SIMULTANEOUS ACTION OF TWO TONES. 909 



the summits and valleys come together. Consequently, if two summits 

 fall together there must be an increase in the strength of the tone, but 

 if the summit of one wave coincides with the valley of another, there 

 must be a diminution in the strength of the tones. In this way there is 

 obtained an impression of variation in tone-intensity that is designated 

 beat or tremor (battements). 



The number of beats is, naturally, equal to the difference between the number 

 of vibrations in the two tones. The beats are most clearly perceived when two 

 deep tones of the same pitch, for example, of organ-pipes, are slightly out of tune. 

 If of two organ-pipes, each of w r hich produces C with 33 vibrations in the second, 

 one is made to yield 34 vibrations, one distinct beat will be heard every second. 

 It is evident, further, that the beats are fewer the less the difference between 

 the two vibration-numbers, and that they are more frequent the greater this 

 difference. Further, with equal relative difference in pitch of two tones, the 

 beats are fewer the deeper the tones, and they are the more frequent the higher 

 the tones. If, for example, the tone c with 66 vibrations is sounded with a second 

 tone with 68 vibrations in the second, two beats must occur in every second 

 (while in the preceding example, with equal relative differences in pitch, only 

 one beat is heard). 



The beats produce widely different impressions upon the ear, accord- 

 ing to the rapidity with which they follow one another. 



When they occur at long intervals, they may be perceived as com- 

 pletely isolated reinforcements of the tone, with subsequent enfeeble- 

 ments; they thus produce the sensation of completely isolated beats. 



If the beats follow one another more rapidly, the inequality pro- 

 duced causes a continuous, disagreeable, whirring impression that is 

 designated a discordant sensation. The highest degree of disagreeable, 

 painful discord is felt when there are 33 beats in the second. 



The intense unpleasantness of this sensation may be well likened to the dis- 

 agreeable impression produced by a nickering light before the eye. It is evident 

 that in order to produce this intense discord, two low tones must have a much 

 greater difference of pitch than two high tones. 



If, by an increase in the difference in the number of vibrations of the 

 tones, the beats follow oftener than 33 in the second, the sensation of 

 harsh discord gradually disappears, as the beats become more frequent. 

 Hence the sensation progresses from moderately inharmonious tone- 

 ratios (which in music demand a resolution in the succeeding chords) 

 to more and more consonant, and finally to completely harmonious 

 ratios. These tone-ratios are successively the second, seventh, minor 

 third, minor sixth, major third, major sixth, fourth, and fifth. 



As 33 beats in the second produce the greatest discord, it is evident that for 

 the production of discord in tones of low pitch, the tones of the scale must lie 

 further apart than when they are of high pitch. In deep tones the major third 

 may easily be discordant; in high tones, on the contrary, even those lying close 

 together sound much less discordant, because the number of beats quickly exceeds 

 33 in the second, on account of the high number of vibrations. In general, there- 

 fore, musical passages that possess but little harmony are much less inharmonious 

 in high notes than in low ones. 



The conditions are exactly the same for two musical tones that are 

 heard at the same time by the ear as for two simple tones. Under such 

 circumstances, however, the overtones come into consideration, as well 

 as the fundamental tones that determine the pitch. The degree of dis- 

 cord of two musical tones is, therefore, all the more prominent the more 

 the two fundamental tones and the overtones (and finally the differen- 



