240 
NATURE 
[| Fan. 11, 1883 
varieties following the indication of Coccinella 10-functata, 
L., and 6 or 8 analogous varieties are appended to many 
other species of Ladybirds. Taking it as a whole, this 
excellent catalogue may serve as a model for compilers of. 
lists of the Beetle (or other entomological) fauna of other 
districts. 
LETTERS TO THE EDITOR 
(The Editor does not hold himself responsible for opinions expressed 
by his correspondents. Neither can he undertake to return, 
or to correspond with the writers of, rejected manuscripts. 
No notice is taken of anonymous communications. 
[The Editor urgently requests correspondents to keep their letters 
as short as possible. The pressure on his space is so great 
that it is impossible otherwise to ensure the appearance even 
of communications containing interesting and novel facts.) 
Equal Temperament of the Scale 
In your number of November 8, 1877, p. 34, Mr. Chappell, 
F.S.A., has intimated that mathematicians who propose to divide 
the octave into twelve equal semitones instead of ‘‘ equally tem- 
pered semitones,” are deficient in musical ear. I have not 
noticed that any mathematician has replied to him. 
Representing (with Mr. Chappell) the number of vibrations in 
the C of my piano by 1, and the octave ¢ therefore by 2, and 
dividing the octave into 12 equal intervals, I obtain for the 
vibration-numbers— 
ot G = 1'4983 = 27* 
cz = 1105904 = © | Gg = 15874 = 912 
D = 1'1224 = N S028 = oe 
Dg = 1'1892 = Bb = 1°7818 = gt 
T2599 a B = 1'8877 = 21? 
F = 1°3348 =: ¢ =2 
Fg = 1°4142 = 
In these equal semitones each is equidistant from the preceding 
and following: as F is to Fg, so is Fg to G, &c. Hence in 
whatever key I play a passage on my piano, the divergence from 
harizonic intervals will be alike at every point ; the keys on my 
piano will have no distinctive character, the key of 3 sharps will 
not be more ‘‘ brilliant ’’ or less ‘‘ plaintive” than that of 4 flats. 
In the key of C, the harmonic third, fifth, and seventh will 
be, according to the above notation, 1°25, 1°5, and 1°75 respec- 
tively. As regards the fifth G it is a remarkable numerical 
coincidence that 21? only differs from 1°5 by sty, z.e. the equal 
temperament G only differs from the harmonic by its $y part, 
a difference so slight that it may be neglected. We tune fiddles 
by fifths therefore. This coincidence is the fundamental fact which 
enables us to modulate into various keys on a piano, and it is the 
reason why the scale must be divided into 12 (and not any other 
number of) semitones ; for it will be found that, until you get to 
the unmanageably high number of 53, no other equal division of 
the seale has any note so near the harmonic G. 
The crucial point of tempering arises on the third. The E of 
1 
my piano is 25 = 32%, whereas the harmonic E is = 333 ; 
E is therefore by its 4 part too sharp, zz che key of C, a per- 
ceptible degree of error, unpleasant to many musicians. In 
ordinary pianoforte tuning, the E (by the plan in Hamilton’s 
pianoforte tuner or some similar compromise) is tuned somewhere 
; my 
between 325 and 44%, say at, and the wolf between this E and 
the upper ¢ is distributed. 
This is all very simple so long as we remain in the key of C; 
indeed if we remain there, we want no tempering. But Gg is 
the third to E, and cis the third to Ah; on the piano Gg and 
Abare one. On my equal-semitone piano I have 
L 
c=1; E= 2° (= 75 nearly; 
Gh = Ab = 25 (= 485 nearly); ¢=2. 
I now ask the champion of ‘‘ equally tempered semitones ” what 
is the numerical value of his E and what of his Ge. If he gives 
j 3 ovted 
bem any other values than 2° and 2° respectively, it is clear 
hat a greater error will be introduced in one part of the scale 
than is saved in another. Instead of algebraic proof I take an 
instance—suppose that Mr. Chappell tunes his E at 125) if he 
100 ” 
equally tempers his Ge in the scale of E, it will be em = ah 
very nearly. Then when he puts down the common chord in 
the key of Ab, his third the c will be by its j; part to» sharp, 
whereas on my equal temperament piano it would only be by its 
3) part too sharp. In other words, though the keys of C and 
E may be somewhat better on Mr. Chappell’s piano than on 
mine, the key of Ah will be very much worse. This is pretty 
nearly what occurs in practice. The point of my argument is 
that Mr. Chappell cannot move his E ever so little from the 
1 
value 2% without introducing a greater error somewhere else. 
The term ‘‘ equally tempered semitone’’ is inaccurate ; the semi- 
tones on my piano are all equal; and no one of them can be 
altered by a disciple of the ‘‘ equally-tempered semitone” with- 
out making them unequal, The ‘‘equally-tempered semitones ” 
are mot equally tempered. Moreover if you ‘‘ temper” at all 
en lose the effect of the harmonics ; by moving E from 33% to 
125 
100 
The simple reason that unequal tempering is ‘practised is 
because all keys are not used equally often. A piano is un- 
equally tempered so that the keys C, G, A, F are fair, E, Bp, Eb 
tolerable, the other keys being very much worse than on my 
equal-semitone piano. On most church organs, being unequally 
tempered, if you modulate even transiently into 4 or 5 flats, the 
effect is unendurable. 
The crucial question in tuning is the question, if your E is not 
1 2 
2° and your Ge 28, what values do you put them at? The 
question of the seventh is more complex ; I may observe that 
though my equal-semitone seventh (1°7818) appears far away 
from the harmonic seventh (1°75), yet that the Bp of tuners on 
the ‘ equally-tempered semitone ” system is not much nearer it. 
Their Bb is 4° or thereabout, or in other words, the sub-sub- 
dominant of C. Therefore, on the piano, you have not got the 
‘‘harmonic-seventh ” at all; the note which replaces it is one 
that suggests overpoweringly the key of F. This is the secret 
which underlies several of our rules in harmony. It is also the 
reason why valve-horn players play Bb (though an open note) 
with valve 7.2, or if they play without a key “‘lip it up” very 
carefully. 
It is often supposed that the ‘‘ wolf” has been introduced into 
music by that most useful though imperfect instrument the piano, 
and that the noble violin or human voice knows it not, except in 
so far as our natural good ear for harmonic intervals has been 
debauched by continually hearing tempered intervals. This is 
not so ; the ‘‘ wolf” is not only in the piano but in the scale. 
It is true that a violin can play in harmonic tune so long as the 
melody runs in one key, or if it modulates into a closely allied 
key, and dack again the same way. But suppose my violin begins 
by rising from C to E harmonically, ze. to 42%; then after 
playing awhile there proceeds to Gz (335)? harmonically, being 
then in 8 sharps; and then, after playing awhile in 8 sharps, 
proceeds to c; the c of the fiddle will then be (33%)? instead of 
2, z.e. it will be ;3< out of tune. In this simple case the fiddle 
is supposed to play alone, unfettered by any harmonics but its 
own; in the case of a string-band, the agreeableness of many 
modulations actually depends upon some chords being harmonic- 
ally out of tune, the note in the chord which performs the duty 
of Ge to its preceding chord, performing the duty of Ab to its 
succeeding chord. 
The practical conclusion is that the best plan of tuning a 
piano for vulgar music and vulgar players is that now ordinarily 
practised by the tuners, and recommended by Mr. Chappell ; 
but if the piano is to be used equally in all keys (or even fre- 
quently in 4 or 5 flats, 5 or 6 sharps) the best plan is to tune it 
in 12 mathematically equal semitones. C, B. CLARKE 
you sacrifice harmonic coincidence. 
Animal Intelligence 
In an excellent paper on “Animal Intelligence” (NATURE, 
vol, xxvi. p. 523), Mr. C. Lloyd Morgan says that ‘‘’The brute 
has to be contented with the experience he inherits or indivi- 
dually acquires. Man, through language spoken or written, 
profits by the experience of his fellows. Even the most savage 
tribe has traditions extending back to the father’s father. May 
there not be, in social animals also, traditions from generation 
