HARMONY IN MUSIC. 91 



combinational tones will have different pitches, and produce 

 faint beats. 



The combinational tones are usually much weaker than 

 the upper partial tones, and hence their beats are much less 

 rough and sensible than those of the latter. They are conse- 

 quently but little observable, except in tones which have scarcely 

 any upper partials, as those produced by flutes or the closed 

 pipes of organs. But it is indisputable that on such instruments- 

 part-music scarcely presents any line of demarcation between 

 harmony and dysharmony, and is consequently deficient both in 

 strength and character. On the contrary, all good musical 

 qualities of tones are comparatively rich in upper partials, 

 possessing the five first, which form the octaves, fifths, and 

 major thirds of the fundamental tone. Hence, in the mixture 

 stops of the organ, additional pipes are used, giving the series 

 of upper partial tones corresponding to the pipe producing the 

 fundamental tone, in order to generate a penetrating, powerful 

 quality of tone to accompany congregational singing. The im- 

 portant part played by the upper partial tones in all artistic 

 musical effects is here also indisputable. 



We have now reached the heart of the theory of harmony. 

 Harmony and dysharmony are distinguished by the undisturbed 

 current of the tones in the former, which are as flowing as when 

 produced separately, and by the disturbances created in the latter, 

 in which the tones split up into separate beats. All that we 

 have considered tends to this end. In the first place the phe- 

 nomenon of beats depends on the interference of waves. Hence 

 they could only occur if sound were due to undulations. Next, 

 the determination of consonant intervals necessitated a capability 

 in the ear of feeling the upper partial tones, and analysing the 

 compound systems of waves into simple undulations, according 

 to Fourier's theorem. It is entirely due to this theorem that 

 the pitches of the upper partial tones of all serviceable musical 

 tones must stand to the pitch of their fundamental tones in the 

 ratios of the whole numbers to 1, and that consequently the 

 ratios of the pitches of concordant intervals must correspond 

 with the smallest possible whole numbers. How essential is. 



