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Full text of "An elementary treatise on musical intervals and 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;"

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An elementary treatise on musical Interv 




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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 



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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 



The original of tiiis book is in 
tine Cornell University Library. 

There are no known copyright restrictions in 
the United States on the use of the text. 



http://www.archive.org/details/cu31924022233088 



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 


2 




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. 






V\f3 . 


. 




10. 


. 






Wbb, . 




9. 




Web, . 




. 




8. 






. Wab. 


, , 




7. 


Wok, 


• 


• 







23 



II. 

Section. 



ic. 



IC" 



EleVftbloTV 



'/ 



\c 



t» -IL. 



Id? 



9 



i'^ 



\df 



^e 



:f» 



\9 



Plan 



\c 




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 


3) J> 


IV. 


60 


» J> 


V. 


72 


J) )> 


VI. 


84 


» if 



' 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.