610 
Proceedings of the Royal Society 
pressure; in the opposite phase-relation there is a slow change before 
and after the time of maximum pressure, and a rapid change before 
and after the time of minimum pressure. In the former case the 
difference of maximum from mean exceeds the difference of mini- 
mum from mean ; in the latter the difference of the minimum from 
mean exceeds the difference of maximum from mean. If the ear 
could not distinguish between two such sounds, but could distinguish 
between either of them and the sounds represented by the first and 
third quarter-period curves, the number of beats would be twice the 
error of frequency of the higher note. But I find that the number 
of beats is quite distinctly equal to the error of frequency of the 
higher note. I have found that the beats on the 1 : 2 harmony are 
most easily perceived when the intensity of the higher note is com- 
paratively faint, as would be the case if they were explained by 
Helmholtz’s theory that they are the beats on the approximation to 
unison which there is between the higher note and the first overtone 
of the lower note. But the simple-harmonic character of the two 
constituent tones at the entrance of the ear precludes the acceptance 
of this theory unless extended, as it has actually been by its author, 
to the interior mechanism of the ear. 
Whatever may be the physiological theory by which the beats 
are to be explained, it is an interesting fact that the ear does dis- 
tinguish, as it were, between push and pull on the tympanum in 
the manner illustrated by the preceding curves, not only for the 
case of approximation to the harmony 1 : 2, but for every other even 
binary harmony. I have heard distinctly the beats on approxima- 
tions to each of the harmonies 2:3, 3:4, 4:5, 5:6, 6:7, 7:8, 
1 : 3, and 3 : 5. The two last mentioned, though sometimes less 
easily heard than the beats on most of the others, are unmistakably 
distinct ; and by counting the numbers of them in ten seconds or in 
twenty seconds, I have ascertained that they, as do all the others, 
fulfil the condition of having the whole period of the imperfection, 
and not any sub-multiple of it for their period of audible beat. 
They are interesting as being cases of odd binary harmonies. Before 
making the experiments, I thought it possible that what is heard in 
the beat might not make distinction between the configuration II. 
and IV. (first quarter phase and third quarter phase) : but a revolving 
character which I perceive in the beat seems to me distinct enough 
to prove that the ear does distinguish between these configurations, 
