ON BEATS OF IMPERFECT HARMONIES. 405 



quency of the beats is to take the difference of the 

 errors of the two notes, each multiplied into the 

 harmonic number of the other. Thus, let n and ri 

 be the harmonic numbers of the perfect harmony 

 to which the given notes approximate, and let e 

 and e' be the excesses of the vibrational frequencies 

 of the two actual notes above two in perfect 

 harmony nearly agreeing with them. The fre- 

 quency of the beat of the actual notes is n'e - ne . 



For example, take the following table of num- 

 bers of vibrations in a perfect diatonic scale, with 

 256 vibrations per second for C, and in the corre- 

 sponding scale of equal temperament (founded on 

 twelve equal semitones, in each of which the 

 interval ratio is 2^) : 



