of Mistimed Consonances. 435 



misses the whole point of my paper. He objects — "the 

 beats are objective ; they can be seen in a manometric flame 

 if the primary tones are sufficiently loud " &c. That is to say, 

 my exposition of the existence in air of the Smith's beats, or 

 variations of resultant displacements, and demonstration by 

 difference of properties that they are not the same as the beats 

 we hear, is all thrown away. 



To take a simple case : suppose the two notes are represented 

 by a (sin A + sin B). Then the Smith's beats, which you can 

 see in the manometric flame, are given by 



a . A + B . A-B 

 2a sin — -— sin — -— ; 



which we may regard as a harmonic curve sin — - — , with a 



A-B 



2 



varying coefficient 2a sin — - — . That is to say, as I have 



pointed out in my paper, the pitch of the note which varies to 

 make these beats is about halfway between the pitches of the 

 primaries; whereas, in the beats we actually hear, the pitch 

 of the note which varies is very nearly that of the lower pri- 

 mary note itself. Further, the very existence of the resultant 

 beats depends on the maintenance of the unvaried primaries ; 

 whereas the beats we hear consist of variations of one of the 

 primaries. This and the argument from resonators are entirely 

 distinct ; they converge on the same result. 



The rest of the note I need not answer ; it appears to de- 

 pend on the use of the word " subjective," about which 1 do 

 not care. I am only solicitous about certain questions of fact, 

 which can be quite well stated without the use of any parti- 

 cular word ; though I maintain that, in the present case, my 

 use of the word subjective is amply justified by authority. 



The other point with which I have to trouble you is again on 

 p. 352, just below the first. It is given as a statement in my 

 paper — " (2) that the beats consist, as Konig discovered in 

 1875, of variations in the intensity of the lower of the two in- 

 terfering tones." 



This is really too bad. The whole of my work was origi- 

 ginated by the fact that Konig never analyzed his beats, and 

 never enunciated any opinion whatever as to the notes of 

 whose variation the beats consist — except in the one case of 

 the octave, which 1 have observed upon in my paper, where he 

 says that fundamental and octave are heard alternately. The 

 law above mentioned was unquestionably first obtained by me. 

 I knew Konig's papers intimately at the time, and there was 

 no vestige of guidance in them on the subject; so I object to 



