] with respect to r, in order to obtain the sum of the beats in all Corti^s organs.
This would require an at least approximate knowledge of the density of Corti's
organs for different values of y, that is, for different parts of the scale, and of that
we know nothing. In sensation, the highest degree of roughness is certainly more
Important than the distribution of a less degree of roughness over many sensitive
organs. Hence I have preferred to take only the maximum of the vibrations from
(7b) into account.
Finally we have to consider that very slow beats cause no roughness, and that
•when the intensity of the beats remains unaltered, and their number increases,
the roughness reaches a maximum and then decreases. To express this, the value
of s must be also multiplied by a factor, which vanishes when the number of beats
is small, attains a maximum for about 30 beats in a second, and then diminishes,
and again vanishes when the number of such beats is infinite. Suppose then that
the roughness r^, due to the j?th partial tone, is expressed by ^
4eva^£f
The factor of Sp reaches its maximum value =s i, when p^^0\ and becomes
=: o, when 0, that is, half the interval between the two tones in the scale, is = o
or 00 . Since h may be indifferently positive or negative, the expression can only
involve even powers of h. The above is the simplest expression which satisfies
the conditions, but it is of course to a certain extent arbitrary.
For 0 we must put half the extent of the interval which at the pitch of the
lower beating tone causes 30 beats to be made in a second.
Since we have taken & with 264 vibrations in a second, as the lower tone, 0
has been put = -^ . Hence we have finally
264
r = 16 B'B- ^^'cy^_
•(/3^+4'r«^»)(«Hi^«t»r %
And from this formula I have calculated the roughness of the intervals, shewn
graphically in the diagrams, fig. 60, A and B, p. 193^, c, and fig. 61, p. 333a.
The roughnesses due to the separate partial tones have been drawn separately
and superimposed on one another in the drawing.
Although the theory leaves much to be desired in the matter of exactness, it at
least serves to shew that the theoretical view we have proposed is really capable
of explaining such a distribution of dissonances and consonances as actually occurs
in nature.
Professor Alfred M. Mayer, of Hoboken, New Jersey,* has instituted experi-
ments on the duration of sensations of sound, and the number of audible beats.
Between a vibrating tuning-fork and its resonator he interposed a revolving disc
witii openings of the same shape as that of the resonator, so that the sound was
heard loudly when an opening in the disc came in front of that in the resonator,
and faintly when the latter was coshered. His results agree essentially with the
assumptions I have made on pp. 143 to 145, and 183 to 185, but are more com- m
plete as they have been pursued throughout the whole scale. His conclusions are
as follows : —
* SiUiman's Journal, ser. iil. vol. viii. tinaity wonld have the ratio 2048 : 2249, giving
October 1874, PhUoaqphicdl Moffazine, May 162 cents, and the interval of maximum disso-
1875, ^<>1* ii* [From the following table, p. nance would be 67 cents, and hence the beats
418a, it is seen that the interval of a minor of b'" c"" should be quite conspicuous, agree-
Third as the limit of continmty applies only ing with observations. In reference to p. 1446,
to the Octave c to c'. For gf"", supposing the Prof. Mayer observes that the law abruptly
law connecting D and N, given on p. 4i&i, to breaks down for vibrations below 40 in a
hold for such a high pitch, the interval of con- second, and thinks that this abrupt breaking
tinaity would have the ratio 3072 : 3072 + 225, down * can only be explained by the highly
or 122 cents, and the interval of maximum probable supposition that co-vibrating bodies
dissonance would be 49 cents, so that the in the ear, tuned to vibrations below 40 per
interval of one Semitone is near the limit of second, do not exist, and therefore ... the
continnity ; hence it is not surprising that no inner ear . . . can only vibrate en masse,* and
beats were heard in the ease referred to on also that such oscillations cannot last ^ sec.
p. 1 73c, note *. But for c"" the interval of con- — Translator,]
Digitized by ^JOOgle
4i8
BEATS OP COMBINATIONAL TONES.
APP. XV. XTl.
Number of the
besto for which
the interruptions
ranish
[Cents in the
Number of beats
[Gents in the
Note
Pitch number
corresponding
interral*
lor the greatest
interrmlf
c
64
16
386
6-4
165 !
c
128
26
308
104
135 1
d
256
47
292
188
123
^^
384
60
251
240
105
c"
512
78
245
31-2
loa
e"
640
90
228
36-0
9S
9"
768
109
230
43-6
§5
c"'
1024
US
214]
540
89]
APPENDIX XVI.
ON BEATS OF COMBINATIONAL TONES, AND ON COMBINATIONAL TONES IN THE
SIREN AND HABMONIUM.
(See pp. 199a and note *, also 155c to 158a.)
Let a, h, c, d, e,/, g, h be whole numbers. Let an and ^n+^ be the pitch nomben
of the primes of two compound tones sounded simultaneously, where B is supposed
to be very small in comparison with n, and a and b are the smallest whole numbers
by which the ratio a : b can be expressed. The pitch numbers of any pair of
partial tones of these two compound tones will be
acn and bdn-^-dd.
These will beat with each other dh times in a second, if
1^ ac^bd or r = ~'
And since the ratio •- is expressed in its lowest terms, the smallest values of
o
d and c are
and their other values are
d^a and c=s&,
d^ha and c = %2>.
Now 0 and d represent the ordinal numbers of the partial tones which beat
together. Hence the lowest partials of this kind will be the bih partial of tiie
compound a n, and the ath partial of the compound bn+L The resulting number
of beats is a 3.
In the same way the 25th partial of the first compound, and the 2ath of the
% second give 2a c beats, and so on.
The first differential combinational tone of the two partials acn and bdn + di
is
±l{bd'^ac)n + dh]
where the + or * sign has to be taken so that the whole expression is positive.
* [The interval is foand as the ratio of the
pitch number to the same increased by the
nomber in the next colomn to it; thus for
C it is 64 : 64+ i6»4 : 5, and for g' it is
384:384 + 60=96:111, and from these I
have calculated the cents as in p. 701 of the
ist edition.
If^l^ be the pitch number, and I>» dura-
tion of residual sensations or Uie reciprocal of
the numbers of vibrations producing a con-
tinuous sound, 16, 26, (tc, as in the next pre-
ceding column, then Prof. Mayer finds—
.4).
seconds. — Tratu-
^^U + 23 +
lator,^
f [The interval is found as the ratio of the
pitch number to the same increased by the last
mentioned number of beats, thus 64 : 644-6-4.
Prof. Mayer draws attention to the fact that
his beats were all tones of the same pitch,
whereas the beats of imperfect consonances
are tones of variable pitch.--7Vafu2aior.]
Digitized by V^jOOQlC
App. XVI. COMBINATIONAL TONES OP SIREN, ETC. 4^9
Two other partials/aTi and gbn-^- g^ give the differential combinational tone
±[(gb-af)n'^gh].
When both sound together they produce (gzfd)^ beats, if
6eZ — ac=±(^6 — a/)
a gz^d
or
b-fzfc
As before, it follows that the least value of ^ + eZ is = a, and the other (greater)
values are = ^a, so that the smallest number of beats is a^.
To find the lowest values of the partials which must be present in order to
beat with the first differential tones, we will take the lower signs for c and d, and
we thus obtain :
gr = eZ = ia, or ^ = i (a + i), and d = i
/=:c = |fc, or/ = |(6 + i), and c = J
according as a and b are even or odd. If b is the larger number, ^b or i(b + i) is
the number of partials which any compound must have in order to produce beats
when the two tones composing the interval are sounded. If the combinational
tones are neglected, about double the number, that is b, are required.
When simple tones are sounded together, the beats arise from the combina-
tional tones of higher orders. The general expression for a differential tone of a
higher order arising from two tones with the vibrational numbers n and m is
± (an —.few), and this tone is then of the (a + 6— i)th order. Let the pitch
number of a combinational tone of the (c + ^ ~ i) order arising from the tones an
and {bn + i) be
±[{bd-ca) . n-^dh],
and of another of the (/+ ^— i)th order be 1
±[{9b-fa),n + g^l
tlien both produce (gz^d) , ^ beats, when
bd — ac=^±.{bg —af)
a Q'ln.d.
or -==^:?^.
b f^LC
The lowest number of beats is therefore again a^, and the lowest values of
c,d,f, g, are found as in the former case, so that the ordinal numbers of combina-
tioncJ tones need not exceed ^(a +6 — 2), if a and b are both odd, or ^(a +6—1),
if only one of them is odd, the other being even.
To what has been said in Chap. VII., pp. 154-159, 1 wiU add the following f
remarks on the origin of combinational tones.
Combinational tones must always arise when the displacement of the vibrating
particles from their position of rest is so large that the force of restitution is no
longer simply proportional to the displacement. The mathematical theory of this
case for a heavy vibrating point is given m App. XII. , pp. 4 1 ie2 to 41 5^. The same
holds for aerial vibrations of finite magnitude. The principles of the theory are
given in my essay on the * Theory of Aerial Vibrations in Tuoes with Open Ends,'
in Crelle's Jowmal fur Mathematik, vol. Ivii. p. 14. I will here draw attention
to a third case, where combinational tones may also arise from infinitely small
vibrations. This has already been mentioned in pp. is^d-isjd. It occurs with
sirens and harmoniums. We have here two openings, periodically altering in
size, and with a greater pressure of air on one side than on the other. Since we
are dealing only with ver^ small differences of pressure, we may assume, that the
mass of the escaping air is jointly proportional to the size of the opening ta, and to
the difference of pressure p, so that
Digitized Syi^OOgle
420 COMBINATIONAL TONES OF SIREN, ETC. app. xvi.
where c is some constant. If we now assume for uf the simplest periodic fdnction
which expresses an alternate shutting and opening, namely
w = i4 . (i — sin 2irnt)f
and consider j? to be constant, that is, suppose uf to be so small and tiie influx of
air so copious, that the periodical loss through the opening does not essentiallj
alter the pressure, q will be of the form
q = B . (i — sm2irnt)
where B=^cAp.
In this case the velocity of the motion of sound at any place of the space filled
with air, must have a similar form, so that only a tone with the vibrational number
n can arise. But if there is a second greater opening of variable size, through
H which there is sufficient escape of air to render the pressure p periodically variable,
instead of being constant, as the air passes out through the other opening, that is,
if |7 is of the form
p=iP . (i — sin 2irmt),
then q will have the form
q=icAP . (i — sin 2irnt) . (i —sin 2wmt)
zsicAP . [i — sin 27rwi — sin 2Trmt - ^ cos 2 ir(w + n)^+i ^^^ 2ir(m — n)fj.
Hence, in addition to the two primary tones n and m, there will be also the
tones m-hn and m ^ n, that is, the two combinational tones of the first order.
In reality the equations will always be much more complicated than those here
selected for shewing the process in its simplest form. The tone n will influence
the pressure p, as well as the tone m ; even the combinational tones will alter p ;
^ and finally the magnitude of the opening may not be expressible by such a simple
periodic Action as we have selected for ta. This will occasion not merely the
tones m, n, and m + n, m — n, to be produced, but also their upper partiais, and
the combinational tones of those upper partiais, as may also be observed in experi-
ments. The complete theory of such a case becomes extraordinarily complicated,
and hence the above account of a very simple case may suffice to shew the nature
of the process.
I will mention another experiment which may be similarly explained. The
lower box of my double siren vibrates strongly in sympathy with the fork a' when
it is held before the lower opening, and the holes are idl covered, but not when the
holes are open. On putting the disc of the siren in rotation so that the holes bib
alternately open and covered, the resonance of the tuning-fork varies periodically.
If n is the vibrational number of the fork, and m the number of times that a single
hole in the box is opened, the strength of the resonance will be a periodic function
of the time, and consequently in its simplest case equal to i — sin 2irmt,
^ Hence the vibrational motion of the air will be of the form
(i — sin 2Trmt) . sin 2?r?t^ = sin 27rnt + ^ cos 2ir(m + n)i — ^ cos 27r(m— n)^,
and consequently we hear the tones m -I- n, and w — n orn — m. If the siren is
rotated slowly, m will be Yery small, and these tones being all nearly the same,
will beat. On rotating the disc rapidly, the ear separates them.*
* [For the whole subject of beats and com> recent discussions in Appendix XX. sect. I«.—
binational tones the reader is referred to the Translator.']
Digitized by
Google
APP. xvu. JUSTLY INTONED INSTRUMENTS WITH ONE MANUAL. 421
APPENDIX XVn.
FLAN FOB JUSTLY INTONED INBTBUMSNTS WITH A SINGLE MANUAL.
(See p. 319c, and note f.)
To arrange an organ or harmonium with twenty-four tones to the octave in such
a way as to phiy in just intonation in all keys, the tones of the instrument must
be separated into four pairs of groups, thus
I a) /
a, ($
2 &i 0
e, 0,?
3«) ?
4 a) d
6, «,)
lb) /.
a cS
3 bi C|
4 b) d,
k t
Each of these groups must have a separate portvent from the bellows, and
valves must be introduced in such a wav that the wind may be driven at pleasure
either to the right or left group of any horizontal series. This would not be diffi-
cult on the organ. On the harmonium the digitals would have to be placed in a
different order from the tongues, and consequently it would, as on iJie organ, be
necessary to have a more complicated arrangement for conducting the effect of
pressing down a digital to the valva
Hence four valves are to be arranged by stops or pedals in a different way for
every key. The following is a table of the arrangement of the stops for the four
horizontal series of the tones named : —
Major keys
Series
Minor keys
X
3
3
4
Cb*
b
a
a
(^,b)
(B,b
Gfb*
b
b
Db*
b
b
(Cx)
A^*
b
b
E\>*
a
b
Bb*
a
a
F
a
a
\
C
a
a
e\
G
b
a
B* orCb
D
b
b
2^,5 ♦ or Gb
A
b
b
C,« ♦ or Db
E
b
b
G^n*oTAb
B
a
b
D,« ♦ or Eb
a
a
A^Z * or Bb
ir
The minor keys which have their names in parentheses, namely Ei\}j Bi\}, i^,,
Ci, <7i, Di, have a true minor Seventh, but too high a leading tona [Their
dominant chord has an impossible Pvthagorean major Third.] For the six keys
marked with (*), the arrangement of the stops is the same both for major and
minor.f
In order to have a complete series of tonics, each with a perfect major and
minor form, it would be necessary to cut off ai|>, £,{>, 6i|>, /i, Ci, £1, from the other
notes, and to allow them to be replaced when needed by ^, e2$, 0$, <{(, bjj^, and
fJKJl^, by means of a fifth stop. We should thus have 30 tones to the octave. By
drawing out this stop we should have the following system of keys : —
t [The series in the first six lines is the same as in the six last— Tmfw2a^.]
Digitized by VjOOQIC
42 2
JUST INTONATION IN SINGING.
APP. XVII. xvin.
H
Series nwrked with accented letters to shew
1
that they are affected by the fifth stop
HAjorkeys
Minor keys |
I
3
3
4
* 1
F
a'
a'
a'
b'
F
C
a'
a'
a'
a'
C
O
b'
a'
a'
a'
G
D
b'
b'
a'
a'
D
A
b'
b'
b'
a'
A
E
b'
b'
b'
b'
E*
B
a'
b'
b'
b'
A«
Fl
a'
a'
b'
b'
A,t
CI
a'
a'
a'
b'
E,l
0%
a'
a'
a'
a'
B,«
D%
b'
a'
a'
a'
F,t%
A%
b'
b'
a'
a'
Cytn
E%
b'
b'
b'
a'
0,9 8
To have a complete series of minor keys, 28 instead of 30 tones to the octave
would be enough. They would suffice for the 12 minor keys oiAx,Ei,Bi, i^ijf or
Gt>, CiJ or Dt>, (?ij or -4|>, Z>iJf or E[}j B, F, G, O and D, and for 17 major keys
from C|> mjgor to Gj major, t
APPENDIX XVIIL
JUST INTONATION IN SINGINQ.
(See p. 3266.)
IT Since the publication of the first edition of this book, I have had an opportunity
of seeing the Enharmonic Organ, constructed by General Perronet Thompson, t
which sdlows of performance in 21 major and minor scales with different tonics
harmonically connected. This instrument is much more complicated than my
harmonium. It contains 40 pipes to the octave, and has three distinct manuals,
with, on the whole, 65 digitals to the octave, as the same note has to be sometimes
struck on two or all of the manuals. This instrument allows of the performance
of much more extensive modulations than my harmonium, without requiring any
enharmonic interchange& It is even possible to execute tolerably quick passages
and ornamentations upon it, notwithstanding its apparently involved fiingenng.
also explained.- Translator.]
X [* On the Principles and Practice of Just
IntomUion, with a view to the Abolition of
Temperament, and embodying the results of
the Tonic Sol-fa Associations, as illustrated
on ihe Enharmonic Organ . . . presenting the
power of performing correctly in 21 keys (with
the minors to the extent of involving not
more than 5 flats), and a correction for changes
of temperature. . . . Calculated for taking the
place of the choir orgaxk in a cathedral, and
learned by the blind in six lessons. With an
Appendix tracing the identity of design with
the Enharmonic of the Ancients.* By T.
Perronet Thompson, FJK.S. Ninth edition,
1866. The exact compass of this organ wiU
be explained in App. XX. sect. F. No. 6.
General Thompson was bom at Hull, in 17S3,
and died at Blackheath, 6 September 1869.
He had been four years in the navy before
joining the army, and was prominent during
the Com Law Abolition agitation. He was
many years editor of the Westminsier Review^
and was first returned to Parliament for HuU
in iS^s,— Translator,]
* [The E minor has the leading note, but
not the minor Seventh. The otiier minor
keys have hoih.^Translator.]
t [As Prof. Helmholtz has retained this
Appendix in his 4th German edition it is
given in the translation. But the scheme
c| explained has never been tried. The plan
for 24 notes is impracticable because of the de-
fective state of the minor keys, and imperfect
modulating power. It could only be used
as an experimental instrument, and for that
the double keyboard as explained on p. 3166
suffices. The vcJve arrangement for 30 notes
would be complicated, and even if it could be
used would still have a very imperfect modu-
lating power. The 53 division of the octave
introduced by Mr. Bosanquet, and subse-
quently by Mr. Paul White, with fingerboards
which have been actually used, as explained
in App. XIX., and also App. XX. sect. F.
Nos. 8 and 9, are so much superior in mani-
pulation, musical effect, and power of modula-
tion, that it is unnecessary to seek further.
In App. XX. sect. F. the other principal
methods that have been actually tried, are
Digitized by VjOOQIC
Aj»p. xvni.
JUST INTONATION IN SINGING.
425
The organ was erected in the Sunday School Chapel, 10 Jewin Street, Aldersgate,
London,* and was built by Messrs. Bobson, loi St. Martin's Lane, London. It
contains only one stop of the usual principal work, has Venetian shutters form-
ing a swell throughout, and is provided with a peculiar mechanism for correcting
the influence of temperature on the intonation.
Mr. H. W. Poole has lately transformed his organf so as to get rid of stops for
changing the intonation, and has constructed a peculiar arrangement of the digitals,
which enables him to play in all keys with the same fingering. His scale contains
not merely the just Fifths and Thirds in the series of major chords, but also the
natural or subminor Sevenths for the tones of both series. There are 78 pipes to
Die octave, and Fh has been identified with J^i, &c., as upon my harmonium.^
Successions of chords on General Thompson's instrument are extraordinarily
harmonious, and, perhaps, on account of their softer quality of tone, even more
surprising in their agreeable character than on my harmonium.S I had an oppor-
tunity, at the same time, of hearing a female singer, who had often sung to it, per-
form a piece to the accompaniment of the enharmonic organ, and her singing gave ^
me a peculiarly satisfactory feehng of perfect certainty in intonation, which is
usually absent when a pianoforte accompanies. There was also a vioUnist** present
^vho had not been much accustomed to play with the organ, and accompanied well-
Ivnown airs by ear. He hit off the intonation exactly as long as the key remained
unchanged, and it was only in some rapid modulations that he was not able to
follow it perfectly.
In London I had also an opportunity of comparing the intonation of this
instrument with the natural intonation of singers who had learned to sing without
any instrumental accompaniment at all, and are accustomed to follow their ear
alone. This was the Society of Tonic Sol-£aists, who are spread in great numbers
(there were 150,000 in i862tt) over the large cities of England, and whose great
* [Shortly before his death General Per-
Tonet Thompson presented this organ to Mr.
.Tohn Garwen, mentioned in note ff, below.
The General's executors had it reconstructed
in a schoolroom at Plaistow, Essex. It was
afterwards exhibited at the Scientific Loan
Exhibition at South Kensington Museum in
1876, and has remained there ever since, at
the top of the staircase leading to room Q of
the Science Collections. — Translator,]
t Silliman's American Journal of Science
and Arts, vol. xliv., July 1867. [In its origi-
nal form the instrument, with an ordinary
keyboard and pedals, was termed the Ett-
hannonic Organ, and is described in Silliman's
Journal, vol. ix. p. 209, for May 1850. The
new fingerboard is figured and described here-
after, App. XX. sect. F. No. 7 .—Translator.]
X [The text is in error. There are 100 not
78 pipes to the octave, and Ei is not identified
with Fb,— Translator.]
§ [* On organs of many stops, one or more
ought certainly to be tuned with mathemati-
cally correct intonation, on account of their
wonderful effect, to be employed (of course
without using any others at the same time)
as the music of the spheres {als Oesang der
Sphdren). It is impossible to form any notion
of the effect of a chord in mathematically just
intonation, without having heard it. I have
such a one to compare with the others. Every
one who hears it expresses his delight and
surprise at a correctness of intonation that it
does one good to hear {Jeder, der thn hOrt,
gpricht sei/n frohes Erstaunen Uber diese
toohllhuende Reinheit aus).'— Scheibler, Ueber
maihematische Stimmung, Temperaturen und
Orgelstimmung nach Vibrations-Differenzen
Oder Stdssen, 1838. I have given the original
words of the last German sentence, as it was
impossible to do justice to its homely force in
any translation. Every one who has heard
just intonation will understand it.— Trans-
lator.]
*♦ [A blind man, who had therefore no ir
notes to guide him. I had the pleasure of
taking Professor Helmholtz to hear the organ
on this occasion (20 April 1864), and can corro-
borate his statements. Unfortunately the
proper blind organist was not present. It is
to this lady that General Thompson dedicates
his little book, already recited, in these words :
*To Miss E. S. Northcote, Organist of St.
Anne and St. Agnes, St. Martin 's-le-Grand.
In conunemoration of the talent by which,
after six lessons, she was able to perform in
public on the enharmonic organ with 40 sounds
to the octave ; thereby settling the question of
the practicability of just intonation on keyed
instruments, and realising the visions of Guido
and Mersenne, and the harmonists of classical
antiquity.'— Translator.]
ff [The 20 years which have elapsed since
Prof. Helmholtz 's first acquaintance with the ^
Tonic Sol-fa movement have made a struggling
system, slowly elaborated by a Congregation-
fUist minister in connection with his ministry,
into a great national system of teaching sing-
ing. And as the system had the cordial
approval of Prof. Helmholtz (see note p. 427^),
I feel justified in adding a short account of
its origin, progress, and present condition.
In 18 1 2 the two Miss Glovers, daughters of
a clergyman of Norwich^ then, young women,
now both dead at a very advanced age, in-
vented and introduced into the schools under
their superintendence a new sol-fa system,
based upon the 'movable doh,* that is, the
use of doh as the name of the keynote, what-
ever that might be. This was little known
beyond the town where it was used, but
was published about 1827 as a Scheme for
Digitized by VjOOQlC
434
JUST INTONATION IN SINGING.
▲PP. xynL
progress is of much importance for the theory of music. The Tonic Sol-fusts re-
present the tones of the major scale by the syllables Do, Be, Mi, Fa, So, La^ Ti,
rendering Psalmody Congregational, and passed
through three editions. About 1841 John
Curwen, then an unmarried GongregationcJist
minister (bom 14 November 1816 at Heck-
mondwike, Yorks), visited the school, and at
once saw that Miss Glover's scheme gave him
the instrument he desired for his own work.
In 1845 ^G married, and he and his wife
struggled — it was a real and severe struggle,
against small means — to make this system
l^own and active. In the course of working
it out various improvements suggested them-
selves, and the Tonic Sol-fa system, as he
termed it, is not precisely the same as Miss
^ Glover*s; it is essentially John Ourwen's.
Thus Miss Glover*8 scheme (as she says in a
MS. preface in 1862 to the 2nd edition of the
description of her Harmonicon, in the Science
Collections, South Kensington Museum) was
founded on temperament ; Gurwen*s on just
intonation ; and the alterations that this
change involved were many and laborious.
Here Curwen was, I believe, much assisted by
the personal friendship of General Perronet
Thompson, whose works he constantly quoted
in the first book he issued, Singing for Schools
and Congregations, 1843-8. A remarkable
power of methodising, systeraatising, and
teaching, of making fnencb and co-workers,
and of utilising suggestions carried everything
before it — at last. But the work was long,
and the opposition strong. There was only an
* Association ' when Prof. Helmholtz made the
^[ acquaintance of the Tonic Sol-fa system. But
the Association grew to be a * College,* which
held its first * summer term ' on 10 July 1876,
having been ' incorporated ' on 26 June 1875,
and there were in 1884, 1420 * Shareholders '
in this College, which opened its * Buildings '
(at Forest Gate, London, E.) on 5 July 1879.
John Curwen lived long enough to see the
opening and to preside at the unveiling of Miss
Glover's portrait in it, never having neglected
to own his obligations to her initiative. On
a stone at the entrance of the present College
building he placed this inscription : * This
stone was laid by John Curwen, May 14, 1879,
in memory of Miss Sarah Glover, on whose
** Scheme for rendering Psalmody Congrega-
tional " the Tonic Sol-fa method was founded.'
John Curwen died 26 May 1880, of weakness
^ of the heart. His eldest son, John Spencer
Curwen, Associate of the Boyal Academy of
Music, has been since that date annually
elected as President of the College. The work
of the College is chiefly examinational, carrying
on classes by post in the various branches of
music, and granting certificates shewing various
degrees of attainment, on the authority of duly
appointed examiners. From 1858 to 1884 the
numbers of these certificates granted have
been: Junior, 52,000; Elementary, 167,000;
Intermediate, 44,000; Matriculation, 3350;
Advanced, 520 ; Musical Theory (including
Harmony, Composition, Form, Expression,
Acoustics, (fee), 8200; total 275,070, as the
Secretary informs me. During the summer
there is always a term for the special vvod voce
instruction of teachers. Of course large classes
are constantly going on everywhere. I qoote
the following from a letter dftted 15 October
1884, written by John Spencer Curwen to the
Editor of the Times :—
At the most modest estimate, daring the y>
years our system has been at work, we have taognt
at least the elements of mnsic to fbur million per-
sons. There are now, in the elementaiy aehooh of
the United Kingdom, about one mUlion children
learning to sing at sight npon our system. The
Tonic Sol-fa College has 28 difierent kinds and
grades of mnsical ezaminationa, and these were
passed last year b^ 18,716 persons. Eveiy ex-
amination includes individmd tests in «tigiwp at
sight. We have between 4000 and 5000 tcAoen
at work, and at the present time they have under
instruction some 200,000 adults^ in addition to the
children already mentioned. I lately inquired of
16 of our most active professional teacMrs how
many pupils, adults and children, they were in-
structing per week in their classes. The nomber
proved to be 61,051. We have a weJl-ornudaed
movement. During the last four yean I hare
attended 166 meetings in the length and breadth
of the kingdom, my travels extending over 13,000
miles, and ranging from Plymouth to LondoniJerry,
firom Inverness to Norwich. These meetings, at
which demonstrations of mnsical education are
invariably given, have been attended b^ at least
zoo/xx> people. I have further travelled m France.
Germany, Austria, and Switzerland, stud3rin|^ the
condition of popular musical instruction in s<£ooU»
singing clubs, &C., so that we may bring our prac-
tice up to the best continental mo'deU. The quan-
tity of music printed in the Tonic Sol-fa notation
is enormous, and is increasing very rapidly. Two-
thirds of our pupils, having been groundiKl in our
noution, go on to learn the ordinaiy staff nota-
tion, and prove themselves excellent readers ef
that notation.
With regard to teaching musio in 8oh<x)iU,
the following is compiled from the papen
issued by the Educational Department in
1884, England and Wales C. 3941, and Soot-
land C. 394^. They refer to 27,330 onbordi-
nate educational departments for England and
Wales. Of these, 21,743 teach mnsio by ear
only ; 1429 by the staff or ordinary notation ;
3871 by Tonic Sol-fa; 32 by both systems;
and 2161 in some other way. For SootlAnd
there are 3403 subordinate departments, of
which 1238 teach by ear only ; 8 by the
late Dr. Hullah's modification of Wilhelm*8
method, 1746 by Tonic Sol-fa; 117 by old
notation with movable doh (for which many
teasers have a strong predilection), and 7 by
more than one system. There are 94 depart-
ments in England, and 277 in Scotland miAii^g
no returns. These returns shew that Toni«
Sol-fa is the national system of teaching
musio by note in the primary sohooLs of Eng-
land and Scotland at the present day.
John Curwen having started his system
from purely philanthropic motives, gladly
placed his notation at the disposal of all who
liked to use it. A strong proof of the soooesa
of his system is furnished by the fact that aJl
the principal London publi^ers have availed
themselves of this permission. Ctonnod's Re-
demption and Mackenzie's Rose of Sharon are
among the latest additions to the Tonio
Sol-fa repertoire. It is estimated that at the
present time there are 40,000 pages of mnsifl
printed and published in this latter notation.
But the ednoatiottid works on musio and the
system are the private property of the firm of
Digitized by VjOOQlC
APP. xvni.
JUST INTONATION IN SINGING.
425
Do. where Do is always the tonic [vowels as in Italian]. Their vocal music is not
written in ordinary musical notation, but is printed with common types, the initial
letters of the above words representing the pitch.*
When the tonic is changed in modulations, the notation is also changed. The
n&w tonic is now called Do, and the change is pointed out in the notation by giving
two different marks to the note on which it occurs, one belonging to the old, and
one to the new key. This notation, therefore, gives the very first place to repre-
senting the relation of every note to the tonic, while the absolute pitch in which
the piece has to be performed is marked at the commencement only. Since the
intervals of the natural major scale are transferred to each new tonic as it arises
in the course of modulation, all keys are performed without tempering the inter-
nals. That in the modulation from C major to Q major, the Mi (or hx) of the
second scale answers precisely to the Ti of the first is not indicated in the nota-
tion, and is only taught in the further course of instruction. Hence the pupil has
no inducement to confase a with a^.f H
John Cnrwen & Sons, and are of such a remark-
able character that a gold medal was awarded
for them at the International Health and Edu-
cation Exhibition at South Kensington in 1884.
It would indeed be difficult to find so much
information on music and the method of
teaching it (in both notations), so succinctly
and plainly given, and at so cheap a rate, as
in the late John Curwen's T6a4iher^8 Manual,
standard Course, Miisical Theory, How to
Observe Hannony, How to Read Music, not to
mention the very large number of books and
music intended for immediate class use. John
Curwen's especial desire was to teach * the
thing music,' as he words it, and the peculiar
means which he elaborated for this purpose,
he valued only because it proved effectual for
that purpose.
As one who was personally acquainted
with John Curwen and his work for a quarter
of a century, I may be permitted to give this
testimony, and to refer all those who would
learn the history of this successful musical
educationaUst to the Memorials of John Cur-
wen, compiled by his son, J. Spencer Curwen,
iSHz,— Translator.]
* [Great care has also been bestowed on
the representation of rhythm, and exercises in
rhythm form an important part of the Stand-
ard Course and the practice of Tonic Sol-fa
teachers. —Translator.]
t [In a footnote to this passage Prof. Helm-
holtz gives a list of the Tonic Sol-fa works,
which is superseded by the note I have in-
serted above, and at the end of it he says:]
In France singing is taught by the Oalin-
Paris-Chcvi system, on similar principles and
with the help of a similar notation. [This
statement is misleading. Neither principles
nor notation are alike. In 1818 P. Galin,
* Instituteur k TEcole des Sourds-Muets de
Bordeaux,' published his Exposition d*une
Nouvelle MUhode pour VEnseignement de la
Musique, It follows from p. 162 of his book
(3rd ed. 1863, reprinted by ]6mile Chev6) that
Galin adopted as his normal intonation Huy-
ghens's cycle of 31 divisions of the octave,
which closely represents the meantone tem-
perament (see App. XX. sect. A. art. 22, ii.),
although Galin did not seem to be acquainted
with it under that name, and seems to an-
nounce as his own discovery {ibid. p. 80, and
especially p. 107) what was in fact Huyghens's
more than 120 years previously : viz. that I
of a whole Tone » j^ major Semitone «« J of a
minor Semitone, but the ourions thing is that
he considers the resulting flat Fifth of 696773
cents to be correct, and &e Fifth with 701*955
cents from the ratio of 2 : 3 to be wrong.
This is enough to shew how widely Galin's
principles differed from Curwen's. The nota-
tion of intervals which Galin used was Bous-
seau's numerical expression of the major scale
as I 2 3 4 5 6 7, indicating a rising Octave by
overdotting and a falling by underdotting,
but calling the figures ut ri mi fa sol la si.
Here the only resemblance is the movable ut
{ = doh), as distinguished from the usual
French custom of making ut^C. In mark-
ing sharps and flats and time the difference is
greater, but need not be pursued. We should
observe, however, that on this system, as Galin
expressly states (p. Si),gZ is flatter than ab. %
Galin was bom 16 Dec. 1786, and died 30
Aug. 1822. His pupUs, and especially Aim6
a^d Nanine Paris (the latter of whom married
Emile Chev6, a surgeon), continued to teach
his system, and supplied it with text-books.
The principal one is MUhode Slementaire par
Mine. Chev^ (Nanine Paris). La partie thio-
rique de cet ouvrage est ridigie par J^mile
Chevi. In this theoretical part, p. 292, 1 find
that Chev6 imagined Galin to have called his
single division half of a minor Second, whereas
he says, as above, that it was half of a minor
Semitone, which is totally different. The con-
sequence is that Chev6 makes Galin's scale a
division of the octave into 29 divisions, in-
stead of 31 , and hence he obtained a sharp Fifth
of 703*46 cents, a very sharp major Third of
413*8 cents, much sharper than the Pytha-
gorean (App. XX. sect. A. art. 23, vi.). If he <r
could have tuned an harmonium to this major
scale and played the major chords, he would
have been scared at the result. He makes gt
much sharper than ab; his ab was indeed
flatter and gt sharper than on the Pytha-
gorean system. It is evident that his pupils
when they sang in chorus could not have
used his theoretical scale. Hence his prin-
ciples were entirely different from Curwen's.
The notation remained the same as with Galin,
sharps and flats being denoted by acute and
grave accents drawn through the stems of the
figures, but their meaning was altogether dif-
ferent. Also he retained the movable ut of
Galin, and on p. 327 he made out a general
table of the relation of modulations, which re-
sembles my Duodenarinm, App. XX. sect E. art.
18. M. Aim^ Paris also introduced a plan for
Digitized by V^jOOQlC
426
JUST INTONATION IN SINGING.
APP. XVlUr
It is impossible not to acknowledge that this method of notatioo has the great
advantage to the singer of giving prominence to what is of the greatest im-
portance to him, namely, the relation of each tone to the tonic. It is odIj a
few persons, unusually gifted, who are able to fix in their mind, and re-discover
absolute pitches, when other tones are sounded at the same time. But the ordinary
notation* gives directly nothing but absolute pitch, and that too only for tempered
intonation. Any one who has frequently sung at sight is aware how much earner
it is to do so from a pianoforte vocal score, in which the harmony is shewn, than
from the separate voice part. In the first case it is easy to see whether the note to
be sung is the root. Third, Fifth, or dissonance of the chord which occurs, and it is
then comparatively easy to find one*s way ; f in the second case the only resource
of the singer is to go up and down by intervals as well as he can, and trust to
the accompanying instruments and the other voices to force his own to the right
pitch.
Now the instruction conveyed to a singer who is familiar with musical theory
IF by an examination of the pianoforte vocal score, is conveyed by the notation itself
of the Tonic Sol-faist even to the uninstructed. I have convinced myself that by
using this notation, it is much easier to sing from the separate part than in
ordinary musical notation, and I had an opportunity when in one of the primary
schools in London, of hearing more than forty children of between eight and
marking modulations which has a great re-
semblance to John Curwen's * bridge tone,'
but both plans were absolutely independent.
The * langue des durtes ' of Aim6 Paris was,
however, avowedly adapted to the Tonic Sol-fa
system by John Curwen. Both M. and Mme.
K. Chev6 are dead, and after some time their
son Amand Chev6 revived the system, which
has had great success in France, and gained
many^ prizes in choral competitions, shewing
that Emile Chevy's theoretical scale could not
have been adopted. From a correspondence I
V had with M. Amand Chev6 I found he did
not hold with his father's 29 division of the
octave, but adopted the 53 division (not
however as representing just intonation with
a major Third of 17 degrees or 384*9 cents,
but) as representing the Pythagorean intona-
tion with a major Third of 18 degrees or 407*5
cents (App. XX. sect. A. art. 22, iii.). As this
would be frightful in part singing, it is pro-
bable that his pupils, although strictly taught
to make g1^ sharper than ab (indeed to make
the intervals ^ to ab and gi io a identical,
each containing 4 degrees or 90*6 cents, with
an interval of i degree or 22*6 cents between
them), in choral singing insensibly use^ the
equal temperament which Galin and Emile
Chev6 for different reasons inveighed against.
At any rate the Galin-Pari8-Chev6 system,
^ clever and successful as it is, is after til and
was from the first a tempered system, and in
its Chev6 form a ^theoretically) very badly
tempered system, ana hence not in the slightest
degree simUar in principle to the Tonic Sol-fa,
which as taught by John Curwen was always a
system of just intonation. Another immense
difference must be noted. Curwen founds
ever}'thing upon the major chord do mi so at
all pitches, then proceeds to its dominant
80 ti re, and finally to its subdominant
fa la do, in. every case drawing attention to
the character of the notes in the scale. The
Chev6 system began by teaching the melody,
ut re mi fa sol, and not advancing till this
melody was thoroughly impressed on the mind
of the pupil for any ut, taken backwards or
forwards, or stopping at any note and begin-
ning again at that note. Afterwards the sys-
tem took the melody tU} si la sol, and treated
it in the same way. Finally the two were
united as ut re mi fa sol la si iit^. On these
melodies all is founded, and the pupil is told
to take any other intervals by imagining the
intermediate notes, without uttering them^
thus (the notes in roman letters being merely
imagined), ut re mi fa sol la si ut. This is
developed in Mme. Chev6*s Science et Art de
Vintonation, th6orie et pratique, systime des
points d^appui, 1868. On the title-page she
says : * Les grands ressorts de notre methods,
pour P^tude de Pintonation, consistent en
ceci: i^ Chercher les sons un a un et les
^mettre aussi un d un, en les d^tachant les
uns des autres. Hors de Ik point de sncc^
possible. 2^ Se servir de deux rapports que
I'on connait, pour trouver un troisi^me rapport
qu^on ignore ; c^est-i-dire, aller du conna i
Pinconnu ; ce qui conduit k penser par degr^
conjoints, en ohantant par degr^s disjoints,*
May 1868. The two systems of Chev6 and
Curwen ar6 therefore distinct in principle,
value of the signs, form of the signs, notation
of rhythm, and mode of teaching. They are
alike in being taught without an instrument,
but for very different reasons; in the Tonic
Sol-fa to allow just intonation to become the
pupiPs guide ; in the ChevS to allow of taldng
gt sharper than ab, and to make « to/ the
same as gZ to a, but different from ab to a.
They are also alike in having a movable do or
ut, a very ancient device. And also alike in
their nomenclature of lengths, * langue des
dur^es,' which was an original invention of
Aim6 Paris. As to priority of invention. Miss
Glover taught her system in 18 12, Galin pub-
lished his in 1818. Both used tempered sys-
tems. -Translator J]
♦ [Usually called * the Staff Notation ' ot
• the Old Notation * by the Tonic Sol-faists by
way of distinction.— Traniiator.]
f [After a pupil has thoroughly acquired
music on the Sol-fa notation, it be«)me8 part
of his duty to learn the other, and a coarse of
instruction has been prepared for this purpose
by Mr. Curwen, which when properly mastered
(a comparatively easy task) puts the pupil in
a concUtion to sing at sight from the old
notation as readily as from the new. — Trang"
lator.]
Digitized by V^OOQIC
APP. XVIII.
JUST INTONATION IN SINGING.
427
twelve years of age, that performed singing exercises in a manner that astonished
me ('mich in Erstaunen setzten*) by the certainty with which they read the
notes, and by the accuracy of their intonation. * Every year the London schools
of Sol-fedsts are accustomed to give a concert of two to three thousand children's
voices in the Crystal Palace at Sydenham, which, I have been assured by persons
who understand music, makes the best impression on the audience by the har-
moniousness and exactness of its execution.f
The Tonic Sol-faists, then, sing by natural, and not by tempered intervals.
When their choirs are accompanied by a tempered organ, there are marked differ-
ences and disturbances, whereas they are in perfect unison with General Thomp-
son's Enharmonic Organ. Many expressions used are very characteristic. A
young girl had to sing a solo in F minor, and took it home to study it at her piano-
forte. When she returned she said that the A\} and IJ\} on her piano were all
wrong. These are the Third and Sixth of the key in which the deviation of
tempered from just intonation is most marked. Another girl was so charmed with
the Enharmonic Organ that she remained practising for three hours in succession, ^
declaring that it was pleasant to be able to play real notes. Generally in a large
number of cases, young people who have learned to sing by the Sol-fa method,
find out by themselves, without any instruction, how to use the complicated
manuals of the Enharmonic Organ, and always select the proper intervals.
Singers find that it is easier to sing to the accompaniment of this organ, and
also that they do not hear the instrument while they are singing, because it is in
perfect harmony with their voice and makes no beats.
* [On 20 April 1864, after we had heard
Gen. Ferronet Thompson's organ, I had the
pleasure of taking Prof. Helmholtz to hear
the singing of the children in the British and
Foreign School here alluded to, which was
situate behind the chapel in Tottenham Court
Boad. The master of the school, Mr. Gardi-
ner, was a very good Tonic Sol-fa teacher, but
the children were those who ordinarily at-
tended (about forty were then present) and
had received only ordinary instruction. After
hearing them sing a few tunes in parts, from
the Tonic Sol-fa notation, Prof. Helmholtz
himself ' pointed ' out an air on the ' modu-
lator ' or scale drawn out large on a chart,
from which the pupils learn to sing (that is,
by means of a pointer shewed the Tonic Sol-fa
names of the tones the children were to sing),
and the class followed in unison at sight.
Then, on the suggestion of Mr. Gardiner, the
class was divided into two sections, and Prof.
Helmholtz pointed a piece in two parts, one
with each hand, while the class took them at
sight Of course the piece was simple, but
the dissonance of a Semitone was purposely
introduced in one place between the parts,
and Prof. Helmholtz was delighted at the
firmness and correctness with which the chil-
dren took it. I recollect his saying to me
afterwards, *We could not do that in Ger-
many 1 ' meaning, as he subsequently ex-
plained, that there was no German system of
teaching to sing which could produce such
results on such materials. The following is
an extract of a letter from Prof. Helmholtz to
Mr. Curwen printed on p. 159 of the Memo-
rials t dated 21 April 1864, the day after his
visit to the class : ' We were really surprised
by the readiness and surety [certainty] with
which the children succeeded in reading music
that they did not know before, and in follow-
ing a series of notes which were indicated to
them on their modulatory board [modulator].
I think that what I saw shewed the complete
success of your system, and I was peculiarly
interested by it, because during my researches
in musical acoustics I came from theoretical
reasons to the conviction that this was the
natural way of learning music, but I did not
know that it had been carried out in England
with such beautiful results.' — Translator.']
t [I am informed by the Secretary of the
Tonic Sol-fa College that the first Crystal
Palace Festival of the Tonic Sol-faists was
held on 2 September 1857, with a choir of ^T
about 3200 children and 300 adults. These
concerts have been continued year by year to
the present time. For many years two con-
certs were given, one juvenile and one adult,
the singers varying in number from 3500 to
5000. Some of these performances were so
popular that a repetition was given a few weeks
later. The plan of testing the great choirs in
sight singing was first tried at the Festival on
14 August 1867, at which I was present, when
an anthem specially written for the occasion
by Mr. (now Sir) G. A. Macfarren (Professor of
Music at the University of Cambridge, and Prin-
cipal of the Royal Academy of Music) was sung
by a choir of 4500 voices. Of the performance
of this anthem Mr. Macfarren wrote a short
time after in the Comhill Magazine thus :
* A piece of music which had been composed
for the occasion, and had not until then been flr
seen by human eyes save those of the writer
and the printers, was handed forth to the mem-
bers of the chorus there present, and then,
before an audience furnished at the same time
with copies to test the accuracy of the per-
formance, forty-five hundred singers sang it at
first sight in a manner to fulfil the highest
requirements of the severest judges.* Mr.
Macfarren was himself present, and publicly
expressed his own satisfaction.
Sight-singing tests have been given almost
every year since, and always with the same
success. They have become a common part of
public concerts intended as * demonstrations,'
and are regarded by Tonic Sol-faists as no
more extraordinary than reading the words
at sight would be considered. — Translator.']
Digitized by VjOOQIC
428 JUST INTONATION IN SINGING. app, xvm.
*
I have myself observed, that singers accustomed to a pianoforte accompani-
ment, when they sang a simple melody to my justly intoned harmonium, san^
natural Thirds and Sixths, not tempered, nor yet Pythagorean. I accompanied
the commencement of the melody, and then paused wlule the singer took the
Third or Sixth of the key. After he had struck it, I touched on the instrument
the natural, or the Pythagorean, or the tempered interval. The first was always
in unison with the singer, the others gave shrill beats.
After this experience, I think that no doubt can remain, if ever any doubt
existed, that the intervals which have been theoretically -determined in the preced-
ing pages, and tliere called natural, are really natv/ral for uncorrupted ears ; that
moreover the deviations of tempered intonation are really perceptible and un-
pleasant to uncorrupted ewrs ; and lastly that, notwithstanding the delicate dis^
tinctions in particular intervals, correct singing by natural intervals is much easier
than singing in tempered intonation. The complicated calculation of intervals
which the natural scale necessitates, and which undoubtedly much increases the
f manual difficulty of performance on instruments with fixed tones, does not exist
for either singer or violinist, if the latter only lets himself be guided by his ear.
For in the natural progression of correctly modulated music they have always and
only to proceed by the intervals of the natural diatonic scale. It is oidy the
theoretician who finds the calculation complicated, when at the end of numerous
such progressions he sums up the result, and compares it with the starting-
point.
That the natural system can be carried out by singers, is proved by the English
Tonic Sol-faists. That it can also be carried out on bowed instruments, and is
really carried out by distinguished players, I have no doubt at all after the experi-
ments of Delezenne already mentioned (p. 325, note *), and what I myself heard
when I was hstening to the violinist who accompanied the Enharmonic Organ.
Among the other orchestral instruments, the brass instruments naturally play in
just intonation, and can only be forced to the tempered system by being blown out
of tune.* The wooden instruments could have their tones sUghtly changed so as
^ to bring them into tune with the rest Hence I do not think that the difficulties
of the natural system are invincible. On the contrary, I think that many of our
best musical perJFormances owe their beauty to an unconscious introduction of the
natural system, and that we should oftener enjoy their charms if that system were
taught pedagogically, and made the foundation of all instruction in music, in place
of the tempered intonation which endeavours to prevent the human voice and
bowed instruments from developing their full harmoniousness, for the sake of not
interfering with the convenience of performers on the pianoforte and the organ.
Musicians have contested, in a very dogmatic manner, the correctness of the
propositions here advanced. I do not doubt for a moment that many of these
antagonists of mine really perform very good music, because their ear forces
them to play better than they intended, better than would really be the case if they
actually carried out the regulations of the school, and played exactly in Pytha-
gorean or tempered intonation. On the other hand, it is generally possible to con-
vince oneself from their very writings, that these writers have never taken the
trouble to make a methodical comparison of just and tempered intonation. I can
^ only once more invite them to hear, before uttering judgments, founded on an im-
pei^ect school-theory, concerning matters which are not within their own personal
experience. Those who have no time for such observations, should at any rate
glance over the hterature of the period during which equal temperament was
introduced. When the organ took the lead among musical instruments it was
not yet tempered. And the pianoforte is doubtless a very useful instrument for
making the acquaintance of musical literature, or for domestic amusement, or for
accompanying singers. But for artistic purposes its importance is not such as to
require its mechanism to be made the basis of the whole system of music, t
* [On this sentence Mr. Blaikley observes f [This last paragraph, from * Mnsidans
(Proceedings of the Musical Association, vol. have contested ' to * the whole system of
iv. p. 56) : ' It seemed to me worth attention mosic,' is an addition to the 4th Oerman edi-
that this must be talcen as being particularly tion. The remainder of this Appendix, whioh
and not generally true : that is, that though concludes the work in the 3rd German edition,
the ideal brass instrument has such character- was occupied with a description of the madeal
istics, this ideal is not necessarily attained to notation which I employed in my footnotes
in practice.' See pp. 99 and iQO,—Translator.] and Appendix to the ist English edition ; bnt
Digitized by V^jOOQlC
APP. XIX.
PLAN OF MR. BOSANQUET'S MANUAL.
429
APPENDIX XIX.
PLAN OF MR. BOSANQUET'S MANUAL.
(See p. 328c.)
The accompanying figure 66 [taken by permission from p. 23 of Mr. R. H. M.
Bosanquet's Elementary Treatise on Musical Intervals and Temperament^ 1^7 6]
Pio. 66.
Section
\c
Elevation
cT^ch
i^
\C
_vg_
\y
\d ^
plan
\c
shews the arrangement of a part of this manual for 53 equal divisions of the
octave. The upper division gives a longitudinal section of two digitals standing
as I have long since abandoned this notation p. 277, I have thought it right to omit the
in favour of that introduced in Chapter XIV., description.— TraTwZator.]
Digitized by VjOOQ IC
430 ADDITIONS BY THE TRANSLATOR, afp. xix. xx.
one above the other. All the digitals are of the same form and differ only in
colour. The middle part of the figure presents an elevation of the front ends of
these digitals. In the lower part, there is a view as seen from above [plan]. Pro-
ceeding firom one of the tones, as c, upwards and backwards, leaping over an in-
termediate digital in the way, we pass to d and e [on white digitals], and then
continuing by major Seconds we pass tof% ^, c& [on black digitals], and finally
h^ or /c [on a white digital again]. * The sign / means, as has been explained
in the text (p. 329^), sharpening by a comma \i [or one of the 53 degrees = 2 2'642
cents] and is very nearly equivalent to our superior ^ [which means sharpening by
a comma of Didymus J^=2i*5 cents, for which 22 cents is usually employed].
Between the members of this series are inserted, on the digitals leapt over, those
of another series proceeding by major Seconds, d\^, ^ [both black],/, g, a, h, (!$
[all five white].
The series which lie just above one another differ from each other by a comma
^ of the same kind, the upper being ;khe sharper.
In playing the scale of c major * as c, d, \ e, /, S^, \ a, \ b, d, observe that
a horizontal line drawn through the points where d and g are printed in the
figure will just pass through all the required keys. At \ e, \ a, \ 6, we thus
come on a deeper intermediate series.f Every major scale is fingered in precisely
the same way no matter with what note it begins.
The harmonium constructed by Mr. Bosanquet distributes the 53 tones over
84 digitals, some of those at the upper part of the manual being identical with
some of those at the lower part, in order to avoid having frequently to jump from
upper to lower digitals. In the system of 53 divisions ///6 = \(;, since five
smallest degrees represent a diatonic Semitone. [For a further account of Mr.
Bosanquet's notation see App. XX. sect. A. art. 27. For a more detailed plan of
his generalised fingerboard see ihid, sect. F. No. 8, and for his methods of tuning
see ibid, sect, G. art 16.]
*^* \T}m concludes the work in tJie German. Appendix XX. has been en-
f tirely written by tJie Translator j and Prof, Helmholtz is in no respect responsible
for its contents.]
APPENDIX XX.
ADDITIONS BY THE TRANSLATOR.
SECTION A.
ON TEHPERAUENT.
(See notes pp. 30, 281, 315, and 329.)
Art. Art.
1. Object of Temperament, p. 431. 12. Generation of the Metcatorial, p. 432.
2. Equal Temperament and Cents defined, 13. Notation adopted and Fondamental rela-
p. 431. tions between tempered Fifths, major
3. No recarrenoe possible of notes tuned by Thirds, Commas, and Skhismas, p. 432.
just Fifths and just major Thirds, p. 43 r . 14. Linear and Cydlic Temperaments, p. 432.
4. Generation of the Comma, p. 431. Linear TemperamentSt pp. 433-435.
5. Generation of Meantone Fifths, p. 431. 15. The Pythagorean, with its asoal 27 notes
6. Generation of the Skhisma, p. 432. tabulated, p. 433.
7. Generation of Hebnholtzian Fifths, p. 432. 16. The Meantone Temperament, with its nsual
8. Generation of Skhismic major lairds, 27 notes tabulated and their pitch nom*
p. 432. bers calculated for 4 pitches, p. 433.
9. Generation of the Pythagorean Comma, 17. The Skhismic Temperament, p. 435.
p. 432. 18. The Helmholtzian Temperament, p. 435.
10. Generation of the equal Fifth, p. 432. 19. Unequal Temperaments, p. 435.
11. Generation of the Great Diesis, p. 432. Cyclic Temperaments, pp. 435-441.
* [In the German edition a cross was altered accordingly.— TrofuZator.]
placed on the digitals of the plan which were f [A horizontal line through 6 in the
played in this key, but the German copy figure will pass through cdefgabc, and
could not be used here because it followed thus give the Pythagorean major scale. -
German mnsical notation. The text has been Translator.]
Digitized by^OOQlC
BBCT. A. ON TEMPERAMENT. 43^
Art. Art.
20. Conception of a Cyclic Temperament, p. Est^ve's musician's cycle of 55; in ix.
435. as 8 : 5, Henfling^s cycle of 50, p. 436.
21. Equations and conditions for Cyclic Tern- 24. Paper Cycles for calcodation, x. of 30103;
perament, p. 435. xi. of 3010; xii. of 301; xiii. of 1200,
22. Cycles, i. of 12 (equal); ii. of 31 (Huy- P* 437.
ghens) ; iii. and iv. of 53 (Mercator and 25. Equal Temperament tabulated in various
Bosanquet), p. 436. pitches, p. 437.
23. Cycles derived from the ratio of the inter- 26. Synonymity of Equal Temperament nota-
vals of a Tone to a Semitone in i. as tion, p. 438.
2:1, equal cycle of 12; in ii. as 5 : 3, 27. Notation of Bosanquet's cycle anditstabu-
Huyghens^s cycle of 31 ; in iii. as 9 : 4, lation, p. 438.
Mercator*s cycle of 53 ; in v. as 3 : 2, 28. Expression of just intonation by the cycle
Woolhouse's cycle of 19 ; in vi. as 5 : 2, of 1200, p. 439.
Chevy's cycle of 29; in vii. as 7:4, 29. Beferences, p. 441.
Sauveur's cycle of 43 ; in viii. as 9 : 5,
Art. I. — The object of temperament (literally * tmiing '), is to render possible
the expression of an indefinite number of intervals by means of a limited nmnber ^
of tones without distressing the ear too much by the imperfections of the conso-
nances. The general practice has been from the earliest invention of the key-
board of the organ to the present day to make 12 notes in the Octave suffice. This
number has been in a very few instances increased to 14, 16, 19, and even to 31
and 53, but such instruments have never come into general use.
Art. 2. — The system which tuners at the present day intend to follow, though
none of them absolutely succeed in so doing (see infrk, sect. G.), is to produce 12
notes reckoned from any tone exclusive to its Octave inclusive, such that the Octave
should be just and the interval between any two consecutive notes, that is, the
ratio of their pitch numbers, should be always the same. This is known as Equal
Temperament (see suprib, pp. 320^ to 327c). The interval between any two notes
is an Equal Semitone, and its ratio is i : *5/2=i : i •0594631, or very nearly
84 : 89. If we farther supposed that 99 other notes were introduced so as to make
100 equal intervals between each pair of equal notes, these intervals would be those
here termed Cents, having the common ratio i : ^"^2=1 : 1*0005778, or very^
nearly 1730 : 173 1. As the human ear is, except in very rare cases, insensible to
the interval of a cent, we need not divide further, except occasionally for purely
theoretical purposes, to avoid errors of accumulation, as in this section, when even
the thousandth part of a cent may have to be dealt with. In practice the errors
of tuning would soon far exceed the errors arising from systematically neglecting
amounts of less than half a cent. The mode of finding cents from the ratios of
pitch numbers, wave lengths, or vibrating lengths, is given in sect. C, and the
values of most of the usually recognised intervals are represented in cents in
sect. D. From these we take, up to 3 places of decimals.
One just Fifth =7oi*955 cents
„ „ major Third =386-314 „
„ „ Comma = 21*506 „
„ „ Skhisma = 1-954 „
Art. 3. — No recurrence of notes formed by taking intervals of Fifths, major ^
Thirds and Octaves is possible because no powers of the numbers 7, f , 2, or of any
combination of them, however often repeated, can produce a power of any single
one of them. When we only proceed to 3 places of decimals of cents (then of
course using multiples for powers), there would be a recurrence, but so remote that
it would be practically at an infinite distance, and would after all only arise from
our not having carried the decimals fia.r enough. The nearest approximations of
any practical value are given in arts. 4, 6, 9, 11, 12.
Art. 4. — Four Fifths up and two Octaves down, together with a major Third
down, give the Comma (of Didymus ratio, 80 : 81, which is always intended when
no qualification is added), that is, in cents,
4X7oi*95S — 2x1200— 386'3i4=2i"5o6.
Art. 5. — Consequently if we used Fifths di- precisely two Octaves more than an exact
minished by the small but sensible interval of a major Third. These (for a reason given in
quarter of a Comma, that is 701*955 —} x 21*506 art. 16) were called meantone Fifths, and were
or 696-578 cents, four of these Fifths would be long in use.
Digitized by
Google
432 ADDITIONS BY THE TRANSLATOR. app. xx.
Art. 6. — Eight Fifths up and also a major Third up, with five Octaves doinrn,
give the Skhisma, that is, in cents,
8 X7or955 + 386-3i4-5X 1200=1-954.
From this we can deduce two different usage&
Art. 7.— If we employed Fifths diminished by a Skhisma, giving 386*314— 1'954« 384*360
by the insensible interval of i x 1*954, or cents, which may be called a Skhlrauo major
701*955 -'24425-* 701-71075, eight of these Third, then this major Third added to five
Fifths added to a just major Third would give Octaves will give exactly eight Fifths. This
exactly five Octaves. These are the Fifths is the relation which Prof. Helmholtz pointed
proposed supril, p. 316a, and may hence be out (as existing, but not designed) in medieval
called Helmholtz's. Arabic scales, supri, p. 281a, and will be called
Art. 8.— But if we diminished the major Third Skhismic.
Art 9. — Twelve Fifths up and seven Octaves down give the sum of a Comma
1 and a Skhisma, known as the Pythagorean Comma, that is, in cents,
12x701-955— 7 xi2oo=23'46o=2i-5o6 + 1-954.
Art. 10. — ^Hence if we used Fifths dimin- Skhisma, which, more fully calculated, has
ished by the scarcely perceptible interval of i '95372 cents. The difference between these
^x 23*460= 1*9550, or 701-955-1*955 = 700 two intervals is far beyond all powers of ap-
cents, twelve of tiiese Fifths, known as equal preciation by any acoustical contrivance. The
Fifths, would give exactly seven Octaves. Skhisma will therefore be considered as the
These are the Fifths now in general use. The twelfth part of a Pythagorean Comma, and
amount subtracted, 1*95500, is very nearly the also as the error of an equal Fifth. See p. 316^.
Art. II. — One Octave up and three major Thirds down give the difference
between two Commas and a Skhisma, known as ' the Great Didsis,' that is, in
cents,
1200-3 X 386 •3i4=:4i 059=2 X 2r5o6— 1-954.
f Art. 12. — Fifty- three Fifths up and thirty-one Octaves down give what may be
called a Mercatorial, because on it depends the advantage arising from the use of
Mercator's 53 division of the Octave. It is less than two Skhismas by about one-
third of a cent, that is, in cents,
53x701955-31 xi2oo=3-6i5=:2xi-954--293 (5)
Consequently, as ^x 3-61 5= -068, if we used Fifths which were too flat by this
imperceptible interval, or had 701-955 — -068 =5701 887 cents (which may be called
Mercator's Fifths, from their inventor), we should have precisely 53 Mercator's
Fifths=3i Octaves (6)
On these relations depend all systems of temperament which are worth con-
sideration.
Art. 13. — Let us suppose that, measured in cents, in any system of temperament,
V represents the Fifth adopted, T the major Third adopted, and K and S the
Comma and Skhisma adopted, so that j^-f S will correspond to the Pythagorean
f Comma, and 2 K—S to the Great Didsis. Then as the four first relations must
hold for these tempered intervals, and an Octave has 1200 cents, we must have
from art. 9; 12^-8400= -iT-l-S, whence F= 700+ X (Z'-hS)
from art. 11; 1200-31" =2Z^-iS, whence T=4oo-^ (2J5r— 5)
And deducing the values of K and S from these equations,
ir=4F— T- 2400, which is the relation in art 4,
;S=s8F+T— 6000, which is the relation in art 6.
So that there are only 2 independent equations connecting the four intervals
F, r, K^ S. Hence, on assuming values for any two of them we may find cor-
responding values for the other two. But no results are of any European interest
unless F and T both approximate very closely to ihe just values 701*955 and
386-314 cents.
Art. 14.— There are two quite different kinds of temperament, the Linear and
the Cyclic. The Linear contains an endless series of notes which never recur in
Digitized by V^jOOQlC
SBGT. A.
ON TEMPERAMENT.
433
pitch. The Cyclic contains also an endless series of notes which, however, do
recur in pitch, although usually under different names. Hence in Cyclic tempera-
ments all the intervals are made up of aliquot parts of an Octave, or 1200 cents,
which is not the case in Linear temperaments. In both of them the main object
is to substitute a series of tempered Fifths for the several series of Fifths and
major Thirds introduced, supr&, p. 276a, and exhibited at fall in the Duodenarium,
infr^, sect. E. art. 18. The advantage of the Cychc over the Linear temperaments
consists chiefly in a power of endless modulation — a very questionable advantage
when harmoniousness is sacrificed to it.
LiNEAB Temperaments.
Art. 15.— 27w Pythagorean or Ancient Greek Temperament.
Assume F=7oi*955 and J?=o,
Then from art. 16, 5= 1 2 F— 8400=23*460
and T= 400 4-^.5=407 '820
=386-3i4+2r5o6
The major Third is a whole Comma too
sharp, and hence this system is quite unfit for
harmony. It was the theoretical Greek scale,
and is still much used by violinists. See Gomu
and Meroadier*s experiments, infr&, sect. G.
art. 6 and 7. The following are the 27 tones
which this temperament would require for
ordinary modulations, with the cents in the
intervals from the lowest note, the logarithms
of those interval ratios, and the pitch numbers
to c' 264.
PytJuigorean
Intonation.
No.
I
Note
Cents
Log
Pitch
No.
Note
Gents
Log
Pitch
0
0
264-0
15
/«
611-7
15346
375'9
2
b%
23-5
00589
267-6
16
a'bb
678-5
1 702 1
390-7
3
d'b
90*2
02263
278-1
17
^
7020
17609
3960
4
cn
1 137
02852
2819
18
m
725-4
18198
401-4
5
e'bb
180-5
04527
293-0
19
a'b
792-2
19873
417-2
6
d'
2039
05115
297-0
20
b'bb
!J5-6
20461
422-9
7
c««
227-4
05704
3011
21
882-4
22136
439*5
8
e'b
2941
07379
312-9
22
a'
905-9
22724
445*5
9
d't
317-6
07967
317-2
23
g'nn
929-3
23313
4516
10
fy>
384-4
09642
3296
24
i/b
9961
24988
469-3
II
e
407-8
10231
334-1
25
a'n
1019-6
25576
475*7
12
f
4980
12494
352-0
26
&b
10863
1109-8
27251
494-4
13
n
5^J'5
13082
356-8
27
V
27840
501-2
14
l/b
588-3
14757
370-8
I'
c"
1200*0
30103
528-0
IT
These can be all exhibited and calculated as a series of 26 perfect Fifths up,
namely,
«t>t> «bb ^\>\>y Jb c\} gb clb a\} e\} b\}, f c g d a e b,
fi 4 dit ^i^ <4 4 % M<M 9U '
The 17 notes of the medieval Arabic scale (supr^, p. 281c') are those in the first
line of this series, adding d\}\} at tlie beginning, and omitting b at the end, with
all the j{l and jj^jj^ notes in the Table.
Art. 16. — The Meantone Temperament,
The major Thirds are assumed to be perfect, and the Comma is left out of
account. Hence,
T=386*3i4, K=o. Whence.by art. 13
7=696*578 as in art. 5, and S=— 41059.
€k)n8equently the Second of the scale, which
is always two Fifths less an Octave, will
have 193*157 cents or be half a Comma or
10*753 cents flatter than the just major Tone
of 203-910 cents too flat, and hence by the
same amount sharper than the just minor
Tone of 182-404 cents. From this mean value
of the Tofie the temperament receives its
name. This was the temperament which pre-
vailed all over the Continent and in England
for centuries, and for this, and the Pythagorean,
our musical staff notation was invented, with
Digitized by U'CfOgie
434
ADDITIONS BY THE TRANSLATOR.
APP. XL
a distinct difference of meaning between sharps
and flats, although that difiFerence was different
in each of the two cases. For the history of
its invention see infri, sect. N. No. 3. This
temperament disappeared from pianofortes in
England between 1840 and 1846. (See infr&,
sect. N. No. 4.) But at the Great Exhibition
of 1 85 1 all English organs were thus tuned.
If carried out to 27 notes bearing the same
names as in art. 15, but having the different
values in the following table, it would pro-
bably have still remained in use. Handel in
his Foundling Hospital organ had 16 notes,
tuned from dbtoat in the series of Fifths in
art. 15. Father Smith on Durham Cathedral
and the Temple organ had 14 notes from ab to
dZ , and the modem English concertina has
the same compass and uses the same tem-
if perament, and the same number of notes.
The only objection to this temperament was
that the organ-builders, with rare exceptions,
such as those just mentioned (see also 320c
and note §), used only 12 notes to the Octave,
eb bbf eg dae bfZ cU gZ • The consequence
was that in place of the chords ab c eb, f ab
c, <fec., organists had to play gUc ebt f gZc,
where gU was a Great DiSsis (41*059 cents) too
flat, and the horrible effect was familiarly
compared to the howling of ' wolves.* Simi-
larly for bdZftit was necessary to use b eb
/K , eb being a Great Di§sis too sharp, with
similar excruciating effects. In modem mose
it is quite customary to use keys requiriog
more than two flats and three sharps, and
hence this temperament was first stjled * un-
equal ' (whereas the organ, not the Umpera-
ment was — ^not unequsJ, but — defeiUive) sod
then abandoned. But with the 27 notes here
given there would have been nothiii^^ to offend
the ears of Handel and Mozart. At the pre-
sent day, ears accustomed to the sharp lead-
ing note of the equal temperament (where b : t!
has 100 cents) are shocked at the flat leadisx
note of the meantone temperament when b : (f
has 117*1 cents. But played with 27 or 36
digitals on Mr. Bosanquet's generalised key-
board (Appendix XIX. and also XX., secL F.
No. 8) it is the only temperament suitable to
the organ. In my examination of 50 tempos-
men ts {Proc. Boyal Soc,, vol. xiii. p. 404) I
found tiiat this was decidedly the best for
harmonic purposes. For simple melody per-
haps the Pythagorean is preferred by violiniBte,
but that was always absolutely impossible for
harmony.
Meantone Intonation.
No.
Notes
Cents
Logs
Handel
Smart
Helmholti
I>aTham
I
&
0
0
252*7
259-1
2640
283-6
'2
c'U
76-1
01908
264*1
272*0
275-9
296-3
3
d'b
117*1
02938
270*4
277-3
2825
303-4 i
4
c'nt
1 52* I
03816
275-9
282*9
2882
309-6 !
5
d'
1932
04846
2825
2897
295*2
3171
6
e'bb
234-2
o«;876
2893
296-7
302*2
3247
7
d'%
269*2
06753
295*2
302*7
308*4
331-3
8
e'b
310*3
07783
302*8
3100
315-8
339-2
9
ef
386*3
09691
315-9
328*5
3239
330-0
354-5
ID
/b
427*4
10721
3317
337-9
363-0
II
e%
462*4
1 1599
3301
338-4
344-8
370-4
12
f
503-4
12629
338-0
346-6
353-1
379*2
13
ft
579-5
14537
353-2
362-1
369-0
396-3
H
l/b
620*5
15567
361*7
370-8
377-8
405-8
15
/««
655-5
16444
369-0
378-4
385-5
414-1 '
16
9"
6966
17474
377-9
387-5
394-8
424-0
17
a'bb
737-6
18504
3870
396-8
404-2
434-2
18
^n
772*6
19382
394-9
404-9
412*5
443-1
19
alb
813-7
20412
404-3
414-6
422*4
4537
20
nt
848*7
21290
412*6
423-0
431-0
463-0 1
21
a'
8897
22320
422-5
433-2
441-4
474-1 '
22
i/bb
9308
23350
432-6
443-6
452-0
485-5
23
o'«
965-8
24228
441-5
4527
461*2
495-4
24
b'b
1006*8
25258
452-1
463-5
4723
507-3
25
b'
1082*9
27165
472-4
484-3
493-5
530-1
26
c'b
1 1240
28195
483-7
495-8
508-4
505-3
542-8
27
6'«
1158*9
29073
493-6
515-6
528*0
553-9 1
I'
c"
I200-0
30103
505-4
518*2
56r2
On account of the great historical interest
attaching to this temperament, I give the
whole 27 notes, shewing their value in cents
and logarithms, whence the pitch numbers
can be calculated out for any pitch, and I
have actually calculated them out for 4 pitches.
That headed 'Handel' has A 422*5, the
pitch of Handel's own fork, the common
pitch of Europe for two centuries (see infr^,
sect. H.). The piano of the London Phil-
harmonic Society was tuned to A 423*7 or
very nearly this pitch when that Society was
founded in 18 13. But about 1828 Sir George
Smart, the conductor of that Society's con-
certs, raised the pitch to A 433*2, as I have
determined from his own fork, and oolumn
* Smart ' gives the notes for this pitch. As
Sir George considered the fork C 518 to eorre-
apond to his A 433*2 (it is only -2 vib. too
flat), he manifestly used meantone temper*-
ment even so late as this, for the equal C to
A 433*2 would have been much flatter, namely
C 515*1. This was a very curious anticipa-
tion of the French pitch of 1859. The next
Digitized by V^jOOQlC
SBCT.A. ON TEMPERAMENT. 435
pitch which takes Helmholtz*s d 264 for com- sured by me. Now omitting Smart's pitch,
parison, gives o! 441*4, and Father Smith's which does not belong to the old organ period,
pitch for the Hampton Court Palace organ, a curious relation will be found to connect the
determined from an unaltered pipe, 1690, was other three. Handel's dt is Helmholtz's c',
a! 4417. Hence this was a regular meantone and this pitch was therefore a * small Semi-
pitch. The last column shews the extra- tone' of 76*1 cents sharper, and Handel's d.'
ordinarily high pitch used by Father Smith 282*5 is practically the Durham d 283*6, which
for the Durham Cathedral organ, determined was therefore a Tone of 193*2 cents sharper
from an original ^8 pipe, found with all the than Handel's pitch,
others by the organist Dr. Armes and mea-
Art. 17. — The Skhismic Temperament.
The condition is that the Fifths should be perfect and the Skhisma should be
disregarded. This gives
F=7or'955, S=o, and hence by art, 13, ^=23*460,
T=4oo— ?\Zr=384'36o=5386*3i4-- 1'954 as in art. 8. m
3
That is the major Third is too flat by a of Fifths given below the Table, we see that
Skhisma, whence the name of the tempera- /b is the eighth Fifth below c, which follows
znent (see supr&, p. 281a). The effect of this from art. 13 whenever S ^ o. Hence gene-
flat major Third is very good indeed. On rally the notes in the top line will be major
looking at the Table in art. 15 we see that c' l/b Thirds above those in she bottom line re-
is such a major Third, and looking to the lists spectively —
abb ebb bbb fb cb gb db ab eb bb f c g d a e b ft ct
eb bb f c g d a e b fl ct gt d% at el bt fit clt git
Having an English concertina (which has 14 beats 16 times in 10 seconds, which is scarcely
notes) tuned in perfect Fifths from gb to cS in perceptible and far from disagreeable. But it
the series in art. 15, 1 have been able to verify is evident that if this system of tuning were
this result for six of the major Thirds, and to adopted, a different musical notation would
determine that although a cl e^ e gl &, are be necessary, and a convenient typographical
horrible chords, adbe^eab b are quite smooth modification of Mr. Bosanquet's will be ex-
and pleasant. The major Third d :fb of art. 15 plained on p. 438(2.
Art. 18. — The Helmholtzian Temperament. ^
In this case the major Thirds are taken perfect, and the Skhisma is disregarded.
Then by art. 13,
K=i 2o'534= 2i'5o6 — 1*07^
and 7=70171 i=7oi*955—ix i'954 as in art. 7.
The Comma and the Fifth are therefore imperceptibly flattened. In this case
also the major Third would be the eighth Fifth down. And the same reason for
altering the notation would hold as for art. 17.
Art. 19. — In their endeavours to avoid the * wolves ' of meantone temperament
musicians invented numerous really unequal temperaments, which it would be
uncharitable to resuscitate. There is, however, a really practicable unequal
temperament which I call Unequally Jv^t, but it cannot well be explained till the
Duodenarium has been developed. (See infr&, sect. E. art. 25.) I proceed, there-
fore, to the consideration of the
Cyclic Tempebaments. IF
Art. 20.— The Octave of 1 200 cents being divided into different sets of aliquot
parts called t^^ree^, certain numbers of those degrees may approach to the value of
the just Fifth 701-955 and major Third 386*314, and from these the whole scale
may be constructed, each interval being more or less well represented by a certain
number of degrees. There would then be this advantage, that the number of
values of notes (whatever happened to their names) would be strictly limited by the
number of degrees in the Octave, and hence values would recur, and the whole
scale could always be expressed by a limited number of cyclic Fifths.
Art. 21. — The equations for finding such cycles may be immediately derived
from those in art. 13, thus :
Let m be the number of degrees, and ?^=:I20o-^m be the number of cents in
one degree, so that i2oo=7wn. Put F=wt?, T—nt, K=nk, S^^ns in the four
equations art. 13, and divide out by n. Then i2V— 7m=/:+«, m— 3^=2^;— 5,
whence k^^v—t - 2m, 5=8t; + ^— 5w. On assuming values for m, beginning at
12, and putting first A;=o, and then 5=1, 2, 3, &c., or—i, — 2, — 3, &c., and next
Digitized by VjDlDQlC
43^
ADDITIONS BY THE TRANSLATOR.
APP. XX.
s=o, and then ifc=i, 2, 3, or— 1,-2, — 3, &c., we get the corresponding values of
V and t, whence the scale. Most such scales would be useless. For practical
tuning m should be small, and v : rriy t : m should be nearly the ratios of the
numbers of cents in the just Fifth and major Third to 1200 (or of the logarithms
of their interval ratios to log 2='3oio3). Now the approximate values of
7or955
1200
are
'- -2'
7
— »
12
31
53'
&c..
and of
28 31 '59 53 '
1200
are -=
3 12
&c.
Next take the Pythagorean major Third from art 15.
values of
3' 50 53
The approximate
407 'Sio I
are -»
1200 2
&c
Finally, take the meantone Fifth from art 16. The approximate values of
696-578 „, , I 4 S _9 14 23 .
are i, -> — » > » — » — > ""^
1200 •"" *' 2 5 7 12 19 31
Art 22. — These numbers suggest cycles of 12, 31, and 53 degrees.
i. The cycle of 12 with a Fifth of 7 and a
major Third of 4 degrees would imitate Pytha-
gorean intonation well and just intonation in-
difiFerently. It is the equal temperament of
to-day. Here w«i2, v = 7, ^ = 4, k^s—o.
In cents, one degree =100, F=700, r=4cx>,
K=S = o. See art. 25.
ii. The cycle of 31, with a Fifth of 23 and
«r a major Third of 10 degrees would imitate
meantone temperament very closely. It is the
Harmonic Cycle of Huyghens. Here w=3i,
t?=i8, ^=10, fc«=o, a=i — I. In cents, one
degree* 38710, 7« 696773, T= 387-097, J5:=o,
§=-38710.
iii. The cycle of 53, with a Fifth of 31 and
a major Third of 18 degrees, is an extremely
close approximation to Pythagorean tempera-
ment. It is Mercator's cycle. Here m = 53,
v = 3 1, i = j8, /c = o, s = I. In cents, one degree
» 22-642, 7=701-886, !r = 407-547, ir=o,
<S« 22*642.
iv. The cycle, of 53, with the same Fifth
of 31, but a major Third of 17 degrees would
give a sufficiently dose approximation to just
intonation. (See p. 328<2.) As it seems to
have been first supplied with a fingerboard
by Mr. Bosanquet, it is properly called Boean-
quet's cycle, but, as will be seen (infri, sect. F.
No. 9), Mr. J. Paul White has also invented
a keyboard for it. Long previously to either
Gen. T. Perronet Thompson used it extensively
in his works on the Enharmonic QttUar and
Just JrUonation as a convenient approximation
to just intonation, and from his works it was
introduced into the Tonic Sol-fa books for the
same purpose. Here m«53, t? = 3i, <«I7,
A;=.i, 8-0, In cents, one degree » 22*642,
7=701-886, 7 = 384-905. Observe that in
art. 21, Skhismic 7=701-955 is only the im-
perceptible interval -069 cents sharper, and
Skhismic T= 384-360 is only the imperceptible
interval -565 cents flatter. Hence Skhismic
intonation and Bosanquet 's cycle are andibly
interchangeable within the limits of a few
keys. It is only when the modulation beyond
53 degrees is required that the* cyclic intona-
tion has the advantage. See art. 27.
Art 23. — ^Besides these the following temperaments have been at least proposed.
^ They chiefly depend upon assigning imaginary or arbitrary evaluations of the ratio of
a Tone to a Semitone, the Comma being neglected, and the Octave supposed to con-
sist of 5 Tones and 2 Semitones, so that if Tone : Semitone =p : q, the number of
degrees of the cycle will be 5J?-f 27, and v^^p-^-q, i=2p, ^•=o, 2k— s, the Diesis
= — s variable. This apphes to all temperaments where K—Oy thus (art. 22) in i.,
p : g=2 : I ; in h., p : q=$ : 3 ; in iii., p : q—g : 4.
We thus obtain, among others,
V. Woolhouse's cycle of 19, i> : 2 » 3 : 2,
«»ii,i«6, fc=o, « = — i,2A;— fi«i. In cents,
onedegree«63-i6, F-694-76, 7=378-96.
vi. Chev6*8 cycle of 29, p : g-5 : 2, t7:=i7,
<ssio, fc«o, a-i, 2*— »==-!. In cents, one
degree «4i'38o. 7=703-460, 7=413-80.
vii. Sauveur's cycle of 43 m^rides, p : q
= 7 : 4, v«25, /=I4, A;«o, «= -I, 2/c-s= I.
In cents, one degree = 27-907, 7=697-674,
r« 390698.
viii. The Musician's cycle of 55, in 1755,
according to Sauveur and E8t^ve,p ' ff = 9 : 5.
u-32, i = i8, /c = o,»=-i,2/c-s«i. In cents,
one degree =21 -8 1 8, 7=698-176, 7=392*724.
ix. Henfling^s cycle of 50, in 1710, j? : g
«8 : 5, v«29, < = i6, A;=:o, s= — 2. In cents,
one degree = 24, 7=696, 7=384, JC«o,
5* -48.
Both Fifth and Third are much too flat
in V. and too sharp in vi. Both vii. and viii.
were decent approximations, and convenient
on paper. It would not have been worth
while to produce them on instruments.
Digitized by V^jOOQlC
SECT. A:
ON TEMPEBAMENT.
437
Art. 24. — But paper cycles are sometimes extremely useful for the purposes of
calculation, as in the following cases.
X. Cycle of 30103 jots (de Morgan's name),
17 « 17609, t^gSgi, A; = 539, « = 48, 2k-8
= 1030. In cents, one degree = '039863,
y= 701-950, r= 386-3135, £-= 21-4862, S^
1-91343. ,
This is therefore an exceedingly accurate
representation of just intonation. It is
derived from looooxlog 2, using 5 place
logarithms, and the only reason for the differ-
ences from just intonation is that it is taken
strictly to the nearest integer. Thus loooo x
log |s 17609*13, and this would have made
V correct. Mr. John Curwen used this in his
Musical Statics, to avoid logarithms.
xi. Cycle of 3010 degrees, t;= 1761, ^ = 969,
fc = 55» s=»7, 2^—5 = 103. In cents, one de-
gree = -3987i, F«702-o6o, ^=386-3135, ^-
21-9269, 5 = 2-7907. This is derived from
1000 X log 2 to 4 places, and is consequently
not quite so accurate as the last.
xii. Cycle of 301 degrees, ^=176, ^ = 97,
^ =" 5» * ■■ o, 2& - « = 10. In cents, one degree =
3-9866, 7« 701-661, r=386-7n, 2^-19-93355,
iS«o.
This was the cycle used by Sauveur {Mim.
de VAcaditnie, 1 701, p. 310) as a finer division
than was given by his cycle of 43 m6rides (see
vii.). As 301 a 7 X 43, he called each degree a
heptam6ride, which he made =-03987 of an
(equal) Semitone. He also gives a rule for
finding the number of heptamSrides in any
interval under 6:7a 267 cents, which is the
equivalent of my rule for finding cents (infril,
sect. C, I. 4, note), only my rule extends to
498 cents. Sauveur's rule is : multiply 875 by
the differences of the interval numbers and
divide by their sum. For 6 : 7, this gives 67
7
heptam6rides, and as log z"'o67 to three
places, this is correct. It is the earliest in-
stance I have met with of the bimodular
method of finding logarithms. Sauveur's 875
is an augmented bimodulus for 869, for the
same reason as I selected 3477 in place of
3462, p. 447c'.
I have here taken the values of v and t as
they ought to be, but judging by vii. Sauveur
took t7--i75 and £^98, and hence got the
results there given, which agree better with
meantone intonation.
Observe that the Helmholtzian 7=701*711 ^
is only -050 cents, or imperceptibly larger, and
the cyclic T is only -397 cents, also imper-
ceptibly larger than just. Hence the Helm-
holtzian intonation and Sauveur's cycle of 301
are interchangeable within 301 degrees.
xiii. Cycle of 1200, or the Centesimal Cycle.
If we refrain from using decimals of cents, we
really use a cycle where one degree » i cent,
17=: 7=702, <=r=386, A = ^=22, a = 5 = 2,
2k—s-*^2. These are quite imperceptibly
different from the just for a single key, but
when modulation is extended, the relative
value of distant notes to the starting note will
be slightly altered. See infr^, art. 28. In the
body of this work and after this section
* cyclic cents,' as they may be called, will be
used, unless accumulated fractions of a cent
become sensible. But in the investigation of
this section it was necessary to shew differ- f
ences much more minute than a single cent.
Art. 25. — The only cycles requiring further attention, then, are i., the Equal,
and iv., Bosanquet's. The method of tuning Equal Temperament is given infr^,
sect. G. art. 10. The pitch of the notes used is very variable. Six principal
pitches are tabulated below.
Eqtial Intonation.
Italian
French
Scheibler's
Society
English
Schnitger,
No.
Notes
Gents
Logs
Military
Normal
Stattgard
of Artfl
Band
1688
i.
ii.
iii.
iy.
▼.
Tl.
I
c'
0
0
255-9
258-6
2616
264-0
26875
290-9
2
c'«
100
02509
2711
274-0
2772
2797
284-7
3082
3
d'
200
05017
287-3
290-3
293-7
296-3
301-7
3264
4
d'n
300
07526
304-3
3076
3iri
314-0
319-6
3459
5
e'
400
I0034
322-4
3259
3296
332-6
33!1
366-5
6
/
500
12543
34>-6
345-3
349-2
352-4
3588
388-3
7
^J
600
15052
361-9
36S-8
370-0
373*4
380-1
41 1-4
8
if
700
17560
383-4
387-6
392-0
395-5
402-7
435-8
9
^}
800
20069
406-2
410-6
415-3
419-1
4266
461-8
10
a'
900
22577
430-4
435-0
4400
444-0
4520
4892
II
a'5
1000
25086
456-0
4609
466*2
470-4
478-9
518-3
12
6'
1 100
27594
483-1
488-3
493-9
498-4
507-4
519J
I'
c"
1200
30103
51 1-8
517-2
523-2
528-0
537-5
581-8
L The pitch offioially adopted for Italian
military bands in August 1884. The standard
was a jBb456, because Bb can be produced
on the brass instruments without using the
valves. It is really the neai-est approach in
whole numbers to the old arithmetical pitch
of C"5I2.
ii. French diapason normal intentionally
<k'435* gl'^'ii^ eqnal c"5i7*3i is practically the
same as Smart's pitch, which is the lowest
that has been used for equal temperament in
England, and was contemporary with its in-
troduction there. On 19 March 1885 this was
also officially adopted as the pitch of Belgian
military bands, which had hitherto used il45 1 *7,
or say ^45 2, as given in col. v.
Digitized by V^OOQ IC
438 ADDITIONS BY THE TRANSLATOR. Ai»p. «.
iii. Soheibler's pitch, proposed at Btutt- t. The highest asual English pitch, known
gardt, often called the German pitch, having as * band pitch,* or * Kneller HsJl pitch,* to
^440. which military brass instruments are toned in
iv. The pitch proposed by the Society of England, adopted as the pitch of musical in-
Arts, known in Germany as English pitch, struments at the International Inventions and
having £"528, and a'444. Unfortunately the Music Exhibition, London, 1885.
original fork tuned for the Society of Arts by vi. The original pitch of the St. Jaoobi
Griesbaoh proved to be c"534'46, equivalent to organ at Hamburg built by Schnitger 16S8,
equal a'449'4, and commercial copies vary from the oldest as well as the sharpest example of
c''533*3 to c''535'5. equal temperament I have found. On these
Helmholtz's just c"528 a'440, may be con- pitches, see the abstract of my History cf
sidered as represented by iii. and iv. jointly. Musical Pitehj infr4, sect. H.
Ari 26. — The notation of music (see art. 16) was adapted to either the
Pythagorean or Meantone intonation, in which there was a Diesis or interval
between a sharp and a flat, not to the equal where the Diesis disappears and sharp
coalesces with flat. In the table only the sharps are noted as usual, but they
H imply flats. If we arrange the notes in three lines, in the same order of Fifths as
in art. 15, but continue them to 36 tones, we shall have
F C O DA E B ik CJjL Q% Dt A%
c» ^ fM <Mmt^^m^M ftm <Wt mm
The middle line indicates the ordinary notes, but those in the upper and lower
line are (at least from a[>t> to ^'jf^ occasionally met with in modulations. Now the
three notes in each column have the same meaning precisely in equal tempera-
ment. They are absolutely identical. But in the Pythagorean and Meantone tem-
peraments they have three different meanings, as shewn in the tables of arts.
15 and 16. This confusion arises from equal temperament being cyclic. If we
begin at F and proceed by Fifths to A^ we have exhausted all our 12 valuea
The Fifth above A^ is played by F^ but it would be considered * bad spelling ' to
write it so, for 5|> to / is called a Fifth, and aj{l to/ a diminished Sixth, although they
^ make the same interval precisely. This arises from history. It has been pro-
posed to alter the notation, but the objections to changing are so great that the
matter is mentioned here chiefly to explain how the apparently absurd synonymity
of equal temperament arose, and also because this synonymity is a cardinal point
in Mr. Bosanquet's notation.
Art. 27. — ^In Mr. Bosanquet's cycle, art. 22, iv., there are 53 notes to be supplied
with names, and moreover after the 53 notes have been exhausted the values
recur, but, if the old notation is to be in any way preserved, the old names do not
recur. Hence there will be here also another and a different kind of synonymity,
which will affect the position of the notes. In the following tables I have fii^
arranged the notes by Fifths and then by regular ascent. The large letters may
be considered regular. They are each supposed to have all the synonyms of equal
temperament already explained, and hence are written only as the line of capitals
in art. 26. They are divided into ' sets 'of 12, each set being distinguished by a
superior or inferior number, because each is one degree (22*642 cents, and hence
very nearly a real Comma of 21*506 cents) sharper or flatter respectively. Under
% these capital letters is written the number of the note in the cycle according to
Mr. Bosanquet's arrangement, dictated by practical convenience in performance.
Now there are 12 notes in each line, and hence after 4 lines and 5 Fifths, indicated
by II, we have exhausted all the 53 values. The names of the letters are, however,
continued on the same plan, but they are now synonymous with those at the
beginning of the series, and hence the name of the 54th Fifth, or jP', numbered
27, is written in small letters under the first note FJ}^^ which is also numbered
27, and the series after F^ in capitals is written after /^ in small letters. These
small letters have the same value as the large ones above them. This synonymity
forms the chief difiiculty of the instrument when modulations oblige the player
to proceed beyond the first 53 notes. For this reason, partly, Mr. Bosanquet has
in practice extended his keyboard to contain 7 sets or 84 notes. The inferior and
superior numbers are my typographical contrivances, not Mr. Bosanquet's. He
uses sloping lines Uke those supr^, p. 220c, last bar, ascending for my superiors,
descending for my inferiors, and repeated twice, three times, &c., for my 2, 3,
&c. These are very convenient in musical notation, and, on account of using the
Digitized by V^jOOQlC
BCT. A.
ON TEMPERAMENT.
439
qua! temperament synonyms, are the only alterations required to adapt ordinary
dusic for performance in this cycle.
Jfr.
BosanqueVs
Notes-^n Fifths.
'■'.I
c.«
o,n
I>Ji
A^^
^.
c.
^.
A
A,
E,
B^
7
5
36
14
45
23
I
32
10
41
19
50
I
c»
^'
d»
a'
6»
6»
f't
c^n
j7*«
d-^ff
a«5
o««
c^
^Jt
A«
^Jt
^,
c.
0,
A
A.
^,
^,
^
6
37
15
46
24
2
33
II
42
20
SI
3
c*
f
d^
a»
e»
6«
/*«
c»«
1^«
df
a««
^.«
c.«
^j»
A«
ilj
^.
c.
(^t
A
^,
-B,
^1
'9
7
3«
16
47
25
3
34
12
43
21
52
%
c«
IT*
d»
a»
6«
&•
/'«
c*Z
g*ti
d*«
0^5
>2
8
39
17*
17
il5
48
a*
F
26
6«
C
4
6«
35
13
44
^5
E
22
53 m
R'lJ
C'«
G'«
D'«
m
F»
C»
G»
D»
ii»
E'
-B»
JI
9
40
18
49
27
1^
36
14
45
23
I
!»
c»
g"
d»
a»
e*
n
clJ
g%
d-«
a«»
10
41
19
d*
50
28
6
37
15
ft
46
E*
24
d^5
2
Jtfr. Bosanquet*s Cycle of 53.
Cyc!Ic
Names and
Synonyms
Centa
Logs
Pitch
Numbers
Cyclic
Names and
Synonyms
Cents
Logs
Pitch
Nombers
I
C. 6> a««
1132-1
28399
507-69
28
l^C ^
543-4
13632
36134
2
C, 6* a'5
"547
28967
514-37
29
F^tr
566-0
14200
366-10
3
C, 6»
1 1774
29535
521-14
30
l^ f'
588-7
14768
370-92
4
C b*
I200»0
0
264-00
31
F'lP
611-3
15335
375-80
5
C,«c» 6»
22-6
00568
267-48
32
0. nr
6340
15903
380-75
6
cjrc« &•
45-3
01 136
27100
33
0, />»
6566
16471
385-76
7
C,Jc«
679
01704
274-56
34
<?. /*«
679-2
17039
39084
8
C» c*
90*6
02272
278-18
35
9. ^^^
701-9
17607
395-98
9
C'»c»
113-2
02840
281-84
36
0,zg' J*Z
7245
18175
401-19
10
D. c««c«
135-8
158-5
03408
285-55
37
G^g^ ft
747-2
18744
40648
II
D, c»»
03976
289-31
38
o,tg*
7698
19311
411-83
12
D, c*n
i8i-i
04544
29312
39
Gt g*
792-5
19879
417-25
13
D <n
2038
05112
297-00
40
Q'tf
8151
20447
422-74
14
D.«d» c»5
226-4
05680
300-89
41
A, g'ng^
837-7
21015
42831
15
D^d"^ c'5
249-1
06248
304-85
42
A, ft
860-4
21583
433-95
16
A«^'
2717
06816
308-86
43
A, ft
883-0
22151
439*66
17
DJ d*
294-3
07384
312-93
44
A ft
905-7
22719
445*45
18
D'»d»
317-0
07952
317-05
45
A,ta' ft
9283
23287
451-31
19
^, d'^Jt^i*
339-6
08520
321-22
46
A,ta^ ft
950-9
23855
457-25
20
^, d««
362-3
09088
325-45
47
A,ta*
973-6
24423
4^3-27
21
E, d^l
384-9
09656
329-73
48
At a'
9962
24991
469-37
22
E d't
407-5
10224
334-07
49
A'ta>
1018-9
25559
475-55
23
F, 6' d'JJ
4302
10792
338-47
50
J3, a^ta*
10415
26127
481-81
24
F, 6« d'5
4528
11360
34293
5'
3, a't
^"^ft
26695
488-15
25
F, a»
475-5
11905
347-44
52
B, a't
1086-8
27263
494-58
26
F 6«
4981
12496
35201
53
B a't
1109-4
27831
501-09
27
Fjt/' 6*
520-8
13064
35665
i'
B» a««c.
1132-1
28399
50709
Art. 28. — The cycle of 1200 is especially used for indicating the relations of
just notes and the mode in which the notes of tempered and inharmonic scales
generally fit in among these just notes. The series of 11 7 just notes to the Octave
is developed in sect. E. infr^, and the value of each note is given {ibid. art. 18) by
the numbers of the corresponding note in the cycles of 53 and 1200. The number
in the cycle of 53, by means of the table in art 27, gives all the information
required as to these substitutes, including Mr. Bosanquet's names, which are
different from those assigned to just intonation on the principles of Chap. XIV.
pp. 2766 to 277a. To shew the difference between cyclic and just cents the
following table is added, in which all the names of the 117 just notes in the
Digitized by V^OOQ IC
440
ADDITIONS BY THE TRANSLATOB.
APP. IDL
Doodenariimi of sect. E. p. 463, are placed in alphabetical order for easy reference,
with their cyclic and just cents, logarithms, and pitch. It must be remembered
that the inferior and superior numbers in this table refer to differences of a Comma
of 22 cyclic cents, or one of 21-5 just cents, and in that of the cycle of 53 on p. 439,
to a degree of 22*6 cents, hence the same name has distinctly different meanings.
Thus Bosanquet*s No. 5, or C^fi^c^, has 22-6 cents, and just c^ has 21-5 cents,
which agrees well with 22*6, but just C^fi has 49*2 cents and agrees more nearly
with Bosanquet*B No. 6, or Cjj^ with 45*3 cents.
Expression of Just Intonation in ths Cycle of 1200.
Note
Cyclic
Cents
Jnst
Cents
LogB
Pitch
A
906
9059
22724
445*5
A^
928
927-4
23264
451*1
A,
884
8844
22185
440-0
^»
862
862-9
21645
434*6
A,t
998
998-0
25037
469-9
A,l
976
976-5
24497
464*1
A,t
954
955*1
23958
458-3
A,tt
1068
I068-7
26809
480-4
A.%t
1046
10472
26270
483-4
A\>
792
7920
19873
417-2
A'b
814
8137
20412
4224
A^b
836
835-2
20952
427*7
A^bb
722
721-5
I8I00
400-5
A^bb
744
743-0
18639
405*5
A*bb
766
764-5
I9I79
410*6
A*bbb
652
650-8
16327
27840
384*5
B
IIIO
1109-8
501-2
"B.
1088
1088-3
27300
495-0
5,
1066
I066-8
26761
488-9
BS
1 180
ii8o'4
29613
522- 1
B,%
1 158
1158-9
29073
515-6
54
1 136
1 1374
28534
5093
B,t%
72
72-5
01822
275*3
B,l%
50
504
01282
271-8
Bb
996
996-1
24988
469-3
-B'b
1018
1017-6
25527
475*2
B^b
1040
1039-1
26067
481-1
B'bb
904
9039
22675
445.0
B'bb
926
925-4
23215
450-6
B»bb
948
946-9
23754
4562
B<bb
970
968-4
24294
461-9
B»bbb
834
833-2
20902
427-2
jB'bbb
856
8547
21442
432-5
C
0
0
0
2640
C»
22
21-5
00540
267-3
0,
1178
1 178-5
29564
02852
521-5
CI
114
"37
281-9
c,«
92
92-2
02312
278-4
CA
^2
70-7
01770
275*0
CJI
48
49-2
01233
271-6
G^t
184
1844
04625
2937
c.««
162
162-9
04085
c,tt
140
141*3
03546
286*6
c,tti
254
255*0
06398
305*9
C'b
1108
1107-8
27791
5006
C*b
1 130
4293
28330
5069
C'b
1152
1150-0
28870
5'3*2
C»bb
1038
1037-1
26018
480-7
C»bb
1060
1058-7
26557
486-6
C*bbb
946
945-0
23705
455*7
D
204
203-9
o5"S
297-0
D'
226
2254
05655
300-7
A
182
182-4
04576
293*3
A«
296
296-1
07427
313*2
JD,«
274
2746
06888
3094
A5
252
253*1
06349
305-6
AS 8
366
366-8
09201
3263
ajj
344
345*3
458-9 '
08661
3223
A5«5
458
"513
344- 1
Kote
Db
2)»b
2)«b
JD'b
D'bb
D»bb
2)*bb
D*bbb
E
E,
E,
E,t
E^t
^,«
E,t
Eb
E'b
E^b
E'bb
E*bb
E*bb
E^bbb
jB^bbb
F
Ft
F,t
Fjit
F,tZ
F,Zt
F,Ztt
F'b
F'b
F*b
F'bb
F*bb
G
G'
G.
G.«
Ojl
G,«
(?,««
0,8 s
o,tzt
G'b
Q'b
(Pb
G'bb
0*bb
G'bb
G'bbb
Cyclic
Cents
90
112
134
156
20
42
64
1 150
408
386
364
5^
478
456
434
570
548
294
338
224
246
268
132
154
498
520
542
476
612
568
682
660
638
752
406
428
450
336
358
702
724
680
794
772
750
886
864
842
956
610
632
654
518
540
562
448
Just
Cents
90-2
111-7
132-5
154-7
19-6
4I-I
62-6
1 148-9
4078
386-3
364-8
500-0
478-5
457-0
435*5
570-7
549*2
294-1
3>5*6
337*1
223-5
245-0
266-5
131-3
1528
498-0
519*6
54I-I
476-5
611-7
590-2
568-7
682-4
660*9
6394
753-1
405-9
427*4
448-9
335*2
336*7
702-0
723*5
680*4
794*1
772-6
751*1
886*3
8648
843-3
957-0
6098
631*3
652*8
517-6
i^
446*9
Logs
DiyiLized by v
02263
02803
03342
03882
00490
01030
01569
28821
1023 1
09691
09152
12542
12003
1 1464
10924
I4316
13776
07379
07918
08458
05606
06145
06688
03293
03833
12493
13033
13573
1 1954
15346
14806
14267
17119
16579
16040
18892
I0181
I0721
1 1 261
08409
08948
17609
18149
17070
19922
19382
18843
22234
21694
21 15s
24007
15297
15836
16376
12984
13524
14063
II2I0
H^€-
BBCT.B. ON THE DETERMINATION OF PITCH NUMBERS.
441
and pitch number 445*5. Then for ^A, Bosan-
qaet'8 number is 43 (having 883 cents, log
'22151, pitch number 4397)* the cyclic cents
are 906— 27 = 879, the just cents 905-9-27'3
= 878*6, log = '22724 — -00584 = '22140, the
pitch number = 445-5 - A x 445-5 = 445-5 - 6'96
= 438*5, shewing that Bosanquet's substitute
is a trifle too sharp.
If it is wished to introduce the series of
natural harmonic Sevenths as in Poole (infri,
sect. F. No. 7), X being any note, 'X will have
the number of X in Bosanquet's cycle dimi-
nished by I, the cyclic cents of X diminished
by 27, the just cents of X by 27-3, the log of
X by '00584, and the pitch number of X by ^
of its value. Thus A has Bosanquet^s number
44, cyclic cents 906, just cents 905*9, log -22724,
Art. 29. — Those who require more information are referred to my paper on
* Temperament,' Proc. B. S., vol. xiii. pp. 404-422 (where the subject is treated
more in detail and in an entirely different manner), and to the memoirs and
essays of Salinas, Zarlino, Huyghens, Sauveur, Henflins, K. Smith, Marpurg,
Est^ve, Cavallo, Romieu, Lambert, T. Young, Eobison, Farey, Delezenne, Wool-
house, De Morgan, Drobisch, Naumann, there cited. Also to Mr. Bosanquet's
treatise on Mtisical Intervals and Temperament , 1876, to his papers on * Tempera- IF
ment * in the Proc. of the Musical Association, first year, pp. 4-17, 112-54, and his '
article on * Temperament ' in Stainer and Barrett's Dictionary of Mtisical Terms,
Also to Mr. Lecky's article on * Temperament * in Grove's Dictionary of Music,
SECTION B.
ON THE nSTEBlUNATION OF PITCH NUUBEBfi.
(See notes ppr. 11, 56, 168, 176.)
No.
I
No.
5.
6.
The Clock (Eoenig, Lord Bayleigh), p. 442.
Harmonium Beeds (Appunn, Lord Bay>
leigh), p. 443.
Tuning-forks (Scheibler), how to form and
use a tuning-fork tonometer, p. 44.3.
The String rEuler and Bernouilli, Thomp-
son, Griesoach, Delezenne), p. 441.
2. The Siren, p. 442.
3. Optical Method (McLeod and Clarke), p. 442.
4. Electrical Methods (Mayer, Glazebrook),
p. 442.
The determination of the pitch number of any note heard is a very difficult
problem to solve. The following methods have been used. 1[
I. The String, Supposing that a heavy string of uniform density, perfect elas-
ticity, of no thickness, but capable of bearing a considerable strain, could have its
vibrating length determined with perfect accuracy — none of which conditions
can be more than roughly fulfilled — then the pitch numbers of its parts would be
inversely proportional to their lengths. The formula has been worked out by
Euler and Bernouilli, and amounts to this.
Let L be the vibrating length of a suspended string in English inches, I the
same in French millimetres, TFthe stretching weight in any unit, w the weight of
the vibrating length of the string in the same unit, V the pitch number. Then
2 log 7 = 1-98485 + log Tr-(log w + log L)
= 3*38968 + log W-{log w + log I).
This was used for careful measures by Euler, Dr. Eobert Smith, Marpurg,
Fischer, and De Prony. Probably on account of the necessary thickness of the
string the results could not be trusted within 5 vib. 5I
The work is also extremely difiScnlt, and
depends ultimately on determining a unison
between two tones of very different qualities.
General T. Perronet Thompson used such an
instrument for tuning his organ (described in
Just Intanationt 7th ed. p. 69). As his string
was No. 20 (i'i65 mm. in diam.) no reliance
could be placed on the perfect exactness of his
results. Mr. J. H. Griesbach in i860 tuned a
string 5*17 mm. in diam. till one quarter
of its length was in unison with a given note,
and then counted the yibrations of the whole
string automatically. The instrument is in
the South Kensington Museum, and was de-
scribed in the Journal of the Society of Arts,
6 April i860, p. 353. The results were 3 to 6
vibrations wrong.
Delezenne {M&m, de la Soc. des Sciences d
- Lille, 1854, p. i) made the best use of the
string. He stretched 700 millimetres of wire
on a violoncello body, and tuned it to Mar-
loye's 128 vib. (which was probably accurate,
as Marloye*s 256 vib. certainly was), and
then by a movable bridge cut off the length,
which when bowed was in unison with the
fork. This fork had been adjusted by slid-
ing weights to the pitch of a note heard.
Then measuring this length in millimetres, he
divided 128 x 700^89,600 by it, to find the vib.
This assumed that lengths were inversely as
the vib. or pitch numbers. But he found that
the same fork was in unison with 203*8 mm.
of a string '6154 mm. thick, and 198*9 mm.
of a wire -1280 mm. thick. The former gave
439*6, the latter 450*5 vib. The thick wire
therefore gave a pitch 42 cents flatter than
Digitized by V^jOOQlC
442 ADDITIONS BY THE TEANSLATOR. app. xx.
the thin. Hence he confined his obeervations was pressed down on it by a knife-edge. The
to the thinnest wire that woald bear the strain, intervals between the lowest and each of the
In conjunction with Mr. Hipkins of Broad- other forks, and the longest and each of the
woods* and the foreman tuner, Mr. Hartan, other lengths (assuming them to be invers^y
I made some experiments on 2 April 1885 as the number of vibrations), were calculated
with the monochord at Broadwoods*, having in cents. The former were generally sharper
a string of '98 mm. in thickness. Nineteen by the following cents, the minus sign shew-
tuning-forks about 20 vib. apart, with pitches ing when they were flatter: 076133 9 ^^
from 22377 to 578-40 vib. accurately known, 2 23 22 91 118955-^-4. These irregular
were brought into unison with lengths of the differences demonstrate that such a monochord
monochord limited by a movable bridge touch- gives very uncertain results, even when the
ing, but sliding easily under the string, whioh unisons are estimated by very sensitive <
2. The Siren, This has been described in the text, p. 12. But only the most
carefully constructed sirens with bellows of constant pressure, as that described in
App. I., or the * Soufflerie de precision ' of M. Cavaill6-CoU, worked by well prac-
tised operators, can give good results. Here also a unison between tones of very
^ different qualities, one of which is fixed and the other variable, has to be deter-
mined. The best work that I know with the siren was that done by M. Lissajons
(who used M. Cavaill6-Coirs bellows, as the latter tells me) in determining the
pitch of the ' Diapason Normal ' at Paris, which was meant to give 435 vib. at
15® C.= 59'' F., and actually gave 435'45 vib.
3. The Optical Method of Professor Herbert McLeod, F.R.S., and Lieut. R. G.
Clarke, E.E. (Proceedings ofBoyal Society, Jan. 1879, vol. xxviii. p. 291, BudPhilo-
sophidtil Transaction's, vol. clxxi. pp. 1-14, plates i to 3), consisted in viewing white
lines on a rotating cylinder through the shadow of a vibrating fork. The machine
is difficult to manipulate, but in the hands of its inventors gave extremely accurate
results.
4. The Electrographic Method, invented by Prof. A. Mayer, of Stevens Institute,
Hoboken, New Jersey, U.S., consisted in causing a tuning-fork by means of a
copper-foil point to scribe its vibrations on the camphor-smoked paper cover of a
brass rotating cylinder, and marking seconds by passing a strong induction spark
through the fork, scribing point, and paper cover at the passage of a seconds pen-
IT dulum through a spot of mercury, and then counting the sinuosities at leisure.
The weight of the scribing point had to be allowed for, but the results were very
accurate.
For another means of determining the frequency of a fork used as an inter-
rupter of electricity, see Mr. B. T. Glazebrook's paper in Philosophical Magazine^
Aug. 1884, vol. xviii. pp. 98-105.
5. The Clock. Dr. Koenig (in Wiedemann's, late PuggendorflTs Annals, 1880,
pp. 394-417) describes a large tuning-fork having the pitch number 64, which was
made to act on a clock at a constant temperature of 20° C.=68** F., functioning as
a pendulum, so that every single vibration was registered for many hours, and he
was thus enabled to determine one standard pitch with extreme accuracy. He also
found from this that his old well-known forks of nominally 256 double vib. had been
tuned at too high a temperature, and at 20** C. gave 256'i774, and at 15** C.=59** F.,
256*28 d. vib. He also determined the pitch of the Diapason Normal at 15^ G. as
435*45 d. vib. A rise in temperature of 1° F. flattens tuning-forks by i vib. in
f from 16,000 (Eoenig) or 20,000 (Scheibler) or 22,000 (Mayer) double vibrations in
a second (see my ' Notes of Observations on Musical Beats,' Proc, B, S., 28 May
1880, vol. XXX. p. 523), and flattens harmonium reeds by about i in 10,000 vib.
(See my paper * On the Influence of Temperature on the Musical Pitch of Har-
monium Reeds,' Proc. B. S., Jan. 1881, p. 413.)
hordBAyleigji (PhUosophical Transactions, placed a screen perforated by a Bomewhal
1883, Part I., pp. 316-321) describes another narrow vertical slit. If the period of the pen>
method of determining the frequency of a dulum were a precise multiple of that of the
standard fork by means of a clock. * The ob- fork, the flash of light, whioh to ordinary ob-
server looking over a plate carried by the upper servers would be visible at each passage,
prong of the fork [of intentionally 32 vib.] ob- would either be visible, or be obscured, in %
tained 32 views per second, i^. 64 views of permanent manner. If, as in practice, the
the pendulum, in one complete vibration. The coincidence be not perfect, the flashes appear
immediate subject of observation is a silvered and disappear in a regular cycle, whose period
bead attached to the bottom of the pendulum, is the time in which the fork gains (or loses)
upon which as it passes the position of equili- one complete vibration. This period can be
brium the light of a para£Bn lamp is concen- determined with any degree of precision by a
trated. Close in front of the pendulum is suflicient prolongation of the observations.*
Digitized by V^jOOQlC
SECT. B. ON THE DETERMINATION OF PITCH NUMBERS.
443
6. Harmonium Beeds,
HeiT Georg Appunn, of Hanau, invented tonometers of 65, 33, and 57 reeds,
serving for many important experiments (see for example supr^, p. 176, footnote f).
Copies of all three are at the South Kensington Museum, and of the two first at the
Museum of King's College, London. These reeds were tuned to make 4 heats in
a second with each other, so that the 65th reed made with the first, which was
an Octave flatter, 4x64=256 heats, and consequently, according to the theory of
beats (see text, Chap. VIU.), the pitch numher of the lowest note should be 256.
Unfortunately, the condensed air in which the beats took place accelerated the beats
by 76 in 10,000, as I determined by long-continued observations and experiments
(described generally in my first paper already cited, Proc, B, S., vol. xxx. pp. 527-
532), and consequently the results had to be lessened by that amount, making the
pitch number of the lowest reed about 254. These reeds, too, were not sufficiently
permanent in pitch. Hence this instrument, though otherwise very useful, failed II
in determining pitch with sufficient accuracy.
Lord Bayleigh (Proc Mu$. Aasn, vol. v.,
1878-9, p. 15) disoovered a way of determin-
ing the pitch of two low harmoniom reeds,
the lowest C and D on bis harmonium. Keep-
ing the wind for 10 minutes or 600 seconds
as constant as possible and using resonators
tuned by partially covering with the finger to
about the 9th and 10th partials of the low C,
two observers counted the beats, one between
the 9th partial of C and the 8th of D, and the
other between the loth partial of C and the
9ih of JD. They thus found
gC— 8 2) = 2392+6oo
9i>-io 0=» 2341 +600
Whence
C = (9 X 2392 + 8 X 2341) ^600
= (21 528 + 18728) -r 600 = 67-09
and similarly
i) = (io X 2392 + 9 X 2341) -1-600= 74*98
As these notes make an interval of 192*5
cents with each other, Lord Bayleigh had
evidently (as he suggested) altered the interval
to about a meantone 193*2 cents. His object
was to determine the pitch of a fork of
Koenig's, supposed to vibrate 64 times in a
second. Now as Koenig's 256 is really 256*28
at 59^ F., this 64 should be 64*07 at the same
temperature. Lord Bayleigh, to take ac-
count of the effect of the simultaneous beating
of the two reeds, sounded both of them at the
same time with the fork, and on different days
obtained the following results (temperature
not named) :—
Harmonium 67*09
67*04
67*17
67*19
Tnning-fork 64*06
>» 6407
„ n 6417
,> 6398
Which were wonderfully accurate.
7. Tuning-forks.
All the above methods have one important fault. The measuring instruments
are not easily portable and not readily applicable to all kinds of sustained tones,
while the two first required trained ears to discriminate unisons between tones
of different qualities, with great accuracy, that is to say, at least i vib. in 10 sec. All
these faults are obviated by the Tuning-fork Tonometer invented by J. Heinrich
Scheibler (b. 1777, e2. 1837), a silk manufacturer of Crefeld in Ehenish Prussia.
The simplest process of making such a Tonometer, although not the one used by
Scheibler (see his pamphlet cited in note t, p. iggd), is as follows. I quote prin-
cipally from my ' Notes on Musical Beats,' already cited.
Obtain a set of about 70 good forks with
parallel prongs, and of a tolerably large size ;
tone the lowest to about the cf (or 6 for English
high pitch) and tune the rest roughly each
about four beats in a second sharper than the
preceding. Then fit them with wooden collars
or handles, and allow them to rest for three
months, if possible in the same temperature
at which they will be counted, and never
alter their pitch again by filing, but count
the beats between each set most carefully, at
a temperature which remains as uniform as
possible. It may be necessary to use a high
temperature ; thus Scheibler's was from 15^ K.
to 18° R.« 65*75 to 72°-5 F., which I reckon
at 69^ F. as a mean ; and Koenig now works
at 20° G. =s 68° F. Count on one day the beats
between forks i and 2, 3 and 4, 5 and 6, (fee,
and on another between forks 2 and 3, 4 and 5,
Ac, so that the same fork is not used for ^
two counts on the same day. Excite by strik>
ing with a soft ball of fine flannel wound round
the end of a piece of whalebone, as a bow is
not convenient unless the forks are tightly
fixed. Each blow or bowing heats, and hence
flattens, and this tells if the experiments on
any one fork are long continued. Count each
set of beats for 40 seconds if possible, and
many times over, registering the temperature
and the beats, and taking the mean. Having
counted all, observe those forks which are
near the Octave of the lowest fork. Find two
such, beating with the Octave (that is, the
second partial) of the lowest fork less than
they beat with each other. Then the sum of
all the beats from the lowest fork to the lower
of the two forks, added to the beats of the
Octave (that is, the second partial tone of the
Digitized by VjOOQlC
444
ADDITIONS BY THE TRANSLATOR.
APP. XX.
lower fork) with that fork, is the pitch of the
lowest fork. Hence the pitch of all the forks
is known. The extra high forks are for verify-
ing by the Octaves of several low forks, and
for the purpose of subsequently measuring.
From such a tonometer any other can be
made, and the value of each fork at another
temperature calculated.
Scheibler made a 52-fork tonometer with
infinite trouble, on another plan, described in
his book and counted it with marvellous accu-
racy. This tonometer, which I have made
many efforts to find, has absolutely disap-
peared and his family knows nothing of it.
But he left behind him a 56-fork tonometer,
believed to proceed from 220 to 440 vibra-
tions, by steps of 4 vibrations, and through
the kindness of Herr Amels, an old friend of
^ the Scheibler family, who obtained it from
Scheibler 's grandson, I had the use of it for
a year. I had to count it as well as I could,
just as if it had been a set of forks such as I
have described, and I found it was not what
was thought, but that only 32 sets of beats
were 4 in a second, and the other 23 sets
varied from 38 to 42 in 10 seconds. I found
also that the extremes were probably of the
same pitch as in the original 52-fork tono-
meter. After then counting it as well as I
could, and obtaining 219-27 vibrations in place
of 219*67, at 69° F., I distributed the error of
4 beats in 10 seconds, as 2 in 100 seconds,
among 20 of the 23 sets which were not
exactly 4 beats in 10 seconds, leaving the
first 3 sets, which I had repeatedly counted
and felt sure of, imaltered. Then I reduced
all the values from 69** to 59° F. Finally
U to verify my result I measured by beats
with Scheibler's forks as thus determined;
first 5 large forks of various pitches, which
I had had made for me in Paris, and then
4 forks of Eoenig's belonging to Professor
McLeod. Professor McLeod himself kindly
measured all of them, also, by his machine,
and Professor Mayer also obligingly measured
the first 5 forks by his electrograpMc method,
both with the greatest care and precaution.
The three sets of measurements agreed to
less than i beat in 10 seconds, and more
often less than i beat in 20 seconds, when
reduced to the same temperature. Thus the
value of the tonometrioal measurement by
beats only, and the possibility of counting.
a tonometer sufficiently, was fully esta-
blished. Eoenig's measurements of his own
forks reduced to 59^ F., and of the actual
% Diapason Normal at the Ck)nservatoire, Paris,
intended to be used at the same tempera-
ture, also agree with mine within less than
the same limits.
The pitch of all sufficiently sustained tones
can thus be determined mechanically. We
find two forks, whose pitch is known, with
each of which the new tone beats more slowly
than the forks beat with each other, and we
verify the count, by seeing that the sum of
the beats with both forks is the number of
beats of the forks with one another. The
pitch number is then that of the lower fork
increased by the number of beats made with
it in a second, and that of the upper fork
diminished by the number of beats made
with it in a second. The following notes
are the result of much ezperienoe.
Tuning-forks are comparatively simple in
quality of tone but always possess an audible
second partial or Octave, and sometimes higher
partials still, capable of being so reinforced
by resonance jars properly tuned to them,
that beats can be separately obtained from
them and counted. This, as we have seen, is
a matter of great importance in the con-
struction of a tuning-fork tonometer. When
the tone is very compound, as in the ease of
bass reeds (especially tliose of Appunn's tono-
meter, furnished with a bellows giving, -when
properly managed, a perfectly steady blast
for an indefinite lengih of time), beats can
be obtained and counted from the 20th to the
30th and even the 40th partial, without any re*
inforcement by a resonance jar. (Seep. I76c2'.)
Taking tuning-forks first, I find it advan-
tageous to hold the beating forks over one or
two resonance jars, tuned, by pouring in
water, to the pitch of the partial to be ob-
served, whether it be the prime of both or
the prime of one and the second partial (or
Octave) of the other. There may be smaU
differences, but I have not found any diffei^
ence appreciable by my methods of observa-
tion in the number of beats in a second,
whether the resonance jar is the same or
different for the two forks, and whether it is
exactly or very indifferently tuned to each
fork, but a tolerably accurate tuning mneh
' improves the tone and length of the beat. In
that case the resonance jar practically quenches
all other partial tones, and the beats are
distinctly heard as loudnesses separated by
silences. If no jar is used, the other par-
tials are heard. In the case of the Octave, the
low prime becomes a drone and fills up the
silences. In the case of beating primes, the
Octaves, which are beating twice as fast, tend
to confuse the ear. Sometimes the second
partial of a fork is so much stronger than the
prime, that when the fork is applied to a
sounding-board, only the Octave is heard,
which is often inconvenient to the fork tuner.
This is entirely avoided by the resonance jar.
Beats being a case of interference, the ampli-
tude of the beating partials should be equa-
lised as much as possible. With two forks of
very different size and power, it is easy to
regulate the amplitude by holding the louder
fork further from the jar. Otherwise the
beats become blurred and indistinct. For
powerful reeds or organ pipes, beating with
forks, it is best to go to a considerable <lis»
tance from the reed or pipe and hold the fork
close to the ear or over a jar. I find 50 or 40
feet necessary for organs ; in Durham Cathe-
dral, where the pressure of wind was strong
and my forks weak, I found 60 or 70 feet dis-
tance much better. As I was not iJ>le latterly
to go to a distance from Appunn's reed tono-
meter, having to pump it myself, I foond it
impossible to count the primes of the upper
reeds by the Octaves of my forks, which were
completely drowned by the reeds.
I find beats of all kinds most easy la
count when about 3 or 4 in a second. They
can be counted well from 2 to 5 in a second.
Above 5 they are too rapid for aconiacy;
below 2, and certainly below i, they are too
slow, so that it is extremely difficult to teU
from what part of the swell of sound the bort
should be reckoned. Partly from this reason.
Digitized by V^jOOQlC
EOT. B.
ON THE DETEBMINATION OP PITCH NUMBERS.
445
erhaps, I have found great variety in connt-
ag suooeBsiye sets of such slow beats. I
lever use beats of less than one in a second,
I I can avoid it. When the beats are slow it
9 difficult to discover by ear which of the
wo beating tones is the sharper; and even
Lne ears are often deceived. It is easy to
liscover, however, by putting one of the forks
tnder the arm for a minute. This heats and
lattens it by 2 or 3 beats in 10 seconds.
HLence if the beats with the heated fork are
ilower, that fork was sharper, because it has
>een brought nearer the other ; if faster, it
ras flatter and has been brought further away.
>)unt for 10, 20, or 40 seconds, according to
he fork. Up to 20 or 30 beats in 10 seconds
t is easy to count in ones, but from 30 to 50
t is best to count in twos, as one-ee, two-ee,
fee., beginning with one, and hence throwing
>ff one at the end. When counting for 20
seconds I always count in twos, and for 40
seconds in fours, as one-ee>ah-tee, two-ee-ah-
tee, <ftc., because I have to divide the result by
zo or 40 ; and this division is avoided by the
30unt itself. As my counting was never for
more than 40 seconds, small errors of the
olock or pendulum were imperceptible. I
B^enerally used a marine or pocket chrono-
meter when making the principal count for
40 seconds, but for merely determining pitch
from a fork of known pitch, 10 seconds of
time, and any ordinary seconds watch suffice.
For Prof. McLeod's observations, which lasted
5 minutes or more, extreme accuracy in rating
an astronomical clock was necessary. Sup-
pose a watch to gain 5 minutes or 300 seconds
in 24 mean hours, which is an extreme case,
80 that 86,700 watch seconds = 86,400 mean
seconds, then 10 watch seconds = 9*9654 mean
seconds, and 40 watch seconds = 39*8616 mean
seconds. Hence no perceptible error will arise
from identifying watch and mean seconds.
But 300 watch seconds = 298*962 mean seconds,
and the error would have to be allowed for.
Scheibler used a metronome corrected daily by
an astronomical clock, and graduated. On
this a movable weight enabled him to make
one swing of the pendulum take place in the
same time as 4 beats were heard, and then
from the graduation he read off the rate.
But counting by the seconds hand of a watch
is much easier and altogether more conve-
nient, while it is probably as accurate. Owing
to difficulties in beginning and ending the
count, I find the possible error per second
to be 2 divided by the number of seconds
through which the count extends; and that
it is best to take a mean of 5 to 10 counts
for each set of beats. As most persons, in-
cluding myself, begin to count from one and
not from nought, it must be remembered that
the last number uttered on counting the last
beat, is one in excess of the real number of
beats. Thus if in counting for 10 seconds
we end with 37, the number of beats was 36
in 10 seconds. If in counting by twos, one-ee,
two-ee, &G., the last was 19, then we have
counted only 18 pairs, and hence there were
also 36 beats in 10 seconds. If we end with
19-ee, there were 18^ pairs or 37 beats in a
second. It is best to count the same set of
beats in one, twos and fours to realise these
corrections, which are extremely iiAportaftt
Temperature must never be neglcNcted.
Forks should not be touched with the un-
protected hand ; they otherwise easily flatten
by 2 beats in 10 seconds. Interpose folds of
paper. I use two folds of brown paper
stitched between two pieces of wash-leather.
Large forks are generally on resonance boxes
and need not be touched, otherwise the same
precautions should be used, as they are very
sensitive, and retain the heat longer than
small forks. Scheibler's forks are fitted with
wooden handles. In tuning, the file heats and
flattens; the result of tuning, therefore, can
seldom be known for a day or two, when the
forks have cooled and 'settled,* as they will
be sure to * jump up.* I find it best to leave
off filing when the forks are two or three
tenths of a vibration too flat. In sharpening
there is, therefore, great danger of doing too
much, as the fork remains apparently at the ^
same pitch, the flattening by heat balancing
the sharpening by filing. Hence all copies
should be compared some days after, by
means of a third fork about four vibrations
flatter or sharper than each, to avoid the
slow beats of approximate unisons. The
filing also seems to interfere with the mole-
cular arrangement of the forks.
The thermometer should be always con-
sulted when beats are taken. But if the beats
are between two forks, of which the pitch of
one at a given temperature is known, and
both forks may be assumed to be altered in
the same ratio by heat, then the temperature
need not be observed ; but the unknown fork
may be presumed to be as many vibrations
sharper (or flatter) than the measured fork at
the temperature at which the latter was
measured, as beats in a second were observed ^
to take place. This is because the alteration
is very small, and would be quite inappreci-
able for the few vibrations between them.
But for tonometrioal purposes an allowance
must be made.
When forks are counted without a reson-
ance jar, they should not be applied to a
sounding-board, or held one to one ear and
one to the other, but should both be held
about six inches from the same ear, and
their strengths should be equalised by hold-
ing the weaker fork closer to the ear than
the stronger.
When the forks are screwed on and off a
sounding-board or resonance box, there is
great danger of wrenching the prongs, unless
they are held below the bend, but I have con-
stantly seen this precaution neglected. A cr
wrench immediately affects the pitch and
duration of sound of a fork, and renders
it comparatively worthless. Such cases have
come within my observation. To prevent
wrenching when filing forks, only one prong
should be inserted in the vice. And for even-
ness file the same quantity off the inside of
each prong, counting the number of strokes
with the file, near the tips for sharpening,
and near (not at) the bend for flattening.
The next enemy to be guarded against is
rust. Forks should be kept dry, and occasion-
ally oiled with gun-lock oil. Bust towards the
tip affects the fork much less than rust at
the bend. My observations and experiments
shew that errors from rust can scarcely exceed
a flattening of i vibration in 250, and are
generally very much less. But as the amount
Digitized by
^.joogle
446
ADDITIONS BY THE TRANSLATOR.
APP. XX.
is uncertain, rust spoils a fork for accurate
tonometrical purposes. With care, however,
the pitch of a tuning-fork remains remarkably
permanent. Scheibler's had eyidently not
altered in more than forty years.
When the pitch to be ascertained is of a
very short sounding tone, as that of a glass
or wood harmonicon, or very high, it requires
an extremely delicate ear indeed to be able to
determine between which two forks, or the
octaves of which two forks the tone lies.
In this case I have been fortunate in having
the kind assistance of Mr. A. J. Hipkins of
Broadwoods', without whose accurate apprecia-
tion of differences of pitch I should have
frequently been altogether at a loss.
Dr. Eoenig (27 Quai d'Anjou, Paris) makes
^ tuning-fork tonometers of beautiful workman-
ship, but they are necessarily very expensive.
The largest, proceeding from 64 to 2 1 845 double
vibrations at 20*^ F. with sliding weights, costs
1200Z. The medium has 67 forks from 256
to 512 double vibrations at intervals of four
beats each, with / and a! which fall between
pairs of forks, mounted on resonance boxes, and
costs i2o2. A smaller set without resonators
costs 6o2. A small set of 13 forks in a case,
giving the equally tempered Octave c' to c'' for
a! 435 double vibrations, costs only *]l. 4^., and
for the same price another set 4 double vibra-
tions lower, for tuning by beats to French
pitch, can be obtained. This apparently high
price arises from the great difficulty of taxiing
such a succession of forks with perfect acco-
racy to particular pitches. This accuracy is,
however, not necessary, provided the coont be
accurate. Any tuning-fork maker would make
a set of forks such as has been described.
The count must be made by the investigator
himself, and he should verify by a set of
Eoenig*s forks of c\ e', ^, c", such as may be
found in many places, remembering that all
the older sets when reduced for temperature
give at 15° C. = 59<» F., d 256-3, «' 320-3, ^
384*4, d' 512*6. Prof. McLeod*s determina*
tions by his machine at this temperature, as
the mean of many measurements, were c'
256*310, another copy 256*306, d 320-372, tf
3S4'437f d' 512-351. Eoenig considered them
correct at 26-2<* C. = 79*i6° F.
For my own use after returning Scheibler's
forks to Mr. Amels, Ihad 105 forks constructed
proceeding from 223-77 to 586*12 vib., which,
after being tuned partly by Scheibler's maker
in Crefeld by the forks already described, and
partly constructed to differences of about 4
beats by the late Mr. Valantine of Sheffield,
were all very carefully counted again by me
with Scheibler's own forks, the means of manj
determinations up to two places of decimals
being assumed as correct. With this perfectly
unique set of forks I have, since that time,
made all the determinations of pitch mentioned
in this book.
ir
Art.
I.
SECTION 0.
THB CAIiCUIATION OF CENTS FROM INTEBVAL lUnOS.
(See notes pp. I3i 4I1 and 70.)
Art.
16
Nature of Cents, and necessity for using
them, p. 446.
I. Firzi Method, without Tables or
Logarithms^ p. 447.
2. When the interval exceeds an Octave,
p. 447.
3. When the interval lies between a Fourth
and Fifth, p. 447.
4. When the interval is less than a Fourth,
p. 447.
U. Second Method, with Tables but
without Logarithms^ p. 447.
5. Previous Reduction, p. 447.
IT 6, 7. Rule and example, p. 447-^»
III. Third Method, by Five Place Logarithms,
p. 448.
8. Reason for previous methods, p. 448.
I. When the ' interval numbers,* that is the pitch numbers of two notes, have
been found (or the ' interval ratio,* that is ratio of those numbers given theoretically
by means of pitch numbers, or of numbers in proportion to them, or of lengths of
strings assumed to be perfect, or of wave-lengths), it is necessary, in order to have
a proper conception of the interval itself by comparison with a piano or other
instrument tuned in intentionally equal temperament, to determine the number
of cents, or hundredtlis of an equal Semitone, in that interval. Such cents have
been extensively used in the notes, and occasionally introduced into the text, of
this translation, see pp. 4id, 50a, 56a, &c., and supri, sect A. art. 2, p. 4316.
9, 10. Rule and examples, p. 448.
11. To find to the tenth of a cent, p. 448.
IV. Fourth Method, by Seven Place
Logarithms,
12. Rule and examples finding to the thoa^
sandth of a cent, p. 449.
y. Finding Interval Ratio from Cents.
13. Without Logarithms, p. 449.
14. By Five Place Logarithms, p. 449.
15. By Seven Place Logarithms, p. 450.
Given a note of any pitch number to find
the pitch number of a note which makes
with it an interval expressed in oenis^
p. 451.
Principal Table, p. 450.
AuxiUary Tables, I. IL IIL IV., p. 45i»
Digitized by
Google
SECT. O.
ON THE CALCULATION OP CENTS.
447
I. First Method, without either Tables or Logarithms*
2. If the greater number of the ratio be more than twice the smaller, divide the
greater (or else multiply the less) by 2 mitil the greater nmnber is not more than
twice the smaller. This is equivalent to lowering the higher or raising the lower
tone by so many Octaves. Hence for each division or multiplication by 2 add 1200
to the result.
Ex. To find the cents in 47th harmonic.
Interval ratio i : 47. Multiplying the smaller
number 5 times by 2, the result is 32 : 47, for
which 'reduced' interval ratio we have to
determine the cents as ander and then add
5 X 1200=^6000 to the result.
3. If the reduced interval ratio be such that 3 times the larger number is
greater than 4 times the smaller, but twice the larger number is less than 3 times
the smaller number, then multiply the larger number by 3, and the smaller by 4,
for a new interval ratio, and add 498 cents to the result. m
Ex. For 32 : 47, then 3 x 47 » 141 is greater
than 4 X 32 - 128, but 2 x 47 » 94 is less than
3 X 32 » 96. Hence we use the interval ratio
128 : 141 and add 498 cents to the result. If
however as in 32 : 49, twice the larger number
or 2x49^98, is not less than 3 times the
smaller or 3 x 32 = 96, we use this interval
ratio 96 : 98 or its equivalent 48 : 49 and add
702 cents to the result. In the first case the
4. Multiply 3477 by the difference of the numbers of the reduced interval ratio,
and divide the result by their sum to the nearest whole number, and if the quotient
is more than 450 add i. To the result add the numbers of cents from arta i and
2. The result is correct to i cent.
given interval having the ratio 32 : 47 lies
between a Fourth and a Fifth, in the second
case it is greater than a Fifth, but in both
cases the reduced interval ratio 128 : 141 or
48 : 49 is less than a Fourth. The object
of this reduction, which is seldom necessary,
is to have to deal with ratios less than a
Fourth.
Ex. I. Interval ratio 128 : 141
13 di£Ferenc6
10431
3477
Sam
269)45201(168
269 498 from art. 3
6000 from art. 2
1830
1614 6666 cents result
2161 or 5 Octaves
2152 6 Semitones
and } Semitone
in the interval of 47th harmonic from the fun-
damental.
Ex. 2. Interval ratio 48 : 49, difference « i
Sum
97) 3477 ( 36
291 702
567 738 centa in the interval
588 32 : 49, or about Ji
Semitones.
♦»♦ The number 3477 depends on the
principles explained in my paper * On an C*
Improved Bimodular Method of Computing
Logarithms,' Proc. R. 5., Feb. 1881, vol. xxxi.
p. 382. Cents are in fact a system of logs in
which cent log 2=1200, and its bimodulus is
2 X cent log 2 -I- nat log 2-1 2400 -i- '69315 »
3462*4. But if this number had been selected
there would have been constant additive cor-
rections from the first. Hence an augmented
bimodulus 3477 has been selected » 9 x 386^,
or 9 times the cents in the ratio 4:5. The
result is that the rule is exact for intervals of
a major Third. For less intervals it gives too
great a result, but never by more than '6
cent, which may be neglected. Between a
major Third and a Fourth it gives too small
a result, but only after 450 by about i cent.
For small numbers, few calculations, and in
the absence of tables this method is very
convenient. For Sauveur's previous use of m
an augmented bimodulus, see supri, p. 4376,
sect. A. art. 24, xii.
II. Second Method, with Tables but without Logarithms,
5. The reduction for Octaves is always supposed to be made as in art. 2, and
hence only intervals of less than an Octave will be considered.
If the interval numbers (art. t) contain
decimals they must be multiplied by 10 till
the decimals disappear. Thus 264 : 4785 is
6. Bule. Annex 0000 to the larger number and divide by the smaller to the
nearest whole number. If the quotient is less 1 1290, take the nearest ' quotient '
Digitized by V^jOOQlC
taken at 2640 : 4785, so that the rule applies
to whole numbers only.
448 ADDITIONS BY THE TRANSLATOR. - app. xx.
in the Principal Table below. The corresponding nearest whole number of * cents '
is on the same line.
Ex. 48 : 49. 12 I 490000 Nearest quotient in Table 10210 for oents 36 as
• — in art. 4, ex. 2.
Since 48 » 4 x 12 use short division. 4 1 40833
10208
7. If the quotient exceeds 11290, look for the next less quotient in Auxiliary
Table I., multiply and divide by the numbers in the col. of multipliers and
divisors, and thus reduce the quotient to one in the table. Then proce^ as heiore
and add the number of cents on the line with the next least quotient in Aaxi-
liary Table I.
Ex. Batio 32 : 47 as in art 3. Since 32 » 4 x 8 proceed thus :
4 1470000
81117500
14688
3 Next less quotient 13333, mult.3, div. 4, oents 498.
4:44064
11016 Nearest 11019 cents 168
add 498
as before, 666 cents for the 47th harmonic.
Actually 11016 lies exactly half-way be- the error of i cent in the final result is always
tween 11013 and 11019, and in that case the possible, and is here disregarded. It is avoided
rule is to tiJ^e the larger number. On account by the following methods,
of the approximatiye nature of the calculatioa
fl in. Third Method, by Five Place Logarithms,
8. This is by far the best, most exact, and at the same time easiest method,
but as many musicians are not familiar with logarithms, and it is important that
they should be able to reduce interval ratios to cents, the two precedmg methods
have been inserted.
9. After reducing for Octaves as in art. 2, subtract the log of the smaller
number from that of the larger. If the resulting log is less than '05 268, the
number of cents is given opposite to the nearest log in the Principal Table. The
decimal point, zeroes after it, and characteristic are omitted in practice but are
used here for completeness.
Ex. Interval ratio 48 : 49. The difference between any two loga-
log 49=1*69020 rithms in the table is 25 or 26, hence the
log 48 a 1*68124 nearest is that which differs by less than
13. In this case it differs only by 7.
diff. -00896
m nearest log '00903, cents 36.
10. If the log exceeds -05268, find the nearest log, in Aux. Table 11., subtract
it, take the cents from the Principal Table for this diif. as in art. 9, and add the
cents opposite the next least log in Aux. Table 11.
Ex. Interval ratio 32 : 47.
log 47- I '67210
log 32= 1-50515
ist diff. '16695
Next least, Aux. T. II. 'i 5051, cents 600
2nd diff. '01644
Nearest log '01656, cents 66
result 666, as before for the 47th hannonic.
zi. If it is desired to find the number of cents to the nearest tenth of a cent.
Digitized by V^OOQIC
SEOT. C.
ON THE CALCULATION OF CENTS.
449
take the next least number in the Principal Table, find the difference, and add the
tenths of cents from Aux. Table UI. Thus —
Ex. in art. lo.
2nd diff. -01644
Next least '01631 cents 65
3rd diff. 13 >t S
cents 65-5 resalt.
IV. Fourth Method, by Seven Place Logarithms.
12. As a general rule the approximation in art. 10 is amply sufficient, and is
generally used here. But occasionally it is advisable to proceed to three places of
decimals of cents, as in the whole of sect. A. on * Temperament.' The process is
then conducted by Aux. Table IV., the method of using which will appear by the ^
following examples : —
Ex. I. Interval ratio 32 : 47.
log 47= 1-672 0979
log 32 -1-505 1500
difference
600 cents
•166 9479
•150 5150
60 „
•016 4329
•015 0515
S M
•001 3814
•001 2543
•5 n
•000 1271
•000 1254
•007 „
'000 0017
665507 cents result.
Ex. 2. Interval ratio 264 : 478*5.
log 4785 = 2679 8819
log 264 «= 2-421 6039
difference
1000 cents
•258 2780
•250 8583
20
•007 4197
•005 0172
9
•002 4025
•002 2577
•5
•000 1448
•000 1254
•07 ..
•000 0184
•000 0176
•007 „
•000 0018
1029-5707 cents result.
V. Method of finding the Interval Batio from the Cents.
73. Without Logarithms. If the cents are less than 210, the ratio is that of
iocx)o to the quotient opposite the cents in the Principal Table. If the cents are
greater than 210, subtract the next least number of cents in Aux. Table I., and
multiply and divide the quotient opposite the diff. of cents in the Principal Table
by the corresponding divisor and multipher respectively (observe this inversion,
multiply by divisor, and divide by multiplier) in Aux. TaUe L
Thus, given 666 cents,
Bubtr. next least 498 in Aux. Table I.; take 3 as
— div. and 4 as mult.
168
quo. to 168 cts. 1 1019 in Principal Table.
4
This is the correct ratio for 666 cents, it
is larger than 1*4688 obtained from 32 : 47
in art. 7, because the cents were in excess, IT
but the difference is quite unimportant.
3 I 44076
14692, ratio 1*4692.
14. By Five Place Logarithms.
Given 665-5 ^^o^* as results from art. 11—
Genta hog%
600 •15051
65 •0163 I
•5 -00013
Aux. Table 11.
Principal Table
Aux. TaUc III.
Bum * 16695 » log i*4^7i which is now 'oooi smaller than in
art. 7, and is the correct value of 665*5 cents.
Digitized by
^*bogle
45®
ADDITIONS BY THE TBANSLATOR.
APP. XX.
12.
IS' ^y Seven Place Logarithms,
From Aux. Table IV. Cents 665507, art,
600 cents
•150 5150
60
•ois 0515
5
•001 2543
•5
•000 1254
•007 „
*ooo 0018
-which is the more correct valtie of 47-^32.
obtained by carrying the diyision one alep
further than in art. 7.
•166 9480 = log 1-46875,
Principal Table for the Calculation of Cents,
Cents Quotients i Logs
Gents
Quotients
Logs
Gents
Quotients
10638
Logs
02684
Gents
160
Quotients
10968
1 I^8»
I
10006
00025
54
IO317
''111
'°Z
04014
IF
2
12
50
55
23
108
44
709
3
17
75
56
29
405
109
SO
734
161
10974
04039
4
23
OOIOO
57
35
430
no
56
760
162
81
064
5
29
125
58
41
455
163
87
089
6
35
'5!
59
47
III
10662
02785
164
94
114
7
40
176
60
53
505
112
69
810
165
.11000
139
8
46
201
"3
75
835
166
06
164 1
9
H
226
61
10359
01530
114
81
860
167
13
i«9 I
10
58
251
62
65
555
"5
87
885
168
19
214
63
71
116
93
910
169
26
240 1
II
10064
00276
64
77
605
117
99
935
170
32
265
12
69
301
65
83
63'
118
10705
960
13
l^
326
66
89
656
119
11
985
171
11038
04290
14
81
351
67
95
681
120
18
03010
172
45
315
15
87
376
68
10401
706
173
51
340
16
93
401
69
07
731
121
10724
03035
174
V
365
17
99
427
70
13
756
122
30
061
175
64
390
18
10105
452
123
37
086
176
70
415
19
10
477
71
10419
01781
124
43
III
177
V"
440
IT
20
16
502
72
25
806
125
49
136
178
!3
46s
73
3»
831
126
S5
161
179
89
490
21
10122
00527
74
37
II?
127
61
186
180
96
516
22
28
552
75
43
128
67
211
23
34
577
76
49
907
129
74
236
181
11102
04541
24
39
602
77
55
932
130
80
261
182
09
566
25
45
627
78
61
^
183
15
591
26
SI
652
79
67
131
10786
03286
184
22
616
27
57
677
80
73
02007
132
92
3"
185
28
641
28
63
702
133
99
337
186
34
666
29
69
728
81
10479
02032
'34
10805
362
'f^
41
691
30
75
753
82
85
057
>35
II
387
188
47
716
83
91
082
136
17
412
189
53*
741
31
10181
00778
84
97
107
137
23
437
190
60
766
32
87
803
85
10503
132
138
30
462
33
93
828
86
09
157
183
139
36
487
191
11166
04791
34
98
853
87
15
140
43
512
192
73
816 1
35
10204
878
88
21
208
193
79
842 1
36
10
903
89
28
233
141
10849
03537
194
86
867
H
H
16
928
90
34
258
142
55
562
195
92
892
38
22
953
143
61
587
196
99
917 ,
39
28 978
91
10540
02283
144
67
612
197
11205
942 1
40
33
01003
92
46
308
145
74
637
198
12
967 .
93
52
333
146
So
663
688
199
18
992 .
41
10240
01029
94
58
358
147
86
200
25
05017
42
46
054
95
64
383
148
93
713
43
S2
079
96
70
40S
149
99
738
20i
11331
05042
44
57
104
97
76
433
150
10906
763
202
37
067
45
63
128
98
82
458
203
44
092
46
69
154
99
88
484
151
10912
03788
204
50
118
47
75
179
100
95
509
152
18
813
205
57
'12
48
81
204
153
24
838
206
64
16S
49
87
229
lOI
10601
02534
154
30
^l
207
70
'93
50
93
254
102
07
S<9
155
37
888
208
V
218
103
13
584
156
43
913
209
83
a
SI
10299
01279
104
19
609
157
49
939
aio
90
52
10305
305
105
25
634
158
964
1
1
1
53
II
330
106
32
659
159
62
989
1
\
Digitized by V^OO^K^
SECT. D.
MUSICAL INTERVALS.
451
16. Hence given a note of any pitch and the interval in cents between that and
uiother note, it is easy to determine the pitch of this second note.
Ex. Bequired the reduced 47th harmonio to
A. 453*9, the concert organ pitch of Mr. H.
Willis, to which is tuned the great organ at
ihe Albert Hall.
log 4S3'9 = 2-65696
Bents 665*5, art. 14, give log=» -16695
log 666*7 = 2*8239 1
Auxiliary Table I.
Mnltiplien
Gents
Quotients
and
Divlsoni
204
1 1 250
x8+ 9
3i6
12000
X5-5- 6
386
12500
X4+ 5
498
13333
^3-5- 4
702
15000
X2+ 3
884
16667
x6-rIO
1018
18000
x5^ 9
1200
20000
X I -i- 2
Aax. Table
Aux. Table
n.
ni.
Cente
Logs
Cents
Log.
100
02509
•I
•00003
200
05017
•2
^
300
07526
•3
400
10034
•4
10
500
12543
•5
13
600
15051
•6
15
700
17560
7
18
800
20069
•8
20
900
22577
•9
23
1000
25086
1*0
25
I ICO
27594
1200
30103
Hence 666*7 ^^ ^^^ pitch nomber of the
note required. Thus it is possible, for any
given pitch of the tuning note, to calculate the
pitch of the notes for any given temperament,
and hence, as will be shewn, to tune in that
temperament.
Auxiliary Table IV.
Cents
Logs
Cents
Log-
100
025 0858
•I
000 0251
200
050 I7I7
•2
0502
300
075 2575
•3
0753
400
100 3433
•4
1003
500
125 4292
•5
1254
600
150 5150
6
1505
700
175 6008
7
• 1756
800
2006867
•8
2007
900
225 7725
•9
2258
1000
250 8583
IIOO
275 9442
1200
301 0300
10
002 5086
•01
000 0025
20
05 0172
•02
050
30
07 5258
•03
075
40
10 0343
•04
100
50
12 5429
•05
125
60
15 0515
06
»5i
70
17 5601
•07
176
80
20 0687
•08
201
90
22 5773
09
226
I
000 2509
•001
000 0003
2
5017
*002
1
3
7526
•003
4
001 0034
•004
000 0010
5
2543
•005
13
6
5052
•006
15
7
7560
•007
18
8
002 0069
008
20
9
2577
•009
23
SECTION D.
MUSICAL INTEBVALS, NOT EXCEXDINO AN OCTAVE, ARBAKOED IN ORDER OF WIDTH.
(See notes pp. 13, 187, 213, 264, and 333.)
IT
I.
t.
Width of an interval, p. 451.
Cyclic and actual Fifths and major Thirds,
p. 452.
{. Cyclic and Exact cents, p. 452.
|. Interval ratio, p. 452.
^. £x>garithms, p. 452.
J. Theoretical and practical intervals, p. 452.
^ Calculation of just intervals by Fifths and
major Thirds up and down, p. 452.
$. Calculation of interval ratios in the same
way, p. 452.
Art.
9. Harmonics, p. 452.
10. Intervals in an Octave, p. 453.
Table I. Intervals not exceeding an Octave,
p. 453. ,
Table U. The unevenly numbered harmo-
nics of C7 66 up to the 63rd, p. 457.
Table III. Number of any interval, not ex-
ceeding the Tritone, contained in an
Octave, p. 457.
I . An interval was defined supr^, p. i ^d, note J. The tindth of an interval is
(neasiired by the number of cents it contains. Beside the usual diatonic intervals,
a, large number of others occasionally occur, which it is convenient to have arranged
Digitized by^eOgle
452
ADDITIONS BY THE TRANSLATOR.
APP. XX.
according to their widths as measured in cents,
the following table.
Many of these are famished in
2. The oents used are cyclic cents, as de-
fined sapri, sect. A. art. 24, xiii. p. 4376', that is,
those intervals found by taking a certain num-
ber of Fifths and major Thirds up and down
and reducing the result to the same octave,
are assumed to have cyclic Fifths and major
Thirds of 702 and 386 cents respectively. But
as the actual Fifths and major Thirds have
70''955 ^^^ 3^6*314 cents respectively, a slight
error in excess is made in every Fifth up and
every major Third down, and in defect in every
Fifth down and major Third up, which, when
a great many are supposed to be taken for
theoretical purposes, may reach to a sensible
5[ amount. These errors are of no consequence
for ordinary purposes, but a means of correct-
ing them is given in sect. E., p. 463d, and here
it has been tiiought better to add the result to
three places of decimals in many cases, and
this is put in the last column, preceded by the
letters * ex.,* meaning * more exact cents.'
3. Other intervals are given to the nearest
whole number of cents, determined as in sect.
C, which therefore belong to the cycle of 1200,
and hence are rightly called cyclic. Here also
is added the result to three places of decimals,
when it appeared advisable for theoretical
purposes. For ordinary purposes cyclic oents
always suffice.
4. The interval ratios^ being of historical
interest, are always given, although they are
of no assistance to the eye in recognising the
width of an interval. In these ratios the
smaller number is always placed first. In the
case of tempered intervals an approximate
ratio, with f prefixed, is given in the second
column, and the * ex.* or more exact ' ratio' it
given in the last column.
5. The (five place) logarithm of each inter-
val ratio, considered as a fraction of which the
larger number is the numerator, is added in
each case, to enable those who understand
logarithms to deal with them inmiediately in
calculating pitch numbers, Ac. The loga-
rithms always give the exact intervals. The
decimal point is omitted.
6. In the last colunm is given a variety of
information. The name of the interval when
any usual name exists, or the instrument on
which it is found. The Greek and Arabic in-
tervals were theoretical, and given in terms of
lengths of string. As we see from sect. B.
No. I, p. 44 id, there is every reason to suppose
that the real intervals tuned differed from
them materially. Some further information is
occasionally added.
7. If the interval is found in the Duode-
narium (sect. E., p. 463), then a mode of obtain-
ing it by Fifths up and down with major
Thirds up and down is annexed. Here
Vu s one Fifth up <= 702 cents, 2Vu^2x 702 cents, &o.
Yd — one Fifth down « 498 cents up, 2 7(i = 2 x 498 cents, <feo.
Tu = one major Third up = 386 cents, 2rii = 2 x 386 cents, <fto.
Tt^Bone major Third down -=814 cents, 2Td » 2 x 814 oents, Ac
When such additions are made, 1200 or mul-
tiples of 1200 must be subtracted till the result
is less than 1200. That result will be the
number of cyclic cents in the first column. Of
course, if we take the value to three places of
decimals.
Ftt = 7oi-95S. Fd-498-045» T«*-386-3i4, r<f=8i3-686.
and then the result will be correct to at least
two places of decimals.
8. If we put Ftt«?, 2Ftt=3* ^0, pa, a,
2 2« 3
2Yd = ?, Ac. Tu = 5, 2Tu = i* Ac. Td^^,
3* 4 4» 5
2Tda^, Ac. and multiply instead of adding,
and finally multiply or divide by 2 until the
c result lies between i and 2, these formulsB give
the exact ratio. Thus 2 7(2 •i'2Tu» cyclically
2 X 498 + 2 X 386 = 996 + 772 = 1768 - 1200 = 568
cents as in the table. Or to three places of
decimals, 2x498-045 + 2x386-3 14 =996090 +
772-628 = 1768718 -1200= 568-718, which is
correct. Or again, ^x 5!.iiL?|«?| x 2 = ?-5
3« 4« 9 X 16 36 18
the correct result. See Table I. under 568.
9. In Table 11. are given all the unevenly
numbered harmonics up to the 63rd in order of
occurrence. The first column gives the number
of the harmonic, in which those marked * will
be found on the Harmonical as harmonics of
both C 66 and C 132, and those marked f
as harmonics of C 132 only. In the second
column are the pitch numbers of all the har-
monics of C 66. In the third column Mog'
are the logarithms of the harmonics of I, pre-
ceded by a T^lvA sign + , so that if to each of
these be added the logarithm of the pitch
number of the fundamental note, the result is
the logarithm of the pitch number of the har-
monic. Thus Iqg 66=1-81954, which added
to 1*36173, the log opposite 23rd harmonic,
gives 3-i8i27»log 15 18, the pitch number (in
the table) of the 23rd harmonic of C 66. The
column is divided into octaves by cross lines,
at the beginning of which, preceded by a minus
sign — , is the number to be subtracted from
the log given in order to find the log of the
harmonic reduced to one octave as given in
Table I. Thus for 23rd harmonic 1-36173—
1*20412 = '1 576 1, which is the log opposite
628 oents in Table I. In the fourth column is
given the cyclic cents in the ratio of the funda-
mental note to the harmonic reduced to the
same octave, the same as given in Table \^
where will be found the more exact number
of cents. But to each octave in prefixed the
number of cents, followed by a j^lm sign + »
which have to be added in order to find the
unreduced interval. Thus for 23rd harmonic it
is 4800 + 628 = 5428 cents. Finally in the last
column there is given the ntareni equally tem>
pered tone, supposing the fundamental note is
C, and the number of cents to be added to or
subtracted from that note in order to prodnoe
Digitized by V^jOOQlC
EOT. D.
MUSICAL INTERVALS.
453
le harmonic. Thus the 23rd harmonic is
iiarper than \\f"% by 28 cents. These com-
arisonB are readily made from the column of
yolio cents, and can be easily applied to any
uidamental note. Thas the 23rd harmonic
f B,b must be 4 Octayes and 6 Semitones
nd 28 cents sharper than JB^b, and hence
lust be e^" + 28. The marking of the differ-
cLces of the harmonics from equally tempered
otes is convenient for repeating the experi-
lents in pect. N. No. 2.
10. Table III. is constructed to shew how
f ten each principal interval, not exceeding a
^ritone, is contained in the Octave. The first
column gives the cyclic cents in the interval
for easy reference to Table I. The second
column contains the names of the intervals.
The third contains, up to one place of deci-
mals, the number of times that the interval is
contained in the Octave, found by dividing
log 2 by the logarithm of the interval as given
in Table I. This is therefore not always the
same as the number arrived at by dividing
1200 by the number of q/cUc cents, but only
by the number of precise cents, as given in
Table I. Thus, takmg the skhisma of 2 cyclic
and 1*953 ex. cents, 1200+ 2 » 600, too small,
but i20o-ri'953 = 614-4 as in Table III.
I. Table of Intervals not exceeding one Octave,
Interral Batios
tAppTozimatiye
Logs
Name, &o.
I : I
1730 : 1731
32768 : 32805
25s : 256
95:96
2025 : 2048
80:81
524288 : 531441
63:64
3072 : 3125
48 -.49
125 : 128
39:40
38:39
37:38
36:37
35 : 36
\ 239 ; 246
32:33
31 :32
30 : 31
24:25
\ 67 : 70
20 : 21
19 : 20
243 : 256
128 : 135
o
00025
00049
00170
00455
00490
00540
00589
00684
00743
00896
01030
01 100
01 128
01 1 58
01 190
01223
01254
01336
01379
01424
01773
01908
021 19
02228
02263
02312
Fundamental note of the open string, assumed as C 66
Cent, hundredth of an equal Semitone, nearest approxi-
mate ratio, ex. i : 1*0005755
Skhisma, SVu+Tu=^C : B,5 , ex. 1*953
Ex. 6775, the ratio =i| • i|, and the result is the 17th
10 10
harmonic of 2>'b, a diatonic Semitone above C
Ex. 18*128, the ratio is - • — , or the interval by which
the 19th harmonic is flatter than the minor Third
Diaskhisma, 4Vd + 2Td « C : Dbb = C,J : D'b, ex,
19-553
Comma of Didymus, which is always meant by Comma
when no qualification is added, ^Vu+Td^C : C\ ex.
21-506
Pythagorean Comma, 12 7tt » C : Bff « D b : C8 , ex.
23*460
Septimal Comma, or interval by which the 7th harmonic,
969 cents, is flatter than the minor Seventh, 996 cents,
''Bt> : Bb t ex. 27264
Small Diesis, Vd^ sTu^C-b : Bjl , ex. 29*614. In equal
temperament this last interval would be represented (as
||JB ; c) by a Semitone of 100 cents
Interval of Al Farabi's improved Kabab
Great Diesis, the defect of 3 major Thirds from an Octave,
the interval between CZ and Db in the meantone tem-
perament, 3rd = Ca5 : -D'b , ex. 41*059
First interval on the Tambur of Bagdad, the interval by
which the 13th harmonic of 840*528 cents is flatter than
the just major Sixth of 2^84*359, ex. 43-831
Second interval on the Tambur of Bagdad
Third
Fourth
Fifth „ „ „ „
Quartertone of Meshaqah, the quarter of an equal Tone,
ex. ratio -» I : 1*0293022, ||0 : Cq
33rd harmonic, interval by which the nth harmonic ex-
ceeds the just Fourth, F : *'^, ex. 53*273
Greek Enharmonic Quartertone, supr4, p. 265a
Another Greek Enharmonic Quartertone, supri, p. 265a
Small Semitone, Vd + iTu^C : C^ff , ex. 70*673
Meantone Small Semitone, meantone C : CS 1 and hence
the 9 of that system, ex. 76*050, ex. ratio i : 1*0449
Snbminor Second, Greek intervcd, sapr^, p. 264a ; A^ : 'Bb,
on the harmonical
Interval from open string to second fret on the Tambur of
Bagdad
Pythagorean Limma, the * defect ' of two major Tones, 408
cents, from a Fourth, 498 cents, sVd • C : JDb, ex.
90*225
Larger Limma, the defect of a Fourth, 498 cents, increased
by a diatonic Semitone, 112 cents (total 610 cents), from
a Fifth, 702 cents, and hence the interval by which the
IT
Digitized by
^.joogle
454
ADDITIONS BY THE TRANSLATOR.
Tablk of iKTEuviOiB MOT BZOBEDiNa ONB OcTAVB — conUntied,
APP. XX,
C>o.io InterviU Hallos
Cento tApprozinuktiye
IT
94
99
100
105
112
114
117
128
134
13s
139
145
146
150
155
165
168
180
182
193
200
204
224
231
240
246
250
251
267
274
281
i 294
18 : 19
17 : 18
84:89
16 : 17
15 : 16
2048 : 2187
100 : 107
13: 14
25:27
37:40
12 : 13
149 : 162
4235 : 4608
221 : 241
II : 12
32:35
10 : II
49:54
59049 : 65536
9 : 10
t 161 : 180
400:449
8:9
225 : 256
7:8
74:85
125 : 144
200 : 231
32 : 37
108 : 125
6:7
64:7s
17 : 20
27 : 32
02348
02480
02509
02633
02803
02852
02938
03219
03342
03386
03476
03633
03666
03763
03779
03892
04139
04219
04527
04576
04846
05017
0511S
05606
05799
06021
06145
06271
06305
06349
06695
06888
07158
07379
Name, Ao.
Fourth must be sharpened to be a diatonic Semitone
below (i.e. the * leading note ' to) the Fifth, and hence
the interval by which the Fourth is sharpened on moda-
lating into the dominant, 3VU+ Tu^ C : C,8 » ^ . J^,| ,
ex. 92*179. This was used as the meaning of 8 in the
first English edition, for which in the present 114 cents,
or the Apotome, has been substituted, to agree with
Prof. Helmboltz's notation for just intonation
Greek interval in Al Farabi, interval between the i8th and
19th harmonics, d'" : ef"\> on Harmonical
Arabic interval
Ex. 100-099, ^e nearest approximate in small numbers to
the ratio of the interval of an equal Semitone, ex. ratio
I : 1-059461
17th harmonic »C : "d'"b on the Harmonical, ex. 104*955
Diatonic or just Semitone, ex. 111731 cents, Vd-i^Td=^
B : c^E : JF*
Pythagorean ApotomS, 'off-cut,' or what is left of the
major Tone, 204 cents, after * cutting off ' the Limma or
90 cents, used for t in this edition, 7F«»C : Cf «=C,
: C,«,ex. 113685
Meantone great Semitone, meantone C I Z>b« ex. 117-108
Interval between 13th and 14th harmonics, ex. 128*298
Great Limma, a Ck)mma greater than the diatonic Semi-
tone, 112 cents, ex. 133*237, 3Ft*+ 2Td«C : D'b. E^ : F
in the Phrygian tetrachord, supri, p. 263<f\ No. 6
Interval from open string to the thhni fxet of the Tambur
of Bagdad
Interval between the 12th and 13th harmonics, ex. 138*573
Persian * near the forefinger ' lute interval
Arabic * near the forefinger ' lute interval
Meshaqah's 3 Quartertones, imitation of 151 cents, ex.
ratio I : V2»= I : 1*0905
The interval between the nth and 12th harmonics on
the trumpet scale, used in Ptolemy's equal diatonic
mode, snpr&, p. 2646, used by Zalzal in Arabic Inte scale
as ' middle finger,* see infri, sect. K., Persia, Arabia, and
Syria
The 35th harmonic, septimal or submajor Second, snpri,
p. 2I2C, ex. 155*140
A trumpet interval, used in Ptolemy's equal diatonic scale,
supra, p. 2646
Zahsal's * near the forefinger ' on Arabic lute
Abdulqadir's substitute for Zalzal's 168 cents
The minor Tone of just intonation, the * grave Seoond * of
the major scale, 2F(2+ Tu« C : /)i, ex. 182-404
The mean Tone, the Tone of the meantone system, C I D,
the mean between a major Tone of 204 cents, and a minor
Tone of 182 cents, ex. ratio i : ^^/s^^i : 1*1180340
An equal Tone, ex. ratio i : '9'2s^ i : 1*22462
The 9th harmonic, major Tone, 2FttaC : D, ex. 203*910
Diminished minor Third, 2Vd + 2Td =^B^ : D* b, ex. 223-463
Supersecond, or septimal Second, 'Bb : c, on the Har-
monioal, ex. 231*174
The Pentatone, or fifth part of an Octave, ex. ratio
I : {/2«i I : 1*1487 in the Salendro scale, see infri, sect.
E., Java
Acute diminished minor Third, 2F« + 3T(2-B, : i>*b, ex.
244*968
Five Quartertones, on Mesh&qah's scale, ex. ratio i : '^2*
«i : 115535
The 37th harmonic, ex. 251-344
Grave augmented Tone, 3 V3 + sTu « C : Djl , ex. 253*076
Septimal or subminor Third, O : ^Bb on the Harmonical,
Poole's minim Third, ex. 266*871
The 75th harmonic, augmented Tone, Vu t iTw^ C : Djg ,
ex. 274*583
Interval on Tambfir of Bagdad
Pythagorean minor Third, ancient 'middle finger' 00
Arabic lute, 3Va»C : JSb, ex. 294135
Digitized by
\^oogle
MUSICAL INTEEVALS.
Tabli 01 IimtBVAiiii NOT ixcBBDiNO OMS OoTATX — eontinutd.
455
Intenral Batlos
tApproximatire
Logs
Name, &o.
512
520
550 It
551
568
583
16 : 19
37:44
68:81
5:6
14: 17
32 : 39
59:73
"S : 153
22 : 27
6561 :'8i92
4:5
50:63
64:81
25 : 32
33 : 41
7:9
27:35
10: 13
96: 125
16 : 21
243 : 320
25:33
3:4
227 : 303
80: 107
32:43
20 : 27
500 : 687
8: II
18 : 25
5:7
32:45
99: 140
45:64
512 : 729
16 : 23
25:36
90: 131
24:35
32:47
49:72
177147 : 262144
27:40
109 : 163
t 289 : 433
2:3
07463
07526
07598
07918
08432
08591
08648
08780
08894
09642
09691
10034
1023 1
1072 1
10763
10914
1 1288
"394
11464
11810
1 1954
I204i
12494
12543
12629
12832
13033
13797
13833
14267
I46I3
14806
15052
15297
15346
1 5761
15836
16305
16486
16695
I67I3
17070
17474
17560
17609
The 19th hannonio, ex. 297*513
The equal minor Third, pL : C
Persian * middle finger ' on late
JuBt minor Third, Vu+Td^Ail O'^O : E^b^ex, 315*641
Wide or superminor Third in the chord of diminished
Seventh, 'b"b I "d'"b on Harmonical, ex. 336-129
The 39th harmonic, ex. 342-483
Arabic late open string to string of the mean of the
lengths for 204 and 498 cents, practical sabstitute for
355 cents
Meshaqah's 7 Qaartertones, tempered form of 355, ex.
ratio I : ^^2'= i : 1*2241
Zalzal's * middle finger,* or wostd, mean length of strings
for 303 and 408 cents
Abdalqadir's substitute for 355 cents
The 5th harmonic, just major Third, Tu^C'.E^, ex.
386-314
Equal major Third, ex. ratio i : 1^2«= i : 1-25992 10
Pythagorean major Third, or Ditone, as it consists of two
major Tonesa2 x 204, ex. 407*820
Diminished Fourth, iTd^E^ : A^bt ex. 427*342
The 41st harmonic, ex. 429*062
Septimal or supermajor Third, Poole's maxim Third,
^Bb : don Harmonical, ex. 435*084
Meahaqah's 9 Qaartertones, ex. ratio 'Vs'^ 1*297
One of Prof. Preyer's trial intervals
Superfluous Third, Vd + ^Tu^C : E^ ^^A^b : C,8 , ex.
456-986
The 2 1st harmonic, the septimal or Bubfourth, F i^Bb on
Harmonical, ex. 470*781
Grave Fourth, $Vd + Tu=0 : F^, ex, 476*539
Two Pentatones, the representative of the Fourth in the
Salendro scale, see infr4, sect. K., Java, ex. ratio i : Ji^^
= 1:1*3195
Just and Pythagorean Fourth, Vd^'^C : Fj ex. 498*045
Equal Fourth, ||C : J'', ex. ratio i : »|/2»= i : 1*3348
Meantone Fourth, meantone C : F, ex. 503-422, ex. ratio
I : Jx (^^;-A=« : 1-3375
The 43rd harmonic, ex. 511*5x8
Acute Fourth, 3Vu+Td = C : F^ = A^ : D, ex. 519*552
Meshaqah*8 ii Quartertones, ex. ratio I : 'V2'*ai : 1*374
The nth harmonic, ex. 551*318
Superfluous Fourth, 2Vd+2Tu^C : Fjl , ex. 568*718
Septimal or subminor Fifth, E : ^Bb on Harmonical, ex.
582512
The 45th harmonic, Tritone, false, sharp, augmented, or
pluperfect Fourth, 2 Ft* + Tt* = F ; B, = C ; F,5 , ex.
590*224, the Fourth C : J^ as widened for passing into
the key of the dominant, 498 + 92 » 590
Equal Tritone, ^F : B, ex. ratio i : V2^ i : 1*4142
Diminished Pifth,2Fd+ !r(i = C : G*b =-F>5 : c, ex. 609-777
Pythagorean Tritone, 6Vu=C iFt^FiB, ex. 611-731
The 23rd harmonic, ex. 628*274
Acute diminished Fifth, 2Vu + 2Td»C : O^b »ii| : e^b,
ex. 631*283
Meshaqah's 13 Quartertones, ex. ratio i : 'V2'*=i : 1*4556
Septimal, or Subfifth, ^*b : ^-Bb on Harmonical, ex.
653-184
The 47th harmonic, ex. 665*507
Arabic lute, 2nd string, a Fourth above 168 cents, ex.
666*258, and hence *75i cents sharper than the last, the
confusion with the former is due to approximations
Abdulqadir's substitute for 666 cents, being a Fourth above
his 180 cents
Grave Fifth, ^Vd+Tu^C : G„ ex. 680-449
Meantone Fifth, meantone C : G, a quarter of a Comma
too flat, ex. 696*579, ex. ratio i : ] x 4^{}- 1 : 1*4954
Equal Fifth, |iC : O
Just and Pythagorean Fifth, Vu=C:0, ex. 701*955
IT
ir
Digitized by VjOOQIC
4S6
ADDITIONS BY THE TRANSLATOR.
Tabls or iNTERVjOiB NOT BzciEDiNa ONE OcTkYE— conHfiiied.
APP. XX.
Cyclic
OentB
Interval Batios
tApproximatiTe
720
724
738
772
792
794
800
t 95 : 144
160 : 243
32:49
59:91
16 : 25
81 : 128
256 : 40s
63 : 100
807
814
841
850
874
882
900
906
919
926
933
938
947
950
954
960
969
976
996
999
1000
1018
1030
1050
1059
1067
1088
1 100
1 1 10
nil
1117
1 129
1145
1150
1158
"73
1 180
1200
i!
30:
II :
32:
19683:
3:
22 :
16:
10 :
75:
7:
32:
125:
93:
72:
85:
: 13
49
18
32768
5
37
27
17
128
12
55
; 216
161
148
4:7
128 : 225
9 : 16
32 : 57
55:98
5:9
16 : 29
6: II
32: 59
27:50
8:15
89: 168
128 : 243
10 : 19
32 : 61
25:48
16; 31
35:68
64: 125
32:63
1024 : 2025
I : 2
Log!
18062
18149
;K
19382
19873
19922
20069
30242
20412
21085
21323
21388
21913
22136
22185
22577
22724
23045
23215
23408
23521
23754
23831
23958
24082
24304
24497
24988
25072
25086
25527
25828
26340
26570
26761
27300
27594
27840
27875
28018
28330
28724
28848
29073
29419
29613
30103
Kame^ Ac.
Three Pentatones, giving an aoate Fifth, as in SalendiOi
Bee infri, sect. K., Java, ex. ratio i : i^8 = i : 1*5157
Acute Fifth, sVu+Td = C :G\ ex. 723-461
The 49th harxnonio, ex. 737*652
Meshaqah's 15 Quartertones, ex. ratio i ! 'V2**» I : 1*5424
Grave superfluoos Fifth, 2Tu = C : (7,8 , the 25th harmonic,
ex. 772*627
Pythagorean minor Sixth, ^Vd^C : ^b, ex. 792-180
Extreme sharp Filth, 4Vu + Tu=C : O^t , ex. 794*134
Equal superfluous Fifth, ||C : (?S , and also equal minor
Sixth, |)C : ^b, the same notes differently written, ex.
ratio I : ■^4^1 : 1*5874
The 5 1 St harmonic, ex. 807*304
Just minor Sixth, Td»C : A^b, ex. 813*687
The 13th harmonic, ex. 840*528
Meshaqah*s 17 Quartertones, his tempered substitute for
853 cents, ex. ratio i : ■^2"=i : 1*6339
Arabic lute, the Fourth above Zalzal's 355 cents
The 53rd harmonic, ex. 873*504
Abdulqadir*B substitute for Zalzal*8 853 cents, being a
Fourth above 384 cents
Just major Sixth, Vd + Tu^C : -4,, ex. 884-359
Equal major Sixth, ||C : ^, and also equal diminished
Seventh, ||C : Bbb'^A : Gb, ex. ratio i : ^8=1 : 1-6818
The 27th harmonic, Pythagorean major Sixth, ^Vu^C :A,
ex. 905865
Batio of the loth : 17th harmonic, the harmonic dimin-
ished Seventh, e/' : "d"' on the Harmonical, ex. 918*641
Just diminished Seventh, Vd + 2Td^C : B'bb -^ : G'b.
ex. 925*416
Septimal or supermajor Sixth, ^Bb Iff, ex. 933*129
The 55ih harmonic, ex. 937*632
Acute diminished Seventh, ^'^u+S^^'^O : B*bb ^A:
0*bj ex. 946*924
Meshaqah's 19 Quartertones, ex. ratio i : ^^2"si : 1*7311
Just superfluous Sixth, sVd + ^Tu^C : Ajt , ex. 955*031
Four Pentatonea, the fourth note in the Salencbo soUe,
see infri, sect. E., Java, a close approximation to 969
cents, ex. ratio i : •^16 ai : 1*7411
The 7th harmonic, natural, harmonic, or subminor Seventh,
C : 'Bb on the Harmonical, ex. 968*826
Extreme sharp Sixth, 2F« + 2Ttt»C : ^^ ,ex. 976-537
Minor Seventh, used in the subdominant, 2Vd^C : B^,
ex. 996*091
The SJih harmonic, ex. 999*468
Equal superfluous, or extreme sharp Sixth, i|C : ilt , or
minor Seventh, ||0 : Bb
Acute minor Seventh, used in descending minor scales,
2Vtt+Td-C:B'b,ex. 1017*597
The 29th harmonic, ex. 1029*577
Meshaqah*s 21 Quartertones, ex. ratio i : '^2'* = i : 1*8340
The 59th harmonic, ex. 1059-172
Grave major Seventh, ^Vd + 2Tu^C : B„ ex. 1066-762
Just major Seventh, Ftt+ T^»C : B|, the 15th harmonio,
ex. 1088*269
Equal major Seventh
Pythagorean major Seventh, $Vu^C : B^ol. 1109*775
One of Prof. Preyer's trial intervals
The 6iBt harmonic, ex. 1116*884
Diminished Octave, Vu + 2Td » C I C*b * ex. 1 129-327
The 31st harmonio, ex. 1145*036
Meshaqah's 23 Quartertones, ex. ratio i : 'V2*'» i : 1*943
Superfluous Seventh, sTu^C : Bjt , ex. 11 58-941
The 63rd harmonio, ex. 1172*736
The double Tritone, 4F« + 2Tii» C : jB^ , ex. 1180*447
The Octave, C : c
Digitized by
Google
.8X:CT. E.
MUSICAL DUODENES.
457
Table II. The Unevenly Numbered Harmonics of C 66 up to the 6^rd,
No.
•9
*i5
•19
21
23
*25
27
•29
31
Pitch
66
198
330
462
594
726
858
990
1 122
1254
1386
1518
i6|o
1782
1914
2046
Log
-•30103
+ •47712
-•60206
+ •69897
+ •84510
-•90306
+ 0-95424
+ 1-04139
+ 111394
+ 1-17609
-1-20412
+ 1-23045
+ 1-27875
+ 1-32222
+ 1-36173
+ 1-39794
+ 1-43136
+ 1*46240
+ 1-49136
^^ Bqaal notes
1200 +
702
2400 +
386
969
3600 +
204
841
1088
4800 +
298
471
628
772
906
1030
1 145
+ 2
tl -14
a'l -31
r
a"
h"
+ 4
+ 51
-59
- 12
1
e"i
+ 5
d"'t
- 2
r
-29
/'"«+28 1
f'l
-27
a"*
+ 6
o'"«+30 1
t/"
+ 45 1
No.
Pitch
nos.
33
2178
35
2310
37
2442
39
2574
41
2706
43
2838
45
2970
47
3102
49
3234
51
3366
53
3498
55
3630
57
3762
59
3894
61
4026
63
4158
Log
Cyolio
Equal notes
-1-50515
+ 1-5x851
+ 1-54407
+ 1-56820
+ 1-59106
+ 1-61278
+ 1-63347
+ 1-65321
+ 1-67210
+ 1-69020
+ 1-70757
+ 1-72428
+ 1-74036
+ 1*75587
+ 1-77085
+ 178533
+ 1*79934
6000 +
53
155
251
342
429
512
590
666
738
807
874
938
999
1059
1117
"73
c~«
-47
d"'
-45
<rj
-49
dr-t
+ 42
«'"
+ 29
r
+ 12
rt
-10
f
-34
^38
9"l + 7
a''' -26
a'^ +38
a"'« - I
6^ -41
b'" +17
c^ -27
Table HE. Number of amy Interval, not exceeding the Tritons, contained in an
Octave.
Cyclic
Number
Cyclic
Number
cents in
Name of Interval
in an
cents in
Name of Interval
in an
interral
Octare
interval
Octave
2
Skhisma
614-3
200
Equal Tone .
6-0
20
Diaskhisma .
61-4
204
Major Tone .
5*9
32
Comma ....
55-8
231
Snpersecond .
5-2
24
Pyth. Comma
51-1
240
Pentatone . •
S-o
27
Septimal Comma .
440
267
Subminor Third .
4*5
28
Small Diesis •
40-5
294
Pyth. minor Third .
4*>
42
Great Digsis .
29-2
300
Equal minor Third
4-0
50
Qnartertone .
24-0
316
Just minor Third .
3-8
70
Small Semitone
17-0
336
Superminor Third .
3-6
76
Meantone small Semitone
15-8
355
Zalzal's wostd
3*4
85
Sabminor Second .
14-2
400
Equal major Third
4-0
90
Limma ....
13-3
408
Pyth. major Third .
2-9
92
13-0
t^
Supermajor Third .
2-8
100
Eqoal Semitone
I2-0
Just Fourth .
2-4
112
Just Diatonic Semitone .
IO-7
500
Equal Fourth .
2-4
114
Apotome
10-6
503
Meantone Fourth .
2-4
117
Meantone great Semitone
IO-3
590
JustTritone .
2-0
111
Great Limma
9*o
600
Equal Tritone
2-0
Minor Tone .
6-6
612
Pyth. Tritone
3-0
193
Mean Tone .
6-3
SECTION E.
on KUBICAL DU0DBNS8 OB THS DBVELOPMEXIT OW JUST INTONATION FOB HABUONT.
(See notes pp. 3o8, 309, 211, 269, 272, 293, 298, 299, 301, 302, 304. 305, 306, 310, 333, 338. 345,
346, 352, and 363.)
Art. •
1. Introduction, p. 458.
2. Harmonic Elements, p. 458.
3. Construction of the Schemes, p. 458.
4. Harmonic Cell or Unit of Concord, p. 458.
5. Harmonic Heptad or Unit of Chord Rela-
tionship, p. 458.
Art.
6. Harmonic Decad or Unit of Harmony,
7. ChoMs of the Decad, p. 459.
8. Interyals of the Decad, p. 459.
9. Harmonic Trichordals and Scales, p. 460.
10. Principal Trichordal Scales, p. 460.
Digitized by V^jOOQ IC
4S3 ADDITIONS BY THE TEANSLATOR. app. xx.
Art. ^ Art.
11. Hannonlc Daodene or Unit of Modulation, i8. The Duodenarium, p. 463.
p. 461. 19. Construction of the Duodenarinm, p. 463.
12. Modulation into the Dominant Duodene, 20. Just Intervals reduced to steps of Fifths
p. 461. and major Thirds, p. 464.
13. Modulation into the Suhdominant Duo- 21. The oolunm of Fifths, p. 464.
dene, p. 462. 22. The Limits of the Duodenarium, p. 464.
14. Modulation into the Mediant Duodene, 23. Introduction of the Seventh and Seven-
p. 462. teenth Harmonic, p. 464*
15. Modulation into the Minor Submediant 24. Need of Reduction of the number of Just
Duodene, p. 462. Tones, p. 464.
16. Modulation into Relative and Ck>rrelative, 25. Omission of the Skhismas. Unequally
p. 462. Just Intonation. The oyde of 53, p. 465.
17. Duodenation, p. 462. 26. Duodenals, p. 465.
1. Introduction. The following sketch is founded on my paper on the same
subject in the Proceedings of the Royal Society for Nov. 19, 1874, vol. xxiii. p. 3.
^ It is an attempt to develop the trichordal principles of suprit, pp. 293d and 3093.
This, of course, is an inversion of what actually occurred. But the introduction
of the harmonic principle has completely changed the nature of music, and its plaii
consequently requires reconstruction. Harmony alone is considered. Melody is
made dependent on harmony. The harmony is tertian, that is, it includes perfect
Thirds, major and minor ; but not septimialy that is, it does not include the 7th
harmonic of the base of a chord. But this may be superadded, see art 23. The
plan here pursued also has the advantage of showing the precise tertian relation of
the notes of a chord written in the usual notation, by merely superscribing a letter,
called the duodenal, without any change whatever in the ordinary notation itself.
The notes affected by the other harmonies can then be easily indicated (art. 26)
2. The Harmonic Elements are the intervals of a Fifth, major and minor Third,
with all their extensions, inversions, and extensions of their inversions, that is all
the forms in p. 191&, c, which are here assumed. Capital letters will therefore
indicate notes without regard to octave, and even allow of reduplication, or added
octaves. The notation is otherwise the same as for my variation of Herr A. v.
% Oettingen's notation, explained on p. 277c, note*, and used through the rest of
this work. The notation of intervals is used as in p. 276^, notet, so that + is
386, — is 316, ± is 702, I is 294 cents, and (...) is replaced by the proper
number of cents in the interval.
3. In the construction of the schemes notes forming ascending Fifths are
written over one another vertically ; notes forming ascending major Thirds are
written to the right horizontally. Against each note is written the number of
cycHc cents (supri, p. 452a) in its interval from G or the root, reduced to the same
octave. A notation in Solfeggio terms (modified from that used by the Tonic
Solfedsts with Italian pronunciation of the vowels) is also supplied, in which Do
stands for the root whatever the note itself may be.
4. Harmonic Cell or Unit of Concord.
Letter Notation, Solfeggio Notation.
E^\} 316 G 702 Mo 316 So 70a
% C o El 386 Do o Mi 386
This consists of the three harmonic elements, the Fifth G±G (or Do±So) and
major Thirds C+E^ , ^*[>+G (or Do-^ Mi and Mo'\-So) being placed as already
explained, so that the minor Thirds G—E^\} (or Do—Mo) and E^^G (or 1ft— So)
lie obliquely upwards to the left. Such a cell is called the Cell of C (or Do) its
root.
The student should construct such cells on The cell contains therefore all forms of the
any root. C has been adopted simply because major and minor chords, triad or tetrad, given
it is most usual, and because it is suited to the in Chap. XII. above.
Harmonical, on which its effect should be tried.
5. The Hoflrmonic Heptad or Unit of Chord BelationsMp.
Letter Notation. Solfeggio Notation.
E^h 316 G 702 Mo 316 So 702
A^\} 814 Co Ex 386 Lo 814 Do o Mi 386
i^ 498 A^ 884 Fa 498 La 884
Digitized by VjOOQIC
Solfeggio Notation.
To 1018 Re 204
o88
Mo 316 So 702 Ti 1088
386
Lo 814 Do 0 Jlfi 386
884
Fa 498 La 884
E€T. B. MUSICAL DUODENES. 45^
The heptad possesses seven notes, whence its name. It is formed by subjoining
tie cell of F (or Fa) to the cell of G (or Do), so that the Fifth of the lower cell is
lie root of the upper cell. This is called the Heptad of C (or Do) its central note.
t contains not only the 4 cell triads, major O+^i — G. -F+uli-C (or Do'\-
^li—So, Fa^La-Do), minor (7-E'[>+G^, F A^\}-^C (or Do-Mo'\'So, Fa-*
'^o-\-Do), but also two union triads, major i4*t>+0— JS7*b (or Lo+Do— Jfo), and
ainor A^^C-^Ei (or La— Do + Mi), which result from the union of the two cells.
t therefore possesses all the six consonant chords which contain the same note G
or Do), and can hence pass readily into one another, as should be verified on the
larmonical. It possesses also the seven condissonant triads (p. 338, note f) con-
aining G, the major Trine i4*[>+CH-jB?, (or Lo+Do-^Mi) containing two major
[thirds, and the ndnor Trine Ai — G^EJi (pxLa—Do-^Mo) containing two minor
["hirds, the jpure quintal Trine F±:G±G (or Fa±Do±So) containing two Fifths,
he major quintal Trines u4»p+0±G (or Lo-^Do±So) and F±G+E^ (or Fa±
Do-^Mi) consisting of a major Third and a Fifth, and the minor quintal Trines
li — 0±G (or La—Do±So) and i''±C'— J^*t> (or Fa±Do—Mo) consisting of a ^
oinor Third and a Fifth.
All of these should be stadied on the Har- consonant triad containing the same note G
lonioal, and the readiness with which their should be felt.
issonanoe may be removed by passing into a
6. The Harmonic Decad or Unit of Hamu>ny.
Letter Notation,
B»|> 1018 D 204
E^\} 316 a 702 5,
il»b 814 Go E^
F 498 Ay
The Decad possesses ten notes, whence its name. It consists of the heptad
»f G (or So) superimposed on the heptad of G (or Do). These two heptads have
\ common cell, that of G (or Do). Hence the decad of G (or Do) may also be ^
considered as three cells, the tonic or that of G (or Do) in the middle ; the domi-
lant or that of G [ox So) above, and the subdominant or that of i^ (or Fa) below.
Che decad of G (or Do) is the complete development of the cell of G (or Do), for
he root of the upper cell is the Fifth of the root of the middle cell, while that
*oot itself is the fmh of the root of the lower cell.
7. The Ghords of the Decad. The vertical axis is the column of Fifths F±G±.
T:tD (or Fa±Do^So±Be). Those are two horizontal axes of major Trines.
Phe decad contains 3 cell ?na;or triads F-^A^^G, G-^Ei — G, G-^B^^D (or
?a + La— Do, Do + 3ft— So, §0+ Ti—Be) on the right, and 2 union major triads
i>b+C7--B*b» ^'b+^-^*b (or Lo-\'Do-Mo, Mo-¥So-To) on the left. The
lecadalso contains 3 cell minor triads F-A^^^+G, G-E^\}'tG, G-5^b + Z) (or
ya-Lo -^^ Do, Do— Mo + So, So — To+Be) on the left, and 2 union minor triads
ii-C+-&,, Ei-G-^Bi (or La-Do+Mi, Mi-So^Ti) on the right. It has
Jso the dissonance of the dominant Seventh, G-^ B — D \ F (or So + Ti —
He I Fa) and mimyr Ninth G+B-D \ F—A^\} (or So-k-Ti-Be \Fa-Lo), andf
lence of the diminished Seventh (the same less G or So), and of the added Sixth
f^+u4-C 204 D, or 2^+4 520 D {Fa+La-Do 204 Be, or Fa+La 520 Be), but
lot the minor triad 2>,— JF*+u4, (or Ba—Fa-k-La, see Ba in art. 11, p. 461),
^hich is confused with it in tempered intonation. And it has also not the chords
>f the extreme diarp Sixth, D^\}'\'F 204 (t+5,, p. 2866, or/* +a 590 d,J, p. 308&.
8. The Intervals of the Decad. The relative position of the principal intervals
ihould be observed in addition to the vertical Fifths (including Fourths), the hori-
:ontal migor Thirds (including minor Sixths), and oblique minor Thirds (including
najor Sixths), on which the scheme is founded.
Major Tone 204, two Filths vertically np, Diatonic Semitone 112, obliquely down to
aC D (or Do Re). the left in the next line as B, C (or Ti Do).
Minor Tone 182, obliquely down to the right Small Semitone 70, obliquely down to the
D the next line but one as G il| (or So La). right in the next column but one, and in the
Defective Fifth 680, obliquely down to the next line, as i^'b B, (or To Ti^.
ight in the next line but two as D Ai (or This is the smallest interval occurring in a
3e La). Decad.
Digitized by V^OOQ IC
46a ADDITIONS BY THE TRANSLATOR. app. xx.
9. Harmonic Trichordals, A trichordal consists of one triad from each of the
three cells of a decad. Eight such trichordals may be formed from the three
major and three minor cell triads. Abbreviate the names ' major triad ' and ' minor
^triad ' into their first syllables, ma,, mu (wliich however may, if preferred, be read
more at fall as ' major ' and ' minor '), and name them in the order of Subdominant,
Tonic and Dominant triads, then the eight trichordals are ma^nia.ma., mi.ma.ma.,
via,mi.7na,, mi,mi,ma,, ma.ma.mi,,, mi,ma,mi,, ma,rm.mi,, mi,ini,im. The seven
tones in each trichordal reduced to a single octave constitute an harmonic scale, thai
is a scale in which each note belongs to a triad in the scale. We may begin the
scale with any one of the notes of any one of the 3 generating triads. These notes
may be numbered in order of sharpness when reduced to the same octave, as 4, 6, i
in the subdominant, 1,3, 5 in the tonic, and 5, 7, 2 in the dominant. These
numbers may be simply prefixed to the trichordal they affect, to shew on what
note the scale begins. We thus obtain 7 x 8=56 trichordal scales, of which 8 are
% generically different, each genus having 7 species. In some accounts of the modes
they are all represented in faxst as ma.ma.ma., differing only as to the note of the
scale with which they begin. This is, of course, thoroughly erroneous. The student
is recommended to make out on the Harmonical every one of these 56 scales.
10. Principal Trichordal Scales, The following gives a list of the principal
scales thus generated referring to the places where they have been noted in the
text, and the scale is noted as beginning with C,
The figures between two oonseoutive notes the Harmonical a form is given which can be
indicate the interval between them in cents, played. As each note forms part of a cell
The number prefixed to the root of the decad triad, and mostly also of a union triad, each
indicates the note of the chord with which the scale can be harmonised, and the stadent
scale begins, reckoned in the way just men- should therefore harmonise all of them on the
tioned. Beferenoes to the text fpllow. Where Harmonical. See also p. 277, note f. No ex-
any one of these scales cannot be played on amples of the unusual VI. Mi.ica.mi. are given.
I,
m I 0-C 204 D 182 J^i 112 J^ 204 O 182 A^ 204 B| 112 c, No. I of p. 2746 and note. Bfajor
Mode of p. 2986. Ordinary C major.
K F^C 182 D| 204 JE7| 112 F 204 G 182 A^ 112 £b 204 c, No. 5 of p. 275a, there called the
mode of the Fourth. This must be played on the Harmonical as 5 C ma.majaia., which has the
same intervals, namely: O 182 Ai 204 B^ 112 e 204 d 182 e^ 112 / 204 y. It is No. 5 ol
p. 275a, there called the mode of the Fourth.
II. MZ.1IA.MA.
I CsC 204 D 182 E^ 112 F 204 G 112 A^b 274 B 112 c. The minor-major mode of
pp. 3056 and 309^.
III. Ma.mi.ma.
I C = C 204 D 112 J^'b 182 ^ 204 G 182 ill 204 B, 112 c, No. 2 of p. 2746. The mode of th«
minor Seventh, with the leading note, or major Seventh, substituted for ihe minor Seventh, aa
in p. 303d. An ordinary form of the ascending minor scale p. 288a.
lY. M1.MI.11A.
I C^C 204 D ii2£7*b 182 JP 204(7 112 A^b 274 B, 112 c. The * instrumental ' minor seal*
of p. 2886 ; the * modem ' ascending minor scale of p. ^ood,
IF V. Ma.ma.mi.
I C-C 204 D 182 £>, 112 F 204 G 182 Ai 134 B^b 182 c. Although this is called the mode
of the Fourth on pp. 2986 and 309^2, it is a different scale from that called the mode of the
Fourth on p. 2756, which is 5 J^ ma.ma.ma. above, under I., because the Seventh in this case is
JB'b and in that Bbt a comma lower. See p. 277, footnote t, on this and similar confusions.
5 ^a C 182 Di 134 ^^ b 182 F204G iSzA^iii Bb 204 c. This must be played on the Har-
monical as 5 C-G 182 ^, 134 B^b 182 c 204 d 182 e, 112 /204 pr. This is No. 6 of p. 275^
and there considered as a variant of the mode of the minor Seventh, which is really the diffeieni
scale, 3 C majnioni., next immediately following.
VII. Ma.mi.mi.
I C«C 204 D 112 E^b 182 ^ 204 G 182 ill 134 -S^b 182 c. This is No. 4 of p. 2750 taken
upwards, instead of downwards as there. The mode of the minor Seventh of p. 2986 withoni
the leading tone of p. 303c.
VIU. M1.MI.MI.
I C'^C 204 D 112 E^b 182 F 204 G 112 il'b 204 B^b 182 c. No. 3 or descending minor
scale of p. 274c, the mode of the minor Third of p. 294a, No. 4.
5 F» C 112 jD'b 204 £*b 182 F 204 G 112 A^b 182 Bb 204 c. This must be played on tha
Digitized by V^jOOQlC
SECT. E..
MUSICAL DUODENES.
Harmonica! as 5 C^G 112 A^b 204 B*b 182 C 204 D 112 E^b 182 F 204 {7. It is No.
p. 2756, the mode of the minor Sixth of pp. 294a No. 5, 298c No. 5, 305c to 308^.
461
7 of
II. The Harmonic Dtwdene, or Unit of Modulation.
Letter Notation, Solfeggio Notation.
2Pb
134
F^ 520
^ 906
0.«
92
^^1478
0«b
632
1130
B^b 1018
E'b 316
i) 204
F,%
590
^^976
e«b
G 702
Bi
1088
I>JB^274
ip^b
428
A'b 814
C 0
E,
386
OJI772
.B*bb
926
D»b 112
F 498
^.
884
c,t . 70
^«bb
224
G'b 610
Bb 996
^1
182
F4568
rtt
So
U
\di
me
8U
to
mo
re
\fl
li
du
\ 80
\U
ri
fu
lo
1 do
I mi
«e
tu
ro
1 fa
la
de
mu
1-
\ ^
ra
fe
In referring to the figure of the decad in art. 6 two gaps will be noticed, one to
the right at the top, the other to the left at the bottom. On filling these up accord-
ing to the same laws by making F^jf^ a Fifth above ^i and a major Third above
D, and D'|> a Fifth below A^\} and a major Third below F, we obtain the scheme
in the central rectangle of the above figures, which is called a Dtiodene, because it
consists of ttvelve notes bearing to each other the relation of the twelve notes on
the piano, which, by omitting the marks of commas ^1, may be supposed to be
represented by the same letters. The Duodene then consists of 3 columns or
quaternions of Fifths, and four lines or major trines of Thirds, and its root, which
is tiiat of the corresponding decad, is in the centre of the lowest line but one, so
that it is easy to construct a duodene on any note as a root.
The duodene thus completed by these extreme tones possesses two additional
union Thirds, major D^\}'\'F—A^\} (or Bo-^-Fa^Lo) and minor 5,— D+i^iJf (or
Tt—Be-^Fi), and consequently besides the 8 genera of scales of the decad contains T
the major scale of il*|> (major chords D»t?+^-^*bM*b+<^--25;»b. E^h-^-G-B^l})
and the descending minor scale of Ei (minor chords ul| ^C-^-E, E^ — (r+^n -Bi —
D-\-Filf)j and it also gives us the chords of the extreme sharp Sixth (976 cents, a
yexj near approach to the 7th harmonic of 969 cents) in its three forms,
Italian Sixth A^\} 386 C 590 i^i^ scarcely a dissonance,
French Sixth A^\} 386 C 204 D 386 FA,
German Sixth A^\} 386 C 316 E^\} 274 F^%.
These three last chords cannot be played on the Harmonical. They arose in the days of
xneantone temi)eTament. The chord of the dominant Seventh omitting the Fifth E^b 386
O 610 ^'b had then to be played with tempered notes as Eb 386 O 579 cff , because there was no
db on the instrument, and as the 7th harmonic would have been E^b 386 Q 583 'd*b the effect
was so good that the chord was adopted in writing and distinguished from the chord of the
dominant Seventh, by resolving upwards instead of downwards.
These new notes have also introduced two new Semitones of 92 cents, i^ 92 F, JJ 1[
and D^\} gi D. But the smallest interval between any two notes remains the
small Semitone of 70 cents, A^\} 70 -4 1, E^\} 70 E,, B^[} 70 5,,
12. Modulation into the Dominant Duodene.
It is obvious from the construction of the
duodene that the transition from any duo-
dene to an adjacent one is very easy. Suppose
(see scheme in art. 11) that we omit the lowest
line D*b + F + ^, (or Ro-^Fa-^La) and take
in the line i^' + ^ •«- C,S at the top, we shall
have the duodene of G, which has three lines
in common with that of O. The three new
notes introduce 2 commas F 22 F\ Ai22A and
one diaskhisma 0,8 20 D'b. These minute
distinctions neglected in tempered music have,
however, a powerful effect on the harmony of
justly intoned instruments. The C|S is indeed
one of the extra notes and does not occur in
the decad of O or its scales. On the otiier
hand F,S , which was an extra note in the duo-
dene of Cy becomes a substantive note in the
decad, as well as duodene, of G, and we find
then that the F of the C decad becomes the
Fit , 92 cents higher in the G decad. This
difference is so large that it cannot be disre-
garded in tempered music, and it is, accord-
ingly, there represented by an interval of 100
cents, and forms the distinguishing mark for
major scales of what is termed the modulatum
into the dominant as just explained.
Digitized by V^jOOQlC
462
ADDITIONS BY THE TRANSLATOR.
APP. XX.
13. Modulation into the Subdominant Duodene.
Omit the top line B^b + D + F^t (or To
+ Re + Fi) of the daodene of O in scheme
art. II, and take a new line G'b + Bb + A (o^
Sa + Ta + Ra) at the bottom to obtain the
duodene of F (or Fa), The changes made are
the reverse of those for modulations into the
dominant. The notes B^ b, D are depressed by
a comma of 22 cents to JBb» i>i, and the extra
note F^% of the C daodene is raised by a
diaskhisma of 20 cents to O^b of the ^daodene.
These changes are neglected in tempered in-
tonation. Bat the most important change
is that B^ (which is in the daodene, bat not
the decad of F) is depressed by a Semitone of
92 cents from B, to Bb» and this being noticed
in tempered masic, becomes the distingoish-
ing mark in the modulation of the major aeale
of C into that of the subdominant F,
14. Modulation into the Mediant Duodene,
Betuming again to the daodene of C, we
might omit tibe left column D^b±A^b±E^b±
B* b (or ro ±lo ±mo ±to) of the duodene of C
[ (see scheme in art. ii) and introduce a new
column on the right CjS db Q^ ± D^ ±
A^ (or de±8e±ri±t%)y in which each new
note is a great diesis of 42 cents lower than
the note in the column omitted. This differ-
ence is ignored in playing tempered mnsie,
although the distinction is preseired in writ-
ing as Db and C% , <Jkc., bat it is of great im-
portance in just intonation. This is termed
modulation from the root C (or Ikf^ into the
Mediant^ E^ (or Mi) as the root of the new
daodene is the major Third or Mediant of the
root C (or Do) of the old daodene.
15, Modulation into the Minor Svhmediant Duodene,
Similarly we might omit the right column
A^±E^±By±F^% (or la mi ti fi) of the duo-
dene of C (see scheme in art. 11) and in-
troduce a new column B-'bb ± F"b ±C-b± G*b
^or ta fu du su) on the left, thus forming the
duodene of A^ b (or Lo) or minor subtnedianL
16. Modulation into the Belative and- Correlative Duodenes.
But it is usual to combine these two modu-
lations with others into the subdominant of
the mediant (that is the submediant) A^ (or
La), generally called the relative on the one
m hand, and the dominant of the minor sub-
mediant, that is E^b (or Mo)y sometimes called
the correlative^ on the other. In each case the
root of the new duodene differs by a minor
Third from the old root. The change for these
two last modulations is considerable if we take
the whole duodene into consideration, as may
be seen by the schemes in art. 11, where the
dotted lines mark off the new duodenes. But in
the cases which occur in practice the changes
are very small, especially when the difference
of a comma is neglected as in tempered music.
The modulation into the relative is generaUy
from the major scale of the decad of C or chords
F+Ai-CC + Ei-O.O + B^-DinioHieminor
scale of the relative decad of il„ either consist-
ing of the chords D, - F+ i4„ -4, - C + ^,,-E?, -
O + Bi (in which there is only the use of D, for
D, to indicate this modulation), or else con-
In sistingof the chords jD,-F+-4„ -4,-C + -B„
VE?,4Gjr-B„ or even D^ + FJt-At, A^-
C-^Et !&, — Gjf— B„ these being the three
recognised forms of the minor scale. As the
two latter forms are also acknowledged in
tempered intonation (which, however, confuses
F,5 and F.J ), the change of O into G^ (or so
into se, that is the sharpening of the Fifth by
properly 70 cents) has generally been con-
sidered the mark of modulation from a major
scale into its relative minor,— one of the com-
monest in music.
The modulation into the correlative is in
the same way generally thought of as the
change of the tonic major scale chords F+
^,-C., C-*-E^-0, G + Bi-D (or Fai^La
-Do, Do + Mi-So, So+Ti-Re) into the
tonic minor scale chords F—A^b+C, C-
E^b + G,G-B^b + D (or Fa-Lo^Do, Do-
Mo -¥ So, So — To + Re), in which however the
chord G + B^-D (or 80+Ti-Re) may re-
main, and sometimes the chord F+A^ — O
(or Fa La Do), This is not a modulation at
all in the sense just explained, because there
is no change of duodene or even decad. It is
merely a change of scale within the same
decad, that is, a new triohordal scale moving
from I C ma.ma.ma. to I Cmijni jni., mijnLma..
or ma.mi.ma., art. 9. It would, however, be
more consonant to ancient practice to restrict
the term modulation to such changes of tri-
chordal within the same duodene, and to use
a new term for the more general operation.
17. Dux)denation,
This is the term I propose to substitute
for modulation when it means passing from
one duodene to another which bears a known
relation to the first. This relation may be very
close, as in the cases just considered, or so
remote that the two duodenes have only one
note in common. Thus the duodene of D'b
and C have only the note B'b in common.
The annexed figure, called the Duod^ndi^um,
probably contains all such duodenations which
occur even in modem music, though it is im-
possible to be certain how far the ambiguities
of tempered intonation may mislead the com-
poser to consider as related, chords and scales
which are really very far apart. It contains
therefore an approximative estimate of 117,
for the number of tones in aa Octave which
would be required to play in jast intonation,
and are roughly represented by the 12 tones of
equal temperament.
Digitized by VjOOQIC
BCT. B.
MUSICAL DUODENES.
463
18. The Diiodendrium.
The large figures give the cychc cents in the interval of each note from C. See Table
440. The small figures give the corresponding nunberB of the cycle of 53 with the nearest
bole number of cents. See Table p. 439.
I
2
3
i
*
S
6
7^
8
9
-1-26
"•94
-63
—
•31
0
+ •31
+ •63
+ •94
+ 1-26
B*bb
970
D*b 156
JP« 542
A'
928
c«
114
JS7Jsoo:G,t«886
B4S
72D,«««458|
E^bb
^
zz 158
a»b 654
aS 5431 45
B*b IP40,Z)»
9a8
226
Fi
612
96 498) 43 883
^»998Cjm84
„7 681 34 453I
^4« 570 G4«« 95^1
16
A*bb
766
33 657
so 104a X4
aa6
3X
6x1
48 996, xa 181
89
566
40 95 1
Cb 1152
E^b 338
G»
724
B
1 1 10
A«296
F4«682il4«io68
C4«««254
38
2>*bb
770
64
a IIS5
F*b 450
19 340
4»b 836
36
7*5
22
53=
so XX09
408
17 894
34 679
5x
X064
X5 349
F4««752
G,t 794
Bjl ii8oD,«t 366
G^bb
<»
562
B^bb 94^
D»b 134
F.'
"3
520
aa
A
9^
^39 79a
C|« 92
3 11771 ao
E^ 478Gf,tt
8^6^
37 749
B4« 50
«9
C^bb
566
io6n
46 951
10 136
«7
5a«
44
906
8 91
85 475. 4a
4JJ 976C,«J
86a
162
6 45
E,nt 548
£;»bb 246
a*b 632
B«b
IOI8Z)
204
F,« 590
F?bb
1064
3S2
15 a49
4«bb 744
C*b I 130
^t'b
r.'^'a'
904
702
bTi<^
47 974' "
D^ 274F««jr
6)5^
a8 543
4,«« 1046
B*bbb
i^
I^bb 42
X xi3a
F'b 428
z8
4'b
^^c"
70a
0
E^ 3t^
_i6 a7a 33
QS 772 B,5
w'^
50 104a
D,nt 344
E*bbb
860
i';4
6 45
a»bb 540
as 430
B'bb926
Dt^b
8x5 4
112F
498
/: 8l1
38 770 a
c^ 70 ^,«
XI58
456
o]tt ^°
IK X58
A^bbb 652
a8 543
C"bb 1038
45 9*8
£;«bb224
9
XX3, a6
49B
43 883
7 68
F^ 568
a4
4.»
453
954
41 838
C4« 140
a»b
610
Bb
996
D, 182
33
6S7
so 104a
14 a26
3«
6xx
..^
996
xa z8z
89 566 46
95"
xo X36
2>bbb
1 150
F'bb 336
i4«bb722
C'b
1 108
Eb
294
G| 680
B, io66D,S
252
F, J 8 63 J
• X155
G*bbb 448
X9 340
36 7a5
53=
l?^b
0 ZZ09
406
17
894
34 679
51 X064
««
849
3a 634
B,t I 136
B»bbb834
D«bb 20
iib
792
a, I 178
S, 364'<?,«
750
«4
0*bbb
«1
946
41 838
5 »3
aa
408
39
79a
3 "77
ao 362 37
747
z XZ33
^4« 434
£;>bbbi32G«bb5x8
B«bb
904'Db
90
F, 476\A^ 862,C,«
48
46
951
xo 136 87 sai
44
906 8
91
as 47Sj 4a 860
6
45
83 430
-126
"•94 "•63
—
31
0
+ •31
+ •63
+ •<
H
+ I'26
I
2 3
i
1
5
6
7
8
9
-•32
-•27
-•23
-•18 IT
•14
•09
-05
o
+ •05
+•09
+•14
+ i8
f
+ •23
This is the first table of modulations adapted to Just Intonation that has been constructed.
But .the table in (Gottfried Weber's Versuch Hner geordneten Theorie der Tonaeizkunst (Attempt
%i systematic theory of musical composition, 1830-2, vol. ii., § 180, p. 86), although only
Gbdapted to equal temperament, was of much assistance to me.
19. Constniction of the Duodenarium, The arrangement is that of all the
previous schemes, proceeding from bottom to top by intervals of a Fifth 702 cents
(or from top to bottom by intervals of a Fourth 498 cents), and from left to right by
intervals of a major Third 386 cents (or from right to left by intervals of a minor
Bixth 814 cents). The number written against any note shews the cyclic intervals
of the note from (7, when all are reduced to the same Octave, see App. XX. sect. A.
art. 24, xiii. p. 4376'.
But as a Fifth is 701*955 cents, and a major Third 386-314, errors of accumulation occur,
and hence the cyclic numbers require corrections if the precise numbers are wanted; apply
those given at top or bottom of the column, or at either end of the line containing the number.
Thus F28 S682 has the column correction + -63, and the line correction --23, and its true dis-
tance from C is therefore 682-4 cents. On referring to the name of the note in the Table,
Beet. A. art. 28, p. 440, the precise number of cents to one place of decimals, the logarithm and
pitch of the note will be found in addition.
The interval between any two notes, reduced to the same Octave, is the dif-
ference of the number of cyclic cents assigned ; corrected if required. The number
of the note by which the just note would be represented in Bosanquet*s cycle of
53, is added in smaller figures under the just note, and the nearest whole number
of cents is annexed. Referring to that number in the Table in sect. A. art. 27,
Digitized by V^OOQ IC
1[
4^4'
ADDITIONS BY THE TRANSLATOR.
APP. XX.
p. 439&, the name given by Mr. Bosanquet, the precise number of cents, the loga-
rithm and the pitch will be found.
20. Jvst Intervals Bediiced to Steps of Fifths and Major Thirds, On account
of the construction by Fifths and major Thirds, we can proceed from any note to
any other by taking a certain number of Fifths up or down, and then ms^'or
Thirds up or down, and reducing to the same Octave. See suprii, sect. D. arts.
7 and 9, p. 452&^ c, where the process is described.
21. The Column of Fifths.
The central column of Fifths has no sape-
nor or inferior indexes. The superior indexes
*,*,*,* to the left not only serve to distinguish
the columns, but indicate that the note bear-
ing it is I, 2, 3, 4 commas of 22 cents sharper
I or higher than the note of the same name in
' the column of Fifths. Thus Dbin the oolumn
of Fifths has 90 cents, D*bhas therefore 90 +
3 X 22 = 1 56 cents as there marked. The in-
ferior indexes p «, ti 4 to the right also not
merely distinguish &e columns, but indicate
that the notes are i, 2, 3, 4 commas flatter or
lower than a note of the same in the oolmnn
of Fifths. Thus Ct has 1 14 cents, but C^ has
114-3x22 = 48 cents. It is thus quite easy
to continue the line of Fifths up at least to
-DS S 8 546 from the table by adding the ap-
propriate number of commas, thus Z>8 8t
= ^4««« +45<22 = 458t88-546 cents and
down to Cbbb 858 by subtracting the
as shewn by C^bbb-4x 22 = 946-88=858.
22. Limits of the Ihiodenarium.
These were determined thus : The central
dark oblong is the duodene of C. Within
the next cUkrk oblong are all the duodenes
which have at least one note in common with
the duodene of C The extremes are the duo-
denes of D^b (with the note B»b), of A^bb
(with the note i>>b), of E^l (with the note
FyU ), and of B^ (with the note ^1,). Then the
outer black oblong contains all Uie duodenes
whose roots are notes in the intermediate
black oblong. Supposing the original dno*
dene, then, to be one which had its root in
the duodene of C (which may always be con-
sidered as the case), the limits allow of modu-
lation into any duodene containing that note,
and thence into duodenes which have at least
one note in common with the last named. We
thus obtain 9 x 13= 117 notes, forming 7 x 10
= 70 duodenes.
23. Introdnction of the Seventh and Seventeenth Harmonics.
If it is desired to proceed beyond tertian
to septimal harmony to inftroduce the har-
monic form of the phord of the dominant
Seventh, with the ratios 4:5:6:7 as Mr.
Poole has done (see sect. F. No. 7), or even
to septendecimal harmony to introduce the
harmonic form of the chord of the minor
Ninth 8 : 10 : 12 : 14 : 17 (see p. 346c, note ♦),
the number of the notes will be nearly tripled.
Taking the root of the chord as C, to each
minor Seventh Bbwe should have to add 'Pb,
which is 27 cyclic cents flatter than this
minor Seventh Bb (as shewn in the duo-
denary arrangement of Mr. Poole's notes,
I sect. F. No. 7), and to each minor Ninth as
i>ib we must add *^i>*bf which is seven cyclic
cents flatter than this minor Ninth. The cents
in the tertian chord of the minor Ninth C JB7,
G Bb -D'b are o, 386, 702, 996, 1200+ 112.
Hence the cents in the harmonic septimal
chord of tiie dominant Seventh, or C E^ G
*Bb, are o, 386, 702, 969, and the cents in the
septendeeimal chord of the minor Ninth, or
C E,G 'Bb"i>»b, are o, 386, 702, 969, 1200
+ 105. This form can be played in all its
inversions on the Harmonica), see sect. P.
No. I. If the root be omitted in the cJiord
of the minor Ninth, we obtain the chord of
the diminished Seventh, which in its har-
monic form is 10 : 12 : 14 : 17, or E^ G ^Bh
*'jD'b, in cents o, 316, 583, 919, which can also
be played on the Harmonical in all its inver-
sions. In Mr. Bosanquet*B cyde of 53, the
chord of the dominant Seventh is played by
the degrees 4, 21, 35, 47, or cents o, 385, 702,
974, of which the last note is 5 cents too
sharp, but the effect is good. The chord of
the diminished Seventh must be played bj
degrees 4, 18, 30. 45, or cents o, 317, 589. 929.
the last of which is 10 cents too sharp, and
the result would not be improved by taking it
one degree or 23 cents flatter. Altogether it
is only a slight improvement on the imitatioa
of the tertian form, degrees 4, 18, 31, 45, or
cents o, 317, 611, 929.
24. Need of Beduction of the Number of Just Tones,
Of course it is quite out of the question
that any attempt should be made to deal with
such numbers of tones differing often by only
2 cents from each other. No ear could appre-
ciate tiie multitude of distinctions. No in-
strument, even if once correctly toned, would
keep its intonation sufficiently well to preserve
such niceties. No keyboard could be invented
for playing the notes even if they eooid bs
tuned, although, as will be seen in art. 26, it
is very easy to mark a piece of otdinazy
music so as to indicate the piedse notes to
be struck. Hence some compromise is needed,
such as the following.
Digitized by VjOOQlC
SECT. E.
MUSICAL DUODENES.
46.«;
25. The Omissicni of the Skhisma. UneqiiaUy Jiist Intonation.
The Cycle of 53.
The first compromise is to consider all
tones differing by 2 cents (a skhisma) as iden-
tical. The dotted lines in the Duodenarium
inclose 7 x 8 = 56 tones which differ from each
other by more than 2 cents. Any note in the
line just above the upper dotted line differs
only by 2 cents from the note just above the
lower dotted line in the preceding column.
We may proceed then by perfect Fifths of
702 cents up from D3S 252, the extreme note
in the right-hand bottom comer of this oblong,
to D,t 1 366, at the top, and thence by an im-
perfect Fifth of 700 cents (the same as in the
equally tempered scale) to B^ 1066 at the
bottom of the next column to the left. Then
again we may go by perfect Fifths to Bjt 1 180,
and then by another Fifth of 700 cents to G,
680 J= 1180 + 700- 1200) and so on till we
had by these alternating Fifths of 702 and
7cx> cents reached the 56th note and 55th
Fifth F*b 450. In this way, at the 53rd Fifth
or 54th note we should have reached E*[>b
246, over which a short line is drawn. Now
this is lower than the initial note D^ 252 by
only 6 cents. Hence if on the three last occa-
26. The Duodenal, The duodenal is the letter name of the root of any
dnodene. By placing it over any note or chord we indicate that that note and all
which follow till a new dttod^nal is giveii are to have such values only as tJiey
would have in the duodene of which the tone indicated by the duodenal is the root.
This prevents all ambiguity by restoring in fsbci the notation of commas higher or
lower, which alone is wanting for the representation of tertian harmony in the
ordinary staff notation. If the 7th and 17th harmonics have to be introduced they %
will have sloping lines placed before them as in chord 1 7 below. The examples
given are not intended as specimens of desirable harmony, but of the means of
representing differences of just intonation. The first 16 chords are from God save
the Queen ; the four last are merely examples of notation.
sions where Fifths of 700 cents were to have
been used, we had taJcen the perfect Fifths
of 702 cents we should have made C*b^ mo
cents, ii*bb»726 cents and ^bb»342 cents,
and consequently E*\)b 252 cents. This would
have become identical with the starting note
Dji 252. This mode of tuning, which if accu-
rately executed no ear could distinguish from
just intonation, forms the unequally jtist in-
tonation mentioned in sect. A. art. 19, p. 435c.
It is also the foundation of substituting for
the perfect Fifth another of 31 x 1200-7-53 =
701-886 cents, 80 that on repeating it 53 times,
and deducting 31 Octaves we should come back
to the starting note. And this gives the cycle %'
of 53 already described, sect. A. arts. 22 and 27,
to which reference is made on the Duoden-
arium itself, shewing exactly the mode in
which it can be substituted for Just Intona-
tion without perceptible injury to the har-
monic effect. For this and other less happy
but more handy attempts, see sect. A. The
mode of fingering this cycle is explained infr4,
sect. F. Nos. 8 and 9, and of tuning it in sect.
G. arts. 19 and 20.
C G C
I I I, I
123 4 56 7 S 9 10 II 12 13 14 15 16
17 18
19 20
The chords are numbered for convenience
of reference, and only the treble is given for
brevity. When the bass is added, the duo-
denal should be repeated in the bass or merely
placed between treble and bass. Observe the
chords 3, 9, 13, which introduce the ambi-
guous chord on the second. The duodenal
C over chord i shews that we begin in the
duodene of c, so that the first chord is e' 316
g^ 498 e'. But G over chord 3 shews that there
is a dnodenation into the dominant, and that
the chord is the true minor /'' 386 a' 498 d" and
not the dissonant chord of tiie added Sixth
/ 386 a/ 520 d". The d" must be retained for
the voice to descend by a perfect minor Third
to 6/ in chord 4, and be the true Octave of d'
in that chord wliere C shews that the duodene
of C is again reached. Hence it is not allow-
able to tak^ chord 3 in the dnodene of F as
/386 a/ 498 dy". The following chords 5, 6, 7,
8 are also in the duodene of C, as there is no
change of duodenal. But chord 0 is in the
duodene of F, because a/ is retained from
chord 8, and d"j f* must harmonise with it.
In chord 10 the duodene of C is again reached.
As purposely written in this example chord 13
is the dissonant added Sixth ^386 a,' 520 d'\ ^
which is resolved on chord 14, but the reten-
tion of a{ would make it more natural to take
the duodene of F, as /'386 a,' 498 d" and then
return immediately to the duodene of C. In
chord 15, d'267 ^f 2^1 ^ 386 6/ the method is
shewn by which the septimal 7 is indicated.
The duodenal C would make / without the
mark before it, to be true Fourth of the root
c. But this Fourth is 27 cents too sharp for
the 7th harmonic of the dominant g, and
hence the line sloping down to the right in-
dicates that the Fourth has to be taken 27
cents flatter in s^timal harmony. In ordi-
nary tertian harmony as indicated by the duo-
denal only, the Fourth would remain unaltered.
In chord 17, the duodenal C would shew that
the a'b must be a^'b, the minor Sixth Of the
root c, or a diatonic Semitone above th^ domi-
nant g. But this is 7 cents too sharp for the
Digitized by V560gle
466 ADDITIONS BY THE TRANSLATOR. app. xx.
17th harmonic of the dominant, and henoe served if this minor Sixth werieoBedin place
the line sloping down to the right indicates of the 17th haimonic provided only the 7th
that it is to be 7 cents flatter. The sloping harmonic of the dominant were retained,
line, therefore, indicates different degrees of Equal temperament of course not recognis*
flattening according as it is applied to the ing the difference of a comma, so far as sound
Fourth or minor Sixth of the root expressed is concerned, retains the same tempered duo-
by the duodenal. If, therefore, we wished to dene throughout, although there is a difference
have the chord «,' 316 fir'267 '6'b336 *'d*"b we in writing it, as would be shewn in the Duo-
must write the duodenal as F to get the right denarium (p. 463) if the indices were omitted,
intonation, as in chord 19. Since the 17th Such an omission reduces the Duodenarium
harmonic of the dominant is so nearly the to a table of modulations in any temperament
minor Sixth of the root, and the chord is which neglects the comma,
dissonant, much of the effect would be pre-
SECTION F.
m BZPBBIXEMTAI* ZNSTBUMBNT8 TOB EXHIBITIMO THS BFFECTS OV JUST OTTONATIOM.
(See notes pp. 6, 17, 217, 218, 222, 256, 329, and 346.)
No. No.
Introduction, p. 466. 5. Rev. H. Listen's Organ, p. 473.
1. The Harmonical, p. 466. 6. General Thompson's Organ, p. 473.
2. The Just Harmonium, p. 470. 7. Mr. H. W. Poole's Organ, p. 474.
3. The Just English Concertina, p. 470. 8. Mr. Bosanquet's Generalised Fingerboard
4. BIr. Colin Brown's Voice Harmoninm, p. and Harmonium, p. 479.
470. 9. Mr. Paul White's Hiumon, p. 481.
Inteoduction.
At the present day ordinary masical instruments are intended to be toned in
accordance with equal temperament (see pp. 313a, 432^, art. 10 ; 4366, art. 22, i. ;
437c, art. 25 ; sect. G. arts. 11 and following). The English concertina, which has
14 keys for the Octave, is still usually tuned in the older Meantone temperament
H (P- 433^» art* ^6» ^^^ ^^^' ^* ^' ^^)' ^^* neither system gives the only inter^
which will allow chords in the middle part of the scale to be played without giving
rise to beats. In order, then, that the ear may learn what is the meaning of ' just
intonation,' it is necessary for it to have special instruments, or at least instru-
ments specially tuned. Prof. Helmholtz has for this purpose invented a tuning
for an harmonium with two rows of ordinary keys, explained on pp. 31 62) to 320a.
Others, as Colin Brown, Liston, Poole, and Perronet Thompson, have invented
harmoniums or organs with novel fingerboards ; and others, as Bosanquet and
J. P. White, have invented means for using the division of the Octave into 53 parts,
which, as is seen in sect. E., p. 463, is practically almost identical with just intona-
tion. A brief account of these instruments (with the exception of Prof. Helm-
holtz's, which is fully described in the text) will here be given. But none of them
meet the wants of the student. They are all too expensive and require so much
special education to use, that (with the exception of Mr. Colin Brown's) they have
remained musical curiosities, some of them entirely unique. But there are two
^ instruments which are cheap and which can be tuned so as to illustrate almost
every point of theory, though they of course remain experimental instruments
intended only to shew the nature of musical intervals, chords, and scales, and not
to play pieces of music except especially composed exercises. These two I shall
take first. They are a specially tuned harmonium and English concertina. Beed
instruments are far the best for experiments, because they give sustained notes
possessing a large number of powerfid upper partial tones, so that any deviations
from just intonation are extremely conspicuous, painfully evident indeed on any
harmonium tuned in equal temperament.
I. The Harmonical.
The scale of the Harmonical and the number of vibrations for every note in &•
first four octaves will be found on p. 17, note. The instrument has been con^
stantly referred to in the Translator's notes to the preceding pages. It is an
harmonium with one row of vibrators extending over five octaves. The toning of
tiie fifth octave wiU be explained further on.
Digitized by
Google
SECT. F.
EXPERIMENTAL JUST INTONATION.
467
Any one baying such an hannoniun of
Ifessra. Moore A Moore, pianoforte and har-
monium makers, 104 and 105 Bishopsgate
Street, London, for 165s. net, may have it
tuned as an Hannonical, by my forks, and
provided with an *harmonical bar' as pre-
sently explained, both without extra charge.
I am sure that all musical students, as well
as myself, must feel greatly indebted to this
firm, who at the instance of Mr. H. Keatley
Moore, Mus. B., a student of the first edition
of this work, have so kindly undertaken to
furnish this almost indispensable aid to the
Btudy of music on Hehnholtz's principles at
Buch a very moderate cost.
On the first four octaves this instrument
contains all the 10 notes of the Decad of C
(p. 459&), and hence all its chords (p. 459c),
and allows of playing and harmonising all the
56 trichordal scales (p. 460) contained in that
decad. Its 10 notes, C D E'b E^ F O A^\) A^
B'b ^1, are placed on their usual digitals.
Hence so far there is no new fingering to
learn. The remaining two digitals are em-
ployed to furnish two notes of great theoreti-
cal importance, the grave Second D,, which is
of course placed on the Db or C% digital, and
the natural or harmonic Seventh 'Bb, which
had to be placed, rather out of order, on the
6 b or ^ digital, the only one at liberty.
Hence, using small letters to represent the
short black keys, the keyboard for each of the
first four octaves is
C D E^
vib. in two-foot Octave 264 { "^^* ^o; ^'^ 330 352
'6b a*b 6*b
F G A, B^ C
46a 49»f 475J
396 440 495 528
and its scheme in the Decad form with the
two additional notes is
B'b
E'\>
A'b
D
O
C
F
'Bb
^1
In this form (...) in the second column
indicates the absence of Bb, and ^Bb forms a
column by itself. The scheme is seen filled
up on p. 474c, d. The addition of ^Bb gives
an opportunity of playing the first sixteen
harmonics of C with the exception of the
I ith and 13th (whence the name Harmonical),
thus:
Note C c g c' e^' g' '6'b c" d" c/'
Harmonic 123456789 10
12 14 15 16
By means of the *harmonical bar' pro-
vided with the instrument, these harmonics,
except the 7th and I4ih, can be pressed down
at the same time, and then the 7th and 14th,
being on short keys, can be added with the
fingers of the hands which press down the bar.
The pegs which press the notes are arranged
on different lines, so that the first 8 harmonics
can be played by themselves, and then the
effect of adding the higher Octave can be tried.
It is thus possible to play the harmonics
simultaneously with or without the 7th and
14th, and thus to estimate their presumed
dissonant effect. To my own feeling these'
harmonics greatly enrich and improve the
quality of the very compound tone produced.
It is evident therefore that the effects of
all the intervals depending on the numbers
I to 16 (omitting ii and 13) can be immedi-
ately produced, and hence all the intervals on
p. 2126, c, induding the septimal intervals, aris-
ing from 'Bb, which are of special importance
and interest because they can be so rarely
heard.
The existence of higher upper partials of
the low notes can easily be made evident by
beats. If we press down one of the digitals
for the shortest distance that will allow the
note to sound at all, we flatten it slightly, and
hence put it out of tune. Keeping then C
sounding fully, and slightly flattening its har-
monics, one by one in tills way (indicated by a
prefixed grave accent) we easily obtain the
beats from Ccy C% Ccf, C'e,', Cg\ C"6'b,
CTc". Cd", CTe/', CTg", CTb^b, Cl^\ Cc"\
making evident the existence of 13 out of 15
of the upper partials of C In the same way
by slightly flattening the upper or lower notes
of any of the consonant intervals, as e : (7, we
IT
can produce the beats which shew that the
consonance has been disturbed. These are
some of the most striking illustrations of
Helmholtz's discoveries.
Beats between the primes of two notes are
well shewn by DDp <W„ d'd{, d"d^\ which
should beat about 9, 18, 37, 73 times in
10 seconds, the number of beats doubling for
each ascent of an Octave. The very impure
character of the beats of i)/>„ arising from
our hearing at the same time the beats of the
upper pairs of notes as partials, is instructive.
We can also hear the beats faJl given for 10
seconds and fractions omittea) in D£>b 50,
'Bb B>b 33t ^'b E^ 33, il'b A, 44, B»b B, 50,
but the higher Octaves of these notes beat too
rapidly to be counted.
Combinational tones are easily heard. Any
two consecutive harmonics of C give C, and
by sounding two of them strongly and slightly %
flattening the C, the beats of this flattened
^C with the combinational tone may be heard,
but much care and attention are necessary for
this purpose. On pUying 6," c'" the rattle of
the 66 beats in a second may be heard, as
well as the combinational C of 66 vib. Simi-
larly for 6," and 6"'b the rattle of the 39-6
beats in a second, and also the deep combina-
tional tone ^'^b of 39-6 vib. And if all three
keys 6»"b, 6,", and c" be held down together,
the low-pitched beat of the two combinational
tones may also be heard with proper care and
attention. If we play <2/'/' we have a beat-
note of 1 17*3 vib., very nearly B>b. If we play
d"/' we have the beat-note il| of 1 10 vib. If we
play all three together the two beat-notes beat
73*3 times in 10 seconds. This must be care-
fully listened for, but the beats being so much
lower in pitch cannot be confused with the
Digitized by V^tibgie
468
ADDITIONS BY THE TEANSLATOR,
APP. XX.
higher beats of d/' d", although their fre-
quency is the same.
All the forms of the major and minor
triads and tetrads on pp. 2 i 86 to 224a oan be
played and appreciated, and in many cases
the combinational tones can be played as sub-
stantive notes with them ; see my footnotes to
these pages.
The e£Pect of the analyses of dyads in my
footnotes on pp. 188 to 191 can all be studied,
and much of the diagrams on pp. 193 and 333
can be verified.
Most of the old Greek tetrachords on
p. 263^2' can be played as there pointed out.
The analysis of scales on pp. 274-278 can
be illustrated.
The discords in Chap. XVII. oan be mostly
^ illustrated, as pointed out in my footnotes.
As shewn by the table on p. 17, note, the
intervals 80 : 81, or comma, the minor Tone
9 : 10, and major Tone 8 : 9, the diatonic
Semitone 15 : 16, and small Semitone 24 : 25,
and other important intervals, can all be illus-
trated. Again, 'Bb : £'b = 3S : 36 is 49 cents,
and hence almost precisely a quarter of a
Tone or 50 cents, and A^ : 'Bb = 20 : 21 is 85
cents, or very nearly the Pythagorean Limma
of 90 cents. The imperfect Fifth of just in-
tonation Z) : iip or 680 cents, may be con-
trasted with the perfect Fifth D| : il„ or 702
cents. The Pythagorean minor Third D : F^
or 294 cents, can be contrasted with the just
minor Third D^ : J?',-or 316 cents.
But it is also necessary to note what the
Harmonical cannot do. It has no Pythagorean
conmia, and no Pythagorean major Third, nor
c| can it play a Pythagorean scale. It cannot
play the chord of the extreme sharp Sixth,
nor can it modulate into the dominant or sub-
dominant, or relative minor (except in the
descending form), but it can distinguish
/ 386 «! 520 <2, the chord of the added Sixth,
from the minor chord/ 386 a, 498 <2„ and oan
modulate from C major to C minor.
It is also able to play Mr. Poole's dichordal
scale F A C,C E O'Bb D with the peculiar
minor chord O : 'Bb : D = 6 : 7 : 9, and the
full natural chord of the major Ninth.
Method of Tutiing, To be sure about the
pitches, I tuned c"528, a'440, a"b422-4,
'6'b462 on forks with great accuracy, by means
of my tuning-forks mentioned on p. 4466'. I
tuned also a second set of forks each two
beats flatter than the above, which I found
very useful in determining the accuracy of the
m tuning by unisons. In fact the note of the
reed is so much more powerful than that of the
fork, that the latter was quite drowned when
near the unison, so that the pitch could not
be determined within 3 to 5 beats in 10
seconds, and this difficulty was entirely obvi-
ated by the flat forks. After these notes then
had been tuned on the Harmonical, the rest of
the notes in the two-foot Octave were tuned by
Fifths or Fourths, namely flrst a»'b to e^%
e»'b to 6»'b, secondly c" to ^, gf' to d'\ cT to/,
thirdly a,' to «/', «," to fe/, and a,' to d/. The
other notes were obtained by Octaves. The
verification is by the perfect major chords
FA^C, CEfi, GB,D, A'bCE^b, JT'bOB'b;
the perfect minor chord D^FA^ and the per-
fect chord of the harmonic Seventh CEfi^B)>,
all without beats in the two-foot Octave.
Pitch. The pitch (^'528 was choeen to
agree with the pitch adopted by Prof. Helm-
holtz in the text; a'440 was the pitch pro-
posed by Scheibler ; a''b422-4 is within -i vib.
of the pitch of HandePs own A fork 422*5, now
in the possession of Bev. G. T. Driffield, Bector
of Old, near Northampton. In the notes not
tuned by forks there may be a very slight but
not perceptible error, so that the Hannonieal
presents a series of trustworthy pitches.
Eacercises, Besides numerous short airs,
and special exercises, the following pieces
may be played with full harmonies, and will
serve to illustrate the meaning of just intona-
tion, especially if they are contrasted with the
same airs immediately afterwards played on
an ordinarily tuned harmonium.
Ood save the Queen (in C major with its
minor chord on the Second of the scale, alter-
nating with the chord of the added Sixth).
The Heavens are telling {C major with the
modulation into C minor).
OUyrious Apollo (altering the brief modula-
tion into the dominant).
The Old Hundredth (C major).
John Anderson (0 nunor).
Adeste Fideles (avoiding the modulation
into the dominant).
Auld Lang Syne (in C major).
Dies ir€Bt in part (C minor modulating
into C major).
Leisej leise (the prayer in Der FreyschSti
in Poole's dichordal scale FAfi, CEfiB\>D,
altering the harmonies to suit the new scale).
Crudel perchA {NoMMe di Figa/ro^ in C
minor, altered, but preserving the burst into
C major).
Wanderer's NachtUed (Schubert).
The Manly Heart (ZauberflOU).
So much relates to the lower four Octaves of the Harmonical, which suffice to
illustrate all the principal peculiarities of just intonation. Advantage has been
taken of the Fifth or 6-inch Octave to exhibit some of the higher harmonics of
C 66, and to give a complete series of the first 16 harmonics of C 132, inclndhig
the nth and 13th. These notes are as follows :
Harmonics
Black digitals
White digitals
Pitch numbers
16
17
18 19
»V"b
20
1056 1 122 1 188 1254 1320
Of course with such high pitches there has
been great difficulty in tuning, and there are
probably several slight errors, but none that
will interfere with the general effect. I pro-
ceeded thus. The harmonics 16, 18, 20, 28,
^4> 30t 32 were the Octaves of harmonics 8, 9,
22 28 24 25 26 29 30 33
'l/"b »a'"b »6"b
»»/" g"f »«te I/** (T
1452 1848 1584 1650 1 716 1914 1980 21 13
10, 14, 12, 15, 16 already tuned for the lower
Octave. Hence only 17, 19, 22, 25, 26, 29 re-
mained to be treated, but they were in them-
selves far too high for me to tune forks for.
I tuned therefore ^^d"b with 561 vib., the 17th
harmonic, to which I had the Octave made I7
Digitized by V^jOOQlC
SECT. F.
EXPERIMENTAL JUST INTONATION.
469
Messrs. Valantine A Carr, music smiths, of
76 Milton Street, Sheffield. Then I tuned
»Vb3i3"S. 'V363* '•a'b4i2-j, »a'429, *»6'b478-5
all harmonics of C„i6'$ and hence two Octaves
too low. From these Messrs. Valantine A Carr
made me forks giving the Octaves with great
accuracy, and afterwards the Octaves of these
forks, which so far as I could test them also
appeared accurate, but it was very difficult to
form an accurate judgment of the pitch of
these high tuning-forks. From the forks thus
made the remainder of the fifth octave was
tuned. But as the tone of the reed drowned
that of the fork« I had here also a second
series of flatter forks constructed, beating twice
in a second with the former. Of course I have
not been able personally to check the tuning
of all the Harmonicals, but I worked with the
toner at first and saw that he perfectly well
understood what was to be done, so that I con-
fidently hope the Harmonicals he turns out
will answer their purpose. One of them was
exhibited in the International Inventions
Exhibition of 1885, Division n.. Music.
By means of Uiis fifth octave the instru-
10 : 12 : 14 : 17 = «," : gf" :
12 : 14 : 17 : 20 = g'* :
14 : 17 : 20 : 24 ^
17 : 20 : 24 : 28
The extreme height of the pitch of these notes, however, will prevent a due appreciation of
these chorda as compared with the usual forms, which can only be played at a lower pitch thus :
20 « d' *
ment has now all the first 32 harmonics of
C66, except 6, namely 11, 13, 21, 23, 27, 31 ;
and has all the first 16 harmonics of c 132 with-
out exception. There are additional loose pegs
to the harmonical bar, which can be inserted,
in order to play all these 16 harmonics at once,
with the exception of the 7th and 14th, which,
being on black digitals, most be struck with
the finger as before.
The fifth octave therefore gives the
trumpet scale 8, 9, 10, ii, 12, 13, 14, 15, 16,
all, with exception of 14, on the white digitals.
These give the peculiar intervals 10 : 1 1 := 165
cents, II : 12^^151 cents, 12 : 13=139 cents,
13 : 14 » 128 cents. The inharmonic character
of these intervals is, however, not well brought
out, owing to the weakness of the upper par-
tials in this region. Other intervals of interest m
are the approximations to the tempered Semi-
tone, 16 : 17 s 105 cents, 17 : 18 = 99 cents.
The 17th harmonic allows of playing the
harmonic form of the chord of the diminished
Seventh in its direct form and all its inver-
sions as
: '6"b : "d'"b
: *6"b : "<i"'b : «,'"
6"b : ''d'"b : «/" : a"'
= "d'"b : «/" : /" : 'I/"b
: i4| : 17
: 14I : 17
14S : 17
iv.20:
7^:20:
:/':a>'b:6/
24 - /' : a"b : W : ^"
24: 28J = a"b:6/:d":/"
of which only the first shews the full harshness of the chord.
It is thus seen that the Harmonical is the only instrument yet tuned which
brings out the full nature of just intonation for the 7th and 17th harmonics.
The difficulty in tuning the Harmonical without forks may be to a great
extent avoided by the following means, which will enable any possessor of a cheap
harmonium which he is willing to sacrifice as an experimental instrument, to get
it tuned by a professional tuner. It will, however, be necessary to give up the
peculiar arrangement of the fifth octave, and when it exists on any harmonium
to have it tuned simply as an Octave higher than the fourth octave.
First tune the 11 notes C D^ D E^\> E^ F
O A^b ^1 B*b Bi thus. Take C to the exist-
ing pitch on the instrument. Tune the Fifths
cf ig'/g^ : d'\ /' : c" till they leave no trace of
beats. Then take &' : e/' and a*'b : c'' to be
as perfect major Thirds without beats as the
tuner can make them, verifying by the major
chord c'e,V» M^d minor chord /'a*'bc^ which
should both be without beats. The combina-
tional tones (which should be C for c" : e/' and
A^b for a"b *. c") will also be a guide to the
ear. But there is very little chance of perfect
accuracy, the ears of tuners having been spoiled
by the sharp major Thirds of equal tempera-
ment. It is best to begin by tuning these
Thirds decidedly too flat, beating 10 or 20
times in 10 seconds, and then gradually to
sharpen tiU the beats apparently vanish. The
Thirds may thus remain very slightly flat, like
the skhismic Thirds on p. 2Sid\ and they
will give very good results. The point is to
avoid sharp major Thirds. Then tune the
Fifths «/ : 6/, a, : a/, d,' : o/, and a"b : e"%
e^'b : 6*'b» the necessary Octaves having been
previously tuned. Verify by the major chords
ray\ c'e.'g^, g^^d"; a'b c^e'% e^'b g'b'X
and the minor chords d/f^a^', a/c"«,", e/g'b' ;
/'a»'b c", (/e"b g', g'b^'b d'\ all of which should
be perfect without sensible beats. Then only
'6 b remains to be tuned. To this we may ap-
proximate very closely thus. In the first place «-
it is 49 cents, or say a quarter of a Tone (that
is, half a tempered Semitone), flatter than 6'b
which has already been tuned, and many toners
can approximate to this interval. Next, in the
lowest octave £'b and 'Bb will beat. If the
pitch happens to be ^"528, then the beats are
33 in 10 seconds. For c"540, which is sharp
band pitch, the beats would be not quite 34 in
I o seconds. For c"5 1 8, which may be taken as
French pitch, the beats would be almost ex-
actly 32 in 10 seconds. Hence by taking them
as 33 in 10 seconds for any pitch, the tuner
will come very near the truth. After tuning
the Octaves, he will verify with the chord
efe^'g'^Vbt which should be without any sen-
sible beats and have merely a slight roughness.
Even a rough approximation to the true value
of ^Bbt as on the 53 division of the Octave,
will gratify most ears.
Digitized by VjOOQlC
£'t>
2)
£'b
a
B,
^'b
G
Ex
i^-b
F
^.
470 ADDITIONS BY THE TRANSLATOR. app. xx.
2. The Just Habmqnium.
This I used for some years. The 'Bl> is sacrificed, as also the D, ; and the 12
notes are taken as in the margin, F^ being put on the F% digital. F^
It therefore contains the duodene of C, with the exception of i^ijj,
and with the addition of F^. This was tuned by an ordinary tuner
in my presence in two hours from the following directions. Make
the 7 major chords CE^Q, GB^D, FA^C, A^\^CE^\^, E^\^GB%
B^\}DF\ D^\}FA^\^ perfect without beats.
This gives more power of playing, as it contains the decad of C complete, and
hence C major without the grave second, and G minor in all its forms, with
all the 56 modes ; E^\} major with the grave second F, and A^\} major without the
grave second. But the harmonic Seventh ^-Eb, and grave second Di are much
missed in C major. There is power of modulating from E^\} major to its sub-
dominant A^\} major (without the grave second) and also into its relative minor C
IT Hence this plan of tuning has many advantages, especially in being easily effected
by any tuner without forks.
3. The Just English Concbbtina,
For many years I have been in the habit of making my experiments on one of
these instruments tuned thus :
Black studs Cjjj! d d^^ f\% g^lf^ a h^
White studs G D, e[ F G A^ B\}
having the following duodenary arrangement :
A
D Fj
G bTdS
If G E, g4
F A, CS
Hence it contains the decad of E^ and four additional notes Ay F, B\}, D,.
This furnished the power of playing in G and G major with the grave seconds ^1,
Di, and in F migor without the grave second G^. Also in Ex major without the
grave second F^^. It can modulate perfectly from G major to the dominant G
major, and from G major to its relative minor J^,. Also from J^i major to its
tonic minor E^. But it cannot modulate perfectly from C major to its relative
minor Ax^ because of the absence of Fjj^ occasionally used in the subdominant. It
can modulate perfectly from G major to its subdominant C major, and thence to
its subdominant F major, less the grave second Gp A considerable variety of
harmony and modulation therefore lies open to it, but most pieces require special
arrangement.
It has been found advisable to put Di, A^ on the white studs and D, A on the
V black studs, that is D on the D^ stud, and A on the A\} stud, and then I put Dj^
on the E\} stud. In other respects the fingering is unaltered.
Tufimg. The tuners of Laohenal's conoer- numbers except for D| ^„ whioh had to be
tins factory (4 Little James Street, Bedford distinguished from D, A. In writing mosie
Bow, London, W.G.) were able to tmie with for it I generally assume the valnes of D and
sufficient oorreotness from the following direo- ^, as in the key of C, and distingnish Dj bj a
tions. downstroke (as '6'b in the diagram p. 22c),
Make the 8 major chords CE^O, GB,X>, and il by an apstroke (as *'/' in the same dia-
DF^tA\ FAfi, B\>D^F\ Afi^^E^, E^G^B^, gram), but occasionally to prevent amUgnity
B,Z)^ F^t perfect without beats. In giving I also use the up strdLe to A u>d the down
these directions I avoided using the inferior stroke to A^,
4. Mb. Colin Bbown's Voice Habhonium.
Mr. Colin Brown, Euing Lecturer on the Science, Theory, and History of
Music in the Andersonian University at Glasgow (see p. 259^2, note ^), has invented
the following keyboard, a full sized model of which is in the Science Collections at
the South Kensington Museum, Room Q. By the kindnose.of Mr. Brown I am
Digitized by V^OOQIC
SECT. P.
EXPERIMENTAL JUST INTONATION.
471
enabled to give a perspective view of his keyboard in fig. 67. He calls his instru-
xxient < The Voice Harmonium/
PLAN OF KBYBOABD.
1
i
ejlt
2S,«
at
gjit
B.n
ct
A«5
^,t
CI
M
A«
Ft
G,«
B
M
c,t
E
*".«
A
a,«
c.«
d^
D
gjt
^1
0
c^
E,
a
^x
c
fa
6.
A
0,
J5b
h.
c,
Eb
d.
Fi
A\>
<h
c.
D\>
9t
B.b
JB,b
Ob
Cb
A,\>
Cb
Fb
Fio. 67.
DUODBNABT ABBANaXMBNT.
White
Digitals
Oolottred
Digitals
Peg
Digitals
c«
E,t
MS
n
Alt
cjf«
B
A«
/^«
E
«.l
M
A
c.«
a
D
Fit
a^
Q
B|
^
C
E,
M
F
Ai
c,«
Bb
A
/^
Eb
G,
ft.
Ab
Cx
«t
Db
Fx
««
Ob
B,b
d.
Cb
E,b
9t
The notation used by Mr. CJolin
Brown, as shewn in fig. 67, is dilfe- m
rent from that used above. The
colamn of Fifths, so far as it is there
shewn, is
'Dby Ab, Eb, Bb, F, C, G. D,
A',E\B^Ft.
The rest of his notation need not be
specified, because it does not appear
in fig. 67.
By the duodenary arrangement it is seen that the onlv duodenation contem-
plated was by Fifths up and down from the duodene of Bi\} to that of DA. But
it was not viewed in this light. It was rather considered as a series ot major
scales modulating into the dominant or subdominant, and also into the relative
minor. The tonic minor was always taken one comma flatter Uian the tonic
Digitized by V^jOOQlC
472 ADDITIONS BY THE TRANSLATOR- app. xx.
major ; thus the tonio minor of C major was considered to be the relative minor of
E\} major, that is Ci minor, which, as shewn by Prof. Hehnholtz's theories and the
annexed duodenary arrangement, has not a single tone in common with C major.*
The third column was considered merely as containing the major Sevenths and
major Sixths of the relative minor scale and accounted of subordinate interest.
The arrangement of the keyboard is highlv ingenious. Observing that in the
major scale there are four notes in the column of Fifths, and three in the
column of Thirds, it became evident that each note of the first would last during four
successive modulations into the dominant, whereas each of the latter would last only
through three modulations. Hence the digitals containing the former were made
four parts long, and those containing the latter three parts long. In going up a
series of Fifths each digital advanced one part. In the plan the longer digitals with '
only one note name were four parts long, and were left white. It wiU be seen
that F is one part lower than 0, G than O, O ilian Z>, and so on. Then D stands
two parts higher than C. E, a major tone above D, stands also two parts above
H D. The long white digitals, read diagonally, give therefore the first colunm in the
duodenary arrangement. Immediately below each is a short coloured digital, dis-
tinguished in the plan by having two names of notes on it. The lower is that of the
note corresponding to the digital, and it is exacUy one comma flatter than that of
the long white digital above it. By this means the diagonal series of coloured
digitals give the column of Thirds. Any white long digital is separated from the
white digital next below it by a short coloured digital, and hence it corresponds
to a rise of seven Fifths from the note of the lower white digital, and this rise
gives the Pythagorean sharp, or 114 cents. Consequently the coloured digital
which separates two white ones, being a comma of 22 cents flatter than the upper
white one, is 92 cents sharper than the lower one. This gives the complete
order, white C, coloured Cx f^ above it, and white G& above that ; coloured C|
below G, and white C\} below C^ Then each digitfd on the right begins two
parts higher, corresponding to a major Second, or two Fifths less an Octave;
and the fingerboard is complete for the first two colunms of the duodenary
^ arrangement. In the fingerboard itself, as shewn in fig. 67, the lower and
upper digitals are out through at the dotted line, but they have been continued in
the plan to shew the arrangement. Beginning then with any white digital as
G we play the major scale in a horizontal line passing through the letters D, Bi
on the plan, and giving C, D (both white), JSr, (coloured), F, G (both white), Jj,
^1 (both coloured). The fingering is absolutely the same for all major scales
whatever note is used as the white digital to commence with. The grave second
D, is furnished by the coloured digital below the usual second.
For the relative minor, suppose the descending form with three minor chords
is used ; another line not quite horizontal through Di,AiBi gives AiBi G Di Ei
F G Ai. To make the dominant chord major, ukL hence change g into gjj^^ touch
the small peg which rises out of the left-hand comer of the Ai digital, and it gives
a diatonic Semitone of 112 cents below A^, that is, the leading note to ^|. This
peg is immediately to the right of the G digital. In the same way to make the
subdominant chord major, and hence change / into /^, use the peg/^ immedi-
H ately to the right of the white digital F. The names of these pegs are written in
small letters on the plan. We could by introducing the Cg j|l peg next to the G
digital, play the complete major scale of J[|, as Ai B^ c^fi J^i E^ f^% ^gj]! il,, and
all major scales beginning with a coloured digital would be fingered m the same
way. Thus we could play the major scale of A^ and modulate into the tonic
minor scale of A^. Similarly we could play the major scale of c^ and modulate
into the tonic minor scale of Ci. But the fingering was not intended for this,
and hence it is not so convenient.
* Mr. Brown considers (Music in Common upon the same tone of absolute piteh.' Bai
Things, p. 35) that C major and C, minor on his own fingerboard we have the major
have one tone in common, F. This makes his scale of A^^ and what is, according to ProL
descending form of the scale of C, minor read Hehnholts, the relative minor of C, both com-
upwards c^ d^ eb / Qi ab bb c/, where I should menoing with A^, the one having the three
use /|, which is ready to hand on the instru- minor chords d^fa^^ a, C6,,6| fjrdi and the other
ment, if desired. Mr. Brown asserts (ibid, p. 35) the three corresponding major chords d^/jl a.^
that * it is impossible to build a major and a, CjS 0p «, g.it dj.
its tonic minor scale in trus key relationship^
Digitized by
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SECT. F,
EXPERIMENTAL JUST INTONATION.
473
For the sake of perfect uniformity in fingering the difference of a skhisma
is tnned, or intended so to be. Observing the two dotted lines in the duodenary
arrangement, we see that no note between them differs from any other by less
than a comma, but the 8 notes A\}, D\}, G\}, C\}, and G^, F^, B^)}, E^\} below
them, and 6 notes Di% Ax% ^i$ and f^%% c^jtff* 9^ above them, differ from
notes between them by a skhisma only, the first eight being respectively a skhisma
flatter than 0^% Ci^, Fx% Bi and b^, ejf^, ajj^, dj^, and the last six re-
spectively a skhisma sharper than £[>, ^, F and Gi, D^, Aiy which notes all lie
between the dotted lines.
The tuning was effected, as the duodenary arrangement shews, by Fiflhs and
major Thirds, and to overcome the difficulty of the latter the combinational tone
was employed*
5. Rev. Henby Liston's Obgan.
The pitch of Mr. Liston's notes has been calculated from the data furnished by %
Mr. Farey, Philosophical Magazvtie, vol. xxxix. p. 418. Mr. Liston's Essay on
Perfect Intonation was published in 1812. The following is the duodenary
arrangement of his notes : —
__
_^
c«
E,n
_
—
—
D»
Ft
A^Z
C,«|
—
—
(3f»
B
A«
y,5l
lp»b
A*\,
C
E
G.«
Bji
—
jD»b
F^
A
0.5
Ejt
JJ»btl
0«b
J5'b
D
F,Z
^.1
cjin
—
C*b
£*b
0
-Bi
Dtt
F,tt
—
F*b
A'\>
C
E,
Gjt
B.n
—
B»bb
D'\>
F
A,
Ojf
E^
—
—
G»b
Bb
A
PJl
A^
—
—
C»b
Jg?b
Gx
—
—
JP»b
Ab
c\
.^
—
—
B»bb
—
—
The two isolated tones on the left were, I believe, added for the sake of tuning.
It is evident that Mr. Listen contemplated considerable modulation, and provided
for the tonic as well as relative minors.
6. Oen. Pebbonbt Thompson's Enhabmonio Organ.
This was the organ constructed by Robson for Gen. T. Perronet* Thompson,
which I took Prof. Helmholtz to hear, as described in App. XVIQ. p. 423. It had
three manuals, each with a complicated fingerboard, and is completely described
and figured in the General's Principles and Practice of Just Intonationy which
is also full of curious musical information. It contained, on the whole, 40 tones
to the Octave, and had considerable power of modulation, as shewn by the follow-
ing table. But the gaps left indicate that the problem had not been completely
grasped*
Duodenary Arrangement.
c
E
._
-
_
F'
A
o,t
B>b
D
i^.«
^a
E'b
G
•Bi
DJt
^at
A^b
C
E,
0^
Ba
2>'b
F
^.
CJI
Ea
G'b
Bb
A
P.t
^a
—
Eb
0,
B,
A«
—
Ab
0,
E,
eta
—
^.
^
Oji
7
9
9
8
7
f
Ko. of tones
* On the organ iieell the name is painted as Peronet with one r, bat the General printed
and wrote his name with rr.
Digitized by V^OOQIC
474
ADDITIONS BY THE TRANSLATOR.
APP.
7. Mr. Henbt Ward Poole's Organ,
See p. 323d, note t» and App. XVIII. p. 423a, for references to Mr. Poole's
keyboards. His papers are in Silliman's Americcm Journal of Arts and Sciences,
1850, vol. ix. pp. 68-83, 199-216; 1867, ^o^« ^^- PP* i~2> (which contains thm
diagrams of his keyboard, here reproduced as figs. 68, 69, 70, p. 475, by the pholo-
graphic processes of the Typo-etching Company) ; 1868, vol. xlv. p. 289. The
first papers contain Mr. Poole's theory and an acconnt of his Euharmonic Organ,
constructed by himself and Mr. Joseph Abbey, of Newburyport, Massachusetts,
having an ordinary fingerboard and a pedal to change the pipes that it affected, and
playing from the major key of I)\) with 5 flats to that of B with c sharps. The 12
digitals brought into action by each pedal produced the 7 notes of the major scale, the
leading note of the relative minor scale, the perfect Seventh, and three others
belonging to adjoining scales, of which only one (the grave Second) is specified.
This arrangement, which was actually used in Boston, was abandoned, because it
was found advisable to have all the notes under command of the hand ^th-
out pedal action, and to use pedals for the bass only. But the new keyboard in
its complete form does not appear from Mr. Poole's papers to have advanced
beyond the stage of a cardboard model, although more recent simplifications,
with 24 and 48 tones to the Octave, have been practically worked out. To these
reference will be made at the conclusion of this notice.
In his theory of this keyboard, to which all subsequent remarks refer, Mr. Poole
recognised 5 series of Fifths, namely those in cols. 5, 6, 7 of the Duodenarium,
p. 463, and two others interposed which may be numbered as ^5 and ^6, because they
contun notes which are a septimal conmia (63 : 64, or 27 cents) flatter than the
corresponding notes in cols. 5 and 6. These are given in the following duodenary
arrangement of his notes. But instead of my superior and inferior numbers he
used varieties of type, as shewi^ in the letterpress below fi^g. 70, which was photo-
graphed from the original at the same time as the fig. itself.
IF
Dttodena/ry Arrangement of Mr. Poolers 100 Tones.
Cols.
No. of tones
Thus ool. 5, or * key notes,* was represented
by Roman capitals, as G, D, and had white
digitals ; col. 6, or * Thirds,* by Roman small
letters, as b, e (this was, in fact, Hauptmann's
5
'5
6
'6
7
Et
522
_
_
_
AZ
1020
-
c,ti
206
—
-
—
m
318
816
-
F,nt
704
—
-
A^tlO^Q
01
»Gf«"
789
B,n
2
—
-
D^t 388
c%
114
»C«
87
E,Z
s<»
'E,t
473
Qjil 886
n
612
'n
58s
M
998
M.ff
971
C^J 184
B
mo
»J3
1083
A«
296
'A«
269
F^Z 682
E
408
'E
381
o,t
794
'Gi«
767
B^ 1 180
A
906
'A
879
c,«
92
'0,5
65
E^ 478
D
204
'D
177
^,«
590
'F,t
563
A^ 976
Q
702
'Q
675
-Bi
1088
'B,
io6i
I>J 274
C
0
'C
"73
E,
386
'Ey
359
Ott 772
F
498
'F
471
^,
884
'A,
857
c^ 70
Bb
996
'J5b
969
^.
182
'A
155
F^ 568
E\>
294
'Eb
267
Gi
680
'G.
653
B, 1066
A\>
792
'A\>
„
c,
1 178
'C,
"51
Er 364
D\>
90
Wb
63
Fr
476
'F,
449
A^ 862
Q\>
588
'Gb
561
B,b
974
'B,b
947
D, 160
C\>
1086
'Cb
I0S9
E,b
272
%b
245
Q, 658
F\>
882
'Fb
357
A^b
770
M,b
743
C, I 156
JBbb
'Bbb
85s
D,b
68
'Ab
41
—
E\>\^
150
'Ebb
153
0,b
566
'G.b
539
—
'Abb
652
—
'C,b
1037
—
22
20
21
19
18
original plan in 1853, snpr^ p. 2766), and had
black digitals rising 0*4 inch ; ooL 7, or * domi-
nant Thirds, minor,* by italic small letten,
as d% , and had flat blue digitals rising 0*1 inch
Digitized by V^jOOQlC
B^CT. F. EXPERIMENTAL JUST INTONATION.
Pio. 68.
475
PBBSPBOnVB VIBW Of MB. P00I.|'B KBTBOABD.
7ta. 69,
m
Fio. 70.
1
h
Sot*
Ejl,
Do
2b
5 s|^
•0
V4
8m
n
p-Zk S« 1 ri 1 Do
IB mir T viwiw
• tf filQ U.U S:« ff:10 8:« if:U
SOVBLB »IA«Oai« •O&I.B.
T m w I ■ m IT T
t:ie U<U S:«
PLAN OF MB. POOLB'S KBTBOABD.
SECTION THBOUOH AB FIO. 69.
above the white keys, marked with heraldic
horizontal cross lines in fig. 68; col. *5, or
' Sevenths,* by antique Boman capitals, with
the index ', as p, and had red digitals, marked
with heraldic vertical cross lines in fig. 68, and
rising only 0*05 inch above the white digitals ;
col. '6, or * dominant Sevenths, minor,' had
antique small letters and an index as d', and
ycXUm keys marked with heraldic dots in
fig. 68, rising 0*15 inch above the white keys.
The length of a white digital for col. 5 being
taken as 4 inches, that of the black digital for
coL 6 was 3 inches, and that of each of the
coloured keys i inch. Fig. 68 gives a per-
spective view of this arrangement, with Mr.
Poole's names of the notes on the digitals for
the key of C major and some adjacent notes.
Fig. 69 gives a plan of this arrangement with
solfeggio names, except for two notes (pro-
nounced with ItaJian vowels), to shew that it
serves for any key beginning with a white
digital and a cross line A B * through the
centre of the third quarter (inch) of the key-
note.' Fig. 70 gives the solfeggio names of the
notes thus cut, with perspective views above of
the remaining parts of the digitals furthest
from the player. Underneath the solfeggio
names are the relative number of vibrations ^
of each note, taking do as 48. Below this
again are the numbers of the notes in Mr,
Poole's * triple diatonic* or trichordal scale,
with the ratios of their intervals, and also the
numbers of the notes in his * double diatonic '
or dichordal scale (see p. 344c, note *). And
finally the three last lines give the names of
the notes as he writes them, supposing the
first white digitals to give O (key of iff or one
sharp), C (natural key, where the mysterious
symbol may be meant for an n— that is,
* natural,' but seems to have been reversed by
the wood-engraver), and F (key of i b or one
fiat). Interpreted into our symbols, with the
interval in cents from the lowest note in each
line, these will be :
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476
ADDITIONS BY THE TRANSLATOR.
APP. XX-
rel. vib.
Kotes
in
fig. 70
cents
Solfeggio
colours
48
54
a
52i
60
ft.
64
72
d'
75
d,'n
80
84
90
96
9
'«!
c'
^\
'/'
'i'.
iT
c
d
'di
X
/
g
9i^
«i
'6b
c'
F
0
G
'0,
Bb
c
c^
rf.
'eb
<H
/
204
155
386
498
702
772
884
969
1088
laoo
Do
Be
Mi
Fa
Sol
La
Se
Si
Do
white
white
yellow
black
white
white
blue
black
red
black
white
If we took only the black and white digitals,
the arrangement of the keyboard would be
like Mr. Colin Brown's ; but this was an acci-
dent, Mr. Brown having never seen the draw-
ings of Mr. Poole's keyboard. Both arrange-
% nients arose from the column of Fifths in the
Decad (p. 4596) containing four, and the column
of Thirds oiUy three notes.
The great peculiarity of Mr. Poole's board,
where Mr. Brown differs from it entirely, is in
the introduction of the columns '5 and 6,
both containing natural Sevenths, and tiieir
amalgamation, as it were, with the col. 7
(which Mr. Brown alone uses) in three short
flat digitals, placed beside the black, and hence
of the same length. They are placed from
front to back, as red, yellow, blue— that is, as
two Sevenths and a Third, which belong to
three different keys. Thus in fig. 68, to the
left of the black digital for E^ (marked e), lie
three coloured digitals, i) the red 'jE7b (of which
the name is not marked in the fig.), the natural
Seventh of the tonic in the key olF\ 2) the
yellow 'Di (marked d'), the natural Seventh of
^ the dominant of the relative minor in the key
of C, which is e^ gfi 6, '(2, ; 3) the red D^
(marked (2]t ), the leading note (or major Third
of the dominant &, d^ /, 'a,) of the relative
minor in the key of Q, The situation of th^e
digitals is such that the lowest (or red) digital
gives the natural seventh of the white digital
immediately adjoining (below in the figure,
compare C : 'C, F : 'F). The middle (or
yellow) digital gives the natural Seventh of
the black digital, the right-hand top comer of
which touches its left-hand bottom caroa
(compare D^ : *i),, printed d d^). The upper-
most (or blue) digital gives a note which is a
small Semitone (24 : 25 = 70 cents) sharper
than the note of the white digital on the left
(compare D : Z).J , marked D : ds), and a
diatonic Semitone (15 : 16 or 112 cents) flatter
than the note of the black digital on the ri^
(compare D^S : JEr„ marked d% : e). The only
oigital placed out of ascending order from left
to right is the yellow one, which should have
come between the two white digitids for so
and re, but has been displaced from motives
of convenience.
Mr. Poole, as Mr. Colin Brown afterwards, provided only for modulations from
major keys into the dominant major, subdominant major, and relative mimn'.
For the modulation into the tonic minor, therefore, he had to flatten by a comma.
Thus C minor was considered to be the relative minor of E[} major instead of E^\^
major. And although, in deference to Mr. Listen and Gen. Perronet Thompson,
he also made a provision for temporarily introducing the tonic minor if desired,
giving col. 4 of the Duodenarium {Silliman, vol. xhv. p. 18, art. 35), he did not
require it himself. ' In the theory I have advocated,' he says, ' the major keys are
based on the first series of sounds,' p. 474c, col. 5, ' and the minor keys on the
Sixths of the major keys,' ibid. col. 6. ' That there must be such a relation and
order is inevitable.' But this does not exclude taking minor keys also upon notes
of col. 5 as a base, considered, if desired, as the Sixths of major keys on the not^
of col. 4 of the Duodenarium. Otherwise tonality is destroyed by constant shifts
of a comma severely felt on justlv intoned instruments.
[ Mr. Poole was £Jso aware of the alteration by a skhisma, and of the consequent
reduction of the number of pipes. He also refers to the 53 division, but he does
not seem to adopt either, and is not distinct enough on these points for me to
state his conclusions with certainty. In the duodenary arrangement, I have by
dotted lines marked the places where tlie skhisma comes into play, and by affixing
tiie cents to each note have shewn how it acts.
It will be seen that Mr. Poole had 100 notes to the octave, of which 39 arose
from the harmonic Sevenths. If the skhisma were neglected there would remain
only 36 tertian and 20 septimal, or in all 56 tones to the octave. The duodenary
arrangement has been taken from Mr. Poole's Enharmonic table (SiUinian^ xliv.
p. 13), consisting of 19 lines similar to the 3 at the bottom of fig. 70. He adds
the following example of the fingering of chords upon his keyboard, the double
numbers indicating ' that the key is touched with one finger and immediatelj
changed for another.* The duodenals and mark of the natural Seventh are accord-
ing to my notation, sect. E. art. 26, p. 465c. The upper figures refer to the Notes
which follow.
Digitized by V^OOQIC
«ECT. F.
EXPERIMENTAL JUST INTONATION.
477
C.
«3
A,.
— . ■ ■ ■ ^ M ^^ — —
"2sr
3
+
^* 3
3
I
+
4
2
+
3
I
+
3
I
+
3
+ 1
+
4
3*
1 +
3
I
+
4
I
+
34
I
+
i
3z^g>.gg^-g:^.^ I ^~^
?i:
S
321
m
=^=f3^
4 3
I I
+ +
4
I
+
34
I
+
3
I
+
2 3 4
I 2
+ +
3
I
+
* ^o^. — I. Sabdominant chord/ and a/.
* 2. Dominant with Seventh /.
* 3. Same with Ninth a! [not a/, and hence
causing a dnodenation into the dominant (?,
but forming the second chord in Poolers di-
chordal scale of C. Of course, ''f itself is not
in the duodene of G, but when these natural
hievenths are introduced the special marks are
used. See supri, p. 349a].
* 4. Dominant Seventh.
' 5. Dominant of the relative minor, the
iBeventh 'd, may be added [it is added here,
but to secure the intonation a duodenation
into the relative is marked].
'6. Subdominant with Seventh [duodena-
tion into F, therefore].
' 7. Grave second or Sixth of subdomi-
nant [as Uie duodenal gives the root F^ the
<2/' is sufficiently marked].
*8. The flattened note should be made
natural in the next chord* [meaning *next
chord but one,* namely at 8, so that there is
again a duodenation from i^ to C as marked].
The above examples will shew how Mr.
Poole treats the chord of the dominant Seventh
and the major Ninth. The three last chords
are added to shew his treatment of the chord
of the diminished Seventh (SiZZtman, vol. ix.
pp. 78-80). He considers the first of these
chords to be merely the chord of Q with the
dominant Seventh, </ 6/ d" '/', which is of
course in the major scale of G, but this lies
within the duodene of ^p as I have marked it, ir
including the ^F within that duodene, as shewn
in the daodenary arrangement p. 4740. Then
he supposes that in order to resolve the chord
a/ c" e/' (the last chord), the ^ is altered in
the second chord by ' a chromatic Semitone *
(that is, the small Semitone 24 : 25 » 70 cents),
to git , which is necessarily in the duodene of
JS7„ bat this gi % serves merely as ' a passing
note * to the following a\ and therefore, he
says, *must be thrown out when we reckon
the harmony.* But this will not explain the
present use of this chord, which is now intro-
duced without preparation, and as a means of
modulation. The ratios of the chord Mr.
Poole gives are 25 : 30 : 36 : 42, or taking
the ^,'8 an Octave higher, to compare with
my form, it becomes 30 : 36 : 42 : 50, that is
10 : 12 : 14 : i6§, or in cents o, 316, 583, 884,
Mr. J. Paul White (see below No. 9) makes the %
ratios of the chord 30 : 35 : 42 : 50, that is
10 : II* : 14 : i6f, or in cents o, 267, 583, 884.
The individual intervals in the first are 316,
267, 301, and in the second 267, 316, 301', so
that the two first intervals are transposed.
But in both the interval of the extreme notes
is 3 : 5 = 884 cents, so that in neither have we
a chord of a diminished Seventh at all, which
must have 919 or 926 cents. It is only equal
temperament which confuses the major Sixth
and diminished Seventh together by using 900
cents for either of them.
With regard to the double diatonic or dichordal scale, which Mr. Poole always
Bolfas as fah sol la se do re mi fah (where ae is the harmonic Seventh to do),
80 that do is the dominant, he says that ' the most beautifol, varied, and ornate
compositions are made from the elements it contains. It has the capacity in oer- «r
tain styles of music of using with much grace accidentals, or chromatics as they
are called ; for example, the si, the regular leading note to do, and the soljt, a
diatonic Semitone to below la, or the leading note to the relative minor ; these
chromatics always ascending a diatonic Senutone (15 : 16) to the notes above.'
In an example given he also admits se to be raised by 27 cents, that is to be the
regular Fourth of the triple or trichordal scale, and also allows the introduction of the
Sixth of this scale. Hence if we use the duodenary form and represent the dichordal
scale of F by capitals and these permissive additions by small letters we shall have
the scheme in the margin. This gives the trichordal
scale of C major complete with its grave second, and
gfjjj also one form of its relative il, minor complete, but
both without the harmonic Seventh of the dominants,
which of course he would be ready to add when the
harmony in his view required it. There is also the complete trichordal scale of F
major without the grave second. Hence his dichordal scale resolves itself into a
means of bringing these three scales into close connection, chiefly by help of the
Digitized by V^jOOQ IC
D
G
ii
G
JS?.
F
^1
bt>
^B\} d,
478
ADDITIONS BY THE TRANSLATOR.
APP. XX-
chord of the Ninth CE^G ^J?|>Z> in the above scheme. The example that he gives
of its use is the accompaniment to Figaro's Numero quindici from Rossini's
Barbieret afterwards smig as the air Ah I che d'amore by Ahnaviva. This is written
in G major. He gives the scale thus» using my notation and indicating accidentals
by small letters :
Double Diatonic Scale in G, with Accidentals.
G A ajjf Bx ^C, c cj D JE? JP,#
''t-ZT~T-^r-^^
The Dnodenals are mine, but as the Ninth
is not in a single duodena, it can be marked
only by giving the duodene containing all but
the natural Seventh and indicating that by a
sloping line in the usual way. The notes in
inverted commas are from Mr. Poole, except
the bracketed portions, which are mine.
* I. This may be «," [in that case the three
first notes are in the duodene of G].
* 2 and 3. These may be c" as well as V'
[in the latter case the whole run would be in
D, the c" being marked as V ; in the former,
the run would be in G].
* 4. This may be e," [this will be only if 3
« is c", so that the whole run is in (?J.
* 5. This note is clearly and necessarily 0".
[In this case 3 certainly should be V and 4
should also be e", but Mr. Poole does not ob-
ject to gf' a' 6/ c" d" e," followed by e", saying]
the enharmonic change from e" to ^', a rise of
a conmia, is often required, and I have proved
that it can readily be made, for my singers,
who know this change of a comma as well as
others know the Tone or Semitone, will give
it, even without accompaniment, with perfect
accuracy, as proved by the harmony after-
wards supplied as a test. All this variety
Mr. Poole has also devised an enharmonic keyboard pedal for the bass of his
organ, but then confines himself entirely to cols. 5» ^5, 6. The 'keynotes ' cor-
responding to the white digitals are in front in
order of f^hs from left to right. Behind them,
^ at a higher elevation, are the major Thirds lying
between them. The Sevenths are in a back row
behind their Fifths. This is indicated by the letters in the margin.
Mr. H. W. Poole is a native of Salem (afterwards Danvers, now Peabody),
Massachusetts, U.S., and is now Professor of Public Instruction in the Oovem-
ment Institute of the city of Mexico, whence he kindly wrote to me on 9 March
1885, describing one of his new keyboards. From this I take the following
summary and extracts : —
within the limits of musical laws — which only
forbid what is disorderly, complicated, or what
the ear will not distinguish — adds to the plea-
sure of vocal music, and it is the exact render-
ing of all the melodies and harmonies which
gives the charm to a good singer. [A little
difficulty arises as to tonality.] When acutely
perceptive of such accuracy, I had the good
fortune to listen to Alboni on all the occasions
when it was possible to do so. I thought her
then, am still of opinion, that she was the
best singer I have ever heard. It is certain
that she had a wonderful exactness in execating
whatever she undertook. There was no '* tem-
perament " in her scales, and what the strictest
theory requires in intonation she understood
and gave. She sang music whose analysis
would alarm a student with its apparent £ffi-
culties ; but the delighted auditors perceived
only a delicious and "easy " flow of melody.'
[This has been quoted at length to add to the
examples supr&, p. 325a.]
6. Mr. Poole says nothing about this (f*^
but I presume he would take it regularly as
v. In that case the whole of this would be
in D major.
red
black
white
7^b
^1
^-Bt>
^1
G
* I send you a stereograph of a simple form
of one of them [his new manuals] , which v\
easier to comprehend than a larger one with
48 levers to the octave. This may be called a
working model, and suffices for an organ or
pianoforte for instruction or study in effect
of chords and fingering. It is solidly con-
structed in wood, ebony, and ivory, and works
as fre<»ly as a common one. These 24 levers
are a quarter of an inch wide, and can pJay
a pianoforte with hammers half the common
width, with single strings, but larger and lightly
strained so as to yield the wifiTimTiTn of tone,
tension nearly to breaking-point giving bad
tone.' The &ager-key8 for each Fifth rise ^
inch, BO 0 is ^ inch above Ab> The white
digitals have the same shape as in fig. 69, but
from each projects a narrow black finger-kry.
Digitized by V^jOOQlC
IBCT. P. MR. BOBANQUET'S GENERALISED FINGERBOARD. 479
Krith a note one comma flatter, giving the (x x ^ inch each) was allotted.' They fill up
najor Thirds, and fitting into the left-hand tne space on the left-hand side of the black
lick of the next lower white digital. What- Thirds, and are of the shape of the white digi-
»ver white digital the player begins with, the tals in fig. 69, only very much narrower, half
ingeiing is the same, and for major scales the thin part being separated, and for ex-
mach Kke that for the key of A on the usual ample given to A.Jj^ , leading note to B^,
manual. For the Seventh and Ninth of the while the rest, including the wide part, is
Dichordal system separate digitals must be given to '^b, harmonic Seventh to Bb. * My
kouched. Mr. Poole can arrange for a minor keyboard admits of eiqual facility in execution
on the same tonic, but thinks it an extrava- and in taking the chords, with the common
ganoe. * The diatonic scales with the broad one of 12. I think its first utility will be for
ivory keys (larger than on the common board) teaching singing, accompanying violin players
are of first importance ; next the raised ebony and students of harmony. For this I recom-
digitals for Thirds. The Sevenths are well mend the simple form with less outlay of
provided for and convenient. The leading money.' This form of Mr. Poole's keyboard
notes to major Thirds are introduced as dia- is therefore equivalent to Mr. Colin Brown's
tonio Semitones below these black notes, and (No. 4) with the addition of the natural
eerve them as the black ones do the white. Seventh. V
On my model an equal space of two measures
8. Mb. Bosanquet'b Generalised Fingbbboabd and Harmonium.
Mr. R. H. M. BoBanquet's harmomuxn is partly described in the text, p. ^-iSc,
and its keyboard is figured and briefly explained in App. XIX., p. 429. In App. XX.
sect. A. art. 27, the nature of Mr. Bosanquet's cycle of 53 and his notation, and
the value of every one of his notes are explained. In App. XX. sect. E. art. 18,
there is an elaborate comparison of this cycle with just intonation giving the
number and pitch of every note, and, ibid, art. 25, it is shewn how such a cycle
might have been suggested by just intonation. In sect. G. arts. 16 and 17, the
methods of tuning the cycle of 53 adopted by Mr. Bosanquet and Mr. White are
described. In the South Kensington Museum, Science Collections, Room Q, the
harmonium itself may be inspected, Mr. Bosanquet having presented it to the
Museum, as he generally employs for his own use an organ with the same finger-
board, and two sets of pipes, one set for 48 notes of the temperament advocated
by Prof. Helmholtz (p. 432a), with perfect major Thirds and Fifths imperceptibly ^
flattened by ^ Skhisma, answering to the notes written with capital letters on the
digitals in the following plan ; and the other set for 36 notes of the meantone tem-
perament, brought into separate action by a stop. The pipes are stopped, with a
screw plug, so that they are more readily tuned.
It remains in this place to give the plan of the fingerboard, shewing the dis-
position of the notes upon it both for the 53 division and meantone temperament,
and to describe its arrangement, referring especially to App. XIX., p. 429, fig. 66.
In the present plan of the keyboard, all the digitals are represented as of the
same length, corresponding to that from tip to tip. This is 3 inches in the original
and is here only i inch. At each side runs a column of figures i to 12 con-
tinually repeated. It will be observed that in the first column the lines terminating
the oblongs come against 2, and that 2 is at the head pf the column. In this case
the end of each oblong gives a form of c, and in passing from one form to another,
as c to c^ we have gained a Pythagorean comma, which results from taking 12
Fifths reduced to the same Octave. In the g column headed; 3 the lines are ^
opposite 3 ; in the d column, headed 4, opposite 4 ; and so on ; each Fifth corre-
sponding to a rise of ]^ inch from tip to tip of the digitals, and to a vertical rise
of iV i^<)h &om level to level. Hence in going from one degree to another, as d
to c* or 4 to 5, we go backwards 12x^ = 3 inches, and rise 12 x 1^= i inch.
Mr. Bosanquet says {Mus. Int. and Temp. p. 20) : —
' The most important practical point about the keyboard arises from its sym-
metry ; that is to say, from the faict that every key is surrounded by the same
definite arrangement of keys, and that a pair of keys in a given relative position
corresponds always to the same interval. From this it follows that any passage,
chord, or combination of any kind, has exactly the same form under the fingers
in whatever key it is played. And more than this, a common chord, for instance,
has always the same form, no matter what view be taken of its key relationship.
Some simplification of this kind is a necessity if these complex phenomena are to
be brought within the reach of persons of average ability ; and witib this particular
simplification, the child or beginner finds the work reduced to the acquirement of
one thing, where twelve have to be learned on the ordinary keyboard/
Digitized by VjOOQIC
480
ADDITIONS BY THE TEANSLATOB.
APP. XI.
IT
2
PLAN OP MR. BOSANQUET'S GENERALISED KEYBOARD.
9 4 II 6 I 8 3 10 5 12 7
BLACK
3 10
BLACK
c>
10
IIS9
C
♦4
e
3
dbb
41
2
35
D'b
*9
76
8
cff_
"db
117
7
BLACK
^bb
08 S
152
*i3
d
d
193
A
12
d,
ebb
234
II
88
228
^'b
♦18
d'U
"di"
269
E\}
dn
eb
310
16
d.8^
A8
15
<i4
2?^b
23
d8 8
345
22
0
386
^1
♦21
fb
427
D,85
20
d8
88
421
2^1
27
08
462
»26
/
f
503
J^,8
= ^i
25
/.
^38
24
/t_
WHITB
G'b
-2^'8
32
/•8
08 8
538
G^b
-JP8
31
ft
«G'
36
9'
f8 8
656
f8
579
-^18
♦30 i
_/8_'
~gb
621
♦35
9
697
29
34
9i
abb
738
F,8 8
33
9-2
A*b
-G'8
41
Sr'8
f8
88
732
ii»b
= G8
♦40
g'8
ff8
773
Gf,8
39
y8
ab
814
0,Z
38
^i8
B'bb
45
98 8
849
a
890
^1
•43
bbb
931
42
a.
BLACK
925
S'b
♦49
a»8
a8
966
Bb
48
ja8
bb
1007
=B,b
47
«.8
^.8
=Bjb
46
C7«b
-B»
I
6«
a8 8
1042
Cb
53
b
b
1083
•52
2»i
Ob
1 124
51
C»
b8
1 159
C
«B,I
*4
B.8
3
dbb
41
B^
2
7
to
12
16
21
2S
30
34
38
43
47
52
3
7
to
to
to
to
to
to
to
to
to
to
to
to
I
6
10
15
19
24
28
32
37
41
46
50
1
The numbers in the oblongs and in the lines at the bottom of the Table are the nnmben
assiimed to the tones of Bosanquet's cycle of 53, in sect. A. art. 27, and sect E. art 18, and
shew their distribution on this keyboard, which was invented for playing them. The smaU
italic letters under the numbers at the bottom of each oblong are the transcriptions of one of
Mr Bosanquet*s names, of which all are given in sect. A. ait. 27, against the number ci
the note. The stars preceding the numbers shew those which constitute the duodene of C.
The notes of all other duodenes stand in the same relative position to their root- The capital
letters are 48 out of the 56 tones between the dotted and thick lines in the Duodenarinm,
sect E art. 18, and those following the sign = are other tones diflfering from them by a
skhisma, which are purposely identified with them, see sect. E. art. 25. These are the 48 tones
used by Mr Bosanquet for his organ, but he names them as at the bottom of each digital. The
small thick Roman letters at the top of each oblong are 36 tones of the meantone temperam«&i.
and the numbers below them are the cents in the intervals from c to these notes.
This keyboard is applicable to any intonation in which all notes, by the neglect
of the comma or skhisma, are reduced to one set of Fifths, no matter whether pe*-
fect or imperfect, as the flat Fifths of the meantone temperament. On referring
to the plan, we see how the 53 division is placed on the notes. Mr. Bosanqnct
Digitized by V^jOOQlC
SECT. P.
MR. PAUL WHITE'S HARMON.
481
finds it convenient to use 7 x 12 = 84 digitals, so that there are repetitions as
shewn by the figures at the bottom of the plan, each of the 12 columns contain-
ing 7 digitals. The position of the meantone notes is shewn by a small thick
Roman letter at the top of the digital Having 36 digitals at his disposal, Mr.
Bosanquet has used 36 notes of the meantone scale in place of only 27. They
are disposed in 12 rows with three digitals in each row. In the plan, thick lines
limit tiie three digitals thus placed at the disposal of the meantone notes, and
nnder the name of each note is inserted the number of cents in the interval be-
tween it and c. It will thus be seen that each note differs from the one above it
in the plan by a Didsis of 41 cents. Thus the first row has bjf 1159, c o = 1200,
dt^ 41. Also each digital lies against two to the right and two to the left. The
apper one to the right is 76 cents or a small meantone Semitone higher, the
Lower is 117 cents or a great meantone Semitone higher. The upper one to the
Left is, on the contrary, 117 cents lower, and the lower one to the left is 76 cents
Lower. The sum of the two Semitones is 193 cents = ^(204 + 182), a Meantone, f
uid their difference 41 cents a Diesis.
The use of this fingerboard is easily acquired by any pianist, the fingering for
idl major keys resembling that for A major on ordinary instruments.
9. Mb. J. Paul White's Habmon.
Mr. James Paul White, of Springfield, Massachusetts, U.S. America, a tuner
t)y profession, having been much impressed by Mr. Poole's papers in Silliman*s
Tou/mal, cited under No. 7, determined to realise them so far as possible by means
)f the 53 division of the Octave. Now on examining this division by the tables in
sect. A. art. 27, and sect. D. table i, we find that the number of degrees by which
uiy interval is represented can always be expressed by multiples, or the sums or
lifferences of multiples, of 2, 5, 7 (which may therefore be called iridices), as in the
following table : —
Name of Interval
ItsCenta
Bepresented In the 53 dlylsion by
IT
Cents
Degrees
Comma
221
24/
Pythagorean Ck>mma
23
I
=2x4-7
Great Diesis . . . .
43
n
2
= 2
Small Semitone
70
3
=5x2-7
Limma
Greater Limma
92/
91
4
= 2x2
Diatonic Semitone .
112
113
5
«5
Minor Tone ....
182
181
8
= 2x4
Major Tone ....
204
204
9
■5+2x2
* Snpermajor Tone .
231
226
10
= 5x2
t Snbminor Third
267
272
12
«7 + 5
Pythagorean minor Third
294
294
13
=7+2x3
Jnst minor Third .
316
317
14
«7X2
Just major Third .
386
s
17
-.7 + 5>^a
Pythagorean major Third ' .
408
18
-7x2+2x2
* Supermajor Third .
III
430
19
-7x2+5
f
Foturth
498
22
=7+5x3
JastTritone • . . .
590
589
26
=7x3+5
Pythagorean Tritone
612
611
27
=7+5x4
Grave Fifth . . . .
680
679
30
-5x6=7x5-5
Just Fifth . . . .
702
702
31
= 7x3 + 5'<2
Pythagorean minor Sixth
792 \
794/
793
Extreme Sharp Fifth
35
«7x5
Jnst minor Sixth .
814
815
36
=7x3+5x3=53-5x2-7
Jast major Sixth .
884
883
39
-7x5+2x2-53-7x2
Jnst diininished Seventh
• Supermajor Sixth .
926\
933/
928
4X
= 53-7-5
Jnst snperflnous Sixth .
951
42
-7x6
t Snbminor Seventh .
Extreme sharp Sixth
Minor Seventh
976}
974
43
'-53-5x2
996
996
44
=7x6+a
Acate minor Seventh
1018
1019
45
= 5x9=53-2x4
Jnst major Seventh .
1088
1087
48
= 53-5
Pythagorean major Seventh .
mo
1 109
49
= 7x7-53-2x2
Octave ......
1200
1200
53
-7x4+5x5
?Ie
ninitboH '^>' ^ -»00
482 ADDITIONS BY THE TEANSLATOR. app.xx.
It is really Biirprising how accurately the intervals of jnst intonation are ihtis
represented. Only those marked * and t depending on the 7th harmonic are
5 cents too flat, or sharp respectively, which is barely perceptible.
Influenced no doubt by such a calculation as the above, Mr. Paul White con-
ceived and executed a fingerboard of which the typographical plan below will
give some conception. And this conception will be much improved by drawing
pencil lines on the diagram parallel to the rows of figures sloping up (as 48 i 6
II 16 21 26 31 36 41 46) and down (as 41 48 2 9 16 23 30 37 44) to the right.
These lines will divide the plan into a number of parallelograms or irregdar
lozenges, each of which represents a digital of nearly the same shape, but fitting
loosely into its place. These are pieces of wood all diamond-shaped and all of
the same height, variously marked to assist the player, and all bearing upon them
the numbers printed in the plan. The typographical plan is, of course, only
approximatively correct. In reality the vertical lines are not quite vertical, and the
lines parallel to the numbers differing by 12 as 39— 51 — 10— 22— 34— 46— 5 at
U the top (which may be connected by pencil lines as just shewn) are more nearly
horizontal or rather slope slightly downwards instead of up. But as the design
has not been published, it was desirable to give only a conception and not an
accurate plan of the arrangement with the curious slopes of the actual lines.
TYPOGRAPHICAL PLAN OP MB. J. PAUL WHITE'S FINGEKBOABD.
II39 . 5
||27 46 12
!|iS 34 83
22 41 "^^ 7 ■ ,
10 29 48 14
51 fi7 36 2
H 39 5 24 *t43 *t9
46 12 t31 50 x6
t53 19 38 4
41 ~° 7 *26 45 "
4^ 14 33 *52 18
2 ♦21 ♦« 6
43 *8 28 47 *I3
50 16 ♦35 1 20
♦4 23 42 8
45 . n *30 **9 15
52 *18 37 3
6 25 44
f 47 *i3 32
I 20 39
8 27
49 15
3
The thick figures represent white digitals and serve as land-marks, forming the
53 c, gdy 18 e, 22/, 27/tt, 31 gr, 40 a, 496.
The numbers marked * are the same as those in the plan of Mr. Bosanquet's,
supr&, p. 480, and represent the duodene of note 4. The numbers marked % or
53» ^7» 3^> 43»9» represent the chord of the major Ninth c «, gr^i[> d.
In the columns the numbers as they proceed from top to bottom increase
by 2, of course taken in the reverse direction they decrease by 2. When necefisary
53 is subtracted or added here and elsewhere, as the numbers must not exceed S3-
Digitized by V^JiOOQlC a
SECT. (I. ON TUNING AND INTONATION. 483
In the lines which slope wp to the right the numhers increase upwards hy 5, and
of course decrease downwards by 5.
In the lines which slope down to the right the numbers increase downwards
by 7, and of course decrease upwards by 7.
It is thus seen that the three indices 2, 5, 7 are represented by nearly vertical
and sloping lines, and it becomes easy by the preceding table to pick out any intervaL
Thus to take the just minor Sixth from 18 (one of the thick figures) ; we have by
the table, 36 =7x3 + 5x3, so that we go down 3 steps on the line of 7's, and then up
3 steps on the line of 5's, and thus reach i, the right degree for 18 + 36=54=53 + 1.
!Biit in the table we also find 36=53 — 5x2 — 7, hence we may also go down
2 steps on the line of 5's and then up 1 step on the line of 7's, reaching i as before,
but not the same i. It is now the Octave below, and if from this new i, we de-
scend 4 steps on the line of 7's and then ascend 5 steps on the line of 5's, we reach
the old I, for 7x4 + 5x5=28+25=53, or the Octave.
The body of each digital is a block of wood 2^ inches high and not far from ^
I inch square on the top. The grain of the wood is vertical so as to feusilitate the
action of the key on its two steel guide pins, which are driven firmly into a board
as wide as the manual. The valve is opened by a pin under the key in the usual
way. Of course the fingering is entirely different from ordinary fingering, but is
the same in all possible keys. Contrasting his board with Mr. Bosanquet's, which
he admits is admirable and would be probably regarded with more favour by
musicians than his, Mr. Paul White (in a private letter to me) says i) that his
board combines the advantages of Mr. Poole's with Mr. Bosanquet's, and has
digitals of a simpler construction than either, shewing also the Pythagorean tones
conspicuously for every note, and having the complete cycle of tones. 2) The
chords are all easy for the fingers, including those depending on the 7th harmonic.
3) Digitals differing by one comma are far apart, so that there is no danger of
playing too sharp or too flat by a comma. The fingers can easily make the just
chords, but to make them false by a comma is difficult. 4) This fingerboard can
be made more compact than any other. The extreme width of the present in-
strument (the third made) is only 1 1 inches, or twice the ordinary width. 'If
Mr. Paul White uses only 56 digitals to the Octave, Nos. 15, 27, 39, marked [|,
being the only repeats. He was kind enough to send me two photographs of his
instrimient, (which he calls the Harmon, and which he constructed almost entirely
with his own hands,) one giving a bird's-eye view of the digitals, and the other their
connection with the rods that open the valves. He has as yet not arranged any
system of notation and does not himself play on his instrument from notes.
SECTION G.
ON TUNING AND INTONATION.
(See notes pp. 256, 287, 31 1, 325.)
Art. Art.
1. Difficulties of tuning, p. 483. 10. The ' Tuning Octave * for harmonium,
2. Specimens of tuning in meantone tempera- organ, and piano, p. 488.
ment, p. 484. 11. The Translator's Rule for tuning in equal f
3. The Fifths and major Thirds in the same, temperament beginning with c', p. 489.
p. 484. 12. Modification of the same for beginning
4. Specimens of tuning in equal tempera- with a', p. 489.
ment, p. 485. 13. Proof of the Bule, p. 490.
5. Examination of the Fifths and Fourths in 14. Bule for checking the tuning of Octaves,
four of the same, p. 485. p. 491.
6. Violin Intonation according to the observa- 15* The Translator's Bule for tuning in mean-
tions of Comu and Mercadier, p. 486. tone temperament, p. 491.
7. Observations on the same, p. 487. 16. Mr. Bosanquet's method of tuning of the
8. Scheibler's method of tuning, p. 488. 53 division, p. 492.
9. The Translator's approximative method 17. Mr. J. Paul White's two metliods of tuning
and counting beats, p. 488. the same, p. 492.
Art. I. — We have seen in sect. E. that just tertian harmony requires the dis-
crimination of 1 17 different tones within the Octave. They all indeed depend upon
just Fifths, Fourths, Thirds, and Sixths. But very few ears could be trusted to
tune a succession of perfect Fifths and Fourths. Herr G. Appunn told me that it
cost him an immense labour to tune 36 notes forming perfect Fifths and Fourthsj
Digitized by WQOQ IC
484
ADDITIONS BY THE TRANSLATOR.
APP. XI.
upon an experimental harmonium, and he had the finest ear for appreciating
intervals that I have ever heard of. The accumulation of ahnost insensible into
intolerable errors besets all attempts to tune by a long series of similar intervals.
Even Octaves are rarely tuned accurately through the compass of a grand pianoforte.
But for major Thirds and minor Sixths there is no chance at all (except by a real
piece of haphazard luck) to get even one interval tuned with absolute correctness
by mere appreciation of ear. Hence to attempt to tune the Duodenarium of
sect. E. art. 18, p. 463, merely by Fifths and major Thirds is quite hopeless.
But if we cannot tune just intervals with sufficient correctness, how can we expect
to tune all the variously tempered intervals mentioned in sect. A.. (and these are
only a few of the most important) sufficiently well to discriminate their qualities
and appreciate their merits ? No ear knows d priori what result it has to expect,
or has any means of judging whether the result obtained is correct. It follows
that all attempts to tune by ear must have grievously failed, wherever they de-
pended upon considerable alterations of just intervals, and that even the laborious
IF and careM training of modem tuners for obtaining the very slightly altered Fifths
and Fourths of equal temperament can only lead tibem to absolute correctness ' by
accident.'
Art. 2. — To ascertain whether these theoretical views were confirmed in prac-
tice, I have made some observations on the tuning of the old meantone and the
recent equal temperaments. It is easy (from the data in sect. A.) to determine
the cents which should be contained in each interval, and (by measuring the actual
pitch of each note with the forks described on p. 4466')* ^ ^^^ what the interval
obtained in any particular case really is. For brevity I give only the names of the
notes in the octave, and the interval in cents from the lowest note. But every
such figure is the result of a careful observation.
Line i in the following Table gives the
theoretical number of cents.
Line 2 gives the cents observed on a pitch-
pipe of 1730 belonging to the bellfonndry
Golbaochini at Padua, blown with the least
^ force of wind possible to bring out the tone,
on an organ bellows at Mr. T. Hill's, the organ-
builder's.
Line 3 gives those observed on another
pitch-pipe of 1780, belonging to the same and
similarly blown. The mean value of a' on
both was 425*2 vib.
Line 4 gives those observed on accurate
copies of a set of timing-forks (V and &' miss-
ing) belonging to the beUfoundry Gavedini at
Verona, supposed to be a century old, and pre-
served with great care, having a' 423-2.
Line 5 gives the cents from an octave oi
pipes on Green's organ at St. Katharine's.
Begent's Park, from the pitches determined b;
me in 1878, up to which time it was one oi
the few organs tuned in meantone tempera-
ment. Of course in this case the tuning was
modem.
Specimens of Tuning
in Meantone Temperament.
Notes
I
c
0
CZ
D
E\>
E
F
n
G
n
A
B\>
B
c
76
193
310
386
503
579
697
in
890
1007
10^3
1200
2
0
102
216
308
414
516
619
723
827
920
103 1
1 126
1232
3
0
134
236
329
443
561
626
701
814
900
982
I07S
"75
4
0
"7
229
301
439
507
633
733
864
934
1042
—
5
0
72
198
300
3«2
509
586
699
78s
898
IQ20
1096
1209
Art. 3. Meantone Fifths, if properly tuned,
should have 696*6 cents, and the major Thirds
386*3 cents. The old tuners did not use the
Fourth in tuning, but took, for example, d to
(f^ then ^ to d'\ and then the Octave down to
d\ thence the Fifths d! to a\ a! to e", and the
Octave down to e', so that c' to e' ought to be
a just major Third. The Fifths and major
Thirds actually obtained can be calculated
from the Table in Art. 2 by subtraction, taking
care to increase the minuend by 1200 when it
is less tiian the subtrahend. We thus find the
following values. The figures placed between
the names of any two notes give the cents in
the interval between them, which, neglecting
decimals, should be 697 for Fifths, and 386 for
major Thirds.
Line 2. Old Pitch-pipe. Fifths: J?b723Bb
685 F 684 C 723 G^ 693 D 704 4 694 JB 712 B
693 -^ 683 c« 725 <n •
Major Thirds : E\> 415 (? 403 B ; Bb 385 D
403 1?^ ; F404 ^ 382 CI ; C 414 -B 413 Gl -
Line 3. Another old Pit<^-pipe. Fifths:
E\> 653 Bb 779 ^ 614 C 701 G 735 D 664 J
743 E 632 B 751 Ft 708 Cf 780 Ql .
Major Thirds: JSb 372 G 374 B; Bb 454 1>
390 Fl ; ^ 339 ii 434 C« ; C 443 ^ 471 G5 .
Line 4. Old Forks. Fifths : £7b 741 fib
665 JP 693 C 733 Gf 696 D 705 A 705 K\
(B missing) ; ^ 684 C]( 747 <it •
Major Thirds : J?b 432 O ; (B missing) ; fib
387 B4042?'«;lr'427 .438305; CaV^E^^(^.
These old tunings are very imperfect. BoUi
Fifths and major Thirds would make dreadful
harmony. The forks are if anything
than the pitch-pipes.
Digitized by V^jOOQlC
^TECT. O.
ON TUNING AND INTONATION.
435
Line 5. St Katharine's Organ. Fifths: Eb
^20 JBb 6891^7000 699 G 699 D 700.1 684 -fi?
^ 14 B 690 i?Tr 686 Cff 713 On .
Major Thirds : ^b 399 G 397 B ; Bb 378 D
3S8 Ft; Fz^A 374 C« ; C 382 ^ 403 Gfff .
Thia modem tuning is better, but not good.
Art. 4. — It takes a quick man three years to learn bow to tune a piano well
ixL equal temperament by estimation of ear, as I learn from Mr. A. J. Hipkins.
Timers bave not time for any otber metbod. Tbe following are good examples : —
liine I. The theoretical intervals, all ezaot
bundreds of cents.
liine 2. My own piano, tuned by one of
JBroadwoods* usual tuners, and let stand un-
used for a fortnight.
liines 3, 4, 5. Three grand pianos by
Sroadwoods' best tuners, prepared for examin-
flktion through the kindness of Mr. A. J. Hip-
luns, of that house.
Line 6. An organ tuned a week previously
t>y one of Mr. T. Bill's tuners, and used only
The Fifths, which should be a quarter of a
comma 5*4 cents flat, are often sharp; the
major Thirds are unequal. But the errors
here are not more than might be reasonably
expected from tuning by ear.
once, examined by the kind permission of Mr.
G. Hickson, treasurer of South Place Chapel,
Finsbury, where the organ stood.
Line 7. An harmonium tuned by one of
Messrs. Moore A Moore's tuners, kindly pre-
pared for mj examination.
Line 8. An harmoniimi, used as a standard ^
of "pitch, tuned a year previously by Mr. D. J.
Blaikley (p. 97^), by means of accurately
counted beats, <&c., with a constant blast, put
at my disposal for examination by Mr. Blaikley.
Specimens of Tuning in Equal Temperament.
Notes
<^
cz
D
200
300
E
400
F
Ft
G
('^
A
At
B
c
I
0
100
500
600
700
800
900
1000
1 100
1200
2
0
96
197
297
392
49«
S90
700
797
894
990
1089
1 201
3
0
99
200
305
4x1
497
602
707
805
902
XOO3
1 102
1206
4
0
100
200
300
395
502
599
702
800
«97
999
IIOO
1200
5
0
lOI
199
299
399
500
598
696
800
899
999
1 100
1200
6
0
lOI
192
297
399
502
601
702
806
898
1005
1099
I20I
7
0
98
200
298
396
49«
599
702
800
898
999
1099
1199
8
0
100
200
300
399
499
600
700
800
900
1001
1099
X200
These were all tuned by the modem way of
Fifths up and Fourths down, and the object is
to make the Fifth up 2 cents too close, and
the Fourth down 2 cents too open. As this
interval of 2 cents lies on the very boundary of
perception by ear, the difficulty of tuning thus
without attending to the beats is enormous.
The above figures in lines 2, 3, 4, 5 shew how
very close an approximation is now possible in
pianofortes.
Art. 5. — The order of tuning differs in dif-
ferent houses. Messrs. Moore & Moore's tuners
set </ by a c'* fork, and then tune in order :
tf g d' aef bft dt gt d't • Then begin again
and go on as c' / 6 b e'b. The proof of the work
is that e'b and d% are identical. Messrs.
Broadwoods' tuners also set & from c, but then
proceed thus : cfgd'ae'bfZ c'% gt d*Z atfc^,
the proof being that the final agrees with the
initial cf. In this case a%f is taken m bbf
f
or at et t that is a Fourth down. Observe that
the tuning in both cases takes place in the
Octave /to/, for which the beats of disturbed
Fifths and even of disturbed Fourths are very
slow. This arises from the great prominence
of the second partial tone in this region on
pianoforte notes. In taking the pitch of each
note, I found that d', d't , e' taken as disturbed
unisons, beat with my forks much less dis-
tinctly than /, ft , &c., to eft fts disturbed
Octaves. Now the above table enables us to
calculate the cents in the Fifths and Fourths
actually tuned, which were the intervals esti-
mated by ear. I take only line i as containing
the theoretical intervals, and lines 2 to 5 as
being by Broadwoods' tuners, so that the order
is certain. The numbers of cents placed be-
tween two notes shews the interval, all the ^
Fourths being ti^en down and the Fifths up.
Pianoforte Tuning — Fourths and Fifths.
I c'500 0700 d'soo ayoo e'500 6500 /« 700 c'ffsoo (75700 d'JJ 500 ©5500 /700 c'
2
500
697
503
1698
3
493
693
498
1709
4
498
698
503
;698
5
504
703
500
I700
503
499
706
499
700
507
497
703
509
506
697
594
708
502
506
709
495
501
701
500
700
50X
*97
698
500
502
703
501
700
SOI
49«
700
These examples must probably be con-
sidered the best that pianofoi-te tuning by ear
can accomplish. But even in line 5, which is
the best, there are only five intervals abso-
lutely correct, two others are only an inappre-
ciable I cent in error, two are a just appreciable
2 cents wrong, two are 3 cents out, and one
wrong by the very pjerceptible interval of 4
cents. Now if this is the work of a clever
tuner in constant practice for many hours daily
for many years, in tuning one kind of tem-
perament only, what are we to expect from
those who attempt to realise new intervals ?
Digitized by
Google
486
ADDITIONS BY THE TRANSLATOR.
APP. XX,
Art. 6. — ^Tlie vocalist does not, properly spealdng, tnne at all. It is with him
a matter of ear, that is, sense of pitch, which guides the muscles to alter the tension
of his vocal chords, and make them produce tones of various pitch. The ease and
rapidity with which this can b^ done are matters of careful training, followed by
long practice. They can never be acquired by those who have not the proper
cerebral organisation. The extreme mobility of the voice and the difficulty of sus-
taining a pitch, or of exactly reacliing it again after a pause, throw great impedi-
ments in the way of testing unaccompanied singers. The habit of choral singing
leads to just intonation (App. XVIII.), but an accompanying instrument is quite
sufficient to lead the voice astray (p. 207, note f). Hence I pass by voices alto-
gether. The violinist apparently tunes only four strings to make three perfect
Fifths, and in doing so he is assisted by an audible combinational tone, which should
be just one Octave below the lower note of the pair he is tuning, and hence a Twelfth
below the upper note. But he really tunes every note in the compass of his instro-
ment g to e"", by his method of stopping, as much as the pianoforte tuner by
H increasing or diminishing the tension of his strings. And according to the
' school,' he should tune the notes in equal temperament. How then does be
tune ? Or, to put the question in more usual language, what intonation does he
use ? Messrs. Comu and Mercadier (as mentioned on p. z^S^> ^^^^ '^) instituted
a series of experiments on voices and organs by the phonautograph {Comptei
Rendus, vol. Ixviii. pp. 301 and 427), and on violins and violoncellos by means of a
tin plate placed under the bridge, which was connected with a wire that conducted
the vibrations to the inscribing style (see Comptes Benchis, 17 July 1871, vol.
Ixxiii. p. 178, from which, and vol. Ixxiv. p. 321, vol. Ixxvi. p. 432, 1 obtained the
data which I have here reduced to cents). I give the results only for different indi-
vidual pljiyers on the violin and violoncello. Some of the scales are fragmentary.
I prefix the just, Pythagorean, and equal number of cents for comparison. The
scales are first major and then minor. 'The root is omitted as unnecessary. The
numbers in one line refer to a single trial
Scale of G Major.
Notes
J>
498
0
A \ B c j
Notes
J>
B
F a \ A
i^ 1 < 1
Jnst oents
204
702
884'io88,i20O
Just oents
204
386|498i702!884
I088I200
Pyth. „
204
408 498 702 906 1 1 10 1200;
Pyth. „
204408 498 702 906 1 1 10 1200
Equal ,♦
200
400*500
700900
701
1 100
1200
Equal „
200
400500
700
900! 1 100
1200
VioUn
401 505
Violin
41149^
Amateurs
212
4121498,709
Professional
417495714]
212
399,499710'
(M. Leonard,
417
198
396 1708
Belgian)
411
2iO;4o6'5o5'7o2
207
910
1 128
188 401 1 I702
216
912
1 122
201
4064901711
II28
1
411
490I 1
207
407
II23
201
396
495 70S
411
194
411
411
404
704
207
401
401
401
899
199
407
696
209
404
1
1
Violoncello
415
710,
Violoncello
393
487702
Amatenrs
39S
1
Professional
407I 1
208397
507
710
(M. S^ligmann)
419,496,716
408
411
1
411
399
491 702
.401
712
496,
407
705
414
695892
404
1
407
696896
204
415
5"
701
1 109] 1 202
403
697907
201
5"
702
1 105'
i98;4i5;5oo
I
j
207'4i4J498j7oi| '11201202
202'403'499j7i2j >iii7i20i
1
Digitized by VjOOQlC
SECT. O.
ON TUNING AND INTONATION.
Scale of C Minor.
487
Kotea
1^
e\,
/
9
ab
b
1200
{ Notes
d
eb
1/
9
702
814
b
1088
1200
Just cents
204
3161498702
8141088
Just cents
204[3i6;498
Pyth. „
204J294!498|702
792 1 1 10
1200
Pyth. ..
204I294 498^702
792
mo
1 200
Equal „
aoo
300
500
700
697
800 I IOC
1
1200
1208
Equal „
200
300 500 700
295494700
80c
8x0
IIOO
1200
Violin
292
Violin
199
Amateora
281
708
Professional
2071311499
701
793
i"3
295
702
I2IO
(M. Ferrand, of
2041298503
698
793
1116
1201
295
the Op^ra
206
306492
700
794
294
1 196
Gomique)
204
298;
301
70s
1206
298
702
1203
202
300
701
196
300
s
195
292
707
292
303
705782
783
709
802
III8
II96
696798
nil
1209
712
704
It
III8
IIOI
1 197
712
783
1 102
Art. 7. — Messrs. Cornu and Mercadier conclude finaUy (ibid. vol. IxxvL p. 434)
that :—
* Musical intervals belong to at least two different systems of different values :
* i) The intervals employed in melodies which have no modulations agree
with those of the Pythagorean scale. ^
* 2) The intervals between two notes sounded together in chords, the basis of
harmony, have for their ratios the following numbers : 2 for Octave, ^ Fifth,
J Fourth, f major Third, 4 minor Third, f major Sixth, | minor Sixth, and
J Seventh, where the Fourtn and Sixths were deduced firom observation of the
Fifth and Thirds, and the Seventh from the dominant chord.'
Thus for nnaocompanied harmony of two
tones (chords more than two tones were not
tried) just intonation dUme was used. For
melody the major Thirds, major Sixths, and
especially the major Sevenths (leading notes)
were much sharpened, and the minor Thirds
and minor Sixths generally much flattened.
Bat did this arise from the custom of equal
temperament? (as M. GuSroult thinks, i6u2.
9 May 1870, vol. Ixx.p. 1037, to which Messrs.
Coma and Mercadier replied, on 30 May 1870,
vol. Ixx. p. 1 1 70) or really from the feeling of
Fifths ? The latter was impossible for the lead-
ing note, which is sometimes much sharper
than in Pythagorean intonation, and the Fifths
played were by ho means always true. Messrs.
Coma and Mercadier say that the divergence
from the mean only reaches ^ of a comma,
that is, about 7 cents; but as the Pytha-
gorean major Sixth, major Third, and major
Seventh differ from the corresponding equal
tempered intervals by only 6, 8, and 10 cents
respectively, this uncertainty renders it im-
possible to decide whether the scale played
was intentionally equal or Pythagorean, or
whether even it did not vary with the feeling
of the moment. Taking into consideration
that the pitches actually shewn in the tables
vary considerably, that they very rarely repeat
themselves, that the notes are sometimes
flatter and sometimes sharjier than either just
or Pythagorean intonation, and that this un-
certainty pervades even such intervals as the
Fourth, Fifth, and Octave, I am inclined to
adopt the hypothesis of an intentionally vari-
able intonation. Whether founded on the
feeling of Pythagorean or equal temperament,
it is cUfficnlt to decide. But it is certainly not
founded on any feeling of just intonation for
harmony. If then these players, as Messrs.
Cornu and Mercadier assert from first to last
in the unmistakable terms already cited, adopt
just intonation of intervals for harmony, a 5[
serious question arises as to how they treat
the relations of tonality. The first part may
lead and the others may be adapted to it, or
the bass may determine the intonation of the
other parts. In either case there would be ^
great variability, through which modulation,
and even the adjustment of parts without
much previous combined practice, would be-
come extremely difficult; see pp. 208c and
note *, 324^), c, d. But how about the return
to the same key after modulations (p. 3286) ?
Huyghens (Cosmothedros, lib. i. p. 77, as
cited by Dr. Smith, Harmonics^ 2nd ed.
p. 228) suggests that as * erring from the pitch
first assumed . . . would greatly offend the
ear of the musician, he naturally avoids it by
his memory of pitch, and by tempering the
intervals of the intermediate sounds, so as to
return to it again.' But ho^ accurately does
Digitized by V^jOOQlC
488 ADDITIONS BY THE TBANSLATOB. app. xx.
he remember the original pitch ? In some a series of modulations be noticed ? We knofw
oases in the above observations of Coma and indeed that unaccompanied singers constantly
Mercadier the Octave (and hence the original * flatten * by much more than a comma. The
pitch) will be seen to be sharpened or flattened Duodenarium simply shews what sounds ought
by more than 2 cents. In the course of a few to be played in modulations, and would be
modulations this 2 cents might easily become played on instruments with fixed tones pio-
22 or a comma. Where is the guarantee for perly arranged ; not what intervals are nom
remembering the original pitch? Would an played and sung, and mere memoj^ of ear
alteration of a whole comma in passing through does not suffice.
Art. 8. — Sclieibler*s method of timing instruments was theoretically perfect.
It consisted in tuning by his fork tonometer a series of forks 4 vib. flatter than the
pitches required. The string, pipe, or reed was then tuned sharper than the fork by
4 beats in a second. (See p. 4466 for Eoenig's forks for this purpose.) This only
applied to one octave, and perhaps the octave below ; the others were tuned from
them by estimation of Octave. The errors thus made were a minimum, but
^ there was the obvious disadvantage of having to tune a new set of forks for each
pitch desired. This Scheibler overcame by tuning auxiliary octaves on organs, ajid
counting beats by rather a troublesome rule, entailing the need of an accurate
metronome. Even then he only taught how to tune in equal temperament. For
practical purposes we require not only the equal but the meantone temperament,
and also the 53 division of the octave. The only sure way is by calculating the
pitch number of each note, and thus determining the beats between the note and
the forks of a tuning-fork tonometer. This method may be dismissed as generally
impracticable. 80 may any method which depends upon accurately knowing the
pitch number of the tuning-note. What is required is a method of tuning at any
unknown pitch which an organ or piano may happen to possess at the moment
within the limits of, say, 0^256 and c'270 without determining exactly what that
pitch really is. This would save the great trouble of entirely altering the pitch
of the piano (never very certain in its results), and the still greater trouble and
expense of enturely altering the pitch of an organ or harmonium.
Art. 9. — For this purpose I invented an approximative method (given on p. 785
IT of the ist edition of this translation and subsequently communicated to the
Musical Times), which I here subjoin in an improved form. It is based on the
result of Prof. Preyer's investigations (supn^ p. 147^^') that errors of ^ or '2 of a
vibration cannot be heard in any part of the scale, so that any attempt to tone
more accurately is labour thrown away. Moreover, even at high pitches '3, *4, and
•5 vib. are scarcely perceptible in melody and quite inoffensive in harmony. It
will be found very difficult when the beats are less than 4 in 10 seconds, that is,
when the error is less than '4 vib., to count them with any approach to accuracy.
But it is only by beats that we can work effectively.
Any one who undertakes tuning should only, there are 10 doable swings in 10 seeonds.
learn to estimate the meaning of 6, 10, 15, 20, By watching and counting this, say for half &n
30 beats in 10 seconds. This is best done by hour, the tuner will learn to feel the rate of
short pendulums eonstructed of a piece of these two sets of swings. Then make another
thread with one end tied to a curtain ring, and pendulum with a length of string of 4} inches
the oth^ passed through a sUt in a piece of measured as before. This vibrates much more
firewood, round which it is ultimately tied, quickly, making 180 single and hence 90
% The stick is put under a book by the side of a double swings in 60 seconds, and consequently
table, so that the pendulum swings freely. 30 single and 15 double swings in 10 seconda.
Measure the length of the string from the Finally make the length 27^ inches, always
centre of the ring to the beginning of the from tiie centre of the ring to the stick, and
wood, which aUows of an easy alteration of the pendulum will make 6 'double and 12
the length by drawing the string through the single vibrations in 10 seconds. Bemember
slit. Make this length 9| inches. The pendu- that if you begin counting with one, you will
lum swings backwards and forwards 120 times end with seven for 6, eleven for 10, uxteen
in 60 seconds, and hence 20 times in 10 for 15, and so on, so that you will always have
seconds. Adjust it more accurately by a to throw oft one from your count,
seconds w«tchl Counting the swings one way
Art. 10. — The rule has to be arranged in several £orms according to the custom
of tuning instruments. Harmoniums are best tuned from c* to c", that is, in the
two-foot octave. Organs are- generally tuned in the principal stop, so that on
touching the keys from & to c", the sounds are from c" to &'\ in the one-foot
octave, and hence the beats are twice as fast. But for pianos it is the custom to
tune from/ to/ (see art. 5). Most tuners in England begin with &\ from which
Digitized by VjOOQlC
BBCT. O.
ON TUNING AND INTONATION.
489
a' is ' set/ and then the tuning commences. Some tmiers in England and all
al)road begin with a!. This makes no difference in the rule, provided the tuning
octave remains the same.
Absolately the beats arising from imper-
fect Fifths and Sixths vary for every difference
of pitch of the lower note. As the Fifth is
always too dose and the Fourth too open, the
reader can find the beats from the numbers in
tlie table, p. 437c, <2, by subtracting twice the
larger from three times the smaller pitch
nximber for Fifths (thus d \ ^ in col. ii. of the
table is 258*6 : 387*6, whence 3x258*6— 2 x
387*6 = 775*8 -775*2 -'6, giving 6 beats in 10
seconds), and four times the smaller pitch
from three times the larger for Fourths (thus
^ '.d'" 387-6 : 290*3, whence 3 x 387*6-4 x
274B 1162*8 — 1161*2- 1*6 or 16 in 10 seconds).
But for the purposes of the rule all the beats
of Fifths are supposed to be the same through-
out the tuning octave, and similarly all the
beats of Fourths are assumed to be tiie same.
The errors will be found to oorrect each other,
and in no case to exceed the permissible limits.
Art. 1 1 . — Bide for tuning m equal temperament at any pitch between c'256 and
c' 2 70-4.
Tune in the following order, making the Fifths closer and the Fourths wider f
than perfect. The numbers between the names of the notes indicate the beats
in 10 seconds.
For harmoniums :
& 10 g^ 15 d' 10 a' 15 e' 10 b' 15 /# 15 c% 10 g% 15 d'# 10 a% 15/
For organs, using the metal principal, sounding thus an Octave higher than
the digitals shew :
c" 20 ^" 30 (i" 20 a" 30 e'' 20 b" 30 f% 30 &t 20 ^"Jf 30 d'f 20 a"# 30/'
For pianofortes :
c' 10 g 6 d' 10 a 6 e* 10 b 10 fjj^ 6 c% 10 ^ 6 d% 10 ajj! 10/
On the piano the beats can often be heard
for only 5 seconds, and then the beats will be
3 and 5 in 5 seconds, in place of 6 and 10 in
10 seconds.
In the two first cases the intervals beating
10 in 10 seconds are all Fifths up, those beat-
ing 15 in 10 seconds are Fourths down; in the
last case the Fifths up beat only 6 times in 10
seconds, and the Fourths down beat 10 times
in 10 seconds.
Tune each Fifth as accurately just^ or
-without beats, as possible, and then make the
interval closer by flattening the upper note
-very slightly indeed till 10 beats are heard in
10 seconds. Then from the Fifth thus reached
tune a Fourth down as accurately just^ or
-without beats, as possible, and then make the
interval opener by flattening the lower note
very slightly till 15 beats are heard in 10
seconds. From the note thus gained proceed
to the next until / is reached. The Fourth / tr
to d is not tuned, as both notes have been
determined. It never beats faster than 15 in
10 seconds.
If the Fifths and Fourths are not brought
to be as nearly as possible just in the first in-
stance the tuner can never be sure whether
the second note of the interval is too sharp or
too flat, because the error itself is too small
to be judged of with accuracy on merely
sounding the notes in succession, and the
same number of beats would result whether
we had sharpened or flattened the note, but
the whole scheme would be entirely frustrated
if the interval were rendered opener instead of
doser or conversely.
Art. 12. — If it is preferred to commence on a', set a' to fork and proceed to e\
b' up to/', &c., as in the regular scheme. Stop at /' and begin again at a', and
time the Fifth a' to d' down, first making the interval just, and then making it %
closer by sharpemng the lower note till 10 beats in 10 seconds result. Next take
d^ to g' a just Fourth down, and then make the interval opener hy flattening the
lower note till 15 beats are heard in a second. Lastly, from this g^ take & a Fifth
lower, and after making it just, render the interval closer by sharpening the lower
note till 10 beats in 10 seconds are heard. This modification is merely an adapta-
tion of the general principle that Fifths are to be closer than just, and Fourths
opener than just.
The tuner should carefully familiarise him-
self with tuning just Fifths and Fourths, with
recognising them as just by their total absence
of beats in this 2-foot Octave, and by feeling
how beats arise by altering either of the notes
either way. On the harmonium this is easy,
when the just interval has been tuned. It is
only necessary to press down the digital of the
upper note slightly, so as just to hear the
note ; this process flattens it and renders the
interval closer; or to do the same with the
lower note, which renders the interval opener.
In either case beats ensue. As many just
Fourths and Fifths exist on the Harmonical,
this experiment is ready to hand.
Art. 13. — The proof of my rule consists in
shewing that for c'256, a'435, a'440, c'27o*4, it
leads to results which no ear could distinguish
Digitized by V^jOOQlC
490
ADDITIONS BY THE TRANSLATOR.
APP. XX.
ir
from perfectly correct tuning, or which are
equal at least to the very best in art. 4. For
it is clear that if it holds for all these pitches,
of which one is practically the lowest and one
practically the highest in use, and the other
two are as nearly as may be halfway between
them, it must hold for all intermediate pitches.
Now the results obtained by the rule are easily
calculated for a given c or a\ Using c', c'8 ,
d,\ ifec, for the pitch numbers of these notes,
and remembering that the rule gives i beat in
a second for flat Fifths and 1-5 beats in a
second for sharp Fourths, we have, for har-
moniums or organs, supposing d known, and
the above rule accurately followed,
2/ «3c' -I
4^' =3/ -1*5
2a! =3d' -I
4c' =3a' -1-5
26' = 3e' - 1
4/ff =3^>' -1*5
4c'ff =3/«-i-5
2(7'tt =3c'5 -I
A^'l = 3^'tt - 1'5
2a'«=3^'«-i
4/ =3flt'tt-r5.
If we begin with a\ then the three first equa-
tions will give
3d' = 2a'+i, 3^' = 4d'+i-5, 3c' = 2/+i.
In the case of the pianoforte, using the Octave
/ to /, Uie equations for calculating the pitches
of the notes from beats will be, if we begin
with c',
4g -3c' -I
2d =3gf - -6
4a = 3d — I
zd ^Tfl, — -6
46 ^yl - 1
4/B «32» -I
2c'« =3/tt - -6
4{75 ^V^l -*
2d% «3^ - -6
4flff =3i'ff -I
4/ =3«« -I-
But if we begin with a, the first three equations
will give
3(i = 4a+i, 3^«2d + '6, 3c'«4<?+i.
In the following calculations the rule is
necessarily supposed to be carried out with
perfect accuracy. Of course in practice, espe-
cially when the beats are estimated instead of
being accurately counted, this is impossible.
But the results will be found much more accu-
rate than in the ordinary way of tuning en-
tirely by e8tin[iations of ear, and the rule much
more easy to manipulate than the ordinary
method of tuning.
'Proof of Bute for Tuning in Equal Temperament.
Fob Hasuoniuiis and Oboanb.
f
Notes
e
n
271-2
271-1
274-0
273-9
277-2
277-2
286-5
286-5
d'
287-4
287-3
290-3
290-3
2937
293-6
d%
304-4
304-3
307-6
307-4
311-1
3II-I
322-6
322-4
325-9
3259
3296
3296
340-7
3406
/
341-7
341-5
345-3
345-0
349-2
349-2
360-9
36x0
ft
g*
n
a'
430-5
430-4
435-0
435-0
440-0
440-0
454-8
454-7
456-1
455-9
4609
460-6
466-2
466-1
481-8
481-8
V
c*
C'256
Equal .
By Bule
256-0
256-0
258-7
358-7
261-6
26x6
362-0
362-0
365-8
365-8
370-0
370-1
383-6
383-5
387-6
387-6
392-0
391*9
406-4
406-1
410*6
410-3
415-3
415-2
4292
4292
483-3
4831
4883
488-1
493*9
493-9
510-4
510-4
512-0
5120
517-3
517-3
523*3
5233
540-8
54D-8,
Equal .
By Bule
a'440
Equal .
By Bule
c'270-4
Equal .
By Bule .
270-4
270-4
303-5
303-5
321-6
321-5
3824
382-5
405-2
405-1
Fob Pianos.
Notes
/
170-9
170-9
180-5
180-6
i8i-o
181-0
191*2
191-3
9
X9I-8
191-8
202-6
202-6
9Z
2032
2032
214-6
214-7
a
215-3
215-2
227-4
2274
air
228-1
2282
240-9
24I-I
h
241-7
24X'7
255-2
255*4
256
256
270-4
270-4
271-2
271-2
286*5
2866
d'
287-4
2873
303*5
3035
d'l
304-4
304-6
-
321-6
321-8
322-6
322-6
340^7
340-8
341-7
341-7
361*0
361-2
Equal .
By Bule
Equal .
By Bule
Hence it is apparent that the rule ^ever
makes an error exceeding -3 vib., and generally
keeps below this limit. Now at 256 an error
of -3 vib. amounts to 2 cents, and at 540 to
less than 1 cent. The rule, therefore, properly
handled will give results equal if not superior
to the specimens in Art. 4.
Digitized by
Google
BSCT. G.
ON TUNING AND INTONATION.
491
Art. 14. — The rule applies only to one octave and gives what are known as
• the bearings,' whence the other notes must be derived by taking Octaves in the
usual way.
Tuners so frequently get out in taking the
Octaves that it is convenient to have a (dieck
on the estimation of ear. This is furnished
by the fact that any note will beat the same
number of times in a second with an imperfect
Fourth below and its Octave (that is, an im-
perfect Fifth) above. Thus if the note have
401 vib., its imperfect Fourth below 300, and
the Octave above that Fourth below 600, the
beats of the Fourth are 3 x 401 -4 x 300=3,
and the beats of the Fifth are 3 x 401 — 2 x 600
= 3 also. Now the imperfect Fourths are fur-
nished by the bearings themselves. Thus,
going upwards, we have c'/ c", c% ft c^ff , d*
tf d", <fcc. Going downwards, the tuner takes
a Fifth and then a Fourth, as h' ef h,Vbe^bh\),
a' d* a, and so on from octave to octave. Mr.
Hermann Smith prefers to insert the octaves
above when tuning the original bearings. Thus,
if the bearings were taken in the two-foot oc-
tave d c" as a' xo d' 15 gr* loc/ 15/ 15 a't \od't
15 fl^ff lodt 15 /ff 15 fe' 10 e', he would intro-
duce the octaves in this order, a a* d' d" ^ g
d d'f fal a!% d'l d"% ^ff g\ dl d'% fl fl
hU d d' a\ But the method I have proposed
seems simpler.
The principle of the check applies to the
inversions of other imperfect intervals, and
may serve as additional verifications. Thus
a note beats equally with an imperfect minor
Sixth below and its Octave the imperfect major
Third above. Thus 500 : 801 beat 5 x 801
-8x500 = 8, and 801 : 1000 beat 5x801-4
x 1000=8 also. Again, a note beats equally
with an imperfect minor Third below and its
Octave the major Sixth above. Thus 500 : 601
beat 5x601— 6x500 » 5, and 601 : 1000 beat
5 X 601 —3 X 1000= 5 also. If in each case we
inverted tiie order, we should double the beats.
Thus for the Fifth, 200 : 301 will beat 2 x 301
— 3 X 200= 2 ; but 301 : 400 will beat 4 x 301 — 3
X 400 = 4. For the major Third, 400 : 501 will ^
beat 4 X 501 - 5 X 400 = 4 ; but 501 : 800 will
beat 8 X 501 — 5 X 800 = 8. For the major Sixth,
300 : 501 will beat 3 x 501 - 5 x 300 = 3 ; but
501 : 600 will beat 6 x 501 — 5 x 600=6. The
reason is obvious. The fractions expressing
the minor intervals f , f , f have odd denomi-
nators and even numerators, and hence their
inversions reduce by dividing by 2, but this is
not the case for the major intervals, f , f , |.
So much of the beauty of tuning pianos,
harmoniums, and organs depends on the per-
fection of the Octave, that tuners would do
wen to apply the first test with the Fourth
below and Fifth above, as a matter of course.
Art. 15. — Hule for tuning in meantone temperament from d 2527, HandeVs
pitch, to & 283-6, Father Smith's pitch for the Durham organ.
As will be seen by the table p. 4346, c'264
or Helmholtz's pitch is a small meantone
Semitone, and the Durham pitch is a mean-
tone Tone, sharper than HandePs. The great
flatness of the Fifths in the meantone intona-
tion makes it necessary to divide ike tunings
into three classes, sufficiently ascertainable by
a fork in Helmholt^'s or even in French pitch.
The first is from rather less than a Semitone
to about a Quartertone flatter than French
pitch ; the second is French pitch and from a
Quartertone flatter to a Quartertone sharper ;
the ^hird is from a Quartertone to a Semitone IT
sharper than French.
The rule would extend to 27 notes, but on
the proof it will be carried out only for 14
notes, as on the English concertina, for which
this intonation is still used. And for this in-
strument the ' tuning octave ' may be taken as
d to d*. For the few organs that still use this
intonation the same Octave must be tuned,
and hence, when taken on the * principal,' the
digitals must be fingered from c to d, because
the beats would be otherwise too rapid to count
Tune in the following order the numbers 25-6-7 and 40-1-2, meaning that
the beats are to be 25 and 40 for the low, 26 and 41 for the medium, and 27 and 42
for the high pitch in 10 sec., according to the three grades already laid down.
c' 25-6-7 g' 40-1-2 d' 25-6-7 a' 40-1-2 e' 25-6-7 b' 40-1-2 /J 40-1-2 c'J 25-6-7
^'#40-1-2. If
d 40-1-2/ 40-1-2 6't> 25-6-7 e'\} 40-1-2 a]}.
The tuning is conducted in the same way 2^ •
as before, only in two series, from d to g'tt and 4d'
from d to a'b, making the Fifths and Fourths 2a'
at first perfect, and then the Fifths closer and 4d >
the Fourths wider. But in two cases df^fb^b 26' -•
the Fourth is taken upwards, and then the 4/^9
upper note has to be sharpened. And there is 4di ■■
an additional verification after tuning to e', for 2/5 ■■
the major Thirds cV, gf'6', d'ft , a'dZ , difZ , Ad'Z
and also fa' and minor Sixths d'b'b^ da'b
should be all sensibly perfect. Octaves can 3/ •
be verified by imperfect Fourths more easily 3yb =
than in equal temperament. 3e'b
The equations for determining the pitch 3a' b =
irom the beats are
'3c' -2-5.-6-7
-39' -4'o--i-*2
.3d' -2-5.-6-7
»3a' — 40--X--2
■•3d -2-5--6-7
= 36 — 4-0-' I -'2
-3f9 -4*o-'i-'2
= 3c'ff -2-S--6-7
■'Ztft -4*o--i--2
•4c' +4*o--i-'2
= 4/* +4-0--I--2
-.2b'b +2-5--6-7
■4db +4-0--I--2
Digitized by VjOOQIC
492
ADDITIONS BY THE TRANSLATOR.
AFP. XX«
'Proof of the Bulefor Tuning in Meantone Temperament.
Notes
Handel .
By Bole .
2527
2527
Helmholtz
By Rule .
Durham .
By Rule .
264*0
264-0
283-6
283-6
264-1
264-1
d' d'l
282-5 295-2
282-41295-I
302-8
302
275-9 295-2^308-4
27S*8|295-o 308-3
296-3]3i7-i
296-6317-0
331-3
331-6
^b
315-9
9
431S
315-8
3159
339-2
339-2
338-0
3383
330-0
329-9
354-5
354-6
353-2
353-4
353-1
353-4
379-2
379-5
ft
369-0
3691
377-9394-9404-3
377-8 ^
394-8
394*7
3963
396-9424-1
g't
a'b
_. ._ 422-5
394-8 4046 422-5
b'b
412-5
412-5
424-0443-1
443-5
453-7
453-7
422-4441-4
422-6441*2
472-3
472-
474-1
474-2
472-4505-4
505-4
452-1
452-3472-5
_ 493-5
6493*6
507-3
507-4
530-1
530-6567
528-0
528-0
5672
2
The results are seen to be nearly as good
OB before, only the Durham ft is 396*9 in
place of 396-3, an error of *6 vib. and 2*6 oents,
which is scarcely perceptible.
Art. 16. — ^In the Froceedings of the Mibsical Association for 1874-5, vol. L
^ pp. 141-145, Mr. Bosanquet gives the process that he followed in toning his
cycle of 53 (see sect. F., p. 479), but it is too complicated to be abstracted with
a reasonable hope of its being understood. It depended mainly on taking the
beats of the differential tones in the major chord, with the major Third either
in the middle or highest. In this case the Fifth was presumed to be accurately
tuned to begin with. Taking the numbers in sect. A. art. 27, but doubling them
for the Octave higher and supposing the Fourth to be perfect, we have
c"528, /'704, a"879'32 in the cycle.
Differentials 176 175*32
Beats per second *68 , that is, beats per minute 40*8
and the beats were apparently counted for a minute. Mr. Bosanquet, however,
does not recommend the process for his harmonium because the differential tones
were not distinct enough.
Art. 17. — Mr. Paul White has two methods of tuning the cycle of 53 by means
^ of beats. We may suppose that for the given pitch of the initial c, aJl the pitch
numbers have been calculated, as in sect. A, art. 27, for No. 4, Mr. Bosanquet's
initial 0=264 vib. Two places of decimals are required for ijiis purpose, as in
both methods it is necessary to rely upon the slow beats of the Thirds and Sixths
and check the result at every few steps.
First method. Tune 5 minor Thirds of the cycle up, alternating with major
Sixths down, to keep within the same octave. Since a minor Third has 14 degrees
this gives 5 x^=i^, or an Octave and 17 degrees, that is an Octave above the
major Third of the cycle, which will beat slowly with the note on which we
started. Thus beginning with 264 and taking first 3 cyclic minor Thirds up, and
then a cyclic major Bixth down, followed by a cyclic minor Third up, we have
vib 264
beats in 70 sec.
np np up down np
317*05 380-75 457'25 274-56 329-73
12 14-5 17-5 io'5 12-9
and as 264 : 329*73 is a cyclic and not a just msgor Third, which would be 264 : 330,
^ it will beat 10*8 times in 10 seconds, and this will be the verification of the work.
Observe that the interval of the minor Third must be too wide, and hence of the
major Sixth too close, so that when we tune up, the uppemoi& must be made sharp,
and when we tune down, the lower note must be made sharp. Also that as the
5th and 6th partials are involved in the beats, the method will suit only qualities
of tone, like reed-tones, with strong upper partials. Observe also that in equal
temperament 5 equal minor Thirds are an Octave and an equal minor (not a major)
Third. Having completed one set proceed with the next set of 5 minor Thirds (or
major Sixths) until the whole cycle is complete for one octave and then tune by
Octaves.
Second method, which Mr. Paul White prefers. Tune 7 cyclic major Thirds
down (alternating with minor Sixths up to keep within the same octave). The
result will be a cyclic Pythagorean minor Third of 13 degrees down, or 40 degrees
up, for 3-7xH=3-W=3-2-if=i-|J=f5. And this can be verified by
3 cyclic Fifths up, for 3 x f J=ff =1 J§, such Fifths being practically perfect. Thus
beginning at 528 vib. we obtain, taking major Thirds down, and minor Sixths up :
Digitized by V^jOOQl€
«BCT. It. THE HISTORY OF MUSICAL PITCH IN EUROPE. 493
down down down np down down up
^b 528 42274 338-47 271-00 433*95 347'44 278-18 445*45
beatsin losec. 17 13-9 11-2 17-5 14 11-4 17-9
a Fourth down another Fourth down a Fifth up
^b 528 396 297 445-5
Ihe Fourths and Fifths are taken just, and the> result agrees to -05 vib. It must
he remembered that the cyclic major Thirds are too close, hence in tuning down
the lower note must be sharpened. On the contrary the cyclic minor Sixths will
be too wide, and hence in tuning up, the upper note has to be sharpened. Having
completed this set of 7 proceed to another, till the cycle is complete. This method
also only suits qualities of tone, like reed-tones, with powerful 5th and 8th partials.
The process thus carried out would of course be tedious, and Mr. Paul White
seems to assume a tolerably uniform beat, perhaps of 15 in 10 seconds, for he says:
' The beats cannot of course be made, or be made to remain uniform, but if they
are nearly so, or if a few do not beat at all, the temperament is still good. 1 1|
have found that the Fifbhs can be kept almost entirely free from beats by talong good
care of the very slow beats of the Thirds. I have long been convinced that beats
in the middle octave do much more good than harm in a musical cycle, for it would
be impossible to tune a musical cycle of any size correctly without them. The least
scratch on a reed will change a beat, while it often takes quite a scrape to cause
a beat where none existed.' The processes Mr. Paul White has worked out with
the ingenious system of checks, shew that he is a thorough master of the whole
art of tuning, and, a rare thing to be met with among professional tuners or even
musicians, perfectly understands its rationale.
Art. 18. — A succession of just Fifths, as mentioned in art. i, is very difficult to
tune ; and one of just major Thirds is still more difficult. Hence an auxiliary stop
on an organ or an auxiliary harmonium is required when just intervals have to be
tuned.
It is not difficult to ascertain by ear whether tuned i beat in a second sharper than the
a Fifth or major Third is considerably too flat, auxiliary d\ And in this way by a laborious m
Suppose we start with c', then tune an auxiliary double process the succession of Fifths could
g' (indicated by a roman letter) decidedly flat be tuned with great accuracy. For the major
beating 40 times in 10 seconds with cf. Then Thirds, tune an auxiliary e' decidedly flat, and
3c' — 2g' « 4, so that |c' = g' + 2 , but fc' is the beating 4 in a second with c'. Then $(/ - 40' == 4,
perfect Fifth to </, hence we must tune the re- and true e/Bfc'»e'+ 1. In the same way we
quired Fifth 9' » g' + 2, that is, sharper than g' could get g2Jt and b^'t » But for a' b, /'b, ^'b b
by 2 beats in a second. For the next Fifth in we must tune auxiliary minor Sixths, which is
order to remain in the same octave we should troublesome and not feasible except on reed
take the Fourth down. Tune the auxiliary d' instruments. Tune an auxiliary a'b flat, so as
so that it should be too flat, and beat 4 times to beat 5 times in a second with &. Then
in a second with the correct &. Then 3^ 8c'— 5a'b = 5,and truea"b«Sc' = a'6+ 1. And
— 4d' = 4, and f^ = d' + i . But Jgr is the correct so on.
d', or Fourth below ^. Hence it must be
It appears, then, that tempered intervals which present beats of their own are
more easy to tune than just intervals for which an auxiliary beating tone has to be
supplied. The only satisfactory way, however, of tuning perfect and tempered
intervals is by a fork tonometer, one of which suffices for every possible case that
can arise, when once the pitch numbers of the notes have been calculated as in f
sect. A.
SECTION H.
THE HIBTORT OF ITDSZCAL PITCH IN EUROPE,
(See note p. 16.)
Art. Table I. — continued,
1. Pitch of a Note, p. 494. 4. Mean Pitch of Europe for Two Cen-
2. Musical Pitch, p. 494. turies, p. 495.
3. Early Pitch, p. 494. 5. The Compromise Pitch, p. 497.
4. Materials and Authorities, p. 494. 6. Modem Orchestral Pitch, and
5. Description of the Tables, p. 494. ♦Church Pitch Medium, p. 499.
Table I. Historical Pitches in order from the 7. Church Pitch, high, p. 503.
Lowest to the Highest, p. 495- 8. Church Pitch, highest, p. 503.
1. Church Pitch, lowest, p. 495. 9. Chamber Pitch, highest, and Church
2. Church Pitch, low, p. 495. Pitch, extreme, p. 504.
3. Chamber Pitch, low, p. 495.
Digitized by VjOOQlC
494
ADDITIONS BY THE TRANSLATOR.
APP. XX.
Table II. Classified Index to Table I.,
p. 504.
I. Austro-Hungary, p. 504.
II. Belgium, p. 504.
III. England, Scotland, and Ireland,
p. 505-
IV. France, p. 508.
V. Germany, p. 509.
VI. Holland, p. 510.
VII. Italy, p. 510.
VIII. Russia, p. 510.
IX. Spain, p. 511.
X. United States of America, p. 511.
GoMGLnsioNS, p. 511.
Art.
6. The History of Musical Pitch in Europe
for 500 years, p. 511.
^ 7. Original Motives for determination of
Church Pitch, p. 511.
Art.
8.
Effect of the foot measure of different
countries on the pitch of organs, p. 511.
Origin of Chamber Pitch, p. 512.
Evolution of the Mean Pitch, its great
extent geographically and chronologi-
cally, p. 512.
Difficulties arising from singing at hi^
pitch classical music written for mean
pitch, p. 512.
How the rise in pitch commenced and
spread, p. 512.
The Compromise Pitch in France, England,
and Germany, p. 512.
14. Variations in English Organ Pitch, p. 513.
1 5. Rise in pitch connected with wind instru-
ments, p. 513.
16. What must be done, p. 513.
12.
13
Art. I. — The pitch number of a note has been already defined as the number of
double vibrations which the sonoroos body producing it makes and commnmcates
in one second (p. iia).
Art. 2. — The pitch number of a musical instrument, or briefly its musiccd
pitch, is taken to be the pitch number of the tuning note at a temperature of
59°F.=i5°C.=i2°R.
The tuning note is here assumed to be the
a' of the violin, from which the pitch number
of all the other notes in the scale must be cal-
culated, or determined approximately by ear
from the temperament (sect. A.) and system
of tuning (sect. G.) in use. By taking a' as
the tuning note, the inquiry is practically
limited to European music within the last 500
years.
Art. 3. — The following passage from Syntagmatis musid, Michaelis Pr^tobh
C, Tomus Secundus, de Organographia, 1619, p. 14, explains the condition of
^ early pitch.
* In the first place it must be known that
the pitch, both of organs and other musical
instruments, varies greatly. Since the ancients
were not accustomed to play in concerfc with
all kinds of instruments at the same time,
.wind instruments were very differently made
and intoned by instrument makers, some high
and some low. For the higher an instrument
is intoned in its own kind and manner, as
trumpets, shawms, and treble viols, the more
freshly it sounds and resounds. On the con-
trary, the deeper trombones, bassoons, bassa-
neldi, bombards, and bass viols are tuned, the
more majestic and magnificent is their stately
march. Hence when the organs, positives,
olavioymbals, and other wind instruments are
not in the same pitch with each other the
musician is much plagued.'
Art. 4. — The authorities on whom 1 rely are minutely specified in my * History
of Musical Pitch ' in the Journal of the Society of Arts for 5 March and 2 April
1880, and 7 Jan. 1881. The two last papers contain indispensable corrections
and additions. Li the privately printed copies there was an addendum on U.S.
America from Messrs. C. K. Cross and W. T. Miller, American Journal of Otology,
Oct. 1880.
f Here it must suffice to say that after learn-
ing to determine pitch to ^ vib. (p. 444) I
obtained the loan of authentic forks from the
Society of Arts, Mr. A. J. Hipkins, Rev. G. T.
DriflSeld (Handel's), Fran Naeke of Dresden,
Prof. Rossetti of Padua, Mr. Blaikley, and Dr.
W. H. Stone. I procured compared copies of
forks in the Conservatoire at Paris, and others
tuned at known temperatures to remarkable
organs at Vienna, Dresden, Hamburg, Stras-
burg, and Seville. Then, with the assistance
of many organists, I measured numerous
organs in England of which the pitch had not
been changed, or, with the kind help of Beveral
organ-builders, obtained untouched pipes of
altered organs. When these failed I had
models made of pipes of which the dimensions
were given by Schlick 1511, PrsRtorius 1619,
Mersenne 1636, Tomkins 1668, Bi'-dos 1766,
and others, which were obligingly presented to
me by Mr. T. Hill, the organ -builder, on whose
bellows I measured them. These constituted
my own materials. Then I had recourse to
the measurements and lists of Cagniard de la
Tour, CavailU-Goll, de la Fage, Delezenne, de
Prony, Euler, Fischer, French Commission on
Pitch, Eoenig, Lissajous, McLeod, Marpuig,
Naeke, Sauveur, Scheibler, Schmahl, Dr. R.
Smith, and others. From these I constructed
the lists which follow. In my original papers
each pitch is accompanied with full details.
Here I give the smallest possible account.
Art.. 5. — The pitch given is always that of
a', where possible at 59° F. But this was not
always the note measured. When it was not,
a' was calculated on the assumption of either
meantone or equal temperament. Assuming
a lowest ideal pitch of a'370, which has never
Digitized by V^jOOQIC
SECT. H. THE HISTOEY OF MUSICAL PITCH IN EUEOPE.
495
yet been found, I g^ve the cents by which any
other pitch exceeds this, so that the interviJ
between any two pitches is immediately deter-
mined by subtracting the cents. I give also
the date, adding occasionally a for ante^ before,
;p for 2)05^ after, and c for circa^ about ; and
the authority, or observer, where £. means
that I am responsible for the measurement,
directly or indirectly. Finally, I add a list,
classified by countries, stating the kind of
pitch. I hav^ not thought it necessary to give
absolutely every fork and pitch entered in my
* History,* but have reported a large number of
these entries, and especially all the most inter-
esting of them. A complete Grerman trans-
lation of my paper is in preparation, and will
be published at Vienna.
Table I. — Historical Pitches in obdeb fbom Lowest to Highest.
Cents I a' | Date
Obeerrer
Place and other particalars
370
—
373-1
—
3737
1648
374-2
I7c»a
376-6
1766
377
1511
384-3
3846
3922
1 700c
1851
1739
393-2
395-2
17x3
1759
395-8
1720
1789
396-4
161S
3987
1854a
199 415 "754
I. Church Fitch, Lowest
E.
Delezenne
B.
Delezenne
E.
E.
4029
1648
403-9
1730
406-6
407-3
407-9
1704
1854
1762
409
1783
410
—
41 1-4
1688
4133
414-4
1776
Ideal lowest pitch or zero point
Calculated from D.'s measurements of an open
wooden pipe 1-3 metres long, taken as c
Paris, from a model after Mersenne
Lille, organ of I'Hospice Comtesse
Paris, from a model after B§dos
Heidelberg, from a model after Arnold Schlick
(see 535 cents)
2. Church Pitch f Low:
Delezenne .
Euler .
Stockhausen & E.
Dr. B. Smith
McLeod & E.
E.
Delezenne •
Lille, old fork found 1854a by M. Mazingue
Lille, organ of St. Sauveur, rebuilt, with old pitch
St. Petersburg, a clavichord according to Marpurg,
but Euler gives no particulars
Strassburg Minster, great organ by A. Silbermann
Cambridge, Bernhardt Schmidt's organ at Trinity
College, 1708, after being new voiced and
* shifted' in 1759
Bome, pitch-pipes observed by Dr. B. Smith
France, Versailles, copy of fork No. 410 at the
MuB^e du Conservatoire, Paris, compared with
the original by Cavaill6-.Coll
Palatinate of the Bhine, from a model of pipe de-
scribed by Salomon de Cans
Lille, old organ of La Madeleine restored
3. Chamber Pitch^ Low.
E. . .
E.
Sauveur
Delezenne .
Schmahl & E.
Lissajous
Schmahl & E.
Naeke .
Marpurg
Paris, Mersenne's Spinet, from his statement that
Bq = B^dos's 4-foot c (see 31 cents)
Padua, from copy sent by Prof. Bossetti of the old
lower /' fork of the bellf oundry of Colbacchini
Paris, result of several experiments on an e pipe
Lille, organ of St. Maurice repaired, old pitch kept
Hamburf?, organ of St. Michaelis Kirche, buUt
by Hildebrand of Dresden, imder the direction
of Handel's friend, J. Mattheson (1681-1764),
in the chamber pitch of the period, still pre-
served; now, and probably always, in equal
temperament
Paris, Court clavecins, fork of Pascal Taskin,
their tuner
Paris, 1 8th century pitch-pipe found in the cabinet
of the Faculty of Sciences
EEamburg, chamber pitch on the former 8-foot
Qedact of the St. Jacobi organ (see 484 cents)
* Schneider's Oboe,' date and place unknown
Breslau, clavichords
4. Mean Pitch of Ewropc for Two Centuries,
E.
IT
IT
Dresden, organ of the Boman Catholic Church by
Gottfried Silbermann, pitch of the chained fork
placed there by King August der Orechte,
1 763-1827, who would not allow the pitch to be
changed ; the fork was lent me by Frau Naeke
Digitized by V^jOOQlC
496 ADDITIONS BY THE TRANSLATOB. app.xx.
Tablb I. — Historical Pijcrkb in obdiek fbom Lowest to Hiobist — ecmiinued.
f
IT
Oenta
Date
Obeerrer
Place and other particulan
20I 1
415-5
211
212
418
4181
215
419
217
218
4»9*5
4196
»»
II
219
4199
220
420-1
224
421-2
225
421-3
226
421-6
229
422-3
>i
II
230
422-5
*i
II
»i
II
II
II
II
422*6
II
II
II
II
231
422-7
23*
423
233
4232
1}
II
II
4233
4. Mean Pitch
1722 Naeke .
1780
1878
1 700c
1714
1858
1785
Euler .
E.
E.
Naeke .
de la Fage
E.
1715c E.
1780
i860
1780
1780
1780
17800
1751
1820a
E.
E.
Naeke .
Naeke & E
Naeke .
E.
E.
E.
1838
E.
I877P
E. . .
1790a
E. . .
1754c
Delezenne .
1 780c
E.
18000
E. . .
1820
MoLeod & E.
1778
E.
1815-
1821
E.
1813
E.
of Europe for Two Centuries — oontinaed.
Dresden, organ of St. Sophie, built by G. Silber-
mann
St. Petersburg, organs ; no particnlars
Dresden, present pitch of the organ of the Roman
Catholic Church, from a fork tuned for me there
London, Benatas Harris's organ at St. John's,
Clerkenwell
Freiberg, Saxony, G. Silbermann's organ
Madrid, ton de ohapelle, calculated
Seville, Spain, pitch of the old organ of Torje
Bosch, from a fork said by the organist Don
Yniguez to be in exact unison with its a' at a
mean temperature
England, rude tenor a fork, belonging to Rev. G. T.
Driffield, who held it to have been made by John
Shore, the inventor of tuning-forks
Winchester College organ, from one of the pipes
added by Green when repairing B. Harris's
organ of 168 1
Bussian Imperial Court church band from fork
lent by Frau Naeke
Vienna, fork of the Saxon organ-builder Schulze,
who lived at Vienna in Mozart's time
Vienna, copy of fork of Stein, who made Mozart's
clavichords and pianos, lent me by Frau Naeke
Dresden, fork of former Court organist Kirsten
Verona, from a copy of a c' fork believed to be the
Boman pitch of 1780, preserved at the bell-
foundry of Cavedini, procured by Prof. Bossetti
of Padua
England, Handel's fork belonging to Bev. G. T.
Driffield. The organ at Cannons in the private
chapel of the Duke of Chandos, bmlt by Jordans,
and afterwards bought by Trinity Church, Gos-
port, has been recently (in 1884) examined by
the organist, Mr. Hewlett, and found to have
had in Handel's time, when he used to play on
it, a Bq (now Bb) pipe of 12-3 inches long,
and I inch in diameter ; this shews that its pit<^
was then ^1423-5, or practically the same as
Handel's fork .
Westminster Abbey, as originally tuned by
Schreider and Jordans, from indications by Mr.
T. Hill, who retuned it to a'44i*7. It had been
altered by Greatorex to 0^433*2, Smart's pitch
Bath Abbey Church, as rebuilt by Smith of Bristol,
from indications by Mr. T. HUl
England, Mr. J. Curwen's Tonic Solfa standard
c''507, using the just a' only
Eew Parish Church, Green's organ, untouched
and in meantone temperament when measived
in 1878, built as a ohunber organ for George IIL
Lille, very old fork found in workshops of M.
FranQois, musical instrument maker there
Padua, from copy of the higher /' fork of the bell-
foundry of Colbacohini (see 152 cents)
England, from old fork, c"505-7, belonging to
Messrs. Broadwoods
Paris, Th64tre Feydeau, Op^a Comique, from copy
of fork at the Conservatoire, Paris, compared
with the original by Cavaill6-Coll
London, Green's organ at St. Katharine's, Begent's
Park, still (when I measured it) in meantone
temperament (see sect G., p. 484CO
Dresden, band of the opera while C. M. von
Weber (i 786.1826) was conductor {KdpM-
meister)
London, second copy of Peppercorn's fori: by
which the pianofortes of the Philharmonic
Digitized by V^jOOQlC
6CT. H. THE HISTORY OP MUSICAL PITCH IN EUROPE. 497
Table I. — Historical Pitches in order frou Lowest to Highest — continued.
Date
Observer
Place and other particulaia
4.
4237
I8I3
4241
1740-
I8I2
4242
I6I9
l»
1823
424-3
1750a
II
1749
424-4
1833
4246
I800C
424-9
1805
249
250
250
251
255
260
262
425-2
425-5
4256
425-8
425-9
426-5
427
4272
427-5
427-6
4277
4278
428-7
430
430-4
1 800c
1 730c
.I780C
1829
1740-
1780
1764
1824
1839
I8IOC
I70I
Mean Pitch of Europe for Two Centuries — continued.
Society were originally tuned; this copy was
prepared for the Society of Arts in i860, and is
now in the possession of Messrs. Broadwoods
London, first copy of Peppercorn's fork made be-
fore i860, belonging to Mr. Hlpkins; see last
entry, the original is lost, and it is impossible to
say which was correct. The difference, 2 cents,
is utterly insignificant
Naeke . . . Eutin (18 miles N. of Liibeck), fork of Franz
Anton von Weber, father of Carl Maria von
Weber
E. . . . Brunswick, from a model made from Prastorius's
drawing of an organ pipe at a * suitable * church %
pitch
Blscher . Paris, Italian Opera, mean of twenty measure-
ments of a fork given by Spontini
E. . . . London, old forks formerly belonging to Prof.
Faraday, lent me by Mr. D. J. Blaikley
E. . . . London, organ at All Hallows the Great and Less,
Upper Thames Street, built by Glyn & Parker,
by whom Handel's Foundling Hospital organ
was built
E. . . . Weimar, from a model of Tdpfer's wide principal
c"-pipe
E. . . England, old fork said to have been used in Ply-
mouth Theatre, lent me by Dr. Stainer
E. ' . . . London, old D fork of Elliott's, by which he tuned
the organ built for the Ancient Concerts at the
Hanover Square Booms, lent me by his suc-
cessor, Mr. T. Hill
Naeke . . . Germany, fork of the bassoonist Eummer
E. . . . Padua, mean of two ancient pitch-pipes belonging
to the bellf oundry of Colbacchini, lent me at
the request of Prof. F. Bossetti there
Lissajous . Paris, pitch of opera piano as distinct from the
orchestra, verified by Monneron for de la Fage
E. . . . England, Schnetzler's organ at the German Chapel
Boyal, St. James's Palace
E. . . . Halifax, Schnetzler's organ, from indications by
Mr. T. Hill
Lissajous . . Paris, pitch of opera, suddenly lowered on 31
March for Mme. Branchu, whose voice was fail-
ing. The piano for rehearsals was also lowered,
and was not raised inmiediately when the or-
chestra was raised ; this was called opera pitch
de la Fage . . Bologna, Italy, pitch of fork of Tadolini, the best
tuner in the town
Great Yarmouth, St. George's Chapel, by Byfield,
Jordan & Bridge
Wimbledon Church, organ built by Messrs. Walker at
Paris, Grand Opera
Norwich Cathedral organ before it was altered by
Bryceson, supposed to be by B. Harris
Tonic Solfa pitch to 1877, afterwards 422-5
Paris, Th6dtre Feydeau, fork given by Spontini
London, old organ built by B. Harris, a pipe of St.
Andrew Undershaft, from Green's organ, pre-
served by Mr. T. Hill
E. . . . St. George's Chapel, Windsor, measured in Feb.
1880, while stiU in meantone temperament
Ions . . Newcastle-on-Tyne, St. Nicholas Church organ
built by Benatus Harris, frequently altered ex-
cept in pitch
5. The Comjpromiee Pitch.
Lissajous . . Paris, fork of M. Lemoine, a celebrated amateur
E. . . . Fulliam Parish Church organ, built by Jordans.
This pitch was officially adopted in Italy in 1884
K K T
Digitized by V^jOOQ IC
1740 ; Tunbridge & E.
E.
Scheibler
E.
1843
1811
1878a
1877a E.
1823 ' Fischer
1696 E.
1788
1670
498 ADDITIONS BY THE TRANSLATOE. app. xxT
Table I.— Historical Pitches ik order from Lowest to BiQBXffr^continued.
Cents
Date
Observer
Place and other particulars
264
267
269
270
272
f
273
275
276
278
»»
279
280
»»
i82
^: S
285
286
287
431-3
4317
4322
4323
433
433-2
433-6
433'9
434
434*3
434"5
4347
435
>»
435-2
435*4
435-9
436
4361
436-5
4367
436-8
288 , 4369
1625
1826
1854a
1846c
1 820c
Lewis .
Fischer
Delezenne
E.
E.
1828
1847
1834
1829
18340
1818
1869
1826
1859
1834a
1859
1868
1802
1846P
1878
1878
1834c
X845
1740-
1780
1869
E.
Byolin & E.
Scheibler
Cagniard
Tour
de la
Scheibler
McLeod & E.
E.
E.
Naeke
Fr. Com.
Scheibler
Eoenig & E
Cross & Miller
Sarti .
E.
E.
E.
Scheibler
Delezenne
E.
R.
TJie Compromise Pi£cA— continued.
as the pitch of the Italian army brass bands,
giving £b456, the nearest whole number to
equal £b456'i3, which would correspond to the
* arithmetical ' pitch C5 1 2
Lavenham (16^ miles W.N.W. of Ipswich), from a
famous old tenor bell sounding (2288*4
Paris, Grand Opera, fork given by Spontini
Lille, organ of St. Andr6 repaired
England, old fork which belonged to the father of
Messrs. Bryceson, organ-builders, and had not
been tuned since 1848, when it had been
sharpened slightly
London, fork approved of by Sir Qeorge Smart,
conductor of the Philharmonic Concerts, in pos-
session of Mr. Hipkins, |prom c"5i8 using mean-
tone temperament ; if equal temperament were
used it would give 0^435*4 and be a 30 years'
anticipation of French pitch. Used in this way
it is Broadwoods' lowest pitch. Long sold in
shops as * London Philharmonic '
London, Sir G. Smart's own Philharmonic fork.
Sir G. Smart considered this a' fork of his to
agree with c''5i8 (see last entry). This shews
that he used meantone temperament
Shrewsbury, St. Mary's, built 1729 by John Harris
and John Byfield, pitch altered in 1847 ^7 Gtkj
& Davison
Vienna, fork I., Delezenne's Vienna minimum
Paris, opera, verified by M. Montal, after the opera
had recovered its pitch, the opera piano remain-
ing at a'425'5, which see, and also a'425*8
Paris Opera, fork by Petitbout, luthier de I'opdra
Paris, Chapelle des Tuileries, from a copy com-
pared by Cavaill^-Coll of fork No. 493 in the
Conservatoire
Baden, fork sent officially to Society of Arts
London, from a model of pipe representing 6'486*i,
one foot long and one inch diameter, on Benatus
Harris's organ at All Hallows, Barking
Dresden, opera, fork of Kapellmeister Beissiger,
successor to C. M. von Weber. Naeke considers
this to have been Dresden pitch from 1825 to 1830
Carlsruhe, opera, the fork which determined the
French Diapason Normal
Paris, Conservatoire, fork made by Gand, luthier
du Conservatoire
Paris, the Diapason Normal in the Conservatoire,
used extensively in Germany, officially adopted
for the Belgian army in 1885. The various im-
perfect copies used are not cited
U. S. America, E. S. Ritchie's standard pitch
St. Petersburg, five-foot organ pipes
London * Philharmonic,' from Mr. Hipkins's
vocal pitch, c"5i8*5, which for equal tempera-
ment gives a'436, but on meantone temperament,
for which it was first used, gave a'433'5 ; the fork
with which Mr. E. J. Hopkins compmd the pitch
of the organs at Liibeck, Hamburg, and Strass-
burg, see his The Organ ed. 1870, art. 791, p. 189
London, Messrs. Bishop's standard for church
organs
London, fork to which Messrs. Bryceson tuned the
organ at Her Majesty's Theatre
Vienna, opera, fork U.
Florence, fork lent by M. Marloye
Dublin, Green's organ in the B^ectory of Trinity
College, probably sharpened
Wiirtemberg, fork sent officially to the Society of
Arts
Digitized by V^jOOQlC
SECT. H. THE HISTORY OP MUSICAL PITCH IN EUROPE. 499
Tabus I.— Historical Pitches in order from Lowest to Kiqkebt— continued.
Oenta
Obserrer
Placo and other partioalars
6. Modem Orchestral Pitchy and * Chwrch Pitch Medium,
288
2S9
437
4371
1859
1666
tl
437-3
1872
II
4374
1854a
II
II
1744
291
437-8
1862
295
438-9
1696
297
439*4
—
»l
II
II
II
18340
1878
298
439-5
1813
II
II
1855
299
439-9
440
1845
1829
II
II
440-2
1878
1834
♦ I
II
302
303
II
440-3
440-5
4409
1879
1834c
1878
18340
304
44I-0
1836-
1839
II
II
1836
II
II
1859
tl
»»
1879
II
441*10
1834
305
441-2
1878
II
441*3
1842
307
441*7
1690
l»
II
1660
"
II
1878
Fr. Com. .
E.
Fischer
Delezenne .
Streatfield & £.
E.
E.
Delezenne .
Scheibler
E.
MoLeod <& E.
Delezenne
Lissajons
E.
Scheibler
E.
Scheibler
E.
Scheibler
Delezenne
Gagniard de
Tour
Fr. Com.
E.
Scheibler
E.
E.
E.
E.
E.
Tooloase, Conservatoire
^Worcester, cathedral organ built by Thomas and
Benatas Harris, from a pipe at Mr. T. Hill's
Berlin, from a fork furnished by Pichler, who
tnned the piano of the opera
Paris, opera, from four forks purchased before
1854, and found to be in unison
^Maidstone, Old Parish Churdi, built by Jordans,
altered, but not in pitch, in 1878 in meantone
temperament
Dresden, fork given by the direction of the Court
Theatre to its librarian, Herr Moritz Furstenau,
after the conference on pitch held there, by whom
it was lent me to measure, meant for a'440
'Boston, England, organ built by Christian Smith,
from a pipe preserved by Mr. T. Hill
Lille, old fork formerly belonging to the Marquis
d'Aligre
Vienna, opera, fork in.
Dresden, opera pitch at date, from a fork specially
prepared for me by the Court organ-builder,
Jehmlich, and sent by Herr Moritz Fiirstenau,
librarian of the theatre
Paris, Conservatoire, from copy of a fork preserved
there, verified by Cavaill6-Coll
England, Barking, Essex, Parish Church organ
(probably originally a'474'i), built by By field &
Green, 1770, after alterations by Messrs. Walker
Turin, fork lent by Marloye
Paris, opera orchestra, verified by Monneron for
de la Fage
London, Messrs. Gray A Davison's standard pipe
Stuttgart pitch, =440 at 69^ F., Lissajous mea-
sured it as 440*3 to French Diapason Normal^
reckoned as 435, which then when corrected to
435*4 gives 4407
London, Messrs. Walker & Sons* standard pipe
Vienna Opera, fork IV.
London, Messrs. Bevington's standard pipe
Paris Conservatoire, not trusted so much by
Scheibler as 435'2
Paris Opera, fork of M. Leibner, who kept the
pianos to pitch of orchestra, verified by
Meyerbeer
Paris, Op^ra Comique
Dresden, fork sent to Fr. Com. by the Kapell-
meister Beissiger
London, church organ pitch of Messrs. Lewis &
Co.
Vienna Opera, fork V., given by Prof. Blahetka as
trustworthy ; in 1879 this fork was found and
lent to me, and then from rust and ill>treatment
measured only 439*9, the greatest loss of pitch
I have found in any fork
London, Covent Garden Opera, f6rk for Messrs.
Bryceson to tune the organ to
London, the equal a' corresponding to the late
Dr. John Hullah's standard fork, c"524*8, pur-
porting to be c"5i2 ; J. H. Griesbach measured
it as 521*6
Hampton Court Palace, Bernhardt Schmidt's
organ from an original pipe, 12 inches long and ;
I '2 inch in diameter, giving 1/ b472 *6
Whitehall, Chapel Boyal, organ by Bernhardt
Schmidt, according to indications by Mr. T. Hill .
London, standard pipe of Messrs. Hill and Sons,
from c"525-3
H
ir
Digitized by
v55bgie
500 ADDITIONS BY THE TRANSLATOR. app. m.
Table I. — Historical Pitchkb in ordbb pbom Lowest to BiaHEvr— continued.
Date
Observer
Place and other partioalars
6. Modem Orchestral Pitchy and * Church Pitch Medium — oontintied.
307
310
441-8
442-5
1834c
1859
*3ii
n
4427
1878
312
443-0
1859
443-1
1815c
1869
313
443-2
1878
»*
n
n
314
315
443-3
443-4
443-5
443*9
1836
III?
»»
444
i860
316
444-2
1880
317
444-3
1840
»»
»i
1880
318
444-5
1858
»»
444-6
1877
»»
444-7
1879
319
444-8
1859
t«
ti
»l
»»
»t
320
»i
321
444-9
445-0
It
445-1
1862
1834c
»i
445-2
1878
322
it
445-4
445-5
1845
1879
It
445-6
"
*•
445-8
1867
1856
Scheibler
Fr. Com.
t»
E.
Fr. Com.
♦»
E.
E.
E.
W61fel .
Fr. Com.
E.
Fr. Com.
Cross & Miller
Cavaill6-Coll
E.
Lissajous
E.
E.
Fr. Com.
LissajoQS
Hipkins
Naeke .
Sohmahl
Scheibler
E.
Delezemie .
Hipkins & E.
E.
E.
Lissajous
Berlin opera
Toulouse opera
Brussels, opera under direction of Bender
*Vienna, small Franciscan organ kept at modem
pitch, from a fork tuned for me by the organ-
builder UUmann
Bordeaux opera
Stuttgart opera
*Durham organ, as altered by shifting from a'474* i ;
a'4447, the present pitch of new organ, is by
Willis
Bologna, Italy, Liceo Musicale, from fork sent
officially to Soc. of Arts
^Vienna, St. Stefan cathedral organ, from a fork
tuned for me by organ-builder Ullmann.
Paris, Wdlfel's pianos
Gotha, opera
London, from Messrs. Bryceson's standard pipe
Brunswick, opera
U.S. America, Boston, organ of Church of the
Immaculate Conception
Intended but unexecuted standard of Society of
Arts to c"528
U.S. America, from c"528, the • low organ pitch *
of Hutchings, Plaisted A Co.
^France, St Denis Cathedral, organ built by
Cavaill^-CoU
^London, Temple Church organ after rebuilding
by Messrs. Forster and Andrews, who retained
the pitch which they found, which was Bobom's,
originally built by Bernhardt Schmidt, with
both Eb and Dff , and both ^b and G8 keys,
and perhaps then having a'4417
Madrid, Theatre Royal, fork sent to de la Page by
the Maitre de Chapelle. French pitch was
adopted on 18 March 1879
♦London, St. PauPs, after rebuilding by Willis,
from a fork belonging to Mr. Hipkins at
57°-5
♦Durham Cathedral organ, rebuilt by Willis ; for
its original state, see a'474'i
Turin opera
Weimar opera
Wiirtemberg concerts
Naples, San Carlo opera, Gxiillaume*s fork
London, Her Majesty's opera, fork of the theatre
Vienna, piano of Kapellmeister Proch
Hamburg * old pitch,' date unknown
Vienna opera, fork VI., *a monstrous growth*
(Auswitchs) in Scheibler's opinion
♦London {from c"529*4), Mr. H. Willis's church
pitch, to which he tuned the. organs of the
cathedrals of St. Paul's (London), Durham,
Salisbury, Glasgow (established), St. Mary
(Edinburgh)
Vienna Conservatorium, fork lent by Marloye
London, Her Majesty's opera during pOTform-
anoe
London, Covent Garden opera, fork in possession
of Mr. Pitman, organist, and Sig. Vianed, con-
ductor. Mr. Pitman said the pitch was thus in
1878 because oboe, bassoon, and flute would not
play lower
London, Exeter Hall, both organs as originally
built, from a pipe at the makers', Messrs. >
Walker ; since sharpened to a'447'3
Paris opera, from the fork of M. Bodin, professor
of the piano and music
Digitized by V^jOOQlC
$ECT. H. THE HISTORY OF MUSICAL PITCH IN EUROPE. 501
Table I.— Historical Pitches in order from Lowest to Hiquzst— continued.
Oento
Date
Obseryer
Place and other particulars
6. Modem Orchestral Pitch, and * Church Pitch Medium— continued.
445*9
446
4462
446*6
446-8
1849-
1854
1859
1856
1859
1845
1851
1878
447-0 1859
447*3 1879
447*4
447*5
4477
448
448-1
1856
1878
1877
1854
1839
1840
1859
448-2; 1869
448-4
448-5
448-8
449
449*2
449-4
449*7
449*8
449*9
450*3
450*5
450-6
450-9
451*5
451*7
451*9
452
1857
i860
1880
1859
1855
1877
i860
1879
1859
1877
1856
1848
&1854
1877
1880
1858
1874
1867
1880
1878
1885
£. . . . ; London, from Broadwoods* original mediom pitch
of c"530-6, fork of the tuner Finlayson ; since
1854 Messrs. Broadwoods use a'446-2 as their me-
dium pitch. This pitch was chosen empirically
Pr. Com. . . Pesth, opera
Lissajous . . Paris, opera and Ck)n8ervatoire
Fr. Com. . . Holland, the Hague at the Conservatoire
Delezenne . . Milan, fork lent by Marloye
„ Lille, festival organ, fork of the tuner Mazingue
E. . . . Vienna opera, from a fork sent me by the organ-
builder, Ullmann, who had charge of the organ
there
Fr. Com. . . Marseilles Conservatoire
E. . . London, Exeter Hall organ, from a pipe of the
makers, Messrs. Walker, see 445*8
Lissajous . Paris, Italian opera, Bodin's fork
Hipkins . London, Covent Garden opera harmonium
E. . . . Gloucester Festival organ, built by Messrs. Walker ;
from the fork to which it was tuned at 64° F.,
the temperature of the pipe being reduced to 59°
Lissajous . Paris, Grand Opera— also at Lyons and Lidge
Schmahl . . Hamburg, opera, under Erebs
Fr. Com. . . Munich, opera
E. . . Leipzig, Qewandhaus Concerts, from fork sent offi-
cially to the Society of Arts
Lissajous . . Berlin, opera, fork of the conductor Taubert
E. . . . London, from Gramer*s c"533*3, purporting to be
the Society of Arts' pitch, intended for c''528
Cross & Miller Boston, Nichol's fork of Germania Orchestra, as
corrected to 59® F.
Fr. Com. . . Leipzig Conservatoire
Lissajous . Paris opera, experiments by Lissajous and Fer-
rand, the first violin
Hipkins . Covent Garden Opera, pitch of the harmonium
£. . . . London, from Griesbach's 0^534*5, tuned for the
Society of Arts as c''528 ; he tuned a' as 4457
Hipkins . . London, Covent Garden opera, taken from organ
a* during performance
Fr. Com. . . Prague, opera •
E. . . . London, firom copy of CoUard's standard fork
Lissajous . . Milan, opera
Delezenne . . Lille, from forks tuned by the oboist Colin, during
the performances of Robert le Diahle, 27 April
1854, between the acts, and carefully venfied
E. • . . Glasgow Public Halls organ, from fork settled by
the organist W. T. Best and the late H. Smart,
lent me by the builder Lewis
Cross & Miller U.S. America, Boston Music Hall, reduced from
pipe 0271-2 at 70** F.
Fr. Com. . . Bussian opera, from a cf' fork, probably miscalcu-
lated, as the a' from Broadwoods* c" forks were
E. . • . Belgian army pitch, reduced from Eoenig*s 451
vib. by his old standard, and also measured
from copy sent by Mahillon. On 19 March 1885
the Belgian Ck)vernment adopted French pitch,
^435
Lissajous Milan, Scala Theatre
Cross & Miller U.S. America, New York, from Chickering's 0268*5
standard fork
E. . . . British Army regulation, from fork lent by Dr.
W. H. Stone
E. . . The International Inventions and Music Exhibi-
tion of 1885 adopted this as the pitch of all
instruments for the exhibition, being the near-
est whole number to the next preceding and
next following. The fork was verified by myself
Digitized by V^OOQ IC
502 ADDITIONS BY THE TKANSLATOR- app. tt
Tablx I. — HisTomcAi. Pitches in obdxu fbom Lowest to Hiohbst — continued.
Cents
Date
Obeeirer
Place and other parttcnlars
6. Modem Orchestral Pitcht and * Church Pitch Medium — continaed.
349 452-5
f
350
354
355
357
358
359
362
452*9
453
4539
454
4541
1852-
1874
1880
1878
1645
1878
1862
1877
454*2 1 71 50
4547
4551
455*2
455*3
455'5
455*9
456-1
366
369
380
457*2
458-0
460-8
E.
Chambers & £.
E.
Sohmahl
E.
Naeke
E.
1874
1879
1878
1877
1749
1879
1859
1877
1880
1859a
1879
1880
Hipkins & £.
Sohmahl A E,
E.
Pr. CJom.
E. • •
Cross & Miller
B,
Cross & Miller
>i ft
London, mean of the pitch of the Philharmonic
Band onder the direction of Sir Michael Costa
1846-54, tuned during that period by Mr. J.
Black of Broadwoods*, approved by Sir Bfichael
Costa, and recorded by Mr. Hipkins, who lent
me the fork. Used as Broadwoods' highest till
1874, 1^0. 3 of French Conunission
NewcasUe-on-Tyne, Schulze's Tynedock organ,
from a fork tuned by Bir. Ch. Chambers, Mus. B.
Eneller Hall Training School for Military Music,
from a fork lent by Dr. W. H. Stone
*Holstein, Gliickstadt organ bcult 1645, improved
by Schnitger 1665, measured 1879
London, Wi]lis*B concert organ pitch, to which he
tuned the large organs in the Albert Hall and
Alexandra Palace, from pipe c''543*2 at 65^ F.,
and 54r2 at 61*5** F.
Vienna, piano of EapeUmeister Esser, while the
orchestra was at a'466, the regular fork at
a'456*i, and the piano of the other Kapellmeister
Prooh at a'445
Crystal Palace, from a fork cf'SAO lent by Mr.
Hipkins, to which the piano for concerts was
tuned
London, very old fork found at Brixton 1878 of
the same make as Bev. G. T. Driffield's tenor a,
see a'4i9*9
London, from 0^540*8, a fork representing the
highest pitch of the London Philharmonic ob-
served by Mr. Hipkins since 1874 ; at the sng-
gestion of Mr. Charles Hall6, used as Broad-
wood's highest pitch
London, Messrs. Steinway*s London pitch
London, Messrs. Bryoeson's band pitch, to which
they tuned their organ in St. Michael's, Corn-
hill, London
London, Wagner Festival at Albert Hall, tempe-
rature probably 61*5** F., see above ^'453*9
Hamburg, old positiv or chamber organ, buiU by
Lehnert, in possession of Herr Schmahl
London, Erard's concert pitch, from their fork
Belgium, band of Guides ; probably no such fork
existed. M. Bender used to give the pitch on a
small darinet, from which M. Mahillon has a
fork of at least a'456
London, fork used by one of Chappell's toners,
lent me by Dr. Stone
U.S. America, Cincinnati, pitch used in Thomas's
orchestra. [This is said by de la Fage to have
been the pitch sent by Bettini in 1857 for the
London Italian opera— evidently an error}
Vienna, fork tuned for me by the pianoforte
makers Streicher in Vienna from a fork in
their possession, giving the celebrated 'sharp
Vienna pitch ' before the introduction of the
French Diapaeon Normal, Naeke says he heard
a'466 in the actual playing of the orchestra
U.S. America, New Tork, from a fork obtained for
me bv Messrs. Steinway as representing their
American pitch
U.S. America, New Tork, from a fork furnished
by B. Spice as Steinway's pitch
U.S. America, highest New York pitch, from a
fork furnished by B. Spice ; these two last are
sharper than the next, but they are put first
because they belong to modem orchestral or
pianoforte pitch.
Digitized by
Google
ffiCT. H. THE HISTOBY OF MUSICAL PITCH IN EUEOPE. 503
Tablb I. — Historical Pitches in order prom Lowest to Highest — contiwued.
Gents
Observer
Place and other particulars
7. Church Pitch, High,
368
429
454
465
484
457-6
474*1
I 640c E,
1668
»»
1683
»»
1708
»l
1748
480*8
1879
484-1
1878
4892
1688
E.
Armes & E.
E.
Degcnhardt & E.
Jimmerthal
Sohmahl & E.
Vienna, Great Franciscan organ, stated by organ-
builder Ullmann to be 240 years old in 1878,
and to possess its original pitch ; only used for
leading the ecclesiastical chants
England, in the Pars Organica of Tomkins's
Musica Deo Sacra as quoted in Sir F. A. Gore
Ouseley's Collectum of the Compositions of
Orlando Oibhons, 1873, makes the / pipe
2^ feet long
Durnam, Bernhardt Schmidt's original organ at
Durham, which had both ab and gZ • The
pipe I measured in Feb. 1879 ^ ^'443*1 ^^ ^^^
shifted, and was orginally g% , which gives the
above pitch. This results from an examina-
tion of the original pipes by Dr. Armes, the
organist
Chapel Boyal, St. James's, Bernhardt Schmidt's
organ, now in Mercers' Hall, which I found on
examination had had the pipes shifted a great
Semitone. Handel played on this organ, and
hence his note ordering the voice parts of an
anthem written for the Chapel Boyal to be
transposed one Tone, and the organ part ttvo
Tones, referred to this organ
The Jordans' organ, Botolph Lane, from indica-
tions by Mr. T. Hill
Hamburg, St. Catherine Eirche, built by Hans
Stellwagen in 1543, and frequently repaired.
Herr Degenhardt, the organist, declares that
even at the last repairing, 1867-9, the pitch was
not altered. The original pitch, however, is
doubtful, and Herr Sohmahl thinks it was
altered formerly
Lubeok Cathedral, small organ, which according
to the organist Jimmerthal has its g' in unison
with the pipe on Schulze's new great organ
there, whidi gives French a' in summer at 68® F. ;
whence the above was calculated at 59° F.
Hamburg, St. Jacob! Eirche, built by Schnitger
of Harburg originally in equal temperament,
played on and approved by J. Sebastian Bach ;
pitch determined from an old pipe preserved in
the organ case. Herr Schmahl the organist is
accustomed to transpose all music at sight one
Tone lower, which brings it to French pitch
8. Church Pitch, Highest.
502
494*5
1879
506
495*5
1700
534
5037
1636
535
504-2
1511
541
505-8
1 361
Schmahl & E.
Schmahl
E.
E.
Hamburg, St. Jaoobi Eirche, present pitch, used
since 1866 in order to agree with Sdieibler's
forks, taking his a'440 for g'
Holstein, B6ndsburg,a large organ recently broken
up
Paris, Mersenne's ton de chapelle with G 11 2*6 on
the Frendi four-foot pipe, this being the lowest
note of his own voice
Heidelberg, from a model after Arnold Schlick,
who recommends that his 6^-foot Bhenish pipe,
having 301*6 vib., should give i^ or c. If it gives
F we have a'377, if it gives c we have the pre-
sent pitch
Halberstadt organ, built 1361, repaired 1495, de-
. scribed by PrsBtorius, who gives the dimensions
of the largest pipe IB,,,, whence constructing a
model I arrived at the above pitch, confirmed
by the four preceding pitches
1[
Digitized by
^.joogle
5^4
Cents
ADDITIONS BY THE TRANSLATOR. app. xx,
Table I. — Historical Pitches in order from Highest to Lowest — continued.
Date
Observer
Place and other particulars
9. Cha/mber Pitchy Highest^ and Chu/rck Pitchy Extreme,
726
740
563-1 1636 E.
567-3
1619
Paris, Mersenne's chamber pitch calculated from
F being the pipe of 4 French feet giving 1 12*6
vib. See Harmonie UnwerseUe, liv. 3, p. 143,
but from faulty measurement Mersenne makes
this pipe to have only 96 vib. But even with
that assumption the pitch would be a'48cri, as
at Hamburg, St. Catherine Kirche ; but compare
the next entry
North German church pitch, called by Pnetorius
chamber pitch, taken as a meantone Fourth
(503 cents) above Prstorius's * suitable pitch '
a'424'2, which see
Table II. — Classified Index to Table I.
The countries are arranged in alphabetical order: L Austro-Hungary; £1. Belgium;
IIL England, including Scotland and Lreland ; IV. France; V. Germany; YI. Holland; VH.
Italy ; YIH. Bussia ; IX. Spain ; X. United States of America, which for musical purposes
are included in Europe.
Under each country the pitches are classified as: i. Standards; 2, Old Forks; 3. Church
Organs; 4. Concert Organs; 5. Operas; 6. Concerts, including Conservatoriums ; 7. Piano-
fortes ; 8. Military Music ; 9. Other instruments.
The cents and pitch are as in the former table, to which, therefore, immediate reference can
be made.
Within each division the pitches are arranged first geographidally and then ohronologieaUy,
but for England the organs by the same makers are generally put together.
The mark „ means that the number or date above is to be repeated, and — that the date
or place is unknown.
I The pitches are cited with the greatest brevity which will allow of identification.
Date
Place
Pitch
Cento
164OC
1780
1878
1834*
1878
1859
1845
1780
1862
1879
Vienna
Pesth
Prague
Vienna
Brussels
I. AUSTBO-HUNOABY.
3. Chu/rch Organs,
Large Franciscan organ
Organ-builder Schulz
St. Stefan
368
225
313
Small Franciscan organ 311
5. Opera,
Scheibler, fork I. | 276
n.
„ in. . , .
„ IV. . . .
„ V. (Blahetka)
„ „ VI. (* monstrosity ')
Vienna Old Sharp Pitch .
Ullmann ......
Fr. Com
6. Concerts,
Marloye (Conservatoire) .
7. Pianofortes.
Stein, for Mozart
Esser, per Naeke
Proch, „
286
298
301
304
321
362
327
323
338
3^1
226
354
320
457-6
421*3
443*2
442*7
433*9
436-5
439*4
440*3
441-1
445*1
4561
446-8
446-0
449-8
445*4
421-6
454-0
445-0
II. Belgium.
I. Standards, I |
Mahillon's Army Standard • . . . .1 345 | 451*7
Digitized by V^OOQIC
SECT. H. THE HISTOBY OF MUSICAL PITCH IN EUROPE.
Tablb n. — Classifibd Indbx to Tabue l.—ooniinutd.
5«>S
Place
Cents
1859
1842
i860
1877a
I877P
17150
18000
18460
1625
1668ft
1660
1683
181SP
1879a
1690
♦»
1708
1759
1683
1879
1666
1670
1696
1700
1878a
1778
1780
1788
1790
1696
1749
1730
1820a
BrnsBels
BniBsels
London
II. Bbloium- oon^nued.
5. Opera.
Bender's pitoh
6. Concerts,
Conservatoire ....
8. Military Instruments.
Band of GoldeB (Fr. Ck>m.)
III. England, Scotland, and Ireland.
I. Standards.
Hullah'Bc"5i2, really 524-8 .
Society of Arts intended c"528
Oriesbach's attempt at c"528 a 534*5
Griesbaoh's a' to his c" .
Cramer's a' and c"
Tonic Solf a College . . . .
Plymouth
London
Lavenham
London
Durham
Hampton Court
t»
London
Cambridge
Temple
Worcester
Newcastle
London
»i
Norwich
Londoii
Winchester
Windsor
Kew
Dublin
Boston, Line.
London
Westminster
2. Old Forks.
Faraday's ....
Bev. G. T. DriflSeld's a .
Fork found buried at Brixton, a
Handel's own fork .
Broadwoods' c" . . .
Dr. Stainer's a' . . .
Bryceson's c" , " .
Church Organs and BellSt and Organ-builders^
Church Standards.
Church Bell (2'288-4
Tomkins's Bule ....
Bernhardt Schmidt :
Whitehall, original .
„ altered .
Original
(Altered)
(New, by Willis)
Chapel
Old pipe of original .
St. James's Chapel Boyal, original
Trinity College, after shifting .
Original
Altered
T. A R. Harris :
Cathedral ....
Renatus Harris :
St. Nicholas ....
St. Andrew Undershaft .
St. John's, derkenwell .
(?) Cathedral ....
Green:
St. Katharine's, Regent's Park
Restoration of College organ .
St. George's Chapel .
Parish Church ....
Trinity CoUege (altered ?)
Christian Smith :
Parish Church (restored ?)
Olyn & Parker :
All Hallows the Great and Less
Schreider d Jordans :
Original
(Altered)
310
331
359
317
280
295
^Z7
230
442-5
448*0
455-5
305
441-3
310
444-0
337
4495
322
4457
333
448*4
250
427-5
230
422-5
237
424-3
219
419-9
355
454-2
230
4225
231
422-7
23X
4246
270
432-3
265
431-3
429
474-1
3^
44X-7
429
474-1
312
443-1
318
444-7
308
442*0
307
441-7
429
474-1
114
395-2
444-3
437-1
25s
428*7
251
427*7
215
419-0
249
4272
233
423*2
220
420*1
251
427*8
230
422-6
287
436*8
438-9
424-3
422*5
Digitized by V^OOQIC
5o6
ADDITIONS BY THE TRANSLATOR.
Table £1.— Clabsifibd Index to Table L— continued.
AFP. XX.
Date
Pitch
Cents
I740P
1764
London
Halifax
1855
Barking
It
1826
1847
Shrewsbury
»i
II
1740
Gt. Yarmouth
I74«
Maidstone
London
1838
Bath
1843
Wimbledon
1878
London
—
Newcastle
1879a
»»
It
It
Salisbury
Glasgow
Edinburgh
London
1878
m.
England, Scotland, and Isslajhd— continued.
3. Church Organs, rfc—continued.
Schnetsler :
German Chapel Royal
Parish Church
By field d Green :
Original probably . . . . * .
(Restored by Walker)
/. Byfield d J, Harris :
Original
(Altered by Blythe)
(Altered by Gray & Davison) ....
Byfield, Jordan, d Bridge :
St. Gorge's Chapel
Jordans:
Old Parish Church
St. George's, Botolph Lane ....
Parish Church (altered?) . . . .
Smith of Bristol :
Abbey Church
Walker :
Parish Church
Bryceson :
St. Michael's, Comhill
Schulze :
Tynedock
H. Willis :
Cathedral
Established Church Cathedral
Episcopalian Cathedral
St. Paul's, present state (like the other three
at 59° F., but) at S7°'5 P
Organ-builders* Standard Pipe,
Bishop, c"5 18-5
Gray <& Davison, c"523*2
Walker, c"523-6
Bevington, c"5237
Lewis, c"524*4
Hill.c"S25-3
Bryceson, c"527*3
H. Willis (church), c"S29-4
Experimental English i-foot Pipes.
Diam. 1*2 inch ; wind 2| inch. ; vib. 477-0
taken as c" gives]
" *l/b " \ ^ meantone temperament ■
»t n »' »• )
Same diam. ; wind 3^ inch. ; vib. 478*7
taken as c" gives
in meantone temperament
1/
6'b
a'
Bernhardt Schmidt's, same dimensions; wind 2^
inch. ; vib. 472*9
taken as &' gives
It n ^
II tt ^b
,. ,1 a'
Diam. '95 inch ; wind 3^ inch,
taken as c" gives
ti It o ,,
tt It ^b ti
It It ^ It
Diam. *75 inch. ; wind 3^ inch. ; vib. 498*6
taken as c'' gives)
in meantone temperament
vib. 488*7
in meantone temperament
b'
b'b
a'
in meantone temperament
Digitized by
Goo^
242
429
298
275
244
289
424
262
230
246
357
350
320
II
ti
318
284
300
301
302
304
307
313
320
133
247
323
440
136
253
329
446
"5
231
308
425
171
289
482
206
323
400
516
lECT. H. THE fflSTORY OP MUSICAL PITCH IN EUROPE.
Table II. — Classitisd Indxx to Tabus L—contiwued.
507
Date
Place
Pitch
Cents
1805
1867
1879
1877
1877a
1877
1857
1877
1878
London
GloQoester
Glasgow
London
»»
Sydenham
London
1877
>»
1878
l>
St
If
1879
»»
1880
If
ft
1878
ft
1879
If
1880
ft
'^11-
ft
If
1826
ft
1846-
f»
1854
1874
ft
1877
Sydenham
»»
London
1826
It
1849-
ft
1854
I854P
>•
i860
ft
1852-
tf
1874
I874P
ft
1846a
f>
I846P
If
1877
ft
1879
If
ni. Enolamd, Scotland, and Ibeland — continued,
4. Concert Organs.
ElUott:
Ancient Concerts from d"$6&'2
Walker :
Exeter Hall, original
„ „ sharpened
Festival organ
Lewis:
Public Halls
H. WilUs :
Concert Standard at Albert Hall and Alex-
andra Palace
Albert Hall, observed at 61-5^ F. .
Gray A Davison :
Crystal Palace
Bryceson:
Band pitch
5. Opera,
Opera, Bettini's fork (correct?) . . . .
Covent Qa/rden, :
Harmoniom
Organ (Bryceson's fork)
Harmoniom
Organ (heard)
Band (performing)
Theatre fork (season 1880) ....
Her Majesty^s :
Organ
Band (performing)
Theatre fork
6. Concerts.
Philharmonic :
Copy of original fork
Another copy
Approved by Sir G. Smart ....
Mean pitch while the concerts were under the
direction of Sir M. Costa ....
Highest
Crystal Palace band
Wagner Festival at Albert Hall
7. Pianofortes.
Broadwoods' lowest, London No. i of Fr. Com.
medium, London, No. 2 of Fr. Com.
copy now used
copy made for Society of Arts ....
highest, London No. 3 of Fr. Com. (which
calculated all these forks wrongly)
present highest
Hiplans's Vocal pitch (meantone) ....
ft If (equal)
Collard
Erard
Steinway (in England)
Chappell
8. Military Music.
British Army regulation
Kneller Hall Training School ....
239
323
328
329
341
358
355
357
362
336
305
329
322
282
285
320
319
235
233
272
349
357
355
358
272
323
324
321
349
357
274
284
339
359
357
362
346
350
4299
445-8
447*3
4477
4506
453-9
455-1
454-1
454-7
456-1
449*2
441-2
447-5
445-6
449'7
435-4
4361
445-5
444-9
423-7
423-3
4330
452*5
454-7
454*1
455*1
433-0
445-9
4462
445*5
452*5
454-7
433-5
436-0
449-9
455*3
454*7
4559
451-9
452-9
f
Digitized by
^.joogle
5o8
ADDITIONS BY THE TRANSLATOR.
Table IL— CiiAssiFiKD Index to Tabu L-H^ontinued,
APP. XX.
Divte
Place
Pitch
1 Cents,
1648
PariB
1766
I*
1854
♦1
I700P
»»
1832
If
1834
»»
1858
II
1859
II
II
1 700c
Lille
1754
II
18000
1854a
II
1859a
It
I8IOC
Paris
1636
«i
1700a
LiUe
1789
Versailles
I8I8
Paris
1840
II
1851a
liiUe
I85I
I8II
I8I9
1822
1824
1829
18340
1836-
1839
1854a
1855
1856
1858
1823
1856
1820
1823
1836
1859
1838-
54
1859
1836
1812
Paris
Bordeaux
LiUe
Lyons
Toulouse
Paris
IV. France.
1. Standards.
One French foot pipe :
Mersenne c"447
Dom BMos c"45o-5 .
Delezenne, c"446'4 .
Pitch-pipe at Faculty of Sciences .
de Prony's proposal
Marloye's „ ...
CavaiU^-Coll's proposal .
Fr. Com. „ ...
Diapason Normal, at Conservatoire
2. Old Forks,
Mazingue's
Fran^ois^B
Cohen's
Delezenne*8
Marquis d'AHgre's ....
Lemoine's
3. Church Organs,
Mersenne's Um de chapelU
L'Hospice Comtesse
Palace Chapel, fork at Conservatoire
Tuileries Chapel ....
St. Denis (Ctfvaill6-Coll) .
St. Sauveur
La Madeleine (restored) .
St. Andr6
4. Concert Organ,
Festival organ ....
5. Opera,
Grand Opera :
Boheibler
Cagniard de la Tour
Fischer
lowered for Branohu
recovered pitch
orchestral pitch
Scheibler's Petitbout
Delezenne's Leibner
„ forks
Lissajous and Ferrand
Bodin
Fr. Com
Italian Opera :
Fischer
Bodin
Op4ra Comiquet or Feydeau.
fork at Conservatoire
Fischer
Cagniard de la Tour
Provincial Opera ;
Fr. Com
Delezenne ....
Fr. Com
ij . . . , ,
6. Concerts,
Merscnne's ton de chambre
Conservatoire, fork there
17
31
15
178
307
262
316
280
282
66
230
255
272
297
260
534
19
117
278
317
69
129
269
327
248
276
267
243
276
300
276
304
289
335
323
331
237
329
232
250
304
312
340
331
310
726
298
Digitized by V^OOQIC
SECT. H. THE HISTORY OF MUSICAL PITCH IN EUROPE.
Table II. — Clasbifibd Index to Table I,— continued.
509
Place
Pitch
Cent«t
1822
1834
1815-
1821
1859
1878
1859
Paris
Toulouse
Marseilles
Paris
Brunswiok
Stuttgard
Eutin
Dresden
N. Germany
Saxony
Heidelberg
»>
Hamburg
Palatinate
Holstein
Hamburg
»i
Holstein
Saxony
Strassburg
Saxony
Hamburg
Dresden
Hamburg
Weimar
Dresden
Lubeck
Hamburg
Berlin
Dresden
Brunswiok
Carlsruhe
Gotha
Weimar
Stuttgard
IV. YtLkscE^amtinued,
6. Concerts — continued.
Conservatoire, Scheibler I.
.. II. . .
„ III. (Gand)
de la Page
Fr. Com. .
I 7. Pianofortes y Spinets, rfc.
Mersenne's spinet ....
' Saaveur
' Pascal Taskin ....
' Piano of opera ....
I WSlfePs
y. Gerxant.
1. Standards.
Praetorius's suitable pitch
Soheibler's pitch (reduced to 59° F.) adopted at the
Coilgress of Physicists
2. Old Forks.
F. Anton von Weber's
Eirsten's .
Rummer's
3. Church Organs (in order of date).
Pretorius (called by him chamber pitch) highest
recorded
Halberstadt
Schlick, high jpitch
„ low pitch
St. Catherine (in 1879)
Salomon de Cans
Gliickstadt
St. Jacobi, low stop, old pitch . . . .
„ high stops, „
Bendsburg
Freiberg Cathedral, Silbermann . . . .
Minster, A. Silbermann
Dresden, St. Sophie . . . . <
Lehnert's jTon^ti;
Chained fork of the Boman Catholic Church
Mattheson's St. Michaelis
Topfer's pipe .
Boman Catholic Church .
Cathedral, old organ
St. Jacobi, modem pitch
5. Opera (arranged by towns).
Fischer's Pichler's fork ....
Scheibler, ' trustwortiiy ' ...
282
303
282
324
288
327
148
163
174
242
313
237
301
236
Naeke's fork of Weber's time 233
Fr. Com. .
Jehmlich's fork
Fr. Com. .
4353
4409
435-2
4462
437-0
447
402-9
4066
4090
425-5
4433
4242
440-2
424-1
229
4223
239
4249
740
5673
541
505-8
535
504-2
33
377-0
454
480-8
119
3964
35°
453-0
184
411-4
484
4892
506
495-5
217
419-5
104
393-2
201
415-5
351
455-2
199
4150
169
407-9
237
424-4
212
418-1
465
484-1
500
494-5
289
437*3
307
441-8
233
4232
304
441-0
297
439*4
278
443*5
280
435-0
313
4433
319
444-8
312
443
Digitized by V^OOQ IC
5IO
ADDITIONS BY THE TRANSLATOK.
Table £1.— Classified Index to Table I. — continued.
APP. XT-
Date
Place
Pitch
Oenta
1859
1869
1879
1859
»»
1869
1776
1859
1720
1 730c
1 780c
1 730c
17800
1845
1856
1857
1859
1869
1839
1781
i860
1802
1858
Munich
Baden
Wurtemberg
Hamburg
Leipzig
Wurtemberg
Leipzig
Brealau
The Hague
Rome
[ Padua
Florence
Milan
Turin
Milan
»»
Naples
Turin
Bologna
St. Petersburg
V. Gebmany — contmued.
Opera (arranged by towns)— continued.
Fr. Com
Sent to Society of Arts
»f If .•••..
Similar forks sent from Berlin and Munich, which
had adopted French pitch
Opera under Erebs
6. Concerts.
Old orchestral pitch
Conservatoire Fr. Com
Fr. Com
Oewandhaas, sent to Society of Arts
9. Instruments.
Marpurg ....
Naeke's Schneider's oboe
VI. Holland.
3. Church Organs.
The old celebrated Church organs had all been al-
tered, and I have not succeeded in recovering
their ancient pitch
6. Concerts.
Fr. Com
VII. Italy.
I. Standards.
Pitch-pipes of Dr. R. Smith
Mean of pitch-pipes of the bell-fonndry of Col-
bacchini
2. Old Forks.
From Colbacchini's low/"
high/" , ... . .
5. Opera.
Marloye
Fr. Com. .
La Scala (de la Fage)
San Carlo (Guillaume)
Fr. Com. .
6. Concerts.
Liceo Musicale (Society of Arts)
7. Pianofortes.
Tadolini's fork . . . .
VIII. Russia.
3. Church Organs.
Euler
332
278
288
331
320
334
3*9
332
196
191
4481
434-5
436-9
448-0
445-0 I
448-8 I
444-8
448-2 I
414-4
413-3
334
114
241
152
230
287
326
299
349
345
319
319
312
243
211
224
284
345
Fr. Com. (French pitch was afterwards adopted)
Digitized by V^OOQIC
446-2
395-2
425-2
403-9
422*6
436-7
446*6
439-4
450*3
45«7
444-9
444*8
4431 !
425-8
4iS-o
421*2
436-0
4S'-5
iECT. H. THE HISTORY OF MUSICAL PITCH IN EUROPE.
Table II. — Classitibd Index to Table I. — conliniied.
5^1
Date
Place
Pitch
Cents
1785
1858
1868
1880
1879
1880
Seville
Madrid
New York
Boston
New York
It
Boston
Cincinnati
New York
IX. Spain.
3. Church Organs.
T. Bosch's organ
Ton de Ghapelle
5. Opera.
Theatre (French pitch adopted in 1879) •
X. United States of America.
£. S. Bitohie's standard, and Mason & Hamlin's
French pitch
Church of Immaculate Conception .
Hutchings, Plaisted & Co., * low organ pitch '
NichoPs Fork, Germania orchestra .
Music Hall organ (from 1863 to 1871 at French
pitch)
Organ tuned to Thomas's orchestra
Steinway's American pitch, from a fork furnished
by Steinway
Steinway's, from a fork furnished by B. Spice
Highest New York pitch, from a fork furnished by
B. Spice
218
218
318
283
315
316
333
342
362
366
369
380
419-6
•419-6
444*5
435*9
443*9
444*2
448-5
450*9
456-1
457*2
4580
460*8
Conclusions.
Art. 6. The two preceding tables contain the &icts of the histoiy of musical
dtch in Europe since 1361, the date of the Halberstadt organ, that is for 500 years,
o fiftr as I have been, able to collect information, and I have been fortunate enough
o bring together such an amount of historical evidence that probably no new IT
acts could be ascertained which would materially change the conclusions to which
have been led. These are very briefly as follows.
Art. 7. The organ was originally a mere collection of pitch-pipes, each with a
ixed tone, to steady the voice of the singers of ecclesiastical chants, replacing the
ingle pitch-pipe with a movable piston or some instrument like the flageolet
whistle) and oboe, which subsequently gave rise to the two distinct series of flue and
eed pipes. But when thus collected it was necessary to fix a pitch. The guiding
»rinciples were the compass of the male voice, the rules of ecclesiastical song, the
lase of the performer, to avoid introducing chromatics as much as possible (Schlick),
knd the standard measure or foot rule of the country. The latter suggested a
rhole number of feet for the length of the standard pipe, generally four feet, about
he lowest note of the tenor voice, and the question thus rose what note should
his tone represent ? Here the answer came from ecclesiastical use, — either F or
. Schhck recommends both, thus giving pitches for any given note a whole
rourth apart. Bchlick's high pitch, arising from giving a 6^-foot Rhenish pipe ^
o c, made a^5o4'2. (All pitches named should be referred to in Table i.) His
ow pitch arising from giving the same pipe to F, made a'377. These are a Mean-
one Fourth apart.
Art. 8. — The foot had very different lengths
Q different countries. If we suppose the ' scale *
or ratio of diameter to length of pipe) and the
orce of wind to remain the same (both in fact
aried much), then the influence of the length
•f the foot on the pitch of the organ, suppos-
og the four-foot or one-foot pipe to be given
o the same note, may be appreciated from
he table on p. 512a. In this we see a differ-
nce of more than a Tone, nearly a minor
niird, between the pitch of a i-foot pipe in
l*rance and in Saxony. The difference be-
ween the pitches of pipes of the lengths of
the English foot and French foot is more than
an equal Semitone. Hence probably it hap-
pened that the lowest French pitch measured,
a'374'2, is a Semitone flatter than the lowest
English pitch measured a'395*2. Length of
foot alone would therefore account for great
variety of organ-pitch, to which we must add
force of wind (see the notes on experimental
English I -foot pipes, p. 506c) and different
methods of voicing. The low pitches were (and
still are on old organs) prevalent in France
and Spain, the high pitches were at home in
North Germany (see Table II.).
Digitized by VjOOQIC
5"
ADDITIONS BY THE TRANSLATOR.
AFP. XX.
Names of Feet
Intcrral
Long old French foot, or pied deroi .
Long Austrian foot
Long German, or Bhenish foot .
English foot
Old Niirnberg foot
Old Augsburg foot . ^ ^ . .
Old Roman foot (medieval)
Bavarian foot
Short Hamburg and Danish foot
Short Brunswick and Frankfurt foot .
Short Saxon foot
mm.
325
316
314
305
304
296
295
292
286
28S
283
oenta
0
49
60
109
116
162
168
185
221
227
239
Art. 9. — The solo instmiments were tuned very variously. But it became the
^ custom to have a band to play with the organ, and the princes and petty dukes used
the same bands to play in their private apartments or * chamber.* The very high
and very low pitch were generaUy found unsuitable for non-ecclesiastical music.
Hence the instruments usually adopted a pitch lower than the high and higher than
the low, and this was called * chamber pitch,' the other being distir^ruighed as
' church pitch.' But the same instruments had also to play with &e organ.
Hence the difference had to be a definite number of degrees of the scale, a Semi-
tone, a Tone, or a minor Third. See a'407-9, and especially a'4ii'4, which com-
pare with a'480'8, and a'484-i respectively. This was, however, not always the
case, for the very high church pitch, a'5037 had a still higher chamber pitch
a'563-i.
Art. 10. — But this great variety occasioned much trouble, and the chamber
pitch below the high and above the low church pitch seems to have suggested
PrsBtorius's * suitable pitch ' of a'424-2 in 1619. This was in fact a * mean pitch,'
and as such rapidly found such favour that it spread over all Europe and, with
H insignificant varieties (from a' 41$ to a'428'7 at the extremes, an interval of 54
cents, or a quarter of a Tone), prevailed for two centuries. Handers own fork,
a'422'5 in 1 75 1, quite a common pitch at the time, and the London Philharmonic
fork, a'423*3 from its foundation to 1820, are conspicuous examples, but an inspec-
tion of the numerous pitches cited in Table I. sect. 4 (pp- 495^7), will prove tiie
fact beyond doubt.
Art. II. — As this was the period of the great musical masters, and as their
music is still sung, and sung frequently, it is a great pity that the pitch should
have been raised, and that Handel, Haydn, Mozart, Beethoven, and Weber, for
example, should be sung at a pitch more than a Semitone higher than they in-
tended. The high pitch strains the voices and hence deteriorates from the effect
of the music, when applied to compositions not intended for it. Of course for
music written for a high pitch the compass of the human voice is properly studied
(see App. XX. sect. N. No. i), and so much music has in the last fifty years been
written for a high pitch, that to perform both properly two sets of instruments
would be required. Two sets are actually in use at Dresden, one for the theatre
If a'439'4, and one for the Roman Cathohc Church having a'415, diff^erence 98 cents,
or about a Semitone.
Art. 12. — The rise in pitch began at the great Congress of Vienna, 181 4, when
the Emperor of Eussia presented new and sharper wind instruments to an Austrian
regiment of which he was colonel. The band of this regiment became noted for
the brilhancy of its tones. In 1820 another Austrian regiment received even
sharper instruments, and as the theatres were greatly dependent upon the bands
of the home regiments, they were obliged to adopt their pitch. Gradually at
Vienna, pitch rose from a'42i*6 (Mozart's pitch) to a'456'i, that is, 136 cents, or
nearly three-quarters of a Tone. The mania spread throughout Europe, but at
very different rates. The pitch reached a'448 at the Paris Opera in 1858, and the
musical world took fright.
Art. 13. — The Emperor of the French appointed a commission to select a pitch,
and this determined on a'435, but made a fork called Diapason normal, now
found to be a'435'4, which is preserved at the Mus^e du Conservatoire, and is the
only standard pitch in the world. This pitch was widely adopted, but it i{5 56 cent^,
Digitized by VjOOQlC
SECT. H. THE HISTORY OF MUSICAL PITCH IN EUROPE. 513
or over a quarter of a Tone, sharper than Mozart's pitch, although it was 80 cents,
fully three-quarters of a Semitone, flatter than the old Vienna sharp pitch a' 456*1,
and 49 cents, or a quarter of a Tone, flatter than the then French opera pitch
a' 448. This pitch had been reached independently in many places, and the French
commission had been twitted at taking a Carlsruhe pitch. But it is not generally
known that Sir George Smart's pitch a! 433, adopted with much hesitation for the
London Philharmonic Society about 1820, and extensively sold in London as the
' London Philharmonic * for many years before the French Commission of 1859,
was in fact an anticipation of the French piitch. Both were compromises, a partial
yielding to the new without entirely disregarding the old. The pitches a' 430 to
a' 436-9, therefore (interval 28 cents, or about \ Tone), forming Table I. Sect. 5,
pp. 497-8, are termed the * compromise pitch.' As instruments exist for this
pitch it is the only one that has a chance of being used beside the present
sharp pitch of England. Several attempts have been made to restore it« notably
at Covent Garden Opera in 1880. But the expense of new instruments for a
band, about i,oooZ., renders any alteration extremely difficult to carry out. The ^
tendency in England has been to sharpen, and our orchestral and pianoforte pitch
is now from a' 449*7 to a' 454*7, a difference of only 19 cents, not quite a comma.
In the United States, however, the pitch has reached a' 460*8, that is 23 cents, or
about a comma more. In Germany the compromise pitch adopted was a' 440*2
as proposed by Scheibler, and it is curious that the standard pipes of the English
church organ builders vary from al 436 to a' 445*2, 36 cents, but are mostly
between 440 and 441 '7, an interval of only 7 cents. The concert organs, of course,
follow orchestral pitch. (See Postscript, p. 555.)
Art. 14. — In England the pitch of organs varied with the note on which the
four-foot or one-foot pipe was placed. We have only one record that the one-
foot pipe was placed on c" giving a' 395*2, whereas the same pipe made to give b'
produced a' 423, the mean pitch, which so long prevailed. Put on b^} it produced
a^ 442, which as a' 441*7 was Bernard Schmidt's low pitch, and is still the pitch
of Mr. T. Hill, the organ-builder. Placed on a' it gave a' 472*9, which as a' 474*1
was the highest church pitch used in England, just a Tone above mean pitch.
(See p. 505c, III. 3, for details.) ^
Art. 15. — If we look into the secrets of the rise of pitch we find it always con-
nected with wind instruments. The first rise was from a military band, and the wind
and the brass have constantly rebelled against a low pitch. The singers have not
prevailed against them except for a very short time. The great violin school of
Oremona in Italy lived in the time of mean pitch with a higher chamber pitch,
and the resonance of the boxes of their violins seems to shew traces of the action
of both pitches (suprd., p. 87, note *), but their great object was to insure tolerable
uniformity of reinforcement, and hence they are a treasure for aU time.
Art. 16. — The only possible conclusion seems to be that to sing music written
for pitches different from our own, we must either transpose a Semitone (always a
difficulty, and for some instruments an impossibility) or adopt a new compromise
pitch, the French, already once firmly rooted in England as Sir George Smart's, and
standing half-way between the extremes. On the continent, as formerly shewn in
France, and quite recently in Belgium and Italy, the government has a certain
power in fixing musical pitch, by refusing to subsidise conservatories and theatres
which do not adopt the pitch ordered, and commanding the regimental bands to %
make the change. But beyond this their power does not extend, and the various
regulations which have been made in the two countries last named shew the
great difficulties that have to be overcome in introducing a new pitch even within
the area under government control. In England, however, there are no subsidised
operas or musical conservatories, and even the instruments of the military bands
are not provided by government. Hence the change must be left to the gradual
action of musical feeling. We have already changed in England almost imper-
ceptibly. The raising of English pitch from Sir George Smart's a' 433 was to a
great extent due to the individual action of the late Sir Michael Costa while con-
ductor of the Philharmonic concerts 1846-45 (mean a' 452*5, extreme a' 454*7),
to whose insistence is also due the high pitch of the Albert Hall concert organ,
a' 453*9. Perhaps a similar energetic conductor will arise to turn the tide of
musical opinion in the opposite direction.
Digitized by
GcJogle
5M
ADDITIONS BY THE TRANSLATOR.
APP. XI.
SECTION K.
NON-HARUONIC SCALES.
(See Notes, pp. 7i» 95. 237. 253» 255, 257, 258, 264, 272.)
Art.
I. iDtroduotion, p. 514.
n. Table of Non-Harmonic Scales, p. 514.
III. Annotations to the Table, p. 519.
IV. How these Divisions of the OetaTe maj
have arisen, p. 522.
V. Besalts of the inquiry, p. 524.
I. Inteoduction.
For particulars of my researches into non-harmonic scales, see my two papers,
first : * Tonometrical Observations on some existing non-harmonic Scales * (Proc.
of the B. Society for Nov. 20, 1884, vol. xxxvii. p. 368) and second * On the Musical
Scales of Various Nations * (Journal of the Society of Arts for March 27, 1885,
m vol. xxxiii. p. 485), in both of which I was most materially assisted by Mr. Alfred
James Hiplans of Messrs. J. Broad wood and Sons*.
Properly speaking there is only one harmonic scale, that is, a scale which
allows the musician to produce chords without beats, and therefore has notes with
pitch numbers composed of products and multiples of the powers of 2, 3, 5, 7, 17,
as shewn in Sect. E. But the term harmonic may be extended to all tempered
imitations of such scales as are not worse than equal intonation. If we did not
extend the use of the term thus far, we should find absolutely no harmonic scale in
practical use, except by the Tonic Sol-faists when unaccompanied (App. XVIII.)
Even with this extension of meaning, non-harmonic scales are greatly more
numerous than harmonic. Harmony was a European discovery of a few centmiei
back, and it has not penetrated beyond Europe and its colonies.
scale is then found by subtraction. The
number of the note in the scale is usually
placed at the top, so that the eye can, at a
glance, compare the different usages. The
ratios represented by these cents may g«ii»-
rally be found from the table in Sect. D.
Each scale is numbered, and in the annoCs-
tions immediately following the table, sevecsl
particulars are given. It was not, howeter,
possible to include every case in this arrange-
ment. The complete ancient and medienl
Arabian lute, Bab&b, and Tambour scales, and
the complete Indian scales both in the old and
modem form, and some others are therefoR
differently ordered, preserving, however, tbe
expression of notes by cents as above ei*
plained. See Nos. 66 to 75.
In order to obtain a bird's-eye glance over
the scales given theoretically by ancient Greek
writers (as interpreted in the text), by ancient
m and medieval Arabic writers (as interpreted
by Professor Land); by modem Arabic
theorists (as reported by Eli Smith); by
Indian musicians (as reported by Bajah
Sourindro Mohun Tagore) ; and those which
I have deduced from Javese, Chinese, and
Japanese instruments, with those of other
countries, examined by Mr. Hipkins and my-
self, I have constracted the following table.
The scale is represented by the numbers
of cents in the interval by which any one
of its notes is sharper than the lowest note,
and is generally confined to one octave.
The interval between any two notes in the
II Table of Non-Habmonic Scales.
Old Oreek Tetrachords.
1. Olympos .
2. Old Chromatic .
3. Diatonic
4. IHdymns .
5. Doric
6. Phrygian .
7. Lydian
8. Helmholtz
9. Soft Diatonic .
10. Ptolemy's equal diatonic
XI. Enharmonic
I.
II.
in.
IV.
0
112
4^
0
112
182
498
0
112
316
«!
0
112
294
<9!
0
90
294
498
0
182
316
498
0
182
386
498
0
112
386
49S
0
85
267
498 ,
0
151
316
498 '
0
55
113
498
Greek Tetrachords after Al Fa/rabi reported by Prof. Land.
I. Oenus molle, ordinatttm.
a, continuum : —
12. laxum I o I 386 I 441
13. mediocre \ ^ 3'^ ' 4^5
14- acre 1 o 267 j 386 ,
498
498
498
Digitized by VjOOQlC
BCT. K.
NON-HARMONIC SCALES.
II. Tabls of Non-Habmonio Scales — continued.
515
Greek Tetrachorde after Al Fardbi reported by Prof. Land— continaed.
I. Genua molley ordinatum — oont.
b. non oontinuum.
15. lazum (enharmonic) .
16. mediocre (soft chromatic)
17. acre (syntonically chromatic) .
II. Genua forte.
a. duplicatom: —
18. primum
19. secundum
2a tertinm
6. conjonctnm: —
21. primnm (entonically diatonic)
22. seonndum (syntonically diatonic)
23. tertinm (equally diatonic)
6. disjunctum:—
24. primum (soft diatonic)
o
o
a
o
o
o
o
o
o
n.
386
316
267
231
204
182
231
204
182
231
TIL
460
435
418
462
408
365
435
386
347
413
IV.
Most Ancient Form of Oreek Scales with 7 Tones and Octave,
VI. viL vin.
25. Lydian
26. Phrygian .
27. Doric
28. Hypolydian
29. Hypophxygian (Ionic)
30. Hypodoric (Eolic)
31. Mixolydian
I.
11.
III.
IV.
V.
0
182
386
498
702
0
182
316
498
702
0
90
294
498
70a
0
204
386
590
702
0
204
386
498
702
0
204
294
498
702
0
112
294
498
610
884
884
792
884
884
792
814
Later Oreek Scales toith Pythagorean Intonation,
32. Lydian
33. Hypophrygian (Ionic)
34. Phrygian .
35. Eolio
36. Doric (same as No. 27)
37. Siixolydian
38. Syntonolydian .
0
204
408
498
702
906
IIIO
0
204
408
498
702
906
996
0
204
294
498
702
906
996
0
204
294
498
702
792
996
0
90
294
498
702
792
996
0
90
294
498
588
792
996
0
204
408
612
702
906
IIIO
1200
1200
I203
1200
1200
1200
1200
Al Farahi's Oreek Scales as reported by Prof. Land,
39. Qenus conjunotum medium
40. Genus duplicatum medium, or dito-
num (same as No. 38) .
41. Qenxxs conjunctum primum
42. Genus forte duplicatum primum
43. Genus conjunotum tertium, or forte
flsquatum
44. Qenus forte disjunctum primum
45. Genus non continuum acre
46. Genus non continuum mediocre
47. Gbnus non oontinuum laxum .
48. Genus ohromaticum forte .
49. Genus ohromaticum moUissimum
50. Genus moUissimum ordinantium
0
204
408
590
702
0
204
408
612
702
0
204
435
639
702
0
204
435
666
702
0
204
386
551
702
0
204
435
6i7
702
0
204
471
629
702
0
204
520
639
702
0
204
590
664
702
0
204
471
690
702
0
204
520
613
702
0
204
590
647
702
906 , 1088 ; 1200
906
933
933
884
933
969
1018
1088
969
1018
1088
I no 1 200
II37 I 1200
1 164 I 1200
i
1049 I 1200
III5 I 1200
I 120 I I20O
"37
1 162
1088
nil
1 145
1200
1200
1200
1200
1200
H
Arabic and Persian Scales as reported by Prof. Land.
51. Zalzal. see No. 66 . . . . | o | 204 | 355 | 498 | 702 | 853 1 996 | 1200
Highland Bagpipe made by Macdonald of Edinburgh.
52. Observed 1 o | 197 | 341 I 495 | 703 I 853 1 1009 | 1200
Modem Arabic Scale as reported by Eli Smith,
53. Meshaqah, theoretical . . ; o 200 ' 350 | 500 | 700 [ 850 | 1000 | 1200 \
Digitized by VjOQQ IC
Si6
ADDITIONS BY THE TEANSLATOE.
II. Tablx of Kon-Habuonxc QckLKB—eontinuid,
APP. XI.
H
Arabic Medieval Scales as reported by Prof. Land with 7 Tones and Octave.
I. II. I III. i IV. V. VI. Vn. VHL
54. *Oohaq (same as No. 33)
55. Nawa (same as No. 34)
56. BoasUik (same as No. 37)
57. Bast .
58. Zenkonleh
59. Bahawi
60. Hhosaini .
61. Hhidjazi .
II.
III. 1
204
408
204
294
90
294
204
3«4
204
380
180
384
180
294
180
294
IV.
498
V.
702
498
702
498
588
498
702
498
678
498
678
498
678
498
678
906
906
792
882
882
792
906
882
996
996
996
996
996
996
996
996
1200
1200
1200
1200
1200
1200 ;
1200
1200
Arabic Medieval Scales with 8 Tones and Octave.
62. 'Iraq
63. Isfahan
64. Zirafkend
65. Boozoork ,
II.
180
180
180
180
III.
384
384
294
384
IV.
498 678
498
498
498
702
678
678
VI.
882
882
792
702
VII. vin. IX.
996 1 176
996 1176
882 I10861200
906 ,10861200
1200
1200
66. Earlier Notes on the Arabic Lute as reported by Prof. Land.
*^^* First, Sxoond, Ac, refer to the strings. The notes are named from the fingers — ^indez,
middle, ring, little — by which they were played.
Notes
C
D
Fb
E
F
Gb
G
Ab
Bbb
A
Bb
Cb
First OctaTe
Fibst: open
ancient near index
Persian near index
Zalzal's near index
index
ancient middle .
Persian middle .
Zalzal's middle
Second OctaTe
Cents
Oct.]
Third : index ,
ancient middle .
Persian middle
Zalxal's middle
ring ....
little a. Fourth: open
nng
little « Second : open ^ .
Second : ancient near index
Persian near index
ZalzaPs near index
index
ancient middle .
Persian middle .
Zalzal*s middle
nng
little = Third: open .
Third : ancient near index
Persian near index
Zalzal*s near index
index
Fourth : ancient near index
Persian near index
Zalzal's near index
index
ancient middle .
Persian middle .
2Uzal*s middle
ring ....
little aFiTTH: open .
Fifth : ancient near index
o
90
145
168
204
294
303
355
408
498
588
Oct. a
IMO-t-
O
90
99
151
204
294
384
439
462
498
588
597
643 -
Persian near index
Zalzal's near index
Fifth : ancient middle
Persian middle .
Zalzal's middle
ring .
666
702
792
801
853
906
996
1086
1 141
1 164
1200
649
702
792
882
937
960
1086
1095
1147
Digitized by V^OOQIC
SECT. K.
NON-HARMONIC SCALES.
II. Table of Non-Habmonio BoAiMa^~<ontinued.
517
67. Medieval Arabic Scales as reported by Prof. Land.
*^* Names of stringB as in No. 66, names of notes as altered by the Arabic medieval writers.
Kt^
Notes
I
c
2
Db
3
^t>b
4
D
5
Eb
6
Fb
7
E
8
F
9
Ob
10
Abb
II
G
12
Ab
13
Bbb
14
A
15
Bb
16
Cb
17
Dbb
1'
e
First Ootare
First: open
remnant
near .
index .
Persian
Zalzal
ring .
UtU« .
S]sooin>: remnant
near .
index .
Persian
Zalzal
ring .
Uttle .
Thibd: remnant
near .
index
Cents
Second Octaye
Thibd: index
Persian
Zalzal
ring •
Uttle . .
FouBTH : renmant
near { = Obb)
index .
Persian
Zalzal
ring .
Uttle .
Fifth : remnant
near(»Cbl>)
index .
Persian
Zalzal
ring .
68. Northern Tambour, or that of Khorassan, as reported by Prof Land,
Co. Db 90, Ebb 180, ♦D204, £b294. Fb384, £7408, ♦F498, Gb 588, ^«6i2, ♦G702,
til b 792, GS816, i4 9o6, £b996, filQ 1020, JB iiio, '^ci200, 691224, c9 13141 *^ 1404
cents.
* Fixed tones. t Aitxiliary tones.
Babdb or 2'Stringed viol, after Prof, Land,
69. first I o
70. second jo
71. third I o
204
316
408
520
590
632
724
204
316
408
590
612
724
816
204
316
408
590
794
906
998
906
998
XI80
Southern Tamboury or that of Bagdad^ as reported by Prof Land,
72. theoretical . . . . | o | 44 | 89 | 135 | 182 | 231 | 275 | 320 | 366 | 413 | 462
Indian Chromatic Scale,
*«* Arranged according to the older and more modem division as inferred from indications
by Bajah Sourindro Mohan Tagore.
Degrees
Notes
73. Old .
74. New .
Dbb
49
£bb \i^
36411325*386
259 1316 37+
9 ; zo' XI I Z3 I 13 '14
Bt\F\ — \Ft\F%t V
442 498 549J60O , 651 j7M
4351498 543'589 637 I685
15 16
17 18 19
30
31
AbbAb
- AiB.b
80
B
753 804
736 787
855 9o6;966|
841,896, 95a
xoa7i
lOII
1088
1070
33
Bt
"44
"35
Indian Semdtonic Scale as inferred from Measurement of a Madras Vina,
75. First Octave. . . I ol 89: 178! 2691 373I 4751 S96| 6841 781 8791 99610811199
Second Octave • . I1199I1280I1376I1466I156711681I177611891I19842090J2187I2298I2398
♦^* The Indian partial scales enumerated by Rajah S. M. Tagore, as made up from the 19
inotes in Nos. 73 or 74, 32 of them with 7 notes, 112 vdth 6 notes, and 160 with 5 notes each,
are not given because he does not distinguish the minor variations of one degree.
Indian Partial Scales as played by Bajah Bdm Pdl Singh,
76. First
77. Fourth
78. Second
79. Third
80. Fifth
I.
0
II.
III.
IV. 1 Y.
VI.
VII.
1074
183
342
533
68s
0
174
350
477
697
908
1079
0
183
271
534
686
872
983
0
III
314
534
686
828
1017
0
90
366
493
707
781
1080
VIII.
1230
II8I
1232
IJ98
1087
IT
Digitized by V^OOQIC
5i8
ADDITIONS BY THE TRANSLATOR.
APP. XX.
II. Tablb op Non-Harmonic Scales — conHnued,
Various Wood Harmonicons.
8i. Balafong from Patna
82. Balafong from Singapore
83. Patala from Btirmah
84. Balafong from the same .
85. Banat from Siam, See p. 556
86. Balafong from Western Africa
87. Gen. Pitt-Biyers'B Balaiong from
the same
n.
III.
lY.
V.
672
VI.
VII.
VllL
187
356
526
856
985
1222
169
350
543
709
894
1040
1205
176
350
533
707
899
1053
1246
114
350
550
687
838
1032
1 196
129
277
508
726
771
1029
1254
152
287
533
724
890
1039
1200
195
28g
513
686
796
1008
1209
Pbntatonic Scales.
The Black Digitals of a Pianoforte.
88. beginning with Ct .
90. n „ FU .
91. .. M O^ .
92. » „ AU .
I.
n.
0
200
0
300
0
200
0
200
0
300
in.
IV.
V.
VI.
500
700
900
1200
500
700
ICX)0
1200
400
700
900
1200
500
V^
1000
1200
500
800
1000
1200
93. Balafong .
South Pacific.
. i o I 202 I 370 1 685 I 903 i 1200
Javese Scales, as observed from Instruments and Musicians.
94. Salendro, observed
95. „ assumed
0
228
484.
728
960
1200
0
240
480
720
960
1200
Javese Pehg, Chromatic Scale, from which tJie others are selected.
96. The seven notes
I.
II.
III.
0
137
446
IV.
V.
VI.
575
687
820
VII.
1098
VIII.
1200
The Five-NoU Scales Selected,
97. Pelog
98. Dangsoe {oe as in shotf)
99. Bem
100. Barang .
loi. Miring .
102. Menjoera (/o««- English you)
0
—
446
575
687
—
1098
0
1.37
687
820
1098
0
137
—
575
687
—
1098
0
137
—
575
687
820
—
0
446
575
820
1098
0
137
446
575
—
—
1098
1200 ,
1200
1200
1200
1200
1200
Chinese mdxed Pentatonic and Heptatonic Scales, as observed.
*«* Notes marked * introdnoed for heptatonic playing.
103. Flute (Ti-tsu) .
104. Oboe (So-na) .
105. Mouth-organ (ShAng)
106. Grong-ehime (Yun-lo)
107. Dulcimer (Tang-ohin)
108. Tamboura ^Sien-tsu)
109. Balloon Omtar (p*i-p'a)
Digitized by V^jOOQIC
0
178
*339
448
662
888
♦1103
0
145
297
440
637
8i3
1014
0
210
338
498
715
908
1040
0
169
367
586
674
775
1062
0
169
•274
491
661
878
♦996
0
189
386
702
S^3
0
145
35 »
—
647
874
—
1 196
I2I6
1199
I208
Ii9«
1200
"95
SISCT. K.
NON-HABMONIC SCALES.
5*9
II. Tablk op Non -Harmonic Scales — continued.
Pbntatonic Scales — continued,
Japanese, chiefly Pentatonic, but with extra notes marked *
Koto Tuning^ Popular Scales,
I. II. ' III.
IV.
V.
no.
Hiradioshi, theoretical
0
204
316 1
702
814 ;
1200
III.
„ female player
0
193
357 :
719
801 ,
1 199
112.
„ musio-master
0
i8S
337
683
790
1200
"3-
Akebono I., theore
tical .
0
200
300 !
700
900
1200
114.
Akebono II.
0
100
500 ;
700
800
1200
115.
Eumoi I. „
0
100
500 1
700
8oo
1200
116.
Kumoi II. „
0
100
500 1
700
800
1200
117.
Han-Kumoi „
: 0
200
500
700
800 ;
1200
118.
Kata-Kumoi «,
0
200
300
700
800 ,
1200
119.
Sakura „
0
100 j 500
700
800 ;
1200
120.
Iwato
0
ICO 500
600
1000
1200
121.
Han-Iwato „
0
ICO ' 500
700
1000
1200
122.
Kata-Iwato
0
100 1 500
600
1000
1200
123.
Kumoi „
0
100 ; 500
700
800
1200
Koto Tuning, Classical Scales,
124.
Ichikotsu-Chio, theoretical . 1 0
200
500
700
900
I200*
"5.
Hio-Dio
0
200
500
700
900 j
1200
126.
Sou-Dio ,,
0
200
500
700
900 1
1200
127.
Wausiki-Chio
0
200
500
70J
9DO
1200
128.
Sui-Dio
0
200 ■ 500
700
lOOO
1200
129.
Bausiki-Chio „
0
300 500
700
1000
1200
Heptatonii
J Scal4!s.
I.
I_n^j m.
IV.
V.
VI. VII.
1
VIII.
130.
Classical reosen, theoretical .
0
203 400
♦600 700 1
900
*iioo
1200
131-
,, ritsasen „
0
200 300
♦500 1 700
2°°
♦1000
1200
132.
Popular I. „ • -
0
100 300
500
700 '
800
1000
1200
'33.
Popular U.
n
.
0
100 1 .
500
500
600 1
800
1000
1200
Japanese Bvwa, Classical Instrument, Tetrachords observed on different Strings.
I. I II. • III. t IV. I V.
I
134. Lowest string .
135. Second lowest .
1 36. Second highest
1 37. Highest string
138. Mean
139. Theoretically assumed as
225 i
33?
416
5'2
223
338
429
500
195
320
407
496
212
321
414
503
214 1
328
416
503
200
300
400
500
III. Annotations to the Table.
N08. I to II are those given in the text,
pp. 262-5, ^^^ ^o* 8 was merely suggested
by Prof. Helmholtz.
N08. 12 to 24 are from App. III. to Prof. Land's
paper, Over de Toonladders der Arabische
Musiek (on the Arabic musical scales), and
contain his corrections of the very faulty
MS. of Al Farabi ; the numbers are also
given by Eosegarteu, p. 55. After 24, the
numbers suddenly ^ease in the MS.
Nos. 25 to 31 are the old theoretical form of
the Greek scales with the old tetrachords,
see supr^, p. 268 c, d',
Nos. 32 to 38 are taken from supra, p. 269 a.
Nos. 39 to 50 are from Prof. Land, ibid. p. 38,
corrected from the MS. at Leyden; for
No. 45 the copyist had repeated No. 44, and
Prof. Land has supplied the numbers by
analog}'.
No. 51 is inferred from the complete set of ^
notes nsed on the Arabic lute at dififerent
times as shewn in No. 66,
No. 52. The Highland bagpipe representing
that scale has been inserted immediately
afterwards to show its practical identity. It
was played to Mr. Hipkins and myself by
Mr. C. Keene, the artist.
No. 53. The very modem survival of the
same scale has been put next. It is de-
scribed, supri, p. 264 note **. In practice
each note might be sharpened by one or
more quartertones.
Nos. 54 to 65 are the twelve scales given,
supra, p. 2S4, from Prof. Land, but the four
which employ 8 notes are now placed last.
No. 61, Hhidjazi, is, in fact, more harmonic
than the nsual equal temperament. If we
begin on the note vii., and reckon the
Digitized by V^jOOQlC
520
ADDITIONS BY THE TRANSLATOR.
APP. XX.
intervals from it through an octave, after-
wards subtracting 996, it gives the scale
0 204 384 498 702 882 1086 1200, and if
384 882 1086 were each increased by 2
cents, this would be our just major scale.
The difference is not felt even in chords, as
1 have ascertained by actually playing them
on a properly tuned concertina.
No. 66 is the complete collection of the notes
on the old Arabic lute, as used at different
times, reported by Prof. Land. Of course
the Persian and ZalzaPs notes could not be
used together, and when Zalzal's 355 and
853 were used, both 294 and 408, and also
botii 792 and 906 had to be discontinued,
producing No. 51.
No. 67 gives the complete 17 medieval Arabic
notes as determined by Prof. Land, with the
^ 2 extra ones which appear in the second
Octave. Yilluteau (op. cit. suprd, p. 257,
note t, ed. 1809, folio, vol. i.) declared, as
is well known, that the most generally re-
ceived Arabic division of the Octave is into
thirds of a Tone (op. cit. p. 613). Prof.
Land has demonstrated (Gamme Arabe,
p. 62) that this is not the case. Villoteau,
ftn excellent musician, sent to Egypt by
the French Government to study the native
music, had every facility given to him, and
had native musicians at his beck and calL
How did he arrive at this opinion ? After
an attentive study of his book I consider
the following hypothesis probable. The
greater number of theorists gave 17 notes
to the Octave. This was the medieval
Arabic scale, No. 67. Villoteau was not
used to just intervals, and he was a very
^ poor arithmetician (see the remarkable note
op. cit. p. 668). He was used to the * musi-
cians' cycle ' of 55 degrees (supr&, p. 436d,
viii.), in which tiie Tone contained 9, the
major Semitone 5, and the minor 4 degrees
(op. cit. pp. 667, 678). When he heard the
scale of liaat played (see p. 284 and No. 57,
for the medieval form o 204 384 498 702
882 996 1200 cents), which was the principal
Egyptian scale, he tried to sing it as ^1 £
CK , &c., in his 55 degrees, but was imme-
diately told that his CZ ~392 cents was
too sharp. This would hardly have been
the case if the true medieval 384 had been
played to him. He next tried A B C = 305
cents, but then C was too flat. Now the
interval Cix> Ct - 87 cents was his minor
Semitone of 4 degrees. Hence he concluded
that 3 degrees =65 cents in his tempera-
^ ment would be right, and that satisfied the
natives (op. cit. p. 679). This, however, was
one third of his Tone. But he found also
17 tabaqat or transpositions of each scale,
proceeding by Fourths, called by him per-
fect, but as he really considered 1 7 of these
Fourths to make up 7 Octaves, he arrived at
a cycle of 17 degrees, each having 71 cents
(ex. 70-588233) or being almost precisely a
small Semitone 24 : 25 ( = ex. 70*673 cents).
To the nearest cent, using his symbols,
where x means increased by one third, and
t increased by two thirds of a Tone, and
i, b mean diminished by the same amount),
the notes of this cycle in cents were, 1^0,
2 Ax «Sb 71, 3 A^ =jBU 141, 4 B 212,
5 C 282, 6 Cx =i)b 353. 7 Ct^^Dl 424,
8 D 494, gDx^Eb 565. 10 Dn "El 635,
IX E 706, 12 F 776, 13 Fx «Gb 847, 14
FZ -Gl 918, 15 G988, 16 Gx =^b io5».
17 Ot ''Al 1 129, i' A 1200. Then b«
writes the Bast as he heard it, as ii B C x
D E Fx G Ay which therefore gave the
cents o 212 353 494 706 847 988 1200. Now
it is difficult to conceive that he could have
heard the medieval Bast in this way, even
though the intervals were determined purely
by estimation of ear, apparently his only
method of estimation. But the probability
is that medieval Bast^ like the other me-
dieval scales, had become a thing of the
past, and that what Villoteau heard in
Egypt in 1800 was what Eli Smith in 1849
tells us Meshaqah (supr&, p. 2646) laid down
at Damascus, namely No. 53, that is, the
normal scale o 200 350 500 700 850 1000
1200 cents, a survived of Zalzal's with the
neutral Third and Sixth, and this is very
accurately represented by the above scale of
Kast, as Villoteau notes it. At the very
outset Villoteau says that some divide the
Octave into Tones, Semitones, and Quarter-
tones (op. cit. p. 6x3). This shews that the
24 division was even acknowledged. But
ViUotean was perfectly ignorant of equal
temperament, and hence paid no attention
to this. On the other hand he found the 17
divisions in the theorists, and made them
equal, because he thus seemed to reconcile
theory and practice. But he only obtained
an outline thus, as is evidently shewn by
his speaking (op. oit. p. 612) of * les divi-
sions et subdivisions des tons de la mnsique
arabe en intervalles si petits et si pen na-
turels, que I'ouie ne peut jamais les saisir
avec une pr^ision exacte, ni la voix les
entonner avec une parfaite jnstesse.' It
was evident there were many other Tones
(the Quartertones) not in his list of 17.
Lideed he says (op. oit. p. 673) : * Us savent
anssi qu'il y a d'autres degr^s interm6diaires
aux pr6c6dents, et ils en font usage mtoe
assez fr^quemment, mais ils ne sauraient
dire au juste quelle est la nature et P^tendne
de rintervalle qui s^pare ces degr^ les
uns des autres.' These were possibly the
Quartertones (very uncertainly produced) by
which we learn from Eli Smith that the
Arabs, like the Lidians, continually varied
their scale. If the 17 thirds of tones of
Villoteau, just given in his notation, be
read as i C, 2 Db, 3 ^bb. 4 A 5 ^b, 6 fb,
y E, S Fy 9 Gb, 10 ^bb, 11 O, 12 Ab,
13 Bbb, 14 A, 15 £b, 16 Cb, 17 Dbb, as in
No. 67, his 12 scales will be found to cor-
respond precisely in names of the notes
with those given by Prof. Land (snpri,
p. 284), but the whole of the intervals,
which were originally of 90 or 24 cents, are
now equalised as 71 cents by this confused
temperament of Villotean*s in which medi-
eval Arabic music seems to have been in-
tended, but the modem form was reaDy
misrepresented. To shut up 24 Quarter-
tones into 17 thirds of Tones, at least two
must be given to one note on seven occa-
sions. Thus (in cents) Villoteau*s 7 1 was
50 or 100, his 2S2 was 250 or 300, his 424
was 400 or 450, his 565 was 550 or 600,
his 776 was 750 or 800, his 918 was 900 or
950, and his 11 29 was 1 100 or 1150. This
would fully account for the indistinctness
complained of.
No. 68 is a complete Arabic medieval scale.
Digitized by V^jOOQlC
SECT. K.
NON-HARMONIC SCALES.
521
with additional intervals, 612, 816, mo,
1200+24, 1200+ 114, played on a tambour.
These very long-neoked guitars iJlow of
minute subdivision of the string.
Kos. 69 to 71 are the various notes produced
by the Bab&b, according to the three methods
of tuning the second string as 316, 408, or
590 cents, the intervals between pairs of
notes on both strings being identical. In
No. 69 the note 520 cents is played on the
second string, but is here inserted in order
of pitch.
Ko. 72 gives the most extraordinary and most
limited scale known, produced by using only
the open string and 39, 38, 37, 36, and 35
fortieths of it ; the open second string being
tuned in unison with the sharpest note of
the first string. It is valuable as showing
a primitive method of obtaining scales and
a division of one>eighth of the keyboard
into 5 equal parts.
Kos. 73 and 74 are an attempt to represent
the Indian Chromatic Scale fiom indications
in Bajah Sourindro Mohun Tagore's Musical
Scales of the Hindus^ Calcutta, 1884, and
the Annvaire du Conservatoire de Bruxelles^
1878, pp. 16 1 -1 69, the latter having been
drawn up by Mons. V. Mahillon from in-
formation furnished by the Bajah. As
regards the 7 flxed notes {prahrita) of the C
scale (sharja gr(Una), C, D, JS7, F, G, A (a
comma sharper than our Ay)^ £, there
seems to be no doubt of the theoretical
values. As to the 12 changing notes
{vikrita)t the values given can be con-
sidered only as approximative. The divi-
sion of the intervals of a major Tone of 204
cents into 4 degrees (s'rtUis) ; of a minor
Tone of 182 cents into 3 degrees; and of
a Semitone of 112 cents into 2 degrees, as
indicated by the superscribed numbers, is
also certain. But whether the 4 parts of a
whole Tone were equal and each 51 cents,
and the three parts of a minor Tone were
also equal and each equal to 60} cents, and
the two parts of a Semitone were also equal
and each therefore 56 cents, is quite un-
certain. This, however, was assumed to be
the case in calculating No. 73, and the
results are probably not much out. Nor is
it likely that the alterations by degrees
(produced on increasing the tension of the
string by pressing behind high frets, or
deflecting the string along low frets, or by
arranging the movable frets) were even
approximately constant. In No. 74 we
have the ipodern Bengali division of the
finger-board referred to in the above books.
It seems that the string is first divided into
half and a quarter, giving the Octave and
Ponrth (theoretically). Then the distance
from the First to the Fourth on the finger-
board is divided into 9 equal parts, and that
from the Fourth to the Octave into 13 equal
parts, and each distance represents the
interval of a degree (s'ruti). From these
data the values of No. 74 have been calcu-
lated. It will be seen by subtraction that
the first 9 degrees thus found vary from 49
to 63 cents, and the last 13 from 45 to 64
cents. The cents found, however, from the
inverse ratio of the lengths will differ
slightly from those used practically. The
S and b (tihra and komala) are used in this
scale for deviations of two degrees when a
major tone is divided, and for deviations of
one degree when a minor Tone or Semitone
is divided. This is done by the Bajah in
his translations into ordinary notation. In
addition I have taken the liberty to use bb
to represent only one degree flatter than the
single b, and wish it to be read * very flat '
(ati-komala) ; similarly 8 S is one degree
sharper than % , and should be read * very
sharp* {ati-tibra). The Bajah not having
distinguished the very flat and very sharp
notes from the simply flat and sharp ones
in his 304 scales, I have avoided citing them
at length. Similarly I have not been able
satisfactorily to find the cents for the F
scale or mad*hyama grdma (usually repre-
sented as our just major scale in which
A should be one comma instead of one
degree flatter than in the O scale, as it ^
would appear to be), or for the E scale or
gand*hdra grdma (in which D and A appear
to be one degree flatter, and B one degree
sharper than in the normal C scale or
sharja grdma), and hence I have not given
them in the table. But, using numbers
before the notes for degrees, they may
possibly be for the F scale, iC, 5l>. 8£,
loF, 14G, 17 A, 21B, which would use
degree 17, and the corresponding note may
be called grave A, and written A\ For the
E scale we may possibly have iC, 4i), SE,
loF, 14G, 17 A, 22B, where 4D is now
utilised, and becomes grave D\ But these
A\ D\ are not our il„ Dp and hence the
scales are different from ours. This is,
however, pure conjecture.
No. 75, for the old national Indian instru-
ment, a Vina from Madras, in the South ^
Kensington Museum, gives the value of
24 notes by measuring vibrating lengths of
string from fret to bridge, and is, of course,
very uncertain. It will be found, however,
that the notes agree with No. 74 better than
with No. 73, for the scale C Db D Eb E
F Ft O Ab A Bb B c. The other degrees
could be easily produced by pressing the
string behind the frets, which were about
one inch in height.
Nos. 76 to 80 are five observations of scales
played by Bajah lUun P&l Singh, and ob-
served with forks. These were set by alter-
ing the movable frets of a sit4r. The first
and fourth are placed together in the table,
as they are believed to have been meant for
the same scale, and differed only because
they were set on different days. They seem
to be meant for Bajah Sourindro Mohun ^
Tagore's first scale. The second setting
seems meant for his 13th, the third for his
29th, and the fifth for his 9th.
No. 81. A wood harmonicon in the South Ken-
sington Museum, stated to have come from
Patna, but probably arrived from some hill
tribes. Its scale resembles one which I
deduced by measurements of strings from
a Tar of Cashmere, which was o 175 354
512 720 896 1062 1237, but I thought this
scale too uncertain to put in the table.
No. 82. A wood harmonicon sent direct from
Singapore to Mr. A. J. Hipkins, taking the
central Octave.
Nos. 83 to 87. Wood harmonicons in South
Kensington Museum, of which the last be-
longed to General Pitt-Bivers.
Nos. 88 to 92 are inserted because theyare the
Digitized by V^jDOQlC
522
ADDITIONS BY THE TRANSLATOR.
APP. XX,
examples of pentatonio soales asnally giTen.
They are at any rate now used for penta-
tonic Scotch masic. See supr^, p. 259^2. They
are not, however, by any means usual forms.
No. 93. A balafong from the South Pacific be-
longing to General Pitt-Bivers, seems to be
intended for No. 90, but 370, 685 cents for
400 and 700 cents are both rather flat.
Nob. 94 to 102. Javese instruments examined
at the Aquarium, London, in 1882, and pitch
of notes determined by forks. In Nos. 96
to 100 the order of these notes settled by
Mr. W. Stephen Mitchell, and the order
conflrmed by information received through
Prof. Land of Leyden. Nos. loi and 102 were
inferred from information of missionaries
obtained by Prof. Land. The assumption
in No. 95 of a division of the Octave into
^ five equal parts was confirmed by other
measurements communicated by Prof. Land.
Nos. 103 to 109. These seven scales were taken
from the playing of Chinese musicians at
the InternationaJ Health Exhibition, 1885,
during four private interviews. No. 106 had
two additional tones of 497 and 797 cents
above the lowest note ; these were omitted
in playing by the musician. There was a
second yiin-lo at the South Kensington Mu-
seum, also with 10 tones, which gave o 52
240 266 418 437 586 589 712 738, but these
form no scale. They may be a fund out of
which scales are constructed. The four fol-
lowing may be among such: — To a sharp
Fifth, o 240 418 589 712; to a flat Fifth,
o 188 366 534 686; to a flat Fifth again,
o 214 385 537 686; to a fourth, o 197 349
498. The last is a Meshaqah tetrachord,
il see No. 53. A chime of four small bells, be-
longing to Mr. Hermann Smith, gave o 312
480724.
Nos. no to 139. Japanese scales. Accepting
the statement of native musicians that the
intervals are those of equal temperament,
or at least so near them that Japanese ears
do not perceive tl^ie difference, then the
theory gives Nos. no and Nos. 113 to 128
for * popular* koto tunings. There are 13
strings to the koto, but only 5 give the scale,
the rest being Octaves or onisons. Nos. 1 1 1
and 112, heard at the * Japanese Village'
Knightsbridge, shew how practice sometimea
overrides theory. Several of the scales seem
to be identical, but they are at different
pitches, and henoe no more identical than
our major scales for different keys.
Nos. 124 to 129 are ' elassical ' koto tamngs of
which 124 126 128 are classed as rio$en or
analogous to our major scales, and 125 127
129 as ritsusen or analogous to oar minor
scales. This must have been effected by
sharpening some of the notes by pressure
on the string behind the bridges which limit
their vibrating length.
Nos. 130 and 131 are both * classical ' hepta-
tonic scales, owing to the introdaotion of
the notes marked *.
Nos. 132 and 133 are both * popular ' hepta-
tonic scales, in which however the intro-
duced notes are not pointed out in Mr.
Isawa's Report on Japanese music at the
Health Exhibition of 1885, Educational
Division, from which the scales No. 1 10 and
Nos. 113 to 133 have been taken.
Nos. 134 to 139 result from an examination of
the Biwa, a classical instrument, closely re-
sembling the Arabic lute, fretted only as far
as the Fourth. The strings are tuned to
one another in six different ways, and hence
produce a great variety of notes. By touch-
ing the strings on the frets (taking care not
to press behind them) and determining the
pitch of the notes sounded, Mr.Hipkins and
I found that the tetrachord produced dif-
fered according to the string employed, as
shewn in Nos. 134 to 137. The mean of the
intervals thus determined is given in No. 138,
and accepting equal temperament, as in the
whole of Mr. Isawa's report, the correspond-
ing divisions are given in No. 139. These
divisions were in so far assumed to be cor-
rect that the three Semitones, though ma-
terially differing (having the mean vadnes of
114, 89, and 86 cents), are, in Mr. Isawa's
account of the notes, considered as alike,
and exactly half of the value of first Tone,
which had a mean value of 214 cents.
IV. How THESE Divisions of the Octave may have abiben.
It is impossible to trace such scales to their germs. Singing and playing on
pipes were probably the first music. Striking of bone, wooden, and metal bars was
^ probably also a very early form, and as their notes are tolerably persistent they are
valuable for determining scales (see Nos. 81 to 87 and 93 to 102). But scales them-
selves are a great development, and two or three notes, varied rhythmically, pro-
bably long preceded them. We must be content to commence with strings, certunly
a very late form of musical instrument, on which, however, the chief work of the
older theorists was expended. After the examples in the latter part of the Table
we have no right to assume an accurate musical * ear,* or appreciation of just
intervals. Even in Europe it requires much practice for the majority to sing
accurately in tune or to appreciate small errors. (See suprii-, p. 1476?.) We now
know that on any stringed instrujnent such as the violoncello or guitar, where, to
prevent jarring, the string has to rise further and further from the finger board
as the finger proceeds from the nut towards the bridge, the pressure of the finger
in * stopping ' the string either on the board or fret increases the tension of the
string and hence makes the note sharper than it would be if the string could be
stopped at its natural height, though even tlien, as we have seen (p. 442a), the
results are not absolutely trustworthy. The law that the number of vibrations is
Digitized by V^jOOQlC
SECT. K.
NON-HARMONIC SCALES.
523
inyersely proportional to the length of the string holds hut very roughly for such
instruments. Thus stopping a violoncello string exactly at its middle gives a note
sharper than the Octave, hence the finger has to he placed sensihly nearer the nut.
Moreover, the amount of error depends on the nature of the string. The examples
Nob. 134 to 137 in the tahle are very instructive. The determination with which
those differences are overlooked (see Annotations to these Nos.) is equally
instructive. It is evident that Euclid in his Canon for the Pythagorean notes No.
32, and Ahdulqadir in giving his rule for obtaining the 17 notes of his Ecale No.
67, considered the division to be perfect, and Prof. Land in calculating the value
of the notes had, of course, to assume that it was so. The old intervals were,
therefore, not so accurately tuned as was supposed, and hence when we take them
to be accurately tuned we are ourselves inaccurate.
The fact was strongly impressed on me in
making an instrument on which I could play
any scale expressed in cents. I had a Dichord,
that is, a double monoohord, constructed with
wires 1,200 millimetres long, diameter *3 mm.,
height of nut 7 mm., of bridge 24mm., from
soimd board. Then having a number of laths
5 mm. thick, I used them as moveable finger-
boards, and marked on one the place of the
notes of the just scale as determined by the
theory of inverse ratios. Trying, by my Har-
monical, I found every place much too sharp,
and it was only by marking the places which
gave unisons that I was able to correct the
error. If any one constructs such an instru-
ment, I recommend his setting off the place
where he should stop for each semitone for
two octaves, by a well -tuned pianoforte, and
then dividing each distance representing a
semitone into 10 parts. Each of these parts
will represent 10 cents with quite sufficient ac-
curacy, and can be subdivided by the eye. Thus
a geometrical scale can be constructed by
means of which the places to touch the string
can be marked oft on a new lath, and the scale
played. This geometrical scale was placed
under one string and the finger-board to be
played from under the other. I found it best
for accuracy to stop the string with the side
of my thumb-nail. All the principal scales
above given were thus realised.
Now assuming the usual law of division,
suppose a string divided in half, giving the
Octave, and each half subdivided in half, giving
the Fourth and double Octave. This division
being the simplest possible would naturally
give a preponderance to the Fourth, whence
would arise the tetrachords, the foundation of
Greek, and hence of European, and of Persian
and Arabic music. The Fourth is also recog-
nised in India; but in pentatonic regions,
especially in Java, where the string is not in
use (the rab4b they use is Arabic in name and
origm, Nos. 69 to 71), the Fourth is not cor-
rect. It is clearly, therefore, not a primitive
interval, and the quartering of the string may
really have much to do with its adoption.
The interval was, however, too wide, and it
was necessary to subdivide it. The most ob-
vious plan was again to quarter it. Thus, the
additional distances of ^, ^ » i, ^ of the string
from the nut would be obtained, giving the
vibrating lengths ^|, {, ^, the ratio. The first
gives the diatonic Semitone of 1 1 2 cents, and
its defect from the Fourth, 498 cents, the major
Third of 386 cents, see No. i. This ^ of the
string « 1 12 cents is conspicuous in Nos. i to
4, some of the oldest forms. The J, « 23 1 cents,
we find in No. 72. The Jf = 360 cents did not
come into use, but it is practically Zalzal's
355 cents. No. 51.
Continuing this simplest of all subdivisions ^
by two, we have half of ^^^ » ^^ of the string
from the nut, giving the vibrating length §| of
the string =55 cents. Hence we obtain the
enharmonic division No. 11. At the same
time my observations on p. 265, note *, hold
good, for the errors in coming so near the nut
as ^ of the string, would be too great to ob-
tain anything like accurate results by measure-
ment. On my dichord I found it impossible
to take less than a semitone of 100 cents VTith
any degree of certainty. It is interesting to
observe that this JJ of the string gives very
nearly 50 cents or the Quartertone, and still
more nearly 54^, the 22nd part of an Octave,
corresponding to the Indian degree (Nos. 73
and 74 Annotation), and is really the com-
mencement of the variation of notes by about
a Quartertone.
Then the divisions attempted in the South- %
em Tambour No. 72, and also in forming
the Persian middle-finger note, 303 cents (see
No. 66), by taking a place half-way between that
for 294 cents and 408 cents, and again for Zal-
zars middle-finger note of 355 cents, by taking
a place half-way between 303 and 408 cents^
and finally the modem Bengali division of the
distance occupied by a Fourth on the finger-
board into 9 parts (see No. 74) and that for
the following Fifth into 13 parts, suggest that
the attempts were made to divide the ^ of the
string from the nut to the Fourth, by other
simple numbers beside 2. The division into
three parts would be more difficult, but might
be done very fairly by guess. Now the dis-
tances of the stopping-place from the nut ■^,
IS = a o' *^® string or the vibrating lengths {^
and I of the string corresponding to 151 cents or
and 316 cents, the first, the Threequartertone, '
which is the real parent of all the neutral
intervals to be considered presently, and the
second the minor Third. Both occur in
No. 10, and the minor Third occurs also in
Nos. 3 and 6.
The division of the whole string into thirds
could hardly have taken place, but ^ of the
string from nut gives the vibrating length ^ of
the string « 702 cents, the Fifth, which, as
exceeding the Fourth of 498 cents, would not
be regarded till the tetrachord had been ex-
tended to the Octave. The defect of a Fourth
from a Fifth gave the major Tone, one of the
most important intervals, also obtained di-
rectly by taking ^ of i, or ^ of the string from
the nut, giving { of the strmg as the vibrating
length = 204 cents. This interval finally ab-
sorbed all the others, except the Fourth,
Digitized by VjOOQlC
524
ADDITIONS BY THE TRANSLATOB.
APP. XX.
espeoially after the observation that it was
also the defect of two Fourths from an Octave.
Its direct use is apparent in No. 19, where 204
cents corresponds to i string from nut, and
408 cents to j^ of the vibrating length of the
204 cents from the stop for 204 cents. Thus
if the nut be called A^ and the stopping places
for 204, 408, and 498 be B^ C, D, and the
bridge be Z, we shall have AD'^\ AZ^ whence
i>Z = | AZ\ also AB = l AZ, whence BZ^
% AZ\ also BC = | BZ, whence CZ = g BZ =
jj^ AZ, If we suppose the whole length AZ
divided into 324 parts, then AB = 36, BC « 32,
Ci>» 13. But the whole division is obtained
by taking halves and thirds. Ko. 5 is the
reverse of No. 19. Let the stopping-places
for 90 and 294 be E and F, Then DZ^\ AZ,
FZ-I DZ = DZ-\-\ DZ, so that F is found
f Jrom DZ by adding ^ DZ^ which is obtained
by continual halving. Again EZ = \ FZ =
FZ + i FZ, so that E is found from FZ by
adding ^ FZ, which is again obtained by con-
tinual halving. This made it easier to produce
No. 5 than No. 19. The complicated value of
AE^^AZ would be thus altogether avoided.
Nevertneless it is most probable that Nos. 5
and 19 were both obtained simply * by' ear,'
and that they were never exactly * in tune.'
The next division of the Fourth to be ex-
pected is by 5, as in No. 72, and ^is is there-
fore advanced by Prof. Land as a probable
means of obtaining the Arabic tetrachord.
This gives the lengths of the string from the
»«!*» 551 TO * A» A» A ^^ i» 55 - }» aJ^d the vibrating
lengths fi, A, JJ, ±, 5 of the string giving inter-
vals of 59, 182, 281, 386, 489 cents. Of these
^ 89 is a fair representative of 90 in No. 5,
which has just been otherwise obtained. The
minor Tone, 182 cents, occurs direct in Nos. 6
and 7, and also possibly in No. 2» where it
would be simpler to obtain it from the open
string as ^ its length than as {{ of the vibratiiig
lenjg^ of 112 cents, that is, as |} of ^ of the
string. The 281 cents appears not to have
been used, but it approximates to 294 cents,
which may have been introduced from Gxeeoe
in place of it. This is, however, mere con-
jecture. The 386, or major Third, is in No. 7
only (for No. 8 is not ancient), and there is
very little probability that it was tuned direct.
It might have been got as 182 + 204 or as
498 — 112. Both the 182 and 386 cents wexe
certainly lost at an early time in 204 and 408
cents, so that it is difficult to suppose that
386 at least was ever obtained directly. It
was indirectly produced in Nos. i and 2,
where, judging from Japanese habits, the
tuner tried to get the Semitone by ' feeling/
and left the major Third to arise as the defect
of the Semitone from the Fourth.
The division into 7 parts belongs to a mach
more advanced stage, and never seems to have
come into use. But we may understand No. 9
thus. Stopping at \ the string, we obtain a
vibrating length of J ^ 267 cents. Then taking
I of ^, or ^ the length for the stopping place,
we obtain a vibrating length of ^ string « 85
cents, and thus find both intervals in No. 9.
Of course when the ball had been set tolling,
and there was no harmony to check the
fancies of dividers or musicians, such forms as
Nos. 12 to 24 could be produced. But the
ancient Nos. I to 7 and 9 to 1 1 (No. 8 was not
ancient) are sufficient to have traced.
V. Besults op the Inquiry.
The chief points of interest which the exhibition of these scales afifords appear
to be the following :
I. The predominance of the Fourth, and mere evolution of the Fifth, in Greece,
Arabia, India, and Japan.
These may be only different forms of some
original system. The Chinese may have
imported the principle, but on this the ex-
treme uncertainty pervading all exhibitions of
Chinese scales hitherto made (including Van
Aalst's treatise on Chinese Music, 1884) renders
it difficult to judge. The Fourths actually
heard are uncertain, see Nos. 103 to 109. But
they seem at home in Japan, where they are
m used in tuning, but may have been imported,
and they are occasionally absent. In the im-
portant Javese scales, Nos. 94 to 102, they are
never in tune. Even in Arabia and India they
are apt to be altered. The specimens of ruder
music. Nob. 81 to 87, are not favourable to the
Fourth. The Fifth never had the same pre-
dominance. It is constantly too sharp or too
flat. In modem India generally it is too fiat.
In one set of scales in Java it is too sharp
fNo. 94) ; in the other set as fiat, as in India
Nos. 74 to 79). These differences probably
pervaded also the scales of other countries as
actually used, but we know Greece and Arabia
from theory only. That the Fifth is true on
the bagpipe (No. 52) depends apparently on
the use of tne drone, which would prodace
frightful beats if it were as much out of tone
as in the other cases cited.
2. The use of Tones and Semitones of about 200 and 100 cents depends upon
the Greek tetrachordal system as modified by Pythagorean intonation.
In Zalsal's scale (No. 51), and even in the
medieval Arabic scales, Nos. 57 to 65 (that is,
omitting the three Nos. 54 to 56, which are
identical with the Greek, and the exceptional
scales, Nos. 68 to 72), they do not both exist.
In the Indian scales they are overridden by
the system of 22 degrees, and only un-
designedly come close to our equal intonation.
In the ruder scales Nos. 82 to 87 they cannot
be traced. In Java they do not exist actually.
In China great difficulty was felt by the native
musicians of the Health Exhibition of 1884 in
respect to the Semitones. The Tones were
variable, and in some cases there seemed to
be a liking for the minor Tone as in tuning
the Tamboura, No. 108. In Japan Semitones
and Tones play a great part theoretically, bat
in the only practical cases I have been able to
observe, Nos. iii and 112 both were very un-
certain, and the Fourth was absent.
Digitized by VjOOQlC
8BCT. K.
NON-HARMONIC SCALES.
525
3, Neutral intervals, each lying between two European intervals, and having
the character of neither, but serving for either, abound.
The earliest instance is 151 cents in No. 10,
which is the Threequartertone between the
Semitone of 90 or 112 cents and the tone of
1 82 or 204 cents ; thus i x (90 + 204) = 147 "
^x (112+ 182). In ZalzaPs, No. 51, it was
355- 204 « 151, and 498-355 -H3 cents; in
the bagpipe observed, No. 52, it was 341 —
197 = 144, and 495 - 341 = 1 54» 853 - 703 «= » 50»
1009-853=146 cents, merely variants of
tuning; Meshaqah's theory gave 150 cents.
In bagpipe music it serves indifferently for
what would be a Tone or a Semitone in music
for another instrument. In the Japanese
cases, No. iii represents the two theoretical
Semitones by 357- 193 = 164, and 801 - 719=*
82 cents, and No. 112 by 337-185 = 152, and
790— 683= 107 cents.
Zalzal's neutral Third of 355 cents, No. 51,
is so truly neutral between the just minor and
major Thirds, 316 and 386 cents, that Mr.
Hipkins was quite unable to determine to
which it most nearly approached in character,
but for 345 and 365 cents, as tried on my
dichord (p. 5^26), the minor and major
characters were slightly but decidedly felt.
In the observed bagpipe this interval was 341
cents. In Meshaq^'s Quartertone tempera-
ment it was 350 cents, which may be taken as
its usual tempered form. It is the 374 of the
New Indian, No. 74, as shown in Nos. 76, IT^
and 80. Compare also Nos. 81 to 84, 103,
105, 106, and 109. The correlative neutral
Sixth arises similarly.
The neutral Tritone, 550 cents, is also ^
sometimes found, but it is rare, and as the
Tritone, 600 cents, is itself rare, this neutral
form is not easily observed. See Nos. 73, 74,
76, 78, 79, 82, 83, 84, and perhaps 96.
4. Modem Arabic and Indian scales have changing or alternative intervals,
produced by varying the pitch of one or more of the regular notes in any one scale
by a Quartertone or Degree.
To a smaller extent alternative tones are
known in Europe. Thus the just major scale
borrows its occasional grave second from the
Bubdominant key, where the difference is only
a comma of 22 cents. Nos. 62, 63, 65 have
each two tones which differ by only a comma
of 24 cents, namely 11 76 and 1200 in Nos. 62
and 63, and 678 and 702 in No. 65. The
scales consequently have 8 tones, as our just
major scale of C would have if we inserted
both D and D^, This, however, disappears in
tempered intonation. Again our just ascend-
ing minor scale of A^ has two alternative
notes, F^ and G^S , as well as F and G, and
these reknain in tempered intonation. If we
Inserted these, we might say that our minor
scale had 9 tones. Similarly No. 64 has
8 tones, with three intervals of 114 cents
between 180 and 294, 678 and 792, 1086 and
1200, and one interval of 90 cents between
792 and 882. We might suppose that 792 and
882 are alternative notes, and that we might
play either
o 180 294 498 678 882 1086 1200, or else
o 180 294 498 678 792 1086 1200,
the three final intervals in the first case being
204 204 114, and in the second 114 294 114.
This, however, is only an illustration, and is
not the point raised. Meshaqah's complete
scale consists of 24 Quartertones to the Octave,
for two Octaves, each tone having its own
individual name. Of these, only 7 are selected
to form the normal scale, namely No. 53. But
any one of these 7 notes may be raised (or also
probably depressed) by one or two Quarter-
tonps. And so freely is this variation of the
scale employed, that of the 95 snatches of
melodies which Eli Smith reports from Mesha-
qah, there are only 7 in whi(^ some change is
not occasionally made. Sometimes the change
is in ascending and not in descending or con-
versely. Thus in the air called Remel, 1 find
9 notes, the alternatives 950 and 11 00 being
introduced so that the scale, instead of ending
700 850 1000 1200 ends as 700 850 950 1000 ^
1 100 1200. Something of the kind occurs in
bagpipe-playing at the present day, owing to
the system known as *crossfingering,' which
gives nominally two ways of fingering the
same g"^ but actually produces two sightly
different notes, the sharper being used in
ascending passages. This was observed by
Mr. Bosanquet at a bagpipe competition, and
has been confirmed on inquiry by Messrs.
Glen, the great bagpipe makers of Edinburgh.
A similar thing apparently occurs in In£an
scales, where some of the notes may be de-
pressed one, two, or three degrees, and others
raised by similar amounts as shewn in Nos. 73
and 74. And there seem to be other altera-
tions of the kind not written, but conditioned
by the rdgini or modelet in which the
musician is playing. This is a point which
greatly requires elucidation. ^
These tones changing by a degree are made
by pressing the string behind the fret or de-
flecting it along the fret. A similar thing
occurs in playing the Japanese koto. The
player is copstantly pressing slightly or heavily
on the string beyond the bridge, or pulling
the string towards the bridge, and thus more
or less sharpening or flattening the pitch of
the note.
5. Scales of five tones may be formed by omission from scales of seven tones,
but on the other hand many scales of five tones seem to be entirely independent of
tones of seven tones, neither generating them nor being generated by them.
Though some of Chinese pentatonic scales,
as Nos. 103 and 107, seem to be derived from
heptatonic or conversely, yet all the Javese
pentatonic scales are thoroughly independent
of any heptatonic form. No. 94 and Nos. 97
to 102 could not be expressed as parts of even
Digitized by V^jOOQlC
526
ADDITIONS BY THE TRANSLATOR.
APP. XX.
oar ohromatio sc&le of 12 semitones. Euro-
pean masioians, indeed, persist in hearing and
writing the Salindro scales,
properly o 240 480 720 960 1200,
as o 200 500 700 9cx> 1200,
or o 300 500 700 1000 1200
bnt this mast arise from their not appreciating
240 cents, which is almost a neutral intervid
between a Tone, 200, and a minor Third, 300,
and is hence mistaken by European ears some-
times for one and sometimes for the other.
As for the Pelog scales, I cannot find that
any one has ventured to put their airs into a
European dress, the intervals No. 96 are so
strange. I have however tried to appreciate
them by having a concertina tuned with the
white studs in Saldndro (tuning E and F and
^ also B and C as unisons), and the black studs
. in Pelog, and writing them witH the notes
which would belong to the stud used in the
ordinary tuning. For the Saldndro I used the
airs in Raffles's Java, and in Crawford's paper
in the Tagore collection. For the Pelog I had
to invent airs myself. The characters of the
two sets are quite unlike. But pentatonio
Scotch airs played with the Saldndro scale are
quite recognisable. Whether this scale is the
primitive pentatonio scale it is quite impos-
sible to say.
In Japan the koto timings are all penta-
tonio, and according to theory the * popular *
have intervals of a Semitone, a Tone, and a
major Thirds whereas the * classical ' have
intervals only of a Tone and a minor Thirds
just as on the commonly received black digi-
tals of a piano ; but the classical is said to
come from China. In practice probably these
intervals are varied as in Nos. 11 1 and 112.
6. Pentatonio scales do not necessarily arise from inability to appreciate
Semitones.
This is shewn by the Javese Pelog notes,
which contain intervals of 137, 129, 112, 133,
and 102 cents, and by the Japanese popular
koto tunings, No. no and Nos. 113 to 123, all
pentatonio, and theoretically founded on the
8. There is an entire absence of tonality in our sense of the term and of any
attempt at harmony.
diatonic Semitone. If in practice the diatonic
Semitone sometimes grows to a Threeqnarter-
tone, it also sinks to a small Semitone, see Nos.
Ill and 112.
There is regard to the final cadence, at
least in Meshaqah^s scales, and probably in
all. There is in the Indian a ruler note (vddi),
«r see p. 243c', and minister notes ^am/oddij^
which function as our tonic, dominant and
subdominant in certain respects, and Prof.
Helmholtz thinks he discovers a reference to a
tonic in Aristotle (p. 241). But the European
feeling of tonality is one of very late growth,
and in non-harmonic scales must have been
something quite different, and if we refer it to
the same feeling as our own, it is from want of
power to appreciate the feeling of those who
use non-harmonic scales. This is parallel to
what constantly happens in appreciating the
intervals of these scales.
There is plenty of ensemble playing with
notes of very different qualities of tone, but
but they regularly proceed in unisons and Oc-
taves. In the Indian instruments there are
sympathetic and secondary strings. The for-
mer have their partials evoked by the notes
played. The latter, generally tuned in rela-
^ tions of an Octave Fourth or Fifth, are occa-
sionally thrunmied. But there is nothing like
a chord, or a tissue of harmony. It would not
be possible with the notes at command. There
is also discant playing as in the old polyphony
before harmony proper was invented. Prof.
Land, speaking of the Gamelan, or band of
Javese musicians sent by the independent
prince of Solo to the Amheim Industrial Ex-
hibition in 1879, says : * The musical treat-
ment is this. The rab4b plays the tune in the
character of leader* [at the Aquarium, the
player of the gambang (wooden bar harmo-
nium) seemed to be leader] ; * the others play
the same tune, but figured, and each for him-
self and in his own way ; the s4ron (metal bar
harmonium) resumes the motive or tune. All
this is accompanied by a sort of basso osti-
7iato, and a rhythmical movement of the drum.
and the whole is divided into regular sections
and subsections by the periodical strokes of
the gongs and kenongs [kettles]. The Taria-
tions of the same tune by the different instru-
ments produce a sort of barbarous harmony,
which has, however, its lucid moments,
when the beautiful tone of the instruments
yields a wonderful effect. But the principal
charm is in the quality of the sound, and the
rhythmical accuracy of the playing. The
players know by heart a oouple of hundred
pieces, so as to be able to take any of the
instruments in turn.'
In his report on Japanese music, Mr. S
Isawa, director of the Musical Institute at
Tokio, distinctly claims a species of harmony
for Japan, and gives an arrangement of the
Greek 'Hymn to Apollo' (Chappell, p. 174),
which he had directed 'a Ck>urt musician,
and a member of the [Musical] Institute, to
harmonise purely according to the principles
of Japanese classical music' It was set for
five instruments, the Riuteki (fuye), Hiohiriki,
Sho, Koto, and Biwa. I possess the oopyof
the music in European notation, sent to the
Educational Section of the Health Exhibiti<Mi
in 1884. Though much was in Octaves, the
koto played a figured form, with dissonances,
followed by consonances. A non-professionsl
Japanese gentleman, a student of physics, ac-
quainted with European music, in answering
my questions, says : * Anything like Eniopean
[harmony] cannot be heard in Japan. If it
exist, it is of the rudest possible description.
We have certainly ensemble playing with many
instruments of different sorts; but it seems
to me that we have no idea of such things
as chords. . . . We go generally parallel in
Octaves and in Fifths, rarely in Fourths,
but there are cases where two different tones,
not belonging to the three consonances, are
sounded, but they are not harmonic^ but what
Digitized by V^jOOQlC
SECT.L. RECENT WORK ON BEATS AND COMBINATIONAL TONES, 527
Helmholtz calls polyphonic. We have many they do not seem to distinguish from Earo-
figures for accompaniment. ... In popular pean equally tempered notes, and which will
music, we meet with cases where two instru- probably be soon reduced to that form by the
ments play Octaves or Fifths. With singing labour of Mr. Isawa, there seems to be no
this would also hold, but it is very rare that reason why harmony should not be natural-
people ever sing chorus.' ised, like so many European customs, in the
At the same time, as the Japanese use a wonderfully progressive country of Japan,
system of twelve notes to the Octave, which
It may be added, although it cannot appear from the table of the scales, that in
listening to native Javese, Chinese, and Japanese performers, there seemed to be
a total absence of what we term expression. There was no piano and forte, no
shading or nuance, merely a hard playing of the notes, as on street mechanical
pianos. They appeared to depend principally on gongs, clacks, or accumulation
of various instruments to give rhythm and spirit to the music. But so far as I
could judge by the very little India music I heard from Bajah K4m P41 Singh, it
seems to have some expression, as it certainly has an extremely varied rhythm,
sounding very strange to European ears. (See Siamese scales, Postscript, p. 556.) H
SECTION L.
BECENT WORK ON BEATS AND COMBINATIONAL TONES.
(See notes throughout Part II., pp. 152-233, and especially pp. 43, 55, 126, 151, 152, 155, 156,
157, 159, 167, 199, 202, 204, 205, 226, 229, 231, and 420. The reader is particularly re-
quested to defer any reference to this Section L until he has studied Part II., and become
familiar with the whole phenomenon of beats and combinational tones, and with Prof.
Helmholtz*s theories respecting their origin. Until such familiarity has been gained, much
* of what follows will be unintelligible.)
Art.
1. Papers considered, p. 527.
i.-v. Koenig; yi. Bosanquet; yii. and
viii. Preyer.
2. Koenig's Simple Tones, p. 528.
(a) Simple tones of forks, p. 528.
(5) Simple tones of the wave-sirens,
p. 529.
3. The Phenomena which arise when Two
Tones are Sounded together, p. 529.
(a) The facts as distinct from theory,
p. 529.
(5) Upper and lower beats and beat-
notes, p. 529.
(c) Limits within which either one or
both beat-notes are heard, p. 529.
(d) Beat-notes and differential tones,
p. 530. , ^ ^
(0) Bosanquet's summary of the phe-
nomena, p. 530.
4. Objective Beats and Subjective Beats, Beat-
Notes, and Differential Tones, p. 531.
{a) Objective beats, 531.
(6) Subjective beats and notes, p. 531.
(c) Preyer*s experiments to shew the
subjectivity of differential tones,
p. 531.
(d) Subjectivity of Summational Tones,
p. 532.
Theory of Beats, Beat-Notes, and Combi-
national Tones, p. 532.
{a\ Origin of beats, p. 532. %
\h) Can beats generate tones? First,
beats of intermittence, p. 533.
(c) Can beats generate tones ? Secondly,
beats of interference, p. 533.
(d) Would a tone generated by beats be
louder than its primaries ? p. 534.
(e) Experiments with the wave-siren,
p. 534.
(/) Beats and beat-notes heard together,
P- 535-
(g) Beat-notes and beat-tones, p. 535.
0i) Koenig' s explanation of summational
tones, p. 536.
(i) Eoenig*s theory of the origin of
beat-notes, p. 536.
(k) Lecture-room demonstration of beat-
notes, p. 536.
Influence of Difference of Phase on Quality .
of Tone, p. 537. ^
Influence of Combinational Tones on the
Consonance of Simple Tones, p. 537.
Art I. Papers considered.
The papers here considered are
i. B. Koenig. Ueher den Ztcsammenklang Zweier Tone (On the sounding of
two tones at the same time). Pogg. Annal. Feb. 1876, vol. 157, pp. 177-237.
This paper appeared a year before the 4th
German option of Helmholtz's Tonempfin'
dun§en, and is cursorily referred to, supriL, p.
1596. The other papers of Koenig here men-
tioned appeared subsequent to Prof. Helm-
holtz*s 4th German edition. But this paper
is placed first because it commenced the new
investigations. A translation appeared in the
Philosophical Magazine, June 1876, and sup-
plement of the same date, pp. 417-446, 51 1-
525, under the title * On the Simultaneous
Sounding of Two Notes,* and communicated
by the late W. Spottiswoode, President R. S.,
who also read a paper on ' Beats and Combi-
nation Tones * before the Musical Association
on May 5, 1879 {Proceedings of Mas. A.,
1878-9, pp. 1 18-130), when he exhibited K.'s
apparatus and repeated several of his experi-
ments.
Digitized by
Google
528 ADDITIONS BY THE TRANSLATOK. app. xi.
ii. R. KoENiG. Ueher die Erregung harmonischer Ohertone durch Sckwingungen
eines Grundtones (On the excitement of harmonic upper partials hy the vibrations
of a fundamental tone). Wiedemann, AnnaL 1880, vol. xi., pp. 857-870.
This treats of a subject incidentally mentioned, soprA, p. 159a, in reference to combinational
tones.
iii. R. KoENiQ. Ueher den Ursprung der Stosse und Stosstone bei harmonischen
Intervallen (On the origin of . beats and beat-tones for harmonic intervals).
Wiedemann, Vnnal. 1881, vol. xii. pp. 335-349, introducing an entirely new
method of experimenting by means of the wave-siren.
iv. R. KoENiG. Beschreibung einss Stosstoneapparates fur Vorlesungsversucke
(Description of a beat-tone apparatus for lecture-room experiments), Wiedem., 1881,
immediately after the last paper, vol. xii. pp. 350-353.
V. R. KoENiG. Bemerhungen iiber die Klangfarbe (Remarks on quality of tone)
^ Wied., 1 88 1, vol. xiv., pp. 369-393, more folly describing the wave siren of No. iii.
In drawing up this notice I made use solely p^riences d'AcoustiquSj 1882, to be had at his
of the original Q«rman papers just cited. But present establishment, 27 Quai d'Anjou, Paris,
I find that Dr. Koenig has republished the and I have made some use of the additional
whole of his 16 acoustical papers, of which notes then added. I cordially recommend
those just cited form the 9th, 14th, loth, nth, this collection as a valuable and almost indis-
and 1 6th, respectively, in the French language, pensable supplement to Prof. Helmholtz's
in one volume, with beautifully printed wood- work,
engravings, under the title of Quelques Ex-
vi. R. H. M. BoBANQUBT. On the Beats of Consonances of the Form h: i.
Proceedings of the Physical Society of London, vol. iv., Aug. 1880 to Dec. 1881,
pp. 221-256. This was written before B. had seen No. iiL and iv. above. •
vii. W. Preyee. Ueber die Orenzen der Tonwahmehmung (On the limits
of the perception of tone) containing the sections. I. The Lower Tones. IL The
Highest Tones. IIL Sensitiveness for Difference of Pitch. IV. Sensitiveness
^ for the Sensation of Interval. V. Sensation of Silence. Forming the first part
of the first series of Physiologische Abhandlungen (Physiological Essays) edited
by W. Preyer, M.D. and Ph.D., Prof, of Physiology and Director of the Physio-
logical Institute at Jena, 1876, the year before the publication of the 4th German
edition of Hehnholtz, who quotes it several times. It is here inserted for com-
pleteness.
viiL W. Pbeyeb. Akustische Untersuchungen (Acoustical Investigations) in tlie
same collection, second series, fourth part, containing I. Deepest Tones without
upper partial Tones (supri, see footnote, pp. 176-7). II. Combinational Tones and
upper partial Tones of Tuning Forks. EDL Contributions to the Theory of Con-
sonance. IV. Notice on the Perception of the smallest differences of Pitch,
Jena, 1879.
These will be cited by the initial of the editions of vii. and viii. Prof. Hehnholtz will
author, E. or B. or P. followed by the number be cited as H., generally followed by tiie page
of the paper, and generally by the page, which of this edition,
in the case of P. will refer to the separate
^ Art. 2. Koenig' s Simple Tones,
(a) Simple Tones of Forks, The tones dealt with by K. are as simple as
K could make them. * The forks that I used with resonators,' says K. iii. 337,
* had no recognisable harmonic upper partials at all. The occurrence of harmonic
upper partials in tuning-forks depends not so much on the lowness of their piteli
and the amplitude of their \ibrations as on the relation of the amplitude to the
thickness of the prongs.'
From a c fork (128 d. v.) with prongs opening of the resonator tuned to them, almost
7 mm. ( « '28 inch) thick, K. obtained as many touched the prongs of the fork. The pitch of
as 4 partials. From another c fork with forks varies directly as the thickness, and in-
prongs 15 mm. (»'59 inch) thick and 20 mm. versely as the square of the length, of their
(s 79 inch) wide, only 2 partials were gene- prongs (K. iii. 338). K. proceeds to mention
rally obtained, but extremely violent blows that the forks he used, even the largest, when
brought out a 3rd partial. With prongs placed before properly tuned resonators, had no
29 mm. (1*14 inch) thick and 40 mm. (=1-57 detectable upper partials. Subsequently B.
inch) wide, it was not possible to hear even a repeated K.'s observations in part with the
faint Octave, and a Twelfth, except when the stopped organ-pipes of B.'s experimental
Digitized by V^OOQIC
MOT. L. RECENT WORK ON BEATS & COMBINATIONAL TONES- 529
organ, in which only the Twelfth or 3rd par-
tial was perceptible and could be allowed for.
Also, afterwards, K. iii. 342, used stopped
organ-pipes and tuning-forks. B. used tones
(6) Simple Tones of the Wave Siren.
however, E. invented the wave siren.
An harmonic curve constructed on a large
scale and reduced by photography was cut on
the edge of a wheel. The wheel revolved
under a narrow slit, placed exactly in the
position of a radius of the wheel, through
which wind was driven as the wheel rotated.
The curve alternately cut oS. and let pass the
stream of air, and produced a perfectly simple
tone, the pitch of which depended on the ra-
pidity of rotation. Forms of this wave siren
are figured in E. iii. 346, 347, and E. v. 386.
The last shews 16 harmonic curves which
may be made to act in any groups, producing
all the combinations of perfectly simple tones,
of moderate force, and E. also used weak tones
with the pipe, and not the strong tones of his
tuning-forks mentioned in E. i.
To avoid the suspicion of upper partials,
of which the ratio numbers lie between i and
16. Hence, although upper partials are found
on most tuning-forks, and especially on cer-
tain of E.'s forks, it would be wrong to assume
(as P. ii. 38 apparently assumes) that in all
E.'s cases, at least the Octave was audible.
This waa not the case with stopped organ -
pipes used by both E. and B., and still less so
with the tones of the wave siren. E.'s results •
therefore cannot be explained by upper partial
tones. But when we are dealing with com-
pound tones, each pair of partials forms a li
combination of simple tones to which E.'s
observations apply.
'Art. 3. The Phenomena which arise when two notes are sounded together,
according to Koenig and Bosanquet,
(a) The facts as distinct from theory. We must distinguish the phenomena from
any theoretical explanation of them that may be proposed. The phenomena de-
scribed by such an acoustician as K., so careful in experiments, so amply provided
with the most exact instruments, will, I presume, be generally accepted. The
theory by which he seeks to account for them is a matter for discussion. The
following relates to two simple tones only, and this must be carefully borne in
mind, because H. 1592^ apparently imagined that the tones used really had upper
partials.
[h) Upper and Lower Beats and Beat-notes,
If two simple tones of either very slightly or greatly different pitches, called
generators, be sounded together, then the upper pitch number necessarily Hes ^r
between two multiples of the lower pitch number, one smaller and the other
greater, and the differences between these multiples of the pitch number of the
lower generator and the pitch number of the upper generator give two numbers
which either determine the frequency of the two sets of beats which may be heard
or the pitch of the two beat-notes which may be heard in their place. The term
* beat-notes ' is here used without any theory as the origin of stwh tones^ but only
to shew that they are tones having the same frequency as the beats, which are
sometimes heard simultaneously.
Beferring to the tables in the Translator's
footnote to p. 191 suprd, which relate to com-
pound tones, and therefore contain multiples
of the pitch numbers (or of the numbers which
give the interval ratios) of two generators, we
see from the minor Tenth' 5 : 12, that the
prime 12 of the upper generator lies between
10 and 15, the 2nd and 3rd multiples of the
lower generator, and hence the beat or beat-
note frequencies would be 12— 10= 2, and
15 — 12 = 3. If, then, the two generators are
low enough, say, having the pitch-numbers
. 5 X 6 » 30 and 12 x 6 = 72, the beats heard
would be 2 X 6 » 12 and 3 x 6 » 18, which
But if they were higher, as 5 x 20 « 100 and
12 X 20 « 240, the beats would be 2 x 20 » 40,
and 3 X 20 — 60, which, though far too rapid
to be counted, would be clearly heard as beats,
and at the sanie titne the beat-notes of 40 and
60 vib. would aUo be audible. If, however,
they were much higher, as 5 x 100 » 500 and
12 X 100 = 1200, then only the beat-notes of
2 X 100 = 200 and 3 X 100 = 300 vib. would ir
be heard. The beats heard are then the same
as if the upper generator were simple and the
lower generator compound; but it must be
remembered that both generators are really
simple.
would be plainly distinguishable as beats.
The frequency arising from the lower multiple of the lower generator is called '
the frequency of the lower beat or lower beat-note, that arising from the higher
multiple is called the frequency of the higher beat or beat-note, without at aU
implying that one set of beats should be greater or less than the other, or that
one beat-note should be sharper or flatter than the other. They are in reality
sometimes one way and sometimes the other. <
(c) Limits within lohich either one or both Beat-Notes are heard. Both sets of
beats, or both beat-notes, are not usually heard at the same time. If we divide the
intervals examined into groups (i) from 1:1 to i : 2, (2) from i : 2 to i : 3,
(3) from I : 3 to I : 4, (4) from i : 4 to 1:5, and so on, the lower beats and
Digitized by VuiODQ IC
530
ADDITIONS BY THE TRANSLATOR.
APP. XX.
beat-tones extend over little more than the lower half of each group, and the
upper beats and beat- tones over little more than the upper half. For a short
distance in the middle of each period both sets of beats, or both beat-notes, are
audible, and these beat-notes beat with each other, forming secondary beats, or are
replaced by new or secondary beat-notes.
(d) Beat-Notes and Differential Tones, The lower beats, as long as they are
distinctly audible, and refer to an interval less than 5 : 6, or a minor Third, sgree
with the beats of H. 171a, and when they have a greater frequency than from 16
to 20 there is also heard the beat-note, which then coincides in pitch with the
differential tone of H. 153a. Above a minor Third, H. ijid says the beats are
practically inaudible. K. however hears them — and B. vi. 235-237 also heard
them — passing over into a roll and a confused rattle, as far as the major Sixth
(G : Ai){or the lower beats.
confased when hoard separately, althongh the
frequency was the same. K., so far as I can
see, does not anywhere mention the pitch of
the beats he heard, bat B., tI. 237-9. sajs
that in all cases he has obserred, when the
required partials have been remoyed, 'the
beats . . . consist entirely of variations of in-
tensity of the lower note/ and adds that, * as
he (E.) does not analyse the beats, we cannot
tell whether the variations of the lower note
were produced in his experiments.' Mr. Blaik-
ley {Proc, Mub, Asm, i8Sf-2, p. 25) relates,
however, that when E. exhibited the beats to
him in Paris, E. said : • You hear distinctly —
there can be no doubt about it— that the beat-
ing note is the lower one.' This gives K.*s
opinion, which Mr. Blaikley did not share.
Observe that E. does not deny the exist-
ence of tones having the pitch of differential
(or summational) tones, but, as in this case,
he shews their existence, and that they are
distinct from his beat-notes, having frequently
a different, and only occasionally the same,
pitch. When, therefore, B., vi. 239, talks of
second, third, and fourth combinational tones
having been demonstrated directly by K., he
seems to have identified beat-notes and diffe-
rential tones, which, however, E. distinguishes.
The above observations of E. on differential
tones are taken from the German edition of
E.'s paper in 1876. In the French repablica-
tion in 1882 they still appear, but in paren-
theses, and with a long note (ibid. p. 130), in
which he states that subsequent investigations
have induced him to change his opinion, as he
finds that even very wide harmonic intervals
between extremely weak tones may prodace
distinct beats. Hence (for the case where an
auxiliary fork produced beats with two forks
having the ratio 8 : 15, shewing a very weak
tone 7, that might have arisen from ' the tone
of the lower beats of 8 and 15 '), in his Ger-
man summary of results, E. i. 236, parai^raph
III. 6, admitted the actual existence of differ-
ential tones, though * extraordinarily weaker
than beat-notes ; * but in his French republica-
tion (p. 147) he has altered this paragraph to :
* No experiment has yet proved with certainty
the existence of differential and summationiU
tones.* Observe that the existence of tones .
with the pitch of differential tones is not dis- 1
puted. It is only the theoretical origin of snch 1
tones that is called in question. At present it \
seems impossible to decide that point.
In other respects E.'s beat-notes are differ-
^ ent from H.'s differential tones. Thus for the
minor Tenth 5 : 12, our first example, the
beat-notes are 2 and 3, as just shewn, but the
differential tone is i2-5»7, which is not
obtained by E. i. 216, who says : * These inter-
vals, which are formed by high tones, allow
the beat-notes to be heard quite loudly, but
give no trace of differential tones. Thus
&" : b'" (8 : 15) gives only i and no trace of 7,
c'" : d'" (4 : 9) gives only i and nothing of
e"' (5) ; &" :r (3 : 8) only / and /' and no
a*" (5) at all, hence the differential tones must
be extraordinarily weaker than the beat-notes.
But I was able to establish the actual exist-
ence of these differential tones with certainty
by forming the above intervals with deeper
notes, which, lasting longer, allowed me by
means of auxiliary forks to get a definite
number of beats with the differential tones in
«r question.' This experiment I have repeated
.several times. I made the tone of the generat-
ing forks as loud as possible by holding them
over resonance jars. The auxiliary fork had
to be held at a considerable distance from its
jar in order to reduce its loudness to about
that of the differential tone, to allow the beats
to be counted. Thus the mistuned minor
Tenth 22377 : 539*18 gave the differential
tone 315*41, which, although inaudible, beat
with the fork 319*59 audibly 4*18, which was
counted as 4*2. And so on in other cases.
In the case of a mistuned Octave, it appears
to me that the lower fork acts as this auxiliary
fork to catch the differential tone. Thus the
mistuned Octave 22377 : 451*14 gives the dif-
ferential tone 227*37, which would possibly
have been quite inaudible if it had not been
caught by the lower fork, with which it made
^ 3*6 beats in a second, as I myself counted. In
this case I had to hold the higher fork far
above the resonance jar. The beats were heard
as low beats at the pitch of the lower fork. Also
in this case, on continuing to hold the high fork
over the resonance jar of the upper fork to
weaken its sound, but bringing the low fork
over the higher resonance jar as closely as
possible, the higher Octave of that fork, or
447'54, was produced, which beat with the
higher fork also 3*6 times ; but now the beat
was clearly and distinctly at the pitch of the
npper fork. This was a beat of the 2nd par-
tial of the lower fork with the upper fork, and
was altogether distinct in character from the
lower beat. Hence the beats could not be
ie) BosanqneVs summary of the pJienom^na, B. vi. 228. * As two notes of
equal amplitudes separate from unison, they are at first received by the ear in the
Digitized by V^OOQIC
BECT. ir. EECENT WOBK ON BEATS & COMBINATIONAL TONES. 531
manner of resultant displacements, consisting of the beats of a note whose fre-
quency is midway between the primaries. When the interval reaches about two
commas [say 43 or 50 cents], the ear begins to resolve the resultant displacements,
and the primary notes step in beside the beats. When the interval reaches a
minor Third in the ordinary pans of the scale, neither the beats nor the inter-
mediate pitch of the resultant note are any longer audible, at least as matter of
ordinary perception ; but the resultant displacement which reaches the ear is
decomposed, and produces the sensation of the two primary notes, perfectly distinct
from each other : that is to say, Ohm's law has set in, and is true, for ordinary
perceptions and in the ordinary regions of the scale, for the minor Third and all
greater intervals.' These phenomena are not mentioned by Koenig, and in my
own observations I feel a difficulty in appreciating them.
Art. 4. — Objective Beats and Subjective Beats, Beat-Notes and Differential
Tones.
(a) Objective Beats. Beats of a disturbed unison exist objectively as disturb-
ances in the air before it reaches the ear. They are reinforced by resonators, the}- %
disturb sand, &c. In the case of the beats of harmonium reeds in Appunn's
tonometer, they strongly shook the box containing the reeds. Other beats, beat-
notes, and combinational tones appear not to exist externally to the ear.
(b) Subjective Beats and Notes. K. i. 221 says : * Neither these combinational
tones nor the beat-notes already described are reinforced by resonators.* B. vi.
233-4, after describing his improved resonator, by means of which he can effec-
tually block up both ears against any sound but that coming from a resonance
jar (see p. 4^d\ note J), says : * By means of these arrangements I some time ago
examined the nature of the ordinary first difference-tone, and convinced myself
that it is not capable of exciting a resonator. In short, the difference-tone of H.,
or first [lower] beat-note of K., as ordinarily heard, is not objective in its character.
. . . When the nipples of the resonator-attachment fitted tightly into the ears, no-
thing reached the ear but the uniform vibrations of the resonator sounding C. But
if there was the slightest looseness between the nipple and the passage of either ear,
the second note (c) of the combination got in, and gave rise to the subjective
difference-tone (first [lower] beat-note of K.), by the interference of which with the ^
C 1 explain the beats on that note. These beats are therefore subjective.'
This expression is not meant to imply that again says, H. 216a, that he has * always been
they are the prodnct of the imagination, but able to hear thefleeper combinational tones o!
that they do not exist externally to the ear. the second order, when the tones have been
Hence, when H. 157c says that they are at played on the harmonium and the ear was
least partly objective, although he admits that assisted by proper resonators,' he had possibly
the greater part of the strength of combina- not succeeded in blocking both ears properly
tional tones arises only within the ear, and against the outer air.
{c) Preyer's experiments to shew the subjectivity of Differential Tones. — P. viii.
II. had seven tuning forks of extraordinary delicacy constructed, giving /lyof,
c' 256, /' 341^, a' 426I, c" S^^iJ[" 6S23, gr" 768 vibrations, and hence having the
ratios 2:3:4:5:6:8:9, which were so ready to vibrate on the slightest ex-
citement that they could be experimented on at night only. The three lowest
forks had the following partials.
Fork / had the 2nd/' strong, the 3rd &' strong, and the 4th/" weak.
Fork c' had the 2nd &' strong and the 3rd g^' strong. ^
Fork/' had the 2nd/" strong.
Sounding these forks in pairs to get the differential tones,
c" &/gave/' or 6 — 2=4 ;/" &/gave c" or 8 — 2=6; g'^ & c' gave c" or 9—3=6;
and that these tones were objective enough was shewn by their making the forks
/', c" vibrate sympathetically. But we see that/' and c" are partials of / and c',
which existed already strongly on those forks, and if the forks / and & were
sounded separately, they also made the forks /', c" vibrate sjrmpathetically.
Hence these results did not prove the objective existence of their differential dupli-
cates. On the other hand, the pairs of forks gi\dng the audible differential tones —
/"-c" = / or 8-6 = 2, a'-c' = /or5-3 = 2, c"-/'=/ or 6-4 = 2,
g' -c" = c' or 9-6 = 3, a'-f= c' or 5-2 =3, /"-a'=c' or 8-5 = 3,
gF"-a'=/'or9-5=4, gf"-/ = a' or 9-4 = 5, /"-c'=a' or 8-3 = 5,
utterly failed to produce the slightest effect on the forks having the same pitch, t
Digitized bykJaQUQlC
532 ADDITIONS BY THE TRANSLATOR. app.xx.
(d) Subjectivity of Summational Tones. Again, for summational tones the
combined forks
/+/'=c''or2 + 4=:6, /+c"=/" or 2 + 6 = 8, c' +c" = 5f" or 3 + 6 = 9
gave tones perfectly objective, but then these tones c", /", gf" ahready existed as
partials of one of the two forks excited. On the other hand,
/+c'=a' or 2 + 3 = 5, c'+a'=/" or 3 + 5 = 8, /'+a' = sf" or 4 + 5 = 9
were inaudible, that is, neither existed without nor within the ear. * Perhaps,'
says P., * they might be made audible after properly arming the forks by means of
resonance boxes while sounding. But the observation would not be easy.* Just
as H. could hear the cases he cites (Pogg. Ann. vol. xc. 1856, p. 519) ' only with
great difficulty.' But the forks tried by him each possessed the Octave, as he
states (ibid. pp. 506, 510).
Now, when Octaves exist, and in case of P., * even the oomprehensive investigations of
€r the siren other partials are strongly developed, Koenig do not make the [external objective]
these summational tones — as G. Appunn existence ^of summationiA tones probable/
pointed out to P.— could be conceived as differ- Hence, like the differential tones, they most
ential tones of the second order— that is, be generated within the ear.
differential tones arising from the first differ- E., in the French edition of his papers
ential acting on the partial, if such action is (note, p. 127), says : ' This explanation is not
admitted. Thus H. found b +/ « (2 or 2 + 3 <= 5, admissible, because it assumes that two sounds
but then b and/ included the Octaves 2>'and/' always generate a differential tone, which is
or 4 and 6, and we had, the first differential not correct. For example, take two tones cor-
/— 6 = Bor3 — 2 = 1, and the second f'—B = d" responding to the fundamental tones d and t'
or 6— I =: 5, or, without using B in the formula, [my notation], or 256 : 320=54 : 5. They give
/'—(/— 6) « d" or 6 — (3 - 2) = 5. In this way the beat-note cf-i\ but this sound I does not
P. proceeds to shew that all the cases recorded form the sound 7 with the Octave of c^.that is,
can be explained. Hence he concludes that with c" = 8 ; nor does it form the sound 9 with
the summational tone, if not existing as a the Octave of e/ or e/' » 10, as we can be con-
partial on one of the tones, is entirely gene- vinced by sounding at the same time the
rated within the ear. Thus, according to primary sounds C and c"->i : 8, or G and «/'•
K., i. 220, from c' : ^ = 2 : 3 he heard clearly » i : 10, even when these latter are much
5 = tf", 7=sfl" + c', 8 = e" + gf', 9*d"', io = e'" stronger than the beat-note in question and
m and II, the last by auxiliary forks which beat than the two Octaves of the primary sounds
with the required tones. But there were in cf and e/.' The explanation by differentials of
this case the partials 2, 4, 6, 8 = c', c", ^', e'", the second order given by P. is an adoption of
and 3, 6, 9, 12 = ^, /', d"\ /". Hence from a theory of Appunn ; and, of course, nntil the
the summational tones we have 8 and 9 as reality of the differentials assumed can be
partials, while 5 = 8-3, 7=^—2, 10^12-2, proved, remains a merely theoretical explana-
Ii«i2~(3 — 2), and so jon. * Therefore, * adds tion.
Art. 5. — Theory of Beats, Beat-Notes, and Combinational Tones.
(a) Origin of Beats. * How do the beats of mistuned consonances arise ? ' asks
B. vi. 228, and replies : * They may be regarded as springing from interference of
new notes, which arise by transformation, in the passage of the resultant forms
through the transmitting mechanism of the ear, before the analysis of the sen-
sorium.*
The theory of beats of a disturbed unison on the hypothesis of interference is
given in H. 164. The theory of differential and summational tones is given in
H. 159a and App. XIL pp. 411-413. This, however, extends only to the first
differential and first summational tone. But H. 1586, c, gives a theory for the
f generation of such tones within the ear owing to the non-symmetrical structure of
its drumskin and the looseness of the joint between the hammer and anvil within
the drum. And B. vi. 242-8, by means of some perhaps rather hazardous assump-
tions, succeeds in shewing that the asymmetry of the drumskin acting upon
the waves of air coming to them would, as he terms it in the above extract,
* transform ' the result into one for which the displacement is not relatively
infinitesimal, but in which its higher terms must be taken in consideration. Tlien
proceeding to the fourth order of displacement, he ultimately obtains six summa-
tional and six differential tones * produced by direct transformation of the prima-
ries ' (B. vi. 246), so that he avoids the introduction of numerous differential tones
of various orders (H. 200-203, B. vi. 241), which H. seems to have borrowed fix)m
Scheibler, who, although he did great things with the beats of tuning-forks, was
not a physical authority. Calculation based on the introduction of these entirely
hypothetical, because always inaudible, tones leads, as K. i. 200 shews, to the right
number of beats ; but, as he says, * we are compelled continually to assume the
existence of tones which have not only not been heard themselves, but which are
Digitized by V^J-OOQi€
SECT. L. EECENT WORK ON BEATS & COMBINATIONAL TONES. 533
supposed actually to generate and be generated by other likewise infl.udible tones/
"We have an example in the first differential of a tone which, when it does not
coincide with the lower beat-note, is appreciable only by beats with an auxiliary
tone, and is hence very faint indeed in respect to the generators ; and yet these are
supposed to be the progenitors of others relatively weaker, till at last they produce
one strong enough to be well heard (K. i. 186). The diflBculty is surmounted by B.,
so far as the existence of the ultimate tone, without assuming the action of hypo-
thetical intermediate generators. But we know nothing of the strength of these
xdtimate tones as determined by the formula, and we are constrained to beheve
that what depends upon the higher powers of the displacement, when the latter is
not infinitesimal in respect to the length of the wave, must be extremely small,
not at all comparable with the beat-notes actually heard, and hence must be in-
sufiBcient to explain them. That is, we may admit all the differential and summa-
tional tones of H. and B. without having approached a satisfactory explanation of
the main phenomenon, the beat-note.
(b) Can Beats generate Tones ? First, Beats of Intermittence. Now, the obvious IT
hypothesis is that the beats coming within the frequency of musical notes are
heard as tones. H. 156c, in mentioning this, states three objections, of which
P. viii. 27 says that not one is at present tenable. They are : (i) that this hypo-
thesis does not explain summational but only differential tones. On which P.
remarks that summational tones, which have been heard only when at least the
second partial of the generators was audible, can be explained as differential tones of
the second order, as noted supr^, p. 532 6. (2) That ' under certain conditions the
combinational tones exist objectively,' which is against art. 4, p. 531 ; and P. viii. 25
especially observes that the only experiment which H. has cited (Pogg. AnnaL
vol. xcix. p. 539) to prove the objective existence of summational tones by sand
strewed on a membrane cannot be critically examined because the two generators
are not specified. (3) That ' the only tones which the ear hears correspond to
pendular vibrations of the air.' P. considers this to be disproved by the inter-
mittence tones obtained by K., who rotated a disc perforated with 128 holes before
tuning forks of different pitches, and obtained the same tone of intermittence what-
ever was the pitch of the fork. This tone was accompanied by two variant tones f
having pitch numbers equal to the sum and difference of the frequencies of the
fork and intermittences.
E. i. 230, varied the experiment by oon- qnickly, the 16 periods of the first, then the
fifructing a disc * with three circles, each with 12 of the second, and finally the 8 of the third
96 equidistant holes, the diameters of which circle, passed over into a musical tone. Finally,
increased and diminished on the first circle 16 when the high tone of the 96 holes on revolv-
iimes from i to 6 mm. (a'04 to -24 inch), on ing 8 times in a second had reached </' with
the second 12, and on the third 8 times. On 768 d. vib., the deep tones c, G, C— answering
blowing through a tube 6 mm. (»'24 inch) in to the numbers of the periods 128, 96, 64 d.
diameter, and revolving the disc slowly, the vib. — could be heard loudly and powerfully at
separate periods of holes on each circle gave the same tim^ as y".'
separate beats. On revolving continuaUy more
(c) Can Beats generate Tones f Secondly , Beats of Interference. Now, in
reference to these tones of intermittence, E. i. 231 remarks that although they show
great similarity to beating combinations, as proving the possibility of separate
maxima of intensity passing over into a continuous tone, they were in reality very «
different from such combinations, because in the case of beats there was a change
of sign, a maximum of condensation being followed by a maximum of rarefaction.
This was precisely the objection made by equal to half the difference of the frequencies
liord Bayleigh when Mr. Spottiswoode gave of the primaries.' Hien B. says that if the
his account of Eoenig's experiments (Proc. law held for widely separated notes, as for the
Mu», Ass. 1878-9, p. 128), and he in conse- * Fifth (4:6), the note heard would be the
quence could not understand how beats could major Third, which would beat rapidly . . .
generate tone. B. {ibid, p. 129) raised the but as a matter of fact the note 5 is not heard
same objection, which he developed in B. vi. at all in the above case.' Further, * supposing
223-5. He there shews that in the case of that in some unexplained way the beats whose
two tones of equal strength, less than two speed is ' half the difference of the ^equencies
commas from a unison, * the resultant dis- of the primaries, as just stated, * gave rise to a
placement ' would produce a tone * whose fre- note as supposed by K., then the speed of that
quency is the arithmetical mean between the note does not agree with that required for E.'s
frequencies of the two primaries, and having first [lower] beat-note, which has the same
oscillations of intensity whose frequency is speed as H.'s difference-tone,' or the whole in-
defined by a pendulum vibration of frequency stead of half the difference of the frequencies.
Digitized by VjOOQlC
534
ADDITIONS BY THE TRANSLATOR.
APF. XX.
Now this objection was fully realised by E.
i. 232-3, which paper was before B. when he
wrote the passage just cited, but was possibly
overlooked by him. K. i. 232 says: *If two
tones of 80 and 96 d. vib. are sounded together,
they generate a tone of i. (80 + 96) =88 vibrar-
tions with an intensity increasing and di-
minishing 16 times, and at each passage from
one beat to another there is a change of sign,
so that the maximum of compression of the
first vibration of the following beat is hidf
a vibration behind the maximum of compres-
sion of the last vibration of the preceding
beat.' To meet this case he made two experi-
ments. In the first he divided a circle into
176 parts, and in the five points i, 3, 5, 7, 9 he
drilled five holes, gradually increasing and ihen
^ diminishing in size. Similarly in the points
12, 14, 16, 18, 20, and then in the points
23f 25. 27, 29, and 30, and so on. * When such
a disc was blown upon through a pipe with the
diameter of the largest opening, in addition to
the tone SS and the very powerful tone of the
period 16 both of the tones 80 and 96 could
be heard, but they were very weak, and, on
account of the roughness of the deep tone,
difficult to observe.* In this case the phase
was the same throughout. To imitate the
change of phase, K. i. 233 divided each of
two concentric circles, running parallel to each
other, into 88 parts, and * disposed the holes
which were to represent the successive beats
alternately on each. As 88 holes and 16
periods give 5^ holes to each period, E. took
two periods together, and pierced on the first
circle the divisional points i, 2, 3, 4, 5, 6, and
f on the second 6, 7, 8, 9, 10, 11, then again on
the first 12, 13, 14, 15, 16, 17, and on the
second 17, 18, 19, 20, 21, 22, and so forth.'
The sizes of the holes were alternately in-
creasing and diminishing to represent beats.
* When these circles of holes were blown upon
at the same time through two pipes of the
diameter of the largest opening, and placed on
the same radius, one circle from above and the
other from below, then at each revolution of
the disc there were created SS isochronous
impulses, varying 16 times in intensity, which
changed sign on each transition from one period
of intensity to the other. In this experiment
the two tones 80 and 96 were morcf distinct
than in the first experiment, where the circles
of holes were blown upon from one side only.*
On B.'s objections just quoted (supr&, p.
533), E. observes (French edition, p. 143, note) :
*The change of phase of the separate vibra-
tions of a variable amplitude, forming the
beats, does not cause these maxima of in-
tensity to be produced in contrary directions.
Besides, these maxima remain isochronous,
and consequently fulfil the conditions under
which primary impulses are combined to form
sounds. The only influence which the change
of phase in question exerts on the disposition
of the waves consists in these maxima of in-
tensity not standing apart by a whole number
of complete vibrations, but by an odd number
of half-vibrations. The disc of the siren in
which the resultant compressions of all the
successive vibrations of the complex sound are
represented by holes of a proper size, and, still
better, the disc that has its rim cut oat
according to the curve of a series of successive
beats [art. 5 (e) below], render this mechanism
readily apprehensible, and allow of shewing
that, notwithstanding the change of phase,
the beat-note must always have Uie same fre-
quency as the beats.'
(d) Would a Tone generated by Beats be louder than the Primaries ? K. i. 234
then proceeded to meet Tyndall*s objection {On Sound, 3rd ed. p. 350) that if the
resultant tones (as he calls them) were formed from the beats of the primaries
they would be heard when the primaries were weak, which is not the case. E.
observes that beats would always be more powerful than their primary tones,
' provided that equal amplitudes of vibration produced equal intensities for all
tones,' and proceeds to shew by experiment that this is not the case, and that ' deep
tones must have much larger amplitudes of vibration than high tones in order to
exhibit the same intensity/ .
(e) Experiments with the Wave Siren. Thus the question was left till 188 1,
when K. applied his Wave Siren, originally exhibited in the London International
Exhibition of 1872, already partly described in art. 2 (6), p. 529, to solve the ques>
tion experimentally. The complete form (K. v. 386) was of course applicable to
«|f any pairs of tones with ratios expressible by numbers not exceeding 16. But the
simplest method was to draw out the two harmonic curves, and the result of their
combination, as is done above (H. 30 i, c) on a very large scale, and then reduce
the drawing by photography to the required dimensions. Then the compound curve
t^jus drawn was inverted, so that the high parts became low and the low high, cut,
and affixed to the rim of the wave siren. The reason for inversion in this case was
that the heights on the curve represented greater intensities, but on the siren
would give less intensities.
K. iii. 345 then says : • When a disc with .
such a rim is rotated before a slit fixed over it
in the direction of the radius, and of a length
at least equal to the greatest height of the curve,
the slit will be periodically shortened and length-
ened according to the law of the curve ; and if
wind is blown through the slit, a motion in
the air must be generated corresponding to
the same law. And this motion must be pre-
cisely the same as that produced by the simul-
taneous sounding of two really simple tones
without any admixture of upper partials.* The
beauty of this arrangement thus consists in oar
knowing precisely what tones act, and that
they are undoubtedly simple. The result is
thus described : * The discs for different inter-
vals, when the rotation was slow, gave beats,
and when it was more rapid, beat-notes, exactly
Digitized by V^jOOQlC
BiicT. L. recent: work on beats & COMBINATIONAL TONES. 535
corresponding to those observed when two
'tuning-forks were sounded together. Thus the
major Second 8 : 9 produced the lower beat-
r&ote I ; the major Seventh 8 : is, the upper
l>eat-note i ; the disturbed Twelfth 8 : 23, the
upper beat-note of the second period, which is
again ^ i, loudly and distinctly. In the same
way the ratios 8:11 and 8 : 13 gave quite
distinctly and at the same time the upper and
lower beat-notes 3 and 5 for the first, and 5
and 3 for the second:* 11 -8 = 3, 2 x 8-11 = 5,
and 13-8=5, 2x8-13 = 3.
(/) Beats and Beat-Notes heard together. The preceding experiment shews the
gradual passage of beats into tones, the transitional part being where both beats
and tones are heard together. This occurs where the rotation is sufficiently quick
to generate a tone (see H. 174-9, and especially footnote t to p. 176), but not so
fast as to destroy the distinct perceptions of beat
To this I drew attention in a footnote to
p, 231 of the 1st edition of this translation,
now reproduced in a modified form (suprii, p.
1 530^, note) . This hearing of the two phenomena
K. i. 227 explains by a theory of H. (contained
on pp. 217-^ of the ist English ed., but
omitted in this 2nd English ed., because it was
struck out in the 4th German ed., H. having
altered his opinion) that tones are heard in
the cochlea and noises in other parts of the
ear. In the additions to the 4th Qerman ed.
^8upr&, pp. 1506 to 151(2) H. attributes the hear-
ing of both musical tone and noise to the
cochlea, and reserves the labyrinth for the
sensation of revolution of the head, thus
agreeing with Exner. P. viii. 29-33, thinks
there are many reasons why we should not
accept the theory that all perceptions of noise
are due to the cochlea. If so, he says, * animals
-without a cochlea would be deaf. Fishes cer-
tainly are mostly dumb, and do not hear
acutely, as anglers well know, but they are
not deaf.* On examining Exner's paper
(supr^, p. I Sid, note *), and especially Anna
Tomaszewicz*s * Contributions to the Physi-
ology of the Labyrinth of the Ear,' {BeitrCtgeeur
Physiologie des OhrlabyrinthSy Medic. In-
augural Dissertation, Zurich, 1877), with other
phenomena, he comes to the conclusion that
the cochlea hears only musical tones with a
pitch-number not less than about 16 (the
lowest audible musical tone as usuaUy pro-
duced), and that separate noises are heard by
other parts of the ear — if not in the vestibule,
then in the sacculus. He considers it probable,
as others have also thought, that the function
of the semicircular canals is rather to give a
sensation of the direction whence sound comes.
The point is, however, still undecided.
E., in the French republication of his
paper (p. 137), says:— -'At all events the
simultaneous perception of separate beats and
the sound which results from their succes-
sion is no more in contradiction with the new
hypothesis than with the old, for we can very
well suppose that, beside the general excite-
ment of the basilar membrane due to each
separate beat, the particular parts of this
membrane, whose proper tones correspond to ^
the period of the impulses, are more strongly
shaken, and execute lasting vibrations giving
the perception of sound.'
Lord Bayleigh, in his Presidential Address
to the British Association meeting at Montreal,
Canada, in Aug. 1884, says : — • Every day we
are in the habit of recognising, without much
difficulty, the quarter from which a sound
proceeds, but by what step we attain that end
has not yet been satisfactorily explained. It
has been proved that, when proper precautions
are taken, we are unable to distinguish whether
a pure tone (as from a vibrating tuning-fork
held over a suitable resonator) comes to us
from in front or from behind. This is what
might have been expected from an d priori
point of view ; but what would not have been
expected is, that with almost any other sort
of sound the discrimination is not only pos- f
sible, but easy and instinctive. In these cases
it does not appear how the possession of two
ears helps us, though there is some evidence
that it does ; and even when sounds come to
us from the right or left, the explanation of
the ready discrimination which is then pos-
sible with pure tones is not so easy as might
at first appear. We should be inclined to
think that the sound was heard much more
loudly with the ear that is turned towards
than with the ear which is turned from it, and
that in this way the direction was recognised.
But if we try the experiment we find that— at
any rate with notes near the middle of the
musical scale — the difference of loudness is by
no means so very great. The wave-lengths of
such notes are long enough, in relation to the
dimensions of the head, to forbid the forma-
tion of anything like a sound-shadow in which %
the averted ear might be sheltered.'
ig) Beat-Notes and Beat-Tones. After K.'s final experiment (p. 532^^') on the
passage of beats into tones, we might perhaps disuse the interim term ' beat>note,'
which implied no theory as to its origin, but only a statement as to its frequency,
and use K.'s term ' beat-tone,' implying that the tone is generated by beats. But
just because * beat-note ' does not imply a theory, and because no theory has been
at present generally accepted, nor is sufficiently supported by proofs to be so, it
will be convenient to continue the use of the word * beat-note,' which simply
states that the frequency of the beat is identical with that of the note. At the
same time we must not disuse the terms ' differential and summational tones, of
various orders,' because if they really exist they are a decidedly diflFerent pheno-
menon from beat -notes, and only in the most frequently observed case coincide
in pitch (but not in intensity) with beat-notes. P. viii. 29, however, decides to
identify the two. K., on the other hand, considers the existence of differential and
siumnational tones not proved.
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535
ADDITIONS BY THE TRANSLATOR.
APP. XX.
Messrs. Preece and Strob, referring to their
machines for the synthesis of vowels, noticed
infrd, sect. M. art. 2, p. 542^, say (Proc. R.
8. 27 Feb. 1879, vol. xxviii. p. 366) : * The
eurves arrived at synthetically do .not differ
materially from those arrived at analytically
by H. They principally differ in the promi-
nence of the prime. But the prime can be
dispensed with altogether. Curves produced
by the synthetic machine, compounded of the
different partials without their prime, shew
that there exist beats or resultant sounds. A
vowel sound of the pitch of the prime may
be produced by certain partials alone, with-
out sounding the prime at all. The beat, in
fact, becomes the prime. This point is clearly
illustrated by the automatic phonograph, and
graphically by the sketch drawn by the syn-
thetic curve machine. In fact, every two
partials of numbers indivisible by any common
multiple [divisor?], if sounded alone, reproduce
by their beats the prime itself. Thus, the
3rd and 5th partials, or the 2nd and 3rd,
<&c., will result in the reproduction of the
prime.' Observe that this gives the beat-note,
not the differential tone. The differential
tone of 3 and 5 is 2, but the beat-note is
2x3~5»i. *In fact, the figure illustrates not
only this, but it shews that when the number
of partials introduced is increased, the beats
become more and more pronounced.* Mr.
Stroh from his own experience considers the
the beat-notes thus produced to be generated
in the same way as K. supposes.
H (7^) Koenig's explanation of SummatiovM Tones. In the matter of snimna-
tional tones, P. (see p. 5326) explains them as differential tones of the second
order. K. i. 217-8 thinks that they arise as beat-notes from upper partials. But
P. viii. 24 notes that this explanation fails when very high partials would be
required.
but the explanation proposed by M. Preyer is
absolutely inadmissible, for 496 and 528 d. vib.,
even when they have considerable force, give 32
beats, which do not as yet allow the deep tone
C, to be heard, so that at any rate such tone
must be extremely weak. Now the Octave of
528 (or 1056) is the 33rd harmonic of this
excessively weak sound. But two primary
sounds of 32 and 1056 d. vib., even when
extremely powerful, never produce a sound of
1024 d. vib. The second manner in which M.
Preyer thinks the sound might have been pro-
duced is equally opposed to all that has been
directly observed when two primary tones
sound together. Thus he makes the Octave of
528 (i.e. 1056) produce with 496 d. vib. a dif-
ferential tone of 560 d. vib., and then makes
this tone 560 produce with the Twelfth of 528
(i.e. 1584) a new differential of 1024. Bat
these two sounds of 496 and 1056 ( «= 2 x 496 +
64) give the beat-note 64, and not 560 ; and if
the sound 560 really existed it would give with
1584 («2 X 560 + 464^ 3 X 560-96) the beat-
note 96, and also more faintly 464, but not
1024.*
Thus, to get the summational tone 64 from
31 : 33 we should require the 32nd partial,
which is not heard. So, from the reed tones
496 and 528, P. heard 1024. The 32nd par-
tials would be 16896 and 15872, difference
1024. But such partials are inaudible, * whereas
every term of the acoustical equations
2x528- (528-496) = 1056- 32=1024
3 X 528 - (2 . 528-496) = 1584 - 560 = 1024
is easily proved.' The tones mentioned cer-
tainly exist; the question is only, are they
m powerful enough to produce the result ?
To the above remarks K. replies in the
French edition, p. 127, note, continuing the
passage already quoted (p. 5326') : * M. Preyer
cites in favour of his views that on sounding
together free reeds of 496 and 528 d. vib.
■=31 '- 33fbe heard the sound 1024 d. vib. = 64,
and he thinks that we cannot assume that the
reeds had the 32nd partial, 16896 and 15872
d. vib. If the sound really observed was 64,
and not the Octave of 31 or 33, we might be
really astonished that the 32nd partials were
snfficiently strong in these tones to produce it ;
(i) Koenig's theory for the origin of Beat-Notes. K. i. 186 gives the following
theory for the origin of tones from beats. He says that ' the beats of the harmonic
intervals, as well as of the unison, should be deduced directly from the composition
of waves of sound, and we should assume that they arise from the periodically
«- alternating coincidences of similar maxima of the generating tones, and of the
maxima with opposite signs. The similar maxima for these harmonic intervals, as
in the case of unisons, will either exactly coincide, or else there will be maxima of
condensation in the higher tone lying between two successive vibrations of the
fundamental tone, slightly preceding one and slightly following the other ; but in
both cases the effect on the ear will be the same, for a beat (fluctuation) is no
instantaneous phenomenon, but arises from a gradual increase and diminution
of the intensity of tone.* Then he adds some drawings of the compounded vibra-
tions of two tuning-forks, one of which bore a piece of smoked glass and the other
a style. These are almost precisely the same as the curves drawn by means of
Donkin's harmonograph, and inserted at the end of B. vi., opposite p. 256. That
is, both K. and B., who are strongly opposed in opinion, refer to practically
identical curves in support of their own views. This serves to shew the extremely
difficult and delicate nature of the investigation.
(k) Lecture Demonstration of Beat- Notes. In the beat-notes produced by the
wave siren, K. had the great advantage of producing tones which could be contmaed
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SECT, u EECENT WORK ON BEATS & COMBINATIONAL TONES. 537
for any length of time, whereas those from tuning-forks vanished so rapidly that
they could be with diflBculty recognised. But this did not suffice for lecture
demonstrations. Hence K. invented a machine which produces beat-tones audible
over a whole lecture room.
This consists (E. iy.) of pairs of glass tubes tones and either one or both of the beat-notes
adjusted so as to give notes with definite inter- continnoasly, and load enough to be appre-
vals by longitudinal vibrations. These are held dated by the whole audience. A piece of
at the node by two olamps against the surface paper wrapped round the node and bearing the
of a wheel bearing a thick cloth tire, which con- number of the relative pitch enables the glass
tinually dips into a trough of water, and thus tubes to be selected and changed with the
rubs the tubes sufficiently to produce loud greatest rapidity.
Art. 6. — Infltience of difference of Phase on Quality of Tone.
H. p. 126a finds that dmerence of phase has no effect on quality of tone. But,
on p. 127c, H. points out 'an apparent exception,* on which E. v. 376 remarks
that if quality depends on the relative intensity of the harmonic upper partials, %
and this relative intensity is really altered by difference of phase, the influence of
this difference is ' actual, and not merely apparent.' Then observing on the
difficulties attendant on H.'s rule for finding the differences of phase (supr&,
p. 124c), he proceeds to describe his new experiments with the wave siren (for
which reason they are mentioned in this place), which certainly admit of very
much more precision. They were conducted thus : £. compounded harmonic
curves of various pitches, and with various assumptions of amplitudes, under four
varieties of phase : (i) the beginning of all the waves coinciding ; (2) the first
quarter, (3) the halves, and (4) the third quarter of each wave coinciding ; briefly
said to have a difference of phase of o, -j, ^, |. These were reduced by photo-
graphy, inverted, and placed on the rim of the disc of a wave siren, and then
made to speak. He gives the remarkable curves which resulted in a few cases,
and instructions for repeating the experiments. The following are his conclu-
sions (K. V. 391) : —
* The composition of a number of harmonic tones, including both the evenly
Bnd unevenly numbered partials, generates in all cases, quite independently of the If
relative intensity of these tones, the strongest and acutest quality tone for the
\ difference of phase, and the weakest and softest for | difference of phase, while
the difference o and ^ lie between the others, both as regards intensity and
acuteness.
' When unevenly numbered partials only are compounded, the differences of
phase \ and | give the same quality of tone, as do also the differences o and ^ ;
but the former is stronger and acuter than the latter.
' Hence, although the quahty of tone principally depends on the number and
relative intensity of the harmonic tones compounded, the influence of difference of
tone is not by any means so insignificant as to be entirely negligible. We may
say, in general terms, that the differences in the number and relative intensity of
the haisnonic tones compounded produces those differences in the quality of tone
which are remarked in musical instruments of different famihes, or in the human
voice uttering different vowels. But the alteration of phase between these har-
monic tones can excite at least such differences of quality of tone as are observed
in musical instruments of the same fojnily, or in different voices singing the same If
vowel.*
Of course, as K. v. 392 observes, the complete wave siren figured on K. v. 386
is applicable to numerous other investigations.
Art. 7, — Influence of Combinational Tones on the consonance of Simple Tones »
This is a brief notice of P. viii. HI. It would appear from H.'s theory of con-
sonance (see especially supra, pp. 2ood and 2056) that, if there were no upper
partial or combinational tones, dissonance and consonance could not be distin-
guished— in the Thirds for example. P.'s experiments rendered this doubtful. He
had a series of 1 1 forks made, very accurately tuned to —
Vib. 1000 iioo 1200 1300 1400 1500 1600 1700 1800 1900 2000
Cents 165 151 138 129 119 112 105 99 93 89
Sums o 165 316 454 583 702 814 919 1018 mi 1200
where the upper line gives the numbers of vibrations, the second the cents in the
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538 ADDITIONS BY THE TEANSLATOB. aPf.xx.
intervals between two snccessiye forks, and the bottom the sums of those cents,
or the cents in the intervals between any fork and the lowest, from which the
cents in the intervals between any two forks can be immediately deduced by sub-
traction ; and by a reference to the table in Sect. D. the names of such intervals can
be found. P. selected the difference of loo vib. because it was small enough to allow
of a sensation of roughness when two successive forks were sounded together.
And he selected the pitches looo and 2000 because they precluded hearing upper
partials, while the frequency 1000 to 2000 not being sufficient to have any dOTect
on distinguishing consonance, the absence of power to distinguish it could not be
ascribed to the high pitch.
Both practised and onpractised ears imme- pair of sounds, it loses the disagreeable effect
diately recognised on them that successive forks of dissonance.*
were dissonant to each other; this was due The Octave and Fifth were generally re-
to the small difference of 100 vib. Almost all cognised with certainty, probably from long
other intervals of these x I forks, when the forks practice. This appears to be an excellent
■r were not too loud, were frequently considered proof of H.'s theory. And the less care there
' consonant, especially by musicians— such as was taken to exclude upper partials and com-
10 : 13, iz : 13, 12 : 19, 17 : 20, <&c. Also the binational tones, the more unpleasant became
ratios expressible by small numbers (except the dissonance, and the easier it was for the
8 : 9 and 9 : 10), namely, 5 : 7, 5 : 9, 6 : 7, ear alone to determine the interval immedi-
7 ' 8, 7 : 9, 7 : 10, often passed as consonances; ately. But this is not all. H.*s theory that
and though the i : 2, 2 : 3, 3 : 4, 3 : 5, 4 : 5, dissonances should be recognised only by beats
5:6, 5:8 were genenJly preferred, some of the partials or combinational tones implies
observers found 6:7, 15 : 19, 11 : I3t <&o* that, if these were too far distant in pitch to
more harmonious than the pure Thirds and produce beats, there would be no roughness.
Sixths, 4 : 5} 5 : 6, 5 : 8, 3 : 5, especially than and hence no beats. This did not prove to be
the minor Sixth 5 : 8. The listener was the case. The pair 1400 : 1600 vib. formed a
always kept in ignorance of the numerical dissonance, although all partials and combi-
ratios, and only one person was tried at a national tones differed by 200. The ratio
time. The sum of the different judgments 8 : 9 was universally called a cutting disso-
was therefore : nance, even in the 4 times and 8 times accented
* After aU upper partials and combinational Octave,
tones have been eliminated from a dissonant
^ The explanation of the above phenomena seemed to require a remodelling of
H.'s theory, and P.*s conclusions are stated thus (P. viii. 58) : —
' (i) The larger the least two numbers required to express the ratio between two
tones, the greater the nun^r of combinational tones, which always form an
arithmetical series, and arise, whether upper partials be present or not (H. 155c).
' (2) The greater the number of simple tones which affect the ear simultaneously,
the less distinct is each single tone.
' (3) The more coincidences there are between the tones which might be and
are generated by any interval, the more pleasing is the sensation ; and the fewer
the coincidences the more confusing, and hence unpleasant, the impression.'
And as these conclusions hold for tones which, on account of their own dis-
tance from eadi other and the distance of their partials and combinational tones,
cannot generate sensible beats, P. considers that this is both a formal and an
actual extension of H.'s theory of consonance. But if, with E., we consider
these differential tones absolutely insensible, it would be difficult to see how
they would affect the result, and the facts noted would still require explanation.
V The whole subject of combinational tones and beats evidently requires much more
examination.
SECTION M.
ANALTBIS AKD SYNTHESIS OF VOWEL SOUMBS.
(See Notes, pp. 75, zz8, 124.)
Art. Art.
I. Analysis of Vowel Sounds by means of the 2. Synthetical Production of Yowol Sounds,
Phonograph, p. 538. p. 542.
Art. I. — Analysis of Vowel Sounds by means of the Phonograph. The follow-
ing is a brief account of a paper by Prof. Fleeming Jenkin, F.R.SS. L. and E.,
and Mr. J. A. Ewing, B.Sc, F.R.S.E., On the Harmonic Analysis of certain
Vowel Sounds, * Transactions of the Royal Society of Edinburgh,' vol. xxviii.
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PECT. M. ANALYSIS AND SYNTHESIS OF VOWEL SOUNDS.
539
pp. 745-777, plates 34-40, communicated June 3 and July i, 1878, and published
'With additions to July 19, 1878— that is, subsequently to the appearance of the 4th
German ed. of this work.
Messrs. Jenkin and Ewing make use of a variety of Mr. T. A. Edison's phono-
graph which, by means of a style affixed to a vibrating disc against which words
are spoken or sung, impresses the amplitude of vibration at any time on a piece
of tinfoil passed beneath it by machinery. On repassing the style over these
indentations the vibrations are recommunicated to the disc, and the sounds re-
produced sufficiently, on the form of the instrument used by these gentlemen, for
listeners to understand sentences impressed during their absence. Then the
indented foil was passed under another style in communication with a system of
delicate levers, ending in one of Sir W. Thompson's electrical squirting recorder
tubes, which magnified the depth of the indentations 400 times, and squirted their
form, without friction, on to a telegraph-paper band wound round a cylinder re-
volving at such a speed as to magnKy the length of the indentations 7 times.
Perfect records of the vibrations registered by the phonograph were thus obtained, If
of sufficient size to be measured. The amplitudes of the compound vibrations of
the curves were measured to the 200th part of an inch ('005 inch). Then, as the
apparatus could not properly determine high partials, the curves were assumed to
be compounded of six partials, and the ordinates or amplitudes had to be deter-
mined by Fourier's formula —
y=-4o+'4i sin aj-h-ij sin 2a;-f- . . . +-4,sinna;-f . . •
+-B, C0QX+B2 cos 2X+ . . . +B^ cos nx-jr . . .
The period was taken as the length between two minima of ordinates, and
divided into 1 2 equal parts for successive values of a?, and then the corresponding
values of y were measured. The 1 2 resulting simultaneous equations, giving the
values Aq to A^ and By to -B5, were then solved by Professor Tait's formulaB (given
in the paper), and thus the amplitudes of the six partials for any length of the
ordinate were determined. The Authors say :— • IF
' The experiments were chiefly directed to the two sounds o and u (the vowels in o^ /
and food). Several different voices were employed. Voice No. i was a powerful bari-
tone with a considerable range and good musical training. No. 2 was a high set and
somewhat harsh voice of limited range and without musical training. No. 3 was a rich
and well-trained bass voice of a man of eighty. Nos. 4 and 5 were somewhat alike, being
voices of moderate range and power and with some musical training. No. 6 was a
powerful bass. Generally the vowels were sung in tune with notes given by a piano,*
the pitch of which was supposed to be c^ 256, but was probably much higher.
. Photo-lithographs of the records of the vibratory curves are given in the paper,
and ingeniously arranged tables are added shewing the maximum amplitudes of
the partials for each pitch of the prime. Of these, the following is Table VU. p. 761
slightly re-arranged, with the names of the upper partials inserted : —
Vowel Sound 0 {'oh').
Voice
Pitches and AMPLmrDBB of thb First Sec Pabtiam
No.
I.
II.
III.
IV. 1 V.
VI.
/'«
rz
C"«
rt
a"'t
c-'jf
2
44
32
6
0
4
2
/'
/"
&"
r
a'"
d'"
I
121
71
7
I
5
4
5
53
19
6
2
3
I
V"
c'
e"
h"
• e'"
rt
I
105
69
7
3
2
3
2
51
30
5
2
I
I
3
53
18
3
I
2
I
4
55
34
7
2
2
0
5
52
53
5
6
5
2
Digitized by
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S40
ADDITIONS BY THE TEANSLATOB.
VowBL BoxmD S (* OB ')—eontinwd.
APP. XX.
VOICK
PrrCHBB AND AlfPUTUDBB OF THJt PIR8T SlX PaBTIALS |
No.
I.
II.
in.
IV.
V.
VL
J
d'
d"
a"
d'"
/'"«
a"'
"J
76
5
I
3
2
66
40
4
0
3
2
27
42
6
4
2
I
tf
c"
1^'
c"'
c"'
iT'
no
160
15
10
10
7
37
30
I
4
I
I
54
25
0
3
2
I
1
47
41
5
3
3
2
b
1/
rz
6"
d'"«
/"'«
70
126
15
14
6
I
45
66
7
4
6
2
36
31
2
4
I
0
45
i^
4
2
i
0
47
61
2
14
2
6b
I/b
r
y'\>
d'"
f"
75
185
li
8
II
I
49
104
6
4
2
s
82
5
7
I
2
70
13
7
I
3
a
a'
e"
a"
c'"«
e"'
125
190
25
22
5
2
3a
5!
6
8
6
2
1
40
36
4
4
3
2
16
§i
4
4
I
I
40
68
10
8
3
0
^
g^
d"
(7"
6"
d"'
103
27
6
2
2
23
51
14
3
2
2
46
29
2
2
2
I
33
44
7
2
I
2
32
50
3
6
I
2
/»
f\
c"«
r«
a"J
c"'«
18
58
15
3
I
0
28
62
10
2
3
2
/
/'
c^
r
a"
c^
55
140
45
4
8
I
25
37
11
3
4
2
IT
70
109
58
13
12
5
6
«'
y
c"
0
6"
72
131
73
7
10
4
40
67
35
5
5
2
25
12
21
6
6
0
41
88
64
13
5
2
d
(f
0'
d"
/"»
a"
44
134
82
16
22
10
27
61
38
19
4
2
20
46
21
3
4
I
33
72
56
5
7
2
e
c'
1^
c"
e"
iT
I
18
• 95
61
33
3
0
3
19
48
33
18
2
4
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3ECT. M. ANALYSIS AND SYNTHESIS OP VOWEL SOUNDS.
541
VowH. Souin) S (* OH ') — eontimied.
Voicx
Pitches akd Akplztitdis of thb Fibst Six Fabtialb
No.
I.
11.
IIL
IV.
V.
VI.
B
5
>"!
y
d"«
/"«
I
as
«5
28
31
6
5
3
21
46
29
28
10
0
4
12
34
23
10
4
I
5
6
38
23
25
6
3
J5b
6b
/'
l/b
d"
/"
I
^2
58
61
^I
II
0
3
28
41
25
36
9
0
5
i8
26
15
»S
2
2
6
l8
22
32
75
9
2
.1
a
«'
a'
c"«
c"
I
15
15
18
29
6
3
3
26
^i
35
39
18
2
5
15
8
22
21
4
0
^
9
46
44
80
12
4
G
0^
d'
g"
«/
d"
I
13
0
15
40
8
4
6
34
30
8
45
9
7
J'^
/
c/
r
a'
c"
6
22
10
IS
8
34
I
The following is a table of the results for H for voices i and 5 only, where, for
brevity, I give only the pitches of the primes, for the pitches of the partials are
^ven in the preceding table, and the numbering of the partials is sufficient to
shew the great peculiarity of the jump from one reinforced partial to two, the H
second being then by far the most prominent, and the different pitches at which
different voices make the change. Voice 5 could not get out a clear H at the pitch a.
To these are added the results obtained from voice 5 for the vowels &^ (* awe ') and
a (' ah ')•
Vowel fl ('oo'). Voick 5
Vowel « (*oo '); Voice x
Pitch
Amplitudtjs OF Partlulb
Pitch
Amplttudkh of Pahtials
OF PRIMK
OF FllIHK
I.
136
n.
III.
IV.
V.
VI.
I.
II.
III.
IV.
V.
VI.
d
6
2
4
3
I
d'
94
7
0
2
3
2
d
85
3
0
3
2
2
b ,
287
26
12
3
8
0
bb
250
8'
II
1
I
2
bb
22
189
12
38
2
8
a
13
120
12
12
4
0
g
38
128
9
4
10
II
9
22
136
6
16
3
I
f
23
13s
5
10
3
5
f
21
108
7
13
6
2
e
34
148
18
7
6
3
e
27
127
12
14
2
2
d
33
107
14
4
3
0
c
31
74
41
4
6
3
B
18
50
28
3
5
I
VOWKL SP (* AWK '), VOICK 5
VOWKL a (• ah •), VOICF. S
d'
9
22
14
3
2
2
d
20
48
58
i
10
0
b
20
SI
S6
3
2
a
41
48
48
3
6
2
a
37
62
46
20
4
3
9
24
44
32
15
3
0
9
23
35
24
25
6
0
f
14
18
14
23
2
0
f
24
29
12
24
I
I
e
23
39
32
40
6
2
e
12
23
II
15
9
3
d
19
26
20
30
7
3
Digitized by V^OOQ IC
54a ADDITIONS BY THE TRANSLATOB. app. xx.
On the first table the authors remark : —
' At the pitches ordinarily used in speech, the vowel o consists almost wholly of the
two constituents — a prime and its Octave — the ratio of whose amplitudes may vary
widely. But when the range is extended so as to reach lower pitches, higher purtials
successively appear in such a way as to allow the highest strongly reinforced partial to
remain in the neighbourhood of &'b. . • . Generally we may say, . . . that there is a
wide range of reinforcement, extending over about two Octaves (from /or g to/'O* within
which all tones are more or less strongly reinforced, and that there is a specially strong
reinforcement at the pitch h'b.'
But this last did not appear for the artificial 6*a produced by Prof. Gram
Brown's instrument (described in the paper), and recognised by the ear as o's.
They also draw attention to the sudden alteration of amplitude of the 4th partial
with voice 6, and also of the sth partial, for pitches B\}, A, O, a,a compared with
F ; and to siioilar sudden alterations in the 3rd and also 4th partial with voices
ir 1, 3. 5-
After discussing the results condensed in these tables, the authors review former
vowel theories and give their own conclusions, of which the following may be
noted ; but the whole paper requires careful examination, as a most original and
laborious study of a very difficult subject,
* In distinguishing vowels the ear is guided by two factors, one depending on the har-
mony or group of relative partiaJs, and the other on the absolute pitch of the reinforced
constituents. It seems not a little singular that the ear should attach so distinct a unity
to sounds made up of such very various groups of constituents as we have obtained from
different voices and at different pitches, so as to recognise cdl these sounds as some one
particular vowel. We are forced to the conclusion • . . that the ear recognises the kind of
oral camty by which the reinforcement is produced. . . . The vowel-producing resonance
cavities are clearly distinguished in virtue of two properties — ^first, the absolute pitch at
which they produce a maximum reinforcement ; and, second, the area of pitch over which
reinforcement acts. The latter property, when it is extensive, is very probably due to
the existence of subordinate j^roper tones not far from each other in pitch. . . . W'e
m should . . . describe the u cavity as an adjustable cavity, with a very limited range of
resonance, whose efiect is to reinforce strongly only one partial above the pitch of a. . . .
If we assume that the o cavity is absolutely constant, we must describe it as a cavity re-
inforcing tones throughout nearly two Octaves, or from g to f^\ . . . We are disposed to
regard it as more probable, that in human voices the 0 cavity is slightly tuned or modi-
fied according to the pitch on which the vowel is simg ; . . . the genuine character of 0 is
given by a cavity reinforcing tones over rather more than one Octave, with an upper
proper tone never far from 6' b. ... It is very satisfactory to find that the 5*8 given by
the human voices which we have experimented with are marked by the strong resonance
on &'b which Helmholtz has noticed by quite different methods of observation. It (ends
to shew that our 0 was essentially the same vowel sound as his, and to give us confidence
in the mode of experiment we have adopted.' {Ibid, pp. 772-775.)
Art. 2. — Synthetical Production of Vowel Sounds, A most ingenious method of
producing artificial vowels was invented, and is explained by Messrs. W. H. Preece
and A. Stroh in their paper entitled Studies in Acoustics : On the Synthetic Ex-
amination of Vowel SouTiids, * Proceedings of the Eoyal Society,' Feb. 27, 1879, ^^1.
^ xxviii. No. 193, pp. 358-67. Mr. Stroh, to whom all the machinery is due, was
kind enough, on May 29, 1884, for the purposes of this Appendix, t6 shew me the
machines in action, and to reproduce the results many times over in order that I
should be able to judge of them. Essentially there are four machines. First, one
to produce the curve resulting from compounding 8 harmonic curves, representing
partials, with maximum amplitudes decreasing inversely as the number of the
partial increased, but with arrangements for altering the amplitudes and phases
of composition. The resulting figures are extremely beautiful. Secondly, a
machine for cutting the curve thus produced, but on a reduced scale, on the edge of
a brass disc, so that 30 periods were included in one circumference of this disc, the
curves being automatically transferred from the first machine. Third, a machine
by which an axis on which 8 of the discs thus cut were placed, representing 8 par-
tials. These discs by springs could be brought into action in any combinations, and
could convey the resulting vibration to a style working against a sensitive disc hke
that of the telephone. The sensitive disc on vibrating produced the corresponding
sound audibly. Not being satisfied with these results, Mr. Stroh took the combina-
Digitized by V^jOOQlC
SECT.M. ANALYSIS AND SYNTHESIS OF VOWEL SOUNDS. 543
tions and amplitudes which these experiments shewed were Hkely to sncceed best,
made the corresponding compound curves by the synthetic machine, cut them by
the second machine, mounted them as in the third, and then in a fourth or vowel
machine conducted the vibrations from each compound curve to a disc, which spoke
them. The details and drawings of the first and fourth machine, the speaking
disc, and various compound curves are given in the paper. The curves are also
compared with those resulting from my table of Prof. Helmholtz's results (supdk., p.
1 24c, dy footnote) , which had also appeared in the i st edition of this translation, p. 1 8 1 .
The table of the intensities of the partials given in the paper (on its p. 365) — though
I am not quite sure that they agreed with those I heard — ore as follows, the pitch of
the prime being B\} : —
8 16
Vowels I
2 3
4 5
6
U ff
mf pp
0 mf
f mf
P
A p
p p
mf mf
P
E mf
mf
ff
I mf
P
P
1>
mf
The effect of these vowels on my ear was not like that of human vowels, hence
I found it extremely difficult to place them anywhere in the human vowel
scale. Boughly, I felt that —
U was a sort of 00, tending towards oh t
0 was more like the word awe than oh !
A was a very high ah, tending to the long sound of English a in fat,
E was very imperfect, and had the effect of a hollow low French i mixed*
with EngHsh u in but.
J, was the worst vowel. It had none of the character of ai in air, but was
far from ee. The sound * tootled.'
When taken in rapid succession, the ear at once recognised that these sounds
were meant for 00, oh, ah, and perhaps ay, ee ; but on prolonging the sound of
any one, the character of the vowel became lost, as indeed is frequently the case
in singing. Curious effects resulted from raising and lowering the pitch. The O if
flattened became a very decent 00 (in boot), and the A flattened almost a good oh.
The effect of taking all an Octave higher was not so successful.
The synthesis of Prof. Helmholtz and that of Messrs. Preece and Stroh,
together with the analysis of Messrs. Fleeming Jenkin and Ewing, in art. i, prove
distinctly that difference in the quaUty of tone, taking only harmonic partials, is
the foundation of vowels, and also that difference of phase has, so far as they could
observe, no effect on the ear. (But see supr^, App. XX. Sect. L, art. vi. p. 536.)
Both, however, also prove that there is much more yet to be learned before we
can satisfactorily imitate spoken vowels. Each of these methods of synthesis
necessarily relates to sung vowels, which are quite distinct from spoken vowels, and
indeed never satisfactorily imitate them. It appears to me that the mode of vibra-
tion of the vocal chords is a most important element of vowel character, and that
the resulting effect is modified by the resonance in the ventricles of Morgagni, in
the cartilagenous larynx more or less covered by the epiglottis (acting, possibly,
like the cup mouthpiece of brass instruments, see supr^, p. gSd, note), in the ^
pharynx, and between the pillars of the velum, before it reaches the larger re-
sonance cavities of nose and mouth, with which we are almost solely able to deal.
By the original mode of vibration of the vocal chords for spoken vowels many
inharmonic proper tones are probably produced, which are overcome in singing,
and this is possibly one of the many differences between speaking and singing.
Also, we should bear in mind that each speaker has his personal quality of ' voice *
(that is, mainly, of vowel sound), by which he would be recognised in the dark, and
that in each individual the feeling of the moment varies the pitch and the charac-
teristic quality of his vowels ; so that there are really millions of different qualities
of tone all recognised generically as the same vowel. And yet in the artificial vowels
just considered I could not recognise any exact form of human vowel with which
I was acquainted, although I have made speech sounds an especial study for more
than forty years. We have an analogy in the multiform presentment of the
human countenance, which is nevertheless unhesitatingly recognised as distinct
from that of the anthropoid ape.
Digitized by VjOOQIC
544
ADDITIONS BY THE TEANSLATOB.
APP. XX
SECTION N.
MIBCELLANBOtTS NOTES*
(See pp. 78, 179.)
No.
X.
2.
CompaBs of the Hainan Voice, p. 544.
Harmonics and Partials of a Pianoforte
String, struck at one-eighth of its length,
P- 545.
History of Mean Tone Temperament, p. 546.
History of Equal Temperament, p. 548.
-Prof. Mayer's Analysis of Compound Sounds
and Harmonic Curves, p. 549.
No.
6.
7-
8.
Presumed different characters of Keys, both
Major and Minor, p. 550.
Dr. W. H. Stone's Restoration of i6-fbot C,
to the Orchestra, p. 552.
On the Action of Reeds, from notes by Mr.
Hermann Smith, p. 553.
Postscript, p. 555.
Compass of the Suman Voice,
Instruments can be tuned or manufactured
at almost any required pitch. The human
voice is bom, not manufactured. Although by
skilful training its compass can generally be
somewhat extended, both upwards and down-
wards, yet it must in generid be considered to
be an instrument beyond human control. The
usages of Europe have, however, made it the
principal instrument, and, when it is present,
have reduced all others to an accompaniment.
Hence it is necessary that these other instru-
ments should have their compass and pitch
regulated by that of the human voice. Now
the voice, like the viol family, represents at
least four different instruments — soprano, alto,
tenor, and bass, with two intermediate ones,
m mezzo-soprano, between soprano and alto, and
barytone, between tenor and bass. It is there-
fore as necessary to determine the average and
exceptional compass of these species of voice
as it is to know the compass of any other
instrument, in order that composers may be
certain as to what sounds can be reproduced,
and not demand any other. To do this, the
precise acoustic meaning of each written musi-
cal note should be ascertained. The difficulty
of determining it has been shown by the pre-
ceding history of musical pitch (pp. 494-513)1
from which, combined with the tauies of mean-
tone and equal intonation (pp. 434 and 437), it
is evident that Handel's sustained a" in the
HdlUlvjah chorus had 845 vib., but would
now be sung to 904 vib. ; and that Mozart's
/" in the ZauberJWte would have meant 1349
CI vib., but would now have to be sung at t^SS
vib. The strain that this would put upon
voices is evident, and no composer who wished
his music to be well represented would think
of making such demands on his singers. It
appeared, therefore, necessary to ascertain
more precisely than had been hitherto done,
and to express in numbers of vibrations, the
limits of the different kinds of voices. If the
composer will then only translate his written
notes into numbers of vibrations, by the table
on p. 437, according to the pitch he employs,
he will avoid all danger of straining singers.
Through the kindness and liberality of the
choir conductors Messrs. Henry Leslie, W.
G. McNaught, J. Proudman, Ebenezer Prout,
L. C. & G. J. Venables, and 542 members of
the choirs they conducted, I was able to
examine a sufficient number of singers, in
January, 1S80, to arrive at something like a
trustworthy account of the compass of the
voice. I gave each singer a paper with the
words do re mi fa sol la ti do' printed on them
in four columns up and down to the requisite
extent, and then started them on (io in 4
different pitches, 507, 522*5, 528, 5407 rib.
(representing the just cf^ corresponding to a'
422-5 Handel's pitch, a' 435-4 the French
pitch, a' 440 Scheibler's pitoh, and the equal
c" of a' 4547i the highest Philharmonic pitch
of 1874, respectively). I got them to sing up
and down in chorus under the direction d
the conductor, and to mark with a pencil the
highest and lowest note eaoh one could reach,
first easily t or secondly by an effort (falsetto
of male voices being exduded in the first
case, but not in the last). From these papers
I determined by calculation, on the assump-
tion of just intonation (as being most pro«
bable for unaccompanied singers), the numbers
of vibrations in the limiting notes. These are
contained in the following table, together with
the mean height and depth of all the voices.
The extreme highest limit for male voices, as
it included falsetto, is a mere curiosity. For
writing music, the mean should not be as-
sumed as the limit, for perhaps half the chorus
could not reach it. But it would be perfectly
safe to write from the highest low easy limit
to the lowest high easy limit. Thus, for so-
pranos it would not do to write up to b" 993
and down to / 180, but it would be quite safe
to write up to /' 704 and down to 6 253.
Viewed in this way, my results agree more
nearly with Randegger's, which I add for com-
parison. These last are given in a staff-
notation form in his primer on Singing (No-
vello, 1879) 1 uid as he politely informed me
that he assumed Broadwood's medium pitch
a' 446*2, 1 was able to calculate the vibrations.
All the numbers of vibrations are given to the
nearest integer only, and it is to these numbers
that attention should be especially paid, the
names of the notes being merely guides. Those
letters preceded by a turned period relate to
high pitch in the column * Actual,' and those
not so preceded relate to a medium pitch,
as French or German. But in the column
* Mean ' no precise system at all could be
selected. In Randegger's, of course. Broad-
wood's medium pitch is intended. If, how-
ever, the notes be played on any ordinary piano,
they will seldom be in error to the extent ctf
a quarter of a Tone.
Digitized by V^jOOQlC
,^OT. N.
MISCELLANEOUS NOTES.
Mean akd Actual Coupasb of the Huuam Voice.
545
'VoicEB Observed
Easy Lowkr Limit
VOICM Obseryso
KxTiisMx Lower LiMn*
Mean
Actual
•6 253 to
•^ 13s
ab 211 to
c 132
e 163 to
D 76
Ab 106 to
C 66
Mean
Actual
X46 Sopranos .
91 Altos .
X07 Tenors
X 25 Basses
/ 180
eb 161
Q 98
E 81
173 Sopranos .
108 Altos .
114 Tenors
140 Basses
eb 162
d 147
E 85
C« 72
g 203 to
c 130
g 198 to
B 124
'B 127 to
C 66
'F 90 to
•^. 56
VOICSB Obsertkd
Bast Hiohkr Ldcit
BXTRJCME HlQHKR LnflT
Mean
6- 993
9"« 836
C" S2I
/'« 375
Actual
Mean
Actnal
145 Sopranos .
83 Altos .
114 Tenors
120 Basses
/'" 1408 to
r 704
'd'" 1216 to
•e" 676
d" 608 to
•c" 541 to.
d' 294
173 Sopranos .
105 Altos .
112 Tenors
139 Basses
c"'Z 1 124
.6"b 952
d" 617
6'b 4S3
'a'"b 1690 to
y' 811
i^" 1584 to
•r 721
Y 8uto
^' 396
•c'" io8i to
«' 330
Bandkooeb's Statement of LimnNo Tones.
VOICSB
Reoi
TLAR
V01CE8
BXCKPTIOKAL
Lower Limit
Upper Limit
Lower Limit
Upper Limit
Soprano .
Mezzo Soprano
Alto .
Tenor
Barytone .
Bass .
6 b 236
g 199
$ 167
c 133
A\> 105
F 89
c'" 1061
6"b 945
/" 708
6'b 473
/; 354
eb 316
Soprano .
Mezzo Soprano
Alto .
Tenor
Barytone .
Bass
6b 236
g 199
e 167
F 87
D 75
/'" 1417
c"' io6i
C 795
c^« 562
i 398
/ 354
IT
Harmonics and Partials of a Pianoforte String struck at one-eighth
of its length.
On p. 77, note* , will be found Mr. Hipkins's
observations on the striking-point of piano-
forte strings, shewing that one-seventh of the
length, which seemed to be assumed as usual
by Prof. Helmholtz, was not in use generally,
or (p. y6d') at Steinways*. Prof. Helmholtz
conceived that the origin of this presumed
custom was to get rid of the 7 th partial,
which he also considered likely to injure the
quality of tone. Mr. Hipkins's experiments
were therefore made with the object of deter-
mining whether when the striking-place was
one of the nodes the corresponding partial
disappeared, as results from the mathematical
formula (12a) supdL, p. 3836.
^ Mr. Hipkins's first experiments are de-
tailed in his paper entitled * Observations on
the Harmonics of a String, struck at one-eighth
its length ' {Proc. Royal Society, 20 Nov. 1884,
Yol. zxxvii. p. 363). The main facts are given
supr&, p. 78^. The results were all witnessed
by Dr. Huggins, F.B.S., and myself. The
string was exactly 45 inches long, and was
struck at precisely one-eighth its length from
the wrestplank -bridge (that nearest the player).
When it was touched with a piece of felt at
5*63, i6'88, and 28*13 inches from the belly-
bridge (that farthest from the player), which
are three positions of the nodes for the 8tlv
partial or third Octave higher, selected to avoid
errors (as not being positions of the nodes of
the 2nd or 4th partials), in each case the 8th
harmonic was well heard. It was not so
strong as the 4th, 5th, 6th, 7th, and 9th, all
of which were heard, but quite unmistakable,
and was heard better on removing the felt
immediately after the note had been pro- «r,
duced. The i6th partial was also heard when ^'
the string was touched at its nodes 2-81 and
8'44 inches from the belly-bridge, which are
nodes of the i6th hat not of the 8th partial.
What was heard was the harmonic^ not the
simple partial tone, and it was suggested, that
perhaps touching the string at the node coerced
the string and obliged it to vibrate with these
nodes, notwithstanding that it was struck in
one of the series of such nodes. Mr; Hipkins,
therefore, at my suggestion made a new series
of experiments, detailed in his paper entitled
* Observations on the Upper Partial Tones of
a Pianoforte String, struck at one-eighth its
length * (Proc, of the Royal Society , 15 Jan.
1885, vol. xxxviii. p. 83). These experiments
I also witnessed. The object was to leave the
string perfectly uneoerceid, and to avoid the
use of resonators, on which some suspicion had
Digitized by
(Jbogle
546
ADDITIONS BY THE TRANSLATOE.
APP. XX.
been (wrongly, as I believe) oast by at least
one observer. Calculating the pitch of the
partials, which would be the same as that of
the harmonics, and the interval which the
tempered notes of the piano would make with
them (as in Table II., supriL, p. 457), a string
of the corresponding note was slackened (or
tightened, as convenient ; sometimes both al-
ternately), while the other unison strings were
damped with the usual tuners' wedges ; and in
the same way only one of the three lower
strings was allowed to vibrate. Then the low
note and the high note were struck simul-
taneously. It is evident that the high note
being shghtly out of unison with the upper
partial of the lower note, beats would ensue if
such a partial existed. Now, for the 5th, 6th,
^ 7th, 8th, 9th, loth, and i ith partial such beats
were perfectly audible, but their duration for
ithe nth partial with 1487 vib. was so short
ihat higher ones were not tried. For the 8th
partial the beats were quite distinct, and, on
removing the wedges that damped the unison
strings for the high note, and striking the three
high strings without the lower note, it was
evident that the beats heard were the same in
rapidity and character as when the single
string was sounded with the low one.
The fact, therefore, that the 8th partial
existed was conclusivelv proved. Various
causes have been assigned. On p. 383^, note *,
I have suggested that if terms omitted by the
hypothesis named were introduced, perhaps
there would be a residuum which would account
for hearing the 8th partial. This partial was
mr really much weaker than the 7th. The last,
'' indeed, was quite clear and ringing, so that it
did not seem affected by striking the string so
near its node.
It is curious that when the nodes do not
lie Very close the harmonic could be brought
out by touching the string somewhat near the
proper place. Thus for the 2nd harmonic,
node at 22*5 inches from the belly-bridge ; the
next nodes were 1*2 inch nearer and 1*2 inch
farther from that spot, and on trial the 2nd
harmonic or Octave came out when the string
was touched between 22-1 and 22*95 inches
from the belly-bridge, but not at 22*05 and 23*0,
so there was a * play ' of '85 inch. For the
3rd harmonic there was similarly a * play ' of
'65 inch. A very remarkable fact was that by
stopping within 1*5 inch of the belly-bridge, the
simple prime or lowest partial came out un-
accompanied by any other audible partials.
This was tested by beats of forks, shewing that
the 2nd and 4th partials did not exist.
Various causes for the sounding of the 8th
harmonic have been suggested. One of these
was that the hammer of the pianoforte, being
round and soft, did not strike at one point, and
so excited the string on each side of the node.
To avoid this action the much harder hammer
of the highest note (A in the 3-inch octave) was
used in supplementary experiments made on
2 April 1885. The width of the part of the
hammer that came in contact with the string
did not exceed ^ inch. And again, an ivoty
edge, not more than ^ inch in width, was used
instead of the felt covering of the hammer.
The 8th partial, tried by brats, in both cases
came out much stronger than before, and the
beats could be distinctly heard 10 or 12 feet off.
Again, it was supposed that the string might
not be uniform, and that if the strikio^-plaoe
were slightly moved from the theoretical node,
an actual node would be reached, and the par-
tial quenched. Hence the ivory head of the
hanmier was shifted so as to strike up to ^
inch away from the node on either side. The
partial was heard strongly, but the sound oi
the note was not so pleasant as when the string
was struck at the actual node.
It has been also suggested that the string
moved the points of support, but that we had
no means of testing. The phenomenon, there-
fore, remains unexplained ; but thanks to Mr.
Hipkins, who had the resources of Broadwoods'
establishment and the assistance of expe-
rienced tuners at command, there is no doubt
whatever of the fact, that a pianoforte string
when struck at a node by a hard or soft ham-
mer does not lose the corresponding partial,
and does not materially enfeeble the partials
with adjacent nodes.
3. History of Meantone TemperamenL
This is the temperament usually, but
wrongly, known as * unequal ' (supr&, p. 434a'),
which prevailed so long over Europe, and is
- not yet entirely extruded.
M Arnold Sohlick, Spiegel der Orgelmacher
irh Oraanisten (Mirror of Organ-builders and
Orgamsts), 1511, chap, viii., orders the Fifths
jPC, CO, OD, DA to be tuned as flat as the
ear could bear, so as to make the major Third
FA decent. Then he tunes AE, BD in the
same manner. Beginning again with F, he
tunes the Fifths down FBb, Bb Eb, sharpen-
ing the lower note for the same reason. Then
he tunes EbAb and makes ^ b * not sharp, but
somewhat flatter than the Fifth requires, on
account of the proof (vmb das brifen), although,
however, the Gt thus made is never a good
Third or perfect Sixth to the Fifth E a.nA Bt^
for cadences in A.* He prides himself, how-
ever, on the Ab or G8 , and shews how to dis-
guise inaccuracies. And he refutes those who
would make Ot good for cadences in ii in the
chord E Gff B, by saying this produces weak-
ness, and takes away the effect of good and
strange consonances. For the rest, he tunes
B FZ with the upper note flat, and apparently
FH Ct in the same way. This was really an
unequal temperament, and looks very like the
meantone temperament spoiled, but that sys-
tem was not yet discovered. Schlick's editor
(Bob. Eitner, in the MonaUhefU der Mttsik-
OeschichUt monthly parts of the History of
Music, part I., 1869) says that what Schliok
claims for Ab and Ot was supposed to be the
invention of Barth. Fritz of Brunswick in 1756,
245 years later.
Giuseppe Zarlino of Chioggia, 50 yetrs
after Schlick, in his Le istilvtioni harmoniche
(Institutes of Harmony), Venice, 1562, speaks
of alcuni (some people) who seemed to think
that the interval of the comma should be dis-
tributed among the two nearest intervals, and
the others left in their natural foim (cap. 43«
p. 128). This would give a meantone for the
second of the major scale =}x (204-1- 182)"
193 oentR, but leave the others very dissonant.
Digitized by V^jOOQlC
SECT. N.
MISCELLANEOUS NOTES.
547
and to this Zarlino rightly objects. It would
give the major scale Co, 2>I93, ^3^* ^49^,
O702, ASS^j B1088, CI200 cents, so that the
Fourtii D : O would have 509 cents, and the
Tilth D : A would have 691 cents, which com-
ing in the midst of just intervals would be
intolerable, and beyond the natural key it fails
entirely. Zarlino's remedy (chap. 42, p. 126)
is to diminish every Fifth by two-sevenths of
a comma, and he proceeds to shew how this
affects the tuning. It preserves the small
Semitone 24 : 25 «■ 70*673 cents. He says,
p. 127, * although in instruments thus tem-
pered consonances cannot be given in their
perfect — that is, their true and natural form —
yet they can be used when the chords have
to be given in their true and natural propor-
tions. I say this,' he adds, * because I have
frequently made the experiment on an instru-
ment which I had made for the purpose, and
the effect may be tried on any other instru-
ment, especially the harpsichord and clavi-
chord, which are well adapted for the purpose.*
Then, in chap. 43, he proceeds to shew that
this temperament is rationally constructed,
and that no other is so {che per altro modo
non H possa fare, that is, ragiofieuolmente).
It is quite clear, then, that Zarlino, as has
often been asserted, did not invent the mean-
tone temperament, and did not consider equal
temperament worth mentioning, even if he was
acquainted with it.
Francis Salinas of Burgos in Spain, bom
1 513, died 1590, blind from infancy. Professor
of Music in the University of Salamanca,
Abb6 of St. Pancras do Bocca Scalegno, in
the kingdom of Naples, in 1577 published his
De Muaica libri septem^ of which a very im-
perfect, and, as respects temperament, in-
correct account is given in Bumey & Haw-
kins. Salinas says (lib. iii. cap. zv. p. 143,
I translate his Latin) : —
' From what has been said, in order that
Tones should be rendered equal, the minor
must be increased and the major diminished.
It must be observed that this can be done in
several ways, because the comma, by which
they differ, may be divided in many ways.
Of these, three have been thought out up to
this time, which seem to me most suitable
{aptissimi). Hence arise three ways of tem-
pering imperfect instruments. The first is
to divide the comma into three proportional
parts, giving one to the minor Tone, and tak-
ing two from the major Tone. This gives a
new Tone, larger than the minor and smaller
than the major. The decrement is twice the
increment, and through the maximum in-
eqnality the tone becomes equal.*
The comma has 21*506 cents, hence
^ comma has 7*169, and ) comma has
14*388 cents. Then 182*404-1- 7*169" 189*572
= 203*910-14*338 cents, which is what the
above statement comes to, giving 189*572
cents for the new Tone. Notwi&standing
this very precise statement, Salinas ought to
mean precisely the reverse. His object was to
make tiie Tritone perfect, and to make it con-
sist of three new Tones. Now a Tritone F : B
consists of 2 major Tones and i minor Tone,
— that is, 3 minor Tones and 2 commas, or
590*224 cents, ^ of which is 196*741 cents,
which is 182*404+14*338 and 203*910-7*169
—that is, the reverse of the former result. By
a singular error perpetuated in a figure (which,
of course, being blind he could not see), Salinas
makes the Tritone in Hhis place consist of
2 minor Tones and i major Tone — that is,
3 minor Tones and i comma, having the ratio
18 : 25, or cents 568-718, which is not the
Tritone, but the superfluous Fourth, and may
here be called the false Tritone. This mis-
take seems to have arisen thus. The Octave,
as he rightly says, has 6 minor Tones, 2
commas, and a great DiSsis. * The comma
being divided into 3 proportional parts, if one
is added to each minor Tone, 2 commas will
be added to the six Tones, and one to three,
equally distributed among them. From which
distribution it will follow in this constitution
of the temperament that the Tritone consists
of three minor Tones and one comma, or f
2 minor Tones and one major,* whence he
deduces the ratio 18 : 25. But he thus alto-
gether loses sight of the great DiSsis, and
considers a Tritone to be half of an Octave
after it has been diminished by a Digsis. On
p. 155 he again notices the Tritone as 32 : 45,
the correct ratio. The false Tritonic tempera-
ment therefore makes the Tone 189*572, the
Fifth 694-786, and the false Tritone 568*718
cents. But the true Tritonic system gives the
Tone of 196-74, the Fifth of 698*37 and the
true Tritone 590^22 cents.
Salinas continues his account of the three
temperaments thus : ' The second [tempera-
ment] divides the comma into 7 proportional
parts, giving 3 to the minor and taking 4 from
the major Tone.* This is Zarlino's tempera-
ment already described, and preserves the m
small Semitone 24 : 25 » 70*673 cents. ' The
third will arise from halving the comma,
giving half to the minor and taJcing half from
the major Tone.* Then he adds (p. 164) :
* Wherefore any one of these three tempsraments
seems most suitable for artificial instruments ;
nor have any more been as yet thought out
(neque plwra adhuc excogitata sunt) ; * that is,
Salinas, like Zarlino, utterly ignores the equal
temperament. *The first, so far as I know,
has been laid down by no one.* From which
it is to be inferred that it was his own inven-
tion. * The second I have also found in the
harmonic institutions of Joseph Zarlino of
Ghioggia,* as already given. * The third was
commenced, but not perfected, by Luigi or
Ludovico Folliano of M6dena,* who must have
been Zarlino's * some people * (alcuni), * And ^
Joseph Zarlino has properly considered it in
his harmonic demonstrations. But no one
has previously acknowledged all three, nor
observed upon their relation and mutuiU
order.*
It was Salinas who finished Folliano*s
work, and in chaps. 22 to 25 he describes
the result thoroughly. As, therefore, we con-
sider Watt, and not the Marquis of Worces-
ter, to have invented the steam engine, we
must consider Salinas, and not Folliano, to
have invented the meantone temperament. I
^ve a comparative table of all three schemes
m cents to the nearest integer, from Eb to
OZ • distinguishing the true and false Tritonic
and adding the Equal, which will shew the real
rdbibtions of these three temperaments to each
other.
Digitized by
G5bgle
548
ADDITIONS BY THE TRANSLATOB.
AFP. XX.
Notes
True Tritonic
Palse Tritonic
Zflrllno
Ueantone
Equal
c
O
0
0
0
0
c»
89
64
71
76
100
D
197
190
192
193
200
Eb
305
316
313
310
300
E
393
379
383
386
400
F
502
50s
504
503
500
Fti
S90
569
575
580
600
G
698
695
696
697
700
GZ
787
758
766
773
800
A
89s
884
887
890
900
Bb
1003
lOIO
1009
1007
1000
B
1092
1074
1079
1083
1 100
c
1200
1200
1200
1200
1200
ir The true Tritonio, making the Tritone
590-22 cents, necessarily differs very slightly
from equal temperament, which makes it 600
cents, while the false Tritonic, making the
Tritone 568716, or a comma too flat, ap-
proaches very near to Zarlino's and the Mean-
tone, BO that I think Salinas must have
intended to use this one; which he lays down
so clearly, and that he accidentally made a
mistake of a comma in estimating the Tritone,
by hastily neglecting the DiSsiB. For later
usages see suprd,, pp. 320, 321.
4, The History of Equal Temperament,
When once the Pythagorean division of
the Octave had been settled, and it had been
observed that 12 Fifths exceeded 7 Octaves by
the small interval of a Pythagorean comma
(p. 432, art. 9), the idea of distributing this
error among the 12 Fifths was obvious.
Aristoxenus, a pupil of Aristotle, the son of
a musician and a writer on music, is said to
have advocated this. At any rate he stated
that the Fourth consisted of two Tones and a
^ half, which is exactly true only in equal tem-
perament. Amiot reports equal temperament
from China long previously even to Pytha-
goras. In later times Mersenne {Harmonie
Universellej 1636) gives the correct numbers
for the ratios of equal temperament, and says
(Livre 3, prop. xii. ' Des genres de la musique *)
of equal temperament that it * est le pita
usiU et le plus commode, et que tous les prao-
ticiens avoiient que la division de TOctave en
12 demitons leur est plus facile pour toucher
les instruments.* This should imply that
there were numerous instruments in «qual
temperament, but I have not been able to find
any noticed. BMos (L'Art dufacteur d^Orgues,
1766) knows only meantone temperament,
which he gives directions for tuning. In Ger-
many, Werckmeister (Orgelprobe, 2nd edit.
1698) says that he can only recommend equal
«r temperament, and Schnitger of Harburg in
Hanover, and afterwards of Hamburg, an ad-
mirer of Werckmeister, built the organ of St.
Jacobi-Kirche in Hamburg in 1688, and tuned
it in intentionally equal temperament. Herr
Schmahl, who had been the organist there
since 1838, never knew it otherwise tuned,
and could find no record of any change of
intonation in the archives of the church, and
he also could not recollect having ever heard
of any other intonation in North Germany.
His master, Demuth (died 1848) of St. Catha-
rinen-Kirche, whose memory extended back-
wards to iSio, also knew of no other tuning
in North Germany. Of course the tempera-
ment never was thoroughly equal, so that
when Herr Schmahl practised on the St.
Catharine's organ, the usual keys C and G
were not sic good as the unusual keys FU and
D8. Dr. Robert Smith, 1759, must have
heard equal temperament, or else he could
hardly have spoken of *that inharmonious
system of 12 hemitones* producing a 'har-
mony extremely coarse and disagreeable '
{Harmonics f 2nd ed. pp. i66>7), but it may
have been only an experimental instrument of
his own.
As regards the recent introduction of equal
temperament into England, Mr. James Broad-
wood, in the New Monthly Mctganne^ i Sept
z8i I, proposed it, and gave the error of the
Fifths as ^ Semitone ( = 2^ cents), which was
to him the smallest sensible intervaL On
1 Oct. 181 1, Mr. John Farey, sen., shewed that
this was too much (it should be 1*954, or about
2 cents — that is, about -^^ Semitone), and re-
ferred to the article * Equal Temperament ' in
Bees's Cyclopedia. Hereupon, on i Nov. 181 1,
Mr. James Broadwood rejoined that be gave
merely a practical method of producing equal
temperament, * from its being in most general
use, and because of the various systems it has
been pronounced the best deserving that ap-
pellation by Haydn, Mozart, and other masters
of harmony.' ITnfortunately he gives no refer-
ences, and consequently this assertion can be
taken only as an unverified impression. Haydn
died 1808, Mozart 1791, but the Hamburg
organs had equal temperament long before
that time. Sebastian Bach (died 1750) is gene-
rally credited with introducing equal tem-
perament, but M. Bosanquet says * Uiere is no
direct evidence that he ever played upon an
organ tuned according to equal temperament '
(Musical Intervals and Temperament^ 1876,
p. 31). Bitter, however, states, in his life of
Sebastian Bach, that he once played on the St.
Jacobi organ at Hamburg, and expressed his
approval of the tuning, and even applied for
the post of organist. The wohl temperirtes
Clavier^ or well-tuned clavichord, the notes of
which are very fugitive, was the instniment
mentioned by Carl Philip Emanuel Bach, who
died 1788.
As regards Mr. James Broadwood *8 state-
ment that equal temperament was in 181 1 *in
most general use ' —presumably in England--
Digitized by V^jOOQlC
SECT. N.
MISCELLANEOUS NOTES.
549
'h/lr. Hipkins has been at some pains to ascer-
XsLin how far that was the case, and from him
X learn that Mr. Peppercorn, who tuned origin-
aJ.ly for the Philharmonic Society, was concert
'tuner at Broadwoods', and a great favoarite of
Bf r. James Broadwood. His son writes to Mr.
Bjpkins that his father * always tnned so that
will keys can be played in, and neither he nor
I [neither father nor son] ever held with making
some keys sweet and others sour.* Mr. Bailey,
liowever, who succeeded Mr. Peppercorn as
concert tuner, and tuned Mr. James Broad-
VT'ood's own piano at Lyne, his country house,
used the meantone temperament to Mr. Hip-
Icins's own knowledge, and no other. Not one
of the old tuners Mr. Hipkins knew (and some
iiad been favourite tuners of Mr. James Broad-
^'ood) tuned anything like equal temperament.
Ck>llard, the Wilkies, Challenger, Seymour, all
timed the meantone temperament, except that,
like Arnold Schlick, 151 1 (see p. 546<2), they
raised the Gt somewhat to mitigate the *■ wolf '
resulting from the Fifth Eb : GU in place of
£b : Ab. Hence Mr. James Broadwood did
not succeed in introducing equal temperament
permanently even into his own establishment,
and all tradition of it died out long ago. So
far runs Mr. Hipkins's interesting information.
In 181 2 Dr. Crotch (Elements of Musical
Composition^ pp. 134-S) gives the proper
figures for equal temperament, shews how it
arose, that its Fifths are too flat and its major
Thirds too sharp, adding * this will render all
keys equally imperfect,' but says nothing to
recommend it. Yet in 1840 Dr. Crotch (who
died in 1847) had his own chamber organ
tuned in equal temperament, as I have been
informed by Mr. £. J. Hopkins, author of The
Organ^ dtc.
It is one thing to propose equal tempera-
ment, to calculate its ratios, and to have trial
instruments approximately tuned in accord-
ance with it, and another thing to use it com-
mercially in all instruments sold. For pianos
in England it did not become a trade usage
till 1846, at about which time it was intro-
duced into Broadwoods* under the superintend-
ence of Mr. Hipkins himself. At least eight
years more elapsed before equal temperament
was generally used for organs, on which its
defects are more apparent, although not to such
an extent as on the harmonium.
In 1 85 1, at the Great Exhibition, no Eng-
lish organ was tuned in equal temperament,
but the only German organ exhibited (Schulze's)
was so tuned.
In July 1852 Messrs. J. W. Walker & Sons
put their Exeter Hall organ into equal tem-
perament, but it was not used publicly till
November of that year. Meanwhile, in Sept.,
Mr. George Herbert, a barrister and amateur,
then in charge of the organ in the Koman
Catholic Church in Farm Street, Berkeley
Square, London, had that organ tuned equally
by Mr. Hill, the builder. Though much op-
posed, it was visited and approved by many,
and among others by Mr. Cooper, who had the
organ in the hall of Christ's Hospital (the
Bluecoat School) tuned equally in 1853. ^
In 1854 the first organ built and tuned
oiiginally in equal temperament, by Messrs.
Gray & Davison, was niade for Dr. Eraser's
Congregational Chapel at Blackburn (both
chapel and organ have since been burned).
In the same year Messrs. Walker and Mr.
Willis sent out their first equally tempered
organs. This must therefore be considered
as the commercial date for equal temperament
on new organs in England. On old organs
meantone temperament lingered much later.
In 1880, when I wrote my History of Musical
Pitch, from which most of these particulars
have been taken, I found meantone tempera-
ment still general in Spain, and used in Eng-
land on Greene's three organs, at St. George's
Chapel, Windsor (since altered), at St. Katha-
rine's, Regent's Park (see p. 484^/), and at
Kew Parish Church ; and while many others If
had only recently been altered, one (Jordan's
at Maidstone Old Parish Church) was being
altered when I visited it in that year. Hence,
in England, equal temperament, though now
(1885) firmly established, is not yet quite 40
years old on the pianoforte, and only 30
years old on the organ.
The difficulty of tuning in equal tempera-
ment led to the invention of Scheibler's tuning-
fork tonometer. In Sect. G., art. 11, p. 489,
will be found a practical rule for tuning in
sensibly equal temperament at all usual pitches.
5. Professor Mayer's Analysis of Compound Tones and Hariruynic Curves,
The following are two of the numerous
acoustical contrivances of Mr. Alfred M. Mayer,
Pb.D., Member of the National (American)
Academy of Sciences, and Professor of Physics
in the Stevens Institute of Technology, Hobo-
ken, New Jersey, United States (see supr^, p.
417c).
I. New Objective Analysis of Compound
Sounds. The analysis of compound sounds by
resonators has two disadvantages: first, that
it is subjective, inasmuch as but one observer
at a time is capable of hearing the results ;
and, secondly, that the range of pitch reinforce-
able by a resonator is too great for extreme
accuracy in the estimation of the actual com-
ponent sound present. Both of these disad-
vantages were thus overcome (Phil, Mag, Oct.
1874, vol. xlviii. pp. 271-3, with a figure).
A Greni^'s free-reed pipe, of the pitch
C=i28, had part of its wooden chamber re-
moved and replaced by morocco leather, at
one point of which 8 silk cocoon fibres were
attached, having their opposite extremities
attached to 8 tuning-forks tuned to C, c, a, c', ^
ef g't'b'bt c", at the point of the upper node in
each where it divides into segments when
giving its upper harmonic, so that this har-
monic was eliminated. The cocoon fibres were
stretched till they made no visible ventral seg-
ments when vibrating. The reed was tuned
accurately to the C fork (of 64 vibrations) by
means of the g fork. The forks were placed
on proper resonance boxes. When the reed
was sounded each fork * sang out ' loudly, but
if the prongs of any fork were only slightly
loaded the fork was mute, ard was so rapidly
affected that Prof. Mayer estimates (same vol.
p. 519) that the effect of intervals such as
2000 : 2001 (or '87 or not quite i cent) can be
rendered sensible to the ear. On ceasing to
sound the reed the forks continued to sound,
and produced a tone of so nearly the same
Digitized by V^jOOQlC
550
ADDITIONS BY THE TEANSLATOR.
APP. XX.
quality as that of the reed that it was easy to
feel that the difference was due to the absence
of partials higher than the eighth. By this
means, then, the analysis and synthesis of a
compound tone can be shewn to a large
audience at once, and all doubt as to its ob-
jective reality removed. At the same time the
air in the resonance chamber of the reed acts
on the leather cover as in hearing it would
have acted on the drumskin of the ear, and
the conduction of that vibration by the cocoon
fibres replaces the complicated arrangements
in the interior of the drum and the fluid of the
labyrinth of the ear, while the forks themselves
serve as the organs in the cochlea. Prof.
Helmholtz's physiological theory of audition is
thus perfectly exhibited in a * working model.*
^ The action of the resonance chambers of the
forks is simply to make the effects heard at a
distance.
2. Harmonic Curves (see p. 387^). In the
Philosophical Magazine^ Supplement for Jan-
uary 1875, vol. xlviii. pp. 520-525, Prof. Mayer
gives curves compounded by six curves of sines
(p. 23d'), representing six partial tones, where,
for convenience, the amplitudes are taken to
vary as the wave lengths, and to have the same
initial phase. They are combined by taking
the algebraical sum of their ordinates, which
law would of course not hold true for the am-
plitudes chosen (about one-third of the length
of the wave). The resulting figure bears a
most remarkable resemblance to fig. 25, supr^,
p. 846. There is the same sudden rise on the
left and step-like descent on the right, but
the steps are more rounded, and the upper
crest more pointed, and there are five steps in
addition to the crest. Prof. Mayer then com-
bines two such compound curves, and thus
produces the resulting curves of two oompoond
tones forming an Octave (with one high and
one low crest, and also one high and one low
trough, and the steps uneven and reduced in
number), a Fifth (with four crests, two with
long and two with short descents, the shorter
having only one step), and a major Third (with
eight crests, two extremely small, two mode-
rate, three intermediate between the two last
kinds, and one high, the asoents being abrupt
as before, the descents rather wavy than
stepped). These had all been drawn on a
large scale with several hundred ordinates,
and were reduced photographically. They form
an excellent practical illustration of the nature
of harmonic motions.
6. The presumed different Characters of Keys^ both Major and Minor.
See 6upr&, p. 3 loc to 3 1 1 c. It is first neces-
sary to know what is the presumed pheno-
menon to account for. In the discussion of
my paper 'On the Measurement and Settle-
«[ ment of Musical Pitch ' (Journal of the Society
of Arts, 25 May 1877, p. 686), Prof, (now Sir
George) Maofarren, Principal of the Boyal
Academy of Music, spoke of * the difficulty of
representing the compositions of different eras,
which had been written for different standards
of pitch,* and added * it was a marvellous fact
that, while the pitch was felt to be changed,
the impression of the character of the keys
seemed to remain with reference to the nominal
key, not to the number of vibrations of each
particular note. Thus the key of D at the
present day represented the same effect as was
produced by the same key according to one's
earliest recollections ; it did not sound like the
key of JSb, although it might be of the same
pitch. If Mozart*s symphony in C were to
be played a Semitone lower, to bring it to the
original pitch, it would not sound at all the
^ same. How far this result was subjective —
how much depended on the imagination of the
hearer, and how much on the physical facts —
was a deep, perhaps an insoluble question ; but it
was one which really ought to be considered.*
The Chairman (Mr. William Pole, FJt.S.,
Mus. D. Oxon), on the contrary, said : * In a prac-
tical point of view the French did an exceed-
ingly good thing when they fixed on one pitch,
. . . and they had practically done so, not
only for France, but the Continent generally.
He had the gratification some time ago of
hearing Beethoven's Sinfonia Eroica played
at a Conservatoire concert in ^b, as it should
be, but he could not get rid of the idea, when
he heard it played, at the Philharmonic con-
certs, that it was in £q .*
The mention of the performance of sym-
phonies by Sir G. Macfarren and Dr. Pole
takes the whole question out of the action of
individual instruments, in which there is no
doubt of considerable variety depending on the
tonic, but this can be traced in every case to
some defect of the instrument itself, as haa
been considered in the text {loc. cU,). The
point of the long and short keys on a piano-
forte, spoken of by Prof. Helmholtz, has
been well worked out by Mr. G. Johnstone
Stoney, D.Sc., F.B.S. Dublin, in a paper read
before the Boyal Dublin Society on 16 March
1883 {Scientific Proc, R. Dublin Soe. p. 59),
who calls attention to the fact that in il all
the Fifths are on white digitals and the major
Thirds on black digitals, while in ^b the
Fifths are on black and the major Thirds on
white digitals, and argues that this must
make considerable difference in playing. Mr.
Hipkins, however, gives it as his opinion that
it is impossible to tell in the performance of
a first-rate player whether he is striking a
white or a black key. That would relegate
the difference to the degree of skill of the
player. But this does not at all affect the
organ, the harmonium, or the voice. And by
reference to symphonies 'we are constrained
to consider the question independently of any
particular instrument, as a simple acoustical
fact.
In order to ascertain what that fact is sup-
posed to be, according to recognised musicians
of high standing, I give a condensed re-arrange-
ment of the characters attributed to different
keys in Mr. Ernst Pauer's Elements of the
BeauUfulin Music rNovello), p. 23, placing
the major and minor keys in opposite columns,
and proceeding by intervals of a Semitone.
Digitized by
Google
8S0T. N.
MISCELLANEOUS NOTES.
55t
Presumed characters of
Major Keys.
Minor Keys,
C. Expressive of feeling in a pure, certain,
and decisive manner, of innocence, powerful
Tesolve, manly earnestness, deep religious
feeling.
08 . Scarcely used ; as Db it has fulness
of tone, sonorousness, and euphony.
D. Expressive of majesty, grandeur, pomp,
triumph, festivity, stateliness.
Eb» Greatest variety of expression ; emi-
nently masculine, serious and solemn ; expres-
sive of courage and determination, brilHant,
firm, dignified.
JB, Expressive of joy, magnificence, splen-
dour, and highest brilliancy; brightest and
znost powerful key.
F, Expressive of peace and joy, also of
light passing regret and religious sentiment.
Fz. Brilliant and very clear; as Gd
expresses softness and richness.
O, Favourite key of youth ; expresses sin-
cerity of faith, quiet love, calm meditation,
simple grace, pastoral life, and a certain
humour and brightness.
Ab. Full of feeling and dreamy expres-
A, Full of confidence and hope, radiant
-with love, redolent of genuine cheerfulness;
especially expresses sincerity.
Bb. Has an open, frank, clear, and bright
character, admitting of the expression of quiet
contemplation ; favourite classical key.
B, Expresses boldness and pride in for-
tissimo, purity and perfect clearness in
pianissimo ; seldom used.
C. Expressive of softness, longing, sad-
ness, earnestness and passionate intensity,
and of the supernatural.
Db. The most intensely melancholy key.
D. Expressive of subdued melancholy,
grief, anxiety, and solemnity.
Eb. Darkest and most sombre key of all ;
rarely used.
E, Expressive of grief, moumfulness, and
restlessness of spirit. mr
F. Harrowing, full of melancholy, at
times rising into passion.
F^ . Dark, mysterious, spectral, and full
of passion.
O. Expresses sometimes sadness, at
others quiet and sedate joy, with gentle grace
or a slight touch of dreamy melancholy, occa-
sionally rising to a romantic elevation.
Ab, Fit for funeral marches; full of sad,
heartrending expression, as of an oppressed
and sorrowing heart.
A, Expresses tender womanly feeling,
especially the quiet melancholy sentiment of
Northern nations; also fit for Boleros and
Mauresque serenades; and finally for senti-
ments of devotion mingled with pious resigna-
tion.
Bb. Full of gloomy and sombre feeling,
like Eb ; seldom used. If
B. Very melancholy ; tells of quiet expecta-
tion and patient hope.
In reading over this Table it is impossible
not to feel that the character, often contra-
dictory, arises from the reminiscence of pieces
of music in those keys, as the author indeed
admits (ib, p. 22). Such a distinction as that
made between Ft and Gfb» which, in equal
temperament, is a mere matter of notation,
but is here made to yield incompatible results,
shews that the writer was thinking more of
treatment than of actual sound. This is con-
firmed by his saying (t&. p. 26) : ' We shall often
find that the general character of a key may be
changed by peculiarities and idiosyncrasies of
the composer ; and thus a key may appear to
possess a cheerful character in the hands of
one writer, whilst another composer infuses
into it a melancholy expression ; all depends
on the treatment, on the individual feeling of
the composer, and on his acute understanding
of the characteristic qualities of the key he
employs.' The writer then goes on to consider
the effect of rhythm and time, and the different
characters which he assigns to their varieties,
independently of the key employed, clash so
much with the preceding that it is difficult to
know what is supposed to belong to one and
what to the other.
Now the acoustical facts, independently of
any particular instrument or temperament or
any errors of tuning or performance (both
numerous but variable), are these. Whether
we take just intonation, or that of any uni-
form linear or cyclic temperament, carried on
to a sufficient number of tones to prevent the
occurrence of 'wolves' within the piece of
music performed, the one thing aimed at is
to have the intervals between the same notes
of the same scale precisely the same, at what-
ever pitch they are played, or however they
may be conventionally noted. If there is any
difference between the scales of, say, just A^
and ^'b, which have a difference of 70 cents,
or equal A and A b, which differ by 100 cents,
or meantone A and Ab^ which differ by 76 ^
cents, or Pythagorean A and A b, which differ
by 114 cents, this difference must be due
solely to pitch. There is no doubt that on
the piano, the organ, and each instrument of
the orchestra, the difference will be consider-
able and very appreciable, but that does not
enter into oon^eration. What effect does
simple difference of pitch in the tonic pro-
duce ? In the human voice and in all instru-
ments quality of tone varies together with
the pitch. A change of tonic implies a dif-
ferent pitch for the most frequently returning
sounds, and those most important to the
nature of the key. Hence it produces a dif-
ferent quality of tone, with a variation in the
range of partials possessed, and consequently
affects the distinctness of the delimitation of
the principal consonances and dissonances of
Digitized by'
5Sa
ADDITIONS BY THE TRANSLATOR.
APP. XX.
the keys, and by that means alters their audi-
ble effect. For intervals so small as we have
supposed this difference must necessarily be
small, whereas the difference of the keys of A
and Ab is said to be great. If so, it can only
arise from errors of intonation or performance.
In the days of the old meantone temperament
in its defective state of 12 notes only to the
Octave, there was a vast difference between
the keys of A and Ab; the first had all its
chords correct (supposing OU were not sharp-
ened, tnpr«L, p. 546(2), the latter had all the
chords involving Ab and Db (which had to be
represented by OZ and CU) frightfully erro-
neous.
It seems to me that the feeling of a di|-
ferenoe in the character of the keys whose
^ tonics vary but slightly in pitch was esta-
blished at this time (in Sir George Mac-
farren's younger days, b. 1813). Any differ-
ence so slight as a Semitone would have been
strongly felt (except in passing from A ix> Bbt
the two extreme good keys) . Whereas for differ-
ences of a Fifth there would generally not have
been such violent distinctions (except at the ex-
tremes Ato E and Bb to ^b). It would ap-
pear that these mechanical distinctions partly
influenced composers in their choice of a key,
and produced what has become an hereditary
prejudice, for which there is no longer any
ground, and which never ought to have existed
in just unaccompanied singing. But even at
that time we have composers ignoring the
difference. Handel composed his dead march
in Saul in C, and having written an anthem
(' 0 sing unto the Lord a new song *) for St.
% James' Chapel Koyal, in which the organ had
a pitch one Tone higher than his own (sea
sect. H.p. 5036, under a' 474*1), he directed the
singers, as tiie voice parts were too high, to
take them a Tone lower, and the accompany-
ing organ to play tivo Tones lower. Had
Handel any idea of the innate di^erence of
the character of keys? and, if he had not,
what does it all amount to? Beyond those
differences inevitable to varieties of pitch,
already pointed out, and easily perceived by
slowly playing up the major scale on any
instrument from its lowest , note, or ainging
it on any voice from the lowest note it can
reach easily (see p. 544^) — beyond such differ-
ences, all seems to be subjective, or due to
hereditary feeling created by former defective
temperaments, or at present to mechanical
errors of tuning, stopping, or blowing, espe-
cially in unusual keys. Possibly a composer
at the present day would write a piece of a
totally different character, as pointed out in
the table, according as he made the signature
FZ OT Gbf but that must have been a reaction
on his own mind, for the tones he would play
would be precisely the same in both cases.
That tuners of the piano sometimes still
intentionally tune unequally, and hence make
the effect of A and A b really very different, has
nothing whatever to do with the matter. Those
who do so have not learned their profession.
Similarly for players on a pianoforte who
cannot equalise the effect of the long and short
keys. But the singer knows very well when a
piece of music falls upon his bad notes, and
ruthlessly transposes the key, quite reckless
of these presumed varieties of key-character.
7. Dr. W. H. Stone's Bestoration of i6'foot C, to the Orchestra.
See p. 175c on the deepest tones which can
be heard. The following is condensed from
information furnished me by Br. Stone.
Dr. Stone has for some years endeavoured to
restore to the orchestra the lower notes of the
16-foot Octave, which appear to have been
neglected of late. It seems to him a contra-
diction that, while the organ possessed that
Octave, and another, the 32-foot, below it, and
while even an instrument of so comparatively
feeble a tone as the pianoforte could obtain
these deep notes, they should be absent from
the full band. Most of the great composers
have employed them, especially Beethoven and
% Onslow. Many passages of their compositions
had to be partially transposed, often (as in the
C minor Symphony of Beethoven) much to the
detriment of the general effect. In the Trio
of this great work a scale passage occurs
several times for the double basses alone,
beginning on the 16-foot G^ But this note
being entirely absent on the ordinary three-
string basses, as used in England, it was there
customary to take it either ^together or in
part in the Octave above. Some players,
indeed, were in the habit of letting down the
A, or lowest string, by a Tone to O, for this
special passage ; but the resonance of a string
thus slackened was far inferior to what could
be obtained by more legitimate means. The
fine part for the contrafagotto in the same
symphony, descending to C,, was usually
omitted, or played an Octave higher by the
ophicleide. The Pastoral Symphony likewise
frequently contains F, natural, a note quite
unattainable except on the four-string basses,
whose lowest note is E, (p. i8c).
It was obvious, in attempting to remedy
this defect, that of the three modes by which
vibrations in a stretched string may be
slackened, two, namely, length and thidkness,
were inadmissible. The first renders the m-
strument so large as to be unwieldy and out
of the reach of an ordinary arm. The second
was found to cause rotation of the string under
the impulse of the bow acting at its periphery,
and thus to generate false notes. The third
remained, in increasing the specific gravity of
the string without enlarging its diameter. This
was satisfactorily accomplished by covering a
gut string with heavy copper wire saoh as is
used for the lowest strings of pianos. The
note C, was obtained, and an instrument thus
strung was exhibited in London in the Inter-
national Exhibition of 1872.
But it became clear that to give the new
notes full power, and to prevent the danger of
shaking the instrument to pieces, a means of
strengthening the belly in the direction of
strain was required, which should not unduly
increase the weight of the sound-board. This
requisite was ingeniously fulfilled by Mr.
Meeson : - - Four strips of white pine are glued
on to the back of the belly, running its whole
length, one on one side of the ordinary bass-
bar, and three on its other side, thus eoire-
sponding in number, and to a certain degree
in position, to the increased number of strings*
Digitized by VjOOQlC
BEOT. N.
MISCELLANEOUS NOTES.
553
Two of them cross and intercept the usual
/-shaped sound-holes, thus removing a weak
place in the belly, and causing it' to vibrate
more homogeneously. They appeared on trial
to add great power to the instrument through-
out, and to remove the inequality and varying
intensity of vibration which exists on most
old instruments even by celebrated makers,
and which musicians usually designate by
the term • wolf.' The bars are curved to an
olliptical shape to fit the hollow of the belly,
and to give the greatest resistance to com-
pression with the smallest quantity of mate-
rial. Even in a double bass the quantity of
wood required is very small, and from its
lightness when perfectly dry it hardly exceeds
an ounce in weight. From their shape and
function they are termed eUiptical terision
bars.
It appeared from subsequent experiments
that the same system was applicable to the
smaller members of the viol and violin family,
giving an increased sonority and firmness to
the tone. It succeeds best with violins of
sweet but feeble quality, and in some of the
older Itijian instruments, where the progress
of decay had to a certain extent diminished
the volume of sound.
The contrafagotto or double bassoon as
made on Dr. Stone's designs by Herr Haseneier
of Coblenz, consists of a tube i6 feet 4 inches
long, truly conical in its bore, and enlarging
from j^-inch diameter at the reed to 4 inches at
the bell or lower extremity. It is curved on
itself for convenience of manipulation, so
that in actual length it is ,about equal to the
ordinary bassoon. Its extreme compass is
from C, to c', but its ordinary range is to g
only, the other notes being difficult to bring
out. Haydn gives a part to such an instru-
ment in his Creation^ Mozart uses it occa-
sionally, Beethoven frequently, Mendelssohn
sometimes. ^
Both of these instruments I have heard in
use. Their tone is not perfectly continuous,
but is very good, and when played in conjunc-
tion with other instruments, musically effec-
tive.
8. On the Action of Reeds.
(See pp. 95 to 100.) Knowing the long,
patient, and practical attention which Mr.
Hermann Smith had paid to the action of
reeds, I requested him to furnish me with an
account of the results of his experience. He
obligingly sent me a series of elaborate and
extensive notes, which the space at my com-
mand utterly precludes me from giving at
proper length, and which I am therefore forced
to represent by the meagrest possible outline,
with the hope that they will appear elsewhere
in suitable detail. Only passages in inverted
commas contain Mr. Hermann Smith's actual
words, the rest is my own necessarily imperfect
attempt to condense his statements.
Beeds maybe classed as, i. single^ whether
stHking (in clarinet of cane, and reed-pipe of
organ of metal) or free (harmonium and
American organ, both of metal); 2. paired
(bassoon and oboe, both of cane, in action
compressible; horns and larynx, both mem-
branous, in action extensible) ; 3. streaming
(flutes, flageolets, flue-pipes of organ, all of
rushing air, in action abstracting). The last
kind has been partly considered, supr&,
pp. 396-7.
In the clarinet the reed is straight and
very thin at the tip, but the edge of the sec-
tion of the wooden tube against which it
strikes presents a slight curve, whereas in the
organ pipe the reed is curved and the edge
against which it strikes is straight (p. 9^c).
The time of vibration consists (i) of the time
of forward motion, which may vary slightly ;
(2) of the time of rest, which may vary greatly ;
(3) of the time of recoil, which does not vary
* When the reed is placed in the mouth, the
air on both surfaces of the reed is of equal
pressure, and on increase of strength in the
wind, the tendency would be to separate
the reed still further from the edge of the
mouthpiece, were it not that a current of air
quickly passing into a tube exercises suction
at the orifice of entry; therefore the elastic
reed yields in the direction of the place of
suction, so that it is held there. The current
of wind having been sent forward with im-
petus, leaves behind it a partial vacuum, which
is strongest close upon the inner face of the
tongue. There, then, is a region of least
pressure, which continues to exist during the
transit of the pulse of compressed air to the
first found point of outlet. When that point is
reached, the external air rushes in and restores
equilibrium, and in doing so causes the shock
of arrested motion in which the reed recoils,
and forthwith the action commences as before.' f
The clarinet should not be described as a
stopped pipe, though both have unevenly num-
bered partials and give similar pitch for simi-
lar length, because * in the clarinet there is a
propulsive current going through the pipe ; in
the stopped pipe, on the contrary, there is an
abstracting current acting outside by suction.'
Too much has been attributed to the cylin-
drical bore for producing only the unevenly
numbered partials. * An oboe reed fixed
on the clarinet tube gives oboe pitch of tone
and oboe partials.' The Japanese Hichi-riJci
has an inverted conical bore, that is, the dia-
meter is the smallest at the point furthest
from the reed. * Like the clarinet, it g^ves
notes which are an Octave lower in pitch
than would be calculated from its length.
The first note after the fundamental is the
Twelfth. The reed is broad, not single as on ^
the clarinet, but double and as the bassoon
reed, differing, however, in having an enlarged
base where it fixed into the tube. ... On
substituting an oboe reed, the pitches of the
notes correspond to those of the oboe and the
first tone after is the Octave. ... If the end
of the pipe is placed full within the mouth,
and is blown tl^ough without the use of any
reed whatever (and without any action from
the lips), clear and powerful sounds are elicited,
varying as the openings of the holes are varied,
provided one of the upper holes is left open
... it is indifferent whether the end of the
wide diameter or that of the narrow is taken
into the mouth, either way sounds are in this
manner readily produced.' This effect Mr.
Hermann Smith attributed to a stream reed
from the open hole.
Digitized by V^OOQ IC
554
ADDITIONS BY THE TRANSLATOB.
AW. XX,
Bassoons and oboes have paired reeds, \7hich
touch down their oater edge, and do not vibrate
length-wise but cross-wise, so that a transverse
section through them has alternately the form
of the outer lines in fig. 63, p. 387c, when
they are open, and of two parallel lines when
they are dosed. The reeds are sections from
a small hollow reed-plant of particular growth
{arundo donax or saHva)^ niade very min at
the tip, and rendered supple by the moisture
of the mouth. The player's lip restricts the
size of the oval in notes of high pitch. * The
pressure of wind would keep the pair of reeds
apart but for the influence of the suction when
the current is thrown through. The vibration
therefore is produced by the same kind of action
as in the clarinet, but there is a new mechani-
cal method for bringing it about.'
^ The membranous reeds formed by the lips
and vocal chords are reeds of extension^ be-
ginning to vibrate from a state of closurej
contrary to all other reeds. In horns the cup
acts as an exhaust chamber, and when it is
too large the upper notes cannot be well pro-
duced ; * that is to say, the necessary degree of
vacuum cannot be brought about in time to
coincide with the reciprocating return of the
column of air in the tube.' In the larynx
the ventricles of Morgagni between the true and
false chords probably act as exhaust chambers.
The stream reeds have been already con-
sidered (p. 396c'), but Mr. Hermann Smith
has developed his theory of displacement
actio7i, or the actual tone of air under cleav-
age, deduced originally from observations of
the different sounds of wind sweeping through
the branches of leafless trees, in which tone is
5f produced without a vibrating agent. Mr. Her-
mann Smith found the common doctrine of
friction unsatisfactory. In 1870 he had made
a series of rods about 5 ft. long, the sides of
each having same smoothness and of the uni-
form width of i^ inches, with a V-Bhaped or
triangular section, and these he swept swiftly
through the air like swords, sharp edge first.
He found that, although the friction surface
was similar on each, they developed different
notes, which he discovered to be according to
the thickness of the back, the pitches being
inversely as the thickness. C 528 vib. re-
quires a thickness somewhat less than half an
inch. * Covering irregularly the tips of the rods
did not affect the sound. Half an inch thick-
ness of spongy felt fixed on the back of the
rod, the same width being preserved, lowered
the pitch a Fourth. The felt entangled air in
If its pores, so that the vacuum by suction was
less perfect. . . . With less speed of stroke the
pitch is again lowered. ... In organ pipes, &c.,
in all wind instruments, a certain displacement
of air must take place immediately near the
agent of vibration, in which space a right de-
gree of vacuum is requisite, else the right note
will not follow. To this result all the devices
of mouthpieces tend. This work of displace-
ment in the origination of sound raises a ques-
tion distinct from the transmission of sounds
in waves.'
The free reed is supposed to have been
adopted from China. But the European and
Chinese forms are different. The Chinese
reed is stamped out in the same piece as the
frame, with which it lies level. The reeds act
upon tubes which (agreeing with M. W. Weber's
law, though made long before its discovery)
are three-quarters of the half -wave length.
The harmonium reed is placed above the frame,
and the end turns up from it. If it is set level
with the frame it will not vibrate. To produce
vibration a stream of air must pass between
the tongue and the frame, producing a partial
vacuum on the underside of the reed, and the
amount of suction thus caused must be pro-
perly graduated or there will be no action.
The chief peculiarity of the free reed is that
the pitch is only slightly affected by the
cavities with which it is associated, but these
boxes or cavities, according to their dinaensions,
and governed by the operation of partial ocdn-
slon, mainly determine the qttaUty of tone.
The reed is not properly compared to a
vibrating rod, because the reed actually in use
is not uniform. In a series, the low reeds
are thickest at tip and thinnest at root;
high reeds thinnest at tip and thickest at
root. Hence they are affected differently by
different pressures of wind, an^ alter their
pitch differently. Expressive playing there-
fore becomes playing out of tune. Only with
a constant blast will a free reed maintain a
constant pitch. The stronger blast flattens
deep reeds and sharpens high ones, and from
this cause arises much of the painful disso-
nance of series of chords played on these in-
struments.
To remedy this defect the action of the
wind on the tongue in the American organ is
limited by making the frame very thin, which
is * dished out ' underneath till the edge passed
by the tongue is barely thicker than the tongue,
instead of being 8 or 10 times as thick as
on the harmonium. The suction in the har-
monium is longer in time and stronger in de-
gree, the tongue moves a greater distance, and
more intensity of tone is produced. *The
American reed cannot make a deep excursion,
for the suction is spent as soon as the tongue
gets below the edge of the frame.' It is there-
fore not suited for expression, but produces a
smooth and flexible kmd of tone.
Mr. Hermann Smith considers the harsh-
ness of the free reed, resulting from its large
number of partials, to be chiefly due to its pro-
portions. * The tongue is inordinately long in
proportion to its width, and hence under the
stress of the wind (which necessarily shifts its
incidence during the movements of the tongue
and its re-course) there is developed a diagonal
strain or torsion* from one comer at the tip of
the tongue to the opposite comer of the root,
so that a lateral irregular motion is set up
accompanying the longitudinal vibration.'
Hence he concludes that * the long reed is a
wrong reed,' and that, * in view of its liability
to lateral torsion, the rectangular form is about
the worst. . . . Trial of various reeds shews that
a long rectangular reed is strident in tone, that
increase of width in reeds brings in increasing
proportion smoothness of tone, and that the
width may be increased till it equals the
length.'
In * voicing,' a bend is made across the
tongue, turning the point upwards. This some-
what checks lateral vibration. By an early
plan of his, * reeds were yoked toother by a
bar across the middle of the lengUi, and the
improved quality of tone was obviously doe to
the fa6t that the two reeds were equivalent to
one broad reed, and that the bar across hin-
dered the operation of anvdiagoxu^ strain or
Digitized by VjDOv LC
SECT. K..
MISCELLANEOUS NOTES.
555
twist during vibration. . . . When a reed is much
ourved it is slow in speech, and a great amount
of wind passes wastefully, compensated only
by the smoother tone.' Mr. Hermann Smith
says his * best toned reeds have been series in
which, according to his design, the openings
made by the curve given to the reed were filled
up at the sides by arched blocks added to the
top of the frame, following the line of curve.
The discontinuity was therefore sharply de-
fined, yet the tones were mellow and rich. A
reed mounted on wood surface may have its
quality greatly changed by an interposed pad
of leather or felt between the reed frame and
the wood, the extreme harshness disappears
and the tone is smoother altogether, shewing
how much that is unpleasant is due to the
jarring from arrested motion.*
Mr. Hermann Smith's * conclusions are that,
in the making of free reed instruments, broad
reeds should be used, and with broad channels
or boxes or cavities of varied shape ; that within
the large chamber small suction chambers
should be placed below or beyond the reed
tongue, in imitation of the ventricles of the
larynx ; and if these afford areas and cavities
suitable for the displacement accordant with
the pitches, the speech will be quickened and
firmness given to the tones, or, in other words,
the mechanical motion of air and reed will be
steadied. The rectangular form of reed, except
when stops of hard metallic quality are re-
quired, should be abandoned, and broad shapes
substituted, having tips semicircular, semi-oval,
or ovate, or shapes to ensure a central line of
strain. Weighting the tips of reeds should be
avoided as much as possible.*
Mr. Hermann Smith's latest device in the
treatment of reeds is designed to overcome the
difficulties of inordinately long or weighted
reeds. His plan is ' to use metal or material of
uniform thickness, and to get the degrees of
flattening by drilling out or excising portions
of material at or near the root of the reed,
and then to fill such spaces as are thus made
with other fixed pieces of metal that are neu-
tral, and do not enter into vibration. Thus
the sides of the reed tongues remain with the
fibre intact, unweakened by thinning or scrap-
ing, which takes the best vigour out of the
reed. Any degree of fiattening may be at-
tained according as the excision is made to
extend up the tongue. A like treatment of the
half of the tongue forming the tip will pro-
duce the opposite effect, sharpening pitch by
lightening the tip ; the spaces left by the ex-
cised portions are then covered with lighter
material, such as goldbeater's skin.'
Mr. Hermann Smith states that his device or
of * adding to the large cavities small exhaust
chambers or cup-like cavities, fixed just below
the tongue of the reed, causes the most un-
manageable reeds, even those in the 32-foot
Octave, when made broad on the above plan,
to render good musical service, to be free in
speech, and to produce a full pervading quality
of tone, devoid of the harshness of long reeds
having heavily weighted tips.' Mr. Augustus
Stroh (see Sect. M. No. 2, p. 542^) informs me
that he has been led to a similar contrivance
in a machine he has recently constructed.
For further details of Mr. Hermann Smith's
inventions respecting reeds, see Specifications
to his Patents 1878, No. 227 and No. 4942 ;
1880, No. 68 ; and 1884, No. 7777. A large
amount of varied information may also be
found in his treatise entitled ' In the Organ and
in the Orchestra,* now (1884-.5) publi^ing in ^,
Musical Opinion^ a monthly magazine. Of this
treatise 25 chapters have already appeared full
of interesting elucidations of instrumental diffi-
culties.
9. Postscript,
Standard Musical Pitch in England.— la
consequence of a communication from our
Foreign Office, due to the Belgian change of
pitch (p. $oid)t Sir G. A. Macfarren, Prin-
cipal of the Boyal Academy of Music, con-
vened a public meeting of musicians, theorists,
instrument-makers, and their friends, * to con-
sider the desirability of a standard musical
pitch for the United Kingdom.' It took place
on 20 June 1885, and was well attended.
Three resolutions were passed: (i) declaring
uniformity desirable: (2) recommending French
pitch ; (3) declaring it advisable to teSae steps
for its adoption in civil and military bands.
A committee of 4 theorists, 15 musicians, and
4 instrument-makers was appointed to carry
out the resolutions. See the Times^ 22 June
1885, p. 7, col. 2, and p. 9, col. 3, and Musical
Opinion, i July 1885, p. 493.
Addenda to History of Pitch. In 1845 t^e
pitch of the Philharmonic Society was 0^447*1,
according to a fork tuned in the orchestra at
that date by Mr. B. S. Bockstro.
Herr Eduard Strauss, of Vienna, perform-
ing at the Inventions Exhibition 1885, used
a' 452*5, but the bandsmen said the pitch of
the opera was nearly a Quartertone fiatter, say
a' 447.
The band of the Pomeranian (Blucher)
Hussars performing at the same Exhibition
used a' 460-8, or, as the bandmaster said,
'exactly a Semitone sharper than French
pitch.'
Effect of Rust, dc, on Tuning-forks. See
p. 445 d\ Mr. Bockstiro possesses a fork which
in 1859 was at French pitch, and now through
rust (and possibly bad treatment) shews only
a' 424-5, that is, has gone down by 42 cents,
far exceeding any other fork examined.
Flute Intonation. Mr. B. S. Bockstro, in
June 1885, kindly brought me an eight-keyed «r
flute, about 40 years old, an excellent instru- ^'
ment of its kind. This he played so as to
preserve its natural intonation without cor-
rection. Before striking any new note it was
ascertained that g" 404 vib. remained constant.
The a' 461 vib. was a quarter of a Tone too
sharp. The result gave in cents, reckoning g*
as 1200-0, c' 488, c'5564, d' 678, d'J763, «'
906, /' 1000, f% 1098, ^ 1200 = 0, ^Z 92, a'
229, a% 292, 6'43i, c" 513, c"5 566. This seems
meant for meantone intonation, sharpening
many of the sharps to pass as the flats above
them. But even in this case the intonation
was imperfect, and Mr. Bockstro thought it
was rather due to a series of compromises.
Mr. B. S. Bockstro brought at the same
time the ' Bockstro model ' flute, invented by
himself, to have a more correct equal intona-
tion than Boehm's. This was blown in the
same way, but in this case a' was always
brought to 452 vib. Besult in cents, reckon.
Digitized by V^OOQ IC
556
ADDITIONS BY THE TRANSLATOR.
APP. XX.
ing a* as i200«o, was d 312, c'«4o6, d' 506,
d'ft 601, «' 697,/ 801, /JJ 895, g' looi, gr'5 1097,
o' i200;so, a% 102, 2/ 201, c" 304, c"5 404.
This is very good, and may be better than
the above numbers shew, as, on account of the
difficulty of sustaining notes on the flute
without variation, it was not possible to deter-
mine the pitch of each note within less than
I vib. in a second. With regard to the lowest
d from the open end of the flute, Mr. Bockstro
says he leaves it purposely too sharp in relation
to a', because it is easy, by management of
lip, to blow it in tune, but if it were originally
in tune, it would not sound sharp enough in
very soft passages.
Siamese Scales. The King of Siam sent
over his Court Band with their instruments to
^ the London Inventions Ejthibition 1885, and
the Siamese minister obligingly allowed Mr.
Hipkins and myself to determine the musical
scale. Prince Prisdang told us that the in-
tention was to divide the Octave into 7 equal
intervals, each of which would then have
1 7 1 -43 cents. Hence the following comparison.
The scales are given as usual in cents from the
lowest note.
Theoretical scale : — o, 171, 343, 514, 686,
857, 1029, 1200 cents, having a neutral Second
171 lying between 100 and 200, a neutral Third
343 lying between 3cx> and 400, a slightly
sharpened Fourth 514 for 500, a slightly flat-
tened Fifth 686 for 700, a neutral Sixth 857
lying between 800 and 9CX), and a neutral
Seventh 1029 lying between 1000 and iioo,
but much nearer the former. As there is no
harmonic interval but the Octave, and as the
^ Siamese seem to tune by Octaves and single
degrees, there is room for much variation from
the ideal intonation, as shewn in the following
observed scales.
Banat ek or wood harmonicon, first Octave
o, 208, 326, 537, 698, 883, 1048, 1208, second
Octave (pitch of lowest note 382-6 vib.), o, 200,
359.537, 7". 883, 1057, 1222, third (incom-
plete) Octave o, 193, 347, 549, 698 (two more
bars, too high to measure). This instrument
is tuned by lumps of wax mixed with some
heavy substance stuck to the underpart of the
bar. The tuning lump having fallen from the
second bar of the first two Octaves, it was
quite out of tune, and its proper pitch (regis-
tered above) was determined by a comparison
with other instruments. In the Banat, p. 518,
No. 85, all the lumps had been removed,
hence, it was entirely out of tune,
f Banat t'hong or brass harmonicon (pitch
of lowest note 382*6 vib.) scale o, 200, 340,
537i 699, 881, 1043, 1207.
Banat lek or steel harmonicon, first Oc-
tave (the second bar absent), o, 327, 519, 679,
856, 1075, 1202, second Octave (pitch of low-
est note, 385-5 vib.), o, 150, 299, 447, 614, 743,
960, 1 1 79, third (incomplete) Octave o, 90,
222, 430, 609.
Tak*hay or crocodile, a three-stringed in-
strument with high frets, played with a conical
plectrum, o, 198, 362, 528, 720, 890, 1080, 1250.
Hence 52 single degrees were examined,
each of which should have had theoretically
171*43 cents. In reality 5 were less than 132,
8 between 140 and 159, 12 between 160 and
167, 9 between 170 and 179, 3 between 180
and 185, 6 between 190 and 198, and 9 be-
tween 2CO and 219. Hence only 15 approached
to equal Tones, and only 2 approached to eqnal
Semitones, both sets being clearly erroneons,
while the 21 between 160 and 179 were toler-
ably close approximations to the ideal. Bear-
ing these variations in mind, it is probable
that p. 518, Nos. 81, 82, and 83, at least, be-
longed to this system of 7 intentionally equal
heptatones, as they may be called. And this
confirms the conception that Salendro, p. 518,
Nos. 94 and 95, consists ideally of 5 equal
pentatones.
The instruments were beautifully and
artistically ornamented, the execution by the
musicians was florid and musicianly in aoca-
rate and varied rhythm, there was an obser-
vance of light and shade, together with a clear
conception of melody, but none of harmony.
Besides the harmonicons there were kettles or
gongs (k'hong), a three-stringed viol (saw t'hai),
a two-stringed fiddle (saw Chine), the three-
stringed crocodile (tak'hay) ; reed instruments,
flutes, and drums.
Japayiese Scales, see pp. 519 and 522, Nos.
no to 139. In July 1885, Mr. Isawa, Director
of the Musical Institute, Tokio, Japan, sent to
the Inventions Exhibition several tuning-forks
and tables. From the tables it appeared that
the classical 12 Bitsu or Semitones resulted
from tuning 1 1 perfect Fifths up (or Fourths
down), and then a Fifth too flat by a Pytha-
gorean comma, giving the scale : o, 1 14, 204,
318, 408, 522, 612, 702, 816, 906, 1020, mo,
1200 cents. But the 13 forks sent had the
following pitch (as determined by the Trans-
lator), the number and name of the Bitsu and
the name of the nearest European note at
French pitch being prefixed :- -I. Ichikotsa
d' 2927, n. Tangin d'n 305*6, III. Hiyojo
d 326*2, IV. ShOretsu / 343* i, V. Shimoma
fn 3657. VI. Sojo g' 391*5, VII. Fusho ^I 4io-ii
VIII. Waushiki a' 437, IX. Bankei a| 460,
X. Banshiki 6' 491-5, XI. Shinsen d' S^7'3^
XII. Kamimu d'Z 549'5i I'. Ichikotsu d" 585*4,
this gives the scale in cents : o, 75, 188, 275,
385. 503. 583. 693» 782,897, 986, 1091, 1200.
Mr. Isawa also sent forks for tuning the
popular scale Hiradioshi (p. 519a, Nos. no to
112) in two forms, old and new, both different
from those already given. Old style in cents
o, 102, 502, 706, 809, 1x97, evidently meant
for just o, 112, 498, 702, 814, 1200. Nev
style in cents o, 85, 502, 708, 793, 1200, for
Pythagorean o, 90, 498, 702, 792, 1200.
Mr. Isawa also sent a Standard Tuning-
fork giving d 145*45 at 52° F. French pitch
d= 145*2 vib.
There was also a monoohord on which
many scales were indicated, and two sets of
reed pitch-pipes, which cannot be described
for want of space.
Modem Greek Scale, According to Me-
shaqah, in Eli Smith (op. cit p. 264, note §),
the modern Greeks divided the Octave into
4 X 17 « 68 parts, and form the scale by 12, 9,
7, 12, 9, 7, 12 of these divisions. Since, then,
1 200 -^ 68= 17*65 cents, the scale in cents will
be o, 212, 371, 494, 706, 865, 988, 1200, which
again has a neutral Third and Sixth, 371*865.
If the scale had consisted of 12, 8, 8, 12, 8. S,
12 of these divisions, we should have got the
precise scale of Villoieau (p. 520a' 1. 5), which
is a singular additional justification of iiis divi-
sion of the Octave into 1 7 equal parts.
Digitized by V^jOOQlC
INDEX.
The nnmben refer to the pages ; the letters a, b, c, d to tlie quarters of a page, and, when there are donble colomns,
of the first column. In which care a', V^ d, d\ refer to the qoarterw of the rarond column.
* before the number of the pa^ shewn that the title uf a book or paper is there olted.
t ] indicate notes and aiditions by the Translator.
A« X, A", see VoweU
[Aalst, Van, on Chinese Music, 524c]
Abdul Eadir, same as Abdulqadir, Persian, f 4th
cent., 282a. his 17 tones, 282a, 6, and 12
scales, 282 to 283
Abdulqadir, 2816', cf, 282^. [his 16 Fifths,
281c'. his system, 282^, 3646, 523a]
Accented note names, how related to pitch
numbers, i6c
Accidental Scales defined, 267a
r Acoustical facts in change of key, 55 li]
Acoustics, physical and physiological, 16, its
connection with music, ic. has hitherto not
helped musical theory, i^. physical, a section
of the theory of elasticity, 3c. physiological,
investigates processes in the ear, 4a. its
physical part, 46
Added Sixth, chord of, 294^, or imperfect
minor triad, 344c
Addition, algebraical, of waves, of velocities
and displacements, 2^d^ note
[Africa, Western, Balafongs, 5186]
[Air-reed, aerial, or aero-plastic reed, 397c]
[Alboni, her just intonation, 4786]
Alexander the Great, zyid'
Al Farabi, 282a [his Greek scales after Pro-
fessor Land, 515c]
[Alternative Intervals varying by Quartertones,
in modern Arabic and Indian scales, 524c]
[Amati of Cremona, 1 596- 1684, resonance of
his violins, 876]
Ambrose of Milan, a.d. 374-398i his authentic
scales, 243a [doubtful whether they were
really his, z^id'.] his numerical notation of
the modes, 2696. his ecclesiastical scales, to
be regarded as essential, 2716, c
Ambrosch, Chinese melodies, *2$%d\ 26id'
r Amels possesses Scheibler's Tonometer, 444a]
[America, U.S. of, 511a]
[American Organ Reeds, 554^']
[Amiot, ♦95^', 262d, 548c]
Amplitude, loc, 346
Ampullce, see Ear
Analysis of compound and composite tones
into simple vibrations by the ear, 33. this is
independent of power to analyse vibrational
forms by eye, 34fi{. of air into pendular vibra-
tions by sympathetic resonance, independently
of ear, 42^2. objective, 48a, 6. of sensations,
its difficulty, 496. of compound into simple
tones by ear, its theory, 148c
Anche lihre^ or free reed, 956
Ansa, presumed Indian tonic, see Vadi, 2436
Antony, ♦239^
Anvil, see Ear
Appogiatura, always makes a Semitone, 2876
Approximation in pitch, forms a musical con-
nection, 352&
Appunn, late of Hanau, his high pitch from
forks, x8a, used by Preyer, 1 5 ic. his pipes for
artificial vowels, 127^, 128a. suggestion to
Preyer, iSyd, his loaded reeds, 176^, 177^'.
[their pitch, as determined by Translator,
1776. his conical resonators, 373d. [his reed
tonometer, 443a. his difficulty in tuning a
series of perfect Fifths, 483^2]
Aqtueductus vestibuli, its function, 136a
Arabesques compared to music, 252c
[Arabia and Persia, scales after Professor Land,
515^. modem, after Eli Smith, 515^. medieval
scales after Professor Land, with 7 and 8
tones, 516a, 6, 5i9<f. lute, earlier notes after
Professor Land, 5166, 520^. medieval notes
after ditto, 517a, 520a]
Arabic Scales, 282 to 283. modem, of 24
Quarter-tones, 2646. [according to Professor
Land, 284 note]
Arabic and Persian musical system, 28o&to 2856.
Arabs have no pleasure in polyphonic music,
1966
Architecture, its analogy to musical composi-
tion, 2c. the periods of its progress are the
analogues of those of music, 235c
Archytas first settles major l%ird as 4:5,
262c, 3626, d
Aristotle, on Consonance, 237a, />, on variations,
2376. his indications of a tonic, 240c, 24i{2,^',
indications of a downward leading note, 242a,
251C, d, the only writer who indicates a
tonic, 2676. his conception of effects of
music, 251a. makes mese tonic, 268a, d, on
descending leading note, 286c, [d]
Aristoxenus, his twelve Fifths, 271a [5486]
[Armes, Dr., 502a]
Art, works of, must not display their purpose,
366c
Artificial compound tones, 120c to 122^2
Artificial production of vowels, or vowel synthe-
sis, by, 123a to 124. [tabular statement of
results, 1 24^:2.] by organ pipes when the effect
of difference of phase is not under investiga-
tion, 12yd, Appunn*B pipes for this purpose,
128a
Artusi blames Monteverde for using dominant
Seventh without preparation, 248^
Auditory apparatus, its advantage, 1346. the
mechanical problem it has to solve, 134^,
how solved, 135a, &, c. sand, 137a, stones or
otoliths, of fish, 139a, of crastaceans, 149c.
oiliffi of ampu)lsB, in former editions sup-
posed to be hearers of noises, 151a, 6, may be
hearers of squeaking, hissing, <S;c., isid, hairs
Digitized by
^.joogle
558
INDEX.
of Mysis, 150a. nerves, how excited, hypothe-
sis, 5a. ossicles described, 131a, b
[Austro-Hungary, pitch, 504c]
Authentic scales of Ambrose of Milan, 242^2,
243a. Glarean's six, 245c, d, the first, 267a
B
[B natural and B flat, ancient signs for, Si2d]
Bach, C. P. Em., his equal temperament, 321c,
considers equal temperament the most per-
fect intonation, 323c [548^1
Bach, J. Sebastian, down to his time final
chords always major, or without the Thirds,
217a. his suites, 245a. his use of closing
minor chord, 296^2. his use of the major
Sixth in the ecclesiastical Doric or mode of
the minor Seventh, 3046, 305a. [example
analysed by duodenals, 304c, d, c'.] his use of
the mode of the minor Sixth, 307c [503c, S^Sd"]
[Bagdad Tambour, its scale, after Prof. Land,
517c]
[Bailey, 549a]
Bajazet, 282a
Ball, struck up as it falls, its periodic motion,
i^, 21C, and fig. 9
Barrow, *95d', 262d
Basevi, 352c, ♦352^
Basilar membrane, 1 38a. Hensen and Basse's
researches on, 145&. its breadth probably
determines the tuning, 14 Sh. high notes near
round window, low notes near vertex, 146c.
breads easily along radial fibres, not trans-
versely, 146a. consequent mathematical
theory, 1466. its fibres form approximatively a
series of stretched strings, 146c. its behaviour
for noises, mathematicfJly investigated, 403c,
its vibration in the cochlea mathematically
investigated, 4o6d
Bass, figured, shows new view of harmony, 248c
Bass notes with tinkling upper partials, 11 65
Bassoon, its tongue or reed, 966. conical tube,
produces all harmonics, 99a [reeds, 554c]
Bausoh, his violin, 85c
Beats, 5a, of simple tones, how distinguished
from combinational tones, 159^2. their origin,
their frequency := the difference of pitch num-
bers of generators, 164^. diagram of, 165a.
examples, 1656, c. from upper partials as
well as primes, 165c. rendered visible,
1 65(2. require the sympathetic body to be
nearly of the same pitch as itself, i6sd,
what becomes of them when too fast to be
counted, i66d. according to T. Young, they
become the differential tone, i66d to 167a.
objections to this hypothesis, 1676. [cheap
apparatus for shewing 167 d\ use of Har-
monical for shewing, i6Sd,] how best ob-
served, 167c. their character, z68a, jarring
like letter B, 168&. intermittent tones heard
with a reed pipe or tuning-fork and double
siren, i68c. produce intermittent excitement
of auditory nerves, 169&. do not disappear
from rapidity only, but also depend on in-
terval, 170(2. even 132 beats in a second are
audible, 171a. the character of the roughness
alters with the number of beats in a second,
17 ic. beats of a Semitone heard up to 4,000
vib. per second, 171c, of a whole tone to 2,000
vib. per second, ijid. major and minor
Thirds, smooth from 264 to 528 vib., are
rough in bass, 17 id, their roughness does not
depend solely on their frequency, I7i<2, 172a,
but, in a compound manner, on magnitude of
interval and frequency, 172a. on the siren
will determine whether the note heard is the
prime or an upper partial, i74d. from the
upper partials of a single tone, 1786 [I78d'].
of upper partials of two compound tones, i8oa,
examples, i8oc. why consonances produce
no beats, and why if they are slightly altered
beats ensue, 18 16. of disturbed consonances,
how observed with double siren, 182c. of
upper partials, their rapidity has a pre-
ponderating influence on distinctness of
definition, 184&. law for determining them,
with tables, 184c, d, the amount of dis-
turbance of a consonance being the same,
the beats increase with the higher numbers
expressing it, 185a. table of, when consonances
are altered by a Semitone, 185c. due to com-
binational tones, 197c. of combinational
tones, can alone distinguish consonance from
dissonance of simple tones, 199&. of differen-
tial tones cannot occur if consonant interval
ratios are exactly observed, but occur instantly
if they are not, 203c. peculiar character of
those with bowed instruments, 208&. of the
tempered triad, 3226, c. their effect on its
harmoniousness, 322^. variation in the pitch
of the beating tones, 414c. calculation of
their intensity according to the intervals of
the beating tones, 415^2. [how to count,
444^]
[Beats and Ck>mbinational Tones, recent "works
on, sect. L. see table of contents, p. 527]
Beauty, subject to laws dependent on human
intelligence, 366
B6dos, Dom, *i6c [his 4 old French foot pipe,
1 6c note, 494^, 50&1. knows only mean tone
temperament, 548c]
Beethoven [uses pianofortes by Slein, 77(2],
his use of the mode of the minor Sixth, 308ft.
his relation to equal temperament, 32 7(;
[Behnke, Emil, *iood\ *ioid', on registers of
voice, 1 01 6]
[Belgium, pitch, 504^]
[Bell, Graham, inventor of Telephone, finds
and demonstrates double resonance in all
vowels, 107(2. his paper on Vowel Theories^
*io8(2]
[BeU, Melville, *io$d, his vowel system, io5(i,
107(2]
Bellermann, *26sd'
BeUs, large, how set swinging by periodical
efforts, 36d. their inharmonic proper tones,
72c. why they beat, 73a
Bell-shaped glasses, broken by singers, 39(2
[Belly-bridge of piano, old single, 77^. the
divided, was introduced by John Broadwood,
1788, 77cr\
[Bender, 505a]
Bemouilli, Daniel (1700-1782), on law of mo-
tion of strings, 15a [441c]
[Best, W. T., organist, 500c]
[Bettini, 507c]
[Bevington, organ-builder, 5o6e]
[Bishop, organ-builder, 506c]
[Bitter, Life of J. S. Bach, 548(2]
[Black Digital Scales, 518&, 5226^
Blade of air, blade-shaped lamina of air aft
mouth of flue-pipe, its action, 92a, 395a
[Blahetka, on Vienna pitch, 504^]
[Blaikley, D. J., on velocity of sound in tabes,
*90(2. distance of plane of reflexion from end
of flue-pipe, *gid, action of lips in blowing
the horn, &o., 97(2. office of the air in the
tube in relation to the lips, *97d% gSd^ 990,
* c' d\ his account of the clarinet and its
Digitized by V^jOOQlC
INDEX.
559
partials, 996, e. how brass tabes for horns
are shaped and bent, 99^. their shape not
truly conical, 99^. sounds harmonics on
^French horn in exact tune, 99^'. form of
trumpets, looc. two lowest partials out of
tone, looc. length of French horn with its
various crooks, lood. trombone, its shape,
1 00c'. slide trumpets, looc'. keyed horns
obsolete, looc'. piston horns, lood', on side
holes, 105^2, 210(2. on horns being in just
intonation, 327^. on the conical tubes, tried
by Professor Helmholtz, 394^, note, says
there are no ideal brass instruments in prac-
tice, 42Sd, forks, 494d]
[Bodin, so&d]
[Boehm, Theobald, on flutes, *i03d. English,
edited by W. 8. Broadwood, 103d']
Boethius, on the old tuning of the lyre, 355c,
266c
Boltzmann, 93a
[Boston Music Hall, 96^
Bosanquet [his resonator, 43^', 374c. on the
measurement of intensity of sound, *7Sd.
distance of plane of reflexion from end of flue-
pipe, *gid. Becent work on combinational
tones and beats, 152^2, 155^2', 156(2, 157(2,
1^7(2, *322(2. his use of Mercator*s cycle of
53* 328c, (2. his arrangements, 329a, [329^2,
d'], his manual, 429. his cycle, 4366', its
synonymity and intonation, 439. his gene-
ralised fingerboard and harmonium, 4796 to
481a. on beats and combinational tones, 528
to 538. on J. S. Bach's equal temperament,
548(2]
[Bosch, his Seville organ, 511a]
Bourget, J., *73(2'
Bowed instruments, 190. their musical tones,
80c, 2076. their use in harmony, 207c. not
suited for soft melodies or sustained chords,
3o8a. their scraping character, 208c. harsh-
ness in quartetts by good players not accus-
tomed to play together, 208c. [one of its
causes, 2oSd. develop combinational tones
well, 208(21 see Violin, Ac, 2076
[Branchu, Madame, 508c]
Brandt, cited, on Young's law, 536. modifica-
tion of his experiment, 53c, *syd
Brass instruments have sluggish attack, 67a. not
suited to harmonies except out of doors, 210c
Breguet, his watch-key resembles the articu-
lation of anvil and hammer in the ear, 133a
[Broadwood, James, in z8ii, advocates equal
temperament, 548c]
[Broadwood, J., A Sons, 497a, Soyd]
[Broadwood, John, introduces divided belly-
bridge on piano, 1788, and uniform striking-
place, 77c]
[Broadwood, Henry Fowler, present head of
firm of J. Broadwood & Sons, has introduced
the ^-length striking distance on all his
pianos, 77(21
[Broadwood, W. S., his edition of Boehm, 103(21
[Brown, Colin, on characteristics of Scotch
music, 259(2 note, on WiU you go, lassie,
262(2'. his Voice Harmonium described and
figured, 326(2', 47od]
[Browne and Behnke, on Voice, Song% and
Speech, *iood']
Brumel, Anton, 2g6d
[Bryceson, 5056, 5066, c]
[Burmah, Patala and Balafong, 518a]
[Byfield A Green, 506a]
[Byfield<j^ Harris, 506a]
[Byfield, Jordan, A Bridge, 5066]
[Byolin^ 498a]
Caccini in 1600 invents recitative, 248c
Cadence, complete and imperfect or plagal,
2935, examples, 293c
Cairo, Quartertones used there, 2656
[Callcott on extreme sharp Sixth, *^oSd]
Canals, see Semicircular
Canonic imitation developed early in 12th cen-
tury, 244c
Cantus firmus, shews leading note, 287c
Carissimi, 307^^
Catgut strings, their quality of tone, 80&
[Cavaill4-Goll, Aristide, organ-builder, his
rules for finding pitch of flue organ pipes,
*^, (2. tried and abandoned free reeds,
96c. builds Mr. Hopwood's organ, 96(2. on
Boehm's flutes, I03(2'. first draws attention
to the mode in which flue-pipes speak, 397a'.
his * soufilerie de pr^ision,' 4426, 494(2', 508(1
[Cavallo, Signor, helps J. Broadwood with the
divided belly-bridge, 77c']
[Cell, harmonic, or Unit of Concord, 458c]
[Centesimal Cycle, 437&1
[Cents or hundredths of an equal Semitone,
41(2. how to calculate, App. XX. sect. C, see
contents, 446c]
[Challenger, 5496]
Changing or passing notes, 353a
[Chappell, 507(2]
[Character of each tone in the major scale,
according to Curwen, 279c, d]
Characters of keys, are there any absolute ?
3 IOC. differ on pianos and violins, why,
3 1 1 c, and on wind instruments, 311c. may be
influenced by peculiar resonance of ear pas-
sage, 3 1 1 c. [presumed, after E. Pauer, 5506]
[Chev6, Emile, 425(2, marries Nanine Paris,
writes theory of Galin-Paris-Chev6 method,
his 29 division of the Octave, 4250', 436(2. his
system contrasted with Tonic Sol-fa, 426(2]
[Chev6, Mme., formerly Nanine Paris, 425c',
her principles of teaching to sing, 4266I
[Chickering, 5oi(2]
[China, scales from instruments and musicians
at the Health Exhibition, 51 8(2, 5226]
Chinese, their numerical speculations on music,
229c. their pentatonic scales, 257(7. learned
heptatonic scales under Tsay-yu, 258a. their
pentatonic airs are dull, 258(7. 260a. 261 6.
[their equal temperament, 548c. their free
reeds, 554^2]
Chladni, 1756- 1827, his sand figures on elastic
plates, 41 &, 71(2, the proper tones of such
plates, J2a
Chord defined, 24a, 211c. of four parts a
tetrad, 2226. rules for open and close posi-
tions hitherto given had no theory, 224c
Chords, growth of feeling for their relation-
ship, 292(2, 293a, 2966. of the tonal modes
with double intercalary tones, 297c, (2, with
single Do, 2986, c. of Sixth and Fourth J,
and of Sixth and Third J or Sixth only 6,
213a. of extreme sharp Sixth, their Gtreek
Doric cadence, 2866. of the Seventh used
to connect other chords, 3576. see also
Closing Chords, and see Italian, German, and
French Sixth, diminished Seventh, iftc.
Chordal sequences, 355c
Chordal relationship felt in 15th and i6th
centuries, 369a
Chrysanthus of Dyrraohium, Archbishop, de-
clares Greeks have no pleasure in polyphonic
muFio, and leaves it to the West, 196&,
♦196(2
Digitized by V^jOOQlC
56o
INDEX.
CilisB of ampullffi, see Auditory CiliaB
Cithara, five-stringed lute, 2Syd
[' Clang/ used by Prof. Tyndall for compoond
tone, why not so used here, 24d. Webster's
definition of, 24d]
[' Clangtint,' used by Prof. Tyndall for quality
of tone, why not so used here, 246']
[Clark, Lieut., see Macleod]
Clarinet, its tongue or reed, 966. cylindrical,
with unevenly numbered partials, 98c. [D.
J. Blaikley's account of the clarinet as not
wholly cylindrical, 996, c] its peculiar
action in forming chords, 2iod. experiment
with it and just harmonium, 211a. [its
tones, 392c. its reeds, 553c]
[Clavichord, strikes the string at end, ySd']
Closing Chords, their development, 290^.
their five major forms, 291c. major chords
in minor modes, 296^2, 297a
Cochlea, see Ear
Cochlean nerve, its expansion, 139&, e
[Cohen, 5086]
[Colbacchini, 510c]
CoUard, 507^, 5496
[Colour, used by T. Young for quality of tone,
why not so used here, 240^
Coloured lights, mixture of, different from
mixture of pigments, 646
Colourings, Greek, XP^^ t^eir reality, 2656
Colours, primary, scarlet-red, yellow-green,
blue-violet, 646, [d note] never seen pure,
64c. power to distinguish generally absent,
640^. analysis of, into three, by Waller, 94a
Combinational tones, 5a. occur when the vi-
brations of the air are not infinitesimal, 152c.
result from all the partial tones, 153&. most
easily heard when generators are less than
an Octave apart; for harmoniums can be
reinforced by a resonator, but in other cases
not, 153c. [heard simultaneously with rattle
of beats, 153c. from two flageolet fifes, 153d.]
multiple, considered as of different orders,
iS4d, [not all audible, 155^'.] once thought
to be subjective, and to result from beats be-
coming too rapid to be heard separately, 156c.
objections to this theory, 156c. they arise
from the largeness of the vibrations, 156(2.
condition for being well heard on harmonium
and polyphonic siren, 157^1 b, may be gene-
rated in the ear by unsymmetrical form of
drumskin, 1586, and loose joint of hammer
and anvil, 158c. produce tingling in the ear,
and are strong when soprano voices sing
Thirds, 1580. an accessory phenomenon by
which beats are not interrupted, 1676. beat,
197c. delimit consonances when the partials
do not suffice, 201 5. the most general cause
of beats, 2046. important for the harmoni-
ousness of a chord, 214c. those of major and
minor triads, 215a, &, essentially different,
214c. their mathematical theory, 411(2. their
effect in Just, Pythagorean, and Tempered
chords compared, 31 46 [and note], their
beats mathematically investigated, 4186 to
419c. their origin, 41 9(2, in the siren and
harmonium, 419(2, 4^cx2
[Commas higher indicated by superior, lower
by inferior figures, proposed by Translator
and here used, 2776, c]
Compass of instruments, [lydlt i8a, [186, c]
[Compass of the Human Voice, 544^1
[Composite and compound tones distinguished,
33^1
Composition of simple tones, tone and its ,
Octave, 306, c, tone and its Twelfth, 32. ]
artificial, of simple tones how arranged, l2oe
to 122(2
Compound tones, 22a, and see Tone
Concatenation, or musical connection of tones,
direct, 3502, indirect, 3516
[Concertina, Just English, 470&]
Concords are consonant chords, 2ii<2
[Condissonant triads defineci, 2ii(2', 35812
note t, 459«]
Congregational singing, its results, 2460
Connection, musical, or ooncatenation of notes
in the scale, 350(2 to 352
Consecutive Twelfths and Fifths, why forbid-
den, 359(2 to 361C. Octaves, why forbidden,
359c
Consonances result from coinoidenoes of upper
partials, iSzd, tables of such ooincident
partiab, 183a. also in musical notation,
183(2. disturbed by the consonances next
adjoining them in the scale, meaning of this
expression, i86(2. defined, 1946. and disso-
nances, their boundary, 2286, which has not
been constant, 228(7. absolute, Octave, Twelfth,
and double Octave, 1946. perfect. Fifth and
Fourth, 194c. medial, major Sixth and major
Third, 194c. imperfect minor Third and minor
Sixth, 194(2. great diversity of opinion on the
order of their relative harmoniousness, 196&.
order of de Vitry and de Muris, Franco of
Cologne, and Glarean's Dodecachordon, 196c
here it is based on their independent har-
moniousness, 197a. their influencse on each
other, tabular views, 187. separately con-
sidered, iSyd to 1906
Consonant intervals, why so called, 181c.
[their beating partials and the ratios of those
partials compared, 191 6, c]
Consonant triads not exceeding an Octave
examined, only six possible forms, 2 1 2d to
217a. exceeding an Octave, examined, 2176
to 2226. effect of transposition, 2180-2196
Consonants, tenues and fnedue, their charac-
ter, 66a, h. hisses, F V, <tc., R and L,
67(2
Comu and Mercadier, experiments on yiolin
intonation, *32$d
[Correlative Duodene, 4626]
Corti, the Marchese, his formations, or arches,
or rods, 1396, d to I4id, seem most suited
for sympathetic vibration, 1456. they in-
crease in size as they approach the vertex of
the cochlea, 145(2. their alteration of fonu,
146(2. probably play only a secondary part
in the function of the cochlea, 146(2. viewed
as the means of transmitting the vibrations
of the basilar membrane to the terminals of
the nerves, 147a. may be 4500, or, throwing
off 300 for extramusical tones, 4200 in the
octave, 1476. how they may determine pitch
continuously, 147a. mode in which they
analyse tones, 147c to 148&. may vibrate to
two tones, and their vibrations may be com-
pounded of them, 1 66c. attempt to estimate
intensity of sympathetic vibration for in-
creasing intervals of tones, 172c. choice of
hypotheses, 172(2. considered as explaining
consonance and dissonance, 2276
[Costa, Sir M., 502a, 507c, SSSdj
[Couchet, Jan, Antwerp, harpsichord maker,
knew that striking-pliu>e affects quality, 77c'.
Coussemaker, * 196(2, *243(2', *244d
[Cramer, 5056]
[Crawford, 526a]
[Cross and Miller on American pitch, 494c]
[Crotch, Dr., 5496]
Digitized by VjOOQIC
IKDEX.
S6i
Crmnples in vibrational form of vioUn strings,
f^b
Cmstaoea, observations on their auditory ap-
paratns by V. Hensen, 1490
[Corwen, John, names of registers, loie. his
use of the character of a tone in singing,
*279c, d. on major Sixth of the minor scale,
^joid. his work with Tonio Sol-fa, 4246 to
4256. his pitch, 496c]
[Carwen, John Spencer, eldest son of John
Corwen, President of the Tonic Sol-fa College,
424^. his letter to the Timest 424a'. his
ifemoriaU of John Curwen, 425c]
[Cycle of 53, how it arises, 465a]
[C^rclic Temperaments, 435^]
Cjmdbalom of 19 notes to the Octave, men-
tioned by Pnetorius, 320c. value of the
notes, 320(f
[Caezmak, on whispering, *io&i']
d'Alembert, his theory of consonance, 232a to
2336. his Pythagorean Sixth in major scale
not allowed, 2y$d. says Bameau*8 tuning
was common in 1762, 32 1&. his explanation
of the limits of the Greek heptachord, 35 id
[d'Aligre, Marquis, 5086]
Damping, rapid, of tones of air in mouth, 1 12c
to 113a
Damping of vibrations in the ear, 142c to 143d
Danoe music in form of madrigals and motetts
in Aj>. 1529, down to J. S. Ba^ and Handel,
245a
[Deoad, harmonic, or Unit of Harmony, 4596]
[de Cans, Solomon, 509c]
Deep tones require more power to make them
audible than high tones, 174&. experiments
dewing how weak deep tones are, 1756. below
iiovib. are more or less discontinuous on the
siren, 178a. jar on harmonium below 132
vib., 178a. produced by a string weighted in
the middle, 176a. 37^ vib. weak, 29^ vib.
scarcely audible, 1766. Professor Helmholtz,
vrith large forks of 24 to 35, and 35 to 61
vib., found 30 vib. weak, 28 vib. scarcely audi-
ble, 176c. on Appunn's reeds, Preyer's ex-
periments, I76d. Author's conjecture, I76d\
{Translator's experiments, lyed']
XNsepest musical tone, how many vibrations it
has, 174a
Deepest practical orchestral tone, 41} vib.,
t7SC ,
[Degenhardt, 5026]
[de la Fage, 494£f, 509a, 510c]
de la Tour, Cagniard, his siren, 12c. [494^',
SoSc,d]
Delaitre, ssad
de la Motte Fouquet, 105a
Delesenne shows that first-class violinists play
in just intonation, 325a, ^3250, title, [his
monochord, 44id',494d', 508a, d, d]
de Muris, Jean, his consonances, excludes
Fourths, 1960
[Demuth, organist, S4Sd\
[de Prony, 494^', 5080]
de Vitry, Philipp, his consonances, excludes
Fourths, 196c
Diapason normah i6<2, [512^'}
[Diapason work of organs, 93(2']
Diaphony of Huobald, 24^
[Dichord, 5236]
[Diohordal or double diatonic scale of H. W.
Poole, 344«> 477c to 478c]
Didymna included major Third 4 : 5 in the
syntono-diatonic mode, 228c. his tetrachord,
263a
Differential combinational tones (Surge's and
Tartini's) , 1 53a. of usual harmonic intervals,
154a. generated by upper partials, 154c. of
different orders exemplified, 155a, 6, c, [cal-
culated, iSSd], form a complete series of
harmonic partial tones up to the generators,
155c. [influence of this on consonance of
simple tones, iSS^\ 537^*] their reinforce-
ment by resonators, often small and dubious,
I57d, may then arise from vibrations com-
municated to resonator by drumskin, 158a.
of the first order, their beats, 198a, b» beat
only when upper partials beat and with the
same frequency, 199a. of higher orders, their
use in distinguishing Fifths and Fourths of
simple or slightly compound tones, 201 . [530a]
[Digitals a finger-keys, 50^']
Dixninished Seventh, its chord and transform-
ations of the same, 345c, d. [its chord has
the just form, 10 : 12 : 14 : 17, considered,
346c, d. its transformations considered,
346c', d',] its usual just chord, 3496
[Dionysius of Halicamassus, 24fid]
Direct system, its chords, 342c
Direction, the sensation of, is partly due to
muscular sensations, 63d
Discant at end of eleventh century in France
and Flanders, its nature, 2446. develops poly-
phonic music and musical rhythm and ca-
nonic imitation, 244c
Dissonance defined, 194&, 204&. how it arises,
330c, whither it tends, 330(i. different for
different qualities of tone, 205a. how cha-
racterised, 226c. and eonsonanoe, their boun-
dary, 2286. not been always the same, 228c.
unprepared, 3540
Dissonant chords imperfectly represent com-
pound tones, 3466. dissonant notes, ^6c. es-
sentially, why used, 3536. intervals, why so
called, 18 ic. considered, general view, 33 id
to 333d. notes of chords of the Seventh, con-
sidered, 3476 to 3500. tetrads, 341c. tones how
introduced, 3536. triad C E A\> or C E 0%,
213d, [2i4dr]. triads, 3386
Disturbance of consonances by adjacent con-
sonances, meaning of this expression, i86d
Division of small intervals into equal parts by
ear, 256a
Dogs very sensitive to high e"" of violin, i i6d
Dominant. Seventh, its chord not used in 1 6th
century, 246^2. [Duodene, 461^2]
[Doncaster, Schulze's organ at, 96(2]
Donders, first draws attention to noises attend-
ing vowels, 67(2, io6cf, first discovered vowel
resonances, *io86. how he estimated them,
io8c. his vowel resonances compared with
Prof. Hehnholtz's, 1096, [♦i6&2T
Donkin's Acoustics, 377a
Doric, national Greek scale, 242a. its scale,
267c, 305c. Glarean's, or ecclesiastical, 245c.
considered as the mode of the minor Seventh,
303c
Double Octave an absolute consonance, 194&
Double Siren, see Siren double
Dove's polyphonic Siren, 13a, 14a
[Driffield, Bev. G. T., 494(2, 5056]
[DriUed Beeds, 5556]
Drum and Drumskin, set Ear
Du Bois Beymond, sen., his vowel trigram,
1056, ♦io5(2
Ductus ooMearis, lyjc
[Duiffoprugcar (Swiss Tyrol ,Bologna,and Lyons,
1510-1538), resonance of his violins, 876, 0
Digitized by ^oOJOgie
S62
INDEX.
[Duodenal, its xneaiiing and use, 4656]
[Duodenarium, the, 463a. how constructed,
463(2. limits of, 4646J
[Duodenation, 462^2]
[Duodene, hannonio, or Unit of Modulation,
461a]
[Duodenes, musical, or the development of just
intonation for harmony, Sect. £, see con-
tents, 457<2]
E
Ear, its analysis of musical tones according to
the law of simple vibrations, 496, 52a. espe-
cially sensitive from 2640 to 3163 vib., e"" to
g''"y I i6a. consequent effect when bass voices
sing, 1 1 66. how it apprehends and analyses
compound tones, i2Sc. construction of, de-
scribed, 129c to 142a. its labyrinth, I2gdt
135(2 to 137a. fluid of the same, 136&. its
membranous labyrinth, i^6d, its hammer,
or malletis, figured and described, 13 1&. its
anvil, or incuSy 132^2, and its anvil's joint with
the hammer, 133&. its stirrup, 133^2. figured,
134a. its stirrup's attachment to the oval
window, 134a, and excursions, 134&. its sac-
cuius t^Z^' its ampullcBti^Cd, its utriculuSf
136(2. its sand, 137a. its cochlea, 137& to
142a. sensitive tor f"'t and its consequences,
179a. contrasted with eye as to capabilities
of perceiving waves, 29. compared with eye
in its apprehension of compound vibrations,
128c. ear can analyse, eye cannot; eye can
distinguish all forms, ear can only distinguish
those which have different constituents, I28d.
compared with eye in analysis of compound
sensations, I48<2, and for intermittent irrita-
tions, 173a, 6, c. compared with muscles for
intermittent irritations, 173&. [compared
with eye and muscles in timing a transit,
I73<2, (2'.] its windows, see Oval and Bound
Ecclesiastical scales of Glarean, with incorrect
Greek names, 245c, d
[Edison, T. A., his phonograph, 539a]
Egyptian flute, interpreted by F^tis, 2'jid
Eight-stringed Scales, Lydian, Phrygian, Doric,
Hypolydian,Hypophrygian (Ionic), Hypodorio
(Eolio or liocrian), Mixolydian, 2670. [the
same with the intervals in cents, 268<;]
Ekert, •307(2'
Eleventh not so pleasant as Fourth, 1895
[its partials compared with those of the
Fourth, i89(2], 196a
[Elliot, 507a]
Ellis, A. J. [♦i6(2', 17(2, 56(2, 66(2', ♦68(2, ♦68(2',
♦ io5(2', 1 140^, ♦ 147(2. his Alphabet of Nature^
1845, first makes mention of Willis's and
Wheatstone's experiments and theories,
♦11 7(2, ♦191&', 390a]
Energetic tone of voice, how produced, 115c
Engel, G., ♦ii2(2
[England, pitch, 505(2]
Enharmonic confusions occur in just intona-
tion, 327^. [organ of Gen. T. P. Thompson,
473c]
Eolic mode, Glarean's, 245(2
Eolic (Hypodoric) Greek scale, 267c
Equal temperament, circumstances favourable
to it, 3226. first developed on the pianoforte
where much favoured, 323(1^. its defects on the
organ, 323c, and harmonium, 324a, on violins,
3246. not used in double stop passages, 324c.
its influence on musical composition, 3276.
Mozart and Beethoven, 327c. [its cycle, 4366.
its intonation, 437c. its synonymity, 4386.
its history, 548J
[Erard, 507(2]
Eratosthenes, his method of tuning the older
chromatic tetraehord, 262c
Erse have learnt heptatonic Scales, 2586
Essential scales, 2676
[Esser, 504^2]
[Estdve, 436(2']
Esthetics, Musical, i&
Esthetic principles modify physical in the
formation of scales, 234 to 236. analysis of
works of art, 3666
Esthonian treatment of leading note, 288a
Euclid on consonance and dissonance, 226(2,
[523a]
Euler, Leonard (1707- 1783), on law of motion
of strings, 15a, on why simple ratios please,
1 5&. his theory of consonance and dissonance
founded on integers explained, *22gd to 2316.
the gap he left filled by Prof. Helmholtz, 231c
[his determination of pitch numbers by a string
and formula, 441c. 494/2'. 5iO(2]
Eustachian tube, 1306
[Ewing, J. A., 1 1 8(2'. analysis of vowels ^7
Phonograph, 538^542, see Jenkin]
Exner, S., *i5i(2, ^3726
[ExperimentfU Instruments for Just Intonation.
4666 to 483c, see contents, 4666. pipes, 506c]
Extreme sharp Sixth, its chord and Greek
Doric cadence, 2866, 308c
Eye contrasted with ear as to oapabilitiea of
perceiving waves, 29
Fagotto, see Bassoon
False (Midence, 356c. relations forbidden,
36i(2, their meaning, 362a, often found in
J. S. Bach's chorales, 3626
Farabi, same as Al Farabi, (2. 950, 282a
[Faraday, 5056]
[Farey, J., sen., controversy with James Broad-
wood, 548c']
Fessel, I22(2, 377c, his resonance tubes, dimen-
sions of, 377^
F6tis,*239(2,*240(2',*257(2. adopts term &maZ»/y,
240&. on pentatonic scales, 257c 271(2. his
interpretation of an Egyptian flute, 271(2,
2800'. [his story of Lemmens, 28o(2'j
Fifth, 14S, not sensibly disturbed by adjacent
intervals, i88c. [its partials compued, i88(2.]
a perfect consonance, 194^. of simile tones
delimited by beats of differential tones, aoo(2.
repetition in it presents new elementa, 2546.
used in modem music, 254(2. occurs in the
scale, 255a. [indicated by ±, 276(2'.] grave
or imperfect and just, 3356. false or di-
minished, 335c. superfluous or extreme sharp,
335(2. see also Subminor Fifth
Fifths, consecutive, why forbidden, 3590, 36o(2,
3616
fifths and Fourths consonant and dissonant
enumerated and considered, 3356
Fifth and Fourth triad, 3386
Fifth and Sixth triad, 3386
Films of glycerine soap and water for shewing
vibrations of air in a resonator, 374
[Finaly called Umic in text, 267(2j
Final Chords, see Closing Chords
[Finlayson, 5<X)a]
[Fischer, 494(2', 508c, d]
[Flatter or lower tones defined, i id']
Floroke, on vowel resonance, io8c', iogc
Digitized by V^OOQIC
INDEX.
5^3
Flue-pipes of an organ open and stopped, S&bt
c. [their pitch according to Cayaill6-Goll, 89c,
€2. effect of temperature on pitch of, 89(/, d'J]
motion of air inside, for open pipes, 89c to 906,
for stopped pipes, 90c. their reduced length,
91 6. distance of plane of reflexion from the
end of pipe according to Prof. Helmholtz, 91&,
[according to Bosanquet, Loid Bayleigh, and
Blaikley, gid], motion of air at mouth, 926.
'narroii70r stopped cylindrical, have proper tones
corresponding to the unevenly numbered
partials, the toider not so, and henc^ give
prime tone almost alone, 94a, the blowing of
them, 394a to 3966. see Organ pipes, flue
flute pipes, see Flue pipes
Flutes, bad for harmony, joke on a Flute
concerto and concert, good in combination
with other instruments, 20$d, [old and new,
their intonation, ss$d' to 556a]
[Tolliano, L., invento the meantone, but not
the meantone temperament, 547<2]
[Foot, lengths of, in different countries, their
effect on pitch, $120]
force of tone, loc. its measure, lod, of sound,
its mechanical measurement, lod, [75<i']
Forks, tuning, their sympathetic resonance,
39^, 40a. [generally have the second partial,
54^. conditions of not having any partials,
55^'.] purified from secondary tones by ajar,
54£2, or a string, 55c. see also Tuning-fork
Forkel, ♦296^, 32 id, ^32 id'
[Forster A Andrews, 5006]
Forte and PianOt how produced on organs, 94c
Fortlage, 307c
Four-part chords, 2226
Fourier {1768- 1830), his law, 346. its acous-
tical expression, 34c. what it shews and
does not shew, 356. mathematically solves
Pythagoras*s problem, 229a
Fourth, 14^. chiefly disturbed by major
Third, 189&. its precedence over major Third
and major Sixth principally due to its being
the inversion of the Fifth, 189&. a perfect
consonance, 194c. why formerly not con-
sidered as a consonance, 196^. between two
simple tones delimited by beats of diffe-
rential tones, 20ca. mode of, Greek Ionic
Ecclesiastical Mixolydian considered, 302c.
[its predominance, 524^]
[France, pitch, 508a]
Franco of Cologne, end of 12th century, admits
Thirds as imperfect consonances, 190a. his
order of consonances, 196c
[Francis, 508&]
[Fraser, his organ, first commercially issued in
equal temperament, since burned, 5492/]
Free Beeds, 956. [in Chinese Shdng, 6sd.
not used in English organ pipes, 96c.
treated hj Mr. Hermann Smith, 554^]
French pitch, i6d, [Commission on pitch,
494(2'. Sixth, 461c]
French horn, see Horn
Frequency defined, iia, [iid]
Fullah negroes have pentatonic scales, 257c
Fundamental major and minor chords, 21 2d,
bass, 294c
[Furstenau, M., 4996]
Gabrieli Giovanni of Venice, composer, con-
temporary with Palestrina, 247c. what we
miss in him, 2486, 296a
Gaels have learned heptatonio scales, 2586
[Pafori, treatise on music, 1480, *si2d]
Galileo (1564-1649) on laws of motion of
strings, 15a
[Galin, P., his book and system, 425^. adopts
Huyghens's cycle of 31, 425^]
Galin-Paris.Chev6 system of teaching singing,
42 5c. [its history and principles, 425^ note f]
Galleries of cochlea, 137 6
[Gamelan or Javese band, how it plays, 526c]
[Gand, 509a]
[Gardiner, Tonic Sol-fa teacher, 427c]
[Gameri, G., or * Joseph ' (Cremona, 1683-
1745), also called Guamerius, resonance of his
violins, 87c]
[Gameri, P. (Mantua, 1701), resonance of his
violins, 87cQ
Oeigen-principal organ stop, 93a, c, d
Oemshom organ stop, 94a, [94^]
Gerbert, ♦196^
[German peculiarity of consonants, 66d, d',
habit of beginning vowels with the check or
Arabic hamza, I04£2'. Sixth, 461c. pitch,
5096]
[Gewandhaus concerts at Leipzig, pitch, 510&]
Glarean sometimes allows tenor and bass to
be in different keys, 2 \$d, his Dodecachordon
and its order of consonances, *i96c. his six
authentic and six plagal scales with false
Greek names, 245c, d, his names of the
modes, 269a
Glass harmonioon, 71a
[Glazebrook's electric method of determining
pitch, 4426]
Gleitz, organist, on Erfurt bell of 1477 and its
tones, *j2d
Glottis, 98a
[Glover, Miss Sarah, starts the system of
teaching to sing developed as Tonic Sol-^
by John Curwen, 424a]
[Glyn & Parker, maker of HandePs Foundling
Hospital organ, So$d]
Goethe, 1749 -1832, relied on mixtures of pig-
ments, 64^
Goltz, his investigation of the ciliie of am-
pullffi, leads to suppose that they and the
semicircular canals serve to give sensation of
revolution, 151 6
[Got, M., pronounces oi«i without voice, 68d]
Goudimel, Claude, a Huguenot, master of
Palestrina, 2476
Graham, G. F., *258<2', 2&od'y 26id
[Grave harmonics = combinationid tones, 153c
note]
[Gray, Dr., helps J. Broadwood with divided
belly. bridge, 77(f|
[Gray <fe Davison, 506c, 507a. in 1854 first
send out an organ in equal temperament,
549&Q
[Great Exhibition of 1851 had no English
organ in equal temperament, 549a']
[Greatorex, 496c]
[Greece, old tetrachords, 5i2<2, 519(2. ditto
afterAIFarabi, 512^,519(2. scales, 5 14 to 5 15.
most ancient, 51 56. later, and Al Farabi's,
5«5c. Si9d]
Greek Music, 237. tonal system, 262 to 271.
later scale, 270c, &. [scales, 514 to 515a]
Greeks had a certain esthetic feeling for
tonality, but tmdeveloped, 242c
[Green, 505(2]
Gregory, Pope, a.d. 590-604, his settlement of
the Liturgy, little more than established the
Boman school of singing of Pope Sylvester,
239a. inserts accidental scales among Am-
brosian essential, 27 id]
[Griesbaoh, J. H., 499^, 5056] , ^.^^T^
Digitized by Vo:«ILiOy IC
5^4
INDEX.
[Grove, Sir O., ♦87(i']
[Orundton^ fundamental tone or root of chord,
Goadanini, violin by, 85c
[Guamerios, P., Sy&, see Gameri]
Gu6roult, 165^, note and 414c. [4^7^]
Guido d' Arezzo, b. about 990, 351c.
[Guillaume, 510^
Guitar, 746
[Halberstadt organ, 1361, oldest pitch ascer-
tained, 51 1&]
[HalU, Ch., 502c]
HaUstroem considered multiple combinational
tones to be of different orders, * 154^2, 413a
Hammer, moved by water-wheel, its periodic
motion, 19c, 21c. soft and elastic for pianos
complicates tiie problem, 74c, a sharp-edged
metalho, rebounding instantly, excites but one
point and produces numerous partials, some
more intense tiban the prime, 75a. of piano-
forte, why felted, 75c. hard and soft, their
different qualities, 78c. see also Ear
Hammer-Purgstall, von, on Arabian Music,
3810, <2
Bdnt&hu of cochlea, 137&
Handel sometimes concludes a minor piece
with a major chord, 21 76. his suites, 245a.
his use of closing minor chord, 295a, 2g6d,
his use of the mode of minor Sixth, 307a, 6,
[takes a chorus from Carissimi, sojd notef .
his fork, 496c, 5056]
Hanslick, £., on tiie Beautiful in Music, 26,
2Sod
[Harmon, Mr. J. Paul White's, for the 53 divi-
sion of ^e Octave, 481 6]
Harmonic music, modem, how characterised,
246c. inducements for the change, 246c.
distinguished from polyphonic by the in-
dependence of chords, 296a. relationship
began in the middle ages, 368c. Seventh, see
sub-minor Seventh, upper partials produced
by the same peculiarities of construction of
a body as allow combinational tones to be
heard, 15&I, 1590. these always accompany
a powerful simple tone, 159c. upper par-
tiiUs, why they play a leading part in the
sensations of the ear, 204a
[Harmonical, an instrument for musical ob-
servation, 6c. a specially tuned harmonium,
its price and compass, 17&, c, d, useful to
shew the existence of partial tones, 22d,
useful for shewing increasing frequency of
beats with increasing pitch, i$Sd. full de-
scription, 466^2]
Harmonicons of metal, wood, and glass,
7ia,&,c
[Harmonics, how they differ from harmonic
upper partial tones, 24^'. defined, zsd,
of C66, table of partials of the first sixteen,
to shew how they affect each other in con-
sonances, 197c, d, of a string 45 inches long
struck at ) length, 78^. partials of a piano-
forte string struck at one-eighth its length,
545c. of a violin or harp, and fading har-
monics of piano, 24(2'. Seventh and Seven-
teenth introduced into harmony, 464c
Harmony, of the spheres, solely heard by
Pythagoras, 229c, and plays a great part in
middle ages, 22gd. its modern principle,
249a, 249c. not natural, but freely chosen,
2496, gave rise to a richer opening out of |
musical art, 369a. [absent in non-harmonic
scales, 526c]
Harmonium, its reeds, 956, 554a'. with 24 notes
in just intonation, invented by Prof. Helm-
holtz, its system of chords, 316c. its meUiod
of tuning, 316^'. [its duodenary arrangement,
3170.] its system of minor keys, 318a and d,
its contrast with tempered, 319c. [just, de-
scribed, 470a]
Harmoniousness of combinations in different
qualities of tone, 194a
Harp, 74&. with pedal, 3226
[Harper, trumpeter, and his son, had a slide
trumpet, loocT^
Harpsichord [its striking-place, 77^
[Harris, B., arches the upper lip of flue-pipes,
397^', 505^
[Harris, T. and B., 505^]
Harmonisation, the only point in which
modem excels ancient music, 309a
[Hart, violin-maker, assists in finding reso-
nance of violins, Sje]
[Hartan, monochord, 442a]
[Haseneier, maker of Dr. Stone's contrafagotto,
553&Q
Hasse, C, proves that birds and amphibia have
no Gorti's rodb, I45(2', i^Sd
Hauptmann, objects to a theory of consonance
and dissonance by rationiJ numbers that no
sharp line can be drawn, *22yd. Prof. Helm-
holtz's reply, 228a. his Pythagorean Sixth
in minor scale not allowed, *2';$d, his nota-
tion for Fifths and Thirds, 276^2, 277c. his
reason for avoiding closing minor chords,
295c, 295(2', 3ioe2'. his opinion on the Second
of the Scale, 298^2. his minor major mode,
305 6. denies that there is any difference of
character to keys played on an organ, 3iodL
his system of tones, 315c, 340a. on consecu-
tive Fifths, 3602, 3616
Hautbois, see Oboe
[Haweis, Bev. H. B., assists in finding reao-
nance of violins, 87c]
[Haydn, 54801
[Healey, Mr., assists in finding resonance <d
violms, 876', </]
Heidenhain's Utanomdtort its action, 139a
HiUcotrSma of cochlea, 137&
Hehnholtz, Prof., *6c\ i&2. his Optics, iSd',
♦91c, ♦9&2', io5<2, 106c', 109c, nod, d\ •iiid,
117(2, 123(2, *i34d, ♦iS2(2', 153(2, 195(2', 23&2'.
his names for the modes explained, 269c.
[taken to hear Gen. P. Thompson's organ by
the Translator, 423c', and taken by ihe same
to Mr. Gardiner's School to hear Tonic Sol-ta
singing, 427c. his letter to J. Curwen about
it, 427^
[Helmholtzian Temperament, 435c]
[Helmore, Bev. T., on Gregorian modes, ^26^
Heilwag, Gh., loScf, [on vowel resonance, 109c]
Hemony of Zfitphen, 17th century, his require-
ments for bells, 72(2
[Henfling's cycle of 50, 436(2']
Henle found where greatest increase of breadth
of Gorti's rods fell, 146a
Henrici, estimates upper partials of tuning-
forks too low by an Octave, *62a
Hensen, V., researches on the basilar mem-
brane, 145c, * 1 45(2'. auditory apparatus of
crastaceiB, ^1490
Hensen, 4o6d
Heptatonic Scales, 2626
[Herbert, Geo., had organ tuned equally, 5490'^
Herschel, Sir J., iiyd'
Hervert, J., *93(2'. see Mach
Digitized by V^OOQIC
INDEX.
565
Sexaohord of Guido d*Arezzo, 3510
lUichi-riki, Japanese, its reeds, 553^1
Sidden Fifths and Octaves, 361c. often found
in J. S. Bach's chorales, 3626
High voices more agreeable than low, and
why, 179c
fHildebrand of Dresden, 495c]
[Hill, violin-maker, assists in finding resonance
of violins, 876, c]
[Hill, Thomas, organ-bnilder, his father tried
and abandoned free reeds, 96c. 494^', 506c]
[Hipkins, A. J., on harmonics of a Steinway
piano string, 76^2. on striking-place of
pianoforte, hai^sichord, and spinet, 77c, note,
had not met with a striking-place at | leng&,
yydf, his experiments on harmonics of a
string struck at ^ its lemrth, *ySd. and fur-
ther experiments on its partials, *545c.
assists in finding resonance of violins, 87c',
i83{f, 2ogd,d\ monochord, 442a, 507^. assists
in determining non-harmonic scales, 51461 c,
on James Broadwood's equal temperament,
549a. introduces equal temperament at
Broad woods', 1846, 549c]
[Hitchcock's, early, i8th century, spinet, its
striking-place, jyd]
[Holland, pitch, 5106]
Uomophonic music, 237 to 243
[Hopkins, E. J., on the organ, cited, *33<2, Syef,
d', 93c', d, 94d, d', 96d, 49&i, 5496]
[Hopwood, of Kensington, his organ built by
Cavaill6.Goll, 96^2]
Horn, function of the air in its tube in rela-
tion to lips, 97c [after Blaikley, 97<2'], long
nearly conical tube, 99a, [not quite conical,
99^* gives the harmonics only, 99a [upper
ones true, 99^1. action of hand in the bell,
looa. [2 lower harmonics false, looc. its
various lengths for diHerent crooks after
Blaikley, and error in reporting Zamminer,
iQod.] keyed horns, icx)a, [nearly obsolete,
lood']
[Hewlett, 496c]
Hucbald, Flemish monk, at beginning of loth
century, 244a
Hudson's Bay, pentatonic scales, 257c
[Hnggins, Dr., F.B.S., his observations of effect
of increasing tension of hairs in violin bow,
83^2. on position of touch for Octave har-
monic, *84(2. on function of sound post of
violin, *86c, d, assists in finding resonance of
violins, 87c. the resonance of his Stradivari
of 1708. %^d']
[HuUah, 239d, 499^. 505&]
[Hutchings, Plaisted, A Ck>., 51 1&]
Huyghens, 1629- 1695 [knows that striking-
place of string affects quality of tone, 77a.
his harmonic cycle, 436c]
Hypate, uppermost string in position, lowest
in pitch, answered to dominant, 242a. Greek
music ended on it, 2426
Hypodoric (Eolic or Loorian) Scale, 2670
Hypolydian Scale, 2670
Hypophrygian (Tonic) Scale, 267c
InciUt 8ee Ear
[India, Chromatic and Semltonic Scales, 5170.
partial Scales of Bajah B4m P&l Singh,
Siyd]
[Indian Quartertones, how produced, 265^2']
Indirect or Beverted System of Chords, 3420
Instrumental tones accompanied by distinctive
xkoises, 6S6.
Instruments with inharmonic proper tones not
suitable for artistic music, 73c. effect of
bowing and damping them, 73^
Intercalated tones, always Semitones, 3520
Interference of sound, 160
Intermittent excitement of nerves, its effect
on ear and eye, 169& to 170c
Intensity of sound, how measured, y$d [Bosan-
quet's, with Preece and Stroh's opinion upon
this measurement, 75^, d']
Interval, [a sensation, measured by ratios or
cents, 13d, their names distinguished in print
by capital letters, lyd'.] of Fifth and Fourth,
14a, major and minor Third, 146, major and
minor Sixth, 14c. all hitherto considered so
arranged that a pair of their partials shall beat
33 times in a second, 191a, 192a. [all, except
Tnirds and Fifths, indicated by ... or the
number of cents, 276^'. not exceeding an
Octave, expressed in Cents, Sect. D, see pre-
fixed table of contents, 45id. neutral, 525a,
alternative, 5256]
[Intonation, Tempered Pythagorean and Just
compared, 313c. unequally just, 465a. of
vocalists and violinists, 486 to 7. of flutes, 555
d' to 556a]
Inversions, first or chord, of Sixth and
Fourth S, second or chord of Sixth and Third,
.^, 213a. on what their harmonious effect
depends, 214a
Inwards striking reeds, 976
Ionic, Glarean's, 245c. Greek, considered as
mode of the Fourth, 302c. (Hypophrygian)
Scale, 267c
[Ions, 496^]
Irrational intervals, (Uoyo, 264a
[Isawa, director of the Musical Institute, Tokio,
Japan, 522^, 526c', 527a', S5^^]
Italian melodies rich in intercalated tones,
352c. [Sixth, 461c]
[Italy, in 1884 officially adopts l/bAS^ "Pre-
senting the arithmetical pitch c"5i2, 497<2,
49Sa, prior pitch, 510c]
[Japan, koto tunings, heptatonic scales,
Biwa, 519a, 6, c, 522a', V, d. scales, ^S^ih^^
[Java, music, 7 id, 237^2'. its pentatonic scales,
257^2. Salendro Scales, 518c, Pelog Scales,
518c, (2, 5226]
[Jehmlich, 4996, 509(2]
[Jenkin and Ewing, ii&2'. analysis of vowels
by Phonograph, 538 to 542]
[Jimmerthal, 502c]
Joachim uses 4 : 5 major Thirds in melody,
255(2. plays in just intonation, 325a
John XXH., Pope, a-d. 1322, forbids use of lead-
ing note, 287c
Jones, Sir William, presumes antfa to have
been the Indian tonic, ^2430, 6, c. see Y6di
[Jordans, 5066]
[Josquin, 225(2, 296(2]
[Jots, cycle of 30103, 437al
Just intonation instrumentally practicable,
327(2. in singing, 422 to 428. [extreme close-
ness of its representation by the cycle of 53,
3296, c] the protest of musicians against
it arises from their not having methodically
compared Just and Tempered Intonation,428c,
possible in the orchestra, 428c. natural, 428a.
according to Delezenne, 4286. [expressed in
cycle of 1200, 440. experimeDtal instruments
for exhibiting, App. XX. Seot» F, 4666 to 483c.
zee contents, 4666]
Digitized by V^jOOQ IC
566
INDEX.
Jnstly intoned Harmoniums, with two msnnals
arranged by the Author, 316c to 319c. instru-
ments necessary for teaching singing, 327a.
instruments, plan for them, with a single
manual, 421a to 4226
K
Eeppler could not free himself from musical
imagination, 22gd
Kettledrums, their secondary tones noi inves-
tigated, 736
Key of polyphonic composition different for
different voices, 245(2. [acoustical effects of
change of key, 55ic2.] Keynote, see Tonic
Keys, have they special characters? 310c.
[their presumed characters, 550c]
[Khorassan, Tambour of, its scale, after Prof.
Land, 517&]
Kiesewetter, R. G., 236^2, *28i&, c, 282^'
Kirchengesang, or congregational singing,
•287^'
Eircher, Athanasius, finds both macrocosm
and microcosm musical, 2290
[Kirkman's harpsichord, 1773, its striking-
place, 7 yd]
^mberger tunes Bach's major Thirds sharp,
321C
[Kirsten, 5096]
Eissar, five-string lyre of North Africa, penta-
tonic according to Villoteau, 257c
[Klang, 24c]
Koenig, B. [when tuning-forks have no
partials, 55(2', io6df. on vowel resonance,
♦109c', *iogd\ i22d', on influence of dif-
ference of phase on quality of tone, I26<2.]
with short sounding-rods has shewn that
tones with 4000 to 40000 vibrations in a
second can be heard, 151c, iS2d, 15912, d\
1 67(2. recent work on combinational tones
and beats, I52(2.] experiments on forks with
sliding weights, his results, 159&, [i 59(21, 176c.
makes resonators, 372(2. his manometric
flames, 374(]^, 5. [his clock method of deter-
mining pitch, 442c. his tuning-fork tonome-
ters, price of various kinds, 446a, &, 494(2'.
on beats and combinational tones, 527 to 538]
Kosegarten, J. G. L., 282(2'
[Krebs, 510a]
[Eummer, 509&]
[Kfitzing, C, 1884, gives ^ length as suitable
striking-place, but had met with |, *77c'J
[Land, Prof. J. P. N., Gamme Arabe, ♦28o<j',
*28ic. his account of the Arabic scales, 284
note. Al Farabi's Greek Scales, 515c. Arabic
and Persian, 5 1 $d. Ditto Medieval and Ancient
Lute, 5 1 66. Tambours of Khorassan and
Bagdad, 5176, c, 5236^
Larynx, action of, 98a
Later Greek Scales, 270a, 6, e
Leading note, conception of, 2856, between
Seventh and Octave, not between Third
and Fourth, 286a, but between Second and
Third in the minor mode, 2866. not marked,
but sung, even in Protestant churches, to
1 6th and 17th centuries, 287^. found in
Cantus firmuSf 2870. not sung by Esthonians
even when played by the organ, 288a. exists
only in two tonal modes, Greek Lydian and
Hypolydian, 288a. causes alteration of
Greek Phry^an mode to the ascending minor
scale, and Greek Eolic to instrumental minor
scale, 2886. its general introduction leads to
development of feeling for the tonic, 2886, c.
effect of excessively sharpening, 315a
Leaps, when they are not advisable, 3556
[Lehnert, 509c]
[Leibner, 508c]
Leibnitz, 1646- 17 16, perdpirt and appereipirt
» synthetically and analytically perceived,
62(2, [d']
[Lemmens, jMreferred false intervals as a child,
F6tis's story, zSod']
[Lemoine, $oSb]
[Lewis, 5<i6c, 507a]
Lichaon of Samos, 266e
Light and Sound, analogies of their oompass,
186. [light extends over an Octave and a
Fourth, 18c', (21
[Linear temperaments, 433a]
Lips, as membraneous reeds, 97c
[Lissajous, 496(2', and Ferrand, 508c]
[Listen, Rev. Henry, his organ, 4736]
Liturgy, Roman, its singing, 239a
Locrian (Hypodoric) Scale, 267^
Low tones, see Deep tones
[Lupot (France, 1 750-1820), resonanoe ol his
violins, Sjc]
[Lushington, V., and daughter, assist in find-
ing resonance of violins, Syd]
[Lute stop of harpsichord in i8th century, 77<2]
Luther, his feelings on music, 246c
Lydian Scale, Glarean's, 24$d
Lydian Scales, Greek, 267c
M
[Macfarren, Sir G., sees right (just intonation)
through wrong (tempered intonation), 346^^.
writes sight test for Tonic Sol-fa FeBtival,
4270'. on character of keys, 550c, standard
pitch, 555c]
Mach and J. Hervert's experiment with gas
flames before the end of open flue-pipes, 936,
♦93(2'
[Macleod, Prof. H., and Lieut. Clark, their opti-
cal method of finding pitch, ^4426, 494(2^
Madrigals, 244(2
[MahiUon, V., on Boehm's flutes, *io3(2', yxid',
504(2]
Major chords used as a close to minor modes,
296(2, 297a
Major or Ionic mode, its harmony well de-
veloped in 1 6th century, 246(2. gives full ex-
pression to tonality, 293(7. its harmonic
superiority, together with the minor mode,
298c. considered, 302
Major scale, ascending, 2746
Major Seventh, 337a. its chord in the direct
syBtem, 349c
Major Sixth, a medial consonance, 194c
Major Tenth better than major Third, 1956
Major Tetrads, their moet perfect poeitioDa,
223(J
Major Third a medial consonance, 194c not
easily distinguished by differential tones,
200c, [(2']. just, with the ratio 4 : 5 indnded
by Didymus and Ptolenueus in the syntono-
diatonic mode, but not recognised as a con-
sonance, 2280. that it was not considered a
consonance was due to the tonal systems
used, 228(2. not clearly defined and doobtfol
in pure melody, 2$$d [indicated by -h after
276(2.] just, 3346
Major Thirteenth worse than major Sixth,
Major triads, their combinational tones gene-
Digitized by V^jOOQIC
INDEX.
567
rally saiUble to the ehord, 215&. their most
X)erfect positions, 219c. their less perfect
positions, 220c
JlialliuSt see Ear
Marloye makes pipe with additional piece a
half length of wave, for experiment on plane
of reflexion, gid [explanations, gidf, 504£2,
508a, 5 IOC]
Marparg quotes Eimberger on equal tempera-
ment, 32 ic, [494^', 5106]
[Mason and Hemlin, 511a]
Maiheson, his Critica Mtaica, 1752, *32ie.
says Silbermann*s unequal temperament is
best for organs, 323c, [So^d]
[Maxwell, Clarke, fundamental oolours, 64/f]
Mayer, Prof. A., observations on the duration
of sound and numbers of audible beats, 417c
and note, [his eleotrographio me^od of de-
termining pitoh, 4426. his lecture-room ana-
lysis of a reed-tone, 549c. his harmonic
curves, 5506]
[Mazingue, 5086]
[Mazzini (Brescia, 1 560-1640), resonance of his
violins, 87c]
[Meantone Temperament, 433^. its history,
546e. used in 1880, in Spain, in Greene's
organs, Ac, 549cT
Meatus auditorius, 130a
Mechanical problem of transference of the
vibrations of air to labyrinth of ear, 134^2
[Mediant Duodene, 462a]
f Meeson, his elliptical tension bars, 552rf]
Melodic relationship, original development of
the feeling for, 3686
Melodies in modem music supposed to arise
from harmonies, 2536
Melody is not resolved harmony, 289a. in
simple tones, how appreciated, 2896 to 290&.
expresses motion appreciable by immediate
perception, 252a. goes far beyond an imita-
tion of nature, 371a. does not arise from
distinctive cries, 3716
Membranes, circular, their sympathetic re-
sonance, 40c to 41 c
Membranes, stretched, their inharmonic proper
tones, 736
Membranous tongues, how to make, 97a.
[reeds, 5546]
[Mental effects of each degree of the scale,
279^ note]
Mental tune « OemilthsBiimmung, 2$od
Mercadier, see Ck>rnu, *Z2Sd
Mercator*s cycle of 53, 328c, [436c]
[Merkel, io6c'. on whispering, *io84'. his
comparison of all opinions on vowel reso-
nance with comparative table, * 109c, (2, nod]
[Mersenne, 494(2, 508a, 6, d, spinet, 509a. on
equal temperament, 548c]
Mesl^ middle string, answered to tonic, 242c
Meshaqah, ^2646', 26$d. his modem Arabic
scale of 24 Quartertones, 2646, 285^', 5256, d
Metal reeds, 986
Metallic quality of tone, 71 6
[Meyerbeer, 499c]
Middle-tone, see MesH
[Millimetres, their relation to inches, 42^
Minor chord, its treatment by older com-
posers, avoided at close, 217a. its use at the
dose marks the period of modem music, 365a.
its root or fundamental bass ambiguous, 291c.
how avoided, 295a. on the second, a real
modulation into the subdominant, 299a
Minor-Major Mode, Hauptmann*s name, and
its chain of chords, 305&
Minor mode arises fi*om fusion of Doric, Eolic,
and Phrygian in Monteverde's time, 248^2.
does not give full expression to tonality, 2946.
formerly closed with major chords, 296^,
297a. the harmonic superiority of this and
the major mode, 298c
Minor scale, ascending and descending forms,
2746, c. instrumental form, 2886. the effect
on its chords of using the major Seventh, 299c
Minor Seventh of scale always replaceable by
major Seventh, 2996, c. the effect of this on
the chords of minor scale, 299c. acuter or
^der, 336c, d, chord of the, in the direct
system, 349c
Minor Sixth an imperfect consonance, 194^2.
[its partials compared with those of subminor
Seventh, 1956]
[Minor Submediant Duodene, 4626]
Minor system inferior to major in harmonious-
ness, 301 6, c. this nowise depreciates its
value, 302a. its capabilities in consequence,
302a
Minor Tenth, much worse than minor Third,
196a
Minor Tetrads, their best positions, 2246
Minor Third, an imperfect consonance, I94<2.
[indicated by— from 276^.] Pythagorean, 3346*
just, 3346
Minor Thirteenth, much worse than minor
Sixth, 196a
Minor triad, false, 340a. [examples analysed,
340d.] not so harmonious as major triads,
214c. their combinational tones do not belong
to the harmony, 216a. effect of such tones,
2 1 6c. their most perfect and less perfect
positions, 22i5, c
Mixolydian scale, Greek, 267^. Glarean's,
245(2. considered as mode of the Fourth, 302^2
Mode of Fourth ascending form, 275a. of
minor Seventh descending form, 275a, ascend-
ing form, 2756. of minor Sixth descending
form, 2756. its chain of chords, 305c, d. its
relation to the major scale, 306&. traces of
its use by Handel, 307a, &. by Bach, 307c.
by Mozart, 308a, 6. by Beethoven, 30S6
[Modelet = ragiini, 525c^
[Modem Greek scales, 556^']
Modes, ecclesiastical, their system differed
greatly from modem keys, 247^2. Professor
Helmholtz's names for, compared with an-
cient Greek and ecclesiastical, 269a. an-
cient Greek, tabulated with the same initial
or tonic, 269a. the five with variable Seconds
or Sevenths, in the new notation, 277a'. see
Tonal Modes
MddVfilus of cochlea, 137&
Modulation in the modem sense, unknown to
polyphonic music, 246a. rules for, 328a. [into
the Dominant Duodene, 46i(2. into the sub-
domins^, mediant, minor submediant, rela-
tive and correlative Duodenes, 462]
Mongolian pentatonic scales, 257c
Monochord, 14^. [liable to error, 15a], 746.
used to train singers, 326a. [experiments
with, by Messrs. Hipkins and Hartan, with
Translator, 442a]
[Monneron, 497c]
Monteverde, Claudio, a.d. i 565-1649, invents
solo songs with airs, 248c. first composer who
used chords of the dominant Seventh without
preparation, 24842, 296a
[Montal, 4986]
[Moore A Moore, London, makers of Harmo-
nioal, 1 76]
Moore, H. Keatley, ytjd, 31 td, [says Handel
took a chorus from Carissimi, 307(2. explains
Digitized by
^.joogle
568
INDEX.
action of short keys of piano on the character
of the keys, Slid]
Motets, 245a
Motion measures power in the inorganic world,
26. periodic, illustrated, 86. undulatory, ga
Movements, musical, chiefly depend on psycho-
logical action, 2C
Mozart [uses pianofortes by Stein, 77^]. some-
times concludes a minor piece with a major
chord, 2176. his use of chords in his Ave
verum cormu, 225a. his use of closing minor
chord, 29&2, 2970. his Bequiem^ 2976. his
use of the mode of the minor Sixth, 308a, 6.
his Proiegga il giusto cielo seldom sung satis-
factorily, 326a. lived at the commencement
of equal temperament, 327c, [4966, 548<]
Mueller, Johannes (1801-58), starts physio-
logical acoustics, 46. his form of membra-
nous tongues, 976. his theory of the specific
energies of sense, 148c
Music, most closely related to sensation of all
arts, 2d. depends for material on sensation
of tone, 3a. had to shape and select its ma-
terials, 250a. expresses states of sensitive-
ness, 250c, 25 ic
Musical quality of tone, 676
Musical tones, 7c, 106, c, without opper
partials, 69c. with inharmonic upper par-
tials, 70a. of strings, 74a. of bowed instru-
ments, 80c. of reed pipes, 95a. on compound
tones are chords of partials and hence repre-
sented by chords, 3096
[Musician's cycle of 55, 436(2]
Mysis (or opossum shrimp) hears when oto-
liths are extirpated, 149(2. tuning of its audi-
tory hairs, 150a
[Naeke, Herr and Frau, 494^']
[Natural, the symbol Q , whence derived, 3X2(2]
* Naturalness ' of the major chord, according
to Rameau and d*Alembert, and insufficiency
of such assumption, 2326, c
Natural Seventh (see Subminor Seventh)
Nasals, M, N, N', their humming effect, have
peculiarities of U, 117a
Naumann, G. E., 276(2. defends Pythagorean
intonation, 2^40, *3i4d', 328(2
Neidhard, equal temperament, 170&, *32ic, df
Nerve force has only quantitative, not qualita-
tive difFerenoes, the different results depend on
the different terminals, 149c
Netherland system, its harsh polyphony, 2256
Neumann, Clem., simpler way of observing
vibrational form of violin string by a grating,
♦83(2'
[Neutral Intervals, 525a. Third, Prof. Land's
name for ZalzaPs Third, 2Sid]
New Caledonia, pentatonic scales, 257c
New Guinea Papuas, pentatonic scales, 2570
Newton (1642-.1727), on laws of motion of
strings, 15a
[Nichol's Germania Orchestra, 511 6]
Nicomachus, his comparison of the seven tones
to the seven heavenly bodies, 241a, [c\ d']. on
the old tuning of the lyre, 255c, 257(2', 266c
Nodes of strings, how to find ^ose on a piano-
forte, 47(2, 5(3(2, and how to touch, 51a. par-
tials of strings which have no node in a certain
spot are silenced by touching the string at
the node, 52(2. ^exceptions, 78(2, 5406]
Noise- defined, 7c, <2, "80. accompanying in-
strumental notes, 67c. perception of, by
cochlea, i5€)^
[Northcote, Miss, blind organist of Oen. P.
Thompson's Enharmonic organ, 4230^
Note, musical, its construction, 5c. lised for
any musical tone, 24a, d
Notation, new, for distinguishing the relatioDS
of Fifths from those of major Thirds, 276a to
2786 [substitute here used, 277c]
Numbers, what have the ratios of the first six
to do with music ? 2a, see Pythagoras
[Oberzahn, 25(2']
Oboe, its tongues or reeds, 966, 554c. has a
conical tube, and produces all haxmonioa, 99a
Observation, personal, better than best de-
scription, 6a
Octave, a collection of eight notes (uBually
printed with a small letter), onacoenied, or
4-foot, 15(2. once accented or 2-foot, and
twice accented, or i-foot, 16a. great, or 8-
foot, and contra, or 16-foot, and 32-foot, i6b
Octave. [Octave meaning interval (usually
printed with a capital letter), and octave,
meaning set of notes (usually printed with a
small letter), I3(2.] easy to maike the mistake
of, as Tartini, Henrici, and others, 62a.
gives no beats except those from partials in
a single compound, iSyd, same for double
Octave, 1 87(2. greatly distorts adjacent con-
sonances, 1 88a. an absolute consonance,
1946. of simple tones, distinguished by the
first differential tone, 1990. repetition in it
presents nothing but what has already been
heard, 2$yL not allowed in composition,
359c. why its key is identified with that of
the prime, 3296
[Octave divisions, their possible origin, 522(2]
Oettingen, A. von, his notation for relations of
Thirds and Fifths, 277c. its modification by
Translator here used, 277c note, report 00
Esthonian treatment of leading note in minor
scales, ^287(2. his minor system, *3o8c2,
3656
Ohm, G. S., 1 787-1 854, his law, 336, c, 76*.
completion of its proof, 56c. Seebeck's ob-
jections to it, *sSd, his experiment with a
violin to shew fusion of note and Octave, 606.
better form with bottles blown by stream of
air over mouth, 6oe
Olivier, ♦io8c'
Olympos, B.C. 660-620, his pentatonic trans-
formation of the Doric scale, 258a. his
ancient enharmonic tetrachord, 262c
Open pipes, see Organ pipes, open
Opera, one of the most active causes of deve-
lopment of harmony in 1 7th oentnry, 2486
[Organ-builders' measurement of octaves by
feet, l6d]
Organ pipes, wide stopped, unsnited for har-
mony, delimit consonance imperfectly, un-
snited for polyphony, their use, 2056. flue,
delimit Octaves and Fifths by partials, but
require combinational tones for the Thirds,
205(7. open, good for harmony and poly-
phony, 206a
Oigan, its compound stops have fewer pipes in
upper notes, 2106
Organ stops, QuintaUn, 33(2. Twelfth, 3312.
principal register ^ weiiigedacki^ gei^en-regiaUfi
quintaUn^ comet, compound, 57b. ^usqui-
altera^ comet mounted, 57(2'.} The musician
mnst regard all tones as resembling the ecai-
pound ^rgan stops^ s?^
Digitized by V^jOOQlC
INDEX.
569
Organiun of Haebald, 244a
Oscillation, length of, 86. period of, 86
Ossicles, see Auditory
Otoliths, ear stones, see Auditory
[Ouseley, Sir F. A. Gore, 503a]
Outwards striking reeds, 97c
Oval window of labyrinth, 130a, 136&
[Overtones, used by Prof. TyndaU for upper
partial tones, an error of translation, here
avoided ; the term should never be used for
partials in genecal, 2$d']
[Overtooth, 254'J
[Paoifio, South, Balafong, 518c, ^226]
Palestrina, a.i>. i524-*I594, under Pius IV.,
pupil of Goudimel, 2476. carries out sim-
pli£oation of church music, 247a. his use
of chords, especially in the opening of his
Stabat tnateTt 2256, 247c, 2486, 296a, c.
[Paris, Aim6 and Kanine, pupils of Galin,
425c'. Alms's bridge tones, 426&J
Partial tones in general, and upper partial
tones in particular, how distinguished, 22a.
in musical notes, 22c. [partialt&ne^pta-
tial tones, 24c\] partials, contraction for
partial tones, 240'. no illusion of the ear
any more than prismatic colours are of the
eye, 48c. those unevenly numbered are
easier to observe, 49c. methods for observ-
ing by ear, 50c. on piano, 50c. on strings
generally, 50c to 51a. modes of observing on
human voice, 516. [high upper, their exist-
ence proved by beats with forks by Trans-
lator, 56d'.] partials fuse into a compound
tone, shewn by experiments. Ohm's with a
violin. Prof. Helmholtz's with bottles, 606, c.
[Translator's with tuning-forks and resonant
jars, 6 id.] upper, their influence on quality
of tone, 62a. inharmonic upper, 706.
favoured on a piano whose period is nearly
twice the duration of stroke, 76a. of a
string-tone disappear which have a node at
point excited, 76c, 77a. [not always when
struck by a pianoforte hammer, y6dt 78c,
54.66.] ^ewn by flames seen in a revolving
mirror, Koenig*B manometric flames, 374a,
6. [of a pianoforte string struck at one-
eightii its length, 545c.] upper, of human
voice, difficult to recognise but heard by
Kameau and Seller, 1046. upper, perceived
synthetically, even when not anfUytically,
656. by properly directed attention they may
be observed analytically, 65c. at any rate
they effect an alteration of quality of tone, 65c
Passing notes, 353a
[Patna, Balafong from, 518a, 521^2']
[Pauer, Ernst, on presumed character of keys,
S5od\ SSia]
Paul, O., considers that Hncbald invented
the principle of imitation, ^2440
[Pedals - footkeys, 50^1
[Pellisov, see Schafhautl, lo^d]
IPelog scales, 51 8(2, 526a]
Pendular or simple vibrations, 23a. their
law and form, 230^
Pendulum, its periodic motion, 190. how set
swinging by the hand, by periodically moving
the point of suspension, 376. to shew vibration
of membrane due to that of air in bottle, 42a
Pentatonic scales in China, Mongolia, Java,
Sumatra, Hudson's Bay, New Guinea, New
Caledonia, and among Fullah negroes, 257c.
five varieties, 259a, 6, e. [259^. numerous
other, 5186. independent of heptatonic
scales, 525^. do not arise from inability to
appreciate Semitones, 5266]
[Peppercorn, 496^, 497a, 549a]
Perception, synthetical and analytical, 62d
Peri, Giacomo, in 1600 invents recitative,
2456, 248c
Periodic motions of pendulum, 19c. of water-
wheel hammer, 19c. of ball struck up when
falling, igd
Periods of musical composition, three, 236^
Persian, see Arabic
Persian music later develops 12 Semitones,
285a. Eiesewetter's hypothesis, 285a [d]
[Phonograph, Edison's, used to analyse vowel
sounds, by Messrs. Jenkin A Ewing, 539a]
Phase, difference of, 346. its effects on forms
of vibrational curves, iigd. on quality of
tone, 1 20c. in compounded simple tones does
not affect quality of tone, 124c to 127c. as
seen in the vibration microscope, I26(2, 127a.
[its influence on quality of tone according to
Koenig, 537a]
Philolaus, 257^
Phrynis, victor at Panathenaic competitions,
t adds a ninth string to his lyre, 269c
I Phrygian scale, 267c. Greek » mode of the
I minor Seventh, 303c. Glarean's, 245^2, 305c
Pianoforte, echoes vowels, 61 c [izgd,] strings,
where struck, 776 [77c.] takes the first place
among instruments with struck strings, 2o8d.
the quality of its chords arising from the
quality of its tones, 2090, c, d, bears disso-
nances well, 2096. its strings, mathematical
investigation of their vibrational forms, 380a.
[its string struck at one-eighth its length, has
the 8th harmonic and partial, 545c.] see also
Hammer
[Pichler, tuner at Berlin opera, $09(2]
Pipes, their theory, 388-397. theory of blow-
ing them, mathematically treated, 390 to
3966. conical, calculated series of their tones,
393c. [with remarks note * and 394/2 note *,]
see also Organ pipes
Pitch, IOC. number defined, iia [iidJ] de-
pends only on the number of vibrations in
a second, 13c. numbers of just musical
scale, how calculated, 156, e. Scheibler's
1 6c. French, i6<2. numbers of the just
musical scale, tabulated to a'440, 17a. of a
compound tone, is the pitch of its prime,
24a. its definite appreciation begins at 40
vib. 177a. alters by definite intervals and
why, 2506, 2526, d. of tonic undetermined,
depending on compass of voice or instrument,
310a. [numbers, how to determine, App. XX.
sect. B, see contents, 4416. musical, de-
fined, 4946. its history. 495 to 513, sect. H.
see contents, 493(2. Church, lowest, 495a.
low, 4956. medium, 499a. high, 503a.
highest, 503(2. extreme, 524. chamber low,
4950. highest, 504a. mean of Europe for
two centuries, 495(2, 497(2. compromise,
497(2. modem orc^estrid, 499a. when it
began to rise, 512c]
[Pitman, organist at Covent Garden Theatre,
Sood]
[Pitt-Rivers, Gen., his balafong, 5186, 522a]
Pius IV., Pope, AJ). 1559-1565, orders simpli-
fication of church music, 247a
Pizzicato of violin more piercing than piano
tones, 67a, 746
Plagal scales, Glarean's six, 245c. the fourth,
267(», 271(2
Digitized by V^jOOQlC
570
INDEX.
Plastio arts address the eye as poetry does the
ear, 2d
Playford, •atoi'
Plectrum, 74c
Plucking removes the whole string from its
position of rest, 74^. the intensity of prime
is greater than that of any partial, 75a
Plutarch on the Scale, 2626. thinks the later
Greeks had a preference for the surviving
archaic intervals, *2'- $d'
Poetry, its aim and means mainly psychical, 2d
[Pole, Dr. W., on characters of keys, 550c]
Politzer, experiment on the round window,
1366. produces drawings of beats by attach-
ing a style to the columella (auditory ossicle)
of a duck, 1666
Polyphonic music generated by discant, 2446
to 246. siren, description of mechanism for
opening the several series of holes in it, 413c
Polyphonic siren, see Siren, i>olyphonic
[Pomeranian (Blucher) band, its very high
pitch, 555(f|
Poole, H. W. [I95<2', 2I&2, 222^', 22Sd, 323d',
329<i'. his dichordal scale, 344c.] his Euhar-
monic Organ, *423a,[c. his proposed finger-
board, and theories of Seventh Harmonic,
474a to 4/ 9a. writes to Translator from
Mexico about his latest fingerboards, 478^]
Position of stroke that excites a string, 76c
Positions of a chord not hitherto regarded in
musical theory, 224c
Poverty of tone, in what it consists, 756
PrsBtorius mentions a cymbalum with 19 digi-
tals to the octave, 320c, 320(2'. on ' wolves,'
321a. [on early pitch, '^4946, 494i, 49 ;a,
5096, c]
[Preece and Stroh on quantity of sound, *75i',
124^. synthetical production of vowels, 542^
Preparation of dissonant notes, 353d
Preyer, W., high pitch from forks, i8a [♦iS^T],
55(2']. on distinguishable intervals, 147c [*d],
by Appunn's tuning-forks has shewn that
tones with 4000 to 40000 vib. in a second are
audible, 151c, 167J. [his recent work on
combinational tones and beats, 15 2d, '153c,
156^, 167^, ♦i76d, d\ 1776, d', 202d, 204^,
205^. his experiments with two forks of 137
and 18*6 vib., 177^'.] finds the difference
between tones of tuning-forks and reeds dis-
appears at 4224 vib., 179c. his experiments
with Appunn's weighted reeds, l^(>d, says
as low as 15 vib. may be heard, 176^. Prof,
fielmholtz inclined to think the tongues may
have given double their nominal pitch, 176c'.
[the Translator's experiments to determine
the real pitch of these reeds, 176^'. 226^,
229^', 23 id', on beats and combinational
tones, 528 to 538J
Prime of a compound tone defined, 22a
Principal-stimmen of organs, q3c
[Principal work of organs, 93d']
[Proch, 504d]
Ihrogression of Chords of the Seventh, 357d to
358d. by Fifths, 355d. by Thirds, 3566. its
laws are subject to many exceptions, 356d
Protestant congregational singing, its musical
results, 246c
[Provost-Ponsin, Mme., pronounces last sylla-
ble of hachis without voice, 68d]
Proximity in the scale, a new point of connec-
tion between tones, 287a
Ptolemy included the major Third 4 : 5 in the
syntono-diatonic mode, 22SC. his tuning of
the equal diatonic tetrachord, 2646
[Pythagorean minor Thirds indicated typo-
graphically by I , 276d. intonation in 15th
century, 313d. temperament, 433aJ
Pythagoras (fl. b.c. 540-510), his discorety of
law of consonance for strings, id. extended
to pitch numbers, id. the physiological
reason of his numerical law, 56. how he dis-
covered his law of intervals, I4d. was the
sole hearer of the harmony of the spberee,
229c. his enigma, 'why consonance is de-
termined by ratios of small whole numbers,'
solved by discovery of partials, 229a, 249a.
his tetrachord, 2636. first used 8 degrees to
the scale forming an octave, 2666. his dis-
junct tetrachords, 266c. constructed his scales
from a series of seven Fifths, 278d. 'or
rather Fourths, 2796, c, note *.] his system
used as foundation of temperament, 322a
Quality of tone, loe, 18&. defined, 19a, depends
on form of vibration, 19&, 2id. its concep-
tion, 65d. must not be credited with the
noises of attack and release of tones, 66a, nor
with rapidity of dying away, 66c. its strict
meaning, 676. musical, 69a. of reed pipes,
loia, I02d. [of vowels depends not on the
absolute pitch, but on relative force of upper
partials, 113d'.] recapitulation of results of
Chap, v., II 8d to 119c. apprehension of,
I I9d, 1286. independent of difference of phase
in the partials combined, 124c to 127c. appa-
rent exceptions, 127c, [which are thought to
be real by Koenig, I26d, 537a]
Quanten, E. von, his objections to Helmholtz's
vowel theory, 11 5d
Quartertones, Arabic scale of, 2646. [noted by
q a turned b, 264d'. temperament, 525a]
Quartett playing, why it often sounds Ul, 324d.
singing of amateurs often just, 3266
Quincke, *37 7d
Quintatsn (quintam tenentes) organ stop, 946
B
B letter, its beats, 67d, 168&
[Bab4b, scale after Prof. Land, 517c]
[Baffles, 526a]
[Baginl, or modelets, 525d]
[B&m P41 Singh, B&ja, his Quartertones, 265^.
his scales, 5i7d]
Bameau, 1685-1764, hears upper partials of
human voice without apparatus, ^516, 1046.
his theory of consonance, '232a to 2336. com-
plete expression not given to the new harmonic
view till his time, 2496. his assumption of an
* understood ' fundamental bass, 253c. his
chord of the great or added Sixth, 294d. his
fundamental bass, radical tone or root, 309c.
the tuning he defended in 1726, 321a. subee-
quently proposes equal temperament, 321^,
32 ic, 35 id. his law of motion of the funda-
mental bass by Fifths or Thirds, 355d. when
he allows a diatonic progression of the funda-
mental bass, 3566
Bational construction of scales, 272c to 2756.
differs materially from the Pythagorean, 278d
[Bayleigh, Lord, distance of plane of reflexion
from end of flue-pipe, 9 id. his dock method
of determining pitch, 442d. his harmonium
reed method, 4436]
Becitative, invented in 1600, by Peri A Caceini,
248c
Digitized by V^OOQIC
INDEX.
571
Becitation» musioal, 239a
BeedB or vibrators of harmoniumB, how they
ftct, 95a, 6. how their quality of tone is
modified, 95c. on organs and harmoniums
for one note each, 980. on wooden wind in-
struments one reed serves for several notes,
986. of clarinets, 966, oboes and bassoons,
97a. striking inwards and outwards, 976, c.
notes proper to them not used at all on
wooden wind instruments, 98c. pipes, their
musical tones, 95a. of organs, 96a. pipes, their
quality of tone, loia, I02d, experiments to
shew that they are modified by resonance
chambers, 102&. with cylindrical pipes, 391 6.
striking inwards and outwards, 391^. metal,
392a. with conical pipes, 392^. pipes, theory of
blowing them, mathematically treated, 39odt
394a. [action of, by Hermann Smith, 5536
to 555c'. their kinds, 553c. for clarinets,
553c. for Hichi-rikiy SSScf. for bassoons and
oboes, 554a. membranous, 5546. stream,
544c. free reed, Chinese, 554^. harmonium,
554a'. American organ, 5546'. voicing of,
554^2'. suction chambers, 5556. drilled,
555^1
[Begister, used by T. Young for quality of
tone, why not so used here, 24I/]
[Registers of the voice in singing, from Lennox
Browne and Emil Behnke, described, lood' to
loic, d, c' d"]
Beissner's membrane, i27d
Belationship of tones in first and second de-
gree, 2566. differs with the quality of the
tone, 256c. due in melody to memory, in
harmony to immediate sensation, 368^. be-
tween tones more than two Fifths apart im-
perceptible, 2796. of chords, its degrees, 2966
[Relative Duodene, 4626]
Resemblance, unconscious sense of, 369c
Resonance of boxes of violin, violoncello, and
vi61a, 866 [S6d\ 876 to d']. of cavities of
mouth, how to find, I04d to 105a. depends
on vowel uttered independent of age and sex,
105&. [difficulties of determining it, 105c.] of
cavity of mouth, how it affects vowels, i loc.
its influence in reed pipes, mathematically
treated, 388 to 390a
Res'onators for separating the musical tones
in noise, yd. spherical and cylindrical, 436, c.
Bosanquet's, 43d'. advantages of a tuned
aeries of, 446. use in finding partials, 51c.
affect the prime tone of the voice as well as
partials, Uieir effect on reed pipes, 1126.
spherical, their advantages, 372^2. formula
for their pitch, 373a. of glass, their dimen-
sions, 373c. tin or pasteboard in double
cones, 373^. conical, 373^
[Resultant tones. Prof. Tyndall's name for
combinational tones, 153c]
Results of the whole investigation, 362a to 3666
Retrospect of results in Parts I. & II., 226 to :<33
Reverted system, its chords, 342c. serve to
mark the key, 3446
[Beyher, S., on vowel resonance, *io8d, 109c]
Richness of tone, in what it consists, 756
Riemann, Hugo, *^6sdt *4i ic
[Ritchie, E. 8., 511a]
Robson, Messrs., built Gen. Perronet Thomp-
son's Enharmonic Organ, 423a
[Rockstro, W. S., doubts whether the authentic
scales are rightly attributed to Ambrose,
242(2. on Ecclesiastical scales, 266d. on .ne
Hexachord, *35id']
Rockstro, R. S., intonation of his * model ' flute,
555^ to 556a
Rods, effect of their material on the quality of
tone, 71a
Rohrflbte, organ stop, 946, [94^2']
Roman Catholic Church alters its music, 247a
Root or fundamental bass, 294c
[Rossetti, Prof. 494(2]
Roughness of intervals, referred to the same
bass note calculated and constructed, 192^';,
discussed, 193
Round window of labyrinth, 130a, 136&
[Rudall, lo^d]
Rudinger on the semicircular canals, 136^2
Rudolph II. of Prague, his cymbalum of 19
tones to the Octave, 320c
[Ruggieri (Cremona, 1668-1720), resonance of
his violins, 87c]
[Rule for tuning in just intonation, 493c]
[Russia, pitch, 5iOc2]
[Rust, how it affects forks, 555c]
[R, uvular, where localised, 67c2J
S
SacculuSj see Ear
St. Paul's, London, tones of its former bell, 72^
[Salendro Scales, 518c, 526a]
Salicumalf organ stop, 94a, [94(2]
[Salinas, F., 1513-1590, 35i(2'. his tempera-
ments, 5476. completes meantone tempera-
ment, 518]
Salisbury, Prof., at Tale, his MS., 281c
[Samv&di, or * minister ' note, 526c]
Sand, see Auditory
Sand figures on membranes, 4i(2
[Sarti, S^od]
Sauerwald, 16 id'
[Sauveur's cycle of 43, 436 i', 494(2'^ 509&]
Savart on pitch of resonance of violin and
violoncello boxes, 866, 17 $d
Scdla, vesdbilli et iympdni, or gallery of the
vestibule and drum of the ear, 1376
Scales, musical, their construction, $c. major,
division into two tetrachords, 2556. founded on
relationship of tones, 2566. later Greek, with
conjunct and disjunct tetrachord, 270a, ^.
[in the complete notation with intervals in
cents, 274c to d\ harmonic and non-har-
monic, 514&. Greek, most ancient form,
5156. later, 515c. Al Farabi's, 515c. early,
their possible origin, Sizd]
Schafhautl, or Pellisov, ♦72^'. [his theory of
stopped and conical pipes, 103^^
Scheibler, J. H. (i 777-1837), his pitch, i6r.
[why selected by him, i6d\ 87c, 153c, 200/,
d', his tonometer, does not shew that com-
binational tones of higher orders existed,
* 199(2.] rule for tuning the Fifth by the
Octave, 202d. [not found by Translator,
202d\ his method of finding the major
Third on forks, 203^2.] worked out combina-
tional tones for two simple tones only, 227a.
[on the beauty of just intonation, 423(2. his
tuning-fork tonometer, 443c. his theoreti-
cally perfect method of tuning pianos and
organs, and its inconveniences, 4886. 49Ad\
504c, 508c, 509a, 6]
Schiedmayer, J. and P., made Prof. Helm-
holtz's Just Harmonium, Si6d*
[Schlick, 494(2, 509c, 51 1&. his temperament,
♦546(2, 5496]
[Schmahl, 494(2', 548(2]
[Schmidt, Bernhardt, called Father Smith,
organ-builder, 505c]
Schneebeli, on the blade of air in flue-pipes,
*395<2. rhis theory, 3966]
Digitized by V^OOQIC
572
INDEX.
[Schneider, 5106]
[Schnetzler, 506a]
[Schnitger, uses equal temperament, SA-Sd]
[Sohreider, 496c. and Jordans, 505^2]
[Schulze of Paulenzelle, never saw a striking
reed till he came to England in 185 1, but
afterwards scarcely ased any other, 96c, 506&]
Scholtze, Max, hairs on the epithelium am-
pullcB, isSd
Science, musical, 1 6
Scotch handbell ringers, 73 J. Scotch pentatonic
airs bright, 258c. playable on black notes of
a piano, 259c. examples of such airs, 2606,
261a, c. [characteristics of Scotch music, ac-
cording to Colin Brown, 259C2, note %]
[Scotland, bagpipe, 515^^, 519^2]
Scott and Koenig*? Phonautograph, 2od
Scratches of a bow used in exciting bodies
with inharmonic upper partials consist of
those partials, 74a. of a violin, 856
Second of the Scale undetermined, 427. see
Sevenths
Sectional Scales, 266c
Seebeck, A., his siren, lie. his objections to
Ohm's law, did not apply proper means of
hearing the upper partials, *58J, considers
Ohm's definition of a simple tone too limi-
ted, 59a. remarks on Ohm's experiment, 6 la.
agrees with Ohm that upper partials are per-
ceived synthetically, 63a. disputes their
being perceived analytically, 63a, 391^ note*
Seiler, hears partials of a watchman's voice,
1016
Seiler, Mme., finds dogs sensitive to &"' of
violin, ii6d
Semicircular canals of the ear, 136a
Sensations, of hearing belong to physiological
acoustics, 3c. of sound defined, 7c. of
musical tones and noises, how generated, 8c.
compound, problem of their analysis, 626.
easy when usual, otherwise difiicult, 62c
[Septimal harmony, 464c]
[Septendecimal Harmony, 464c]
Seven, notes characterised by, not being ad-
mitted into the scale determine the boundary
between consonance and dissonance, 2286
[Seventeenth harmonic introduced into har-
mony, 464c]
Seventh, see Harmonic, natural, subminor,
major, minor, diminished, mode of the minor
considered, 303c. chords of the, 341c. formed
of two Consonant Trials, 341c. formed of
dissonant Triads, 342c
Sevenths and Seconds enumerated and con-
sidered, 3366
Semitones, Chinese view of, 229c
Seventh dhninished, 3366. chord of the, upon
the Second of a major Scale, 347(2. upon the
Second of a minor Scale, 348c. upon the
Seventh of a major Scale, 3486 to 3496.
chord of the Dominant, 3476, c. chord of
the, on the tonic of the minor scale, 3506
[Seventh harmonic introduced into harmony,
464CJ
[Seymour, 5496]
Sharper or higher tones, iid^
[Sheffield, Sohulze's organ at, 96d]
[ShMg, Chinese, described, gs^
Shirazi, 282a
[Siamese, ranat, 51 86. instruments, music,
scales, and intervals, 556a to 5566']
[Side holes of wind instruments, theory not
worked out, Blaikley, Schafhautl, Boehm,
Mahillon, io3<i, d*]
fSight-singing tests for Tonic Sol-fa, AZjd"]
Silbermann, A., 1678-1733, celebrated cngiB-
builder, his unequal temperament^ 323c.
[495&. 509c]
[Silbermann, G., 495(2, 496a]
Silence may result from two soands, i6oel.
instances of, organ pipes, i6ia, tonixig-foiks,
161&
Simple musical tones, 69c. produced by
resonance mathematically investigated, 3776
to 379(2. see also Tone, simple
Simple Vibrations, 23a
[Smgapore, Balafong, 518a]
Singers, their form'^r caieful training, 326a.
should practise to justly intoned instrumentu,
326(2. their opinion of Gen. P. Thompson*!
organ, 427a. take natural Thirds and Fifths,
428a
Singing, [contrasted with speaking, 68^*.]
voice does not usually distinguish vowels weD,
1 14a. forms the commencement and the natu-
ral school of music, 325c. intonation injured
by pianoforte accompaniment, 3266
Single, distinguished from simple tones, 33(f
Siren, 116, Seebeck's, i ic,Cagniard de la Tour^s,
I2C. action of, 136. of Dove, 13a, 14a. in-
tervals playable on, i62(2, 163d. constmetion
of, and beats on, 163&, c, d. polyphonic or
double, its great use in determining the ratios
of consonances, i82(2. electro-magnetic
driving machine for the, 3726. see also
Polyphonic
Sixth and Fourth, chord of, or first inversion
of major and minor chords, 213a. major,
more harmonious than fundamental, and
these than Sixth and Third, 2 14 6
Sixth and Third, or sixth only, chord of, or
second inversion of major and minor chords,
213a. minor more harmonious than funda-
mental, and these than Sixth and Fourth, 2146
Sixth, not included in consonances till the^
13th or 14th century, 196c. [Italian, French, '
and German, 461c.] superfluous or extreme
sharp, 3376. see also Major Sixth and Minor
Sixth, and Extreme sharp Sixth
Skhisma neglected, its effect in prodadng
identities, 281a [465a]
Skhismic Relation of eight Fifths down to a
major Third, 280&, discovered by Prof. Helm-
holtz, 3 1 6a. [temperament, 28 1 6, note*, 435a]
[Smart, Sir G., 507c. his pitch, 513a]
[Smart, H., 500c]
[Smith, of Bristol, organ-builder, 4960, 5066]
[Smith, Christian, organ-builder, 505(2]
[Smith, Eli, S14C]
[Smith, Hermann, his account of the Shteg,
95(2. his account of Schulse's and Waleker's
and CavaiU^-Coll's organs with scaroely any
free reeds, g6d, his account of the striking
reed, 96(2'.] on the blade of air in flue-pipes,
*395<2* L^iB theory, 396(2. on the Action of
Reeds, 5536 to SSSC^
[Smith, Dr. B., 494^2', 510&, 5482^^
[Society of Arts, 494^
Solidity, sensation of, analysed by stereoscope,
63c
Solo songs vrith airs invented by Monteverdi
and Viadana, 248c
SondhausB, his formula for pitch of resooatonv
3736
Sorge, German organist, 1745, discovers com-
binational tones, * 1 52(2
Sound, velocity of, in open air, god^ [and in
tubes according to Mr. D. J. Blaikley, god]
Soureck on the blade of air in flue-pipet, 3951L
this account, 3970*]
Digitized by V^OOQIC
INDEX.
573
[Spain, pitoh, 511a]
[Speaking contrasted with singing, 6Sd'.]
▼oioe, more jarring than singing voice, i ijd
Speech, its natural intonation, 238c
Spheres, their Pythagorean harmony, 15a
[Spioe, B., 502^, 51 16]
Spinet [its striking-place, yyd]
SpiUflbU, organ stop, 94a, [94x2]
[Spontini, 4976]
[Staff Notation, 426J']
[Stainer, Dr., 4976, 5056]
Stapes^ see Ear
Stark, Prof., 241^
Stefan, ♦37 7^
[Stein of Augsburg, knew nothing of a uniform
striking-place for piano strings, yyd, S04d]
Steinway & Sons of New York, their piano,
76a. [strikes at vV ftncl i the length of string,
ySd'. the 9th harmonic obtained by Mr. Hip-
kins, y6d'. 507(2, 5 1 1 6]
Stereoscope analyses sensation of solidity, 63c
Stirrup, see Ear
[Stockhausen, 4956]
Stokes, Prof., ♦383d
[Stone, astronomer, 1730^
[Stone, Dr. W. H., 494(2. his restoration of
16 foot C, to the orchestra, 5526. his contra-
fagotto, 553aT
[Stoney, Dr. G. J., on characters of keys, SSoc']
Stopped pipes, see Organ pipes stopped
Stops, compound, on organ, their use, 206c.
also see Organ stops
Stradivari, 1644- 1737, his violins, 866. [re-
sonance of the box of Dr. Huggins*s Stradi-
vari violin of 1708, Syc^
Straight lines and acute angles in vibrational
forms how produced, 34^2 35a
[Strauss, E., his pitch, 555(2]
Straw-fiddle, a wood harmonioon, 71a
[Stream reeds, 554c]
[Streatfield, 497a]
[Streicher, $02d]
Striking Beeds, 95c. how constructed, 96c2'
Strings, their forms of vibration, how best
studied, 456, c. number of nodes, 46c. in-
finite number of forms of vibration, 46(2.
different forms excited at tiie same time, 47c.
experiment on, with a flat piano, 47c. [on a
cottage piano, 47(2.] their tones best adapted
for proving the ear's analysis of compound
tones into partials, $2a, motion of, when
deflected by a point, 53(2 to 54c. excited by
striking, 746. their musical tones, 74a. how
to experiment upon, 756. of pianoforte, their
qualities of tone, 7gc. theoretical intensity
for difference of hanmier and duration of
stroke at } length of string, 79a, h, in the
upper octaves prime predominant, in lower
octaves 2nd and 3rd partial louder than prime,
80a. effect of thickness and material, 8oa.
motion of plucked, mathematically investiga-
ted, 374(2 to 377a. of pianofortes, vibrational
forms of, mathematically investigated, 380a.
see also Pianoforte
[Stroh, *7Sd\ see Preece, 124. synthetical
production of vowels, 542(2]
Stroke, for exciting string, its nature, 74(2.
duration of, 75c
Subdominant chord, 293. [Duodene, 462a]
[Snbminor Pifth 5 : 7, its partials examined,
195c]
Subminor Fourteenth 2 : 7 much better than
minor Thirteenth, 196a
Subminor Tenth 3 : 7 much better than minor
Tenth, 196a
Subminor Seventh 4 : 7, 49i'. often more
harmonious than minor Sixth, 195a. why
not used, 195a, 2136, c. its partials ex-
amined, 195&
[Subminor Third 6 : 7, its partials examined,
195c]
[Suction chambers, 5556]
Sumatra, pentatonic scale, 257
Summational combinational tones (Helm-
holtz's), 153a. only heard on harmonium
and polyphonic siren, 155c. exemplified,
156a. are very inharmonic, 156&. from
polyphonic siren act on membranes, 157a.
from harmonium act partly on resonators,
157c. [reasons for doubting the two last con-
clusions, 157(2]
[Supermajor Third 7 : 9, its partials examined,
I95fl
[Superminor Third 14 : 17, its partials exa-
mined, 195(2]
[Supersecond 7 : 8, its partials examined, 195c]
Suspension of dissonant note, 3546
Sylvester, Pope, a.d. 314-335. established Bo-
man school of singing, 239a
Sympathetic oscillation and resonance, its
mechanics, 366. of piano strings, 38(2. of
bodies of small mass, 39c. of tuning-forks,
39(2, 40a. of circular membranes, 40C-41C.
relation between its strength and the length
of time required for the tone to die away,
mathematic!ally investigated, 405c
Sympathetic vibration, the only analogue to
the resolution of compound into simple vibra-
tions by the ear, i2ga, of expansion of
auditory nerves, 1426. relation of amount of,
to difference of pitch, 142c
[Tadolini, 510(2]
[Tagore, Bajah Sourindro Mohun, * 243(2.
514c. Indian scale, 5i7(2]
[Tambour, Northern and Southern, their scales
after Prof. Land, 517a]
[T4r of Cashmere, 522a]
Tartini, 1692- 1770, Italian violinist, dis*
covers combinational tones, 152c*. his theory
of consonance, '232a. estimated all combi-
national tones an octave too high, 62a
[Taskin, Pascal, Court harpsichord tuner, 5096]
Taste, difficulties of perceiving analytically,
636
Taylor, Sedley, his Sound and MusiCj 6c
Tempered fusion of just intervals, 337(2
Temperament, relations leading to it, 312&, c*
[App. XX. sect. A, see contents 430(2]
Terms defined, 236, c, 24a
Terpander, b.c. 700-650, 249a. his seven-
stringed cithara, 257<2. his scale with a
tetrachord and Trichord, 26yd
[Terei suoni^ Tartini*s name for combinational
tones, i$2d:\
Tetrachorids, conjunct and disjunct, 255(1;
i) ancient enharmonic of Ol3m:ipos, 2626;
2) older chromatic, 262c ; (3) diatonic, 262c ;
I4) of Didymus, 263a ; (5) Pythagorean, 2636;
(6) Phrygian, 263c ; (7) Lydian, 263(5 ; (8) un-
used, 263(2; (9) soft diatonic, 264a; (10)
Ptolemy *s equal diatonic, 2646; (11) enhar-
monic, 2656. [old Greek, 51 2(2. Greek after
Al Farabi, 5I2(2]
Tetrads, or four-part chords, when consonant
formed by taking the Octave of one tone of a
triad, 2226, c, 223a. major, their most per-
fect positions, 223c j
Digitized by V^OOQ IC
574
INDEX.
Thebes in Egypt, flutes fonnd there, 27 id
[TheUWne^'gaTi'toneB, or partial tones, 240^
Third, iii, see Major Third and Minor Third
Thirds, Pythagorean, looked on as normal
Thirds to the close of Middle Ages, 190&.
their tempered intonation is the principal
fault of tempered intonation, 31 $6. not ad-
mitted to be consonances till end of twelfth
centary, and then only as imperfect, 190a.
[major and minor, their npper partials com-
pared, i^od]
Third and Fourth triad, 3386
Thirds and Sixths, consonant and dissonant,
enumerated and considered, 334a. triad of
dissimilar, 3386
Thirteenths not so pleasant as Sixths, 1896.
[partials of Sixths and Thirteenths compared,
1906]
[Thomas's orchestra, Cincinnati, U.S., 51 16]
Thompson, Gen. Perronet, [his life, 422dJ] his
enharmonic organ, 422c. its effect, 423(2. a
soprano voice singing to it and a blind man
playing the violin with it, 4236. [his mono-
chord, 44id, notes on his organ, 473^2.]
[Thompson, Sir W., his electrical squirting
recorder, 539a]
Thorough Bass, had formerly no scientific
foundation, 2c
[Threequartertone intervals, 525a]
[Timbre, its proper meaning, why not used
here for quality of tone, 24c ]
Timour, 282<i
Toepler andBoltzman, their experiments on the
state of air inside flue-pipes, *93a
[Tomkins, 494^, 503a, 505c]
Tonal keyp of later times, 270c
Tonal modes, five melodic, 2726. Greek,
formed from a succession of 7 Fifths, 288c.
only two X)ossible for a close connection of
all the chords, 300a. the major is best, 300&.
the minor is second, 300^. as formed from
their three chords, subdominant, tonic, and
dominant, 293(2 to 294a. their chords with
double intercalary tones, 297c, d, with single
ditto, 2986, c
Tonal relationship, unconsoious sense of,
370a
Tonality developed in modem music, 5c. the
relation of all the tones in a piece to the
tonic (F^tis), 240&. Greeks had an unde-
veloped feeling for, 242. complete in major
modes, has to be partly abandoned in other
modes, 2g6d. [absent in non^harmonio scales,
526c]
Tones, (meaning musical sounds), harmonic
upper* partial, 4d, distinguished by force,
pitch, and quality, loc. sharper or higher, and
flatter or lower, 116 [I Ki']. musical, 7c. de-
fined, 8a, c, 236. simple and compound, de-
fined, 236. the term ' tone * used indifferently
for a simple or compound tone, 24a. consti-
tuent, of a chord, 24d\ composite, $6d. com-
pound, 57a. u]^per partial, a general consti-
tuent of all musical tones, 586, c. the difiBculty
of hearing them does not depend on their
weakness, sSd, circumstances favourable for
distinguishing musical tones from different
sources, 59c. old rules of composition were
designed to render the voice parts separable
by ear, 59(2. without upper partials, 69c.
with inharmonic upper partials, 706. of
elastic rods, yod, of bowed instruments, 80c.
with a tolerably loud series of harmonic par-
tials to Sixth indusive, are most harmonious,
119a. their hollow nasal, poor, cutting.
rough character, whence derived, 1 19a, h.
their natural relationship as a basis for
scales, 2566. related to dominant ascending
and descending scales, 2746, c. related to
subdominant ascending and descending scales,
275a, b. of voice, energetic, how produced,
115c. [partial, 246.] proper not generally
determinable, 55(2, but determined in circular
plates and stretched membranes, 56a. simply
compounded, tone and Octave or tone and
Twelfth, 305, 32J. simple, 11 8(2. simple, how
to produce by means of resonance jar, 54<2 to
S$c, combinational {see Combinational tones)
Tones (meaning musical intervals), printed
with a capital letter for distinction, 24c. and
Semitones where used, $24d
Tongue (see Beed)
Tonic, did any exist in homophonic music ?
240&. rules for finding it in the authentic
scales uncertain, 2436. Greeks used any note as
such, 268a. chord represents compound tone
of tonic, 296c. chord, development of feeling
for in 1 6th and 17th centuries, 2966. feeling
for it, weakly developed in homophonic music,
243c
Tonic Sol-fa. singers, 207(2. teaches to sing
by the characters of the tones in the scale,
279(2. system of shewing relation of each
note to the tonic, 352(2. society of, 4236.
[history of, 423(2' to 425c' notes. festiTiOs,
42701
[T6pfer, 509(2]
[Translator, his additions in Appendix XX.,
430 to 556J
[Trautmann, Moritz, on vowel resonance, 109c',
nod]
Triads, how formed, 21 26, e. consonant within
the compass of an octave, 21 2(2. see Major
and Minor triads, of two just major Thirds,
and their transformations, 338(2, 3396. with
^two dissonances, 339c, d. how they limit
the tones of the key, 340&, c, their pos-
sible confusions, 341c. [cell and union, of a
duodene, and condissonant triads, 459a]
Tremor, sonorous, distinct from motion of in-
dividual particles of air, 8(2
Trent, council of, alters music, 247a
Trichordal representation of harmonisabte
modes, ^ogd
[Trichordals, harmonic, 460a]
[Trines, major and minor, pure quintal, major
and minor quintal, 459a]
[Tritonic temperament, true and false, 548a]
Trombones, lengthening, looa. [shape ot
looc'j
Tropes or scales of the best Greek period,
essential, having hypate as tonic, 268e2
[Trumpets, their shape, looc. with slides,
rare, looc^
Tsay-yu, said to have introduced the hepte-
tonic scale into China, 258a
Tso-kiu-ming compares the five Chinese tonel
to the five Chinese elements, 229c
[Tunbridge, 496(2]
Tune, meniAi '^OemUthslimmung, 2$od
Tunes, popular, constructed from the three
constituent major chords of the scale, 292a
[Tuning. Sec. G, see contents, 483(2. its
difficulties, 484(2. examination of various spe-
cimens of, 4B4& to 48^(2. Translator's prac-
tical rule for tuning in equal and meantone
intonation, 488(2 to 491c]
Tuning-fork, its form of vibration, 20a, d. its
tone, 79&. has high inharmonic proper tones,
70c. large, of 64 vib. gat«,J^P93l*M^ '59«-
Digitized by^
LUXUU.IUVUJIU |/«v^fcia I
INDEX.
S7S
[bow to treat and tune, 443a', d\ tonometer,
invented by Scbeibler, 443c. best method of
making, 443^2. how to use, 444a. the Trane-
lator*B, 446^^.] &e€ also Forks, tuning
Twelfth gives no beats, except those from
partials in a single compound tone, iSy^f. an
absolute consonance, 194&. better than Fifth,
1956. repetition in it presents nothing but
what has already been heard, 2536
Twelve Semitones introduced into China,
2566
Tympanum, see Drum
Tyndall, J., *on sound,* 6c'. translated by
Helmholtz and Wiedemann, 6€f. [his * tone,
clang, clangtint, overtone,' 24, footnote. ' re-
sultant tones,* 1 53c. finds beats do not in-
crease the range of power of sounds, *i7gd,
observations on gas jets, 395a, d]
U
U, its resonance cavity, io6a
[TJeberzahn, 2$d']
[XJllmann, 504^']
XJlm,96d
Unconscious apprehension of regularity, 3676
Undertones, harmonic, defined, 44d
Undulatory motion, ga
[Unequally just intonation, 465a]
Unevenly numbered partials, easier to observe
than the evenly numbered, 49c
Unison gives no beats except those from par-
tials in a single compound tone, iSyd, greatly
disturbs adjacent consonances, i88a
Upper partials, see Tones, upper partials, 586
[Upper tooth, 2Sd']
Utficulus, see Ear
[Vddiy chief Indian note, 243c'. or * ruler'
note, 526c, see ans'd]
[Variability of Seconds and Sevenths, its
effect on the modes, 277c', d"]
Testibule of the labyrinth, i^sd
fVianesi, sood]
Viadana invents solo songs with airs, 2480
Vibrating forks, phases of, as compared with
those of the exciting current, mathematically
investigated, 4026
Vibration microscope, dod. curves shewn,
826
Vibrational form or form of vibration, simple,
2ia. for water-wheel hammers and struck-
up balls, 2IC
Vibrations, 86. single, as reckoned in France,
inconvenient for determining pitch, i6d'.
form of, 20a. for a tuning-fork, 206, c.
pendular or simple, 23a
Vibrator, see Beed
[Vieth, G. U. A., first uses the term • combina-
tional tones,* *i53c]
Villoteau, believes the Eissar to be penta-
tonic, ♦257^. [his thirds of Tones, 282^.
origin of his conception of thirds of Tones,
5206, 55W]
[Vince, I35cfl
Vi61a box, pitch of its resonance according to
Zamminer, 865, [S6d' note]
Vidla di Oamba, organ stop, 93a, [93d]
Violin string, its vibrational form, 836. crum-
'pies upon it, 84c. development of Octave,
85c. box, the pitch of its resonance accord-
ing to Savart, reported by Zamminer, 866.
[according to Translator's observations, 876,
c, d,] condition for regularity of vibration,
Ssd. why old ones are good, Ssd, bowing
the most important element, 86a. sound-
post, its function, 866. [according to Dr.
Huggins's experiments to communicate vi-
brations from belly to back, *86c, d.] effect
of the resonance of its box on quality of its
tones, 2ioa. intonation and experiments,
3246, c. Comn and Mercadier, 325^, [486 to
487]. strings, their motion mathematically
investigated, 384 to 387
[Violinists, their intonation as determined by
Comu and Mercadier, 486c. differi&nt for
harmony and melody, 487c]
Vwlofi'bass organ stop, 93a, [93J]
Violoncello organ stop, 930, [g^d]
Violoncello box, pitch of its resonance accord-
ing to Savart and Zamminer, 866, [86d']
Vis viva, insufficient measure of the strength
of tones, 1746, d, [75(f)
Visoher, his * Esthetics,' 26
[Vocalists, their intonation, difficulties in
actual observation, 486a]
Vocal chords or ligaments, 98a. their rate of
vibration not iStered materially by air-
chambers, their watery tissues, and variable
thickness, 1006
Vocal expression, natural means of, 370^
Voice, well suited to harmony, 206c. effect in
certain chords, with different vowels, 2o6d.
[compass of the human, 5446]
[Voice Harmonium, Colin Brown's, 47od]
[Voiceless vowels, 6Sd]
[Voicer, the, his arts, 397iT
[Voicing of reeds, 554^^
Vortical surfaces, 3946
Vowel qualities of Tone, 103a. their charac-
ter, 103a. theory fiTst announced by Wheat-
stone, 103d'. produced by resonance of cavi-
ties of mouth, 104c. trigram of du Bois
Beymond the elder, 1056. resonance recom-
mended to philologists for defining vowels,
io6c. [difficulties in doing so, 106c'. differ*
ences of opinion of Helmholtz, Donders,
Merkel, Koenig, io6d*,] resonances in notes
according to Helmholtz, 1 106. their recogni-
tion by resonators, i loc. their modifications,
ii3«
Vowels, A, 1056. O U, io6a, [io6c, d.] O*
and A**, 106&. the above have only a single
resonance, io6c. X, E, I, double resonance,
107a, 6, c. [Graham Bell finds double reso-
nance in all vowels, lojd,] A, E, I, have
their resonances too high for forks, 1 1 la, 0,
t!, io8a, 5. U, how Helmholtz determined
its resonance, iiO(|i. its degeneration into
Ou, nob. their transitional forms due to
continuous alterations of resonance cavities,
I lie to ^^. better distinguished when
powerfttll^l^^mcnced, ii^d, their quality
of tone, 115a. distinguished preponderantly
by depending on the absolute pitch of the
partials that are reinforced, i i8c. their effect
on harmony, 207a [c, d], practical direc-
tions for their synthesis, 398a to 400c. [their
new analysis by means of the phonograph by
Messrs. Jenkin and Ewing, 538 to 542. Oh
analysed, 539 to 541. Oo, Aw, Ah, analysed,
541. their synthetical production by Messrs.
Preece and Stroh, 542^]. echoed from piano,
61C. vowels only, without consonants, heard
from speakers at a disti^pce, 68c
Digitized by^OOQlC
576
INDEX.
W
Wagner, B., his treatment of the chord of two
major Thirds, 3396. [thinks in equal tem-
perament, 339<2'. his festival, 502c]
[Walcker of Ludwigsburg, had little experience
of striking reeds, 96c]
Waldeyer finds 4500 outer arch fibres of Corti
in the cochlea [giving i for each 2 cents], 1476
Waller, B., 1686, reduces all colours to three
fundamental, 64a
[Walker, J. W. & Sons, organ-builders, never
used free reeds, 96c. 5066^ 506c, 507a. in
July 1852 put the Exeter Hall organ in
equal temperament, 549a']
Water-wheel hammer, its periodic motion, 190,
21C
Waves of water, 96, d. generated by a regular
series of drops, loa. of rope, chain, india-
rubber tubing, brass wire spiral, 9c. their
composition, 256, c, 26a-{2. of water or air,
algebraical addition, 27^, 28c. periodic, re
suiting from composition of simple tones,
y>b to 32^. phases of, caused by resonance,
40(x2, mathematically investigated, 401a,
of the sea, effect of their motion on speota*
tors, 251a
[Weber, Frank Anton von, 497a, 509&]
[Weber, 0. M. von, 498c]
Weber, Dr. Fr. E., on function of the aqtUB-
ductus vesHb&H, 136a
Weber, W., 390*
Weitzmann, *26gc, d, title
Werckmeister (b. 1645), advocated equal tem-
perament in 1 69 1, 321 c, 548c
Wertheim, 3736
Westphal, •265^', ♦268c, d
Wetness, sensation of, compounded of un-
resisting gliding and cold, 63c
Wheatstone, Sir Charles, first announces a
vowel theory, ♦103d. [repeats Willis's ex-
periments on vowel reproduction, 117^]
Whispered vowels, pitch of, io8c. Gzermak
and Merkel on, 108^'
[White, J. Paul, his Harmon, 22&2, 329^'.
described, 481 6. his methods of tuning it,
492c]
Wiedemann, G., ^dc'
[Wilkies, the, 5496I
[Willis, organ-builder, never uses free re«4i,
96c, 5065, 506c, 507a]
Willis, Prof. B., *io2d\ his reproductioii d
vowels by extensible reed pipes, 11 76. iUt
table of vowel resonance, 117c. his ezpiri-
ments with toothed wheels and springs, 1 18a.
his vowel theory, 1186
Winterfeld, von, *24Sd, ♦272^, 287^', *z^
[Withers, violin-maker, assists in findbig
resonance of violin, 876, c]
[W6lfel, 5096]
[Woneggar's abstract of Glarean, 196(2']
Wooden instruments are mobile in tone, 67^.
pipes have softer tone than metal pipes, 94c
reeds, gSb
Wood harmonicon, 71a
[Woolhouse's cycle of, 19, 436(2]
[Ynignez, Don, organist of Seville, 496a]
Young, T., 77d'
Young, Thomas, 1773-1829,^8 law that ezdt-
ing a string at a node destroys the harmoniei
corresponcUng to that node, 526. its proof,
52c. 536 [S^Z^j 5466.] his analysis of eokwr
into 3 primaries, 14BC, 149c. his theory U
differential tones generated by beats, i66i
Z
Zamminer, *62d, on pitch of resonance of violia,
violoncello, and vi61a boxes, 866 [*S6d' note.]
length of horn, 99c. [error in reporting,
lood note.] 323^2, 3906, 394a
[Zaizal, lutist introduces ThreequartertflBO,
264^', 2Sid, d, his two new intervals of 355
and 853 cents, 28i(2, 6^, 5256]
Zarlino assumes the tenor voice part to deter-
mine the key, 245(i, 312a, 326a, t35i(i'. hii
temperament, 546^]
Zillerthal, in Tyrol, scale of its wood haniMi-
icon, 270(2
Zither, 746
Z6nak dmticula'iat i^gd
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