24 A. M. Mayer — Researches in Acoustics. 



sound. The results given only refer to intervals so formed. 

 To obtain them the forks were gently vibrated by strokes of 

 rubber hammers that varied in hardness with the pitch of the 

 forks. The lower the pitch of the fork the softer should be 

 the hammer. A hammer of hard rubber striking low pitched 

 forks will develop the upper partial tones of the forks and so 

 vibrate the experiments that a really consonant interval might 

 be judged as dissonant. 



The results of all the experiments may be summed up as 

 follows : From SOL 2 of 192 v. d. to MI, of 2560 v. d. the 

 smallest consonant intervals are closely given by the formula 



/ 42500 \ 



+ 23J-0001 



^ 



+ 23 



For sounds below SOL 2 the interval as computed by the 

 formula is too small to agree with the true interval. For 

 sounds above MI 6 (2560 v. d.) the intervals computed by the 

 formula, like those below SOL 2 , are too small. That the 

 experimental determination of the smallest consonant interval 

 throughout four octaves, upward from SOL„, or, throughout 

 the tones given by the violin, should agree so closely with the 

 formula indicates the existence of a law connecting the magni- 

 tude of the smallest consonant interval with its position in the 

 musical scale. 



Dr. Koenig has shown that a consonant interval does not 

 exist among simple sounds of pitch below SOL,, (96 v. d.), yet 

 I have found that the sound of UTj (64 v. d.), when inter- 

 rupted by a rotating perforated disk, blends perfectly, to my 

 ear, when these interruptions occur 23 1 times in a second. 

 It may appear strange that although 23*1 interruptions per 

 second of the sound UT 1 blend, yet a consonant interval does 

 not exist throughout the interval of UTj till the interval 

 TJTj : UT 2 is reached ; but the beats produced by the rotating 

 perforated disks are produced by the interruptions of one tone, 

 whereas, when two simple tones are conjoined two sets of beats 

 are produced, inferior and superior ; thus, when UTj forms 

 an interval with UTj+23 v. d., the inferior beats are 23 per 

 second and the superior beats are 41 per second, and the inter- 

 action of these inferior and superior beats produces secondary 

 beats which give to the interval a confused rumbling sound." 

 Of this interval, UTj : UT + 23*!, Dr. Koenig wrote to me as 

 follows : " Your 23 -1 interruptions of UTj correspond, in 

 number, to the inferior beats of the simple tones TJT 1 : UTj + 



*See Quelques Experiences d'Acoustique, par Rudolph Koenig. Paris, 1882, 

 pp. 89, 107, 113. 



