TEMPERAMENT. 



to the common temperate fifth, deficient by ^ of a comma, is 

 56.79763 to 32.38952. The approximating ratios to which 

 axe, 



1. Greater than the true 2 : i, 7 : 4, 19 : 1 1, 50 : 29, &c. 



2. Lefsthan the true i : i, 3 : 2, 5 : 3, 12 : 7, 31 : 18, 

 205 : 119. Where we have the temperaments of 12, 19, 

 31, and 50 parts, before examined. 



And here all ratios greater than the true ought to be re- 

 jefted, becaufe they give the fifth lefs than true, that is, 

 in this cafe, deficient by more than ^ of a comma. 



If we invefligate the approximating ratios to the ratio of 

 the femitones major and minor, or 5.19529 to 3.28612, we 

 ftiall have the ratios i : i, 2 : I, 2 : 2, 5 : 3, whicli re- 

 fpeftively give the temperaments of 12, 19, 31, and 50 

 parts, before defcribed. 



Again, invefligating the approximating ratiosof the fifth to 

 the third major, we fhall find 7 : 4, 9 : 5, 1 1 : 6, 29 : 16, 

 which will alfo give the temperaments 12, 19, 31, 50, as 

 before. 



Laftly, the approximated ratios of the oftave to the true 

 fifth are 1 2 : 7 and 5 3 : 3 1 greater than the true ; tlie others 

 being of no ufe, fince the fifth muil necefTarily be diminiflied. 

 Here we find the temperament of 53 parts. As to the 

 temperaments of 43 and 55, being deflitute of any mufical 

 foundation, it is no wonder they do not appear by this me- 

 thod of invelligation. 



M. Huygens, in his Cofmotheoros, fays that the tone 

 or pitch of the voice cannot be preferved, unlefs the confo- 

 nants be tempered, fo as to deviate a little from the higheil 

 perfe<ftion. For the proof of this aflertion, he brings a 

 melody confining of the foDowing founds, C, F, D, G, C ; 

 where, if the intervals were to be fung perfeft, by taking 

 the interval from C to F a true fourth afcending, from F 

 to D a third minor defcending, from D to G a true fourth 

 afcending, and laflly, from G to C a true fifth defcending, 

 we fhould fall a comma below the C from whence we began. 

 Therefore, if we were to repeat this feriea of notes nine 

 times, we fhould at laft fall near a tone major below our firft 

 found. 



M. Huygens's folution of this difficulty is, that we re- 

 member the note from whence we fet out, and return to it 

 by a fecret temperament, thereby finging the intervals a 

 Kttle imperfeft ; which, he fays, will be found neceffary in 

 slmoft all fongs or melodies. 



A like difficulty is mentioned in the Memoirs of the 

 Royal Academy of Sciences ; and is there urged for the ne- 

 cefliiy of a temperament, even for finging in the fame key. 

 And M. Huygens's folution of the difficulty is there ap- 

 proved of. Ann. 1707, p. 264. 



But the folution of thefe learned gentlemen is, as yet, far 

 from being" decifive. No experiment has yet been brought 

 to fliew that the human voice lings tempered notes ; not even 

 when accompanied by tempered inflruments. It feems to us, 

 on the contrary, that an exercifed voice, guided by a good 

 ear, fings true, even though accompanied by a miftuned 

 ijiftrument, as harpficliords moft frequently are, cfpecially 

 in tranfpofed keys. And were thefe intlruments always as 

 well tuned as art could make them, yet their tones would be 

 equal ; and it feems evident to the ear, that the human voice 

 finging naturally two tones in fucceflion, as C, D, E, never 

 makes them equal : and cannot, without great difficulty, 

 and by means of a variation of liarmony, be brouglit to 

 make them equal. 



Another folution, therefore, of M. Huygens's diffi- 

 culty, mufl be fought for. The truth feems to be, that the 

 fccond of the key muil be the true tone major above the key 

 and therefore the third between the fecond and fourth of the 



fC F D G C7 



key muft be fung deficient by a comma. Thu» in the key e( 

 C, from C to D win be a tone major = -J, and from D to 

 F will be a deficient third = i-i. See Interval. 



M. Huygens's melody, therefore, will lland as follows : 



D 



X ; X 



And the voice would fing the interval F, D, juft as if Uve 

 note E had been interpofcd ; in which cafe the notes would 

 be 



fC F E D G C 7 

 l4x44XiVX4x-J=i S 



Thefe notes all come within the diatonic fcale of C ; and 

 tlie voice naturally falls upon the note from whence it fet 

 out. The fame anfwcr will hold in the example mentioned 

 in the Memoirs of the Academy of Sciences ; where the 

 intervals bB, G, E, C occur. And here tlie interval from 

 bB to G fiiould be taken = ^^ =: 44 x to, as in tke 

 former example ; and for the fame reafon, the key being F. 



There feems, therefore, no repugnancy between the prac- 

 tice and theory of mnfic, while the melody is confiked to 

 one key ; but it mull be owned, that in tranfitions from key 

 to key, efpccially where feveral parts arc to make harmony 

 with each other, there flill remain difficulties, not mentioned 

 by M. Huygens, or any other writer we know of, which 

 might defervc a farther examination. 



We mull not omit mentioning, that the learned Dr. Smithy 

 in his Harmonics, has not only carried the theory of tem- 

 peraments far beyond all the autliors that preceded him ; 

 but has fhewn how to tune an inftrument according to any 

 propofed temperament, by the ear only, wliich is certainly a 

 mofl ingenious difcovery. 



This learned author prefers what he calls the temperament 

 of equal harmony, which differs infcnfibly fron: ihe divifioa 

 of the oftave into fifty parts, to all others ; and infills, that 

 it labours under the fewell defefts, and is of all others the mofl 

 agreeable in praftice. In the fyllem of equal harmony, the 

 temperaments of the fifth, third major and third minor, are 

 refpeftively tV and -i\ a"<l -rr of a comma lefs than the 

 truth. 



It would be impoflible^here to do juflice to the learned 

 author's reafonings on this fubjeft ; we fhall only add, that he 

 eflabhfhes, contrary to the common opinion, that the \ch 

 fimple confonances, generally fpeaking, will not bear fo 

 great temperaments as the Ampler confonances. 



Dr. Smith mentions a temperament communicated to him 

 by the ingenious Mr. Harrifon, which confifts in making 

 the proportion between the oftave and third major equal to 

 that of the circumference of a circle to its diameter. In thii 

 temperament the third major is diminiflied by ?, of 3 comma, 

 but the third minor is very near the ti-uth, and extremely 

 beautiful. 



A late author feems to think the divifion of the oftave 

 into thirty-one parts, not to be of modern invention, but 

 necefTarily imphcd in the doftrine of the ancients. At firfl 

 light, it would feem as if the ancients made hut twenty -four 

 Jiefes or divifions in the oftave, -viz. ten to each fourth, 

 and four to tlie tone; wliich (the oftave being equal to 

 two fourths and a tone) gives twenty. four diefes to the 

 oftave. But the author jufl quoted contends, that this 

 divifion is to be underftood only in one tenfion, that is, 

 either afcending or defcending) but that, accurately fpeak- 

 ing, if we confider all the diefts, or divifions of the fourth, 

 both afcending and defcending, we fhall find thirteen ; five 

 to each tone, and three to the femitonc major : and confe- 

 quently thirty-one divifions in the oftave. Thefe, indeed, 

 are not all naturally equal i but if we make them fo, we 

 R r 3 n>aU 



