PHYSIOLOGICAL ACOUSTICS 291 



modulus of decay of a free vibration. On the above estimate 

 of Helmholtz we shall have 



whence = '018. The intensity is therefore one-half the 

 maximum for 



2 = 1+ -018. 



It will be observed that on the above view we ought in 

 strictness to speak of "simplest" rather than of "simple" 

 sensations of sound, absolutely simple sensations, in the strict 

 physiological meaning, being impossible to excite. 



When two simple-harmonic vibrations, sufficiently far apart 

 in the scale, are in operation, the two groups of resonators 

 which are affected will be practically independent, and the 

 two sensations (of pure tones) will coexist. But when the 

 interval between the frequencies is sufficiently small, the two 

 groups will overlap, and the energy of vibration of those 

 resonators which are common to them will fluctuate in the 

 manner explained in 10. The excitation of the corresponding 

 nerve-endings will therefore be intermittent, with a frequency 

 equal to the difference of those of the two originating vibrations. 

 This is, on the theory, the explanation of beats. As the interval 

 is increased, the beats become more rapid. The " roughness " 

 which is ultimately perceived, in spite of the diminishing 

 amplitude of the fluctuations, has a more remote physiological 

 explanation. According to Helmholtz, there is here an analogy 

 with the painful effect produced by a flickering light, and in 

 other cases where a nerve is stimulated repeatedly at intervals 

 of time which are neither too great nor too small. When the 

 intervals are sufficiently long, the nerve has time to recover 

 its initial sensibility, and so experiences the full effect of each 

 recurring stimulus. When on the other hand the intervals 

 are sufficiently short, the sensation tends to become continuous. 

 It is for this reason that beats exceeding, say, 132 per second 

 cease to produce the sensation of roughness, even although 

 the interval between the beating tones be such as would be 

 perceptibly discordant in a lower part of the scale. 



192 



