TEMrERAMENT. 



the common temperament, the femitones major and mmor ex- 

 ceed the truth by a quarter of a comma, and that the enharmonic 

 dicl'is is preferved true. Hence it follows, that the hype- 

 rochc, or difference between the chromatic and enharmonic 

 ditfis ; for example, the interval between Fb and £«, or Db D 

 and C», &c. will alfo exceed the truth by a quarterof a comma. 

 Now the hyperoche.by our table under Interval, is equal to 

 1.37695, to which adding a quarter of acomma = 0.25000, we 

 have 1. 68695, whichdiffcrs fromtheenharraonicdiefis 1.90917 

 only by 0.28222, or about ,'r of a comma. Neglefting this 

 fmall difference, let us fuppofe all the thirty-one intervals of 

 the oftave equal, it will follow that tranfpofitions to all the 

 notes of the fyftem, whether diatonic, chromatic, or enhar- 

 monic, will be equally good, and differ only in pitch or tone, as 

 they ought, but not in accuracy, which muft next be examined. 



The divifion of the oftave into thirty-one parts may be con- 

 veniently done by logarithms. Under the article Interval, I 

 find the logarithm of the oftave = 55-79763 commas ; confe- 

 quently each diefis, or divifion of the oftave, — 1.79992 

 comma; hence the fifth, being 18 diefes, will be 32.399 

 commas. Now the true fifths being 32.640, the fifth con- 

 sequently in this temperament is deficient by 0.241 parts of 

 a comma, which is lefs than a quarter of a comma byTTn part; 

 and therefore this fifth will, ftriftly fpeaking, be better than 

 that of the vulgar temperament by -rrir of the comma ; but 

 this is infenfible. Next, proceeding to examine the third, 

 we (hall find it equal to 10 diefes or divifions, that is, 17-999 

 commas; and the true third major being 17-963 commas, 

 the difference is 0.036, that is, about A of a comma. Now as 

 the ear can bear a fifth, altered by a quarter of a comma, it will 

 much more eafily bear the alteration of -jV of a comma in a 

 third major. Again, in this temperament the third minor 

 is indeed, flriclly fpeaking, worfe than in the vulgar, which 

 differs from the truth but a quarter of a comma, whereas 

 here it diflFers by about -^ oi i comma more ;. but then this 

 difference is infenfible. 



Thus we have been led from the confideration of the 

 vulgar temperament, to the invention of the temperament 

 which divides the oftave into 3 1 equal intervals, commonly 

 called Huygens's temperament. This great mathematician 

 was, indeed, the firfl who gave a diflinft account of it, and 

 Ihewed its ufe and accuracy. But here, as in many other 

 inventions, we find the hint of the thing much older than the 

 true knowledge of it. See Huygenii Opera omnia, vol. i. 

 p. 748, 749, edit. I. Lugd. Batav. 1724. 



The divifion of the oftave into 31 parts was invented 

 in Italy about 300 years ago, by Don Nicola Vincentino. 

 The title of his book is " L'Antica Mufica Riddotta alia 

 Modema Prattica, &c." Roma, 1555. fol.; and an inftru- 

 ment, called archicembalo, was made upon this fcheme, as 

 Salinas informs us, who at the fame time condemns it, as 

 very difagreeable in praftice. But this could be owing to 

 nothing but its not being tuned according to the intention of 

 the inventor. For if aU the thirds major of this inftrument 

 were made perfeft, and the fifths diminifhed by a quarter of a 

 comma, it is evident that the inftrument would be equally exadl 

 with any tuned according to the vulgar temperament, and 

 would fuffice for tranfpofitions to any diatonic or chromatic 

 notes, though not to all the enharmonic, as D*«, &c. be- 

 caufe we fhould not find its third major. And if the in- 

 ftrument were tuned according to-Jltf. Huygens's fcheme, of 

 making all the divifions equal, it vpould then have all the 

 31 keys equally good, and very near the truth. See 

 Salinas, lib. iii. The title of his work is " Francifci Salinje 

 BurgenCs de Mufica Libri Septem," Salmanticas, 1577, fol. 

 Meri^ennus's work is intitled " Harmonicorum, Libri XII. 

 authore F. M. Merfenno Minimo, Lutetiae Parifiorum," 

 1648, foL He publifhed another book before this, the title 



of which is " Harmonie Univerfelle, contenant la Theorie et 

 la Pratique de la Mufique," Pai-is, 1636, fol. 2 vols. 



Hence it is plain, Salinas and Merfennus had not fuf- 

 ficiently examined this matter. 



The ufe of this temperament of M. Huygens deferves to 

 be introduced into the practice of mufic, as it will facilitate 

 the execution of all the genera of mufic, whether diatonic, 

 chromatic, or enharmonic ; nor does the multiphcity of its 

 parts render it impracticable, the author affuring us that he 

 had a harpfichord made at Paris with fuch divifions, which 

 was approved of and imitated by fome able muficians. 

 Merfennus alfo gives a fcheme for this purpofe ; and Salinas 

 fays he faw and played upon fuch an inftrument. See alfo 

 Don Vincentino before cited, lib. v. p. 99, &c. 



M. Huygens, to facilitate the tuning of inftruments with 

 fuch divifions, has given us a table of the parts of an oftave, 

 according to his fyftem, together with their logaritlims. 

 The table is as follows : 



The fecond column of this table contains the numbers 

 expreffing the length of chords making 31 equal divifions, the 

 longeft, anfwering to C, being fuppofed to be divided into 

 100,000 parts. 



In the third column are the fyllables by which the notes 

 are ufually named in France ; and the afterlfc * /hews fome 

 enharmonic notes, of which that near yo/ is moft necelTary. 



In the fourth column are the letters commonly ufed to 

 denote the found of the odlave. 



The 



