EXAMPLES OF VOWEL ANALYSIS. 151 



130 can be mathematically analyzed into the series of simple sinusoids 

 whose amplitudes and phases are given in the table and in figures 



131 and 132. The original curve can be produced by calculating the 

 ordinates from the table or by adding the ordinates of the components 

 in figure 132. In itself the analysis means nothing more than this; it is 

 a purely formal procedure devoid of physical or physiological meaning. 



A physical interpretation can be given to the results by accepting the 

 fact that the sound which gave the original wave can be closely imitated 

 by adding sounds which would separately give the waves in figure 132, 

 the phases being properly adjusted. We might presumably reproduce the 

 original sound by maintaining a set of tuning-forks in vibration with the 

 periods, ampHtudes, and phases indicated by the curves in 



figure 132. 



An auditory interpretation is often supposed to be 

 given to the results if we accept the Helmholtz theory of 



hearing (p. 125), according to which 



30 

 20 

 10 



2 3 4 5 6 7 8 9 10 II 12 13 14 IS 16 17 18 



the ear physically analyzes a vibra- 

 tion into its harmonic components, 

 Fia. 133.— Plot of inharmonic component to just as WO liavc here analyzed it 

 figure 130. mathematically, but the supposition 



rests on a misunderstanding (p. 126). 



A vowel interpretation must be given if we accept the overtone theory 

 of the voice, according to which the vocal cavities reinforce overtones of 

 the tone from the glottis (p. 109). The vocal cavities then vibrate with 

 the relations of amplitude in column c. This theory we have been forced 

 to reject (Chapter VIII). 



If we take into consideration that the vocal tones may just as well 

 be inharmonic to the fundamental, we have the problem of calculating 

 the inharmonics from the set of harmonic results. Proceeding on the 

 principle (p. 81) that there are as many components present as there 

 are apexes in the harmonic plot, we have to calculate the inharmonic that 

 Ues around 3; the results for the higher harmonics are so small as to be 

 practically 0. Taking the weighted mean for the first 8 results we have 



(lX0.21) + (2x0.71) + (3x3.07)-t-(4X0.50)4-(5x0.30) + (6X0.22) + (7x0.09) + (8X0.07 )_.; 21 

 0.21+0.71+3.07+0.50+0.30+0.22+0.09+0.07 



as the ordinal number of the inharmonic. The characteristic of the wave 

 is thus a tone that has a frequency of 3.21 times the fundamental, or 565.3. 

 The amphtude of this hannonic we can approximate (p. 81) by taking 

 the largest amplitude of the group; we obtain 3.07. The composition of 

 the vowel wave is thus given by the plot in figure 133 and the curve in 

 figure 134. 



