46 Temperaments of different musical Systems. 



substituting the same in the wolf we get 322 -f- 3m ; 

 as in corollary 6; likewise in theorem 3, we have 



—2 -f — m, or — c, the sharp temperament of the 



3 3 3 



major third also, as is consistent with cor. 8. The 



II Id and Vih wolves are here each 28£2 + 2*m. (See 



Harmonics, p. 42.) 



This system approaches very near to a division of 



the octave into 19 equal parts, where — 3*(>Q472 is 



the flat temperament of the fifth, (Dr. Smith, Har- 



1 3 47 



monies, p. 158, makes it — -— c --c = — -~j-c 



or — 3-6955Y); — - c above, is — 3-6602S82. See 



o 



M. Sauveur's table. 



The cases of equality of temperaments between the 

 III and VI, and between the V and VI having occur- 

 red, in schol. 1 and 3, I proceed to 



Scholium 4. If a douzeave be required, in which the 

 major fifth shall be as much tempered flat' as the 

 major six th is sharp ; from theorems 1 and 6 we have 



— r lls—3r ; r 11 



= or Hi — 2r: whence — = — ; also 



s s s 2 ' 



— t u—3t , t I a 



= , or u = 2t: whence — = -— : and - 2 



u u u 2 ' 2 



4- — jBj.or— c, is here the temperament of the 

 2 2 



fifth : and by substituting this value in theorem 6, we 



have 2 -J- — m, or -— c, for the temperament of 



222 



the major sixth also. The V and VI wolves are here 



each 48^2 + 4^m. 



Scholium 5. If a douzeave be required, in which the major 



fifth shall be as much tempered flat as the major third 



is tempered flat; from theorems 1 and 3, we have 



— r 4r— lis . r 11 



■ — = , or 5?== 1 15 ; whence — = — : also, 



s s , s 5 ' 



-1 At-u t 1 4 n 



= or bt — iu whence — = — : and - 2 



u u u 5 ' 5 



4- — m,or — c, is here the temperament of the fifth: 



and by substituting this value in theorem 3, we have 



-—2 -f — m, or — c, for the flat temperament of the 

 5 5 5 



major 



