Sounding of two Notes. 431 



2n 

 note if m = — , and therefore in the intervals 3 : 3, 3 : 8. If, 

 o 



however, m = — + 1, then n— m =-— 1, and we again ob- 

 o o 



tain ( -^ + 1 ) — ( -^ — 2 J , z. e. three beats. 



The beat-notes m and m' together form the twelfth if 

 n . 3 



m = 2, in the intervals 2 : 5, 4 : 9; and if m=- in the intervals 



4 : 7, 4 : 11. If ra = - + 1, then m! =>— — 1, and we obtain 

 4 4 



I — + 3 ) — ( - l\,i.e. four beats. 



In general, then, when the higher note deviates by a double 

 vibration from the perfect interval, there ensue two beats in 

 the intervals 2:3, 2 : 5, 2 : 7, three beats in the intervals 3 : 4, 

 3 : 7, and 3:5, 3:8, and finally four beats in the intervals 

 4:5, 4:9, and 4 : 7, 4 : 11. 



By the use of the loud notes at my disposal I was able to 

 make the following observations on these secondary beats 

 arising from beat-notes. 



Near the fifth double E and double B, where the primary 

 notes make a distinct rattle, only one or two secondary beats 

 are audible ; at the fifth double Gr : D (96 : 144 v. s.), where 

 also the primary beats make a distinct rattle, but where, in 

 consequence of the greater intensity of the primary notes, they 

 are much louder, the secondary beats to the number of 8, and 

 above the fifth to the number of 10, can be followed ; in fact, 

 above the fifth they are more distinct, as is also the case in the 

 higher positions, — which may be explained by the fact that in 

 this neighbourhood the intensity of the lower and upper beats 

 must be more nearly equal, because the upper beats, which 

 are weaker even when their number is the same, are not here 

 so numerous as the lower beats m, whilst below the fifth the 

 contrary is the case. With equal intensity of the fundamental 

 note, the secondary beats are most distinct when the higher note 

 is somewhat weaker, while the rattle of the primary beat is 

 loudest when the higher notes are the stronger. 



In the interval with the fundamental note C, I was only 

 able to observe the secondary beats in the interrupted unison 

 m and m', but there as far as the third period. They may be 

 perceived at C : Gr (2 : 3) to the number of 6 or 8, and at 

 C : e (2:5) to the number of 5 or 6. At 2 : 7 two or three 

 may be heard. 



In the intervals double E : double B, double Gr : D, and C : Gr, 



