Temperaments of different musical Systems, 47 



major third also: the temperaments of the major 



22 2 2 



sixths being r 2 -f- — m, or — c. 



5 o o 



This is the system of Mr. John ITolden, since me- 

 tamorphosed into an irregular douzeave in the article 

 Monochord, Enc. Brit. 3d edit. vol. xii. p. 240, and 

 into a still different one, by the Rev. Mr. Hawkes. (See 

 your xxvith volume, p. 171, and xxxih volume, p. 5.) 



It also approaches near to M. Sauveur's system of 

 43 equal parts in the octave, (see his general table 

 above referred to) wherein —^ 11 772 is the fiat tem- 

 perament of the fifth ; --c above, being —2*2015732. 



One other case of this kind, viz. where the major 

 thirds are tempered flat as much as the major sixths 

 are tempered sharp, will be found to arise from dif- 

 ferent considerations in scholium 11. 



Scholium 6. If a douzeave be required, in which the tem- 

 perament and wolf of the fifth shall he equal, we have 

 from theorem 1, — r = 1 \r— 125, or 12r = 125, whence 



r 1 . ; * 



— ±= — : also —t = 11/ — iu or 12/ = ?/, whence — 

 si 7 u 



= — -, and 2 + -~ m, or — c, is the flat tempera- 

 \ & J i> li 



ment of the fifth, in this case: and which substituted 



in theorem 3, either for wolf or temperament, gives 



2 7 



72 4- -^-m, or ■ -c, the sharp temperament of the 



4. 3 



major third; also, in theorem 6, gives 82 -\- ~~tn 9 



8 

 or - c, the sharp temperament of the major sixth. 



This is the Isotonic or Ecjual Temperament Svstem of 

 Mersennus, &c. called bv Mr. Marsh and others,! hough 

 improperly, the Equal Harmony System (see scholium 

 10). See vol. xxix. p. 347 : see also Dr. Smith's Har- 

 monics^. lSSand 1(37. In the latter page, however, ihe 

 temperaments of theVth,VIth and i I Id are mistakenly 



said to be •— , — and 1Q , instead of --, -- and — 



of a comma, lis they are above, very nearly. 



Scholium 7. If a douzeave be required, in which the several 

 wolves shall differ from their respective temtieromnifs, 

 ly the least known Interval or most Minute (m): ue 



have 



