276 Dr. A. M. Mayer's Researclies in Acoustics. 



this increase in the interval between the tones the dissonance 

 becomes harsher, reaching a maximum dissonance when the 

 number of beats are about ^ of the number required to blend? 

 then becoming less dissonant as the interval increases till, at 

 last, the two tones blend into a consonance. These are the 

 phenomena observed from SOLx [96 v.d.] to the highest 

 tones used in music. 



Having the law which gives the number of beats (produced 

 by the interrupted sounds of tones of various pitch), which 

 blend, one might naturally infer that the smallest consonant 

 interval could be computed by that law. Given the pitch of 

 a tone, we compute by the law the number of interruptional 

 beats of this tone which blend, and adding this number to the 

 frequency of the given tone we should apparently have the 

 pitch of the upper tone which makes with the lower the 

 smallest consonant interval *. This, however, is not so. 

 Take, for example, UT 3 [256 v.d.]. The number of inter- 

 ruptional beats of this sound which blend is 62, and 256 + 62 

 = 318, which, according to the law, should make a consonant 

 interval with UT 3 . But experiment shows that a tone of 

 256-1-58 = 314 v.d. forms the nearest consonant interval 

 with UT 3 . 



To render less tedious the comprehension of the results of 

 many experiments on the smallest consonant intervals among 

 simple tones, I shall at once give a table (Table II.) of the 

 results of these experiments, and then give the account of the 

 experiments that furnished the data of the table. 



In column A are given the lowest tones of the consonant 

 intervals which were experimented on. In column B are the 

 number of vibrations to be added to the tones of column A to 

 form the higher note of the smallest consonant interval, as 

 deduced from the experiments on the duration of the residual 

 sensations of interrupted sounds. In column C the numbers 

 of vibrations by which the tones in column A have really to 

 be increased to form the higher notes of the smallest consonant 

 intervals. In column D are the numbers of vibrations to be 

 added to those of column A to form the smallest consonant 

 intervals as computed by the formula 



N:N + 



f 42500 ao \ A ' 

 (nT23 +23 ) ,0 ° 01 



* Rigorously, we should take in the computation the number of beats 

 which blend corresponding to a sound of a pitch which is the mean of the 

 pitch of the lower and upper sounds of the interval. 



