TEMPERAMENT. 



being h »nd the minor 4 of a comma falfc. Whocas 

 HuyVnV. third m.-,jor doc. not d.ffer fenf.bly from the 

 truth, and the minor has nofenfible difference from the third 

 mino^ def.cu-..t by 4 of a co.nma of the common tonperament 

 which ought to be deemed the hm.t of the dimmution of 

 concords. If «e add to this, that the much greater number 

 of parts in M. Sauveur's odave, makes it much more intri- 

 cate th.-in M. Huygcns's, and that thcfc parts would be 

 falfe or ufelefs, even fuppofmg the enharmonic genus rc- 

 ftored, no mufici.in will long hefitate which he ought to 



''Vhe temperament of 50 parts is propofed by Mr. Hen- 

 fline in the Mifccllan. Beroliii. above cited: he takes the 

 proportion of the femitonc as 5 to 3 : hence his tone is 8, 

 the third major 16, the fifth 29, and the oaave 50. 1 he 

 third major and fifth in tliis fyllem will be worfe tli.-in Huy- 

 gcns's, though the third minor be a little better. 1 he third 

 major is here lefs than the true, and the fifth deficient by 

 more than 4 of a comma, which is a fault, not to mention the 

 inconveniency arifing from dividing tlie odave into 50 parts ; 

 befides, 5 : 3, the proportion of the fcmitones here afTumed, 

 although expreffed in greater numbers, is not fo near the 

 truth as M. Huygens's of 3 : 2. See Ratio. 



The temperament of 53 parts is mentioned by Mcrfennus. 



Here the tones will be unequal, 9 being the tone major, and 

 8 the minor. Hence the third major will be 17, and the 

 fifth 31, which laft does not differ from the truth by above 

 -, ^^ part of a comma. The third minor is alfo more perfeft 

 than in M. Huygens's fyllem. But the multiplicity of 

 parts in the odave of this fyflcm renders it too intricate ; 

 and the diflinftion of tones major and minor upon fixed in- 

 llruments is imprafticablc. 



The lall temperament we have to mention is that of 55 

 parts, which M. Sauveur calls the temperament of praftical 

 muficians. Its foundation lies in afTuming the proportion of 

 the fcmitones as 5 to 4 ; fo that the tone will be 9, the third 18, 

 and the fifth 32. The fifth in this fyflcm, as in that which 

 makes the fcmitones equal, is nearer the truth than M. Huy- 

 gens's, but this advantage is not ^V of ^ comma ; and on 

 tlie otlicr hand, the thirds, both major and minor, are here 

 greatly mifluned, as will appear by the annexed table, ex- 

 hibiting the thirds and fifths of thefe feveral temperaments, 

 as alfo the thirds and fifths of the common temperajnent, 

 and two mentioned by Salinas, marked ift Salin. 2d Salin, 

 The letter V. ftands for the fifth ; III. for the third major ; 

 and 3. for the third minor. The fifths are all deficient, but 

 the tliirds are fometimes lefs than the true ; the firft are 

 marked + , the others — . 



Temperaments formed by the divifion of the oftave into 

 <qual parts, may be called geometrical temperaments. The 

 common, and the two mentioned by Salinas, do not pro- 

 ceed upon this foundation ; the intention of the firft in- 

 ventors not having been to make tranfpofitions to every note 

 of the fyftem equally good ; but only to make the moft 

 ufual tranfitions in the courfe of a piece of mufic tolerable. 

 Hence the parts of the oftave, in their fuppofition, were 

 not all equal. 



The common temperament, as we have faid, preferves 

 the third major perfeft. The firft of Salinas preferves the 

 third minor perfeft. In the fecond of Salinas, the femi- 

 tone minor is perfeft. The foundation of his firft tempe- 

 rament is making the temperate tone equal to the tone minor 

 and 4 of a comma, or the tone major lefs i of a comma. 

 Hence his fifth and third major will be deficient by 4^ of a 

 comma ; and the third minor confequently will be true. The 

 ground of his fecond fcheme is, to add i of a comma to the 

 tone minor, or take i from the tone major for his temperate 

 tone. Hence the fifth will be deficient by 4- of a comma, 

 and the thirds major and minor each deficient by -=■ of a 

 comma. Confequently, the femitonc, being their difference, 

 will be preftrved. 



As to Mr. Salmon's fcale in the Philofophical Tranfac- 

 tions, there is nothing true in it, but the diatonic fcale of C. 

 His fcale for A is falfe, the fourth being erroneous by a 

 comma : moft. of his femitoncs are likewife falfe. In 



fhort, it can neither be confidered as a true fcale, nor as a 

 temperament. 



Before we clofe this article, it may be proper to add a few- 

 words about the method of invention of the foregoing geo- 

 metrical temperaments. M. Huygens having had the hint 

 of a divifion of the oftave into 3 1 parts, had nothing far- 

 ther to do but to examine it by logarithms. But fuppofing 

 no fuch hint had been given, he might have inveftigated it 

 direftly, by the method laid down by himfelf, and alfo 

 by Dr. Wallis and Mr. Cotes, for approximating to the 

 value of given ratios in fmaller numbers. We have given 

 Mr. Cotes's method under Ratio. The application of that 

 method to the prefent purpofe is thus : the ratio of the 

 oftave to the third major is 55.79763 to 17.96282, and the 

 approximating ratios will be, 



1. Greater than the true 28 : 9, 87 : 28, &c. 



2. Lefs than the true 3 : i, 31 : 10, 59 : 19, 205 : 66, Sec. 

 The ratios greater than the true muft all be rejefted ; be- 



caufe they give the third major lefs than true, and confe- 

 quently the tone (its half) deficient by above ^ of a comma ; 

 which gives the fifth deficient above .| of a comma : but this 

 ought not to be. The firft of the ratios lefs than true is 

 3 : I , or 1 2 : 4, which is the temperament of 1 2 parts be- 

 fore defcribed, and too inaccurate. The next is 31 : 10, or 

 M. Huygens's. The reil divide the oftave into too many 

 parts. 



The fame may be alfo found thus : the ratio of the aftave 



te 



