524 Dr. li. Konig on the Simultaneous 



it must be held about double as far from the ear in order 

 to produce the same result ; and it follows therefore that the 

 note c f , with the same distance of vibration, is about four times 

 as powerful as the note c. If we try, then, to make both forks vi- 

 brate in such an amplitude that at the same distance from the 

 ear about the same result is produced, it will be found that the 

 amplitude of the c fork must be about four times as great as 

 that of the c' fork. According to this the amplitude of two 

 equally powerful notes in the interval of the fifth, e.g., must be 

 \) and 4, and the sum of these amplitudes would then be 13 ; 

 but the resultant note, which is an octave deeper than the 

 fundamental note of the fifth interval, would require an 

 amplitude of vibration of 36, in order to acquire the same 

 intensity as the primary notes singly possess. 



If the interval of the primary notes is still smaller, the beat- 

 note falls still lower, and must therefore be weaker in propor- 

 tion to the intensity of the primary notes. It stands to reason 

 that I do not give the above-mentioned experiments, nor the 

 numbers in the example as quite exact ; but they are sufficiently 

 so to show what convinced me that deep notes must have a far 

 greater amplitude of vibration than high ones, in order to 

 equal the latter in intensity. I hope to be able to return 

 before long to a closer examination of the intensity of different 

 high notes. 



The most important results of the above-mentioned experi- 

 ments are shortly summed up as follows : — 



(1) The number of beats of two notes n, v! is always equal to 



V 

 the positive and negative remainder of the division — ; that is, 



equal to the numbers m, m! ', which are produced by stating 

 n' = h n+ m= (h + 1) n—m', where n, n' is the number of 

 the double vibrations, and h the quotient of the division which 

 gives the remainder m. It is as if the beats proceeded from 

 the two overtones h and h 4- 1 of the lower note n, between 

 which the higher note n f lies. The cause of the beat-notes 

 is simply the periodical coincidence of the common maxima of 

 the two sound-waves. 



(2) The beats of the pure harmonic intervals can be heard 

 in the relations 1 : 8 and even 1 : 10, and may, as well as the 

 beats of the unison, be regarded as resulting directly from 

 the composition of the vibrations of the primary notes, without 

 the help of resultant intermediate notes, whose existence cannot 

 be proved. 



(3) Both the beats m and the beats m' } not only of the 



