420 Dr. R. Konig on the Simultaneous 



confused character, and changes into a still rapid but simple 

 roll, which becomes so much slower between the sixth and 

 the seventh that at 233 and 236 v. s. twelve and ten single 

 beats can already be counted, which at the seventh, B = 240 v. s., 

 decrease to eight, at 244 v. s. to six, and become gradually 

 fewer in number, till at the octave of c=256 v. s. they at last 

 cease altogether. 



As the number of vibrations of the primary notes can be 

 directly read off from the tuning-forks, it will be found that 

 the number of single distinguishable beats near the unison is 

 equal to the difference of the double vibrations of the two 

 primary notes, and that of the beats near the octave is equal 

 to the difference of the double vibrations of the higher of the 

 two primary and of the octave of the lower note. 



The above result can be shortly expressed in the following 



manner. Each interval n : n 1 (less than the octave) exhibits 



two sorts of beats, whose number is equal to the positive and 



n' 

 negative remainder of the division — , that is to say, equal to 



the two numbers m and ?vi / = n—?n, which we obtain by stating 

 n / — n-\-m = 2n—m / . I shall in future, for the sake of brevity, 

 call the beats m lower beats, and the beats m! upper beats. 

 If we increase the interval between two notes from the unison 

 to the octave, the number of lower beats increases from o to n, 

 and that of the upper beats diminishes from n to o. At the 



fifth the number of both kinds of beats is = -. If m is much 



less than -, only the lower beats are audible ; if m is much 



z 



greater than -, we hear only the upper beats; and if m is 



nearly equal to -, both kinds of beats, m and n^-m, may be 



distinguished at the same time. 



The lower beats are more powerful than the upper beats ; 

 and their audibleness extends consequently further beyond the 

 fifth than that of the upper beats below the same note. 



In the octave from C to c, which we have now been con- 

 sidering, it is very difficult to distinguish, through the loud 

 and confused rattle of the upper and lower beats above and 

 below the fifth, the rhythm which belongs to both these kinds 

 of beats, as the number both of lower and upper beats is 

 always so great that heard alone they would produce a very 

 rapid roll. I only succeeded, therefore, in bringing to special 

 and absolutely clear proof both kinds of beats during their 



