Chap, xxxn.] DISSONANCE. 433 



ear, because of their increased number, and give a 

 harsh grating character to the sound, and with a still 

 greater number a cutting character. The rising and 

 falling of the sound, within very narrow limits and 

 under special circumstances, may produce a pleasing 

 or effective impression, but beyond these limits and 

 circumstances they are disagreeable to the ear, giving 

 rise, as they do, to the same sort of nervous impression 

 that a flickering light has upon the eye. The 

 sensation produced by the beats is that of dissonance. 

 When beats reach the number of thirty-three per 

 second, according to Helmholtz, the dissonance is 

 intolerable ; as the number increases the roughness 

 lessens, though it is still present when they occur at 

 the rate of one hundred per second, but when they 

 reach 132 per second the roughness disappears. 

 When two notes are sounded together as for the 

 production of beats, a new note may be heard, 

 different from either, whose pitch is due to a number 

 of vibrations equal to the difference between the two 

 primary notes. The rate of vibration of this resultant 

 tone, or difference tone as it has been called by Helm- 

 holtz, corresponds to the number of beats produced by 

 the two primaries, and was supposed by Young to be 

 a tone produced by the fusion of beats. Helmholtz, 

 however, explains them to be due not to beats, but to 

 secondary waves caused by the disturbance of the 

 beats. 



As showing how the physical theory of beats and 

 the physiological sensation of dissonance harmonise, a 

 comparison may be made. Given a note to start 

 with, the most perfect combinations are of the funda- 

 mental note with its octave, with its fifth, and with 

 its fourth ; the consonance with its third is less com- 

 plete. Now a note due to 264 vibrations per second 

 has as its octave one due to 528 per second ; the 

 difference, 264, is so great that beats cannot be 

 c c 7 



