A. M. Mayer — Researches in Acoustics. 17 



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 interruptional 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 experi- 

 ment shows that a tone of 256 + 58 = 314 v. d. forms the small- 

 est 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 re- 

 sults 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 

 sensation 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 + 



/42500 \ 



(^ h23)-0001 



\N + 23 / 



in which N = the number of vibrations of the lower tone of 

 of the interval and N+ /Ant , nn — \ = the number of 



/42500 \ 



(^ +23J-0001 



\N+23 / 



vibrations of the higher tone of the interval. In column E 

 are given the differences between the computed values (D) 

 and the observed values (C). The formula gives quite closely 

 the true values from SOL 2 (192 v. d.) to MI 6 "(2560 v. d.). In 

 column F are given the smallest consonant intervals, as deter- 

 mined experimentally, from SOI^ (96 v. d.) to the tone of 

 2806 v. d., expressed in semitones of the equal-tempered scale. 

 In figure 11 these intervals, computed from the numbers in 

 column C, are expressed graphically by the curve 1ST. C. I. 

 The units of ordinates (on the left of figure) are semitones 

 and the units of abscissas are 100 vibrations. This curve 



* Rigorously, we should take iu 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. 



Am. Jour. Sci. — Third Series, Vol. XLVU, No. 277. — Jan., 1894. 

 2 



