ON BE A TS OF IMPERFECT HARMONIES. 41 5 



let us say for example, four uniform beats per 

 second, through the case of four beats per second 

 with every alternate beat somewhat louder, to the 

 case of only every second beat perceptible, or, in 

 all, two beats per second ; but it can be done, and 

 the result is an interesting and instructive illustra- 

 tion of the reduction from the quick beat of 

 the binary harmony to half speed, or to one-third 

 speed, or to one-fifth speed, as the case may be, by 

 the introduction of a third note. In the several 

 cases I have found that I can, by making the added 

 note faint enough, produce a succession of beats of 

 which every second, or every third, or every fifth, 

 as the case may be, is louder than the others, and 

 that, as the intensity of the added note is gradually 

 increased, the fainter beats become imperceptible, 

 and a regular unbroken slow beat is heard distinctly 

 alone, always in the theoretical time of the whole 

 imperfection of the harmony. I have verified this 

 distinctly in the cases of I, 2, 3 ; 2, 3, 4 ; 3, 4, 5 ; 

 4, 5, 6 (as stated above) ; 5, 6, 7 ; and 6, 7, 8. I 

 have not succeeded in hearing the beats on the 

 approximations to the harmonies 8 : 9 and 9 : 10. 



