CARTLS1AN •rEMPEItAMENTS OF MUSICAL SC A I.E. That the versatile genius of M. Descartes early led hint to the consideration of the musical scale, is well known ; but that his speculations on this subject premed not less abortive and inapplicable to the art, than his vortices and other parts of his philosophy were to the phenomena of Nature, has not, we believe, been anv where shewn.
According to the t. Animadversions" of Lord Bro \nicker ?? on the Alusical Compendium" of De scartcs, it appears, that two different systems of musical intervals, but constructed on similar principles, were recommend ed by that author to the adoption of musicians, viz. one wherein a ratio, whose common logarithm is .9754132, (a semitone, the smaller of Descartes, or interval — 49.97718T, +1+ 5m,) is 10 times added to itself, leaving a difference, or wolf, flat, or in defect, on the twelfth or resulting semitone, of about la of a major comma, or more exactly of-12.2186E ; the other, wherein the log..9744813 (ot the larg, r s, mitone of Des( aril s, 51.87864X+ f + 5m) is 11 s rt peat«I, It as ill c a semitonic NVOI :11)0111. of a major I Ultima, or mole xactly + 10.5985X, sharper than tl c other semito u , this system.
. These systems have been supposed by some rsons to approach very nearly to the equal tt nip( lament, or isotonic system, whit h is, however, far from b u g the case ; and they appear so very inapplicable to pr..t to e, that we shall dispense with calculating the brats of the 12 several fifths in each of these systems, accordii g to our usual practice, and rather devote the room t icy would occupy, in explaining some of the most material properties of all systems constructed on similar IA iuciplv s to the above, viz. \vherein eleven semitones are equal : In which case, the most favourable situation for the result ing semitone or wolf seems to be between and A , and yet here it will be found, that only the five keys (•, B13., A, B, and C;;;", can have their Vths, I I Irds, and 3. , unaffected by this semitonic wolf; that I), EV and L, arc the only remaining keys that can have their lllids and 3rds so unaffected; and that F, with a 3rd unaffect4d. 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, tit. t the above are the anomalous key s ; and that those wfich include the resulting semitone, may be made, by apro per apportionment of the wolf, more harmonious than the above keys that exclude it, since it will Le found on oial that G, and are the only three he s that call have the advantage of a sentitonic wolf so cont•i\ d, in their Vths, IIIrds, and 3rds, in each case ; and 1g.ts1 F is
the only remaining key where the Vth and HIM 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 suppositions, no key beating 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 e an be made, as to the value of tin• semitonic wolf as affecting the conceit ds. the fifth in particular,) the natural keys C and A with others in frequent use, must be sacrificed to on.c rs Ic frequent in their occurrence : disath.intages s Mich not attend the more rational system of eleven equal fifths. instead of semitone s, as shewn by Mr Farey, in his count of regular Douzeave Systyns, in the Phil. Mar. vol. xxxvi. p. 40. But we shall proceed to give a fes. theorems relating to Cartesian sy stems : Puttnit; v, for the semitonic wo/f, or difference between the re-suiting semitone and all the eleven others, and r for the tempe rament or imperfection of the fifth resulting t ter( from, though not included therein, and in terms of the small interval schisma or 1', (see Plate XXX. in Vol. II First, whence the se mitone n y readily be calculated if the fifth he given : at d .1i • s also, that when r vanishes, or the fifth is tt be del the resulting semitone is 1.71540i.Y. less t an tic °the; semitone. Second, F= wl.ence, I a\ big the semitonic wolf, or se iMtones tnemselve s. ',be cause V I II-1 2 semitone s =:.a. the tempt rament of the fifths may be hacl : and also it appears, th.t va nishes, 1.0u06552E results as tee isotonic fifth tc n p r. ment, (see Phil. Mag. vol. xxxviii. p. T it V—r being the sagm, of each of the 5 i*.f ••-, ss er V.0 does not enter, V+4.r-1.7154119X will .) • e _ 1 cf each of the 7 quint wolves or fifths. (or I), E, 1 , G, and Gil. in the above case,) wherein the semitonic wolf enters, as will be evident by equating these two classes of fifths, when —r=-1.0006552Z will re sult, as above. Fourth, We have 7.5770452-4r for the sharp temperament of the 8 major thirds, where w does not enter. Fifth, 5.861636X-I-Ir will be the value of the 4 major tierce wolves (or F, Fes, G, and wherein the semitonic wolf enters, which may be proved as above. Sixth, 7.577045T-44r is the flat temperaments of the 9 minor thirds, where w does not enter: And, Seventh, will be the value of the 3 minor tierce wolves, (or Fyf, G, G;;',.,) wherein the semitonic wolf enters; the equating of which last temperament and wolf, will prove r in them to be equal to the isotonic temperament in such case, as before.