TEM
ma ; that the third minor will be deficient by the fame quantity ; that the fixth minor will be perfect, and the fixth major re- dundant by | of a comma ; and laftly, that the femitone ma- jor will exceed the truth by 4, of a comma. If we introduce chromatic notes, or flats and {harps, the femitone minor will alfo exceed the truth by £ of a comma, and confequently the difference between the two femitones, or the dieiis efafaar- monica, will be preferved. *
If then we had a harpfichord or organ, with each feint or halt note divided, we fhould have the following notes or founds, viz. C . C# . D b . D . D# . E* - E . F . fjf . G b . G . G# . A b .A. A#-B b .B.c. in the compafs of an octave. "Vet this fyftem of notes, numerous as they feem, would not be fufficient for all tranfitions and tranfpofitions. For tho' a piece of mufic, tranfpofed to any of the natural keys CD. E . F . G . A . B. and to the flats, as E b and B b , and fome others, would do well ; yet in tranfpofing to fharps^ as to G$. we mould not find a true third major, unlefs we intro- duced E#. and even in flats, as A b and E b we fhould not find a true third major in defcending, or a fixth minor in afcending, unlefs we introduced F b and C b . and in like man- ner tranfpofitions to G4F and E b would oblige us to introduce "B#andC b . Nor would even this fuffice; for if necefficy required a tranfpolition from the key of C to that of D# , we fhould not find a true third major without introducing F## and c. So that at laft we fhould come to a temperate fyftem, where, in afcending, the notes C, D, F, G, A would each have its (harp and double fharp, and the notes B and E each a iingle fharp. In defcending, the notes E, D, B, A, G would each have their flat and double flat, and the notes F and C each afingle flat. And thus the octave would be divided into 31 intervals, whofe defoliations are C . D bb . C# . D b .
1.2.3.4 C##.D.E bb .D# .E b .D^#.E.F b ,E#.F.G bb .F^. 5.6.7 . 8 9 . 10.11.12.13.14.15.16 G b . F## . G . A bb . G# . A b . G## . A . B bb . A f . B 1 17 . 18 . 19 . 20 - 21 . 22 . 23 . 24 . 25 . 26 . . Aff$ . B . O . Bfr . C. Where the letters C . D . E . 27 . 28 .29 . 30 . 31.
F . G . A . B. fignify the common diatonic notes ; thofe marked with a tingle $: or b are the chromatic, and thofe marked with a double =fr-£ or bb are enharmonic notes j fo called, becaufe the interval between them and the next diato- nic note is an enharmonic diefis; for which reafon the notes E# . F b . and B^f- . C b . are alio enharmonic. But even in this divifion of the octave, all the notes would not have a third major in afcending and defcending ; thus, for inftance, Dianas no third major ; for this would beF^^lr, which tsnotin thefcale, nor can any number of additional notes fuffice in all cafes. But this inconvenience is eafily remedied, and the fyftem confiderably improved by making all the 31 intervals equal. We have already obferved, that in the com- mon temperature the femitones major and minor exceed the truth by -i of a comma, ■and that the enharmonic diefis is pre- ferved true. Hence it follows, that the hyperoche, or difference between the chromatic and enharmonic diefis ; for example, the interval between F b and E# or D bb and G#, &c, will alfo exceed the truth by \ comma. Now the hyperoche, by our table, under Interval, is equal to 1 . 37695, to which adding 4 comma =s 0.25000, we have 1.62695, which differs from the enharmonic diefis 1 . 90917 only by 0.28222, or about ^ of a comma. Neglecting this final] difference, let us fuppofe all the 31 intervals of the octave equal, it will follow that tranfpofitions to all the notes of the fyftem, whether diatonic, chromatic or enharmonic, will be equally good, and differ only in pitch or tone, as they ought, but not in accuracy, which mull next be examined. The divifion of the octave into 31 parts may be conveniently done by logarithms. Under the head Interval I find the logarithm of the octave = 55 , 79763 commas, confequently each diefis, or divifion of the octave = 1 . 79992 commas ; hence the fifth, being 18 diefis, will be 32.399 commas. Now the true fifth is 32 . 640, the fifth confequently in this 'Temperature is deficient by o . 241 parts of a com- ma, which is lefs than i of a comma by T ^ part ; and therefore this fifth will, ftrictly fpeaking, be better than that of the vulgar Temperature by T | r of a comma 5 but this is in- fenfible. Next, proceeding to examine the third, we fhall find it equal to 10 diefis or divifions, that is 17 .999 com- mas, and the true third major being 17 . 963 commas, the difference is 0.036, that is about ^ of a comma. Now as the car can bear a fifth, altered by £ of a comma, it will much more eafily bear the alteration of ■£$ of a comma in a third major. Again, in this Temperature, the third minor is indeed, ftrictly fpeaking, worfe than in the vulgar, which differs from the truth but by | comma, whereas here it dif- fers by about ^ of a comma more ; but then this difference is infenfible.
Thus we have been led from the confideration of the vulgar Temperature^ to the invention of the Temperature which di- vides the octave into 31 equal intervals, commonly called, Huygens' s Temperature. This great mathematician was in- deed the firft who gave a diftinct account of it, and fhewed its ufe and accuracy. But here, as in many other inventions, we find the hint of the thing much older than the true knowledge
TEM
of it. See Hugenii Opera Omnia, vol. 1. p. 748, 74c). Edit. I. LugdvBatav. .1724.
The divifion of the octave into 31 parts, was invented in Italy about 200 years ago, by Don Nicola Vincentino. The title of his book is V Antica Mufica ridotta alia moderna prat- tiea, &c. Roma, 1555- Fol. and an inftrument called archl- cembalo was made upon this fcheme, as Salinas informs us, who at the fame time condemns it as very difagreeable 'in. practice. But this could be owing to nothing but its not be- ing tuned according to the intention of the inventor. For if all the thirds major of this inftrument were made perfect:, and the fifths diminished by \ of a comma, it is evident that the inftrument would be equally exact with any tuned accord- ing to the vulgar Temperature, and would fuffice for tranfpo- fitions to- any diatonic or chromatic notes, though not to all the enharmonic, as D##, taY. becaufe we fhould not find its third major. And if the inftrument were tuned according to Mr. Huygens's fcheme, of making all the divifions equal, ic would then have all the 31 keys equally good, and very near the truth. Sec Salinas, lib. 3. The title of his work is Franeifii Saliva: Burgenjis de Mufica Libri Septem, Salmanti- crs, 1577. Folio. Merjetwus's Work is entitled Harmom- corum, Libii XII. autbore F-, M. Merfenno Minima, Luteticc Parifiorum, 1648. Fol. He publifhed another book before this, the title of which is, HarmonieXJniverfelle, contenant la Theor'te (J la Pratique de la Mujique. Paris, 1 636. Fol. 2 vol.
Hence it is plain, Salinas and Merfennus had not fufficiently examined this matter.
The ufe of this Temperature of Mr. Huygens deferves to be introduced into the practice of mufic, as it will facilitate the execution ofall the genera of mufic, whether diatonic, chroma- tic, or enharmonic ; nor does the multiplicity of its parts ren- der it impracticable, the author alluring us, that he had harpfichord made at Paris with fuch divifions, which was ap- proved 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. 5. p. 99, tiff.
Mr. Huygens, to facilitate the tuning of inftruments with fuch divifions, has given us a table of the parts of an octave, according to his fyftem, together with their logarithms. The table is as follows :
III. IV.
The divifion of the octave in- to 31 equal parts.
I.
N. 97106450 4,6989700043 4,7086806493
4,7183912943 4,7281019393 4,7378125843 4'7475232293 4^75723387+3 4,7609445193 4=7766551643 4,7863658093 4,7960764543 4,8057870993
4^154977443 4,8252083893
4,8349190343 4,8446296793 4,8543403243 4,8640509693 4,8737616143 4,8834722593 4,8931829043 4,9028935493 4,9126041943 4,9223148393 4,9320254843 4,9417361293 4,9514467743 4,9611574193 4,970868064-3 4,9805787093 4,9902893543 4,9999999993
The fecond column of this table contains the numbers expref- fing the lengths 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 fyllablcs by which the notes are ufually named in France; and the after ifc * fhews fome en- harmonic notes, of which that near Sol is moft neceflary. In the fourth column are the letters commonly ufed to denote the founds of the octave.
The numbers of the fecond column were found by means of thofe in the firft, which are their refpective logarithms ; and thefe were found by dividing 0.30102999566 the logarithm
of
5COOO
Up
O
50000
5"3 r
52278
53469
Si
B*
53499
54678
559H
Sa
B
55902
57'79
57243
58471
59794
La
A
59814
61146
62528
62500
63942
Sol*
G*
64000
653S8
66866
Sol
G
66874
68378
69924
71506
Fa*
F*
71554
73122
74776
Fa
F
74767
76467
78196
79964
Mi
E
80000
S1772
83621
Ma
E b
83592
85512
85599
87445
89422
Re
D
89443
91444
935 1 *
93459
95627
Ut*
C*
95702
977 8 9
100000
Ut
C
1 00000
The divifion of the octave ac- cording to the common Tem- perament. VI.
4,6989700043
4,7283474859
4,7474250108 4,7577249674
4,7768024924
4,7958800173 4,8061799740
4,8252574989
4,8546349804 4,8737125054
4,9030899870
4,9221675119 4,9324674685
4,9515449935
4,9706225184 4,9809224750
5,0000000000
