T E M
of 2 by 31. The quotient 97106450 is marked N. and be- ing continually added to the logarithm of 50000 that is to 4.6989700043 gives all the logarithms of the firft column to the greateff. 4.9999999993, which being extremely near to 5.0000000000 the logarithm of 100000, fhews the operation to have been rightly performed.
The fifth column fhews the lengths of the chords in the com- mon Temperament ; And the fixth column contains their re- fpective logarithms. V. Huygenii Opera, vol. 1. p. 752, ■753. The learned author of this Temperature has not given the notes anfwering to all the divihons of the octave ; but that may eafily be fupplied from what has been faid above, when we derived this Temperature, from the confederation of the common.
We have already mentioned the advantages of Mr. Huygens's fyftem ; hut its excellency will better appear by comparing it with the febemes of others. We may diftinguifh and name the different Temperatures by the number of equal parts, into which the octave is fuppofed to be divided. The Tempera- tures that occur in books, are Temperatures of 12, 19, 31, 43, 5°» 53 an( ^ 55 P arts > OI " which in order. The Temperature of 12 parts is founded on the fuppofition that the femitones major and minor may be made equal. Hence the octave will be divided into 12 equal femitones, 7 of which will make the fifth, 4 the third, and 3 the third minor. The Temperature of 19 parts goes upon the fuppofition that the femitone major is the double of the femitone minor. Hence the tone will be 3, and the third major 6. The diefis enharmonica will be i, and confequently the octave being 3 thirds, and a diefis, will be 19. The fifth contains u parts. The harpfichord in this fcheme will have every feint cut in two, one for the fharp of the lower note, and the other for the flat of the higher. Between B and C, and between E and F, will be interpofed keys, which rauft ferve for the fharps of B and £, and the flats of C and F refpe£lively. The Temperature of 31 parts is Mr. Huygens's already de- fcribed j here the femitones are as 3 to 2. The third major is 10, and the fifth 18.
The Temperature of 43 is Mr. Sauveur's, and by him very fully defcribed in the memoirs of the Royal Academy of Sciences, An. 1701, 1702. He fuppofes the proportion of the femitones to be as 4 to 3. Hence his tone is 7. The third major 14, the fifth is 25, and the octave 43. What mufical foundation this learned gentleman went upon in the inveftigation of this Temperature, I know not : But it feems liable to infuperable difficulties ; for here the diefis enharmo- nica is but the half of the difference between it and the chro- matic diefis ; whereas, in truth, this difference, inftead of be- ing double of, is really lefs than the enharmonic diefis, as was long ago objected to him by Mr. Henfling, and appears from the table under the head Interval. Mifcel. Berolin. Tom. 1. p. 285, 286.
Befides, his enharmonic diefis falls greatly fhort of the truth, being but 1.27 of a comma, which is an error of 0.64 or |
T E M
near | of a comma. Whereas in Mr. Huygens's Tempera- ture, the error of the diefis is almoft infcniible, being but t's of a comma. Nor arc the piaflical advantages of Mr. Sau- yeur s i fyftem any way comparable to Huygen's. His fifth is indeed Italy (peaking, better; but fo little, that th, dif ferenceisnotfenfible, not bang A of a comma. On he other hand, his thirds are fenfibly worfe, the major beine 1 and the minor f of a comma falfe. Whereas Huygens's third major does not differ fenfibly from the truth, and the minor has no fallible difference from the third minor deficient by ■ comma of the common temperature, which ought to be deemed the limit of the diminution of concords. If we add to tins, that the much greater number of parts in Mr. Sau- veur s odave, makes it vaftly more intricate than Mr. Huy- genss, and that thefe parts would be falfe or ufelefs, evenfup- pohng t h e enharmonic genus rcftored, I believe no mufician will long hefitate which he ought to prefer.
Aj^f'V of 5 ° parts is P r °P°fed ty Mr. Henfling in the Mifcellan. Berolin. above cited ; he takes the proportion of the femitones as 5 to 3 ; hence his tone is 8, the third major
ll'u \ r a 9 ' *"i ? e ° aaVe S°- T,,e third ma j° r ™d firth in this fyftem will be worfe than Huygens's, though the
third minnr hi* a ]itt]* k ....... ^Pl — *l_ ■ i • • . ° . .
third minor be a little better. The third
major is here lefs
than the true, and the fifth deficient by more than | comma which is a fault, not to mention the inconveniency arifine from dividing the oftave into 50 parts ; befides 5 : 3 the pro- portion of the femitones here aflumed, although expreffed in greater numbers, is not fo near the truth as Mr. Huygens's of 3:2. We have given the proof of this under the head Ratio. 1 he temperature of 53 parts is mentioned by Merfennus 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 rb part of a comma. The third minor is alfo more perfeft than in Mr. Huygen's fyftem : But the multiplicity of parts in the ocfave of this fyftem, render it too intricate ; and the diftinflion of tones major and minor upon fixed inftruments is, I doubt, impracticable.
The laft Temperature we have mentioned is that of 5; parts which Mr. Sauveur calls the Temperature of praSical mufi- cians. Its foundation lies in affirming the proportion of the femi-tones, as 5 to 4, fo the tone will be 9, the third 18 and the fifth 32. The fifth in this fyftem, as in that which makes the femitones equal, is nearer the truth than Mr. Huy- gens's, but this advantage is not ^- of a comma ; and on the other hand, the thirds both major and minor are here greatly mif-tuned, as will appear by the annexed Table, exhibiting the thirds and fifths of thefe feveral Temperatures, as alfo the thirds and fifths of the common Temperature, and two men- tioned by Salinas, marked i«. Salin. 2 d . 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 thirds are fometimes greater and fometimes lefs than the true ; the firft are marked -f-, the others — .
Temperatures
V. Commas.
Error.
lll.Commas,
Error.
3. Commas.
Error.
of 12 parts.
32- 549
0. 091
18.599
0.636 +
5 3-95°
0, 727— 0. 007 +
19.
3 2 - 3°4
0.336
17. 620
°- 343—
14.684
3 1 -
32- 399
0. 241
17.999
0.036 +
14.400
0.277—
43-
32.440
0. 200
18. 167
0. 204 +
14. 273
0. 404 —
50.
32. 363
0.277
17.855
0. 108 —
14. 508
0. 169—
53-
32- 6 37
0. 003
17.897
0. 066 —
14. 740
0.063+
55-
32. 464
0. 176
18.261
0.298 +
14. 203
0.474—
0. 250 —
Com. Temp.
32. 390
0.250
•7-963
0. 000
14. 427
it. Salin.
3 2 - 3°7
°- 333
17.630
°- 333—
14.677
0. 000
2 d . Salin. True Scale.
V- 354 32. 640
0.286 0. 000
17.520 ■7-9 6 3
0. 143—
0. 000
! 4- 43+
14.677
0. 143—
0. 000
Temperatures formed by the divifion of the octave into equal parts, may be called geometrical Temperatures. The com- mon and the two mentioned by Salinas, do not proceed upon this foundation. The intention of the firft inventors not hav- ing been to make tranfpolitions to every note of the fyftem equally good ; hut only to make the moft ufual tranfitions in the courie of a piece of mufic tolerable. Hence the parts of the octave, in their fuppofitions, were not all equal. The common Temperature, as we have faid, preferveb the third major perfect. The firft of Salinas, preferves the third minor perfect. In the fecond of Salinas, the femitone minor is per- fect. The foundation of his firft Temperature is making the temperate tone equal to the tone-minor and 4- of a comma, or to the tone-major lefs \ of a comma. Hence his fifth and third major will be deficient by | of a comma ; and the third minor confequently, will be true. The ground of his fecond fcheme is, to add -f of a comma to the tone-minor, or take % from the tone-major, for his temperate tone. Hence the fifth will he deficient by -| of a comma, and the thirds major and minor each deficient by 4- of a comma. Confequently the femitone, being their difference, will be preferved. As to Mr. Salmon's fcale, in the Philofophical Tranfactions, 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 femi-tones are likewife falfe. In fhort, it can neither be confidered as a true fcale, nor as a Temperature.
Before we clofe this article, it may be proper to add a few- words about the method of invention of the foregoing geome- trical Temperatures. Mr. Huygens having had the hint of a divifion of the octave into 31 parts, had nothing farther to do but to examine it by logarithms. But, fuppofing no fiich hint had been given, he might have mveftigated it direaiy by the method kid down by himfelf, and alfo by Dr. Wallis, and Mr. Cotes, for approximating to the value of given ratio's, in fmaller numbers. We have given Mr. Cotes's method, under the head Ratio. The application of that method to the prefent purpofe, is thus : The ratio of the oaave to the third major is 55.79763 to 17.96282, and the approximating ratios will be
i°. greater than the true 28 : 9, 87 : 28, &c.
■2°. lefs than the true 3 : 1, 31 : 10, 59 ; 19, 205 : 66,
C3V.
The ratios greater than the true, muff, all be rejected ; be- caufe they give the third major lefs than true, and confequent- ly the tone, (its half) deficient by above £ 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 : 1, or 12 : 4, which is the Temperature of 12 parts before defcribed, and too inaccurate. The next is 31 : 10, or Mr. Huygens's. The reft divide the octave into too many parts.
The
