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Full text of "On the sensations of tone as a physiological basis for the theory of music"

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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 


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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the Sensations of Tone, which was originally made from the third German edition 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 

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. 


25 Argyll Road, Kexri>:gton: 

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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. 


Hbidelbebg : October 1862. 




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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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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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 

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. 


* [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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\* 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. 


Belation of Musical Science to Acoustics, i 

Distinction between Physical and Physiological Acoustics, 3 

Plan of the Investigation, 4 

PART L (pp. 7-151.) 

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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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 


PART 11. (pp. 152-233.) 

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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PART IIL (pp. 234-371.) 

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- 

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, 

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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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, 
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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[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, 

[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, 

[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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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. 

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, 

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, 

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, 

The Polyphonic Siren, 162 
Diagram of origin of beats, 165a 

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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, 

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 


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, 

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 with accidentals, 478a 

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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, 


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, 

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, 

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, 

[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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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, 

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, 


[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. 

'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, 

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, 

The Duodenarlum, 463a 
Fingerboard of the Harmonical, first four 

Octaves, with scheme, 4676, fifth Octave, 

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, 

Gen. Perronet Thompson's Organ scheme, 

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, 

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, 

Cornu and Mercadirr's observaliou on 

Violin Intonation, 4S6f lo 4X76 

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Scheme for Tuning in Eqaal Temperament, 

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 

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 


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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Digitized by 



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 


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 % 

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 

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 

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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 

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 


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',] 

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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 

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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 

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. 

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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. 

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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 




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. — 

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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 


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 

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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 

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,'] 

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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 

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. 

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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 

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 ): — 




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 : 

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2. The oncC'Ctccented octave (2-foot) : — 





^ c' ci' e' / 
3. The twice-accented octave (i-foot) : — 








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 : — 






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 

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 

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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. 







Great Ootare 













8 foot 

£ to & 



e"' to V" 

&"' to y" 


I foot 









1 188 
















































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. 




8 foot 



I foot 

Note to Note 

C to Note 

Note to Note 








I : I 














































































































t [The following account of the actual tones 
^>Md is adapted from my History of Musical 


Pitch. C„t commencement of the 32-foot oc- 
tave, the lowest tone of very large organs, two 

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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. 

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 

The scale there- 
fore extends to 
about a Fourth 
beyond the oc- 
tave. — Transla- 

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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* ^ 

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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 

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 

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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 



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. 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 

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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 



'^ c g c' e' g' »6't) c" d" «" "/' g" 'V' '6"t> h" 


c' e' g' »6't? c" d" e" "/' g" »V 


456 7 8 9 10 II 12 13 


464330396 462 528594660726 792858 


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.] 

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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- 


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. 

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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 
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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. 



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 

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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. 

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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. \ 

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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 

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- 

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 

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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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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 

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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 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.] 

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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 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. 


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 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. 

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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 

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.] 

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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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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 analysi s has^j jag 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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is or is not contained in a composite mass of musical tones. The exis tence 
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. 



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 

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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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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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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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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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 


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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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 







Notes nearly 



d' + 



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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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 

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CHAP, m. 



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. 

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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 




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. 


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 


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.] 

Digitized by 





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 

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 


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.* 



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 

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. 

Digitized by 







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 

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 


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 

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 

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 


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 


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 


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 

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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) c d 


f g t 


h a 


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 

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 

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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. 

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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 



Prime tone 

Second proper tone 

Third proper tone 

Fourth proper tone 

I -0000 



3600 + 886 


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 

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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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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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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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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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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 

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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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 

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 

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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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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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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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. 



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 

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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:] 

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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.] 

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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.] 

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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. 

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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.] 

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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- 

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 

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of Nodal 


Namber of Diameters 



2 1 3 





9"» + 






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- 

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 . 






Tones ../... 
Cento . . ... 






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. 
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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 













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- 

t See /. Bourgelt L'Institut, xxzviii., 1870, 
pp. 189, 19a 

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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 , 

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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.'} 

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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 

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. 



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 


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 


CHAP. V. 3. 



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 

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 


Excited by 


1 tV 

: the periodic tiix 

e of the prime to 


Struck by a 

perfect hard 



























1 18-9 


















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 


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 



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 


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 


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 



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 


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 


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 



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.] 

Digitized by 




Fig. 27. 



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 


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, 

* [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- 
Digitized by V^ O OQ IC 



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.] 

Digitized by VjOOQIC 


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 





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- 

Digitized by 



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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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.]. 

Digitized by V^jOOQlC 


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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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] 

Digitized by VjOOQIC 

CHAP^ V. 6. 



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 




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. — 

Digitized by V^ O OQ^ IC 

OHAP. V. 6. 




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 

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. 



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:-- 











8, Ac. 




























































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. 



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, 


4. upper thin. 


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- 

* [See footnote f p. S6d^.— Translator,] 
f [The cheap little mouth harmonicons ex- 
hibit this dleci very well.— IVaYuIa^or.] 

Digitized by V^jOOQlC 


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 

% 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 

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 


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 


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- 

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 

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* 



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 



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. 

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 to » sii in French peu 
or else in peuple (different sounds), and tf to 
« tt in French pu,— Translator.] 

Digitized by V^jOOQlC 


enlargmg with tolerable uniformity from the larynx to the lips. For the vowels of 
the lower series, [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 [o in more nearly] the 
resonance gradually rises ; and for a full, ringing, pure the pitch is 6^« The 

m position of the mouth for 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 
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 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 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 

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 

Digitized by 



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 

Digitized by V^OOQIC 


For the third series of vowels from A through [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 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 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] 

Digitized by VjOOQlC 

CHAP. V. 7. 



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 : — 





[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- 

























[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 





A 1 B 



I. Reyher . 

c 131 


a 990 ) 





2. Hellwag 






6b 233 


3. Flttrcke . 









4, Bonden ac- . 
cording to i 
Helmholts .[ 


<f 294 

6'b 466 


c'TT 1109 


a" 880 



S. Bonders ac-x 
cording to I 
Merkel . ) 

y65 J 







6. Helmiioltz. 

6'b 466 



6*" 1976 

if"' 2349 


cTt 1 109 


+f 587 





7. Merkel . . 




or 0' 440 


a' 440 


A', 6 247 


or<f 294 

8.KoenSg, 7th 

harmonics . 

6b 234 

6'b 448 




0. Trautmann . 



/'" 1397 


EV 1760:/"" 2794 







Digitized by V^OOQIC 




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 


f gfe 

'% 9" 


:tr— tr= 

U Ou 

1 E 


I ±: 


^ 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 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 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.} 

Digitized by V^OOQIC 


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 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 — — U series 
and those of the A — — 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 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 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.'] ' 

Digitized by 



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 — — 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 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 



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 
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 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 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 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 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€ 


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 

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 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 


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 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 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 


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. 



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 


In the Word 



Length of Tube 
in Inches 















Part ^ 








Pay ^ 





*■ fjntf 







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 ; 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 

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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 

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.] 

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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. 



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 

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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. 



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 

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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. 


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\>, — 

Digitized by V^ O OQ IC 

* [These being 7th harmonics Vb and 
Vb are 27 cents flatter than the a"\> 




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 


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 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 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 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 

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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 r 

























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 





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 






35*^ 54' 


SO"" 12' 




68<' 54' 


75° 31' 




84° 50' 


87° 42' 


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 


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 


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 


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 





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. 

Digitized by V^jOOQlC 


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- 


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. - 

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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 


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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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.] 

Digitized by 



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 

Digitized by V^OOQIC 


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. 

Digitized by V^OOQIC 


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 

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. 

Digitized by V^OOQIC 


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 
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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 



1.. ^ 


F ^ 




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. 



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. 




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- 

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 

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Pio. 47. 




¥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 





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 




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 


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 



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 

* The mode of calculation in explained in Appendix X. 

Digitized by VjOOQIC 



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 

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 . 








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 


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 




Difleronce of Pitch 

of Sympathetio Vibration 

Difference of Pitch 

of Sympathetic VibraUon 






Whole Tone 




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 



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 









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 





Length of inner rod . 

Length of outer rod .... 

Span of the arch .... 





* [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 


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 


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 



1.T.O \> 

7.1 h 


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 

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 ' 



investigation than those which take place in the nerves of sense. In those of the 
muscle, indeed, we find only qua ntitative 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 


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. 


Digitized by V^jOOQiC 


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. 





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. 

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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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The following table gives the first differential tones of the usual harmonic 
intervals : — 


Ratio of the 



Difference of 
the same 

The combinational tone is deeper than 

Octave . 
Fifth . 
Fourth . 
Major Third 
Minor Third 
Major Sixth . 
Minor Sixth . 

1 : 2 




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. 








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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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.* 





Major Third. 










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.] 

Digitized by 




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,] 

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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 

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 

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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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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. 



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,'\ 

Digitized by VjOOQIC 



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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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 .] 

Digitized by V^jOOQ IC 



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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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,] 

Digitized by VjOOQIC 


^» 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. 

Digitized by 


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 

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 

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. 

Digitized by V^OOQIC 


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 

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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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 


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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 

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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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 
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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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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PABT n. 



[major, 297-264] and S e' [minor 330-297] 




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, 




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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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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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. 

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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 

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 

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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. 




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 

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 

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 

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 

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."' — 

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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.] 

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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. 



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 

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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 : 




u -i 







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. — 

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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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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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composing it. This numerical ratio always results from the ordinal numbers of the 
two coincident upper partial tones. 

Partial Tones 







2 Octaves 

and Fifth 

I :6 



I : 2 



5 ] 

2 Octaves A 

Major Third 





4 { 

Donble Octave 
I :4 

I ; 2 


3 { 

I :3 



2 1 

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 


I : 2 


I ' 3 

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 



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 

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 : — 


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 ' 

Octave . 
Fifth . 
Fourth . 
Major Sixth 
Major Third 
Minor Third 

300 « 
600 = 

450 = 
400 « 
500 = 

375 = 
360 = 


301 = 

301 = 

301 = 






= 903 
= 1204 

= 1505 
« 1505 
= 1806 

Interral downwariLi 

Octave . 
Fifth . 
Fourth . 
Major Sixth 
Major Third 
Minor Third 

Beating Partial Tones 

Number of 

I X 300 = 


I X 301 = 301 


2 X 150 = 


I X 301 = 301 


3 X 200 = 


2 X 301 « 602 


4 X 225 = 


3 X 301 = 903 


5 X 180 = 


3 X 301 = 903 


5 X 240 = 


4 X 301 = 1204 


6 y 250 ^ 


5 ^ 301 1 50s 


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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 



anil girea beats 

Octave . . . . d'd" 
Fifth . . , . d'g" 
Fourth .... cV" 
Major Third . . . d' d' 
Minor Third . . . d' d'b 

b' d'\> 

d"\> d" 
d")> g" 
d"\> d' 
d"\) d'b 






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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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. 


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 

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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 



Ratio of the 
Pitch Numbers 


Oants In the 


of Cents 

of Influence 



I : I 







63": 64 





D + 


48 : 49 




Subminor Third 







Minor Third 







Major Third 







Sapermajor Third 

E + 









20 : 21 



, 8-3 

Sabminor Fifth . 



14: IS 




Fifth .... 



IS : 16 




Minor Sixth 








Major Sixth 



20 : 21 




Sabminor Seventh 







Minor Seventh . 



9 : 10 






I : 2 





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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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 

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 


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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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 

C c g c' €f g^ b'b c" 

F f d f a' d' 

i_ 2 3 4_ 5 6_ 

c g d d g' 6'b c" 
/ f c" 

I 2 3 

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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 





5 6 7 
e! g[ 6'b 
e a' 
3 4 


No. of Partials. 
Lower note 

Major Thi/rteenth or 3 : 
No. of Partials. 

. I 2 

. C c 

3 4 

9 d 



</ ^ h'\> c" 





No. of Partials. 
Lower note 
Minor Sixth or 5 : 8 
No. of Partials . 

I 2 

. C c 




5 6 7 

«' 9" 6'b 
e'b a'b 
3 4 


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. - 

* [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 : 

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 . 


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 


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. 

Octave or I : 2, cents 1200 

Fifth or 2 : 3, cents 702 

Major Tenth or 2 : 5, cents 1386 


4 s 


6 7 

8 9 



Twelfth or I : 3, cents 1902 

Fourth or 3 : 4, cents 498 

Major Sixth or 3 : 5, cents 884 


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, 


ti4, ti6, 



f 2, 4, 6, 84 10, 
•X Si 10, 


•14, *j6, 




/ > 2 3 4 5 
I 3, 


7 8 


6. •9, 


ti5, i8« 21, 
ti64 2O4 

24, t2;, t3o,, 

24a J28, 

/ 3i 6, ♦9, 124 

I 5i *io. 


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, 


Il45i. llSo„ 

Ceb f Si 102 i5t 20, ♦25,30. t3S7 40, t45 

Minor Tenth or 5 : 1 2, cents 1 5 16 \ 12, 



tSOia 55ii 6o,j 
US, 60. 



6, ♦9. 


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 




t27. 30 • 

ti8. Uh 

_ t20^| 

t3o/ t3S7 



t27. 30 • 
t3o. , 

USb tSO;. 


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. — 

Digitized by. VjOOQIC 


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).* 







15, : 16, 8, : 15, 





6. : 7, S, : 7, 

Eg " I ^ 



Ss : Sa 57 : 6^ 


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 


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. 


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 


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 


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 



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 

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. 



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 

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 



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. 

of C 






























ao ' 


r as 


I ^7 


1 3'> 








17 16 

21 12 






1 188 



1 188 



31 12 












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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 





"SS f?^ 




lil 1' 













^ fl^ 






If we arrange these tones by pitch we find the following groups : — 




















800 903 


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- 

i [But the beats of the upper partials are 
always distinguished by their high pitch.— 

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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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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.] 

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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 

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 / 


Number of beats 2 

b. Fourth, Primes 300 and 401, upper partials 900 and 1203. First dif- 
ferential tone * 

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 


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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b. Minor Third. Primes 500 and 601, Octaves 1202, Twelfths 1500 and 1803. 
Differential tones * 

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 

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 


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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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 

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 


— 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 



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 

* [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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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, ^. 

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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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 

Digitized by 



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 

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 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.] 

Digitized by V^OOQIC 


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 

Digitized by 



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 

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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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,^ 

Digitized by Vj 00 QIC 


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 


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. 



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 



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.] 






















































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) 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 


The next two ohords, FA and F A\}, are termed, from their composition, 

chords of the Sixth and Fomth, written [OioF being a Fourth, and to il 

a major, but (7 to il(> a minor sixth]. If we take O, instead of 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 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 than 
it does with itself. The best triads which it can produce are G E''JB\} = 4 : 5 : 7> 
and 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 

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 


E a 


Q C 


C E^ 



a c 



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 

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. 


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, 



^ ^ b^W — ^ '>m m\rjr 







2.) Minor Triads with their Combinational Tones :* 


b b J^ u" W- 

^ , , , ^^ i / k^ - 








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 

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, 

Second Chord, — Notes 12, 15, 20 ; Octaves 
24, 30, 40. 

1) Crotchets, 15-12 = 3, 20-15 = 5, 20- 

2) Quavers, 24-20 = 4, 24-15 = 9, 30-20 

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 


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— 


2) Quavers, 30^24-6, 30-20=10, 40- 

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 


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 


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 

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.] 

Digitized by V^OOQIC 




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 

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 

Digitized by 


CHAP. xn. 



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 

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 

Digitized by 




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 

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- 

Digitized by 


OHAP. xn. 



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.* 






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 

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 

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 


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. ^ 

Digitized by V^jOOQ IC 


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 

Digitized by 





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,* 











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,] 

Digitized by 





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 

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 

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 

Digitized by V^jOOQlC 


<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- 

Digitized by VjOOQIC 

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.] 

Digitized by VjOOQ IC 


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 

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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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 

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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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 

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 


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 s