116 
Proceedings of Poyal Society of Edinburgh. [sess. 
Approximate a- and /2-Resonances of sung o. 
Sung by 
V.D. 
Partials Reinforced. 
Mean Resonance. 
Radical 
Ratio. 
a 
P 
a 
P 
Hermann, . 
132 
2-4 
4-7 
421 
653 
1*553 
3 ? • 
132 
1-3 
4-7 
290 
672 
2*318 
M ‘Kendrick, 
148 
1-4 
4-7 
365 
812 
2*221 
Boeke, 
128 
2-4 
4-6 
391 
615 
1*573 
For making a more exact calculation, two methods suggest them- 
selves. When a resonance spreads over several partials, it tends 
to spread downwards and upwards with about equal force. There- 
fore, by regarding the downward falling-off of the a-resonance, and 
the upward falling-off of the /2-resonance, we may endeavour to 
conjecture the real decline in the contrary direction. But there is 
room for much caprice in such a conjecture, and it would only lead 
to any certain improvement of result when the common partial is 
manifestly higher or lower than either of the several reinforcements 
can be imagined to be. A sounder, though as yet unpractised, 
method would be to take from one voice a sufficient number of o 
analyses to constitute an average which would eliminate the chances 
which cause the common partial to vary so immensely. The 
number need not be very large, if the instances are fairly represen- 
tative of various grades of its strength and weakness. Our four 
available instances give an average of 368 v.d. for the a-resonance, 
of 688 v.d. for the /2-resonance, and of 1*916 for the radical ratio. 
But they are hardly sufficient, either in number or in distribution, 
and they do not come from identical voices, nor even from 
identical nationalities. 
