104 Proceedings of Royal Society of Edinburgh. [sess. 
8 
7 
6 
9 10 
1 
2 
3 
4 
5 
4‘2 8*5 3-2 7-3 2'2 13*9 447 50‘2 13 ’6 14*6 
Fig. 5. 
This analysis displays but one considerable reinforcement; but 
it extends, in varying strength, over the 6th, 7th, 8th, and 9 th 
partials. This reinforcement probably springs, like that of the 
5th partial (875 v.d.) in fig. 3, from the resonance of the 
intra-velar or pharyngeal cavity of these a vowels. But for the 
present it is better not to commit ourselves to any name involving 
a theory of origin, but to call this the a-resonance. How are we, 
from these reinforcements, to evaluate the proper pitch of that 
resonance in each case ? Hermann used what he called a centre- 
of-gravity calculation, i.e., he proceeds as if he were finding the 
centre of gravity of four heavy points situated as above, on the 
axis of x, and each weighing as many units as there are units in 
the respective amplitudes. Having thus discovered the “ mean 
partial,” he multiplies the fundamental by it to get the value of the 
resonance itself. Both operations are combined in the equation 
where H and n are again the pitch-numbers of the sung note and 
of the desired resonance, respectively ; p, p", p" are successive 
integers, the numbers of the reinforced partials ; and a', a ", a' are 
the amplitudes found for these partials respectively. For the 
resonance displayed in fig. 5, the process is as follows : — 
n — 98 x 
(6 x 13-9) + (7 x 44-7) + (8 x 50-2) + (9 x 13*6) 
(13-9 + 44-7 + 50-2 + 13-6) 
= 737 v.d. 
Pipping does not object in principle to a centre-of -gravity cal- 
culation, but he objects to the assumption, embodied in fig. 5, 
