TEMPERAMENT. 



Tl.e numbers of the fccond column were found by means 

 of thofe in tlie firft, whicli arc tlieir rclpedlive logarithms ; 

 and thcfewerc found by dividing 0.30102999566, the loga- 

 rithm of 2, by 31. The quotient 97106450 is marked N, 

 and being continually added to the logarithm of 50000, that 

 is, to 4.6989700043, gives all the logarithms of tlic lirft 

 column to the greatefl 4.9999999993. wliich being ex- 

 tremely near to 5.0000000000, the logarithm of looooo, 

 thews the operation to have been rightly performed. 



The fifth column (hews the lengths of the chords in the 

 common temperament ; and the fixth column contains their 

 refpeftive logarithms. Vide Huygenii Opera, vol. i. 



P- 752. 753- 



The learned author of this temperament has not given the 

 notes anfwering to all the divifions of the oftave ; but that 

 may eafily be fupplied from what lias been faid above when 

 we derived this temperament from the confidcration of the 

 common. 



As Huygens has not given the names of all the intervals 

 that occur in his temperate fcale, we (hall here infert them 

 in the oilave, from C to c, with their refpeftive meafures in 

 the commas, and tenths of a comma 



Vol. XXXV. 



The temperate diefis cnharmonica of Huygens iieiiig i.R 

 coniina, nearly, which is eafdy remembered,' the meafurc of 

 any interval in the oftave may be found by multiplying it 

 by the luimber denoting the place of that mterval. Thus 

 the fixth minor, being the twenty-firfl; interval, will be = 

 1.8 X 21 = 37.8. The odave, being the thirty-firft, will 

 be = 31 X 1.8 = 55.8, which does not differ from the 

 truth by more than 0.00237, that is, not by t.-<t of a comma, 

 and therefore perfedly infenfible. See Intehvai,. 



All the intervals in the foregoing table, cither have rr- 

 ceived names, or at leaft might receive them, from a perfed 

 analogy to the names in ufe among praftical muficians ; but 

 many of thefe intervals are as yet unheard of among prac- 

 titioners. Perhaps, if all the genera of ancient mufic were 

 reftored, every interval here mentioned might be of ufe, 

 either in melody or harmony, and thereby greatly add to the 

 variety of compofition. 



We have already mentioned the advantages of M. 

 Huygens's fydem ; but its excellency will better appear by 

 comparing it with the fchemes of others. 'We may dil- 

 tinguifti and name the different temperaments by llu.- number 

 of equal parts into which the oftave is fuppofcd to be 

 divided. The temperaments that occur in books are tem- 

 peraments of 12, 19,31, 43, 50, 53, and ^^ parts, of whicli 

 in order. 



The temperament of 12 parte is founded on the fuppofi- 

 tion that the fcmitones major and minor may be made equal. 

 Hence the o£lave will be divided into 12 equal femitones, 

 feven of which will make the 5th, four the 3d, and three 

 the 3d minor. 



The temperament of 19 parts goes upon the fuppofition 

 that ihs femitone major is the double of the femitone minor. 

 Hence the tone will be 3, and the third major 6. The diefis 

 enharmouica will be i, and confequcntly the oftave, being 

 three thirds major and a diefis, will be 19. The fifth contains 

 1 1 parts. The harpfichord, in this fcheme, will have every 

 feint cut in two, one for the (harp of the lower note, and 

 the other for the flat of the higher. Between B and C, and 

 between E and F, will be iuLerpofed keys, which mull ferve 

 for the (harps of B and E, and the flats of C and F re- 

 fpeftively. 



The temperament of 31 parts is M. Huygens's, already 

 defcribed : here the femitones arc as 3 to 2. The third 

 major is 10, and the fifth 18. 



T'he temperament of 43 is M. Sauveur's, and by him 

 very fully dcfcnbed in the Memoirs of the Royal Academy 

 of Sciences, A.D. 1701, 1702. He fuppofes ihc propor- 

 tion of the femitones to be as 4 to 3. Hence his tone is 7, 

 the third major 14, the fifth 25, and the oftave 43. What 

 mufical foundation this learned gentleman went upon in the 

 invefi-igation of this temperament, is not known : but it 

 fecms liable to infupei-ablc difficulties ; for here the diefis en- 

 harmonica is but the half of the difference between it and the 

 chromatic diefis : whereas, in truth, this difference, inflead 

 of being double of, is really lefs than the enharmonic dieCs, 

 as was long ago objedlcd to him by Mr. Henfling, and ap- 

 pears from the table under Intf.kval, Mifccl. Bcrolin, 

 tom. i. p. 285, 286. 



Befidcs, his enharmonic diefis falls gre.itly (hon of the 

 truth, being but 1.27 of a comma, w-hich is an error of 

 0.64, or nearly 4 of a comma. Whereas, in M. Huygens's 

 temperament, the error of the diffis is alm,oll infenfible, being 

 but ^TT of a comma. Nor are the praAical advantages of 

 M. Sauveur's fyftem any ways comparable to Huygens's. 

 His fifth is indeed, ftric\ly fpeaking, belter ; but (o little, 

 that the difference is not fenfible, not being 7'. of a comma. 

 On the other hand, his thirds are fenfibly worfe, the major 

 R r being 



