422 Mr. R. H. M. Bosanquet on the Beats 



to half the difference of the frequencies of the primaries. This 

 is what is actually heard in the case of two notes less than two 

 commas apart. 



8. When the interval is greater than two commas, this 

 ceases to represent the whole phenomenon perceived by the 

 ear as it exists. The separate notes step in beside the resultant 

 form represented by the above expression, with its beats and 

 note having the frequency of the arithmetic mean. As the 

 interval increases, the separate notes become more and more 

 prominent, and the beats diminish in loudness and distinctness, 

 till, by the time that a certain interval is reached, which is 

 about a minor third in the middle of the scale, the beats prac- 

 tically disappear and the two notes alone survive. 



9. It has been supposed by some that the beats disappear 

 only in consequence of their rapidity*; and it is clear that 

 under this supposition, as ordinarily made, lies the assumption 

 that the mass of tone continues to be received in the same 

 manner all the time — i. e. that the phenomena of the beats of 

 imperfect consonances and combination-tones are to be ex- 

 plained by reasoning analogous to that of the above formula, 

 which supposes the whole displacement reduced to its resultant 

 on one receptive mechanism. This, for instance, is assumed 

 whenever Smith's or Young's theories of beats are admitted 

 as sufficient explanations of the phenomena. 



10. In such cases, (a) it is forgotten that the fundamental 

 assumption carries another consequence with it than those it 

 was desired to explain; (b) the explanation itself also fails in 

 an important point. 



(a) The other consequence is, that if it were true that the 

 receptive mechanism of the ear received a resultant displace- 

 ment, so that the combination was as represented by the above 

 formula, then the primary notes would not be heard at all, 

 and the note that would be heard would have the arithmetic 

 mean of the frequencies of the primaries. 



E. g., in the case of a fifth (4 : 6) the note heard would be 

 the major third (5), which would beat very rapidly; just as, 

 when I myself hear the resultant of notes two commas apart, 

 it is one note midway between them beating rapidly. But, as 

 a matter of fact, the note 5 is not heard at all in the above 

 case. 



(b) Again, supposing that in some unexplained way the 



beats whose speed is —^ in the above notation gave rise to a 



note, as supposed by Konig. Then the speed of that note 



* This is absolutely disproved by the argument in Helmholtz's 

 Tonempf. p. 286, ed. 4, 



