612 
Proceedings of the Royal Society 
the G gently enough to give the gradual transition from, let us 
say for example, four uniform heats per second, through the case of 
four heats per second with every alternate beat somewhat louder, to 
the case of only every second beat perceptible, or, in all, two beats 
per second ; but it can be done, and the result is an interesting and 
instructive illustration of the slowing down from the quick beat of 
the binary harmony to half speed, or to one-third speed, or to one- 
fifth speed, as the case may be, by the introduction of a third 
note. In the several cases I have foundthat I can, by making 
the added note faint enough, produce a succession of beats of 
which every second, or every third, or every fifth, as the case may 
be, is louder than the others, and that, as the intensity of the added 
note is gradually increased, the fainter beats become imperceptible, 
and a regular unbroken slow beat is heard distinctly alone, always 
in the theoretical time of the whole imperfection of the harmony. 
I have verified this distinctly in the cases of 1, 2, 3; 2, 3, 4 ; 3, 4, 5; 
4, 5, 6 (as stated above) ; 5, 6, 7 ; and 6, 7, 8. I have not succeeded 
in hearing the beats on the approximations to the harmonies 8 : 9 
and 9 : 10. But the slow beat on the 8, 9, 10 (with vibrational 
frequencies 256, 288, 320), with any one of the three notes slightly 
flattened, is very remarkable. The sound is like that of a wheel 
going round with decided roughness of motion in every part of its 
revolution, but much rougher in one part than another, with a loudly 
perceptible periodic return of the roughness in the theoretical period 
of the approximate harmony. 
The beats on the harmony C E G (vibrational frequencies 256, 
320, 384), with any one of the three notes slightly flattened, are 
very perceptible : untrained ears hear them instantly the first time 
without any education, and the beat is heard almost to the very 
end of the sound if three of Koenig’s forks, one of them, the 
C, for example, being slightly flattened by a brass sliding piece 
screwed to it, be caused to sound. The sound dies beating, the 
beats being distinctly heard all through a large room as long as the 
faintest breath of the sound is perceptible. The smooth melodious 
periodic moaning of the beat is particularly beautiful when the beat 
is slow (at the rate, for instance, of one beat in two seconds or 
thereabouts), being, in fact, sometimes the very last sound heard 
when the intensities of the three notes chance at the end to be 
suitably proportioned. 
