407 



is of course to be understood that the degree 

 >f falseness is the same in all the tempered har- 

 monies of the same name (or having the same 

 harmonic numbers) ; and that the different num- 

 bers shown for the frequencies of the beats are 

 (except for the case of the E with the untempered 

 G) in simple proportion to the vibrational fre- 

 quencies of one or other of the constituent notes. 

 The slightness of the imperfectness in the tem- 

 pered fifth (approximately 2:3) is indicated by 

 the slowness of the beats, not so much as one 

 per second on the C G. The imperfectness of the 

 fourth (approximately 3:4) is even less than that 

 of the tempered fifth, so that, notwithstanding the 

 greater harmonic numbers, the beats are scarcely 

 more rapid (ri/) for the C F than (*86) for the 

 C G. But when we go to major and minor thirds 

 of the tempered scale, we take leave of mathe- 

 matical harmony entirely. The beats on the 

 C E (ten per second) are too rapid to be counted, 

 and it is only in virtue of their not being per- 

 ceived, or not being disagreeably perceived, that 

 the combination is agreeable. The same may be 



