3*2 Temperaments of different musical Systems, 



(see Sauveur's table above referred to) than the last, 



2 # 



since — c = —3* 145 IS, and in his system — 3*048112: 



is the temperament of the fifth* 



Scholium 1 2. If a douzeatfe be required, wherein ike mean 



Tone thereof is to its major Limftta as 5 to 3, we have 



P '. . 104-29025 — 2r 



from corollaries 11 and 12, as 5 : 3 : : - : 



. s 



40'-14Q6.« + 5r 



- , whence 230*7 180s + 25r = 312- 8976\? — 



$ 



i r 82-1406 

 or, or 82*14965 = 31 r, and — = — -^~ -• also as 5 : 



O-0272K— 2* 4*0136m + 5* 



3 : : - 2 — - : — , whence 20*0680w + 



u u 3 



Q5t = 27*08 16a — 6/, or 7*01 36m = — 3H and — = 



u 



—7*0136 , 82-1496 7*0136 



— ~ : and -~ -2 - , m is the flat 



31 ' 31 31 



temperament of the fifth. 



This is the system of M. nuygens. (See Dr. 



Smith's Harmonic:, pages 158, lOS, 121, 208, 



£124, Sec.) whose temperament of the fifth, as calcu- 



11 53 



lated by Dr. Smith, is - — c + - Q c or - -- c, «= 



2*65192; mine above being about 2*65182. 



The octave here is supposed to be divided into 31 

 equal parts. See M. Sauveur's table ; Mr. Ambrose 

 Warren, in 1725, pretended to the discovery of this 

 system. See Monthly Magazine, vol. xxix. page413. 



Scholium 13. If a douzeave br* required, wherein the mean 



Tone thereof is to its major Limma as 9 to 5, we have 



c .. . . 104-2QQ2$~2r 



from corollaries 11 anU 12, as 9 * 5 : : ■ : 



5 



46- H965 + 5r 



— - — - — --, whence 415*346l5-f 45r = 521 '49605 — 

 s 



, r 106-1496 

 lOr, or 55r= 106*14965, and — = 7T > a5so 



90272a -~1t 4-0136w-f5< 



as 9 t 5 : : — ; : , whence 



• u u 



36*1224tt-}-45£ = 45*13te— 10/,or9'0136// = 55f and 



/ 9*0136 . 106*1496^ 9*0136 



_«b 2- r j and t~-^ — ■—; — m is 



u 55 ' 55 55 



the flat temperament of the fiith, = —1*93 132 



Thi* 



