CHAP, iv.] HEARING. 231 



corresponding to the interval between two beats, gives rise when 

 the groups follow each other rapidly, by a similar summation, to a 

 continuous sensation. And, though the matter is one which has 

 been much disputed, the evidence seems to shew that the con- 

 tinuous sensation thus produced is a musical sound, a tone, which 

 has been called a " beat-tone," whose pitch is determined by the 

 number of beats repeated in a second. 



The rapidity however with which beats must be repeated in 

 order to give rise to a continuous sensation, is different from that 

 with which single vibrations must be repeated in order to give rise 

 to a musical sound. Beats repeated 30 or 40 times a second are 

 readily distinguished as such ; it is not until they reach a rapidity 

 of repetition of about 132 a second that they cease to be distinctly 

 recognized. Before they disappear or as they disappear, at the 

 time when they can no longer be recognized as separate beats, 

 but have not as yet become fused into a completely continuous 

 sensation, they give to the sound which they accompany a peculiar 

 quality, a particular roughness and harshness. This quality if ex- 

 cessive is disagreeable to the ear ; we speak of it as dissonance. 



From what has been said it is obvious that when a piece 

 of music is played on an instrument and still more when it is 

 played, as in a concert, on several instruments of different kinds, 

 the disturbance in the air, and the consequent vibrations of the 

 tympanic membrane and of the perilymph, are in the highest 

 degree complex. If the disturbance has certain characters, the 

 sound "gives us pleasure, if other characters, we regard the sound 

 as disagreeable ; and it is found that the disagreeable features of 

 music are associated with the presence of beats, and still more 

 with the presence of that ill-defined roughness which, as we said 

 just now, is the characteristic of beats when, through rapidity of 

 repetition, they are about to disappear. At the same time there 

 are reasons for thinking that it is the prominence rather than the 

 mere presence of this element which offends the ear, that the 

 element is a necessary ingredient of effective music, and that 

 even the very quality of a musical sound is dependent in part on 

 a certain minute admixture of vibrations disagreeing in period 

 with the fundamental tone and with the regular partial tones. 

 But this is a matter into which we cannot enter here ; we have 

 referred to it because it illustrates the extreme complexity of 

 the processes which underlie our sensations of sound. 



