HARMONY IN MUSIC. 75 



avoid doing so in the former, where the tone is due to a single 

 source. And this is found to be really the case. 



I have previously mentioned the form of wave with gently 

 rounded crests and hollows, and termed it simple or pure (p. 65). 

 In reference to this form the French mathematician Fourier has 

 established a celebrated and important theorem which may be 

 translated from mathematical into ordinary language thus : Any 

 form of wave whatever can be compounded of a number of 

 simple waves of different lengths. The longest of these simple 

 waves has the same length as that of the given form of wave, 

 the others have lengths one half, one third, one fourth, &c., as 

 great. 



By the different modes of uniting the crests and hollows of 

 these simple waves, an endless multiplicity of wave-forms may 

 be produced. 



FIG. 9. 



For example, the wave-curves A and B, Fig. 9, represent waves 

 of simple tones, B making twice as many vibrations as A in a second 



