Mr DE morgan, ON THE BEATS OF IMPERFECT CONSONANCES. 



139 



thus : — In every consonance of which the lower number is n, every wrong vibration per second 

 in the upper note is n beats per second*. With this theorem as a key, a rationale can be 

 obtained without difficulty; but it does not connect the two beats, and would, I think, be 

 subject to the doubt I have cast on Emerson's method. 



The formula given by Dr Smith are obtained as follows. The comma, or difference of a 

 major and minor tone, being 81 : 80, let 9 correspond to the fraction q : p oi a. comma. Then 



Now (1 - aY = 1 - 



r8o\? 



2wa 



2 + (« - 1) a 



/■80\J 27 



whence a> = — \p= \ nearly. 



\81/ l6lp + q- •' 



Zq 



nearly; a being small: 



And /3 = (1 - x) mN = 

 = nM' 



\6\p + q 

 29 



mN 

 nM 



X l6lp - q 



which are Smith's formulae (2nd ed. p. 82). 



When the upper note is too sharp, q must be made negative, the negative sign of /3 

 being neglected. 



If /J. be the fraction of a mean semitone by which the upper note is flat, we have, for 

 the number of beats in a minute, 



60 (1 - 2~'*) mN, or ■ mmN, or I04u ( —]mN 



^ ' 30 \30 1000/ 



nearly, and more nearly. If the octave be composed of 30103 atoms, of which the upper 

 note is tuned flat by a atoms, the number of beats in a minute will be 



•001381551a (1 - "OOOOl 15129a) miV very nearly, 

 4x8x13 



or 



301000 



amN nearly. 



These formulae are not accurate enough to give the beats in a minute within three or four, 

 unless both terms be used: and, the vibrations being given, mN - nM is much more easy. 



• The passage over the greatest common measure being 

 fairly arrived at, as the time of a beat, the transition to the 

 formula mN—nM may be very briefly made. We know that, 

 m and n being prime to one another, there is, before we arrive 

 at mn, one way and one only in which ym — 971 = 1 ; and one 

 way and one only in which qn—pm = 1. The ratio N : M o( 

 the numbers of vibrations in the erroneous consonance, and also 

 of the lengths of the waves, is not n : m, but 



M 



mN - nM 

 M 



consequently the commencement of the shorter wave gains the 

 of a common measure in every vibration of 



fraction 



mN — nM 

 Af " 



the higher note, than is mN — nM common measures in one 

 second, or in M higher vibrations; and each gain of a com- 

 mon measure is a beat. This demonstration, a little more 

 developed, will be, 1 should think, the best that can be given. 



18—2 



