1897-98.] Dr Lloyd on Analysis of Tracings of Vowels. 105 
that the partials are equidistant from one another. It is true that 
they are equidistant when considered as an arithmetical progression, 
— 1st partial, 98 v.d.; 2nd partial, 196 v.d.; 3rd partial, 294 v.d., 
etc. But musical distances are not measured in arithmetical but 
in geometrical progression ; and the equidistant partials are not the 
1st, 2nd, 3rd, 4th, etc., but the 1st, 2nd, 4th, 8th, etc. The 
abscissae in fig. 5 ought therefore not to be measured off in propor- 
tion to the numbers 1, 2, 3, 4, etc., but to log 2 l, log 2 2, log 2 3, log 2 4, 
etc., or simply to log 1, log 2, log 3, log 4, etc., seeing that the 
base does not matter if the right mutual proportion is maintained. 
The result is shown in fig. 6. 
1st octave. 2nd octave. 3rd octave. 4th. 
I I I I S i I 8 i 1 
1st partial 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 
Fig. 6. 
If we adopt this view, we get the following equation to find p, 
the mean partial : — 
log p — 
a! logp' + a" logp" + a'" logp'" + etc. 
a + a" + a' + etc. 
5 
and then n = Up as before. 
This view seems well founded as far as it goes ; and I adopted 
it in Journ. of Anat. and Phys ., xxxi. 249 ; but there is a 
further consideration which it leaves out of sight. This is the 
consideration of frequency. Take the partials in fig. 5. The 
amplitudes are 4 ‘2, 8 ‘5, 3*2, etc.; but the first-named ampli- 
tude is traversed only once in each period, whilst the second is 
traversed twice, the third three times, and so on. The general 
expression for the whole amplitude traversed in each period is op, 
i.e., amplitude multiplied by frequency. It seems a right principle 
to estimate the strength of the reinforcement of each partial in 
terms of the whole amplitude traversed ; and if so, the multiplied 
amplitude, a'p', a'p", etc., will have to be substituted for the single 
amplitudes a', a", etc., in our last equation. Thus we get 
^ a'p' logp' + a'p" logp" + a'"p'" log p'" + etc. 
°° ^ — a p' + a'p" + a'p" + etc. 
to find the mean partial. 
