60 Mr. S. Taylor on Variations of Pitch in Beats. 



it being understood that the variations_, both of pitch and inten- 

 sity, are to be perfectly continuous between their limits,, and, 

 further, that the pitch-limits do not form an exact semitone. 



Our figures afford the means of obtaining an approximate 

 algebraical expression for the number of vibrations executed per 

 second by the varying tone when its intensity is either a maxi- 

 mum or minimum, assuming its pitch at those moments to be 

 instantaneously stationary. For this purpose w^e will assume 

 that A and B are nearly equal, i. e. the primaries only 

 slightly different in pitch, and that, in determining the point C, 

 the portions of the constituent curves about A and B may be 

 regarded as approximately straight lines coincident with the tan- 

 gents to the curves at those points. If, now, w^e take for the two 

 displacements Aj sin ^irn^t and Ag sin 27rn^t, where Wj and n^ are 

 the number of vibrations per second made by the primaries, and 

 Aj, Ag the corresponding amplitudes, t being, as usual, the 

 time elapsed, w^e may put, for the position of maximum intensity 

 (figs. 1 and 3.); 



0A= J-, tan CAE = 27r«,A, 



2nc 



i^^i > 



OB = ~, tan CBD = 27rn^A^. 



"2 

 From the figures 



AC tan CAE=BC tan CBD; 

 i. e. : — 



Fig. 1. 



(OC - OA) tan CAE = (OB - OC) tan CBD ; 



rig. 3. 



(OA- OC) tan CAE= (OC-OB) tan CBD ; 

 .'. in both cases 



OC (tan CAE + tan CBD) =0A tan CAE + OB tan CBD, 

 i. e. 



OC(27r/iiAi + 27r??2A2) = ~ . 27r;i,A, + ^ . 27rWjA3 : 



20C= t''^\ ' 

 20C is the wave-length of the varying tone ; its number of 

 vibrations per second is therefore equal to 1^7^-,, or 



A\J\J 



Aj+Ag 



