202 On the Stanhope Temperament 



it as 2 VIII + III above C, we shall have 120 X -f x |- 

 = 600, as before. 



I have now to calculate* (by means of the last theorem 

 but one. No. 65, in the Sup. Encij, Brit, before quoted,) 

 what are the number of heats per second made by the tri- 

 equal quint G d ; and, seeing that the same is tempered Jlat 

 -I- of a comma (No. 5 in my Table II), we have q = 1, m 



= 3, N = 1 80, and p = 3 : and per theorem, ^^ ^ ^ ' ° 

 ' ' ^ ' ^ '161x3+1 



= — — = 2*23, the beats per second. Also in the tri-equal 

 484 



quint da we have N = 268*9 {q, m, and p, remaining the 



same as before), and ~ — = 3'33, the beats per 



" Ibi X 3 + I 



second. And in the tri-equal quint at, we have N= 401 •7> 



and — - = 4*98, beats per second. Now these 



161 X 3 + I 



three rates of beating, viz. 2*23, 3*33, and 4-93, per se- 

 cond, must, to support the position of our noble author, be 

 equal 1 ! Are wc to suppose that the monochord used, failed 

 in determining these sounds (calculated by geometric mean 

 proportionals) within these limits; or, that iht beats were no 

 better attended to than to conclude, that vibrations of more 

 than twice the length of each other were equal ? For the 

 theory of beats is too well established to be questioned ; and 

 the same is, I believe, correctly applied in the above calcula- 

 tions : indeed. No. 68, of the article Temperament, above 



* Let the conchord whose perfect ratio is expressed by — {n being the 



least term of the ratio in its lowest terms) be tempered by the fraction — of 



P 

 a comma {q being the least term of this fraction) ; also, let M and N be the 

 number of Tibrations in one second of time, made or excited by the acute and 

 grave notes of the above conchord respectively ; and let b be the number of beats 

 occasioned by this temperamint in one second. Then, if the temperament 



2qmH 2qnM 



be sharp, or the chord greater than perfect, b = or : but 



'^' ° r . I6lp — q W\.p + q 



if the temperament be flat, or the chord less than perfect, b = -—- or 



'^ ^ 161 p ■>rq 



2qnM 



161 p — q 



quoted 



