Theory of the Beats of Mistuned Consonances. 279 



by difference- and summation-notes, since the frequencies do 

 not correspond*. 



The audibility of beats is regarded as depending solely on 

 their number, and on the intensity of the primaries, and as 

 being independent of the magnitude of the interval. (This is 

 certainly not true ; Helmholtz, Tonemp. 4th ed. p. 286.) 



In several other respects beats are identified with primary 

 impulses of the same frequency. 



According to my results, we should correct these statements 

 by saying that beat-notes are developed from beats by the 

 transformation of a small definite portion of the energy of the 

 beats into the beat-notes, in the transmitting mechanism of 

 the ear. 



We now come to a paper of considerable interest, " On 

 Beats of Imperfect Harmonies/' by SirW. Thomson (Proc. 

 Koy. Soc. Edinb. 1877-78, p. 602.) The facts stated in this 

 paper are in accordance with the details of Konig's experi- 

 ments. The phase-differences corresponding with the different 

 portions of the beat are exhibited by figures of harmonic 

 curves. The correspondence of the number of beats with that 

 deducible from the forms of the curves is demonstrated expe- 

 rimentally ; and the existence of beats is demonstrated in the 

 case of the simpler binary harmonies, when mistuned. 



The following is an example of the nature of the description 

 of the sounds heard: — "a revolving character^ which I per- 

 ceive in the beat seems distinct enough to prove that the 

 ear does distinguish between these configurations." .... It 

 will be seen that the author has here to some extent proposed 

 to himself the question — What are the sounds of which the 

 beats consist ? But if my work is right, he has not attained 

 the complete answer to it. 



When I first took up the consideration of Konig's experi- 

 ments, it seemed to me that the most probable solution was 

 that the notes he used were not pure from overtones. In fact, 

 if we admit the existence of overtones in his notes, phenomena 

 would be produced which would correspond to those he de- 

 scribes. 1 now know that the beats thus accounted for would 

 not contain some of the elements which do appear. But, 

 since Konig gives no analysis of the constituent elements 

 of his beats, it was impossible to foresee this. I consequently 

 spent some time in examining the question of the production 



* This supposed difficulty arises out of the restriction to first difference^ 

 tones. In my recent paper the identity of the beat-notes with the various 

 difference-tones is sufficiently established. 



t Italics in original, foot of p. 010. 



