4ft Temperaments of different musical Systems. 



. c „ 12r— 12s 



have from corollary 9, =0, or 12r= 125 J 



, r 1 . 12/— m * O 



whence - = — ; also, = — I, whence — = — . 



s 1 * ■ u ' u 1 



By which we obtain in theorem 1, — 2 as the flat tem- 

 perament of the fifths, and — £ —in its flat wolf: also, 

 in theorem 3, we get 72 -f-m the sharp temperaments 

 of the major thirds, and 72 their wolves ; and by theo- 

 rem 6, we obtain 8S + m the sharp temperaments of 

 the major sixths and SS their wolves. 



This is my Equal Temperament Syste7n, whose tem- 

 pered fifth, and consequently all its other intervals, 

 can be tuned on an Organ by means of perfect intervals 

 only, viz. 5 4ths — 3 Vths — III = V— 2. (See vol. 

 xxviii. p. 65) ; such tuning to be upwards from C as 

 far as bA and downwards from c as far as bE, between 

 which notes, the wolf 2 -f-m, will result. The beats 

 calculated by Mr. Smyth, at page 452 of your last 

 volume, belong in fact to this system, and not to the 

 strict Isotonic above, but the difference in practice 

 would be imperceptible between these two systems. 



Scholium 8. If a douzeave be required, in which the ratio 

 of the temperaments of the major thirds shall be to 

 their wolves as 1| to 3^*, that is, or as 5 to 14, we 



have from theorem 3. As 5 : 14 : : : — — — ; 



s u ' 



whence 1545— 56r =40r— 5s, or 1595 = 96V, and — 



53 . u-4t 8t . 

 «* «« 5 a ' so 5 as 5 : 14 : : : — - . whence 14w-~ 



32 ' ' u u ' 



t 7 53 

 56/ =40/,or 14//=96/, and — = - --: and 2 — 



9 ' » 48 32 



7 

 -r— m is the flat temperament of the fifth, which sub- 



300/ 20 



stituted in the first theorem gives ~^r^ -f -7 -m y 



* Nfr. Marsh, assuming the true major third to be 48 degrees or parts, 

 states, the tempered III to be = 49^, and the wolf or " extended third," (as 

 he elsewhere calls it) to be 51$ parts: in the system which he most ap- 

 proves; I therefore take the excess of these above the !Ud,as giving the ratio 

 of his temperament and wolf, in order to obtain the values of his notes in my 

 theorems. It is not however clear, that such is exactly his meaning ; since, 

 150 being assumed as the measure of the octavr> the values of the major tkud 

 and of the diesis can be no of her than 48-28921, &c. and 5*132378, &c. o? 

 very nearly in the ratios of 6122, 1972 and 212 : and it is not possible for 

 48 and 6 truly to represent the major third and diesis in such octave» or to 

 any other. 



the 



