CARTESIAN TEMPERAMENTS. 



559 



in defect, on the twelfth or resulting semitone, of 

 ' ' K of a major comma, or more- cx.utly of 



I'J.^l.Sfii: ; the other, wherein the log. .Ji?! 



(or the larger semitone of Descartes, r:51.S7M> l-~ 

 lf_/'.|-5;M) is 11 times repeated, leaving a 'semitonic 

 wolf about , g rt T of a major comma, or more exactly 

 -f 10.59851, sharper than the other semitones of this 

 system. 



These systems have been supposed by some per- 

 sons to approach very nearly to the equal tempi ra- 

 ment, or isotonic system, which is, however, far from 

 being the case ; and they appear so very inapplica- 

 ble to practice, that we shall dispense with calcula- 

 ting the beats of the 12 several fifths in each of these 

 systems, according to our usual practice, and rather 

 devote the room they would occupy, in explaining 

 some of the most material properties of all systems 

 constructed on similar principles to the above, viz. 

 wherein eleven semitones are equal : In which case, 

 thr most favourable situation for the resulting semi- 

 tone or wolf seems to be between G> and A ; and 

 yet here it will be found, that only the five keys, C, 

 B[j, A, B, and Cj#, can have their Vths, Illrd?, and 

 Mrds, unaffected by this semitonic wolf ; that D, 

 Et>, and E, are the only remaining keys that can 

 have their Illrds and 3rds so unaffected; and that 

 F, with a 3rd unaffected, is the only remaining key 

 that can have either of its concords uncontaminated 

 by the semitonic wolf of such a system. And it 

 must not be imagined, that the above are the anoma- 

 lous keys ; and that those which include the result- 

 ing semitone, may be made, by a proper apportion- 

 ment of the wolf, more harmonious than the above 

 keys that exclude it, since it will be found on trial 

 that G, F$, and G$, are the only three keys that 

 can have the advantage of a semitonic wolf so con- 

 trived, in their Vths, Illrds, and 3rds, in each case ; 

 and that F is the only remaining key where the Vth 

 and Illrd can be taken to include the semitonic 

 wolf, situated as above. And thus it appears, that 

 in these Cartesian systems, on one of the above sup- 

 positions, no key bearing one sharp or one flat in its 

 signature, and many others not much less frequent 

 in use, can be made tolerable ; while, on the only 

 other supposition that can be made, (as to the va- 

 lue of the semitonic wolf as affecting the concords, 

 the fifth in particular,) the natural keys (C and A) 

 with most others in frequent use, must be sacrificed 

 to others less frequent in their occurrence ; disad- 

 vantages which do not attend the more rational sys- 

 tem of eleven equal Jtjlhs, instead of semitones, as 

 hewn by Mr Farey, in his account of regular Dou 

 zeave Systems, in the Phil. Mag. vol. xxxvi. p. 40. 

 But we shall proceed to give a few theorems relating 



artesian system* : Putting n for the ttfnilonic Carie**B 

 ivolj\ or <!iif. fence between the resulting tcmilouc " 



ani 



though 

 interval 



ll the 



other*, and r for the temperament 

 ii of the fifth resulting therefrom, 

 ludi'd therein, and in t-nns of the unall 

 or XXX. in Vol. II.; 



First, TV y r 1.7 : ..hence the semitone may 



readily br calculated if the fifth be given : and it 

 appears also, that when r vanishes, or the fifth is to be 

 perfect, the rcbulting semitone is 1.715409 2 leas than 

 the other semitone. Second, r= 1 .00065522 4-^,1* ; 

 whence, having the semitonic wolf, or semitones 

 themselves, (because VIII 12 semitones =n>,) the 

 temperament of the fifths may be had : and also it 

 appears, that when rv vanishes, 1.00065522 results 

 as the isotonic fifth temperament, (see Phil. Mag. 

 vol. xxxviii. p. 436.) Third, V r being the value of 

 each of the 5 fifths, where n does not enter, V-J-^r 

 1.7154092 will be the value of each of the 7 

 quint wolves or fifths, (or D, E[?, E, F, F$, G, and 

 G$ in the above case,) wherein the semitonic woll 

 enters, as will be evident by equating these two 

 classes of fifths, when r 1.00065522 will re- 

 sult, as above. Fourth, We have 7.5770452 $r for 

 the sharp temperament of the 8 major thirds, where 

 n does not enter. Fifth, 5.861 6362 -f-4 r *'iU be 

 the value of the 4 major tierce wolves (or F, F$, G, 

 and G$,) wherein the semitonic wolf enters, which 

 may be proved as above. Sixth, 7.5770452 {- \r is 

 the flat temperaments of the 9 minor thirds where rv 

 does not enter: And, Seventh, 9.292454-2 r will be 

 the value of the 3 minor tierce wolves, (or F$, G, 

 G$c,) wherein the semitonic wolf enters; the equa- 

 ting of which last temperament and wolf, will prove 

 r in them to be equal to the isotonic temperament in 

 such case, as before. 



The above temperaments and wolves of the prin- 

 cipal concords in the four last theorems, are very 

 differently related to each other from what they are 

 in regular douzeaves, as will appear on comparing 

 them with the 13th, I6tb, 14th, and 17th of Mr 

 Farcy's Corollaries, in vol. xxxvi. p. 374, 375. of the 

 Magazine above quoted. 



On the whole, it appears that semitonic systems 

 of this kind, by having six more quint wolves than 

 are necessary, and some of them falling in the most 

 material keys, as G, F, D, &c. are unworthy of fur- 

 ther attention, except as an exercise for the musical 

 student, when enquiring into the very curious nature 

 and constitution of the Scale, which they can never 

 too much explore, and which the seven theorems 

 that we have calculated and given above, will be 

 found greatly to facilitate. See our article TEM- 



PERAMKNT. (?) 



