ON BEATS OF IMPERFECT HARMONIES, 403 



sound recurring periodically in these successive 

 cycles is distinctly heard, even by an unpractised 

 ear, unless the duration of the cycle be too long or 

 too short to suit its sensibility. It is this variation 

 which is called the " beat " on the imperfect 

 harmony. 



The period of the beat that is to say, the 

 duration of the cycle described above is most 

 easily found by taking the reciprocal of its fre- 

 quency, calculated by the following rule: The 

 frequency of the beat is equal to the error of 

 frequency of one note multiplied by the harmonic 

 number of the other. When in a harmony of three 

 or four notes all are perfect except one, the beats 

 due to the imperfection of the false one are to be 

 reckoned just as if the harmony were binary, 

 according to the following rule : 



For the two or more notes which are in per- 

 fect harmony imagine one whose period is the 

 period of their harmony. Take this as if it 

 were one tone of an approximate binary harmony, 

 the false note of the given harmony being the 



other. Example : Let the frequencies of the three 



D D 2 



