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TEMPERAMENT
R. H. M. BOSANQUET
AN ELEMENTARY TREATISE |
ON M [\t,\.\ 5
MUSICAL INTERVALS
TEMPERAMENT
WITH AN
ACCOUNT OF AN ENHARMONIC HARMONIUM EXHIBITED IN THE
LOAN COLLECTION OF SCIENTIFIC INSTRUMENTS
SOUTH KENSINGTON, 1876 ^
ALSO OF AN
ENHARMONIC ORGAN EXHIBITED TO THE MUSICAL ASSOCIATION
OF LONDON, MAY, 1875
R. H. M. BOSANQUET
Fellow qf St. yohn's College, Oxford
MACMILLAN AND CO.
1876
\All rights reserved\
f, f:f: f ! ■
OXFORD:
BY 11. PICKARD HALL AHD J. H. STACV,
PRINTERS TO THE UNIVERSITY
OL-'
Freylich wiirde der Gesang noch mehr gewinnen, wenn wir die
enharmonischen Tone in unserm System wiirklich hatten. Alsdenn
wiirden sich die Sanger auch von Jugend auf angewohnen, die
kleinsten enharmonischen Intervalle richtig zu singen, und das Ohr
der Zuhorer, sie zu fassen; und dadurch wiirde in manchen Fallen
der Ausdruck der Leidenschaften sehr viel starker werden konnen.
Kirnherger, vol. i. p. 19.
Greateb certainly would be the gain of Song if we really had
the enharmonic intervals in our system. For then singers would
accustom themselves, from their youth up, to sing correctly the
smallest enharmonic intervals, and the ear of the listener to appreciate
them ; and thereby would it be possible, in many cases, to make the
expression of the passions very much stronger.
Cornell University
Library
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PREFACE.
The investigations of which this work contains an account
have been published during the last two or three years in
various forms*. But these forms were suitable for those who
were already well acquainted with the subject^ and the order
of treatment was that which commended itself as correct in a
scientific point of view ; i.e. the matter being purely deductive,
the general theory was stated first, and everything deduced
from it. For more general purposes this arrangement does
not appear to commend itself; and the general theory has
now been relegated to such a position that the rest of the work
may be read independently of it. The general treatment is,
except in this Chapter (X) and the note to Chapter III,
elementary throughout ; and repetition is introduced as much
as possible instead of reference ; so that it may be hoped that
the difficulty of the study is much reduced. The arrangement
adopted is unsatisfactory from a scientific point of view. But
the different parts of the subject are so intertwined, that if the
correct order is once forsaken, it is impossible to separate them
out into another satisfactory scheme.
The relations of technical music and musicians to this
subject have until lately been, for the most part, of an
almost hostile character. The facts are entirely unknown to
musicians in general, and of the theory the wildest ideas
have been formed. Of the objects which I place before
myself, musicians generally form their own ideas, and stick
to them. The ' Musical Standard,' after the reading of a
* Proc. Royal Society, June, 1875, p. 390; Philosophical Magazine,
Jan. and Sept. 1875; Proc. of the Musical Association, 1874-5;
Stainer and Barrett's Dictionary of Musical Terms, Art. 'Tempera-
ment.'
ba
vin PREFACE.
paper of mine at the Musical Association, solemnly an-
nounced that I proved on a huge black-board that the
equal temperament scale was all wrong, when nothing of
the kind was even said, much less proved on the black-
board.
As emphatically as I disclaim all idea of proving the
equal temperament scale wrong by 'black-board' con-
siderations, so emphatically do I protest against the idea
that it is my object to abolish the equal temperament. It
is not the same thing to say. Let us make something new,
to wit B ; and, Let us abolish something old, to wit A. The
two things, A and B, are not mutually exclusive ; and there
can be no reason why they should not flourish together.
There is no force in the objection to the study of this
subject founded on the idea that its tendency is towards
the exclusion of the ordinary system. There will always
be the proper applications of the one arrangement, and the
proper applications of every other, just as there are the
proper applications of the different instruments in an
orchestra.
Dr. Stainer's* ' Harmony Founded on the Tempered Scale '
is the only work on technical music, so far as I know, which
takes up on this matter a position logically quite unassailable.
The position may be put thus : — our music is rtiade of certain
artificial things, twelve notes dividing the octave equally ; give
us these, and let us examine their combinations and capa-
bilities. This is a perfectly correct reduction of the harmony
of the equal temperament to a science of classification ; the
various combinations are in fact enumerated and classified.
Deeper than classification the work does not go.
The present work will be grounded on principles of a type
precisely analogous. I shall not attempt to enter into the
question of the physiological basis of harmony, or any of the
* Dr. Stainer informs me that this work does not represent his
present opinions : it forms however an excellent text for discussion.
PREFACE. IX
questions discussed by Helmholtz and others in connexion
with this part of the subject. I avoid the controversy, not
because I fail to have definite opinions on these points, but
because they are quite distinct from what I am deaUng with
here.
I shall assume, as matter of experience, independently of
any theory, that notes separated by certain musical intervals
form smooth combinations when sounded together ; and that
the accurate adjustment of an interval admits of variation
within certain narrow limits, without any serious injury to the
effect of the smooth combination. The provision of material
for such smooth or approximately smooth combinations,
or harmonious chords, forms the object of these investigations.
The law of vibration ratios is assumed, as being amply verified
by experience, and on it is founded a reduction of intervals
to equal temperament semitones, the theory of which I have
endeavoured to explain in an appendix, in the hope of making
it clear to those who are unacquainted with logarithms.
The proceeding of this treatise is therefore one of classifica-
tion ; but it is of a wider scope than Dr. Stainer's. That is a
classification of the material of one system, the equal tem-
perament. This is primarily a classification of systems, with
remarks on a few points connected with the separate classi-
fications, corresponding to Dr. Stainer's, of the material of
some of the principal systems. The treatment appropriate
to different systems differs widely. Such as are derived from
perfect or approximately perfect fifths and thirds require
different treatment from either the equal temperament, or the
class of systems of which the mean-tone system is the type,
and although this latter class admits of treatment by means
of the ordinary notation, yet the practical results differ so
far from those of the equal temperament that the best effects
are produced by a different style of handling. For instance,
in writing for the mean-tone system, it is advisable to avoid
the employment of fifths very high in the scale, where their
imperfection is most sensible.
X PREFACE.
The remarks made by musicians on enharmonic systems in
general are for the most part characterised by misapprehension
of the facts. For instance, at the beginning of Dr. Stainer's
preface, it is stated that, if an enharmonic scale were feasible,
doubtful chords could not exist, 'because mathematical
correctness of ratio would make every chord strictly in tune
in one key, instead of allowing it to be somewhat out of tune
in several keys. The whole of our musical literature, from
the works of Bach to those of Wagner, would therefore be
unavailable for instruments with an enharmonic scale.' Now
although it is true that the best applications of enharmonic
scales must be those which are made with reference to the
particular system employed, yet the particular objection made
above is quite unfounded, at least from my point of view.
A chord is in or out of tune quite independently of the key
in which it is supposed to be. And, as a matter of fact,
there is not the least difficulty in playing Bach on the mean-
tone system. The difficulty that arises in adapting music to
the class of systems which have perfect or approximately
perfect fifths and thirds, arises from a different source*; it has
nothing to do with the so-called enharmonic changes ; the
treatment of these rarely, if ever, presents difficulty ; and it is
generally practicable to arrange an enharmonic change so
that suspended notes undergo no change in pitch. In fact
the enharmonic change, has reference to the relation with the
preceding and following harmony, and does not generally
affect the distribution of the chord itself. Enharmonic changes
of pitch can however be occasionally used purposely with
good effect ; and the conjecture of Kirnberger in the motto
on the opening page has been to some extent verified.
Again, in Dr. Stainer's preface we find, ' The tempered
scale is certainly out of tune, and will not bear to have its
proportions exhibited to an audience with better eyes than
ears, on a white screen' [or black-board], 'but its sounds
* The false fifth in diatonic and allied scales ; (d — \ a in the key
ofc.)
PREFACE. xi
have nevertheless been a source of as rea_L pleasure Is^-^A"
great composers, as of imaginary pain to ce'rtain theorists.'
Now my own attention was first directed to this subject in an
entirely practical manner, viz. by taking part in the tuning of
my own organ (an ordinary instrument with two manuals and
pedal). The process of trial and error employed in tuning
the equal temperament on the organ throws into relief the
effect of the equal temperament modification on the chords ;
and this, in the tuning at all events, is very disagreeable.
I have never yet met any musician who was in the habit
of personally taking part in or superintending the complete
tuning of an organ, who did not agree that the imperfections
of the present methods are startling when thus encountered,
and that closer approximations to harmonious chords are a
matter of great interest. I must say that musicians thus
practically familiar with tuning are very rare.
Again : ' When musical mathematicians shall have agreed
amongst themselves upon the exact number of divisions
necessary in the octave ; when mechanists .shall have invented
instruments upon which the new scale can be played ; when
practical musicians shall have framed a new system of nota-
tion which shall point out to the performer the ratio of the
note he is to sound to its generator ; when genius shall have
used all this new material to the glory of art — then it will be
time enough to found a Theory of Harmony on a mathe-
matical basis.'
This admirable passage, which however contains some
confusion of ideas, was of great use to me in directing
attention to the principal points involved. The theory of the
division of the octave has now been completely studied;
a generalised keyboard has been invented and constructed
upon which all the new systems can be played; a notation
has been framed by which, in systems of perfect and approx-
imately perfect fifths and thirds, the exact note required can
be indicated, and it has been shown that other systems require
no new notation. (' The ratio of the note to its generator '
xii PREFACE.
arises from the notion of a harmonic scale ; but I have
not used anything of the kind, and it is incompatible with
derivation by division of the octave.) The new material may
be therefore said to be ready. But the idea that the theory is
to follow the practice is not true here ; for in this case a some-
what extended view of the theory has beeh necessary to
render the practice possible. This is the general course
where a science has a practical side: the practical side is
in advance up to a certain point in the history; the theory
lags behind. But it may always be expected that at some
point the theory may overtake the practice ; and then, and
not till then, is it capable of rendering useful assistance.
Take the example of astronomy. In Newton's time Flam-
steed, the observer, threw cold water on the theoretical
treatment of the moon's motions ; he said it never had been
of any use, and never would be. But now where would our
knowledge of this subject be but for the Lunar Theory?
Theory, and theory only, has succeeded in so far converting
the moon into a clock in the sky, that ships depend on this
means primarily for ascertaining their longitude.
Throughout the foregoing I have employed the word
theory in conformity with Dr. Stainer ; but I do so under
protest. Strictly speaking. Dr. Stainer's part of the subject,
the harmony of any system, is not a theory at all so long
as it is treated in the way in which he (quite correctly in my
opinion) treats it. It is a classification. The word ' theory'
includes the explanation of the facts by natural causes (the
reference to mechanical and physiological laws) ; it would also,
in its usual acceptation, include the deductive process here
employed, the sense of the word in this application being
analogous to its use in the expression ' Theory of Numbers.'
But we cannot call the process of classification of combinations
a theory, any more than we should call the classifications of
botany a Theory of Flowers. It is, however, merely a ques-
tion of the use of a word, and only becomes important when
it conveys the idea that by the process of classification we
PREFACE. xiii
have got to the bottom of the matter, a view which will be
admitted to be erroneous.
On the question of practical application I may summarise
shortly the principal points. I consider that the best applica-
tion of the mean-tone system will be to the organ with a
generalised keyboard of twenty-four keys per octave.
Of the positive systems, or those with approximately
perfect fifths and thirds, which require a new notation, I have
little doubt that the most obvious application, and one from
which we are not far distant, is in the orchestra. Instruments
are now being constructed with comma valves ; in designing
these, it must be always remembered that the comma devia-
tions must not be from equal temperament, but from perfect
fifths or perfect thirds. We have already a comma trumpet ;
some progress has been made with the clarionet, and on the
whole it seems likely that we shall have the instruments
ready before we have the compositions. The notation of this
work is the only one hitherto proposed which is competent
to deal with this question practically*- The example at the
end shows how it may be used. With the violin it will be
only a matter of study. Harmoniums, such as the large one
now in the Exhibition at South Kensington, besides the
interest and beauty of their effects, seem to be needed as
means for the study of the combinations of these systems ;
for it will not be possible to use these intervals effectively
in the orchestra unless the composer has made himself prac-
tically acquainted with their treatment.
The application of these perfect-fifth systems to the organ
would increase the bulk of the instrument too much in
proportion; and the instability of the tuning of organ pipes
renders it very doubtful whether a proportionate advantage
could be in this way obtained for a permanence. The mean-
tone system is more suitable for the organ, as being less
* Mr. Ellis's notation of duodenes, although theoretically a solu-
tion, cannot be said to be practical, in my opinion.
xiv PREFACE. ■
sensitive in the tuning, requiring only double the ordinary
number of notes per octave, and being remarkably easy of
performance on the generalised keyboard.
As to the pianoforte, the only application that it is at
all likely that it would be worth while to make is one of
the mean-tone system with twenty-four keys to the octave.
But it is only on the finest modern grand pianofortes that
the equal temperament is really offensive; and as these
instruments are generally used for purposes of display, when
the quality of the chords is not heard at all, even this
application cannot be regarded as likely to offer great
advantages.
In discussing the origin of the mean-tone system, I have
entered at some length into the history of the subject,
especially as regards its relations with Handel and Bach.
The statements about Bach conflict to some extent with
received opinions, but they will be found to be well supported.
The principal authority I have employed, which is not
generally accessible, is a life by C. L. Hilgenfeldt, published
by Hofmeister at Leipzig, for the centenary of Bach's death,
July 28, 1850; a work of great completeness, frequently
clearing up matters that are left obscure by Forkel as well
as by later biographers. -
In contemplating the imperfections great and small which
the science of acoustics show us beset all our ordinary instru-
ments, it is not unnatural to wonder that we get our music
so tolerable as on the whole we do. An eminent musician,
who objects strongly to acoustics in general, and to my
investigations in particular, is in the habit of saying, that
he never hears a lecture on acoustics without wondering
that we have any music at all. Now what are the facts ?
First who is to be the judge, and according to what
standard ?
As to who is to be the judge, I think that most musicians
will agree that those who have a very high development
of the sense of absolute pitch have their ears altogether
PREFACE. XV
more finely strung, and more acute, than other people. That
is to say, if a man can tell me the exact sound of c and
of any other note as he ordinarily uses them^ without having
any instrument to refer to, I consider that his musical or-
ganisation is such that his verdict on performances may be
accepted without hesitation, so far as their being in or out
of tune according to his standard is concerned.
As to the standard. The standard of all the musicians
of this class with whom I am acquainted is the equal tem-
perament ; and I think that the limit of the distinct per-
ception of error under ordinary circumstances is about —
of an equal temperament semitone. It is said that much
smaller intervals can be distinguished, but I doubt whether
this be the case under the unfavourable circumstances of
public performances.
With my experience of first-hand accounts of performances
from musicians of this character, it is quite as commonly the
case as not that performances, even by artists, orchestras,
or choirs of considerable reputation, are stigmatised as ex-
tremely defective in tune.
When we pass to the consideration of such errors as half
a semitone, which any competent musician can detect, I say
that it is rare to hear any performance of any kind in which
errors of this magnitude are not occasionally committed ;
especially by ordinary string quartetts, ordinary choirs, and
wind instruments in ordinary orchestras.
There are undoubtedly a certain number of fine organisa-
tions, whose instiricts and great technical mastery guide them
to a satisfactory result. And I take it, it is only in con-
sequence of the facilities which exist for employing every-
where those who are thus eminent in their particular lines,
that we have anything that can be called music in our
public performances. Of course I am speaking only of in-
struments where the intonation is either made by the player,
or depends on his care.
XVI PREFACE.
The method of which the present investigations form one
branch, will try to meet the difficulty in question by making
a special study of small variations in interval, with the view
of adopting systematic methods for the attainment of what
has hitherto only been possible with the assistance of great
natural gifts.
The first thing to be attended to in this more general
view of the subject is the effect of temperature.
It is not possible to go in detail into this subject here;
I will only mention one thing. It admits of being proved
that if the relation be assigned between the source of heat
in action, and the means for its dissipation (radiation &c.),
then there will be a temperature to which each portion of
the space considered will rise, and at which it will remain
steady; as much heat being parted with in every instant
as is derived from the source. We may apply the principle
to such an instrument as the clarionet. The steady tem-
perature of this instrument must be but little below that
of the breath. But whatever the steady temperature is under
given circumstances, the rule should be, raise the instrument
to the steady temperature before the performance begins, or
rather before the tuning is effected. A cupboard at a
regulated temperature would effect this completely.
The actual work done by me, which this treatise is in-
tended to illustrate, consists of the construction of the
harmonium and enharmonic organ, which are mentioned on
the title page and described in their places. I have had
frequent opportunities of letting visitors to the Loan Ex-
hibition at South Kensington hear the harmonium, and the
recognition of its success with respect to the purity of the
chords is all that I could desire. The enharmonic organ
was exhibited to the Musical Association immediately after
it was finished, and has since that time stood in my rooms
at Oxford where it now remains.
St. John's College, Oxford,
1876.
CONTENTS.
CHAPTER I.
HARMONIOUS CHORDS.
PAGE
Intervals defined. Consonant intervals distinguished. The
properties of consonances the basis of harmonious music . i
Ternperament defined. Conception of Major Scale. Con-
ception of Minor Scale 2
Duodenes .... 3
CHAPTER II.
EQUAL TEMPERAMENT.
Equal temperament semitones. Fifths. Thirds ... 4
Departure and Error defined. Note on tuning the equal
temperament 5
Experiment in tuning ........ 6
CHAPTER III.
PERFECT FIFTHS. PYTHAGOREAN SYSTEM.
Experiment on circle of twelve fifths. Major tone . . 7
Pjrthagorean comma. Pythagorean or dissonant third.
Ordinary comma ........ 8
Apotomfe Pythagorica. Pythagorean semitone. Pythagorean
sixth. Note on diatonic and chromatic semitones . . 9
Hebnholtz's Theorem. Note on Relations of semitones,
comma, and skhisraa . . . . . . . 10
Notation for series of fifths. Rule for thirds . . . . 1 1
System of 53 ......... 12
Note on the calculation of Intervals 14
Note on the calculation of Beats 15
Beats of equal temperament fifth. Rule of difference tones . 1 7
Beats of difference tones of equal temperament triad . . 18
xviii CONTENTS.
CHAPTER IV.
THE GENERALISED KEYBOARD.
Arrangement hy equal temperament position, and departure . 19
Dimensions. Symmetry in all keys ..... 20
General applicability to systems
Diagram I. Notes of Thompson's Enharmonic organ
Diagram II. Plan, section, and elevation of a portion of
generalised keyboard 23
CHAPTER V.
24
25
26
27
31
32
MEAN-TONE SYSTEM. OLD UNEQUAL TEMPERAMENT.
Rule of system. History ......
System of Handel. Smith's note on its origin
Additional keys in Temple organ . . • .
Wolf. Bach. The clavichord considered as his instrument
The organ considered with reference to him
Analysis of keys of Bach's organ compositions
CHAPTER VI.
APPLICATIONS OF THE MEAN-TONE SYSTEM.
Rule of system 33
Diagram III, arrangement by fifths and departure . . 34
Diagram IV, arrangement by E. T. semitones and departure . 34
Departure of fifth. False fifths and thirds (Wolf) . . 35
Smith's changeable harpsichord. Ellis's harmonium . . 37
Application of generalised keyboard 37
Rule for sharps and flats. Mean-tone stop on enharmonic
organ .... 38
Tuning 40
CHAPTER VII.
HARMONIC SEVENTH.
Departure of harmonic seventh. Approximation by perfect
fifths ^i
Rule prohibiting suspension. Approximation by mean-tone
system 4a
CONTENTS. XIX
CHAPTER VIII.
APPLICATIONS OF THE SYSTEM OF PERFECT FIFTHS AND ALLIED SYSTEMS.
Approximation of Helmholtz assumed. Perfect fifths most page
important, as most easily tuned. Unsymmetrical arrange-
ments 43
Mersenne's system of eighteen notes. Thompson's En-
harmonic organ 44
Key relationship symmetrical arraiigements, Mr. Poole . . 45
Diagram V. Arrangement of key-notes. Poole and Brown . 46
Diagram VI. Arrangement of auxiliaries. Poole . . 47
Diagram VII. Arrangement of auxiliaries. Brown . . 48
Symmetrical arrangement by intervals. Duodenes. The
generalised keyboard ....... 50
CHAPTER IX.
ENHARMONIC HARMONIUM. ENHARMONIC ORGAN, POSITIVE STOP.
Keyboard of harmonium. Arrangement of system of 53.
Scales made up of semitones 51
Relation between magnitude of semitones formed by perfect
fifths. Number of semitones that make an octave. De-
duction of system of 53 . 52
Rule for identifications in system of 53 . . . . 53
Diagram VIII. Distribution of system of 53 on keyboard • 54
Diagram IX. Succession of notes in the system of 53 . . 55
Enharmonic organ, positive stop 56
Helmholtz's system, or positive system of perfect thirds . . 57
Applications of perfect fifth systems to harmonium and
orchestra rather than organ. Description of enharmonic
organ 58
CHAPTER X.
GENERAL THEORY OF THE DIVISION OF THE OCTAVE.
Definitions. Regular Systems. Regular cyclical systems.
Error. Departure. Equal temperament. Expression of
intervals. Positive. Negative. Order of systems . . 60
Regular Systems : —
Theorem I, On semitones in octave . . . 61
Theorem II. On difference of semitones . . . 61
XX CONTENTS.
Regular Cyclical Systems : — paqe
Theorem III. On difference of semitones . . 62
Cor. Peduction of systems of first and second order . 62
Theorem IV. On departure of fifth. Approximate
identity of system of 31 with mean-tone system . 62
Theorem V. Order condition. Order of system of 301 63
Theorem VI. On total departure of all the fifths of a
system. Deduction of systems of first and second
orders 64
Theorem VII. Number of units in semitones. Re-
sulting expression integral when order condition is
satisfied 65
Theorem VIII. Thirds of negative systems. Deduc-
tion of mean-tone system (negative perfect thirds) . 65
Theorem IX. Thirds of positive systems. Deduction
of Helmholtz's system (positive perfect thirds) . . 66
Theorem X. Helmholtz's Theorem .... 66
Theorem XI. Harmonic seventh in positive systems 66
Theorem XII. Harmonic seventh in negative systems 66
Table of errors of concords of regular and regular
cycUcal systems 67
Theorem XIII. On the complete symmetrical arrange-
ment formed from the system of 612 ... 68
CHAPTER XI.
MUSICAL EMPLOYMENT OF POSITIVE SYSTEMS HAVING PERFECT
OR APPROXIMATELY PERFECT FIFTHS.
Example. Double second of the key . . . . . 69
Minor third 70
Major third. Depressed form of dominant . . . . 71
Minor sixth. Major sixth. Minor seventh. Major seventh . 72
Positive systems require special treatment, Examples of dif-
ficulties in adaptation of ordinary music . . . . 73
Use of elevation and depression marks in the signature . . 74
Discussion of points in example . . . . . . 75
Comma scale. Series of major thirds . . . . . 76
Musical example 77
APPENDIX.
On the theory of the calculation of intervals . . . . 81
CHAPTER I.
HARMONIOUS CHORDS.
When two musical notes are regarded relatively to each
other, they are said to form an interval.
When two musical notes are sounded together, there are
two principal cases which may occur : —
(i) The effect of the combination may be smooth and
continuous ;
(a) The combination may give rise to beats, or alternations
of intensity or quality, more or less rapid.
(i) The principal intervals for which the effect is always
smooth and continuous, whatever be the instrument em-
ployed, are known to musicians under names of the Octave,
Fifth, and Major Third. These are called consonances.
All other consonances can be derived from these. Fourths
to the bass and Harmonic Sevenths may form also smooth
combinations, but are not called consonances by musicians.
(3) If the interval between two notes be nearly but not
exactly a consonance, or a harmonic seventh, beats may be
heard.
The foundation of modern harmonious music may be said
to be the smoothness of the consonant intervals, and of inter-
vals which differ from these only by small magnitudes. The
harmonic seventh has not yet been employed in modern
music.
To provide a material of notes for musical performance it is
theoretically requisite in the first place that to every note used
we should possess octave, fifth, and major third, up and down.
Each of these being a note used we may require the same
accessories to each, and so on.
B
HARMONIOUS CHORDS.
It is universally agreed that the octaves used should be
exact ; but for the most part the fifths and thirds and their
derivatives are made to deviate from their exact values by
small quantities. These deviations constitute ' Temperament.'
The interval of a fifth is easy to tune exactly with all
ordinary qualities of tone; it is also easy to make a fifth
sharp or flat by any given number of beats per minute. For
this reason, as well as others, the relations of fifths are pre-
ferred for discussion.
Conception of a Major Scale.
The Major Scale may be conceived of as consisting of two
sets of notes, first the key-note and notes immediately related
to it by fifths, and secondly notes related to the first set by
thirds. Thus in the key of C we have
F-C-G-D
related by fifths ; and
A-E-B
which form major thirds above the first three.
Conception of a Minor Scale.
The Minor Scale may be conceived of els consisting of the
four first notes, together with the major thirds below the
three C—G—D; namely,
Al' - Eb - Bb
If we write down all these notes so that the fifths are
counted upwards and major thirds sideways, we have the
following scheme, omitting the two notes in brackets : —
Bb
D
[F»]
Eb
G
B
Ab
C
E
[D^]
F
A
HARMONIOUS CHORDS.
If we then fill up the bracketed places, we have a scheme of
twelve notes, corresponding in name to the twelve notes in
ordinary use, but forming a system of consonant chordsj
including the principal chords of the keys of C major and
minor, with some others. Such an assembly of notes Mr. Ellis
calls a duodene * ; and in particular the notes above written
are said to constitute the duodene of C.
It will be seen that, by specifying the duodene in which
any combination of notes is to be taken, the exact notes to be
performed can be indicated ; and the indications can be
obeyed if the notes constituting all the duodenes required are
provided. The system of duodenes forms a practicable
method by which rigorously exact concords can be employed
and controlled.
If however we provide all the notes necessary for an
extended system of duodenes, we have endless series of fifths
running up and down, and endless series of thirds running
horizontally; and it is possible to show that no two of the
notes will ever be exactly the same in pitch. Consequently
in practice various approximations are employed, so as to
reduce the number of notes required.
* Proc. Royal Society, Dec. 1874. The word may be taken to
mean a set of twelve notes.
B %
CHAPTER II.
EQUAL TEMPERAMENT.
This is the method of tuning in ordinary use for keyed
instruments. The simplest way of considering it is to observe
that the interval of any octave on keyed instruments is made
up of twelve equal semitones, thus —
c — c*— d — d' — e— f— f— g— g*— a — a* — b — c.
We shall regard equal temperament semitones simply as
intervals, twelve of which make an octave. And we shall
in future reckon intervals in equal temperament semitones
(E. T. semitones). These are the semitones of the pianoforte
and organ as ordinarily tuned. E. T. is used as an abbre-
viation for the words ' equal temperament.'
Perfect Fifths and Thirds.
It has been long known that octaves have the vibration
ratio 2 : 1, perfect fifths the vibration ratio 3 : 2, perfect thirds
the vibration ratio 5:4; also that the logarithms of vibration
ratios are measures of the corresponding intervals. From
these principles it is possible to show * that —
the perfect fifth is 7.01955 E. T. semitones (say 7—) ;
51 ,
the exact major third is 4 — .13686 E. T. semitones (say
*-7^3)- _
The E. T. fifth is seven semitones, so that the perfect fifth
is a very little greater than the fifth of an ordinary keyed
instrument. The E. T. third is four semitones ; so that the
perfect third is a little less than the third of an ordinary keyed
instrument.
* See note at the end of Chap. III., and Appendix.
EQUAL TEMPERAMENT.
The deviation from E. T. values may be called Departure.
Thus the perfect fifth is said to have the departure — - upwards
51
from E. T. Deviation from exact concords may be called
Error ; thus the error of the E. T. fifth is •— down. So the
51
departure of the exact third is — down ; and the error of the
7.3
E. T. third is — - up.
7.3 ^
The practical effect of these deviations is that the E. T.
fifth has to be made about one beat per second flat in the
lower part of the treble. The beats of a simple third are
generally difl[icult to distinguish clearly as beats; their
number in the same region is about ten per second. The
beats of the first combination tones of an E. T. major triad
in the same region are about five per second*. The error
due to thirds is considerable in the equal temperament ; that
due to fifths is small in comparison.
To obtain some practical idea of the difference in the sound
* For tuning equal temperament with accuracy the following table
may be employed, c'=264 : —
Beats per minute of flat fifths.
c'-g'
634
f»— c"»
75-6
c'lf-g'»
57
g'-d"
80-4
d'— a'
60-3
a'»— e"b
85.2
e'b— b'b
64-2
a'— e"
90
e'— b'
67-5
b'b f"
95-4
f'-c"
71-6
b'-fl
101-4
Proceed in order of fifths, thus ; c'— g' — d", then octave down d" — d',
and so on.
Mr. Ellis has given a useful practical rule, which is more manage-
able than the above, and does not err in its results by much more than
the hundredth part of a semitone. It is— make all the fifths which lie
entirely within the octave c' c" beat once per second ; and make those
which have their upper notes above c" beat three times in two seconds.
Keeping the fifth f — c" to the last, it should beat once in between one
and two seconds. See Ellis's Helmholtz, p. 785. This is a perfectly
practicable rule, and tuners ought to be instructed in the use of it.
There are few tuners who can produce a tolerable equal temperament.
EQUAL TEMPERAMENT.
of equal temperament chords and perfect chords, the simplest
thing to do is perhaps to take an ordinary harmonium and
tune two chords perfect on it. One is scarcely enough for
comparison. To tune the triad of C major first raise the
G a very little, by scraping the end of the reed, till the fifth
C — G is dead in tune. Then flatten the third E, by scraping
the shank, until the triad C — E— G is dead in tune. (When-
ever a third is to be tuned perfect, a perfect fifth ought to
be made, and the third tuned in the middle of the triad.)
Then flatten F till F-C is perfect, and A till F-A-C
is perfect. The notes used are easily restored by tuning to
their octaves. Any small sharp chisel will do to tune with ;
a thin and narrow strip of steel or stiff card is useful to place
under the reed so as to hold it fast. The pure chords ob-
tained by the above process offer a remarkable contrast
to any other chords on the instrument.
This experiment is perhaps the most striking practical
mode of shewing that chords formed by the notes ordinarily
in use are much inferior in excellence to chords which are in
perfect tune.
CHAPTER III.
SYSTEM OF PERFECT FIFTHS. PYTHAGOREAN SYSTEM.
Certain intervals produced by tuning perfect fifths bear
the name Pythagorean. The tuning of exact fifths on the
harmonium is very easy and certain ; and it is recommended
that the observations about to be made be thus verified
experimentally.
Tune the following set of twelve exact fifths or fourths,
gb — db — s,b — el' — bl' — f — c — .g — d — a — e — b — f*. Then
f* will be higher than gl' by a small interval called the
Pythagorean comma.
For this purpose it is convenient to have a harmonium with
two sets of reeds. The f* can then be tuned on the second
set by tuning its b first to the b on the first set. If there
be only one set of reeds, the gl' and f* must be taken an
octave apart.
Major tone.
c — d is a major tone of the theorists. Then six major
tones exceed an octave by the Pythagorean comma.
c — d is arrived at by tuning two fifths up and one
1 2
octave down ; or 2 x 7-— — 12 = 2— -•
51 ol
2
Six major tones =6x2—
12
and — - is the Pythagorean comma.
51
SYSTEM OF PERFECT FIFTHS.
Pythagorean comma.
We can deduce the value of the Pythagorean comma,
or the departure from E. T. of twelve perfect fifths, directly,
by supposing that we tune twelve fifths up and seven octaves
down. 2 J 2
12X7--7X12=-,
or a little less than - of an E. T. semitone. The accurate
4
value is 12 x. 01955 = . 23460.
(Note. — The fifth is accurately , and we must remember
o 1.1 1
that we use — - as an approximation only.)
51
Pythagorean third.
f— a, c — e, g— b, and other thirds arrived at by tuning four
fifths upwards, are very sharp, and are called Pythagorean
thirds, or sometimes ' dissonant ' thirds. Using the chords
c — e — g, c — f — a, we hear the disagreeable effect of the
sharp third.
Ordinary comma.
The comma is defined as the difference between the Pytha-
gorean third and the perfect third.
1 4
The Pythagorean third is 4 x 7— —2 x 12 = 4 — ;
O i. o 1
the perfect third is 4 :
^ 7.3'
so that the comma becomes
4 11
M+^=4:6""^''^y5
or, using the more convenient and accurate decimal values,
4 X. 01955 + .13686 = . 21506.
SYSTEM OF PERFECT FIFTHS.
This number may also be found by the rule at the end of this
— ) of the ordinary comma.
Apofonie Pythagorica.
The semitone formed by tuning seven fifths is given by b''— b.
It is distinguished by some of the older theorists as Apotome
Pythagorica.
Its magnitude is
1 7 1
. 48 = 1 —
51 51
In decimals, 1.13685.
7x7;^- 48 = 1-:^ or 1- nearly.
Pythagorean Semitone.
The semitone formed by tuning five fifths is given by a— b''.
It has been called the Pythagorean semitone.
1 5
Its magnitude is 36— 5x 7 — =1— — - ;
51 51
i. e. it is less than the E. T. semitone by about t^; of a semi-
tone*.
In decimals, 1— .09775.
Pythagorean Sixth.
The interval c — a is commonly called a Pythagorean sixth.
1 3
Its value is 3x7— -12 = 9— ;
51 51
i. e. it exceeds the E. T. sixth by 7- of a semitone.
17
* The diatonic semitone of theorists is the diiference between an
exact third and fourth, or (5— 01955)-(4--13686)=l-11731 = l^
nearly. The chromatic semitone is the difference between this and a
major tone,=2-03910— M1731 = l--0782]=l-r-g nearly.
JO SYSTEM OF PERFECT FIFTHS.
Helmholtzs Theorem.
The following theorem has been brought into notice by
Helmholtz. In a series of perfect fifths, any two notes eight
steps apart determine a major third which is nearly perfect.
Thus if we take an f ' identical in pitch with gl', c' with d'',
g' with a I', and d' with et", then d— f*, a — c*, e — g', b — d',
are very nearly perfect thirds.
For five octaves less eight fifths
= 60-8x7^
51
= 60-56—
51
8
or the third thus obtained is less than the E. T, third by
8 1 ,
— or — nearly.
51 6.4 ^
But the perfect third is less than the E. T. third by
1
The difference of these numbers is — - nearly * ; or nearly the
51
same as the error of one E. T. fifth, and - of the error of
7
the E. T. third. The chords formed with the above thirds
* In decimals 8 X 01955 = -15640
•13686
•01954.
This quantity is sometimes called a skhisma.
Note on Relations of Semitones, Comma, and Skhisma.
Diatonic Semitone = Pythagorean Semitone + Comma
= Apotomb Pythagorica — Skhisma.
Chromatic Semitone = Apotomb Pythagorica — Comma.
= Pythagorean Semitone + Skhisma.
Pythagorean Comma = Comma + Skhisma.
= Apotomfe Pythagorica — Pythagorean
Semitone.
These identities are easily verified by means of the decimal values.
SYSTEM OF PERFECT FIFTHS. 1 1
are therefore very nearly perfect ; and the experiment enables
us to contrast in an effective manner Pythagorean chords with
chords of a good quality.
Notation for series of Fifths.
As the f*, c', g", and d' above mentioned are identical in
pitch with gi', dl?, 2!", and e'', respectively, it is necessary to
adopt some notation to distinguish these notes from those on
the right hand of the series of fifths, which are derived from
rnodulation through upward fifths, and differ in pitch from
the notes last introduced, though bearing the same names.
The notation employed is as follows, if the whole series be
linked by exact fifths, and supposed indefinitely extended
according to the same law in both directions. The E. T. names
(i' or »), are used indifferently. v\b— \\f»— Vk.c*— '\\g«— ■>\dlf
— wbl'— vwf— \\c— \\g— >\d— >\a— We— \\b— \f,*— \c'— \g'
— \d'— \bl'— \f— vc — vg— vd— \a— \e— \b^f.'— c'— g'-
dJ— bl'— f-c— g— d— a— e— b-/f*— /c»— /g'— /d*— /bl'—
/f— /c— /g— /d-/a— /e— /b— //f«.
The notes comprised in any one series of twelve fifths from
f* up to b, all bear the same mark. In the middle there is a
series without marks ; as we pass to the left we have series
with one or more marks of depression (\) ; as we pass to the
right, with one or more marks of elevation (/). The ordinary
names (* or '') are used indifferently; the notation alone marks
the position in the series of fifths.
In each series we have four major thirds such as d— f,
a— c», e-g*, b— d*.
In each pair of adjoining series we have eight major thirds
such as gl'-\bl', d*"— \f, 2>—\c, el"— vg, b*'— \d, f-\a, c— \e,
g-vb.
We may embody this in the rule, that the four notes to the
right of any series form thirds with the four notes to the left ;
but all other thirds lie in adjoining series. The four notes to
the right, which have their thirds in their own series, are the
letters of the word head, which may be employed for the
purpose of remembering them.
12
SYSTEM OF PERFECT FIFTHS.
This notation can be used in the musical staff; and some-
thing of the kind is essential when thirds formed by eight
fifths are employed.
Example.
■ ^^^
System of 53.
It is easy to see that the division of the octave into fifty-
three equal intervals has very nearly perfect fifths, without
going into any general theory. For taking thirty-one units
for the fifth, twelve fifths make 372 units,
and seven octaves = 7x 53 = 371 units ;
or twelve fifths exceed seven octaves by one unit ;
12
and one unit is — - of an E. T. semitone.
53
But the excess of twelve fifths over seven octaves is the
departure of twelve fifths from E. T.
(for in E. T. twelve fifths = seven octaves).
12
Hence the departure of twelve fifths =—;
whence departure of one fifth =— ; a simple and elegant
result.
53
But departure of perfect fifth = — nearly.
Therefore error of fifth of 53 =
51 53
2703
1
1352
nearly,
or less than the one thousandth of a semitone ; an inappreciable
error.
SYSTEM OF PERFECT FIFTHS. 1 3
We may here point out that the diatonic semitone and
apotom^ Pythagorica are both closely represented by five
units of the system of 53, and the chromatic semitone and
Pythagorean semitone by four units of the same system. For
more extensive comparisons of this kind see Stainer and
Barrett's ' Dictionary of Musical Terms,' p. 423.
In 'Hopkins on the Organ,' 2nd edition, p. 160, there are
some small inaccuracies on this subject which it may be as
well to correct. The statement about 'Tempering,' p. 161,
will be alluded to in connection with the mean-tone system.
(1) The comma is identified with the 53rd part of an octave.
This is not correct. Dividing 12 by "21506 (the value of a
comma), we find that 55'8 commas very nearly make an
octave. (2) ' The successive sounds of the diatonic scale have,
by the aid of these commas^ been shown to be separated by
intervals of the following " sizes" or comparative dimensions.'
Then the scale is set out, with the major tone represented by
nine units, the minor tone by eight, and the diatonic semitone
by five. This is the scale of the system of 53, as is easily
seen by counting up the intervals. But it is not the diatonic
scale, only an approximation to it, and the difference of the
thirds and sixths in the two is very sensible. The diatonic
scale is such that thirds fifths and sixths are perfect. The
scale of 53 coincides very closely with that of a system of
perfect fifths, but its thirds and sixths are not very close ap-
proximations to those of the diatonic scale, though sufficient
for many purposes.
In fact, by Helmholtz's theorem, the major third determined
by notes eight steps apart, in a series of perfect fifths, is
too flat by nearly the same quantity as the equal tem-
perament fifth, or .01954. Although we may frequently
neglect this small error, and establish on this neglect practical
approximate methods of importance, yet a fundamental ex-
position in which it is entirely overlooked can only be
regarded as erroneous. The state of things in the system of
53 is very nearly the same as in the system of perfect fifths.
The exact values are easily calculated.
14 NOTE ON THE CALCULATION OF INTERVALS.
Those who are not familiar with the properties of logarithms, are
referred to the Appendix, on the theory of the Calculation of Intervals.
Note on the Calculation of Intervals.
To transform the logarithms of vibration ratios into E. T. semi-
tones.
The vibration ratio of the octave is 2 ; the logarithm of 2 is
•3010300 ; and we admit that E. T. intervals are in effect a system of
logarithms such that 12 is the logarithm of the octave, or of 2. Then,
since different systems of logarithms can always be transformed one
into another by multiplication by a certain factor or modulus, we have
only to find the factor which will convert -3010300 into 12. The
simplest proceeding which embodies this process directly and con-
versely is given in the following rules, which admit of transforming
the logarithms of vibration ratios into E. T. semitones, and vice versd,
with considerable accuracy and facility. For examples worked out at
length, see Proceedings of the Musical Association, 1874-5, p. 7.
Rule I. To find the equivalent of a given vibration ratio in E. T.
semitones.
Take the common logarithm of the given ratio; subtract —
thereof, and call this the first improved value (F. I. V.). From the
original logarithm subtract — of the first improved value, and
of the first improved value. Multiply the remainder by 40. The
result is the interval expressed in E. T. semitones correctly to five
places.
Rule II. To find the vibration ratio of an interval given in E. T.
semitones.
To the given number add -— and of itself. Divide by 40,
oUU lUUUU
The result is the logarithm of the ratio required.
For approximate work a simpler and less accurate form is some-
times useful ; for this purpose the rules can be modified as follows : —
Approximate Rule I.
To find the equivalent of a given vibration ratio in E. T. semitones,
where not more than three places are required to be correct.
Take the common logarithm of the given ratio ; subtract —
NOTE ON THE CALCULATION OF BEATS. 15
thereof, and multiply the remainder by 40. The result is the interval
in E. T. semitones correctly to three places.
Example. To find the approximate value of a perfect fifth, the
3
vibration ratio of which is -, and of a perfect third, the vibration ratio
of which is - :
log. 3 = -47712
log. 5 = -69897
log. 2 = .30103
log. 4=.60206
log. ^=-17609
log. ^=-09601
i-=.00059
300
3^0=-°«°^^
-17550 -09659
40 40
7-02|000 386360
or, 4--136|40
The correct values are 7-0195500, and 4 — -136863.
Approximate Rule II.
To find the vibration ratio of an interval given in E. T. semitones,
where not more than three places are required correct.
To the given number add r— of itself. Divide by 40. The result
oOO
is the logarithm of the ratio required.
Example. The E. T. third is 4 semitones.
J 4-000
300= _^3
40) 4-0i3
-1003=log. 1-259
The correct value is 1-259921.
N.B. The approximate rules are insufficient for the calculation of
beats.
Note on the Calculation of Beats.
It is frequently necessary for the solution of problems in tuning to
calculate the number of beats per second or minute made by imper-
fect unisons, fifths, or thirds.
The following are principles which we shall admit for purposes of
calculation. For their more detailed treatment reference is made to
l6 NOTE ON THE CALCULATION OF BEATS.
Ellis's translation of Helmholtz; or to the elementary works ofTyndall
and Sedley Taylor on the theory of sound.
Musical notes reach us as periodic impulses of the air. The pitch
of the note depends 'on its number of vibrations per second ; and the
interval between two notes depends on the ratio of the vibration
numbers. A note may be regarded as generally containing many
notes of the simplest kind, frequently called simple tones. The
lowest of these is that with which we identify the compound note ; it
is called the fundamental ; the remainder are called harmonics ; and
the forms they can take are as follows, the successive tones which
make up the note being enumerated in a sequence which is called
their order.
Interval from Order and proper-
Fundamental tionate vibration Name.
in semitones.
number.
1
Fundamental.
12.00000
2
Octave.
19.01955
3
Twelfth.
24.00000
4
Fifteenth or double octave,
28— .13686
5
Tierce (octave tenth).
31.01955
6
Octave twelfth.
34— .31174
7
Harmonic seveiith.
36.00000
8
Triple octave.
First Rule of Beats.
When two simple tones are near together in pitch they give rise to
alternations in intensity called beats : the number of beats is the dif-
ference of the vibration numbers ; and the two are said to interfere.
Second Rule of Beats.
When two compound notes form any consonant interval, -two of
their harmonics coincide in pitch ; and if the interval is not exact, the
two harmonics coincide very nearly, and give beats according to the
first rule.
Examples. — Two notes whose vibration numbers are 32 and 34 per
second are sounded together ; resulting beats of fundamentals are two
per second. (Rule 1.)
In the same two notes, the vibration numbers of the twelfths are
96 and 102, and the beats due to the twelfths are six per second
(Rule 2.)
NOTE ON THE CALCULATION OF BEATS. ly
This is easily verified by sounding simultaneously the lowest c and
c* of a Pedal Bourdon on the organ. The Bourdon note contains no
octa,ve ; and the two classes of beats above mentioned combine to
produce an effect like U — u — u — U— u — uin every second^
where U is the beat of the fundamentals, u of the twelfths.
To find the beats per minute of the equal temperament fifth c'g'. (c'=
256.)
The octave of g' will interfere With the twelfth of c', the two notes
being separated by the interval -01955 of a semitone, by which the
equal temperament fifth is flat, since an exact fifth contains 7-01955
semitones.
To find the logarithm of the ratio for the interval -01955, we
proceed by Rule II of the preceding note.
•0195500
•0000652 = 4
19 = -
10000
40>0196171
•0004904
^logarithm of vibration ratio for -01955 semitones.
The vibration number of g" derived from c' is 768.
log. 768 = 2-8853613
log. ratio = -0004904
log^ vibration number of tempered g" = 2-8848709
= log. 767-133
The number of beats per second is the difference of the vibration
numbers
768-000
767-133
-867 = number of beats in one second
60
52-02 =number of beats in one minute.
Rule of Difference Tones.
When two tones sound together a third is produced, whose
vibration number is the difference of those of the first two.
Examples — To find the difference tone of the equal temperament
major third c' — e'.
C
1 8 NOTE ON THE CALCULATION OF BEATS.
By Rule II. of preceding note we find for the correct logarithm of
the ratio of the tempered third,
• 1003433
also log. 256=2-4082400
2-5085833=log. 322-54
which is the vibration number of the tempered e'.
322-54
Whence, for the difference tone — 256-00
66-54
64 would be C ; this is about half a semitone sharper.
Again, to find the difference tone of the equal temperament minor
third, e' — g',
g'is 383.57
e'is 322.54
difference tone 61.03
Beals of Difference Torus.
Difference tones which lie near each other in pitch interfere and
cause beats. %
Example. — To find the beats of the difference tones of the equal
temperament triad c' — e' — g'.
By the two last examples the difference tones of the major and
minor thirds lie near each other ; they are about a semitone apart.
Taking the difference of their vibration numbers, we have
66.54
61.03
5.51
or 5^ beats per second nearly.
CHAPTER IV.
THE GENERALISED KEYBOARD.
In the enharmonic harmonium exhibited by the writer
at the Loan Exhibition of Scientific Instruments, at South
Kensington, 1876, there is a keyboard which can be employed
with all systems of tuning reducible to successions of uniform
fifths ; from this property it has been called the generalised
keyboard. It will be convenient to consider it in the first
instance with reference to perfect fifths ; it is actually applied
in the instrument in question to the division of the octave
into fifty-three equal intervals, which has just been shown
to admit of practical identification with a system of perfect
fifths.
This keyboard is arranged in a symmetrical manner, so that
notes occupying the same relative position always make the
same interval with each other. The requisite minuteness is
secured by providing two separate indications of the position
of each note, the one referring to its position in the E. T.
scale, the other to its departure from the E. T. position.
As to the position in the E. T. scale. Suppose the broad
ends of the white keys of the piano to be removed ; the
distance of the octave from left to right is then occupied by
twelve keys of equal breadth, seven of which are white, and
five black. This is the fundamental division of the new key-
board on any horizontal line. The order of black and white
is the same as usual.
But we have also to express departure from any one E. T.
system ; and this is done by placing the notes at different
distances up and down in these divisions. Apply this to the
series of exact fifths, starting from c.
c-gis7^:
C 3
ao THE GENERALISED KEYBOARD.
7 corresponds to the E. T. g, and would be denoted by
a position in the g division on a level v^fith c.
But our note is — - higher.
51
In the keyboard itself this is denoted by placing the g key
- of an inch further back and — of an inch higher than the c.
Similarly d departs from E. T. by twice as much ; it is .
placed - an inch back and - of an inch higher than c, and so
on ; every note determined by an exact fifth being placed
- of an inch further back and — of an inch higher than the
4 12
note which immediately precedes it in the series of fifths.
Thus after twelve fifths, when we come to /c, we find it dis-
placed three inches back, and one inch upwards ; a position
which admits of its being represented by a key placed behind
and above the c key from which we started. The general
nature of the arrangement will be best gathered from the two
accompanying illustrations. The first is an arrangement of
the notes of Gen. T. Perronet Thompson's Enharmonic Organ,
in a symmetrical form according to the above principle. The
instance is selected as being of historical interest. Each
vertical step from dot to dot may be taken for present
purposes to represent the departure of an exact fifth, or
■— - E. T. S. nearly. Two notes are missing from the
complete scheme, b and >\d ; their importance is well seen.
The second diagram represents a small portion of the
generalised keyboard itself. It will be desirable here to fix
in the mind the conception of the latter as constituting a
mechanical means by which an endless series of uniform fifths
can be controlled.
But the most important practical point about the keyboard
arises from its symmetry ; that is to say, from the fact that
every key is surrounded by the same definite arrangement of
keys, and that a pair of keys in a given relative position
corresponds always to the same interval. From this it follows
that any passage, chord, or combination of any kind, has exactly
THE GENERALISED KEYBOARD. %l
the same form under the fingers in whatever key it is played.
And more than this, a common chord for instance has always
exactly the same form, no matter what view be taken of its
key relationship. Some simplification of this kind is a necessity
if these complex phenomena are to be brought within the
reach of persons of average ability ; and with this particular
simphfication, the child or the beginner finds the work reduced
to the acquirement of one thing, where twelve have to be
learnt on the ordinary keyboard.
Hitherto it has been assumed that we were dealing with
perfect fifths ; but it is clear that this is not a necessary
condition ; the keyboard will serve to represent any con-
tinuous series of fifths which keeps on departing from the
E. T., even though they should be flatter than the E. T. fifths.
As an illustration of the generality of its properties we will
consider its application to the mean-tone system. But as
this system is one of the most important with which we have
to deal, we will first devote some space to an account of
its history and properties. We shall recur later to the
properties of systems of perfect and approximately perfect
fifths.
22
I.
Symmetrical Arrangement of the Notes of Thompsoiis
Enharmonic Organ.
The subscripts „ ^, 3 refer to its three keyboards.
12.
/c. .
/c.
11.
,
.
/f. . . .
10.
, .
/bb, .
9.
/eb,
8.
/ab,
7.
/«e,
,
6.
.
. /fJi .
6.
4.
62
3.
a.1
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2.
d,
3
1.
,
Bi. V s
12.
0„j,s .
^H 2>S
11.
f u 21 3 ■
10.
bb„, .
9.
,
eb„3 .
8.
aba
7.
. oj,
6.
f«l,2 .
6.
. \b,
2t 3
4.
.
. \e..„
3.
. \a,
2> 3 •
2.
. \d,
2, B .
1.
. \gl,2,J .
12.
\o, .
\o.
11.
\f, . . .
10.
,
Vbbj .
9.
Veb, .
8.
. Xab,,,
7.
. Xojj,,
6.
. \f)I., .
5.
,
. \\b
2) >
4.
We,
3.
.
. Was .
2.
•
1.
. W& .
12.
Wo, .
Wo.
11.
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.
10.
.
Wbb, .
9.
Web, .
.
8.
. Wab.
, ,
7.
Wok,
•
•
23
II.
Section.
ic.
IC"
EleVftbloTV
'/
\c
t» -IL.
Id?
9
i'^
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Plan
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CHAPTER V.
MEAN-TONE SYSTEM. OLD UNEQUAL TEMPERAMENT.
We saw that four exact fifths upwards lead to a third
(c — e), a comma sharper than the perfect third (by definition
of comma). If then we make each of the four fifths - of a
4
comma flat, the resulting third is depressed a whole comma,
and coincides with the perfect third. This is the rule of
the mean-tone system ; the fifths are all 7 of a comma flat.
It is called the mean-tone system because its tone is the
arithmetic mean between the major and minor tones of the
diatonic scale, or half a major third. The historical interest
of this system is very great- It can be traced back with
certainty as far as two Italian authors of the sixteenth century,
Zarlino and Salinas, and some claim for it a much higher
antiquity. From this time it spread slowly, and about 1700
was in universal use. The early development has many points
which are historically obscure, but one of special interest, if
anything could be found out about it, is the connection of this
system with our present musical notation. We shall see that
in this system the distinction between such notes as c* and d''
is true and essential * ; so that in the earliest times of what
we can call modern music, we find a system of notes in use
with a notation which exactly represents its properties. It
is impossible to avoid the surmise that the two may have
* We have seen that this distinction is false as applied to systems
of the type of the system of perfect fifths ; for such a note as off has
in those cases two forms, the one of which is practically identical
•with d^, and the other higher in pitch.
OLD UNEQUAL TEMPERAMENT. 35
had a common origin; and perhaps it would not be difficult
to make out a plausible case for Guido d' Arezzo, to whom
both have been ascribed ; but the evidence is too defective
to build much upon.
The historical account of the introduction of ' Tempering '
in Hopkins on the Organ, p. 161, is not quite correct. He
does not allude to the mean-tone system at all. But it is
described clearly by all the principal writers, and there can
be no doubt that it was the usual form of the old unequal
temperament*.
The principal interest of this question for us is the fact
that it was the system employed by Handel and his con-
temporaries under the form known as ' the old unequal
temperament,' and that it kept its ground in this country
until within the last few years. There are still organs re-
maining which are tuned in this manner f- Indeed we may
say with considerable accuracy that this system was the
language of music for nearly two hundred years.
There can be no doubt that, with the musicians of Handel's
time, the good keys of the old unequal temperament, i.e.
the mean-tone system, formed the ideal of the best tuning
* The following note on the origin of the mean-tone system is
quoted from Smith's Harmonics, p. 37.
' Salinas tells us, that when he was at Rome, he found the
musicians used a temperament there, though they understood not
the reason and true measure of it, till he first discovered it, and
Zarlino published it soon after
' After his return into Spain, Salinas applied himself to the Latin
and Greek languages, and caused all the ancient musicians to be read
to him, for he was blind; and in 1577 he published his learned work
upon music of all sorts ; where treating of three different tempera-
ments of a system, he prefers the diminution of the fifth by a quarter
of a comma to the other two
'Dechales says, that Guido Aretinus was the inventor of that
temperament But that opinion wants confirmation '
In Smith's own discussions he generally employs the expression
' system of mean tones,' in speaking of this temperament.
t Instances are, the organ at St. George's Chapel, Windsor ; and
the magnificent instrument in Turvey Church.
36 MEAN-TONE SYSTEM.
attainable. The proof of this is to be found in the fact that
Handel took the trouble to employ an arrangement, by which
the range of good keys available on the ordinary board with
this system could be somewhat extended. It is well known
that he presented to the Foundling Chapel an organ possess-
ing additional keys for this purpose. The organ at the
Temple Church in its original state, as built by Father Smith,
possessed a similar arrangement. The principle will be
explained subsequently. Here it will be sufficient to cite
the following description of the instrument given by Hopkins*,
the well-known organist at the Temple, in his work on the
organ : —
' The fine organ in the Temple Church was built by Father
Smith in 1688. It presents a great peculiarity in regard
to the number of sounds which it contains in the octave.
Most organs have only twelve in that compass, but this
has fourteen : that is, in addition to the common number
of semitones, it possesses an Al' and DS, quite distinct from
the notes G» and ElJ. The general temperament of the
instrument is the same as that of most English instruments —
unequal ; — but the real beauty of the quarter tones is dis-
coverable by playing in the keysf of £*> and Al", where in
consequence of the thirds being so true we have that per-
fection, that cannot be met with in common organs. It gives a
peculiar brilliancy also to the keys of A and E in 3 or 4
sharps J. These quarter tones are produced by the ordinary
G» and EiJ keys being divided crossways in the middle; the
back halves of which rise as much above the front portions,
as do the latter above the naturals.'
These extra keys have long been removed.
The great objection to this system was, that the circle of
fifths deviated widely from the equal temperament, and con-
sequently did not meet at the ends §, and the chords which
* Hopkins, On the Organ, 1st ed., 1855, p. 448; 2nd ed., 1870,
p. 452.
t This passage is slightly modified so as to make the sense clear.
X Reference to Christian Remembrancer, 1833; from which the
greater part of the above appears to be quoted.
§ For details see next chapter.
OLD UNEQUAL TEMPERAMENT. 37
involved notes taken from the two extremities of the circle
were exceedingly bad, their effect being commonly known
as 'the wolf.' This was undoubtedly the only objection
felt to the system by the musicians of Handel's day. The
objection is certainly a good one, so long as efficient means
for avoiding the wolf are not forthcoming ; and no stronger
testimony could be produced to the superiority of the good
portion of the system over the equal temperament, in the
opinion of Handel and his contemporaries, than the fact
that they seem, with few exceptions, to have preferred facing
the wolf of the unequal temperament to abandoning all the
excellences of that system in favour of the equal tem-
perament.
The history of Bach in connection with this subject is of
great importance ; unfortunately little can be ascertained
about it. A few points have however come down to us.
(1) As to clavichord, harpsichord, and piano.
It appears that Bach possessed a clavichord and a harpsi-
chord. The latter was probably the harpsichord with two
manuals and pedal, for which the set of six sonatas, in the
first volume of the Peters edition of the organ works, was
written *. But the clavichord possesses more interest in rela-
tion to Bach. The first peculiarity was that, as he used it, it
was not ' gebunden.' This necessitates an explanation. The
clavichord was an instrument in which a brass wedge attached
to the rising end of the key struck the string and at the
same time performed the office of a bridge, stopping off the .
requisite length of string. Advantage was taken of this in
early times to produce two or more different notes from one
string, and instruments arranged in this manner were called
' gebunden,' or ' tied.' When this was the case, — if c and c",
for instance, were made on the same string, — the interval
between them was determined once for all by the position
of the brass wedges f. From this arrangement Bach freed
* Hilgenfeldt, p. 135.
t Ibid. p. 36. This is the only intelligible explanation of the word
' gebunden ' that the writer has ever seen. The worci is used in
Forkel, but no explanation is given, and the passage is unintelligible.
28 MEAN-TONE SYSTEM.
himself; and we find this stated with an emphasis which
is now unintelligible without explanation.
It is also stated that the harpsichord and clavichord were
the instruments upon which Bach used the equal tempera-
ment ; and that he always tuned them himself.
Bach's favourite instrument was the clavichord. He con-
sidered it the best instrument for the house, and for study.
To appreciate this fully we must obtain some idea of the
effect of the clavichord. It is described as very soft and
expressive in tone : this last quality is said to have been that
which so endeared the instrument to Bach*. It was said
that it could hardly be heard at any distance. It is much
to be wished that some such instrument existed for purposes-
of study in the present day. The qualities of the clavichord
are important with reference to Bach's estimate of the equal
temperament.
Through the kindness of Mr. Carl Engel the writer was
permitted to examine and play upon a clavichord in excellent
order, with Mr. Parratt's assistance ; it was submitted to as
thorough an examination as seemed necessary for making out
its peculiarities. The best tone was produced by a light but
decided pressure of the finger ; so long as the touch was kept
light enough to get a good singing tone, the intensity was
exceedingly faint ; it seems doubtful whether it would be
audible through the least buzz of conversation. With this
singing tone the pitch of the notes was fairly constant ;
but the intensity was far too faint to hear anything of
the quality of the chords ; and the equal-temperament error
certainly could not be objected to on this instrument so far.
But further^ when any considerable energy of hand was em-
ployed, the effect was far from what was expected. The
quality deteriorated and the pitch rose considerably when
According to Hilgenfeldt there were generally six strings per octave
in the treble, and four in the bass. At p. 37 we find it stated that these
instruments were arranged according to the ' Zarlino' sche Temperatur,'
or mean-tone system, ' as were all others at that time.'
* Hilgenfeldt, p. 43. In the following passage ' Flugel ' means
' harpsichord," not ' grand pianoforte.' See also note, p. 42,
OLD UNEQUAL TEMPERAMENT. 39
the pressure passed a certain amount. This variation of the
pitch arose from the stretching of the string directly caused
by the extra pressure ; and its amount was so considerable
that it was impossible to regard the instrument as being really
anything definite in the way of pitch, when any considerable
amount of energy was used. A delicate and beautiful expres-
sion was certainly obtainable from the soft tone, but in leading
out a subject, or anything which called for a noticeable
emphasis, the extra pressure caused a rise in pitch which
might amount to a third of a semitone, or more than half a
semitone if any considerable pressure were used. On an
instrument of this kind, while the wolf of the old unequal
temperament would still be offensive, the errors and variations
of the instrument itself are so great in comparison to the
errors of the equal temperament, that it would seem impossible
to get any substantial advantage by seeking for any better
method of tuning.
Now it is occasionally said, ' Bach preferred the equal
temperament ' ; and his authority is cited against any attempt
to introduce other arrangements. But if it be the case that
his favourite instrument was such as we have described it with
respect to force and accuracy, it cannot be regarded as any
representation of our modern instruments. In particular, any
one accustomed to the varying pitch of this instrument must
have had an ear not to be shocked by small deviations, and
cannot have had the intense feeling' for equal-temperament
intervals which is characteristic of musicians brought up at the
modern piano. The account of Bach's habit of playing on the
unequally tempered organ in its worst keys to annoy Silber-
mann, to which further allusion will be made, confirms this
view. This is the same condition of ear with respect to
melodic intervals which might be expected to be attained
according to the method indicated in the sentence from Kirn-
berger prefixed to this book ; a condition which might well
admit of a power of appreciating the distinctions of diff"erent
systems, and a reference for correctness to the harmonies,
instead of to an arbitrary melodic standard.
So far as Bach's clavier music is concerned therefore, the
appeal to his authority in favour of the equal temperament
30 MEAN-TONE SYSTEM.
falls to the ground. The argument is unfounded in other
respects. Bach compared the equal temperament with the
defective mean-tone system on the ordinary keyboard, and
with nothing else. His objection was to the wolf, and cannot
be counted as of force against arrangements in which the
wolf does not exist.
It is doubtful whether the title ' wohltemperirte Clavier,'
as applied to the 48 preludes and fugues, corresponds to
the mature intention of Bach himself. The two parts were
composed at different times, as independent works. The
second part was regarded as the more important work of
the two, and this did not bear the above title under Bach's
hand, nor when first published in 1799, nearly fifty years
after his death*. The first part was then for the first time
printed together with the second, and the title thus got
carried over. But it is probable that Bach did not intend
the first part to be published at all ; and wrote the second
later in life to take its placet. The title of the first part
is, in the original MSS., ' Das wohltemperirte Clavier, oder
Praeludia and Fuga durch alle Tone und Semitonia,' &c.
It bears the date 1722. The original title of the second
part is — 'xxiv Preludien und Fugen durch alle Ton-Arten
sowohl mit der grossen als kleinen Terz ; verfertiget von
Johann Sebastian Bach.' Hilgenfeldt's autograph of the
second part bears the date 1740 J.
The pianoforte was developed by Silbermann in Bach's
last years. There is evidence that when Bach first became
acquainted with it he disliked it. And although we know
that he occasionally played upon Silbermann's pianos late
in life §, yet we have no evidence as to how they were tuned,
or that Bach ever recommended them for study. We know
only that Silbermann continued to tune the organ according
to the unequal temperament.
In fact it appears that Bach's clavier compositions were
* Hilgenfeldt, p. 85. f Ibid. p. 73. t Ibid. p. 123.
§ There is a well-known story of Frederick the Great taking Bach
round to play on all the new Silbermann Fortepianos in the palace at
Potsdam. Forkel, p. 10.
OLD UNEQUAL TEMPERAMENT. 31
regarded both by himself and others as specially dedicated
to the clavichord ; and when the pianoforte first came into
general use, these compositions were forgotten*. It was an
idea, excellent no doubt, but belonging to a later period,
to take them and apply them to the piano. The step is
a great one, but it is not one that Bach himself contemplated.
If this is done, his authority cannot be adduced as a reason
why further steps in the direction of improvement should
not be taken, if we can find such.
(2) The question as to Bach's point of view of the tempera-
ment of the organ is much more difficult than is supposed.
There is no direct evidence that Bach ever played upon an
organ tuned according to the equal temperament. There
is evidence to show that he thought the unequal temperament
abominable, as anybody would who played as freely as he
did ; and that he expressed himself very strongly on the
matter to Silbermann, who nevertheless continued to tune
the organs unequally. There is a well-known story, how,
when Silbermann came to listen. Bach would strike up in A''
as soon as he saw him, saying, 'you tune the organ as you
please, and I play as I please.' This must have been late
in Bach's life ; Silbermann was not likely to have attended
often unless the performer's reputation was formed f.
The best evidence, however, is that of Bach's compositions
for the organ. There is not a single organ composition of
Bach's published in the key of A I', or any more remote
key. There is one in F minor. Compare this with the
keys in which his clavier works are written. The comparison
furnishes an overwhelming presumption that there was some
potent cause excluding the more remote keys.
If therefore it is said on Bach's authority that his organ
compositions ought to be played on the equal temperament,
it may be answered, that there is no evidence that he played
them so himself. But it must always be admitted that they
should be played without the ' wolf ; that is all that Bach's
authority can be adduced for with certainty.
* In the last third of the eighteenth century. Hilgenfeldt, p. 44.
t Hopkins on the Organ, p. 176.
32 MEAN-TONE SYSTEM,
It is not possible to say witli any exactness when the
change of organ-tuning to equal temperament took place
in Germany. There are considerations which render it
probable that unequal temperament still existed in the time
of Mozart ; and it is possibly owing to this cause that we
possess no true organ compositions by this greatest organ-
player of his day, except the two gigantic fantasias for a
mechanical organ, which are best known as pianoforte duets.
As these are both in F minor, it is probable that the
instrument for which they were written was tuned to the
equal temperament.
All systems which involved ' wolf ' have practically dis-
appeared. We shall now discuss the applications of the
mean-tone system, and endeavour to show how it can be
employed in such a way as to obtain everywhere the ex-
cellence formerly peculiar to a few keys, every inequality
which gave rise to the wolf being got rid of.
Analysis of the number of Bachs published Organ
Compositions in the different Major and Minor
Keys, excluding the Chorales.
Major Keys.
Eb Bb F C G D A
2 1 3 13 7 2 1
Minor Keys.
F C G D A E B
18 5 7 5 4 2
CHAPTER VI.
APPLICATIONS OF THE MEAN-TONE SYSTEM.
We have seen that, in the mean-tone system, a series of
fifths is tuned according to this rule ; — All the fifths are
a. quarter of a comma flat, the thirds formed by four fifths up
being perfect; (for the third formed by four fifths up is a
comma sharp if the fifths are exact, by definition of comma).
We will first consider the condition of things on the ordinary
keyboard when this system was employed, as in the old
unequal temperament, pointing out the nature of the various
contrivances that have been employed with a view to extend
the range of the system in this connection ; and then show
how the problem of its employment is solved by the
generalised keyboard.
To bring clearly before the eye a representation of the
arrangement of the mean-tone system as an unequal tem-
perament, we may adopt a method depending on the same
principle as that which we employed in the case of the system
of perfect fifths. We will arrange the notes first from right
to left in order as a series of fifths, and at the same time
exhibit the deviation or departure from E. T which accrues
as we pass along the series, by displacement downwards for
flattening and upwards for sharpening. The notes placed on
the twelve numbered lines represent those which existed in
the ordinary twelve-keyed unequal temperament : the a I' and
d' at the two extremities represent the notes given by the
additional keys in the Temple organ. Having thus obtained
the departures of the notes up and down, we may now re-
arrange them in the order of the scale, and shall thus obtain
a graphical representation of the intervals commanded by this
arrangement. See Diagrams HI and IV.
D
III.
1
2
3
i
5
6
7
8
9
10
11
12
(ab) .
eb
bb
fj
en
g)l
(d()
IV.
(ab)
eb
bb
9
10
11
12
f«
g»
(dj)
APPLICATIONS OF THE MEAN-TONE SYSTEM. 35
We have now to investigate the departures of these notes
from equal temperament. The amount of this for one fifth
is easily found by remembering that each exact major third
is equivalent to a proceeding through four fifths of the system,
so that the departure of one fifth will be - that of the perfect
third. The latter is — ~ downwards ; so that the departure of
each fifth of the mean-tone system becomes — — downwards*.
The distance between two consecutive dots in a vertical
column represents consequently — - E. T. S. ; and in Diagram
IV the distance between two consecutive dots in a horizontal
line represents one E. T. semitone.
We can now examine the nature of the intervals which give
rise by their dissonant qualities to the term ' wolf; and we
will suppose that we are dealing with the ordinary twelve-
keyed board, i. e. with the unbracketed notes only.
There are two kinds of disturbed intervals ; one fifth and
four major thirds. The false fifth is that made by the
notes marked g*— el' in the diagrams. We see that the
g' is eleven steps below the eW The departure of this fifth
is consequently — — downwards, or considerably more than
29.0
- of a semitone lower than E. T. The error of this fifth is
2
- of a semitone nearly t- This is the worst element of the
5
wolf, on account of the sensitiveness of fifths to tuning.
The four false thirds are those made by the notes marked
b — e^ f— b'', c*— f, g*— c in the diagrams. It will be
In decimals,
.13686_.^3
4
■034215
n
•37636
•01955
•39591
or, .4 nearly.
D 3
36 APPLICATIONS OF THE MEAN-TONE SYSTEM.
seen that there are eight steps between the members
of each pair, the upper note of the third being above.
Consequently the departure of each of these thirds from equal
Q
temperament is — — upwards, or rather more than a quarter of
a semitone sharper than E. T. thirds, which are too sharp
already. The total error of these thirds * is rather more than
2
- of a semitone. The false third b —el' is a terrible annoyance
5
with this system, as it enters into the keys of E major and
minor, which it is hardly possible to keep out of, as the flat
keys beyond B'' are practically unavailable f-
We can now very well understand how great was the gain
obtained by introducing the two notes in brackets (Diagram
IV), as at the Temple and elsewhere. This particular last-
mentioned third was provided for by the (D*) ; and the bad
fifth El"— G', which stopped progress so soon in the flat keys,
was put one remove further back by the introduction of (A'') ;
so that the keys of E and E'' were now fit for use. Considering
the small amount of modulation used in old days, it appears
intelligible enough that Handel should have been pleased
with the arrangement ; and the only wonder is that it did not
obtain a wider currency.
Besides the application of extra keys to the keyboard
other means have been employed for controlling a somewhat
extended series of mean tone fifths. The earliest of these
seems to have been the arrangement of Smith, who was
Master of Trinity College, Cambridge, in the last century, and
wrote the well-known treatise on Harmonics, or the Philosophy
* -034215
8
•27372
•13686
■41058
t It once fell to the writer's lot to play the Wedding March of
Mendelssohn on the organ at Turvey, which is tuned in this manner.
The portion in B major produced a horrible effect which will not
soon be forgotten, chiefly on account of this false third b - elJ.
APPLICATIONS OF THE MEAN-TONE SYSTEM. 37
of Musical Sounds, 1759. It was an arrangement of stops
applied to the harpsichord. The instrument was constructed
by Kirkman ; but the description of it is so mixed up with a
more extended design that it is difficult to say what the
arrangements actually were. Assuming that the design of
the ' Foreboard,' in fig. II of the separate tract on the
changeable harpsichord, represents the actual instrument, it
had the following series of fifths : —
Db Ab I Eb BlJ F C G D A E B F« C» G« I D» A« E» B« F».
Recently Mr. A. J. Ellis, F. R. S., has caused to be
constructed a harmonium, in which an extended series of
mean-tone fifths is controlled by stops. The resources of the
instrument embrace a complete series of keys from seven flats
to seven sharps. There is a short notice of the instrument in
the Proceedings of the Musical Association, 1874-5, p. 41.
Other instances might be adduced ; but the principle does
not commend itself as a desirable one. The perpetual in-
terruption of the performance caused by the necessity of
changing the stops is a great annoyance ; and the method
cannot be considered one of practical value, whether stops or
change pedals are applied.
We will now consider the application of the generalised
keyboard to the mean-tone system. Although a strict
adherence to the principle on which our symmetrical arrange-
ments are constructed gives rise to such a distribution as
Diagram IV, when the notes are placed in the order of the
scale, yet we can by reversing one of our fundamental con-
ventions, reduce this to the same form as that illustrated in
Diagram I. We have only to take distances drawn upwards
to correspond to flattened pitch, and drawn downwards to
sharpened pitch. That is, we represent now a fall by a rise,
and a rise by a fall, instead of rise by rise and fall by fall,
as in the original application to perfect fifths. Or we may
put it thus, — that the distance upwards corresponds simply to
advance along an upward series of fifths, without any regard to
the question whether the fifths are greater or less than equal
temperament fifths, and distance downwards corresponds to a
downward series of fifths.
The advantage of thus reducing the mean-tone system,
38 APPLICATIONS OF THE MEAN-TONE SYSTEM.
where the departure of the fifths from E. T. is downward, to
the same form as the scheme for perfect fifths, where the
departure is upward, is twofold. First, when the symmetrical
arrangement is embodied in a keyboard, the two things admit
of treatment by means of the very same set of keys. Secondly,
the form, which the scale and chords of the mean-tone and
similar systems assume on the keyboard (Diagram II), is in
this case remarkable for facility of execution, and adaptation
to the hand.
The sequence of the white unmarked naturals in Diagram
II is that which constitutes the mean-tone scale of c, when a
series of notes tuned according to the mean-tone rule (fifths
quarter of a comma flat), is placed on the keys.
In passing to other scales than that of c, we must first
remember that in this system the distinction between such
notes as c* and dt" is true and essential. For the major third
formed by four fifths up from A is identical with the true
major third to A, according to the usage of musicians. So
far therefore as thirds and fifths go, we shall dispense with the
employment of the notation for position in the series of fifths,
and rely upon the distinction between sharps and flats to
indicate the key intended to be played.
Since sharps indicate progression through fifths upward,
and flats through fifths downward, we have the following
rule : — Put the finger up for a sharp, and down for a flat.
Recurring to the symmetrical arrangement of the key-
board, and the fact that for a given rule of tuning the relative
position of the notes of a given combination is always the
same, we see that all keys have scales of exactly the same
form as that of c above described ; and the same chord can
be reproduced whatever be its key or key relationship with
the same form of finger.
In the enharmonic organ built by the writer for the meeting
of the Musical Association, May 1, 1875, one of the two stops
was tuned according to the mean-tone system. It is called
the negative stop on the instrument. The term ' negative ' is
applied in the general theory to systems which have fifths
flatter than equal temperament fifths, i. e. to such as are
strictly represented by an arrangement of the form of Diagram
APPLICATIONS OF THE MEAN-TONE SYSTEM. 39
IV. The generalised keyboard has a compass of three octaves,
tenor c to c in alt. ; and there are forty-eight keys per octave,
though only thirty-six were used for the mean-tone stop.
The result of this is a range from dl'i' to dCM, if we start from
the middle of the three c keys as c. Or recurring to the more
intelligible system of denoting position in the series of fifths
by the notation before described, we have a range fromxc to
//f ; vc becoming A^ when translated into ordinary notation,
and Hi becoming d***. The instrument will be further de-
scribed in connection with the other stop p. [56].
It is no exaggeration to say that anything can be played
on this keyboard, with the mean-tone scales. The movements
of the finger required are of the simplest possible character ;
and the uniformity of the fingering in all keys minimises the
necessary study.
To prove the practicability of performance of this kind the
writer performed three of Bach's preludes at the meeting of
the Musical Association where the enharmonic organ was
exhibited, viz. the 1st and 2nd of the first part, and the 9th of
the second part. With three octaves only the fugues were
not practicable.
But it is not in rapid performance that an arrangement of
this kind shews itself specially of value.
The chorale, any massive harmony, not excluding counter-
point, tells well. It is only necessary to remember that we
have here the original system, which belongs from the very
beginning of modern music onward to our musical notation,
to see that by employing it we have the true interpretation of
our notation ; we have the actual sounds that our notation
conveyed to Handel, to all before Bach, and many after him,,
only cured of the wolf, which was the consequence of their
imperfect methods.
It will be unquestionably the case that the modern educated
musician will pronounce these notes out of tune. He will not
complain of the chords ; they are better than equal tempera-
ment chords. On examination it will be found that, all the
intervals employed being of necessity different from equal
temperament intervals,, the ear which is highly educated to
consider equal temperament intervals right, considers all
40 APPLICATIONS OF THE MEAN-TONE SYSTEM.
others wrong ; a result by no means strange. But people
with good ears, who have not been highly educated as to
equal temperament intervals, have no objection to those of
' the mean-tone system. The semitone is perhaps the best
example. The mean-tone semitone is considerably greater
than the equal temperament semitone ; it is about 1 - E. T.
6
semitones. Eminent modern musicians have said that this
semitone was dreadful to them. It was not dreadful to
Handel.
The rationale is, that if people who are taught music are
taught that one thing is right and another wrong, they will
come to believe it. If they were taught the other systems of
interest as well as the equal temperament, they would
appreciate the excellences of all. By the habit of observing
the fine distinctions between them, they would be very much
more accurate in their knowledge of any of them separately ;
and according to the motto from Kirnberger prefixed to this
book, other advantages would be likely to accrue as well.
A mean-tone keyboard sufficient for most practical purposes
would contain twenty-four keys per octave, and would run
from di'i' to e», or fromvc to/f in the notation of the series of
fifths. This would be practicable and interesting as applied
to either organ or harmonium. From experience it is known
that the fingering is easy, and the chords are fine.
The tuning of the mean-tone system is understood by organ
builders. For the notes c — g — d — a — e, c— e is first made a
perfect third, and then the fifths indicated are made equally
flat by trial. The group e— b — f*— c«-g« is then similarly
treated, and so on.
CHAPTER VII.
HARMONIC SEVENTH.
Hitherto we have investigated only chords derived from
octave, fifth, and third ; but in all the approximate systems
to which the generalised keyboard opens a path we can
obtain fine effects of a novel character by the introduction
of an approximation to the harmonic seventh.
It is well known that if we take a minor seventh such
as g— f, and flatten the f by a small interval, we can obtain
a seventh, which presents many of the qualities of a con-
sonance, and in which no beats can be heard.
The note f thus determined is the same as the seventh
harmonic of a string, whose fundamental is two octaves
and the seventh below it ; and the vibration ratio of the
notes g— f is 4:7.
If we compute by Rule I, p. 14, the interval in question,
we find for its downward departure from the equal tempera-
ment f the value .31174, or a little less than - of an equal
temperament semitone.
If we consider the system of perfect fifths, we find that
the f derived from g by two fifths down has a downward
2
departure due to two perfect fifths, = — nearly, or more
accurately,= .03910.
If we pass from f downwards through a Pythagorean comma
to \f (through a circle of twelve perfect fifths), we get a de-
parture from the equal temperament position of .03910 +
.23460, the latter number being the value of the Pythagorean
comma, and the whole departure=. 27370.
Now though this is not quite so great as the required
departure of the harmonic seventh, yet it is sufficiently
near to it to improve the quality of the interval very much
in respect of consonance.
44 HARMONIC SEVENTH.
In certain cases then we can use the note \f instead of
f in the chord of the dominant seventh on g ; and we thereby
obtain a chord of very beautiful quality. There exist certain
limitations on this use of the approximate harmonic seventh.
Rule. The harmonic seventh on the dominant must never
be suspended, so as to form a fourth with the keynote.
For the approximate seventh we can prove this by noticing
that the harmonic seventh to dominant g is\f; and c— \f
forms a fourth, which is a comma flat nearly. The efifect
of the flat fourth is bad.
But the rule applies to the harmonic seventh and all its
approximations. In ratios this stands as follows : — the ratio
of tonic : dominant is 4:3; dominant : harmonic seventh as
4:7; whence ratio of harmonic seventh of dominant to tonic
is 21 : 16, or 63 : 48. But ratio of fourth to tonic is 4 : 3 or
64 : 48 ; whence this fourth differs from the harmonic seventh
to dominant in the ratio 64 : 63, or by more than a comma.
The extreme sharp sixth is susceptible of an improvement
in quality by the introduction of the harmonic seventh. An
example is contained in the illustration on p. 11.
The mean-tone and allied systems also afford approxima-
tions to the harmonic seventh. In these systems the note
employed is that derived by ten fifths up from the root ;
thus the approximate harmonic seventh to g would be e'
in the ordinary notation, or/f in the notation of the series
of fifths. It is convenient to preserve the notation of the
series of fifths for this purpose only, with reference to the
mean-tone and allied systems.
The departure of one fifth of the mean-tone system being
.034215 down*, that of ten fifths becomes .34215, which is
a little greater than the required departure for the harmonic
seventh.
The harmonic seventh must be used with great caution
in the mean-tone and allied systems ; its position is so widely
removed from equal temperament that its employment in
melodic phrases is not generally successful ; it is effective
in full chords.
* Page 35, note.
CHAPTER VIII.
APPLICATIONS OF THE SYSTEM OF PERFECT FIFTHS
AND ALLIED SYSTEMS.
We have seen that the system of 53 is very nearly co-
incident with that of perfect fifths. There are also a few other
systems which may to a certain approximation be treated
as practically the same. We will now consider the practical
application of these systems.
We shall assume for the present simply that the major
third is properly made by eight fifths down, according to
Helmholtz's Theorem ; and we shall disregard the small
error of about — of an E. T. semitone (skhisma) which is
thus introduced in the thirds.
Though some of the systems of this class are theoretically
more perfect than the system of perfect fifths, yet this must
always be the most important member of the class in a
practical point of view, on account of the ease with which
perfect fifths can be tuned.
We will include also in this discussion those practical
applications which deal with perfect fifths and thirds, although
the establishment of a continuous series of fifths is not aimed
at; for these can be conveniently treated by means of the
same methods and notation.
Unsymmetrical arrangements.
The work of Mersenne, dated 1636, is an interesting legacy
from the time before the mean-tone system was generally
44
APPLICATIONS OF
established, although it was then well known and spreading.
Mersenne exerted immense ingenuity in forming arrangements
or systems which should comprise the elements of a number
of perfect concords ; and he devised keyboards of a most
complicated character by which these systems would be
controlled. A number of these keyboards are figured in
the work. Probably none of them were ever constructed.
A detailed account of one of his less complicated systems,
having eighteen intervals in the octave, is given at p. 15 and
p. 1 ] 4, of the ' Proceedings of the Musical Association 1874-5.'
This may be conveniently summarised in the form of Mr.
Ellis's duodenes. Perfect fifths run vertically, major thirds
horizontally. The principal key-note was f.
/ab
g
\b
Ve
\gS
/db
f
\a
\cS
/gb
bb
\d
\\b
eb
\g
This may be expressed shortly as containing the duodenes of
f and \d. Mersenne was very careful about the depressed
second of the key, — \g in the principal key of the above ;
its importance will be discussed presently.
Now directly one examines the possible modulations one
is struck by the completeness of the scheme within certain
limits ; and yet how narrow those limits are when regarded
from the point of view of modern music !
The unsymmetrical keyboards would have been terrible
things to play. We will now consider their modern re-
presentative, Gen. Perronet Thompson's Enharmonic Organ.
In Diagram I, p. 22, a symmetrical arrangement of the
notes of this instrument has been already given : the sub-
scripts refer to the three keyboards.
The lowest board has c for its principal key ; the second
board \e, and the third \d. Each board is tuned so that
the ordinary keys give the diatonic scale of the principal key ;
but on departing from the principal key the notes required
THE SYSTEM OF PERFECT FIFTHS. AS
to produce concords crop up all over the keyboard in three
different shapes, which are called quarrills, flutals, and buttons.
Not only has every key a different form, but in all cases
where keys or parts of them occur on two or more keyboards
the forms are different on the different keyboards. It is
astonishing that so much should have been accomplished
with this instrument as appears to have been the case.
Thompson also paid great attention to the depressed second
of the key.
Key relationship symmetrical arrangements.
The principle of these arrangements is strictly that the
form of a chord of given key relationship is the same in
every key. But the notes are not all symmetrical, and the
same chord may be struck in different forms according to
the view which is taken of its key relationship. The form. of
symmetrical arrangement employed in this treatise may be
said to depend on intervals, not on key relationship. The
arrangement by intervals includes all the symmetrical pro-
perties of the arrangement by key relationship; and much
more besides.
The first attempt in this direction was made by H. W. Poole,
of South Danvers, Mass., U.S. An account of it is given in
'Silliman's Journal,' July 1867. The keyboard does not
appear to have been constructed, but it will be desirable to
notice the principle of arrangement. This appears to have
been based upon the relations of the different notes of the
major and minor scale, but it will be simpler to treat it by
the method of intervals.
We may refer the arrangement primarily to two directions,
one vertical, in which the principal steps proceed upwards
by the apotom^ Pythagorica* and one running upwards to the
right at an angle of about 46°, in which the principal notes, or
key-note series, proceed by whole tones, in the same manner
as the major tones of the generalised keyboard before de-
scribed ; but this law only applies to each row of major
tones separately. The different rows are not related as in
* Semitone of seven exact fifths, p. 9.
46
APPLICATIONS OF
the generalised keyboard ; but according to the rule that
the vertical step is the apotom^ Pythagorica. These two rules
determine the position of the notes called key-notes ; i. e.
those related to any key-note by fifths, thus —
V.
Poole's Keyboard. Colin Browns Keyboard.
Arrangement of Key-notes.
(
E
D
. C
1
J
. Eb
Db
. G
B
A
\
C
. Bb .
Ab
b . . .
These notes are only those which proceed by fifths in each
diatonic scale, i. e. F C G D in the key of C. There are be-
sides four series of auxiliaries ; thirds to the key-notes,
thirds to the thirds, and harmonic sevenths to the key-notes
and first set of thirds. We will omit the detailed considera-
tion of the harmonic sevenths, shewing their places only by
plain circles. The first set of thirds is provided for by an
auxiliary series a comma lower than the principal series ;
these are placed below their principal notes of the same name
in the right-hand half of the division. The left-hand half is
occupied by the auxiliaries of the three remaining series,
those of the second series of thirds being above and the two
harmonic sevenths below. This determines an arrangement
for each principal note and its accessories of the same equal
temperament derivation, which corresponds somewhat in ap-
pearance to a knight's move in chess. In Diagrams VI, VII,
THE SYSTEM OF PERFECT FIFTHS.
47
I
I
►Si
^
48
APPLICATIONS OF
the notes marked E, vE, \\E, exhibit the position relation in
question. Diagram VI contains the notes of the scale of C and
its relative minor, with a few others. The thirds are \E, \A,
\B ; the minor thirds in the \A scale are C, F, G; and the
major thirds occasionally required in \A minor are \C', \F',
\G'. The arrangement is completely determined by the
assumption of the apotom^ Pythagorica between the key-notes
as a vertical step, their oblique arrangement in rows of major
VII.
Knighfs-move arrangement in Colin Brown's Keyboard.
\D(.... .
B
o
OV.E
tones, and the knight's-move arrangement, as we may con-
veniently call it, for the derivation of the auxiliaries.
Noticing that E — \E is a comma, and that the apotom^
Pythagorica = chromatic semitone -f- comma, p. 10, we see
that the intervals between notes of the two first series in a
vertical line are alternately chromatic semitone and comma.
It is clear that this arrangement adapts itself with some
facility to all music in which there is not much modulation,
or in which the modulation is of a simple type.
PERFECT FIFTHS AND ALLIED SYSTEMS. 49
It is however easy to give instances which will at once in-
volve the performer in difficulties. The simple change from
C major to C minor is the first instance that occurs. The
/El' required for this change does not exist on Mr,
Poole's board. He proposes however, 'if musicians decide
that it is desirable to have these minor thirds,' to introduce
them as additional auxiliaries. Now consider a modulation
which may occur at any time, — change to C minor and modu-
late at once into /El'. The performer is lost. The /El' is
present only as an auxiliary, and cannot be used as a key-
note.
Example. ^^^^^gg
Again ; wherever a chord is taken with an auxiliary for its
root, it will have a different form from that which it has when
it has a key-note for root ; though the intervals may be pre-
cisely the same.
The keyboard recently constructed by Mr. Colin Brown is
of considerable interest, as being founded on exactly the same
principles as Mr. Poole's. In fact, if we discard Mr. Poole's
two series of harmonic sevenths, his scheme of position rela-
tions becomes absolutely identical with Mr. Brown's. This
is not so odd as far as the symmetrical arrangement of the
oblique rows of key-notes goes ; for the writer of this treatise
invented this arrangement quite independently of Mr. Poole ;
and as the thing is an obvious step, there is no great wonder
in its having been re-invented. But Mr. Brown also applies
the apotom^ Pythagorica to the upward step between key-notesj
as well as the knight's move arrangement * to the auxiliaries ;
all three arrangements being applied to the same notes, and
in the same way as in Mr. Poole's instrument. The forms of
the keys are a little different, but the position arrangement
is absolutely identical. It appears that Mr. Brown was
unacquainted with Mr. Poole's work, so that the coincidence is
astonishing.
* Diagram VII.
E
50 PERFECT FIFTHS AND ALLIED SYSTEMS.
The objections made to Mr. Poole's keyboard apply with
equal force to Mr. Brown's, except that the size of the keys
in the latter is somewhat smaller, so that a more extended
key-board can be provided ; and it would be possible to get
over the difficulty of modulation from C major to C minor
and so to/El', by making a jump from the place on the key-
board where C occurs as a key-note to the place where it
occurs as an appendage to key-note /C ; in which connection
its minor scale and relative major are conveniently placed,
just as in the case of \ A minor in Diagram VI.
In modern music, however, it is often impossible to say
without hesitation what the exact key relationship of a com-
bination is.
With Poole's key-board five series of notes are required,
with Colin Brown's three, to obtain the command of the com-
binations given by the generalised key-board with one series.
Symmetrical arrangem,ent by Intervals.
This is the simplest principle by means of which the com-
plex combinations of harmonious music can be analysed ; its
effect is to reduce all cases dealt with to a very small number
of simple types.
The simplest form in which this principle is embodied is
in the Duodenes of Mr. Ellis *. In these, fifths make steps
in a vertical line, thirds make steps in a horizontal line. See
illustrations at pp. 3, 44.
The only other form in which this principle has been
applied to the writer's knowledge, is the generalised key-
board which has been already described ; this depends on a
symmetrical arrangement by equal temperament semitones
and departures therefrom. Its principles have been suffi-
ciently explained in connection with Diagrams I and II. We
will now consider the instruments in which the generalised
keyboard has been applied to this class of systems.
* Proc. Royal Soc, Dec. 1874.
CHAPTER IX.
ENHARMONIC HARMONIUM.
ENHARMONIC ORGAN, POSITIVE STOP.
The Enharmonic Harmonium exhibited at South Kensing-
ton, in the Loan Collection of Scientific Instruments 1876,
was built in 1872-3. It possesses a keyboard of four and a
half octaves, containing seven tiers of keys. As each tier con-
tains twelve keys to the octave, there are altogether eighty-
four keys in each octave. These are arranged in the manner
indicated generally by Diagrams I and II, Diagram II giving
the actual detail of a small portion of the keyboard, and Dia-
gram I showing the general nature of the distribution in a
more extended manner.
It has been mentioned that this instrument is tuned accord-
ing to the division of the octave into fifty-three equal intervals,
a system sensibly identical with that of perfect fifths. We will
now investigate the manner in which the system is distributed
on the keyboard. For this purpose we must slightly antici-
pate the general theory, and establish a proposition, the cor-
rectness of which we shall easily recognise.
Writing down the series of fifths about any note c, viz. — ■
db—al'— el'— blJ— f— c— g— d— a— e— b— /f»— /c»,
we see that there is a semitone c— /c" determined by tuning
seven fifths up, and there is a semitone c— dl' determined by
tuning five fifths down. These may be called seven-fifths semi-
tones and five-fifths semitones respectively. Now scales are
made up of tones and semitones, and tones are made up of two
semitones each ; in other words, scales are constructed by
reckoning certain numbers of semitones, whether greater or
less, from a given starting point. The only general way
E 3
5a ENHARMONIC HARMONIUM.
therefore by which all possible scales whatever, in all relations
to each other, can be provided for, is to have an interval for the
unit of the system, which is a common divisor of the different
semitones at our disposal. We have then only to find out
how many and what semitones go to the octave^ and this is
the proposition we shall borrow from the theory.
Now with perfect fifths, the seven-fifths semitone is ap-
7 S
proximately 1 —-, and the five-fifths semitone 1 —77 ; and the
51 o A
ratio of the magnitudes of these two intervals is 58 : 46 or
29 : 23;
and 5 X 23 = 115,
and 4 X 29== 116 ;
so that the seven-fifths semitone is to the five-fifths
semitone nearly as 5 : 4 when made with perfect
fifths.
We may therefore represent the seven-fifths semitone by
five unit intervals, and the five-fifths semitone by four unit
intervals without introducing any serious distortion of the
fifths.
Now the theory will tell us that five seven-fifths semitones
added to seven five-fifths semitones make always an exact
octave. We easily see that this is true.
For five seven-fifths semitones give five E. T. semitones
and the departure of thirty-five fifths upwards,
and seven five-fifths semitones give seven E. T. semi-
tones and the departure of thirty-five fifths down-
wards ;
leaving on the whole 7 + 5=12 E. T. semitones, or an
exact octave.
If we then take five units for the seven-fifths semitone,
and four for the five-fifths semitone,
five seven-fifths semitones make twenty-five units
and seven five-fifths semitones make twenty-eight
units ;
and 25-1-28 = 53;
whence fifty-three such units make an exact octave.
We proceed to construct a symmetrical arrangement, and
attach to the various notes their characteristic numbers in
ENHARMONIC HARMONIUM. 53
the system of 53, according to the rule that
each seven-fifths semitone such as c— /c' is five units,
and
each five-fifths semitone such as c— d^" or c— c' is four
units.
We attach as before no indication of position in the series
of fifths to the different names d>, dl*, but determine this
position entirely by the notation for that purpose.
The note V\c is taken as the first note of the series, and
receives the characteristic number 1. Then c is 4, and the
remaining numbers are assigned by the above process.
Diagram VIII contains a symmetrical arrangement of a
portion about the middle of the keyboard, one octave in
extent. It is continued both upwards and downwards on the
instrument, the highest note in level being m/i, and the
lowest V\c.
It is now seen that a number of notes near the top of the
keyboard are identical in pitch with other notes in the next
division on the right near the bottom. This can be most
simply shown by enumerating successively the characteristic
numbers of the system, and tracing the succession of the notes
of various names which are attached to them.
This enumeration is made in Diagram IX. On inspecting
it, the following simple rule will be seen to hold, a black note
meaning simply an accidental, a sharp or flat.
Rule for identifications in the system ofi 53.
If two notes in adjoining divisions be so situated as to
admit of identification (e. g. a high c and a low c*), they will
be the same if the sum of the elevation and depression marks
is 4 ; unless the lower of the two divisions is black (accidental),
then the sum of the marks of identical notes is 5.
Thus A^c— ^xc* are identical ; also ///c^—\^d.
/ is called a mark of elevation, \ a mark of depression.
The use of these identifications is to permit the infinite
freedom of modulation which is characteristic of cyclical
VIII.
/c.
/o
\0,
/d,
\d,
\o
/eb
/Oa
\eb
/i„
Ve,
>\ej,
/fJi
\f,
/g3.
f».
Vg
VfK:.
/ab.
/a,
a^
ab,
\ab;
/b,
/c.
/bb«
bb
\a.
bb,
\b,
\o,
V\b„
ENHARMONIC HARMONIUM.
55
systems. For in moving upwards on the keyboard we can,
on arriving near the top, change the hands oh to identical
notes near the bottom, and so proceed further in the same
direction, and vice versa. In perfect fifth systems, displace-
ment upwards or downwards on the keyboard takes place
IX.
VXo
1
/b
Wfd
28
//f
Wo
2
//b
\fS
29
///f
\o
3
///b
fit
30
////f
4
/fK
31
/e
5
tin
32
VWg
//o
6
Wo J!
mn
33
Wg
///c
7
\ot
////ft
34
\s
8
of
35
s
9
/off
36
/g
\\\d
10
//ojf
Wab
37
//g
\\d
11
///oJ!
\ab
38
///g
\d
12
////cj
ab
39
d
13
/ab
40
/d
14
//ab
41
W\a
»&
15
Web
///ab
42
Wa
m&
16
\eb
////ab
43
\a
17
eb
44
a
18
/eb
45
/a
WVe
19
//eb
Wbb
46
//a
We
20
///eb
\bb
47
///a
\e
21
////eb
bb
48
e
22
/bb
49
/e
23
//bb
50
W\b
//e
2i
Wf
///bb
51
Wb
///e
25
Vf
////bb
52
\b
26
f
53
b
27
/f
readily by modulation between related major and minor keys,
not, as is commonly assumed, only by modulation round the
circles of fifths. In systems of the mean-tone class, on the
contrary, displacements take place only by modulations of the
latter type. Consequently these last systems require a much
less extended keyboard than perfect fifth systems.
The mechanism of the keyboard consists of seven tiers of
levers, each tier resembling exactly the levers of an ordinary
keyboard. The variations in the position of the notes are
^6 ENHARMONIC ORGAN,
determined by the patterns of the keys which are attached to
the levers. Each of these tiers communicates through a row
of squares with a row of horizontal stickers*. The wind-
chest is vertical, and the valves are arranged on it in seven
horizontal rows. The valves have small tails attached, and
the stickers open the valves by pressing on the tails. There
is no attachment between the stickers and the valves. Thus
the windchest can be lifted out by simply undoing the bolts
which hold it.
The reeds are accessible by opening a large door at the
back of the windchest. Each reed has a separate windchannel ;
and all below treble c have a regulator in the channel by
which the supply of wind can be adjusted for each reed
separately. In this way the usual fault of the predominance
of the bass is completely obviated.
The process by which the instrument was tuned is rather
complicated ; an account of it is given in Proceedings of the
Musical Association, 1874-5, p. 144. As far as the perfect
fifths went the process was simple enough ; but in order to
secure the series meeting at the ends so as to give the system
of 53, an elaborate system of checks was devised, the applica-
tion of which was laborious. Perfect fifths are recommended
by preference for ordinary purposes.
Enharmonic Organ, Positive Stop.
In the enharmonic organ with three octaves of generalised
keyboard, built by the writer for the meeting of the Musical
Association, May 1, 1875, the mean-tone stop of which has
been already described, there is another stop occupying all
the forty-eight keys per octave. It is called the 'positive' stop
on the instrument. This term is applied in the general theory
to systems which have fifths sharper than equal temperament ;
i. e. to systems which are strictly represented by such sym-
metrical arrangements as I or VIII ; for instance perfect fifths,
or the system of 53. The term ' negative' is applied, as has been
* Organ-builder's term for a rod which transmits a pressure.
POSITIVE STOP. 57
mentioned, to the mean-tone class, which have fifths flatter
than equal temperament, and are strictly represented by
Diagram IV.
The positive stop in question was tuned on a system of the
approximately perfect fifth class, the properties of which will
appear in the general theory, but may be easily obtained
independently.
With reference to the theory it is called the positive system
of perfect thirds ; sometimes it is called Helmholtz's system,
as it was brought into notice by him. It differs from the
system of perfect fifths only in that the third by eight fifths
down is made perfect, the fifths being tempered by - t)f the
8
skhisma, or error of the third derived through eight perfect
fifths. The skhisma being about — of a semitone, the tem-
pering of each fifth is — - of a semitone nearly ; an interval
inappreciable' by ordinary means. The result of the tuning
by this system instead of perfect fifths did not repay the
greatly increased trouble ; and in future this stop will be tuned
simply by perfect fifths.
The extent and distribution of the sounds on an octave of
the keyboard can be sufficiently indicated by reference to
Diagram I. If we suppose the two missing notes in this
scheme to be filled up (b,\\d), the /c at the top removed, and
the series continued at the bottom through five more steps
down to v\c, we shall have a representation of the distribution
of the sounds on one octave of this stop.
The result of this stop on this organ is not so satisfactory as
that of the mean-tone stop, and not nearly so satisfactory, in
the writer's judgment, as the result on the 53 harmonium. It
should be mentioned that there is a difference of opinion on
this subject owing to the dislike which some persons entertain
to the somewhat sharp quality of the harmonium reeds. The
large majority of persons- however prefer the harmonium.
The reason is certain : the pipes of the organ are metal
stopped diapasons, and they possess of course the smooth
quality characteristic of such pipes. With this particular
quality of tone, and indeed with diapason tone in general,
5^ ENHARMONIC ORGAN,
little seems to be gained by the degree of additional accuracy
which the perfect-fifth systems possess over the mean-tone
class. It is not possible with the particular stop in question
to illustrate in a striking manner the difference between chords
in and out of tune ; the quality is too smooth to be very
sensitive to tuning, and the general effect is somewhat
monotonous.
General Thompson's organ, which is now* at the Loan
Exhibition, South Kensington, is open to the same remarks.
The quality of tone is somewhat firmer than that of the
stopped pipes of the writer's organ ; but after becoming
acquainted practically with General Thompson's organ, as
well as hearing it performed upon a good deal by persons
who for the most part did not understand it, and merely
flourished about at random on the keyboard, the writer is
prepared to maintain that the gain in the purity of the
chords is hardly enough to make it worth while to face the
enormous cost and demand for space which must be in-
separable conditions of the application of perfect-fifth systems
to the organ on any very considerable scale. The mean-
tone system seems more applicable to this purpose, while the
perfect-fifth systems have special applicability to the har-
monium, and also, in all probability, have a wide field
before them in connection with the orchestra.
The small enharmonic organ which contains the two above-
mentioned stops is designed on a principle which is general,
and susceptible of extension to instruments of greater size.
The keyboard is arranged in four tiers ; the tails of each tier
fit without attachment under the fronts of a row of squares,
the other corners of which hang down, and pull directly on
the principal trackers. In consequence of this arrangement
the entire keyboard can be lifted out and replaced in a couple
of minutes. The principal trackers run from the lower ends
of the squares, parallel with the keys, forming four tiers
corresponding to the four tiers of keys, and engaging at the
other ends in four tiers of rollers ; each roller is parallel to
the width of the keyboard, or at right angles to the length of
* 1876.
POSITIVE STOP. 59
the key and tracker, and the length of the roller is a little
greater than the width of the keyboard. The windchest is
above the roller board ; it is about 2 ft. 6 in. wide, and 6 ft.
from back to front ; the keys are in the narrow front. The
pull-downs are arranged in two rows along the long sides of
the windchest, and come down to the ends of the rollers on
either side alternately. Thus any valve or pull-down can be
got at at once. The stop sliders run parallel to the keys and
trackers, and to the greatest length of the windchest, from
front to back, so that the action is what is called ' direct.'
The stoppers of the metal pipes employed are a novelty.
They are put together in the first instance like square wooden
pipes, with a square channel in the middle of each ; they are
then turned in the lathe to fit the pipes they are to stop.
A square block is fitted tightly into the interior channel, and
a screw fitted into the block. A headpiece through which
the screw passes completes the stopper. The object is to
get a fine adjustment for tuning. When the pipe is nearly
right it can be very minutely adjusted by means of the
screw.
CHAPTER X.
GENERAL THEORY OF THE DIVISION OF THE OCTAVE*.
Definitions.
Regular Systems are such that all their notes can be
arranged in a continuous series of equal fifths.
Regular Cyclical Systems are not only regular, but return
into the same pitch after a certain number of fifths. Every
such system divides the octave into a certain number of equal
inter\'als.
Error is deviation from a perfect concord.
Departure is deviation from an equal-temperament interval.
Equal temperament (E. T.) is the division of the octave
into twelve equal intervals.
Intervals are expressed in terms of equal-temperament
semitones ; so that the octave is written as 1 2, and the E. T.
semitone as 1.
Intervals taken upwards are called positive, taken down-
wards, negative.
Fifths are called positive if they have positive departures,
i.e. if they are greater than E. T. fifths; they are called
negative if they have negative departures, i.e. if they are less
than E. T. fifths. Perfect fifths are more than seven semitones ;
they are therefore positive.
Systems are said to be positive or negative according as
their fifths are positive or negative. (See Diagram VIII or I
for positive systems, IV for negative systems.)
Regular cyclical systems are said to be of the r*'' order,
* This Chapter may be omitted if it is desired to confine the
attention to the practical part of the subject.
THEORY OF THE DIVISION OF THE OCTAVE. 61
positive or negative, when twelve of the approximate fifths of
the system exceed or fall short of seven octaves by r units of
the system.
Thus if thirty-one units of the system of 53 be the fifth,
then 12x31 = 372,
7x53 = 371,
the twelve fifths exceed seven octaves by one unit, and
the system is said to be of the first order positive.
We shall see later that in the system of 1 1 8 the twelve fifths
exceed seven octaves by two units, and the system is said to
be positive of the second order.
In the system of 31, twelve fifths fall short of the octave by
one unit, and the system is said to be negative of the first
order.
In the system of 50, twelve fifths fall short of seven octaves
by two units, and the system is said to be negative of the
second order.
Cor. Hence the departure of twelve fifths is r units of the
system, having regard to sign.
Regular Systems.
Theorem I. In any regular system five seven-fifths semi-
tones and seven five-fifths semitones make up an exact octave.
For the departures from E. T. of the five seven-fifths semi-
tones are due to thirty-five fifths up,
and the departures of the seven five-fifths semitones
are due to thirty-five fifths down,
leaving twelve E. T. semitones, which form an exact
octave.
(This has been proved already in connection with the system
of 53, but it is necessary to repeat it here, as it forms the
foundation of the theory.)
Theorem II. In any regular system the difference be-
tween the seven-fifths semitone and the five-fifths semitone
is the departure of twelve fifths, having regard to sign.
For if we subtract the five-fifths semitone from the seven-
fifths semitone, the E. T. semitones cancel each other;
6% GENERAL THEORY OF
and the departure of the seven-fifths semitone up is
due to seven fifths up,
and that of the five-fifths semitone down is due to five
more fifths up,
making the departure of twelve fifths :
and it is positive if the fifths are positive, and negative
if the fifths are negative.
Regular Cyclical Systems.
Theorem III. In a regular cyclical system of order
+ r, the difi"erence between the seven-fifths semitone and five-
fifths semitone is + r units of the system.
This proposition follows from Th. II, and the Cor. to the
definition of r*'^ order.
Cor. This proposition, taken with Th. I, enables us to
determine the numbers of divisions in the octave in systems
of any order, by introducing the consideration that each
semitone must consist of an integral number of units. The
principal known systems are here enumerated : —
Primary (1st order) Positive.
Seven-fifths semitone Five-fifths semitone ^^^^ ofumts
= PC units. = y units. '" °<='^^| (^''- ^^•
2 1 17
3 2 29
i 3 41
5 4 53
6 5 65
Secondary (2nd order) Positive.
11 9 lis
Primary Negative.
1 2 19
2 3 31
Secondary Negative.
3 5 50
The mode of formation in other cases is obvious.
Theorem IV. In any regular cyclical system, if the
octave be divided into n equal intervals, and r be the order of
THE DIVISION OF THE OCTAVE. 60,
the system, the departure of each fifth of the system is - E. T.
semitones.
Let the departure of each fifth of the system be 8. Then
the departure of twelve fifths =1 28 = r units by definition or
12
its corollary; and the unit = — E. T. semitones (since the
octave, which is twelve semitones, is divided into n equal parts).
Hence— ^^
126 = r. — , or 6 = --
n n
We have seen that the departure of each fifth of the system
of 53 is — of an E. T. semitone; this is a particular case of
the above theorem.
As a consequence of this theorem we can shew that the
system of 31 is nearly the same as the mean- tone system.
For the departure of the fifth of the mean-tone system is t— —
downward (p. 35), or — — -, and by the ' above Theorem
the departure of the fifth of the system of 31, for which
r=—\, is— T7: and the two differ by an amount which is
insensible in practice.
Theorem V. If in a system of the ^*''. order, the octave be
divided into n equal intervals, r-^Tn is a multiple of 12, and
— — — is the number of units in the fifth of the system.
Let (^ be the number of units in the fifth.
12
Then ^. — is the fifth, = 7 + 6, if 6 be the departure of one
fifth; and8 = -byTh. IV.
n
Hence 6. — =7 + -, or * = —-—-,
^ n n ^12
and ^ is an integer by hypothesis ; whence the proposition.
From this proposition we can deduce corresponding values
of n and r. Casting out multiples of 12, where necessary,
from n and r, we have the following relations between the
remainders : —
64 GENERAL THEORY OF
n
1
2
3
4
5
6
7
8
9
10
11
5
10
3
8
1
6
11
4
9
2
7
Y
— 7
— 2
— 9
— 4
-11
— 6
— 1
— 8
— 3
-10
— 5.
Example. It is required to find the order of the system in
which the octave is divided into 301 equal intervals. 300 is
a multiple of 12 ; remainder 1 gives order 5, or —7. 301 is a
system of some interest regarded as a positive system of
order 5, in consequence of its having tolerably good fifths
and thirds, while its intervals are expressed by the first three
places of the logarithms of the vibration ratios, .3010 being
the first four places of log 2. Mr. Ellis has made use of this
system (Proceedings of Royal Society 1874); and Mr. Pole
read a paper about it to the Musical Association 1875-6.
Theorem VI. If a system divide the octave into « equal
intervals, the total departure of all the n fifths of the system
= r E. T. semitones, where r is the order of the system.
For if 8 be the departure of one fifth, then, by Th. IV,
v
8 = - ; whence «8 = r,
n
or the departure of n fifths = r semitones.
This theorem gives rise to a curious mode of deriving the
different systems.
Suppose the notes of an E. T. series arranged on a
horizontal line in the order of a succession of fifths, and
proceeding onwards indefinitely thus : —
c g d a e b f* c' g" d' a' f c g . . .
and so on.
Let a regular system of fifths start from c. If they are
positive, then at each step the pitch rises further from E. T.
It can only return to c by sharpening an E. T. note.
Suppose that b is sharpened one E. T. semitone,, so as
to become c ; then the return may be effected —
at the
first b
in
5 fifths
3»
second b
JJ
1 7 fifths
I)
third
»
29 fifths
)>
fourth
»
41 fifths
fifth
53 fifths,
THE DIVISION OF THE OCTAVE. 65
and so on. Thus we obtain the primary positive systems.
Secondary positive systems may be obtained by sharpening
bl' by two semitones, and so on.
If the fifths are negative, the return may be effected by
depressing c' a semitone in 7, 19, 31 . . . fifths; we thus
obtain the primary negative systems ; or by depressing d
two semitones, by which we get the secondary negative
systems, and so on.
Theorem VII. If n be the number of divisions in the
octave in a system of the ^*'' order, then n + 7r will be
divisible by 1 2, and — — — will be the number of units in the
X a
seven-fifths semitone of the system.
For by the order condition (Th. Y) 7n-\-r is & multiple of
12 ; whence 7{7n + r) = 4:9n + 7r is a multiple of 12; whence,
casting out 48«,
n+7r is a multiple of 12.
Let X be the number of units in the seven-fifths semitone,
then
12 r
^.—=1-1-78=1 + 7-
n n
, n+7r
whence x= ,
and the proposition is proved.
Theorem VIII. Negative systems form their major thirds
by four fifths up.
For the departure of the perfect third is —.13686 or
approximately; that is, it falls short of the E. T. third
7,0
by this fraction of an E. T. semitone. But in negative
systems the fifth is of the form 7— 6 ; and four fifths less two
octaves give 4(7— 8)— 24 = 4 — 48, a third with negative
departure, which can be determined so as to approximate to
the perfect third.
Cor. I, The mean-tone system may be derived from this
result by putting
— 48=— .13686
—6=— .034215
F
66 GENERAL THEORY OF
Cor. II. The departure of a third of a negative cyclical
y
system n of order — ^ is — 4- •
Theorem IX. Positive systems form approximately
perfect thirds by eight fifths down.
The departure of the perfect third is —.13686.
But in positive systems the fifth is of the form 7 4- 8 ; and
five octaves up and eight fifths down give 60 — 8(7 + 8) = 4 — 86,
a third with negative departure, which can be determined so
as to approximate to the perfect third.
Cor. I. The positive system of perfect thirds, or Helm-
holtz's system, can be derived from this result by putting
— 88= —.13686
6= .0171075.
Cor. II. The departure of a third of a positive cyclical
system n of order y is — 8- •
■^ n
Theorem X. Helmholtz's Theorem. The third thus
formed with perfect fifths has an error nearly equal in amount
to the error of the E. T. fifth.
For — 8 X. 01955= —.15640
— .13686
— 01954 which is nearly= — 01955.
The quantity .01954 is called the skhisma.
Theorem XI. In positive systems an approximate
harmonic seventh can be obtained by fourteen fifths down.
The departure of the harmonic seventh is —.31174; and
fourteen fifths down and nine octaves up give,
108 — 14(7 + 6)=10 — 146,
a minor seventh with negative departure.
Theorem XII. In negative systems an approximate
harmonic seventh can be obtained by ten fifths up. For
five octaves down and ten fifths up give
10(7-8) — 60 = 10-106,
a minor seventh with negative departure.
THE DIVISION OF THE OCTAVE.
67
Concords of Regular and Regular Cyclical Systems.
These considerations permit us to calculate the departures
and errors of concords in the various regular and regular
cyclical systems. There is, however, one other quantity which'
may be also conveniently taken Into consideration in all cases,
viz. the departure of twelve fifths of the system. We will call
this A, putting A=126.
We have then the following table of the characteristic
quantities for the more important systems hitherto known.
Q 1
The value of the ordinary comma (— ^ is •21506. It is com-
parable with the values of A, and if introduced in its place
in the table would give rise to a regular non-cyclical system,
lying between the system of 53 and the positive system of
perfect thirds, the condition of which would be that the
departure of twelve fifths = a comma.
Name,
Order,
A = 125,
or 12- •
Error of
fifth,
S -•01955
Error of
third,
•13686-88.
Error of har-
monic seventh,
•31174-148.
17
1
•70688
•03927
-•33373
-•51178
29
1
•41379
•01493
-•27586
-•17101
41
1
•29268
•00484
- •19512
-•02970
Perfect fifths.
•23460
• ■•
-•01954
•03804
63
1
•22642
-•00068
-•01409
•04758
Positive perfect \
thirds. !
[
•20529
-•00244
...
•07223
118
2
•20339
-•00260
•00127
•07446
65
1
■18462
-•00417
■01378
•09635
(»=„-■
is here
negative.)
.13686 + 48
.31174 + 105
43 -1
-•29707
-04431
•03784
■06418
31 -1
-•38710
-•05181
•00783
-■01084
Mean Tone. Nega-
tive perfect thirds.
-•41058
-•05376
-•03041
60 -2
-■48000
-■05955
-•02314
_ ■08826
19 -1
-•63158
-■07218
-■05367
-•21468
F 2
68 THEORY OF DIVISION OF THE OCTAVE.
Theorem XIII. If a symmetrical arrangement like
Diagram I or VIII be constructed, the dots being all con-
sidered as notes, and the vertical distance between two dots
represent — of an E. T. semitone, the whole system will
51
constitute a division of the octave into 612 equal intervals *,
and it will possess both fifths and thirds correct to a high
order of approximation.
For since the octave is twelve semitones, and the semitone
fifty-one units, the octave is 612 units.
Again the system may be regarded as made up of fifty-one
different sets of E. T. notes, each represented by the dots of
a horizontal line. The fifth, the upper note of which is one
step above the lower, will be 7—, and the perfect fifth is
7 , a very small difference.
51.151
The third, which has its upper note seven steps below its
7 1
lower note, is 4—— or 4— and the perfect third is
51 7.286
-, also a very small difference.
7.3064'
A symmetrical arrangement with all the positions filled in
in this manner may be called a complete symmetrical arrange-
ment. It might be constructed with concertina keys.
* The importance of this system was pointed out by Captain
J. Herschel, F.R.S.
CHAPTER XI.
MUSICAL EMPLOYMENT OF POSITIVE SYSTEMS HAVING
PERFECT OR APPROXIMATELY PERFECT FIFTHS.
The following example is repeated here from p. 12, as
containing examples of the principal forms of chords.
The first chord is the major triad of c.
The second chord is the triad of the dominant g, withxf,
the approximate harmonic seventh, as dominant seventh.
The last two crotchets of the first bar are the chords of c
major and minor.
The chord at the beginning of the second bar is the
augmented sixth, rendered peculiarly smooth in its effect by
employment of the approximate harmonic seventh for the
interval (/a''— f).
v^=vJ=vJ;W T* ^ J^.
^
eEe^
:tt^
3i3f
B
-^
I -I
We proceed to notice practical points aff"ecting the employ-
ment of the principal intervals.
Second of the Key. — In any positive system the second of
the key may be derived in two ways : first, as a fifth to the
dominant, in which case the derivation is by two fifths up
from the key-note ; and, secondly, as a major sixth to the
subdominant, in which case the derivation is by ten fifths
down from the key-note. Thus, the first second to c is d ;
the other vd. On account of the importance of this double
form of second, we will consider the derivation of these two
forms by means of the ordinary ratios, in the case, namely, in
which perfect intervals are employed.
70 POSITIVE SYSTEMS HAVING PERFECT
First, two fifths up and an octave down give
when the fifths are perfect.
Secondly, one fifth down gives the subdominant (c— f), and
a sixth up gives the depressed second (\d), or
2 5_10
3 ^ 3~ 9 '
which is the ratio of \d to the keynote, when the fifths and
thirds are perfect.
The ratio of d : \d is then
8 10_81
9"^~9'"'80'
which is an ordinary comnia.
We must remember that our systems only give approxima-
tions to this result, but the best of these approximations are
very close.
In the harmonium, with the system of 53 — which may be
regarded for practical purposes as having perfect fifths, and
very nearly perfect thirds — the exchange of d for \d in the
chord f— \a— \d, or even in the bare sixth, f— \d, produces
an effect of dissonance intolerable to most ears.
Minor Third. — The minor third is not an interval which is
very strictly defined by beats. In chords formed of successions
of minor thirds, almost any form of the interval may be em-
ployed ; and as matter of fact the minor third which comes
below the harmonic seventh in the series of harmonics (7:6),
is one of the smoothest forms of this interval, c— xe'' is an
approximation to such a chord, where the ve'' is derived by
fifteen fifths down. But in minor common chords the con-
dition is that the major third or sixth involved shall be
approximately perfect ; and this gives the triad c— /e''— g
where the /el' is derived by nine fifths up. The intermediate
form, e**, gives a minor third not quite so -smooth as either of
the other two ; but it is capable of being usefully employed in
such combinations as the diminished seventh, and it is pre-
ferred by many listeners, as deviating less from the ordinary
equal temperament note, from which it has only the departure
due to three fifths down. The interval between the harmonic
OR APPROXIMATELY PERFECT FIFTHS.
71
seventh on the dominant and the minor third of the elevated
form on the keynote, is the smallest value of the whole tone
which occurs, the departure from E. T. of such a tone being
due to twenty-two fifths or about two commas ; and although
two chords, involving these notes in succession, may each be
perfectly harmonious, the sequence is generally offensive to
ears accustomed to the equal temperament.
Example.
3!*te2=
W.
Custom makes such passages sound effective, especially
when the succession is slow enough to enable the ear to
realise the fineness of the chords.
Major Third. — This interval has been already discussed ;
the note taken is that formed by eight fifths down.
Fourths and Fifths need no remark.
Depressed Form of the Dominant. — When the dominant is
used in such a combination as the following : —
U^
I
=F
it must be formed by eleven fifths down from the key-note,
unless we regard the key-note as changed for the moment, in
which case, by elevating the subdominant, we may retain the
fifth in its normal position. The most judicious course de-
pends on whether the fifth is" suspended or not. Thus, if the
fifth is suspended, we may write : —
^
J~J-j
^dE
f^-H-^
*j
For if the subdominant be f, its third must be \a, and its
sixth must be vd ; g then makes a fourth with \d, which is
73 POSITIVE SYSTEMS HAVING PERFECT
unbearable to the ear ; the fourth must be made correct, and
the ways of doing so are shown above. The difficulty may
be otherwise got over by writing the passage
1^
U^
s^
&c.
Minor Sixth. — This interval is pretty sharply defined. The
usual form is /a'', which is got by eight fifths up ; the key-
note forms an approximately perfect third with this note by
inversion.
Major Sixth (c— va). — This interval is, as a matter of fact,
more sharply defined than one would expect. This interval
must be kept strictly to its best value. The \a is got by nine
fifths down.
In chords formed of a succession of minor thirds, major
sixths frequently occur. Care must be taken to dispose them
so as to make this interval correct. If a deviation is neces-
sary, it is better, if possible, to extend the interval by an
octave; the resulting major thirteenth (3 : 10) is not very
.sensitive.
Minor Seventh. — There are three forms of the minor
seventh. To fundamental c these are are/b'', b'', and \b'' : —
/bl'; ten fifths up ; the minor third to the dominant.
h^ ; two fifths down ; the fourth to the subdominant.
xbt* ; fourteen fifths down ; approximation to the harmonic
or natural seventh.
Rule. — The natural or harmonic seventh on the dominant'
must not be suspended, so as to form a fourth with the key-
note.
Major Seventh. — There is only one form of major seventh
which can be used in harmony, viz. vb ; this note is got by
seven fifths down ; it forms a major third to the dominant.
In unaccompanied melody the form b produces a good effect.
This is got by five fifths up with perfect fifths. It forms
a dissonant or Pythagorean third to the dominant. The
OR APPROXIMATELY PERFECT FIFTHS. 73
resulting semitone is less than the E. T. semitone by nearly
— of a semitone.
An example of music written for positive systems is ap-
pended, p. IT-
The principal points in the harmony of these systems which
have struck the writer occur in the example. It is to be
specially noticed how certain forms of suspension have to be
avoided — partly because they produce dissonances, partly
because they occasion large displacements up and down the
keyboard. The result of the writer's practical experience is,
distinctly, that there are many passages in ordinary music
which cannot be adapted with good effect to positive systems ;
and that the rich and sweet masses of tone which characterise
these systems, with the delicate shades of intonation which
they have at command, ofifer to the composer a material
hitherto unworked. The character of music adapted for these
systems is that of simple harmony and slow movement ; it is
a waste of resources to attempt rapid music, for the excellence
of the harmonies cannot be heard. The mean-tone system is
more suitable for such purposes.
Some examples of the unsuitability of the positive systems
for ordinary music may be first instanced : —
(1) The opening bars of the first prelude of Bach's 48.
The second bar involves the depressed second (\d), and in
the third bar this changes to d ; the melodic effect is ex-
tremely disagreeable on the harmonium. It does not strike
the ear much with the stopped pipes of the little organ.
(2)
*
^
m,^^^^
:&c.~
The two g's, to which attention is here called by asterisks,
illustrate a difiSculty of constant occurrence in the adaptation
of ordinary music to these systems. The g is here required
to make a fourth to the depressed second of the key (\d), and
also a fifth to the keynote. But the first condition requires
74 POSITIVE SYSTEMS HAVING PERFECT
the note \g, the second g, and it is impossible to avoid the
error of a comma somewhere. It may be said that the first g
is only a passing note ; but with the keen tones of the harmo-
nium such dissonances strike through everything, even on the
least emphasised passing notes. Although the second g seems
to the writer to b^ legitimate, it would be intolerable on the
harmonium. The smoother tones of the organ render such
effects less prominent.
(3) The third phrase of a well-known chant : —
i
--^-
£=5^ g I g ^
f-s=
^ "^- >p- 'iig: ^
To keep in the key of f, the g should fall to Vg at the
second chord ; but this direct descent on the suspended note
would sound bad — consequently, the whole pitch is raised a
comma at this point by the suspension ; and the chant con-
cludes in the key of (i, as it is not possible anywhere to
descend again with good effect. This would be inadmissible
in practice, as the pitch would rise a comma at each repeti-
tion. The resources of the system of 53 admit of the per-
formance of repetitions in this manner, but the case is one in
which the employment of this effect would be unsuitable.
On the organ it might be possible to take the last chord
written above \g^\d— \g— bl', which would get rid of the
difficulty. On the harmonium, however, this drop from the
minor chord of g to that of \g is inadmissible.
In the example of music written for the positive systems, it
is to be noted that the notation-marks are used as signatures,
exactly as flats and sharps are in ordinary music. The sign
adopted for neutralising them is a small circle ( => ), which is
analogous to the ordinary natural. If the general pitch had
to be raised or depressed by a comma, the elevation or de-
pression mark would be written large over the beginning of
the staff :^
Several points in the harmony are regarded as experi-
mental. For instance, in the inversion of the dominant seventh
OR APPROXIMATELY PERFECT FIFTHS. 75
with the seventh in the bass, the employment of the depressed
(harmonic) seventh has on the harmonium an odd effect ;
although, when the chord is dwelt on, it is heard to be de-
cidedly smoother than with the ordinary seventh. The effect
appears less strange on the organ. On this and other points
the judgment of cultivated ears must be sought, after thorough
acquaintance with the systems.
The following points may be noticed in the example at
P-77-
At the beginning xA the seventh bar it would be natural, in
ordinary music, to suspend the a, from the preceding chord,
thus : —
^
r^^-A
y=^
:&c.—
T
As however the first a is \a, and we are modulating into
g, whose dominant is d, the suspension is inadmissible, as it
would lead to the false fifth d— \a.
In bar 14 the ordinary seventh \d to dominant \e is em-
ployed in the bass instead of the harmonic seventh \\d, so as
to avoid the small tone \\d— c. The latter has a bad effect
in the minor key, as before noticed, and this is specially
marked in the bass.
Bar 19. — The use of the tonic as first note in the bass is
prevented by the presence of the harmonic seventh on the
dominant, p. 42.
Bar 24. — This singular change is pleasing in its effect when
judiciously used, but it is advisable to separate the two forms
of the chord by a rest.
Bar 30. — The smoothness of the approximate harmonic
seventh is here applied to the sharp sixth. This effect is the
most splendid which the new systems afford ; nothing like it
is attainable on ordinary instruments.
Bars 34 and 35. — Here the natural course would be to
make the bass : —
^ ^TTT-r^
76
POSITIVE SYSTEMS HAVING PERFECT
the harmony remaining the same. We have however arrived
at our d as the fifth to g, and it is not possible to suspend it
unless we raise the \a to a. It has not a good effect where
a passage is repeated as here, if the repetition is in a slightly-
different pitch. The suspension is therefore avoided.
Bar 37. — This is a very charming effect. The transient
modulation to dominant d gives the depressed key-note, \c, as
harmonic seventh.
Comma Scale.
The following is an example of a novel effect which is at-
tainable in positive systems. If the chord of the harmonic or
natural seventh be sustained, this seventh may be made to
rise and fall again through two or more single commas. The
effect to unaccustomed ears is disagreeable at first ; but the
writer has become so familiar with these small intervals, that
he hears them as separate notes without the sensation they
commonly produce of being one and the same note put out
of tune. There can be no doubt that the reception of such
intervals is a question of education, just as the reception of
semitones was, in the early history of music, a step in advance
from the' early five- note scales. The following passage, as
executed on the enharmonic harmonium, which admits- of a
swell of the tone, has a dramatic effect : —
i
^^^^^^
^
N|?^^olrJ^J^J w^
&c.
J^
m^
:«:
Series of Major Thirds.
i
EE
PE23^^
The chord to which attention is called consists of two per-
fect thirds and the octave. The third Vg'— c has a departure
OR APPROXIMATELY PERFECT FIFTHS.
77
due to sixteen fifths up, and an error from the perfect third of
about two commas. It may be called the ' superdissonant '
third, by analogy from the dissonant or Pythagorean third,
which has an error. of one comma. We have the choice, if we
prefer it, of arranging the chord with two dissonant thirds,
thus : —
c— \e— g(f— c.
The two last thirds are ordinary dissonant thirds ; the writer
prefers the first arrangement. It is a matter of taste.
Example for Systems of approximately Perfect
Fifths, with a Compass of three Octaves.
H = Harmonic or Natural Seventh, or inversion thereof.
J.a J- S
^ m^4^^--^ w=Ft^^^
-Jiej
Imo.,
^
fe^
.U_yAAJ
^^B
^^
--p—p-
2do.
^
I I
^
78
POSITIVE SYSTEMS HAVING PERFECT
r-> y ^ I - ■ I ^ \ '^^ — —
i ;::f^-r-ri-r^
53^
,K ^ ^ -Jx,^■/j_J
i^^
Hj J. _h
^
frp^--F=F=^
-^ r r
19
^^ =4=^r^=^ £?ir-f i^ ^=£ rf^=r^
^
^=i^S^^^^^^^fe^^^E3^
iSE=^
P
5
24
^i^^f^^^^teg ES
:SIIZ33~
s
i
d^^
E&
r-^=^g
SE
=F=«^
-r-r-^5-
I 1^
t^TTrrj
^^?^ g
i
feb^
I
^
! I N
f
*
l£
IE2Z
Hi I
I
W=i^
30
I I
iaf
B^g uxqs:!r-r,^:tP:f-g-r '.frrr f
OR APPROXIMATELY PERFECT FIFTHS. 79
^ife^
i
i
3 P^ r f r> y-y-ff-r'-zog
-4- t : -g- # H
^^^
^
S
-^^Lj^j:^
^ r-piTL
^
^
APPENDIX.
ON THE THEORY OF THE CALCULATION,
OF INTERVALS.
When we consider the interval between two notes, with
reference either to the relation between their vibration num-
bers, or between the lengths of a string which will sound
them, we employ the ratio between the numbers in question,
that is, we divide the one number by the other.
Thus, if we take lengths of a stretched string which are
as 1 : 2, they produce notes an octave apart. If we take
lengths as 2 : 3j they produce a fifth ; and if we take lengths
as 4 : 5, they produce a third. The case is the same with
the vibration numbers.
If we desire to estimate the interval formed by the sum
of two others, retaining for clearness the conception of the
lengths of string, we see that - the string will give us the
2 12 1
octave, and - of that - , or -- x - of the whole length, gives
us the sum of octave and fifth : that is to say, in order to
find the string fraction for the sum of two intervals, we have
to multiply together their separate string fractions.
This refers the principle of multiplication of ratios directly
to our experimental knowledge of the properties of fractional
lengths of a musical string.
G
8a ON. THE THEORY OF
In the same way the same rule for vibration ratios may be
referred directly to our experimental knowledge of the pro-
perties of vibration numbers ; or, more simply, the latter may
be deduced from the fact, known from the laws of mechanics,
that the vibration numbers are inversely as the string lengths
of portions of the same string.
When we perform computations in this manner by multi-
plication and division of ratios, the numbers are apt to be-
come high, the computations troublesome or impossible (e. g.
the division of the octave into 53 equal intervals), and the
appreciation of the magnitude of the intervals in question
difficult. It is, in particular, difficult to interpret the results
of a fractional computation in terms of such intervals as are
in practical use, e. g. equal temperament semitones.
In order to overcome these difficulties, methods are adopted
which are explained at length, in principle and practice, in
the remarks which follow.
If we take the ratio corresponding to any interval (e. g. the
ratio 2, corresponding to an octave), multiply it by itself over
and over again, and then set down the resulting products,
with the number of times the multiplication has been per-
formed, in two columns, we form two corresponding series,
the one in geometrical, the other in arithmetical progression,
thus : —
Geometrical Progression
Arithmetical Progression
of ratios.
of nmnber of ratios.
1
2
1
4
2
8
3
16
4
32
5
64
6
128
7
256
8
and so on.
Again, let x be the ratio corresponding to an equal tem-
perament semitone, that is to say, let x^^, or x multiplied 1 2
THE CALCULATION OF INTERVALS. 83
times into itself, be equal to 2. Then, forming a series like
the above, we have : —
Geometrical Progression
Arithmetical Progression
of ratios.
ofE.
T. semitones.
1
X
1
x"
2
x'
x^
3
4
x'
5
x^
6
X-'
7
x'
8
x'
9
^10
10
^11
11
2=^12
12
and so on.
Now, whenever two sets of numbers form corresponding^
terms in a pair of series of this kind, the numbers in the
arithmetical progression are called logarithms of the corre-
sponding numbers in the geometrical progression; and the
number in the geometrical progression which corresponds to
logarithm 1 is called the base of the system of logarithms.
Equal temperament semitones may therefore be regarded
as logarithms of vibration ratios to base x, where x^ — 2.
Common, logarithms, such as are found in the ordinary
tables, are to base 10.
We can find x independently of ordinary logarithms by
using the ordinary processes of square arid cube root.
For since x^^ = 2
X =V^
= ^^2
The arithmetician may therefore find x for himself by twice
extracting the square root of 2, and then the cube root of
the result ; the numbers x^, x^, x*, and so on, are easily ob-
tained when this operation has been performed.
G a
84 ON THE THEORY OF
In practice we derive these numbers more shortly by-
making use of the labours of those who constructed our
tables of common logarithms, in the manner explained in
the text.
Passing for a moment from the subject of the construction
of such tables as the above, let us see what use can be made
of them. We can only speak of intervals made up of _E. T.
semitones with reference to the above illustration, but this
will be sufficient for the present purpose.
Suppose we have got the numbers x'^ and x^, the first being
the ratio of two semitones, and the second the ratio of five
semitones : first, to find the sum of these two intervals. If
we had only the numbers, we should have to multiply them
together, and interpret the result as best we could ; but having
the table, we have only to add together the nurnbers of semi-
tones, or the logarithms, and we not only learn how many
semitones the resulting interval consists of, but can find oppo-
site that number (7), in the table, the number which x^ and x'^
would give if multiplied out. Instead of multiplying the
ratios we add the logarithms. Similarly, if we wish to divide
one ratio by another, we subtract the logarithms.
Again, we can use these numbers for dividing an interval
into any number of equal parts. •
Thus if we want to divide the octave into two equal parts,
with the ratios we should have to take the square root of 2.
We perform the same process here by simply dividing 1 2 by
2, and noting the ratio opposite the result in the table.
Similarly the multiplication of a ratio by itself any number
of times is reduced to multiplying the number of equal semi-
tones by the number of times the multiplication is desired to
be performed. Thus the E. T. fifth is seven semitones. If
we want to find the value of twelve such fifths we have only
to multiply 7 by 12, which gives 84, or seven octaves.
By means of a system of logarithms then, we reduce —
multiplication of ratios to addition,
division. . ... to subtraction,
extraction of a root . to division, and
raising to a power . . to multiplication.
Practically, for musical purposes, we do not construct a
THE CALCULATION OF INTERVALS. 85
table of this kind. The ratios we have to deal with can be
reduced to two or three, which we turn into E. T. semitones
most simply by employing ordinary logarithms : and when
we once know these equivalents we can form the rest by their
means. Thus if we know the values of the fifth and third in
E. T. semitones, we can form any of the intervals ordinarily
discussed in connection with the diatonic scale, by addition,
subtraction, and multiplication and division by low numbers.
I shall however proceed, for the satisfaction of those who
are not acquainted with logarithms, to develope methods, by
which an arithmetician may perform the computations for
himself.
The first method of proceeding further is very simple in
principle ; and it is interesting as being, in principle, very like
the method which was actually used by the first constructors
of logarithms. It is however so laborious to carry out that
we will dispense with the execution of the calculations.
It consists simply of obtaining equivalent ratios for fractions
of equal temperament semitones by continual extractions of
square roots. Proceeding in this manner, we should obtain
the following equivalents, taking square roots on the left, and
dividing by 2 on the right : —
Ratio.
Decimal of E. T. semitone
l/x
.5
'Jx
•25
^x
■125
v^
•0625
v^
•03125
sy^
•015625
'V^
•0078
""^x
■00390625
""ifx
■001953125
and so on.
For the construction of a table of practical utility it would
be necessary to proceed further.
Now suppose we want to construct the ratio answering to
1-i (one and a tenth) E. T. semitones, we must make this up
out of the terms we have found.
86 _ ON THE THEORY OF
Thus 1 is represented by x
•0625 „ „ iy;tr
•03125 „ „ XI X
•00390625 „ „ ^^^x
•001953126 „ „ '^^l/x
1-099609375
So that a value within of a semitone of the required
1000 ^
value 1-1 is equivalent to the ratio
16/ 32/ 256 /~ 612 /~
X X ^X X sJX X ^X X ^x.
For the practical construction of tables these approxima-
tions require to be carried further.
This process not being suitable for actual use, I proceed to
explain a method analogous to that which would be now
employed for the independent calculation of logarithms.
In treatises on the construction of logarithms, such as occur
in the ordinary books on trigonometry, it is proved that, in
logarithms to any base, if
X be the number whose logarithm is sought,
C a number which is always the same for the same
system of logarithms ; —
then,
, ^(-i^-l , \ ,x-\^ l/;tr-lN5 )
■ ^°s^=^fe+3fe) + 5(^)^ ;•
Where the successive terms become smaller and smaller, and
after a certain point cease to influence the computation.
Suppose we have a ratio 7 to deal with ; then if we put
a
the above becomes ;
(""^ '°S^-^to + 3te)+5to)+ }■
In order to use this formula we have only to determine C;
and this can be always done by making equation (A) satisfy
the law of the given system.
Thus the law of the system of E. T. semitones is, that the
logarithm of 2 must be 12.
THE CALCULATION OF INTERVALS.
87
Putting a=2, 3=1,
a—b
Whence
= - = -333 3333
3
f^y = _L = .037
\a+d/ 3x9
(^V= 1 =.004 1152
/a — b^ _
Va + d/ ~ 3 X 93
\a + (5/ 3x9*
/g-3\"_ 1
Va + 3/ ~ 3x9^
0370
= -000 4572
= -000 0508
= -000 0056
/a — b\ _
\a + b/ ~
/a — b^ _
\^b) "
/a—b^ _
\^+b) ~
/a — b^ _
\J+b) ~
1 / a—b \^ _
9 \a + b) ~
11 \a + b) ~
333 3333
012 3457
000 8230
000 0653
000 0056
000 0005
•346 5734
is the value of the series for the interval of the octave;
putting the logarithm =12, we determine C by means of
the equation (A).
12 = Cx -346 5734
12
whence
C =
•346 5734
= 34-62469
which is therefore the value of the constant C in equation (A),
for the computation of the E. T. semitone system of
logarithms.
88 ON THE THEORY OF
Let us now compute by this method the value of a perfect
3
fifth, whose vibration ratio is - , expressed in E. T. semitones.
a — h 1
a-^b 5
-^3 1
/a—oY _
5x25
b\^ 1
= -008
= -000 32
Whence,
\a + b) ~ 5 X 25^
/a—b\} 1
( r) = 5 =.000 0128
\a + b) 6x25^
\a^b)
= -2
1 m—b^
= -002
6666
1 m-b^
5 V« + /J/
= -000
0640
1 /a-b^
7\a + b)
= -000
0018
•202
7325
iplying this by the constant C,
34-62469
•20 27325
69 24938
692494
242373
10387
692
173
we have 7^0195499 as the value of the
perfect fifth in E. T. semitones. It is within one unit in the
seventh decimal place of the correct value,
7^019 5500
This is one of our fundamental data, procured by an
independent process, in which we have not employed the
labours of those who constructed the tables of common
logarithms.
THE CALCULATION OF INTERVALS. 89
If however we consent to use these, we have a much more
simple process available. It is easy to shew, by the theory of
logarithms, that, in any two systems of logarithms, the
logarithms of given numbers are proportional to one another ;
that is to say, any one system of logarithms can be trans-
formed into any other by multiplication by some factor,
which is called the modulus of transformation.
This relation is easily shewn to exist between E. T. semi-
tones and any other system of logarithms, e. g. the common
system.
For if we refer to the table at p. 83, and form a third
column containing the common logarithms of x, x^, x^, . . . ,
then, if f be the logarithm of x
2 i will be the logarithm of x^
and so on ;
and £,2^,3^
are obviously proportional to 1 , 2, 3 ,
which are the corresponding numbers of the E. T. system.
We have therefore only to find the factor by which com-
mon logarithms must be multiplied, to convert them into
E. T. semitones.
This is easily done by the consideration that the common
logarithm of 2, which is '3010300, must become 12 when
transformed.
The required factor is consequently,
12
— = 39-86314
•3010300
It is only necessary to multiply the common logarithm of
any ratio by this number, to get the equivalent in E. T.
semitones.
We may, if we please, execute the multiplication directly;
or we may divide by -3010300, and multiply by 12, which is
perhaps a little shorter. Or we may adopt the rules given
in the note on p. 14, which perform the process more shortly,
employing an arithmetical artifice.
When we have obtained, by any of these methods, the
values of the fifth and third, all questions connected with the
intervals of the diatonic scale can be solved by means of
90 ON THE THEORY OF
addition, subtraction, and multiplication and division by low
numbers.
By means of the same values, the fifth and third, and inter-
vals derived from their combination, can be compared with
intervals formed by the division of the octave, with great
facility.
Those who are acquainted with the use of common loga-
rithms often employ them, instead of semitones, for these
purposes ; and it has been frequently proposed to use the
division of the octave into 301 equal intervals, by means of
which the common logarithmic tables read into the required
division with considerable accuracy. It is necessary to re-
member however, in using such an approximation as this,
that the solution of problems in beats (p. 17) generally re-
quires five, and sometimes six, significant figures ; any lower
approximation, not having special properties with respect to
the exact measuring of all intervals to be investigated, will
be liable to error. The system of 301 in particular fails to
represent the equal temperament altogether ; its fifths are
not particularly good ; it does not admit of employment for
demonstrating the difference between the different systems
with approximately perfect fifths ; nor can it be employed
at all for the demonstration of the properties of systems of
the mean-tone class. For these purposes at least five places of
logarithms must be taken ; and where the E. T. system has
relations of any interest, the E. T. semitone is vastly superior
as a unit to the logarithm.
There is an approximate method which has been occa-
sionally employed, which it seems worth while to discuss
with the view of obtaining some criterion of its accuracy.
The discussion will lead us, by analogy, to the deduction
from equation (A) of a new approximate formula, of consider-
able accuracy for small intervals.
The old method consists in taking the difference between
the vibration ratio and unity, and treating it as a measure of
the interval.
THE CALCULATION OF INTERVALS. 91
Thus, in a major tone, whose ratio is
8' °^ 'i'
- would be taken as the measure of the interval.
8
We shall show that this is equivalent to taking the first
term of a known logarithmic series, which is less convergent
than the series in equation (A) ; i. e. would require the em-
ployment of more terms to get an accurate result.
The following series is proved in treatises on logarithms : —
(B)...log/^.i/{(^-l)-^b-l)^ + i(j^-l)«-...|
The value of J/ is always half that of C in equation (A) ; so
that for E. T. semitones,
M = 17-31235
9
If we consider the major tone as - , then
8
1
(^-i/=„.
1
8^
and so on.
Q
Again, if we take - , the descending ratio of the major
tone, 1 —y = - , and we can put the above series into the
form.
The value is negative, indicating that the interval is taken
downwards.
Here, in successive terms,
1
and so on. \ -^1 93
93 ON THE THEORY OF
To compare these with equation (A).
9
If we consider the ratio - >
8
then « = 9 b = % ,
a—b 1
and = = —
a + b 17
/a—b^'^
and so on.
The terms with given index are less than those of (B) or
(C) ; and there is only half the number of terms up to any-
given index, since the terms with even powers are not in
series (A). Consequently by twice as many terms of either
(B) or (C) we do not obtain the same accuracy as by any
given number of terms of the series in (A), which we made
use of above.
1
128
/a—b^ _ 1
\a-\-b) ~ 17
The second term of (B) is
these are of such magnitude that, in the present case (major
tone), neither can be neglected in any computation which is
intended to take count of such quantities as a comma.
The second term of (A), on the other hand,
1 /a—b\^
IKa + b)
3x17^
1
14739
a quantity which may be neglected for some purposes.
So long then as the approximation is restricted to small
intervals, we may roughly compute E. T. equivalents by the
approximate formula,
/y h
34-6247 X ; = number of E. T. semitones ;
a-vb
where j is the vibration ratio, and 34-6247 the constant of
equation (A).
■ THE CALCULATION OF INTERVALS. 93
For very small intervals the approximation will possess
considerable accuracy.
Exanlple. To compute approximately the value of a
comma in E. T. semitones :
« = 81 ^ = 80
a-b _ 1
a-\-b ~ 161
34-6247 ^ ^
= -21506 E. T. semitones,
161 '
which is correct to the fifth place of decimals.
3
On the other hand, if we compute a fifth, -, by th is
u
method, we get
-^-^ = 6-9249 E. T. semitones.
5
The correct value is 7-01955
error -09465
and the error of the process amounts to about a tenth of a
semitone, or nearly - a comma.
Determining the octave, we get,
« = 2 b =\
'-^^^ = 11.5412
3
with an error of about half a semitone.
To reverse the process.
^ . a—b - .
Puttmg f = 3, we find,
a _ 1+2
1 ~ \-z'
If then we are given a small interval in semitones {x\ we can
find its ratio by this process.
For ^= 34-6247 (^^)
(by approximate equation (A) ;)
, a—b X
whence
a+b 34-6247
= z
94 THEORY OF CALCULATION.
whence we can find the ratio, from
a
b ~
1 +^
\—z
Remembering
that z =
X
34-6247'
we have,
a
~b^
34-6247 +
34-6247 —
X
■X
This form may be sometimes useful for small intervals.
Example. To find the vibration ratio of one E. T. semi-
, ,, a 35-6247
tone. Kx =\) -. = ■
^ ■' b 33-6247
or, since the accuracy hardly reaches to the last figures,
35-62
33-62
17-81
nearly i
16-81
and the required ratio is a little greater than
18
\7
T. A. JENNINGS, ORGAN BUILDER.
(special experience in pneumatic mechanism.)
Constructor of Bosanquet's Enliarmonic Harmonium and
Organ.
Mr. Jennings is prepared to undertake the construction of Harmo-
niums or Organs with Bosanquet's generalised keyboards for playing
with improved intonation.
With these keyboards the fingering of scales and chords is the same in
all keys.
Where prices are named they are intended as approximate estimates
for total cost.
Organs with 24 keys per Octave, suitable for tlie mean -tone
system.
N.B. The fingering of the mean-tone system is remarkably easy.
Harmoniums with, one reed to each key, compass 4J octaves.
£
36
52
70
90
108
126
I. Suitable for use with the mean-tone system; or for illustration, or
performance of limited extent with perfect-fifth systems.
II. Suitable for a very extended mean-tone system; or for performance
of considerable extent with perfect-fifth systems.
I and II may be constructed with two stops at some increase of cost;
and thus the mean-tone and perfect-fifth systems may be combined in one
instrument.
III. Suitable for an extensive command over perfect-fifth systems.
IV. V, VI. Suitable for such systems as the division of the octave into
53 equal intervals, and extensive experimental work.
I.
24
keys per octave
II.
36
J) »
III.
48
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' No musical lecture room should be considered complete without an instrument
of at least 48 digitals (finger-keys) to the octave, tuned in practically just into-
nation, when it can be obtained at so low a price.' — Mr. A. J. Ellis, F.R.S., Ap-
pendix to Ellis's Helmholtz, p. 696 note.
Address at Mr. Fowler's, 127, Pentonville Road, London.
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