1S8 



Mb DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



the reading very easy, I would desire him to put himself, if possible, into the position of a 

 reader who had had no one but Dr Smith to help him. For of all the difficulties I have ever 

 encountered with any success, I have no hesitation in calling the theory of beats, as presented 

 by its author, the very greatest. I would compromise such another job, if such another there 

 be, by choosing rather to explain to a pupil of reasonable preparation, any fifty pages of the 

 Principia, and of the TMorie des Prohabilitis, and of the Disquisitiones Arithmeticce. 



I now proceed to the formulse which the subject requires. Let the higher note (perfect) 

 jnake m while the lower note makes n vibrations; m : n being > l and in its lowest terms. Let 

 k be what we may call the adjusting factor, that is, let nk and mk be the actual numbers of 

 vibrations in one second of the lower and higher notes. Let ma and na be the actual times 

 of vibration, in seconds, of the lower and higher notes. Then mnka = 1. Let na +9 he the 

 time of vibration of the upper note in the imperfect consonance which gives the beats. When 

 9 is positive, the consonance is tuned flat, the commencements of the more rapid vibrations 

 advance upon those of the less rapid, and the beats may be said to move forwards. The con- 

 trary when 9 is negative. It is the same thing to the ear whether the beats move forwards or 

 backwards. Let as be the ratio of the consonance of the perfect and imperfect upper note; 

 that is, let on = na : na + 6. Thus d?< 1 when the upper note is too flat. And let N and M 

 be the actual numbers of vibrations per second in the lower and higher notes of the imperfect 

 consonance. Hence 



m M ^^ ,, . -^ na 



— .v = — , Nma = 1, M {na + 0) = i , x = 



n N na + 9 



1 - a; 

 9 = na, kmna = I, kn = N. 



w 



Let /3 be the number of beats in one second. A beat, as shown, lasts through as many 



„ , , ., . , . . a . . . . {na + 0) a 



of the shorter vibrations as there are units m - : its time is then ; so that we 



9 9 



have 



)8 = 



e 



{na + 9)a a 



1 — <2? 1 ■" ^ 



= (l - a?) kmn = (l - !c) mN = nM = mN - nM. 



Dr Smith does not elicit * any of these formulae, the last of which is remarkably simple- 

 Thus if a fifth be tuned imperfectly to 200 and 301^ vibrations per second, we have 



200 X 3 - soil X 2 = - 3, 



or the consonance is tuned sharp to 3 beats per second. The number of beats per second 

 depends only on the number of vibrations by which the upper note is wrongly tuned, and 

 the smaller of the two lowest terms of the perfect consonance. Let M' be the proper num- 

 ber of vibrations for the upper note, so that M' : N = m : n, then /3 = {M' - M) n. Or 



• Since this paper was written the article ' Beats ' in the 

 Edinburgh Encycloptedia, attributed to Mr John Farey, has 

 been pointed out to me. This article contains Smith's formula, 



with two varieties arising out of difl'erent modes of expressing 

 the division of the octave, Emerson's method, and tlie formula 

 mN-nM. But no explanution is given. 



