92 



THE STUDY OF SPEECH CURVES. 



may also readily be shown that when the curve consists of the same wave 

 regularly repeated, the amplitude of the harmonic found is independent 

 of the point at which the first ordinate is taken. That the result is not 

 the same when the wave is not regularly repeated can be readily illustrated 

 by replacing some of the end values of the table by others not from the 

 table. This principle marks a fundamental distinction between musical 

 curves and speech curves. 



A curve yielding the adjacent table of products is analyzed into a 

 series with ai = 0, 61= 10, 0^=0, 62 = 5, 03=0, 63 = 5, and all the other 



coefficients = 0. The equation of the curve 



(for T = 36) is therefore 



2_ 

 t/= lO.sin —-t + 5. sin 

 ^ 36 



— f +5.sin — -. 

 18 12 



A curve yields the ordinates 0, 73.4, 5.0, 

 17.1, 8.7, 3.4, 10.0,-2.0, 8.7,-2.9, 5.0, -1.3, 

 0, 1.3, - 5.0, 2.9, - 8.1, 2.0, -10.0,-3.4, 48.7, 

 — 17. 1, — 5.0, —73.4. The analysis with 24 ordi- 

 nates gives 6, = 10.1, 62= 10.0, 63= 10.0, h,= 

 10.1, 65 = 10.0, 66 = 10.0, 67 = 10.1, &8 = 10.1, &9= 10.1, 6,0 = 10.0, and 

 all other coefficients = 0. The equation (tenths of a unit being dropped) 

 is therefore (for T=36) 



y=10 



. 271 



sin ^t. 



A curve yields the ordinates— 7.50, - 3.32, + 1.00, + 4.87, 9.43, 

 + 13 72 + 10 97, + 6.93, + 6.55, + 8.50, + 8.06, + 2.50, - 5.31, - 1.10, 

 - 13.53,' - 15.43, - 13.72, - 9.50, - 2.31, + 3.07, + 2.21, - 3.50 - 8.06, 

 at intervals of l = 5. The analysis gives C = 0, Oi = 0, 6, = 10.0, a, = 2.5, 

 62 = 4.3, ai = 5.0, h = 0, a* = 0, h, = Q,a, = 0, h = 2.0, a, = . . . = Oi 

 ^ 5 =: . . . = &12 = 0. The curve thus has the equation: 



y 



2n^ 



.27t. 



27t 



10.sin gf + 5 sin {f^ t _|) +5.sin {^^t -|) + 2.sin ^^ «• 



The curve and its components are shown in figure 71. 



In the examples hitherto considered, the period of the curve analyzed 

 has coincided with that of one member of the harmonic series used for the 

 analysis. The more general and more important problem is how a sinu- 

 soid curve of any period will show itself in a harmonic analysis. 



