230 BEATS. [BOOK in. 



pitch, the vibratory impulses of the two sounds do not exactly 

 correspond in time. Since the vibration period, the time during 

 which a particle is making an excursion, moving a certain distance 

 in one direction and then returning, is shorter in one sound than 

 in the other, it is obvious that the vibrations belonging to one 

 sound will so to speak get ahead of those belonging to the other ; 

 hence a time will come when, while the impulse of one sound is 

 tending to drive a particle in one direction, say forwards, the 

 impulse of the other sound is tending to drive the same particle 

 in the other direction, backwards. The result is that the particle 

 will not move, or will not move so much as if it were subject 

 to one impulse only, still less to both impulses acting in the same 

 direction ; the vibrations of the particle will be stopped or lessened, 

 and the sensation of sound to which its vibrations are giving rise 

 will be wanting or diminished ; the one sound has more or less 

 completely neutralized or " interfered " with the other, the crest 

 of the wave of one sound has more or less coincided with the trough 

 of the wave of the other sound. Conversely at another time, the 

 two impulses will be acting in the same direction on the same 

 particle, the movements of the particle will be intensified, and the 

 sound will be augmented. And the one condition will pass 

 gradually into the other. The repetitions of increased intensity 

 thus brought about are spoken of as beats. 



The length of the interval at which the beats recur will depend 

 on the difference of period of the two sounds in relation to the 

 actual period or pitch of each. It may be stated generally that 

 the number of beats in a second is equal to the difference between 

 the number of vibrations per second of the two sounds ; thus two 

 very low pitched tuning-forks, vibrating respectively at 64 and 

 72 a second, will give 8 beats a second, and two very high pitched 

 tuning-forks, vibrating respectively at 4224 and 4752 a second wil] 

 give 528 beats a second ; but in this respect there are complications 

 which we cannot consider here. 



Beats are produced when the periods of the coincident sounds 

 are not exact multiples of each other. When the periods are 

 exact multiples no beats occur; two tuning-forks, for instance, the 

 period of one of which is exactly double that of the other, give 

 rise to no beats when sounding together; and so in other instances. 



By beats then a continuous musical sound may be broken up 

 into a series of discontinuous sounds. When the beats are repeated 

 a few times only in a second the discontinuous sounds give rise to 

 discontinuous sensations ; we hear the separate beats. But if the 

 beats are repeated sufficiently rapidly the successive sensations 

 are fused in one, we cease to hear the beats as such, though we 

 have other evidence that the beats continue to be produced. Just 

 as a series of simple vibrations when repeated sufficiently rapidly, 

 say 40 times a second, gives rise, by summation, to a single musical 

 sound, to a tone, so a series of groups of vibrations, each group 



