404 POPULAR LECTURES AND ADDRESSES. 



notes be 257, 320, and 384 (an approximation to the 

 harmony 256, 320, 384, or C, E, G.). The common 

 period of the two last-mentioned is T } f of a second, 

 and we have to calculate the beats on two notes 

 whose frequencies arc 64 and 257. The harmonic 

 numbers of the harmonics to which these notes 

 approximate are I and 4, and the error in fre- 

 quency of the higher note is i per second ; hence 

 the beats are at the rate of I per second. When 

 there is error in two or more notes of a mul- 

 tiple harmony, two or more sets of beats in periods 

 not commensurable with one another arc heard ; 

 but the general effect is apt to be too confused to 

 allow any one of the sets to be distinctly counted. 

 On a multiple harmony with only one note false 

 the beats arc in general exceedingly distinct ; 

 more so in general than in binary harmonies. 



Sometimes, as for distance in reckoning the 

 beats in the imperfect harmonics of a tempered 

 musical scale, it is convenient to regard the two 

 notes of an imperfect harmony as in error from 

 two notes of a perfect harmony differing but little 

 from them ; then the rule for calculating the frc- 



