SYNTHESIS OF VIBRATIONS. 135 



was obtained for each value of e. This set of tables formed the basis of 

 the investigation. 



Combinations of the frictional sinusoids in twos, threes, etc., were 

 made, and the results were plotted and compared with speech curves. 



It was of course impossible to think of carrying out this whole plan 

 up to 10 or 20 components, as the number of curves to be made was 

 unattainable. One assumption after another was made and a few cur\'e3 

 calculated for each. When the results showed incompatibility with the 

 vowel curves, the assumption was abandoned and that Une of computation 

 was dropped. Thus the assumption that vowel curves are composed of 

 simple sinusoids (£ = 0) in a harmonic series could be ruled out at the 

 start, because all such combinations (when all waves begin with the 

 same phase) produce curves symmetrical to the X-axis, whereas vowel 

 curves are never symmetrical. Of course, vowel curves could be counter- 

 feited by adding harmonic simple sinusoids with differences of phase 

 just as any curve can be analyzed into a harmonic series, but the number 

 of elements would be large and the objections to the simple harmonic 

 analysis are also vahd against such a synthesis. The assumption that the 

 friction is the same in all harmonics of a compound (that is, that £,= 

 £j = £,, etc.) produced curves that showed what might be called a dis- 

 torted "symmetry" with the second half of the wave a dwindUng reverse 

 of the first. Vowel curves do not show this phenomenon. 



Results closely resembhng vowel curves were obtained, however, 

 when a comparatively large value was given to Si, and much smaller 

 values to s^, £s, etc. For example, the equation 



y=100.e-°^rsin % +20.e-»-»".sin ?J« + 20.e^''-'^':s{npt 



gives the curs^e in figure 120, which appears to the eye exactly Uke a vowel 

 curve. The pecuUarity of the equation Ues in the comparatively large 

 factor of friction of the first element. What does this fact mean? We may 

 suppose that Si is not a factor of friction at all, but is the factor of sudden- 

 ness d in the glottal puff (p. 121). The first element in the compound curve 

 is — in the writer's opinion — approximately the curve of the glottal puff; 

 the other elements are mainly caAdty vibrations. By using equations 

 of the form just given, and by varying the ampUtudes and periods we 

 can hope to imitate not only the different typical vowel forms but also 

 the personal peculiarities of the speakers. 



This work of producing curves by synthesis is of special importance, 

 not only because it can be used as the test of any theory of the vowels, 



