Temperaments of different musical Systems* 45 



temperament of the major sixths ; which last tem- 

 peraments prove the same as they must do, to fulfil 

 the conditions of cor. 8. 



Though the fifth wolf is here so large, the Illds and 

 Vlths wolves are each 2 only. 



This system approaches very near to that of Mer- 

 cator, wherein the octave is divided into 53 equal 

 parts, and where — -0350942 is the flat temperament 

 of the fifth. See M. Sauveur's general table of tem- 

 pered systems, Mem. de l'Acad. 1711, lomo. p. 416. 



Scholium 2. If a douzeave be required, in which the ma- 

 jor thirds should be perfect, we have in theorem 3 



'lis- 4r s". , r 11 w— At 



= o, or lls = 4r, whence — = — ; also 



s s 4 u 



= 0, or u — At and — = — j whence it appears, that 



the fifth is to be flattened —2 + -ra, or — c: and, 



4 4 4 



by substituting the above values in the wolf, the same 



appears to be 212 -f 2m, as in cor. 3; and by the same 



in theorem 6, we get —2 -f ~r ra ' or "V c * tne sna n? 



temperament of the major sixth ; the same with that 

 of the fifth ; see cor. 8. 



This is the Mean Tone system of Salinas. Zarlino, 

 Aretinus, &c. Dr. Smith's Harmonics, p. 36,41,&c. 

 wherein the adjaceat flat and sharp notes are distant 

 212 -f 2m, or an enharmonic Diesis, as appears by 



r t 



substituting; the above values of — and — in cor. Q. 

 P s u 



It is also nearly the same with a division of the 

 octave into 112 equal parts, (see M. Sauveur's table 

 above quoted), wherein —2*82902 is the flat tempera- 

 ment of the fifth; — c above, being — 2«7519662. 



Scholium 3. If a douzeave be required, in which the major 



sixths should be perfect, we have in theorem 6, 



r 11 . u—Zt 

 = 0, or 1 lj = 3r. whence — = — ; also = o, 



1 l i 11 „ 1 



or, u — Zt\ whence — = -.-, and —2 -f- -— m, or, 



- — c, is here the flat temperament of the fifth : and by 



sub- 



