152 



THE STUDY OF SPEECH CURVES. 



The two presentations of results are direct!}^ opposed to each other; 

 according to one the vowel wave, figure 130, is composed of a set of har- 

 monics (figure 132) ; according to the other it is composed of the single 

 inharmonic (figure 134). 



Both presentations fail in two respects. In the first place no account 

 is taken of the friction, which is known to be present in vowel vibrations, 

 and which is so evident in the curve itself. The first example in this chap- 

 ter and the simpler cases discussed above (p. 102) have shown that the 

 neglect of friction produces erroneous results. In the second place we 

 know (p. 109) that the strongest tone of the vowel is always the glottal 

 tone, or the fundamental. In the wave just analyzed the harmonic inter- 

 pretation represents it as very weak, and the inharmonic one represents 

 it as entirely lacking. We will first consider the method of analysis that 



takes account of the friction, 

 and will then discuss the 

 problem of the fundamental. 

 For an analysis into f ric- 

 tional sinusoids the factor of 

 friction must be found. Using 

 the maxima and minima we 

 perform the adjacent calcula- 

 tion. The first column contains the maxima and minima, the second col- 

 umn gives the differences between the successive ones. To find the natural 

 logarithms each of these values must 

 be multiplied by such a power of 10 

 that the result falls between and 

 1000, in this case by 100. The nat- 

 ural logarithm is then taken from a 

 table (e. g., Ligowski, Taschenbuch 

 der Mathematik). Since we can use 

 only an even number of values for 

 this computation, we must omit one of the five of the list. Omitting 

 the last one, we have n = 4, and (since log^Si — log,s„=log, lOOsi— log^ 100s„) 



Fig. 134. — Component curve of figure 130 according to 

 figure 1 33. 



d=.Q, 3(6.721 ^ - 6. 1312) + (6.620 1 - 6.4457) _q ^^^^^q 



4x15 



Omitting each of the values in succession and computing the factor 

 of friction from the four others, we obtain the five values: 0.19450, 0.22900, 

 0.26045, 0.24301, 0.21262, of which we take the average 0.22792 as the 

 value for d. 



