to fa there is the fame interval as from fi to ut, from fa to fol the fame as from ut to re; from fol to la the fame as from re to mi: this is the reafon why the Greeks diftinguished thefe two tetrachords; yet they joined them by the note mi, which is common to both, and which gave them the name of conjun&ive tetrachords.

31. Moreover, the intervals between any two founds, taken in each tetrachord in particular, are precifely true: thus, in the first tetrachord, the intervals of ut me, and fi re, are thirds, the one major and the other minor, exactly true, as well as the fourth fi mi; it is the fame thing with the tetrachord mi, fa, sol, la, fince this tetrachord is exactly like the former.

52. But the cafe is not the fame when we compare two founds taken each from a different tetrachord; for we have already seen, that the note re in the first tetrachord forms with the note fa in the second a third minor, which is not true. In like manner it will be found, that the fifth from re to la is not exactly true, which is evident; for the third major from fa to la is true, and the third minor from re to fa is not fo: now, in order to form a true fifth, a third major and a third minor, which are both exactly true, are neceffary.

53. Thence it follows, that every confonance is abfolutely perfect in each tetrachord taken by itfelf; but that there is fome alteration in paffing from one tetrachord to the other. This is a new reafon for diftinguishing the fcale into thefe two tetrachords.

54. It may be ascertained by calculation, that in the tetrachord fi, ut, re, mi, the interval, or the tone from re to mi, is a little less than the interval or tone from ut to re. In the fame manner, in the second tetrachord mi, fa, fol, la, which is, as we have proved, perfectly fimilar to the firft, the note from fol to la is a little less than the note from fa to fol. It is for this reafon that they diftinguish two kinds of tones; the greater tone, as from ut to re, from fa to fol, &c.; and the leffer, as from re to mi, from fol to la, &c. See INTERVAL, § III, 1—3.

CHAP. VI. Of the FORMATION of the DIATONIC SCALE among the MODERNS, or the ORDINARY GAMMUT.

55. We have juft fhown how the fcale of the Greeks is formed, fi, ut, re, mi, fa, fol, la, by means of a fundamental bafs compofed of three founds only, fa, ut, fol: but to form the scale ut, re, mi, fa, fol, la, fi, UT, which we ufe at prefent, we must neceffarily add to the fundamental baís the note re, and form, with thefe four founds fa, ut, fol, re, the following fundamental bafs: ut, fol, ut, fa, ut, fol, re, fol, ut; from whence we deduce the modulation or fcale, ut, re, mi, fa, fol, la, fi, UT. In effect, ut in the fcale belongs to the harmony of ut which corresponds with it in the bafs; re, which is the fecond note in the gammut, is included in the harmony of fol, the fecon note of the bafs; me, the third note of the gammut, is a natural harmonic of ut, which is the third found in the bass, &c.

56. Thence it follows, that the diatonic fcale of the Greeks is, at least in fome refpects, more

fimple than ours; fince the feale of the Greeks (chap. v.) may be formed alone from the mode proper to ut; whereas ours is originally and primitively formed, not only from the mode of ut (fa, ut, fol), but likewife from the mode of fol (ut, fol, re.) It likewife appears, that this last fcale confifts of two parts; of which the one, ut, re, mi, fa, fol, is in the mode of ut; and the other, fol, la, fi, ut, in that of fol.

57. For this reafon the note fol is found to be twice repeated in immediate fucceffion in this fcale; once as the fifth of ut, which corresponds with it in the fundamental bafs; and again, as the octave of fol, which immediately follows uz in the fame bafs. As to what remains, these two confecutive fols are otherwise in perfect unifon. For this reafon we are fatisfied with finging only one of them when one modulates the scale ut, re, mi, fa, fol, la, fi, UT: but this does not prevent us from employing a pause or repose, expreffed or underftood, after the found fa. There is no perfon who does not perceive this whilft he himself fings the scale.

58. The fcale of the moderns, then, may be confidered as confifting of two tetrachords, difjunctive indeed, but perfectly fimilar one to the other, ut, re, mi, fa, and jol, la, fi, ut, one in the mode of ut, the other in that of ol. For what remains, we fhall fee in the fequel by what arti fice one may cause the scale ut, re, mi, fa, fol, la, fi, UT, to be regarded as belonging to the mode of ut alone. For this purpofe it is neceflary to make fome changes in the fundamental bass, which we have already affigned: but this shall be explained at large in chap. xiii.

59. The introduction of the mode proper to fol in the fundamental bafs has this happy effect, that. the notes fa, fol, la, f, may immediately fucceed. each other in afcending the fcale, which cannot take place (art. 48.) in the diatonic series of the Greeks, because that feries is formed from the mode of ut alone. From whence it follows: I. That we change the mode at every time when we modulate three notes in fucceffion. 2. That if thefe three notes are fung in fucceffion in the feale ut, re, mi, fa, fol, la, fi, UT, this cannot be done but by the affiftance of a paufe expreffed or underftood after the note fa; infomuch, that the three tones fa, fol, la, fi, (three only because the note fol which is repeated is not enumerated) are fuppofed to belong to to different tetrachords.

60. It ought not then to furprife us, that we feel fome difficulty whilst we afcend the feale in finging-three tones in fucceffion, because this is impracticable without changing the mode; and if one paufes in the fame mode, the fourth found above the firft note will never be higher than a femitone above that which immediately precedes it; as may be seen by ut, re, mi, fa, and by fo!, la, fi, ut, where there is no more than a femitone between mi and fa, and between fi and ut.

61. We may likewife obferve in the feale ut, re, mi, fu, that the third minor from re to fa is not true, for the reafons which have been already given (art. 49.) It is the fame cafe with the third minor from la to ut, and with the third major from fa to la: but each of these founds form otherwife

com

confonances perfe&ly true, with their correfpondent founds in the fundamental bafs.

62. The thirds la ut, fa la, which were true in the former fcale, are falfe in this; because in the former fcale la was the third of fa, and here it is the fifth of re, which corresponds with it in the fundamental bass...

63. Thus it appears, that the scale of the Greeks contains fewer confonances that are altered than ours; and this like wife happens from the introduction of the mode of fol into the fundamental bafs. We fee likewise that the value of la in the diatonic fcale, a value which authors have been divided in afcertaining, folely depends upon the fundamental bafs, and that it must be different according as the note la has fa or re for its bass.

CHAP. VII. Of TEMPERAMENT.

64. THE alterations, which we have obferved in the intervals between particular founds of the diatonic scale, naturally lead us to speak of temperament, To give a clear idea of this, and to render the neceffity of it palpable, let us fuppofe that we have before us an inftrument with keys, a harptichord, for inftance, confting of feveral octaves or scales, of which each includes its twelve femitones. Let us choofe in that harpsichord one of the ftrings which will found the note UT, and let us tune the string SOL to a perfect fifth with UT in afcending; let us afterwards tune to a perfect fifth with this SOL the RE which is above at; we shall evidently perceive that this RE will be in the fcale above that from which we fet out: but it is alfo evident that this RE muft have in the fcale a re which correfponds with it, and which must be tuned a true octave below RE; and between this and SOL there fhould be the interval of a fifth; fo that the rein the firit fcale will be a true fourth below the SOL of the fame fcale. We may afterwards tune the note LA of the firft fcale to a juft fifth with this laft re; then the note MI in the higheft fcale to a true fifth with this new LA, and of confequence the mi in the first scale to a true fourth beneath this fame LA: Having finished this operation, it will be found that the laft mi, thus tuned, will by no means form a just third major from the found UT: that is to fay, that it is impoffible for mi to conftitute at the fame time the third major of UT and the true fifth of LA; or, what is the fame thing, the true fourth of LA in defcending.

65. What is ftill more, if, after having fucceffively and alternately tuned the ftrings UT, SOL, re, LA, mi, in perfect fifths and fourths one from the other, we continue to tune fucceffively by true fifths and fourths the ftrings mi, fi, fat, ut, folk, re, mi, fik; we fhall find, that, though, being a femitone higher than the natural note, should be equivalent to UT natural, it will by no means form a juft octave to the first ut in the fcale, but be confiderably higher; yet this upon the harpsichord ought not to be different from the octave above UT; for every J and every UT is the fame found, fince the octave or the Icale only confifts of twelve femitones. 66. From thence it neceffarily follows, 1. That it is impoflible that all the octaves and all the Miths fhould be just at the fame time, particularly

in inftruments which have keys, where no inter vals lefs than a femitone are admitted. 2. That, of confequence, if the fifths are juftly tuned, fome alteration mult be made in the octaves; now the fympathy or found which fubfifts between any note and its octave, does not permit us to make fuch an alteration: this perfect coalefcence of found is the caufe why the octave should serve as limits to the other intervals, and that all the notes which rife above or fall below the ordinary feale, are no more than replications, i. e. repeti tions, of all that have gone before them. For this reafon, if the octave were altered, there could be no longer any fixed point either in barmony or melody. It is then abfolutely neceffary to tune,the at or fi in a juft octave with the first; whence it follows, that, in the progreffion of fiths, or what is the fame thing, in the alternate feries of fifths and fourths, UT, SOL, re, LA, mi, fi, fakt, ut, folk, rek, lak, mik, jik, it is necellary that all the fifths, fhould be altered, or at least fome of them. Now, fince there is no reafon why one should rather be altered than ano. ther, it follows, that we ought to alter them all equally. By thefe means, as the alteration is made to influence all the fifths, it will be in each of them almoft imperceptible; and thus the fifth, which, after the octave, is the most perfect of all confonances, and which we are under the necef fity of altering, muft only be altered in the leaft degree poffible.

67. It is true, that the thirds will be a little harth: but as the interval of founds which conftitutes the third, produces a lefs perfect coalefcence than that of the fifth, it is necefiary, fays M. Rameau, to facrifice the juftice of that chord to the perfection of the fifth; for the more perfect a chord is in its own nature, the more difpleafing to the ear is any alteration which can be made in it. In the octave the least alteration is infupportable.

68. This change in the intervals of inftruments which have, or even which have not, keys, is that which we call temperament.

69. It refults then from all that we have now faid, that the theory of temperament may be reduced to this question. The alternate fucceffioh of fifths and fourths having been given, UT, SOL, re, LA, mi, hi, fa*, ut*, jol☀, re*, la*, mix, X, in which fix, or ut is not the true octave of the firft UT, it is proposed to alter all the fifths equally, in fuch a manner that the two uts may be in a perfect octave the one to the other.

70. For a folution of this queftion, we must begin with turning the two uts in a perfect octave the one to the other; in confequence of which, we will render all the femitones which compofe the octave as equal as poffible. By thefe means the alteration made in each tifth will be very confiderable, but equal in all of them.

71. In this, then, the theory of temperament confifts: but as it would be difficult in practice to tune a harpfichord or organ by thus rendering all the femitones equal, M. Rameau, in his Gene ration Harmonique, has furnished us with the following method, to alter all the fifths as equally as poffible.

72. Take any key of the harpsichord which

you