Mr. S. Taylor on Variations of Fitch in Beats. 57 



this additional element of variation in beats is thus recognized, 

 dissonance is still, as in the earlier editions, represented as due 

 to alternations of intensity only. The following paper is an 

 attempt to show that the pitch-variations are important consti- 

 tuents of dissonance, and ought, in a complete view of the sub- 

 ject, to be taken into account. 



I am not aware whether M. Gueroult originated the observa- 

 tion the result of which he laid before Professor Helmholtz, but 

 may remark that in a paper read before the Cambridge Philo- 

 sophical Society in 1857, by the late Professor De Morgan*, the 

 existence of alternations of pitch in beats is distinctly asserted 

 as a known fact which he had himself experimentally verified. 



It will be convenient to begin by laying down theoretically 

 the conditions under which pitch-variation occurs, as the results 

 thus to be obtained facilitate the experimental examination of 

 the phenomenon. 



Let two simple tones of different intensities, but of nearly 

 the same pitch, coexist. An assigned molecule of air will be si- 

 multaneously solicited by two sets of sound-waves. Each of 

 these may be represented by a curve of sines, the times elapsed 

 from a given epoch being denoted by distances measured along 

 the axis of abscissse, and the contemporary displacements of the 

 molecule by corresponding ordinates. The problem to be solved 

 is the composition of the two series of vibrations. Two cases 

 must be distinguished, according as the higher or lower of the 

 two tones is the more powerful. We begin with the first of these. 



The waves acting on the molecule may have any degree of 

 difference of phase. The two extreme cases, where they are in 

 complete accordance and in complete opposition, are represented 

 in figs. 1 and 2 respectively. The strong line in each is half a 

 wave of the resultant curve. is the origin at the undisturbed 



Fig. 1. 



position of the molecule; X the axis of abscissae ; OA, OB 



* " On the Beats of Imperfect Consonances," Trans, of Camb. Phil. 

 Soc. vol. X. p. 137. 



