12 24 ON VOCAL SOUNDS. 



base does not matter if the right mutual proportion is maintained. The 

 result is shown as follows : — 



1st Octave. 2nd Octave. 3rd Octave. 4th. 



i_ 



1st Partial. 2nd. 3rd. 4th. 5th. 6th. 7th. 8th. 9th. 10th. 



Fig. 437. — Diagram showing the application of Pipping's method. 



If we adopt this view, we get the following equation to find p, the mean 

 partial : — 



W v = a ' lQ g P' + a " ] °g P" + a '" lQ g P'"* etc - 

 a +a +a , etc. 

 and then n = Np, as before. 



This method of Pipping's has been further developed by Lloyd, 1 because 

 he was of opinion that Pipping's method left frequency out of account. Look 

 again at the above figures. The amplitudes are 4*2, 8 - 5, 3 "2, etc. ; but the 

 first-named amplitude is traversed only once in each period, while the second 

 is traversed twice, the third three times, and so on. The general expression 

 for the whole amplitude traversed in each period is ap, i.e., amplitude multi- 

 plied by frequency. It seems a right principle to estimate the strength of 

 the reinforcements 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 sub- 

 stituted for the single amplitudes (a, a", etc.) in the last equation. Thus — 



, _ ap log p + a p log p + a p log p + etc . 

 pi a'p' + a"p" + a'"p'" + etc. 



to find the mean partial. 



Comparing the three methods of Hermann, Pipping, and Lloyd, it is 

 evident that Pipping's will always give a result somewhat lower, and 

 Lloyd's will give a result somewhat higher, than Hermann's. In the 

 present case, shown in the above figure, the difference between the three 

 calculations is not great. The first (Hermann) gives 737 vibs., the 

 second (Pipping) 732 vibs., while the third (Lloyd) is 741 vibs. With 

 high partials the differences become less and less, but with lower partials 

 the differences are so great as to be, in Lloyd's opinion, " of vital import- 

 ance," the object of these calculations being to find out what is the true 

 pitch of the resonating cavity that may reinforce several or many partials. 



Phonograms. — We are now in a position to examine the methods 

 that have been adopted to obtain graphic tracings of the wave-forms 

 (phonograms) corresponding to vowel tones, so as to submit these to the 

 Pourierian analysis. 



Donders, 2 in 1870, was the first to apply the phonautograph of Leon 

 Scott 3 (invented in 1856) to the investigation of the curves produced by 

 the sounds of vowels. In 1878, Fleeming Jenkin and Ewing 4 succeeded 

 in obtaining tracings of the records of vowel sounds on the tinfoil 



1 Reply to Pipping's review of the author's articles on " Speech Sounds," in Phonetisrhc 

 Stud., 1890-1892 {Z/schr. f. d. franzosische Sprache, Bd. xvi. S. 211) ; also, " The Genesis of 

 Vowels," Jourti. Anat. and Physiol., London, vol. xxxi. p. 233 ; also, "Interpretation of 

 Phonograms of Vowels," ibid., p. 240. 



2 " De physiologie der spraehklanken in het bijzonder van die der nederlandische 

 taal," Utrecht, 1870. 



3 E. Leon Scott, Compt. rend. Acad. d. sc, Paris, tome liii. p. 108. 



4 Fleeming Jenkin and Ewing, "On the Harmonic Analysis of certain Vowel Sounds," 

 Trans. Roy. Soc. Edin., vol. xxviii. p. 745. 



