608 Proceedings of the Royal Society 
Qualities of 
m 
<D 
‘o 
Perfect 
Harmonic 
Imperfect 
Falseness 
a c3 
(X) <x> 
Harmonies. 
Numbers. 
Harmonies. 
of the 
Intervals. 
O 'H 
fH O 
Ph 
Errors. 
{S 
{i 
( ® - 043 
| (2f + 2-54 
| Too small 
17-39 
/ G 0-0 
( ffi + 2-54 
| Too small 
15-24 
(C 
l a 
{! 
/ C' o-o 
+ 3-87 
| Too small 
23-22 
{c 
{3 
/$, + 3-87 
\ c o-o 
| Too large 
11-61 
/ E' 
»G 
{S 
/ & + 5-08 
t <S - 0-43 
| Too large 
17-39 
( E 
f 5 
( <£ + 2-54 
) 
12 
} j G 0-0 
f Too large 
5-08 
( |c 
r 
( \ c o-o 
It is of course to be understood that the degree of falseness is the 
same in all the tempered harmonies of the same name (or having the 
same harmonic numbers) ; and that the different numbers shown for 
the frequencies of the beats are (except for the case of the (0? with 
the untempered G) in simple proportion to the vibrational frequencies 
of one or other of the constituent notes. The slightness of the 
imperfectness in the tempered fifth (approximately 2 : 3) is indicated 
by the slowness of the beats, not so much as one per second on the 
C The imperfectness of the fourth (approximately 3 : 4) is even 
less than that of the tempered fifth, so that, notwithstanding the 
greater harmonic numbers, the beats are scarcely more rapid (1*17) 
for the C Jf than (*86) for the C But when we go to major and 
minor thirds of the tempered scale, we take leave of mathematical har- 
mony entirely. The beats on the C ($ (ten per second) are too rapid to 
be counted, and it is only in virtue of their not being perceived, or 
not being disagreeably perceived, that the combination is agreeable. 
The same may be said still more unqualifiedly of the minor thirds, 
the number of beats on (0? being more than seventeen per second. 
It does not seem easy to explain on any physical or physiological 
