Theory of and the two other sounds which it produces, and with Harmony, which it is accompanied, are, inclusive of its octave, called its harmonics §.

Generator, what.

§ See Harmonics.

Experiment II.

22. There is no person insensible of the resemblance which subsists between any sound and its octave, whether above or below. These two sounds, when heard together, almost entirely coalesce in the organ of sensation. We may besides be convinced (by two facts which are extremely simple) of the facility with which one of these sounds may be taken for the other.

Let it be supposed that any person has an inclination to sing a tune, and having at first begun this air upon a pitch too high or too low for his voice, so that he is obliged, lest he should strain himself too much, to sing the tune in question on a key higher or lower than the first; I affirm, that, without being initiated in the art of music, he will naturally take his new key in the octave below or the octave above the first; and that in order to take this key in any other interval except the octave, he will find it necessary to exert a sensible degree of attention. This is a fact of which we may easily be persuaded by experience.

Another fact. Let any person sing a tune in our presence, and let it be sung in a tone too high or too low for our voice; if we wish to join in singing this air, we naturally take the octave below or above, and frequently, in taking this octave, we imagine it to be the unison (d).

Chap. II. The Origin of the Modes Major and Minor; of the most natural Modulation, and the most perfect Harmony.

23. To render our ideas still more precise and permanent, we shall call the tone produced by the fundamental and rous body ut; it is evident, by the first experiment, harmonics, that this sound is always attended by its 12th and 17th major; that is to say, with the octave of sol, and the double octave of mi.

24. This octave of sol then, and this double octave of mi, produce the most perfect chord which can be joined with ut, since that chord is the work and choice of nature (e).

25. For the same reason, the modulation formed by Harmony ut with the octave of sol and the double octave of mi, reduced to four one after the other, would likewise be the most simple and natural of all modulations which do not descend or ascend directly in the diatonic order, if our voices had sufficient compass to form intervals so great without difficulty; but the ease and freedom with which

Thus the third minor, an harmonic sound which is even found in the protracted and coalescent tones of a sonorous body between the sound mi and sol, an harmonic of the principal sound, may be expressed by the fraction $\frac{6}{5}$.

N.B. One may see by this example, that in order to compare two sounds one with another which are expressed by fractions, it is necessary first to multiply the numerator of the fraction which expresses the first by the denominator of the fraction which expresses the second, which will give a primary number; as here the numerator 5 of the fraction $\frac{6}{5}$, multiplied by 2 of the fraction $\frac{3}{2}$, has given 10. Afterwards may be multiplied the numerator of the second fraction by the denominator of the first, which will give a secondary number, as here 12 is the product of 4 multiplied by 3; and the relation between these two numbers (which in the preceding example are 10 and 12), will express the relation between these sounds, or, what is the same thing, the interval which there is between the one and the other; in such a manner, that the farther the relation between these sounds departs from unity, the greater the interval will be.

Such is the manner in which we may compare two sounds one with another whose numerical value is known. We shall now show the manner how the numerical expression of a sound may be obtained, when the relation which it ought to have with another sound is known whose numerical expression is given.

Let us suppose, for example, that the third major of the fifth $\frac{3}{2}$ is sought. That third major ought to be, by what has been shown above, the $\frac{6}{5}$ of the fifth; for the third major of any sound whatever is the $\frac{6}{5}$ of that sound. We must then look for a fraction which expresses the $\frac{6}{5}$ of $\frac{3}{2}$; which is done by multiplying the numerators and denominators of both fractions one by the other, from whence results the new fraction $\frac{18}{10}$. It will likewise be found that the fifth of the fifth is $\frac{9}{5}$, because the fifth of the fifth is the $\frac{3}{2}$ of $\frac{3}{2}$.

Thus far we have only treated of fifths, fourths, thirds major and minor, in ascending; now it is extremely easy to find by the same rules the fifths, fourths, thirds major and minor in descending. For suppose ut equal to 1, we have seen that its fifth, its fourth, its third, its major and minor in ascending, are $\frac{3}{2}$, $\frac{4}{3}$, $\frac{5}{4}$, $\frac{6}{5}$, $\frac{7}{6}$. To find its fifth, its fourth, its third, its major and minor in descending, nothing more is necessary than to reverse these fractions, which will give $\frac{5}{6}$, $\frac{6}{7}$, $\frac{7}{6}$, $\frac{6}{5}$.

(d) It is not then imagined that we change the value of a sound in multiplying or dividing it by 2, by 4, or by 8, &c. the number which expresses these sounds, since by these operations we do nothing but take the simple, double, or triple octave, &c. of the sound in question, and that a sound coalesces with its octave.

(e) The chord formed with the twelfth and seventeenth major united with the principal sound, being exactly conformed to that which is produced by nature, is likewise for that reason the most agreeable of all; especially when the composer can proportion the voices and instruments together in a proper manner to give this chord its full effect. M. Rameau has executed this with the greatest success in the opera of Pygmalion, page 34, where Pygmalion sings with the chorus, L'amour triomphe, &c.; in this passage of the chorus, the two parts of the vocal and instrumental basses give the principal sound and its octave; the first part above, or treble, and that of the counter-tenor, produce the seventeenth major, and its octave, in descending; and in short, the second part, or tenor above, gives the twelfth... Theory of which we can substitute its octave to any sound, when Harmony, it is more convenient for the voice, afford us the means of representing this modulation.

26. It is on this account that, after having sung the tone ut, we naturally modulate the third mi, and the fifth sol, instead of the double octave of mi, and the octave of sol; from whence we form, by joining the octave of the sound ut, this modulation, ut, mi, sol, ut, which in effect is the simplest and easiest of them all; and which likewise has its origin even in the protracted and compounded tones produced by a sonorous body.

27. The modulation ut, mi, sol, ut, in which the chord ut, mi, is a third major, constitutes that kind of harmony or melody which we call the mode major; from whence it follows, that this mode results from the immediate operation of nature.

28. In the modulation ut, mi, sol, of which we have now been treating, the sounds mi and sol are so proportioned one to the other, that the principal found ut (art. 19.) causes both of them to resound; but the second tone mi does not cause sol to resound, which only forms the interval of a third minor.

29. Let us then imagine, that, instead of this found mi, one should substitute between the founds ut and another note which (as well as the found ut) has the power of causing sol to resound, and which is, however, different from the found ut; the found which we explore ought to be such, by art. 19. that it may have for its 17th major sol, or one of the octaves of sol; of consequence the found which we seek ought to be a 17th major below sol, or, what is the same thing, a third major below the same sol. Now the found mi being a third minor beneath sol, and the third major being (art. 9.) greater by a semitone than the third minor, it follows, that the found of which we are in search shall be a semitone beneath the natural mi, and of consequence mi b.

30. This new arrangement, ut, mi b, sol, in which the founds ut and mi b have both the power of causing sol to resound, though ut does not cause mi b to resound, is not indeed equally perfect with the first arrangement ut, mi, sol; because in this the two founds mi and sol are both the one and the other generated by the principal found ut; whereas, in the other, the found mi b is not generated by the found ut; but this arrangement ut, mi b, sol, is likewise dictated by nature (art. 19.), though less immediately than the former.

(f) The origin which we have here given of the mode minor, is the most simple and natural that can possibly be given. In the first edition of this treatise, I had followed M. Rameau in deducing it from the following experiment.—If you put in vibration a musical string AB, and if there are at the same time contiguous strings CF, LM, of which the first shall be a twelfth below the string AB, and the second LM a seventeenth major below the same AB, the strings CF, LM, will vibrate without being struck as soon as the string AB shall give a sound, and divide themselves by a kind of undulation, the first into three, the last into five equal parts; in such a manner, that, in the vibration of the string CF, you may easily distinguish two points at rest D, E, and in the tremulous motion of the string LM four accentuated points N, O, P, Q, all placed at equal distances from each other, and dividing the strings into three or five equal parts. In this experiment, says M. Rameau, if we represent by ut the tone of the string AB, the two other strings will represent the sounds fa and la b; and from thence M. Rameau deduces the modulation fa, la b, ut, and of consequence the mode minor. The origin which we have assigned to the minor mode in this new edition, appears to me more direct and more simple, because it presupposes no other experiment than that of art. 19., and because also the fundamental found ut is still retained in both the modes, without being obliged, as M. Rameau found himself, to change it into fa. Part I.

Theory of pafs from one found to another which is not immediately contiguous to it; for instance, from ut to re, or from re to ut: for this very simple reason, that the found re is not contained in the found ut, nor the found ut in that of re; and thus these founds have not any alliance the one with the other, which may authorize the transition from one to the other.

37. And as these founds ut and re, by the first experiment, naturally bring along with them the perfect chords consisting of greater intervals ut, mi, sol, ut, re, fa, la, re; hence may be deduced this rule, That two perfect chords, especially if they are major (g), cannot succeed one another diatonically in a fundamental base; we mean, that in a fundamental base two founds cannot be diatonically placed in succession, each of which, with its harmonics, forms a perfect chord, especially if this perfect chord be major in both.

Chap. IV. Of Modes in general.

38. A mode, in music, is nothing else but the order of founds prescribed, as well in harmony as melody, by the series of fifths. Thus the three founds fa, ut, sol, and the harmonics of each of these three founds, that is to say, their thirds major and their fifths, compose all the major modes which are proper to ut.

39. The series of fifths then, or the fundamental base fa, ut, sol, of which ut holds the middle space, may be regarded as representing the mode of ut. One may likewise take the series of fifths, or fundamental base, ut, sol, re, as representing the mode of sol; in the same manner, fa, la, ut, will represent the mode of fa.

By this we may see, that the mode of sol, or rather the fundamental base of that mode, has two founds in common with the fundamental base of the mode of ut. It is the same with the fundamental base of the mode fa.

40. The mode of ut (fa, ut, sol) is called the principal mode with respect to the modes of these two fifths, which are called its two adjuncts.

41. It is then, in some measure, indifferent to the ear whether a transition be made to the one or to the other of these adjuncts, since each of them has equally two founds in common with the principal mode. Yet the mode of sol seems a little more eligible: for sol is heard amongst the harmonics of ut, and of consequence is implied and signified by ut; whereas ut does not cause fa to be heard, though ut is included in the same found fa. It is hence that the ear, affected by the mode of ut, is a little more predisposed for the mode of sol than for that of fa. Nothing likewise is more frequent, nor more natural, than to pass from the mode of ut to that of sol.

42. It is for this reason, as well as to distinguish the two fifths one from the other, that we call the fifth above the generator the dominant found, and the fifth fa beneath the generator the subdominant.

43. It remains to add, as we have seen in the preceding chapter, that, in the series of fifths, we may indifferently pass from one found to that which is contiguous: In the same manner, and for the same reason, our founds, one may pass from the mode of sol to the mode of re, how to be after having made a transition from the mode of ut managed. The mode of sol, as from the mode of fa to the mode of fa. But it is necessary, however, to observe, that the ear which has been immediately affected with the principal mode feels always a strong propensity to return to it. Thus the further the mode to which we make a transition is removed from the principal mode, the less time we ought to dwell upon it; or rather, to speak in the terms of the art, the less ought the phrase (†aa) of that mode to be protracted.

Chap. V. Of the Formation of the Diatonic Scale as used by the Greeks.

44. From this rule, that two founds which are contiguous may be placed in immediate succession in the series of fifths, fa, ut, sol, it follows, that one may form this modulation, or this fundamental base, by fifths,

sol, ut, sol, ut fa, ut, fa.

45. Each of the founds which forms this modulation brings necessarily along with itself its third major, its fifth, and its octave; insomuch that he who, for instance, sings the note sol, may be reckoned to sing at the same time the notes sol, fa, re, sol; in the same manner the found ut in the fundamental base brings by the fundamental base, fa, along with it this modulation, ut, mi, sol, ut; and, in short, the same found fa brings along with it fa, la, ut, fa. This modulation then, or this fundamental base,

sol, ut, sol, ut fa, ut, fa,

gives the following diatonic series,

fa, ut, re, mi, fa, sol, la;

which is precisely the diatonic scale of the Greeks. We are ignorant upon what principles they had formed this scale; but it may be sensibly perceived, that that series arises from the base sol, ut, sol, ut, fa, ut, fa; and that of consequence this base is justly called fundamental, as being the real primitive modulation, that which

(g) I say especially if they are major; for in the major chord re, fa, la, re, besides that the founds ut and re have no common harmonical relation, and are even dissonant between themselves (Art. 18.), it will likewise be found, that fa forms a dissonance with ut. The minor chord re, fa, la, re, would be more tolerable, because the natural fa which occurs in this chord carries along with it its fifth ut, or rather the octave of that fifth: It has likewise been sometimes the practice of composers, though rather by a licence indulged them than strictly agreeable to their art, to place a minor in diatonic succession to a major chord.

(†aa) As the mere English reader, unacquainted with the technical phraseology of music, may be surprised at the use of the word phrase when transferred from language to that art, we have thought proper to insert the definition of Rouffleau.

A phrase, according to him, is in melody a series of modulations, or in harmony a succession of chords, which form without interruption a sense more or less complete, and which terminate in a repose by a cadence more or less perfect. conducts the ear, and which it feels to be implied in the diatonic modulation, \( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \). (n). 46. We shall be still more convinced of this truth by the following remarks.

In the modulation \( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \), the found re and fa form between themselves a third mi- nor, which is not so perfectly true as that between mi and sol (i). Nevertheless, this alteration in the third minor between re and fa gives the ear no pain, be- cause that re and that fa, which do not form between themselves a true third minor, form, each in particu- lar, consonances perfectly just with the founds in the fundamental bas which correspond with them: for re in the scale is the true fifth of sol, which answers to it in the fundamental bas; and fa in the scale is the true octave of fa, which answers to it in the same bas.

47. If, therefore, these founds in the scale form con- sonances perfectly true with the notes which correspond to them in the fundamental bas, the ear gives itself little trouble to investigate the alterations which there may be in the intervals which these founds in the scale form between themselves. This is a new proof that the fundamental bas is the genuine guide of the ear, and the true origin of the diatonic scale.

48. Moreover, this diatonic scale includes only sev- en founds, and goes no higher than fa, which would be the octave of the first: a new singularity, for which a reason may be given by the principles above establish- ed. In reality, in order that the found fa may succeed immediately in the scale to the found la, it is necessary that the note sol, which is the only one from whence fa as a harmonic may be deduced, should immediately succeed to the found fa, in the fundamental bas, which is the only one from whence la can be harmoni- cally deduced. Now, the diatonic succession from fa to sol cannot be admitted in the fundamental bas, ac- cording to what we have remarked (art. 36.). The founds la and fa, then, cannot immediately succeed one another in the scale: we shall see in the sequel why this is not the case in the series ut, re, mi, fa, sol, la, fa, UT, which begins upon ut; whereas the scale in question here begins upon fa.

49. The Greeks likewise, to form an entire octave, Com- pleted below the first fa the note la, which they di- stinguished and separated from the rest of the scale, tave, and which for that reason they called proflambano. See Pro- mena, that is to say, a string or note subadded to the flambano; scale, and put before fa to form the entire octave.

50. The diatonic scale \( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \) is composed of two tetrachords, that is to say, of two com- posed diatonic scales, each consisting of four founds, \( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \). These two tetrachords are exactly similar; for from mi to fa there is the same interval as from fa to ut, from fa to sol the same as from ut to re, from sol to la the same as from re to mi

(n) Nothing is easier than to find in this scale the value or proportions of each found with relation to the found ut, which we call 1; for the two founds sol and fa in the bas are \( \frac{2}{3} \) and \( \frac{3}{4} \); from whence it follows,

1. That ut in the scale is the octave of ut in the bas; that is to say, 2. 2. That fa is the third major of sol; that is to say \( \frac{3}{4} \) of \( \frac{4}{3} \) (note c), and of consequence \( \frac{1}{8} \). 3. That re is the fifth of sol; that is to say \( \frac{1}{4} \) of \( \frac{4}{3} \), and of consequence \( \frac{3}{8} \). 4. That mi is the third major of the octave of ut, and of consequence the double of \( \frac{3}{4} \); that is to say, \( \frac{3}{4} \). 5. That fa is the double octave of fa of the bas, and consequently 3. 6. That sol of the scale is the octave of sol of the bas, and consequently 3. 7. In short, that la in the scale is the third major of fa of the scale; that is to say, \( \frac{3}{4} \) of \( \frac{4}{3} \), or \( \frac{1}{9} \).

Hence then will result the following table, in which each found has its numerical value above or below it.

| Diatonic Scale | Fundamental Bas | |----------------|----------------| | \( \frac{1}{2} \) | \( \frac{1}{2} \) | | \( \frac{2}{3} \) | \( \frac{2}{3} \) | | \( \frac{3}{4} \) | \( \frac{3}{4} \) | | \( \frac{4}{5} \) | \( \frac{4}{5} \) |

And if, for the convenience of calculation, we choose to call the found ut, of the scale 1; in this case there is nothing to do but to divide each of the numbers by 2, which represent the diatonic scale, and we shall have

\( \frac{1}{2} \) \( \frac{1}{2} \) \( \frac{1}{2} \) \( \frac{1}{2} \) \( \frac{1}{2} \)

\( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \).

(1) In order to compare re with fa, we need only compare \( \frac{3}{4} \) with \( \frac{4}{3} \); the relation between these fractions will be (Note c) that of 9 times 3 to 8 times 4; that is to say, of 27 to 32; the third minor, then, from re to fa, is not true; because the proportion of 27 to 32 is not the same with that of 5 to 6, these two proportions being between themselves as 27 times 6 is to 32 times 5, that is to say, as 162 to 160, or as the halves of these two numbers, that is to say, as 81 to 80.

M. Rameau, when he published, in 1726, his New theoretical and practical System of Music, had not as yet found the true reason of the alteration in the consonance which is between re and fa, and of the little attention which the ear pays to it. For he pretends, in the work now quoted, that there are two thirds minor, one in the proportion of 5 to 6, the other in the proportion of 27 to 32. But the opinion which he has afterwards adopted, seems much preferable. In reality, the genuine third minor, is that which is produced by nature between mi and sol, in the continued tone of those famous bodies of which mi and sol are the two harmonics; and that third minor, which is in the proportion of 5 to 6, is likewise that which takes place in the minor mode, and not that third minor which is false and different, being in the proportion of 27 to 32. Part I.

Theory of mi (L); this is the reason why the Greeks distinguished these two tetrachords; yet they joined them by the note mi, which is common to both, and which gave them the name of conjunctive tetrachords.

51. Moreover, the intervals between any two sounds taken in each tetrachord in particular, are precisely true; thus, in the first tetrachord, the intervals of ut mi, and fa re, are thirds, the one major and the other minor, exactly true, as well as the fourth fa mi (m); it is the same thing with the tetrachord mi, fa, sol, la, since this tetrachord is exactly like the former.

52. But the case is not the same when we compare two sounds 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 so; now, in order to form a true fifth, a third major and a third minor, which are both exactly true, are necessary.

53. From thence it follows, that every consonance is absolutely perfect in each tetrachord taken by itself; but that there is some alteration in passing from one tetrachord to the other. This is a new reason for distinguishing the scale into these two tetrachords.

54. It may be ascertained by calculation, that in the tetrachord fa, 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 (n). In the same manner, in the second tetrachord mi, fa, sol, la, which is, as we have proved, perfectly similar to the first, the note from sol to la is a little less than the note from fa to sol. It is for this reason that they distinguish two kinds of tones; the greater tone*, as from ut to re, from fa to sol† Greater &c.; and the lesser ‡, as from re to mi, from sol to la, &c.

CHAP. VI. The formation of the Diatonic Scale among the Moderns, or the ordinary Gammut.

55. We have just shown in the preceding chapter, how the scale of the Greeks is formed, fa, ut, re, mi, sol, la, by means of a fundamental base composed of three founds only, fa, ut, sol: but to form the scale ut, re, mi, fa, sol, la, fi, UT, which we use at present, we must necessarily add to the fundamental base the note re, and form, with these four founds fa, ut, sol, re, the following fundamental base:

ut, sol, ut, fa, ut, sol, re, sol, ut;

from whence we deduce the modulation or scale ut, re, mi, fa, sol, la, fi, UT.

In effect (o), ut in the scale belongs to the harmony of ut which corresponds with it in the base; re, which is the second note in the gammut, is included in the harmony of sol, the second note of the base; mi, the third note of the gammut, is a natural harmonic of ut, which is the third found in the base, &c.

56. From thence it follows, that the diatonic scale of the Greeks is, at least in some respects, more simple than ours; since the scale of the Greeks (chap. v.) may be formed alone from the mode proper to ut; whereas ours, and ours is originally and primitively formed, not only from why, the mode of ut (fa, ut, sol), but likewise from the mode of sol, (ut, sol, re).

It will likewise appear, that this last scale consists of two parts; of which the one, ut, re, mi, fa, sol, is in

---

(L) The proportion of fa to ut is as \( \frac{4}{5} \) to \( \frac{1}{2} \), that is to say as 15 to 16; that between mi and fa is as \( \frac{4}{5} \) to \( \frac{1}{2} \), that is to say (note c), as 5 times 3 to 4 times 4, or as 15 to 16; these two proportions then are equal. In the same manner, the proportion of ut to re is as 1 to \( \frac{3}{4} \), or as 8 to 9; that between fa and sol is as \( \frac{4}{5} \) to \( \frac{1}{2} \), that is to say (note c), as 8 to 9. The proportion of mi to ut is as \( \frac{4}{5} \) to \( \frac{1}{2} \), or as 5 to 4; that between fa and la is as \( \frac{4}{5} \) to \( \frac{1}{2} \), or as 5 to 4; the proportions here then are likewise equal.

(M) The proportion of mi to ut is as \( \frac{4}{5} \) to \( \frac{1}{2} \), or as 5 to 4, which is a true third major; that from re to fa is as \( \frac{4}{5} \) to \( \frac{1}{2} \); that is to say, as 9 times 16 to 15 times 8, or as 9 times 2 to 15, or as 6 to 5. In like manner, we shall find, that the proportion of mi to fa is as \( \frac{4}{5} \) to \( \frac{1}{2} \); that is to say, as 5 times 16 to 15 times 4, or as 4 to 3, which is a true fourth.

(N) The proportion of re to ut is as \( \frac{4}{5} \) to \( \frac{1}{2} \), or as 9 to 8; that of mi to re is as \( \frac{4}{5} \) to \( \frac{1}{2} \), that is to say, as 40 to 36, or as 10 to 9; now \( \frac{4}{5} \) is less removed from unity than \( \frac{4}{5} \); the interval then from re to mi is a little less than that from ut to re.

If any one would wish to know the proportion which \( \frac{4}{5} \) bear to \( \frac{1}{2} \), he will find (note c) that it is as 8 times 10 to 9 times 9, that is to say, as 80 to 81. Thus the proportion of a lesser to a greater tone is as 80 to 81; this difference between the greater and lesser tone is what the Greeks called a comma.

We may remark, that this difference of a comma is found between the third minor when true and harmonical, and the same chord when it suffers alteration re fa, of which we have taken notice in the scale (note i); for we have seen, that this third minor thus altered is in the proportion of 80 to 81 with the true third minor.

(O) The values or estimates of the notes shall be the same in this as in the former scale, excepting only the tone la; for re being represented by \( \frac{4}{5} \), its fifth will be expressed by \( \frac{3}{5} \); so that the scale will be numerically signified thus:

\[ \text{ut, re, mi, fa, sol, la, fi, UT}. \]

Where you may see, that the note la of this scale is different from that in the scale of the Greeks; and that the la in the modern series stands in proportion to that of the Greeks as \( \frac{3}{5} \) to \( \frac{1}{2} \), that is to say, as 81 to 80; these two la's then likewise differ by a comma. Theory of the mode of ut; and the other, sol, la, si, ut, in that Harmony.

57. It is for this reason that the note sol is found to be twice repeated in immediate succession in this scale; once as the fifth of ut, which corresponds with it in the fundamental bas; and again, as the octave of sol, which immediately follows ut in the same bas. As to what remains, these two consecutive sols are otherwise in perfect union. It is for this reason that we are satisfied with fingering only one of them when one modulates the scale ut, re, mi, fa, sol, la, si, UT; but this does not prevent us from employing a pause or repose, expressed or understood, after the sound fa. There is no person who does not perceive this whilst he himself sings the scale.

58. The scale of the moderns, then, may be considered as consisting of two tetrachords, disjunctive indeed, but perfectly similar one to the other, ut, re, mi, fa, and sol, la, si, ut, one in the mode of ut, the other in that of sol. For what remains, we shall find in the sequel by what artifice one may cause the scale ut, re, mi, fa, sol, la, si, UT, to be regarded as belonging to the mode of ut alone. For this purpose it is necessary to make some changes in the fundamental bas, which we have already assigned; but this shall be explained at large in chap. xiii.

59. The introduction of the mode proper to sol in the fundamental bas has this happy effect, that the notes fa, sol, la, si, may immediately succeed each other in ascending the scale, which cannot take place (art. 48.) in the diatonic series of the Greeks, because that series is formed from the mode of ut alone. From whence it follows:

1. That we change the mode at every time when we modulate three notes in succession.

2. That if these three notes are sung in succession in the scale ut, re, mi, fa, sol, la, si, UT, this cannot be done but by the assistance of a pause expressed or understood after the note fa; infomuch, that the three tones fa, sol, la, si, (three only because the note sol which is repeated is not enumerated) are supposed to belong to two different tetrachords.

60. It ought not then any longer to surprise us, that we feel some difficulty whilst we ascend the scale in fingering three tones in succession, because this is impracticable without changing the mode; and if one change of pausae in the same mode, the fourth found above the mode the first note will never be higher than a semitone above that which immediately precedes it; as may be seen by fingering ut, re, mi, fa, and by sol, la, si, ut, where there is no more than a semitone between mi and fa, and between si and ut.

61. We may likewise observe in the scale ut, re, mi, fa, that the third minor from re to fa is not true, for intervals, the reasons which have been already given (art. 49.), though it is the same case with the third minor from la to ut, and with the third major from fa to la: but each of these forms otherwise consonances perfectly true, with their correspondent founds in the fundamental bas.

62. The thirds la ut, fa la, which were true in the fundamental scale la was the third of fa, and here it is the fifth of re, which corresponds with it in the fundamental bas.

63. Thus it appears, that the scale of the Greeks contains fewer consonances that are altered than ours (p); and this likewise happens from the introduction of the mode of sol into the fundamental bas (q).

We see likewise that the value of la in the diatonic scale, a value which authors have been divided in ascertaining, solely depends upon the fundamental bas, and that it must be different according as the note la has fa or re for its bas. See the note (o).

Chap. VII. Of Temperament.

64. The alterations which we have observed in the intervals between particular founds of the diatonic scale, naturally lead us to speak of temperament. To give why need a clear idea of this, and to render the necessity of it far palpable, let us suppose that we have before us an instrument with keys, a harpsichord, for instance, consisting of several octaves or scales, of which each includes its twelve semitones.

Let

(p) In the scale of the Greeks, the note la being a third from fa, there is an altered fifth between la and re; but in ours, la being a fifth to re, produces two altered thirds, fa la, and la ut; and likewise a fifth altered, la mi, as we shall see in the following chapter. Thus there are in our scale two intervals more than in the scale of the Greeks which suffer alteration.

(q) But here it may be with some colour objected: The scale of the Greeks, it may be said, has a fundamental bas more simple than ours; and besides, in it there are fewer chords which will not be found exactly true: why then, notwithstanding this, does ours appear more easy to be sung than that of the Greeks? The Grecian scale begins with a semitone, whereas the intonation prompted by nature seems to impel us to rise by a full tone at once. This objection may be thus answered. The scale of the Greeks is indeed better disposed than ours for the simplicity of the bas, but the arrangement of ours is more suitable to natural intonation. Our scale commences by the fundamental found ut, and it is in reality from that found that we ought to begin; it is from this that all the others naturally arise, and upon this that they depend; nay, if I may speak so, in this they are included: on the contrary, neither the scale of the Greeks, nor its fundamental bas, commences with ut; but it is from this that we must depart, in order to regulate our intonation, whether in rising or descending: now, in ascending from ut, the intonation, even of the Greek scale, gives the series ut, re, mi, fa, sol, la: and so true is it that the fundamental found ut is here the genuine guide of the ear, that if, before we modulate the found ut, we should attempt to rise to it by that note in the scale which is most immediately contiguous, we cannot reach it but by the note fa, and by the semitone from fa to ut. Now to make a transition from fa to ut, by this semitone, the ear must of necessity be predisposed for that modulation, and consequently preoccupied with the mode of ut: if this were not the case, we should naturally rise from fa to ut, and by this operation pass into another mode. Let us choose in that harpsichord one of the strings which will sound the note UT, and let us tune the string SOL to a perfect fifth with UT in ascending; let us afterwards tune to a perfect fifth with this SOL the RE which is above it; we shall evidently perceive that this RE will be in the scale above that from which we set out: but it is also evident that this RE must have in the scale a re which corresponds with it, and which must be tuned a true octave below RE; and between this and SOL there should be the interval of a fifth; so that the re in the first scale will be a true fourth below the SOL of the same scale. We may afterwards tune the note I.A of the first scale to a just fifth with this last re; then the note MI in the highest scale to a true fifth with this new LA, and of consequence the mi in the first scale to a true fourth beneath this same I.A. Having finished this operation, it will be found that the last mi, thus tuned, will by no means form a just third major from the found UT (r); that is to say, that it is impossible for mi to constitute at the same time the third major of UT and the true fifth of LA; or, what is the same thing, the true fourth of LA in descending.

(r) The LA considered as the fifth of re is \( \frac{3}{4} \), and the fourth beneath this LA will constitute \( \frac{1}{2} \) of \( \frac{3}{4} \), that is to say, \( \frac{3}{8} \); \( \frac{3}{8} \) then shall be the value of mi, considered as a true fourth from LA in descending; now mi, considered as the third major of the found UT, is \( \frac{5}{8} \), or \( \frac{5}{8} \): these two mis then are between themselves in the proportion of \( \frac{8}{1} \) to \( \frac{8}{5} \); thus it is impossible that mi should be at the same time a perfect third major from UT, and a true fourth beneath LA.

(s) In effect, if you thus alternately tune the fifth above, and the fourth below, in the same octave, you may here see what will be the process of your operation.

UT, SOL, a fifth; re a fourth; LA a fifth; mi a fourth; \( f \) a fifth; \( f \) a fourth; \( f \) a fifth; \( f \) a fourth; \( R \) a fifth; \( R \) a fourth; \( R \) a fifth; \( R \) a fourth: now it will be found, by a very easy computation, that the first UT being represented by 1, SOL shall be \( \frac{3}{4} \), re \( \frac{3}{8} \), LA \( \frac{3}{8} \), mi \( \frac{3}{8} \), &c., and so of the rest till you arrive at \( f \), which will be found \( \frac{3}{8} \). This fraction is evidently greater than the number 2, which expresses the perfect octave up to its correspondent UT; and the octave below \( f \) would be one half of the same fraction, that is to say \( \frac{3}{8} \), which is evidently greater than UT represented by unity. This last fraction \( \frac{3}{8} \) is composed of two numbers; the numerator of the fraction is nothing else but the number 3 multiplied 11 times in succession by itself, and the denominator is the number 2 multiplied 18 times in succession by itself. Now it is evident, that this fraction, which expresses the value of \( f \), is not equal to the unity which expresses the value of the found UT; though, upon the harpsichord, \( f \) and UT are identical. This fraction rises above unity by \( \frac{3}{8} \), that is to say, by about \( \frac{3}{8} \); and this difference was called the comma of Pythagoras. It is palpable that this comma is much more considerable than that which we have already mentioned (note n), and which is only \( \frac{3}{8} \).

We have already proved that the series of fifths produces an ut different from \( f \), the series of thirds major gives another still more different. For, let us suppose this series of thirds, ut, mi, \( f \), \( f \), we shall have mi equal to \( \frac{3}{8} \), \( f \) to \( \frac{3}{8} \), and \( f \) to \( \frac{3}{8} \), whose octave below is \( \frac{3}{8} \); from whence it appears, that this last \( f \) is less than unity (that is to say, than \( \frac{3}{8} \), by \( \frac{3}{8} \), or by \( \frac{3}{8} \)) or near it: A new comma, much greater than the preceding, and which the Greeks have called apotome major.

It may be observed, that this \( f \), deduced from the series of thirds, is to the \( f \) deduced from the series of fifths, as \( \frac{3}{8} \) is to \( \frac{3}{8} \); that is to say, in multiplying by 524288, as 125 multiplied by 4096 is to 531441, or as 51200 to 531441, that is to say, nearly as 26 is to 27: from whence it may be seen, that these two \( f \)'s are very considerably different one from the other, and even sufficiently different to make the ear sensible of it; because the difference consists almost of a minor semitone, whose value, as will afterwards be seen (art. 139.), is \( \frac{3}{8} \).

Moreover, if, after having found the \( f \) equal to \( \frac{3}{8} \), we then tune by fifths and by fourths, \( f \), \( f \), \( f \), \( f \), \( f \), as we have done with respect to the first series of fifths, we find that the \( f \) must be \( \frac{3}{8} \); its difference, then, from unity, or, in other words, from UT, is \( \frac{3}{8} \), that is to say, about \( \frac{3}{8} \); a comma still less than any of the preceding, and which the Greeks have called apotome minor.

In a word, if, after having found mi equal to \( \frac{3}{8} \) in the progression of thirds, we then tune by fifths and fourths mi, \( f \), \( f \), \( f \), \( f \), \( f \), we shall arrive at a new \( f \), which shall be \( \frac{3}{8} \), and which will not differ from unity but by about \( \frac{3}{8} \), which is the last and smallest of all the commas; but it must be observed, that, in this case, the thirds major from mi to \( f \), from \( f \) to \( f \), or \( f \), &c., are extremely false, and greatly altered. Theory of should serve as limits to the other intervals, and that Harmony, all the notes which rise above or fall below the ordinary scale, are no more than replications, i.e., repetitions, of all that have gone before them. For this reason, if the octave were altered, there could be no longer any fixed point either in harmony or melody. It is then absolutely necessary to tune the ut or fi in a just octave with the first; from whence it follows, that, in the progression of fifths, or what is the same thing, in the alternate series of fifths and fourths, UT, SOL, re, LA, mi, fa, sol, la, mi, fi, it is necessary that all the fifths should be altered, or at least some of them. Now, since there is no reason why one should rather be altered than another, it follows, that we ought to alter them all equally. By these means, as the alteration is made to influence all the fifths, it will be in each of them almost imperceptible; and thus the fifth, which, after the octave, is the most perfect of all consonances, and which we are under the necessity of altering, must only be altered in the least degree possible.

67. It is true, that the thirds will be a little harsh; but as the interval of sounds which constitutes the third, produces a less perfect coalescence than that of the fifth, it is necessary, says M. Rameau, to sacrifice the justice of that chord to the perfection of the fifth; for the more perfect a chord is in its own nature, the more displeasing to the ear is any alteration which can be made in it. In the octave the least alteration is insupportable.

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

69. It results then from all that we have now said, that the theory of temperament may be reduced to Principle this question.—The alternate succession of fifths and whence its fourths having been given, UT, SOL, re, LA, mi, theory may fi, fa, sol, la, mi, fi, in which fi is deduced. If ut is not the true octave of the first UT, it is proposed to alter all the fifths equally, in such a manner that the two ut's may be in a perfect octave the one to the other.

70. For a solution of this question, we must begin Practical with tuning the two ut's in a perfect octave the one to directions the other; in consequence of which, we will render all for tempe the semitones which compose the octave as equal as possible. By this means (τ) the alteration made in each fifth will be very considerable, but equal in all of them.

71. In this, then, the theory of temperament con-Rameau's sists; but as it would be difficult in practice to tune a method of harpsichord or organ by thus rendering all the semi-temperament pro- tones equal, M. Rameau, in his Generation Harmonique, has furnished us with the following method, to alter all the fifths as equally as possible.

72. Take any key of the harpsichord which you please; but let it be towards the middle of the instrument; for instance, UT; then tune the note SOL a fifth above it, at first with as much accuracy as possible; this you may imperceptibly diminish; tune afterwards the fifth to this with equal accuracy, and diminish it in the same manner; and thus proceed from one fifth to another

(τ) All the semitones being equal in the temperament proposed by M. Rameau, it follows, that the twelve semitones ut, ut, re, re, mi, mi, &c., shall form a continued geometrical progression; that is to say, a series in which ut shall be to ut in the same proportion as ut to re, as re to re, &c., and so of the rest.

These twelve semitones are formed by a series of thirteen sounds, of which UT and its octave ut are the first and last. Thus to find by computation the value of each sound in the temperament, which is the present object of our speculations, our scrutiny is limited to the investigation of eleven other numbers between 1 and 2 which may form with the 1 and the 2 a continued geometrical progression.

However little any one is practised in calculation, he will easily find each of these numbers, or at least a number approaching to its value. These are the characters by which they may be expressed, which mathematicians will easily understand, and which others may neglect.

| UT | ut | re | re | mi | fa | fa | sol | sol | |----|----|----|----|----|----|----|-----|-----| | 1 | √2 | √2² | √2³ | √2⁴ | √2⁵ | √2⁶ | √2⁷ | √2⁸ |

It is obvious, that in this temperament all the fifths are equally altered. One may likewise prove, that the alteration of each in particular is very inconsiderable; for it will be found, for instance, that the fifth from ut to sol, which should be 3, ought to be diminished by about 1/15 of 3; that is to say, by 1/15, a quantity almost inconceivably small.

It is true, that the thirds major will be a little more altered; for the third major from ut to mi, for instance, shall be increased in its interval by about 1/15; but it is better, according to M. Rameau, that the alteration should fall upon the third than upon the fifth, which after the octave is the most perfect chord, and from the perfection of which we ought never to degenerate but as little as possible.

Besides, it has appeared from the series of thirds major ut, mi, sol, fi, that this last fi is very different from ut (note s); from whence it follows, that if we would tune this fi in unison with the octave of ut, and alter at the same time each of the thirds major by a degree as small as possible, they must all be equally altered. This is what occurred in the temperament which we propose; and if in it the third be more altered than the fifth, it is a consequence of the difference which we find between the degrees of perfection in these intervals; a difference with which, if we may speak so, the temperament proposed conforms itself. Thus this diversity of alteration is rather advantageous than inconvenient. Theory of another in ascent: and as the ear does not appreciate Harmony, to exactly sounds that are extremely sharp, it is necessary, when by fifths you have risen to notes extremely high, that you should tune in the most perfect manner the octave below the last fifth which you had immediately formed; then you may continue always in the same manner; till in this process you arrive at the last fifth from mi to fa, which should of themselves be in tune; that is to say, they ought to be in such a state, that fa, the highest note of the two which compose the fifth, may be identical with the found UT, with which you began, or at least the octave of that found perfectly just: it will be necessary then to try if this UT, or its octave, forms a just fifth with the last found mi or fa which has been already tuned. If this be the case, we may be certain that the harpsichord is properly tuned. But if this last fifth be not true, in this case it will be too sharp, and it is an indication that the other fifths have been too much diminished, or at least some of them; or it will be too flat, and consequently discover that they have not been sufficiently diminished. We must then begin and proceed as formerly, till we find the last fifth in tune of itself, and without our immediate interposition (v).

(v) All that remains, is to acknowledge, with M. Rameau, that this temperament is far remote from that which is now in practice: you may here see in what this last temperament consists as applied to the organ or harpsichord. They begin with UT in the middle of the keys, and they flatten the four first fifths sol, re, la, mi, till they form a true third major from mi to ut; afterwards, setting out from this mi, they tune the fifths fa, ut, sol, but flattening them still less than the former, so that sol may almost form a true third major with mi. When they have arrived at sol, they stop; they resume the first ut, and tune to it the fifth fa in descending, then the fifth fa, &c., and they heighten a little all the fifths till they have arrived at la, which ought to be the same with the sol already tuned.

If, in the temperament commonly practised, some thirds are found to be less altered than in that prescribed by M. Rameau, in return, the fifths in the first temperament are much more false, and many thirds are likewise false; infomuch, that upon a harpsichord tuned according to the temperament in common use, there are five or six modes which the ear cannot endure, and in which it is impossible to execute any thing. On the contrary, in the temperament suggested by M. Rameau, all the modes are equally perfect; which is a new argument in its favour, since the temperament is peculiarly necessary in passing from one mode to another, without shocking the ear; for instance, from the mode of ut to that of sol, from the mode of sol to that of re, &c. It is true, that this uniformity of modulation will to the greatest number of musicians appear a defect; for they imagine, that, by tuning the semitones of the scale unequal, they give each of the modes a peculiar character; so that, according to them, the scale of ut,

ut, re, mi, fa, sol, la, si, UT,

is not perfectly similar to the gammut or diatonic scale of the mode of mi

mi, fa, sol, la, si, re, mi,

which, in their judgment, renders the modes of ut and mi proper for different manners of expression. But after all that we have said in this treatise on the formation of diatonic intervals, every one should be convinced, that, according to the intention of nature, the diatonic scale ought to be perfectly the same in all its modes:

The contrary opinion, says M. Rameau, is a mere prejudice of musicians. The character of an air arises chiefly from the intermixture of the modes; from the greater or lesser degrees of vivacity in the movement; from the tones, more or less grave, or more or less acute, which are affixed to the generator of the mode; and from the chords more or less beautiful, as they are more or less deep, more or less flat, more or less sharp, which are found in it.

In short, the last advantage of this temperament is, that it will be found conformed, or at least very little different from that which they practise upon instruments without keys; as the bass-viol, the violin, in which true fifths and fourths are preferred to thirds and sixths tuned with equal accuracy; a temperament which appears incompatible with that commonly used in tuning the harpsichord.

Yet we must not suffer our readers to be ignorant, that M. Rameau, in his New System of Music, printed in 1726, had adopted the ordinary temperament. In that work, (as may be seen Chap. XXIV.), he pretends that the alteration of the fifths is much more supportable than that of the thirds major; and that this last interval can hardly suffer a greater alteration than the octave, which, as we know, cannot suffer the slightest alteration. He says, that if three strings are tuned, one by an octave, the other by a fifth, and the next by a third major to a fourth string, and if a sound be produced from the last, the strings tuned by a fifth will vibrate, though a little less true than it ought to have been; but that the octave and the third major, if altered in the least degree, will not vibrate: and he adds, that the temperament which is now practised, is founded upon that principle. M. Rameau goes still farther; and as, in the ordinary temperament, there is a necessity for altering the last thirds major, and to make them a little more sharp, that they may naturally return to the octave of the principal sound, he pretends that this alteration is tolerable, not only because it is almost insensible, but because it is found in modulations not much in use, unless the composer should choose it on purpose to render the expression stronger. "For it is proper to remark (says he), that we receive different impressions from the intervals in proportion to their different alterations: for instance, the third major, which naturally elevates us to joy, in proportion as we feel it, heightens our feelings even to a kind of fury, when it is tuned too sharp; and the third minor, which naturally inspires us with tenderness and serenity, depresses us to melancholy when it is too flat." All this strain, as you may see, is immensely different from that which this celebrated musician afterwards By this method all the twelve sounds which compose one of the scales shall be tuned; nothing is necessary but to tune with the greatest possible exactness their octaves in the other scales, and the harpsichord shall be well tuned.

We have given this rule for temperament from M. Rameau; and it belongs only to disinterested artists to judge of it. However this question be determined, and whatever kind of temperament may be received, the alterations which it produces in harmony will be but very small, or not perceptible to the ear, whose attention is entirely engrossed in attuning itself with the fundamental bass, and which suffers, without uneasiness, these alterations, or rather takes no notice of them, because it supplies from itself what may be wanting to the truth and perfection of the intervals.

Simple and daily experiments confirm what we now advance. Listen to a voice which is accompanied, in singing, by different instruments; though the temperament of the voice, and the temperament of each of the instruments, are all different one from another, yet you will not be in the least affected with the kind of cacophony which ought to result from these diversities, because the ear supposes these intervals true of which it does not appreciate differences.

We may give another experiment. Strike upon an organ the three keys mi, sol, fa, you will hear nothing but the minor perfect chord; though mi, by the construction of that instrument, must cause sol likewise to be heard; though sol should have the same effect upon re, and fa upon fa likewise; insomuch, that the ear is at once affected with all these sounds, re, mi, fa, sol, sol, fa: how many dissonances perceived at the same time, and what a jarring multitude of discordant sensations, would result from thence to the ear, if the perfect chord with which it is preoccupied had not power entirely to abstract its attention from such sounds as might offend!

Chap. VIII. Of Reposes or Cadences (†).

In a fundamental bass whose procedure is by fifths, there always is, or always may be, a repose, or crisis, in which the mind acquiesces in its transition from one sound to another; but a repose may be more or less distinctly signified, and of consequence more or less perfect. If one should rise by fifths; if, for instance, we pass from ut to sol; it is the generator which passes to one of these fifths, and this fifth was already pre-existent in its generator; but the generator exists no longer in this fifth; and the ear, as this generator is the principle of all harmony and of all melody, feels a desire to return to it. Thus the transition from a found to its fifth in ascent, is termed an imperfect repose, or imperfect cadence; but the transition from any found to its fifth in descent, is denominated a perfect cadence, or an absolute repose: it is the offspring which returns to its generator, and as it were recovers its existence once more in that generator itself, with which when founding it refounds (chap. i.)

Amongst absolute reposes, there are some, if Perfect cadences we may be allowed the expression, more absolute, that is to say, more perfect, than others. Thus in the fundamental bass

ut, sol, ut, fa, ut, sol, re, sol, ut,

which forms, as we have seen, the diatonic scale of the moderns, there is an absolute repose from re to sol, as from sol to ut: yet this last absolute repose is more perfect than the preceding, because the ear, prepossessed with the mode of ut by the multiplied impression of the found ut which it has already heard thrice before, feels a desire to return to the generator ut; and it accordingly does so by the absolute repose sol, ut.

We may still add, that what is commonly called Cadence in ed cadence in melody, ought not to be confounded with what we name cadence in harmony.

In the first case, this word only signifies an agreeable and rapid alternation between two contiguous sounds, called likewise a trill or shake; in the second, it signifies a repose or close. It is however true, that this shake implies, or at least frequently enough preludes, a repose, either present or impending, in the fundamental bass (x).

Since there is a repose in passing from one sound to another in the fundamental bass, there is also a fundamental bass necessary in the diatonic scale, which is formed from it, and which this bass represents; and as the absolute repose sol ut, is scale, and of which the most perfect.

terwards exhibited in his Generation Harmonique, and in the performances which followed it. From this we can only conclude, that the reasons which, after him, we have urged for the new temperament, must without doubt have appeared to him very strong, because in his mind they had superceded those which he had formerly adduced in favour of the ordinary temperament.

We do not pretend to give any decision for either the one or the other of these methods of temperament, each of which appears to us to have its particular advantages. We shall only remark, that the choice of the one or the other must be left absolutely to the taste and inclination of the reader; without, however, admitting this choice to have any influence upon the principles of the system of music, which we have followed even till this period, and which must always subsist, whatever temperament we adopt.

(†) That the reader may have a clear idea of the term before he enters upon the subject of this chapter, it may be necessary to caution him against a mistake into which he may be too easily led by the ordinary signification of the word repose. In music, therefore, it is far from being synonymous with the word rest. It is, on the contrary, the termination of a musical phrase which ends in a cadence more or less emphatic, as the sentiment implied in the phrase is more or less complete. Thus a repose in music answers the same purpose as punctuation in language. See Repose in Rousseau's Musical Dictionary.

(x) M. Rousseau, in his letter on French music, has called this alternate undulation of different sounds a trill, from the Italian word trillo, which signifies the same thing; and some French musicians already appear to have adopted this expression. Part I.

Theory of all others the most perfect in the fundamental base, Harmony, the repose from β to ut, which answers to it in the scale, and which is likewise terminated by the generator, is for that reason the most perfect of all others in the diatonic scale ascending.

Definition 77. It is then a law dictated by nature itself, that if you would ascend diatonically to the generator of a mode, you can only do this by means of the third major from the fifth of that very generator. This third major, which with the generator forms a semitone, has for that reason been called the sensible note, as introducing the generator, and preparing us for the most perfect repose.

We have already proved, that the fundamental base is the principle of melody. We shall besides make it appear in the sequel, that the effect of a repose in melody arises solely from the fundamental base.

Chap. IX. Of the Minor Mode and its Diatonic Series.

78. In the second chapter, we have explained (art. 29, 30, 31, and 32.) by what means, and upon what the minor principle, the minor chord ut, mi, sol, ut, may be mode after-formed, which is the characteristic chord of the minor mode. Now what we have there said, taking ut for the principal and fundamental sound, we might likewise have said of any other note in the scale, assumed in the same manner as the principal and fundamental sound; but as in the minor chord ut, mi, sol, ut, there occurs a mi which is not found in the ordinary diatonic scale, we shall immediately substitute, for greater ease and convenience, another chord, which is likewise minor and exactly similar to the former, of which all the notes are found in the scale.

79. The scale affords us three chords of this kind, viz. re, fa, la, re; la, ut, mi, la; and mi, sol, fa, mi. Amongst these three we shall choose la, ut, mi, la; because this chord, without including any sharp or flat, has two sounds in common with the major chord ut, mi, sol, ut; and besides, one of these two sounds is the very same ut; so that this chord appears to have the most immediate, and at the same time the most simple, relation with the chord ut, mi, sol, ut. Concerning this we need only add, that this preference of the chord la, ut, mi, la, to every other minor chord, is by no means in itself necessary for what we have to say in this chapter upon the diatonic scale of the minor mode. We might in the same manner have chosen any other minor chord; and it is only, as we have said, for greater ease and convenience that we fix upon this.

80. Let us now remark, that in every mode, whether major or minor, the principal sound which implies the perfect chord, whether major or minor, may be called the tonic note or key; thus ut is the key in its proper mode, la in the mode of la, &c. Having laid down this principle,

81. We have shown how the three sounds fa, ut, sol, which constitute (art. 38.) the mode of ut, of which scale partake, the first fa and the last sol are the two fifths of ut, one descending the other rising, produce the scale fa, ut, re, mi, fa, sol, la, of the major mode, by means of the fundamental base sol, ut, sol, ut, fa, ut, fa; let us in the same manner take the three sounds re, la, mi, which constitute the mode of la, for the same reason that the sounds fa, ut, sol, constitute the mode of ut; and if we let us form this fundamental base, perfectly like the preceding, mi, la, mi, la, re, la, re; let us afterwards place below each of these sounds one of their harmonics, as we have done (chap. v.) for the first scale of the major mode; with this difference, that we must suppose re and la as implying their thirds minor in the fundamental base to characterize the minor mode; and we shall have the diatonic scale of that mode, sol, la, fa, ut, re, mi, fa.

82. The sol, which corresponds with mi in the fundamental base, forms a third major with that mi, though the mode be minor; for the same reason that a third from the fifth of the fundamental sound ought to be major (art. 77.) when that third rises to the fundamental sound la.

83. It is true, that, in causing mi to imply its third major sol, one might also rise to la by a diatonic progression. But that manner of rising to la would be less perfect than the preceding; for this reason (art. 76.), that the absolute repose or perfect cadence, mi, la, which is found in the fundamental base, ought to be represented in the most perfect manner in the two notes of the diatonic scale which answer to it, especially when one of these two notes is la, the key itself upon which the repose is made. From whence it follows, that the preceding note sol ought rather to be sharp than natural; because sol, being included in mi (art. 19.), much more perfectly represents the note mi in the base, than the natural sol could do, which is not included in mi.

84. We may remark this first difference between the scale

sol, la, fa, ut, re, mi, fa,

and the scale which corresponds with it in the major mode

fa, ut, re, mi, fa, sol, la,

that from mi to fa, which are the two last notes of the former scale, there is only a semitone; whereas from sol to la, which are the two last sounds of the latter series, there is the interval of a complete tone: but this is not the only discrimination which may be found between the scales of the two modes.

85. To investigate these differences, and to discover the reason for which they happen, we shall begin by forming a new diatonic scale of the minor mode, similar to the second scale of the major mode,

ut, re, mi, fa, sol, la, fa, ut.

That last series, as we have seen, was formed by means of the fundamental base fa, ut, sol, re, disposed in this manner,

ut, sol, ut, fa, ut, sol, re, sol, ut.

Let us take in the same manner the fundamental base re la mi fa, and arrange it in the following order,

la, mi, la, re, la, mi, fa, mi, la,

and it will produce the scale immediately subjoined,

la, fa, ut, re, mi, mi, fa, sol, la,

in which ut forms a third minor with la, which in the fundamental base corresponds with it, which designates the minor mode; and, on the contrary, sol rises towards la, (art. 82. and 83.) 86. We see besides a fa*, which does not occur in the former,

\[ \text{sol*, la, fi, ut, re, mi, fa} \]

where fa is natural. It is because, in the first scale, fa is a third minor from re in the bals; and in the second, fa* is the fifth from fi in the bals.

87. Thus the two scales of the minor mode are still between them in this respect more different one from the other than the two scales of the major mode; for we do not remark this difference of a semitone between the two scales of the major mode. We have only observed (art. 63.) some difference in the value of a as it stands in each of these scales, but this amounts to much less than a semitone.

88. From thence it may be seen why fa and sol are sharp when ascending in the minor mode; nay, besides, the fa is only natural in the first scale sol*, la, fi, ut, re, mi, fa, because this fa cannot rise to sol*.

(art. 48.)

89. It is not the same case in descending. For mi, the fifth of the generator, ought not to imply the third major fa*, but in the case when that mi descends to the generator la to form a perfect repose (art. 77. and 83.), and in this case the third major sol* rises to the generator la; but the fundamental bals la mi may, in descending, give the scale la sol natural, provided sol does not rise towards la.

90. It is much more difficult to explain how the fa, which ought to follow this sol in descending, is natural and not sharp; for the fundamental bals la, mi, fi, mi, la, re, la, mi, produces in descending,

la, sol, fa*, mi, mi, re, ut, fi, la.

And it is plain that the fa cannot be otherwise than sharp, since fa* is the fifth of the note fi of the fundamental bals. In the mean time, experience evinces that the fa is natural in descending in the diatonic scale of the major mode of la, especially when the preceding sol is natural; and it must be acknowledged, that here the fundamental bals appears in some measure defective.

M. Rameau has invented the following means for obtaining a solution of this difficulty. According to him, in the diatonic scale of the minor mode in descending, la, sol, fa, mi, re, ut, fi, la, sol, may be regarded simply as a note of passage, merely added to give sweetness to the modulation, and as a diatonic gradation, by which we may descend to fa natural.

It is easily perceived, according to M. Rameau, by this fundamental bals,

\[ \text{la, re, la, re, la, mi, la} \]

which produces

\[ \text{la, fa, mi, re, ut, fi, la} \]

which may be regarded, as he says, as the real scale of the minor mode in descending; to which is added sol natural between la and fa, to preserve the diatonic order.

This answer appears the only one which can be given to the difficulty above proposed: but I know not whether it will fully satisfy the reader; whether he will not feel with regret, that the fundamental bals does not produce, to speak properly, the diatonic scale of the minor mode in descent, when at the same time this same bals so happily produces the diatonic scale of that identical mode in ascending, and the diatonic scale of the major mode whether in rising or descending.

Chap. X. Of Relative Modes.

91. Two modes which are of such a nature that we can pass from the one to the other, are called relative modes. Thus we have already seen, that the major mode of ut is relative to the major mode of fi and to the minor mode of sol. It may likewise appear from what goes before, how many intimate connections there are between the species (+) or major mode of ut, and the species or minor mode of la. For, 1. The perfect chords, one major ut mi sol ut, the other minor la ut mi la, which characterize each of those two kinds of modulation or harmony, have two sounds in common, ut or * See Memo. 2. The diatonic scale of the minor mode of la in descent, absolutely contains the same sounds with the gammut or diatonic scale of the major mode of ut.

It is for this reason that the transition is so natural and easy from the major mode of ut to the minor mode of la, or from the minor mode of la to the major mode of ut, as experience proves.

92. In the minor mode of mi, the minor perfect chord mi sol fi mi, which characterizes it, has likewise two sounds, mi, sol, in common with the perfect chord major ut mi sol ut, which characterizes the major mode of ut. But the minor mode of mi is not so closely related nor allied to the major mode of ut as to the minor mode of la; because the diatonic scale of the minor mode of mi in descent, has not, like the series of the

(v) For what remains, when sol is said to be natural in descending the diatonic scale of the minor mode of la, this only signifies, that this sol is not necessarily sharp in descending as it is in rising; for this sol, besides, may be sharp in descending to the minor mode of la, as may be proved by numberless examples, of which all musical compositions are full. It is true, that when the sound sol is found sharp in descending to the minor mode of la, still we are not sure that the mode is minor till the fa or ut natural is found; both of which impress a peculiar character on the minor mode, viz. ut natural, in rising and in descending, and the fa natural in descending.

(+) Species was the only word which occurred to the translator in English by which he could render the French word genre. It is, according to Rousseau, intended to express the different divisions and dispositions of the intervals which formed the two tetrachords in the ancient diatonic scale; and as the gammut of the modern consists likewise of two tetrachords, though diversified from the former, as our author has shown at large, the genre or species, as the translator has been obliged to express it, must consist in the various divisions and divisions of the different intervals between the notes or semitones which compose the modern scale. Theory of the minor mode of la, all these founds in common with Harmony, the scale of ut. In reality, this scale is mi re ut fi la sol fa mi, where there occurs a fa sharp which is not in the scale of ut. We may add, that though the minor mode of mi is less relative to the major mode of ut than that of la; yet the artist does not hesitate sometimes to pass immediately from the one to the other.

Of this may be seen one instance (among many others) in the prologue des Amours des Dieux, at this passage, Ovide est l'objet de la fête, which is in the minor mode of mi, though what immediately precedes it is in the major mode of ut.

We may see besides, that when we pass from one mode to another by the interval of a third, whether in descending or rising, as from ut to la, or from la to ut, from ut to mi, or from mi to ut, the major mode becomes minor, or the minor mode becomes major.

93. There is still another minor mode, into which an immediate transition may be made in issuing from the major mode of ut. It is the minor mode of ut itself in which the perfect minor chord ut mi sol ut has two founds, ut and sol, in common with the perfect major chord ut mi sol ut. Nor is there anything more common than a transition from the major mode of ut to the minor mode, or from the minor to the major.

Chap. XI. Of Dissonance.

94. We have already observed, that the mode of ut (fa, ut, sol), has two founds in common with the mode of sol (ut, sol, re); and two founds in common with the mode of fa (sol, fa, ut); of consequence, this procedure of the bals ut sol may belong to the mode of ut, or to the mode of sol, as the procedure of the bals fa ut, or ut fa, may belong to the mode of ut or the mode of fa. When any one therefore passes from ut to fa or to sol in a fundamental bass, he is still ignorant even to that crisis what mode he is in. It would be, however, advantageous to know it, and to be able by some means to distinguish the generator from its fifths.

95. This advantage may be obtained by uniting at the same time the founds sol and fa in the same harmony, that is to say, by joining to the harmony sol fa re of the fifth sol, the other fifth fa in this manner, sol fa re fa; this fa which is added, forms a dissonance with sol (art. 18.) It is for that reason that the chord sol fa re fa, is called a dissonant chord, or a chord of the seventh. It serves to distinguish the fifth sol from the generator ut, which always implies, without mixture or alteration, the perfect chord ut mi sol ut, resulting from nature itself (art. 32.) By this we may see, that when we pass from ut to sol, one passes at the same time from ut to fa, because fa is found to be comprehended in the chord of sol; and the mode of ut by these means plainly appears to be determined, because there is none but that mode to which the founds fa and sol at once belong.

96. Let us now see what may be added to the harmony fa la ut, of the fifth fa below the generator, to distinguish this harmony from that of the generator. It seems probable at first, that we should add to it the other fifth sol, so that the generator ut, in passing to fa, may at the same time pass to sol, and that by this the mode should be determined; but this introduction of sol, in the chord fa la ut, would produce two seconds in succession fa sol sol la, that is to say, two dissonances whose union would prove extremely harsh to the ear; an inconvenience which ought carefully to be avoided. For if, to distinguish the mode, we should alter the harmony of the fifth fa in the fundamental bass, it must only be altered in the least degree possible.

97. For this reason, instead of sol, we shall take its fifth re, which is the found that approaches it the nearest; and we shall have, instead of the fifth fa, the chord fa la ut re, which is called a chord of the great fifth.

One may here remark the analogy there is observed between the harmony of the fifth sol and that of the fifth fa.

98. The fifth sol, in rising above the generator, gives a chord entirely consisting of thirds ascending from sol, fa, re, fa; now the fifth fa being below the generator ut in descending, we shall find, as we go lower by thirds from ut towards fa, the same founds ut la fa re, which form the chord fa la ut re, given to the fifth fa.

99. It appears besides, that the alteration of the harmony in the two fifths consists only in the third minor re, fa, which was reciprocally added to the harmony of these two fifths.

Chap. XII. Of the Double Use or Employment of Dissonance.

100. It is evident by the resemblance of founds to Account of their octaves, that the chord fa la ut re, is in effect the double employment of the chord re fa la ut, taken inversely; that the inverse of the chord ut la fa re, has been inverted.

(z) There are likewise other minor modes, into which we may pass in our efforts from the mode major of ut; as that of fa minor, in which the perfect minor chord fa la ut includes the found ut, and whole scale in ascent fa sol la fa ut re mi fa, only includes the two founds la fa, which do not occur in the scale of ut. We find an example of this transition from the mode major of ut to that of fa minor, in the opera of Pygmalion by M. Rameau, where the sarabande is in the minor mode of fa, and the rigadoon in the mode major of ut. This kind of transition, however, is not frequent.

The minor mode of re has only in its scale ascending re mi fa sol la fa ut re, one ut sharp which is not found in the scale of ut. For this reason a transition may likewise be made, without grating the ear, from the mode of ut major to the mode of re minor; but this passage is less immediate than the former, because the chords ut mi sol ut re fa la re, not having a single found in common, one cannot (art. 37.) pass immediately from the one to the other. Theory of found (art. 98.) in descending by thirds from the generator ut (AA).

101. The chord re, fa, la, ut, is a chord of the seventh like the chord sol, fa, re fa; with this only difference, that in this the third sol, fa, is major; whereas in the second, the third re, fa, is minor. If the fa were sharp, the chord re, fa, la, ut, would be a genuine chord of the dominant, like the chord sol, fa, re, fa; and as the dominant fa may descend to ut in the fundamental bass, the dominant re implying or carrying with it the third major fa might in the same manner descend to sol.

102. Now I say, that if the fa should be changed into fa natural, re, the fundamental tone of this chord re, fa, la, ut, might still descend to sol; for the change from fa to fa natural, will have no other effect, than to preserve the impression of the mode of ut, instead of that of the mode of sol, which the fa would have here introduced. For what remains, the note re will always preserve its character as the dominant, on account of the mode of ut, which forms a seventh. Thus in the chord of which we treat, re, fa, la, ut, re, may be considered as an imperfect dominant; I call it imperfect, because it carries with it the third minor fa, instead of the third major fa. It is for this reason that in the sequel I shall call it simply the dominant, to distinguish it from the dominant sol, which shall be named the tonic dominant†.

103. Thus the sounds fa and sol, which cannot succeed each other (art. 36.) in a diatonic bass, when they only carry with them the perfect chords fa, la, ut, sol, fa, re, may succeed one another if you join re to the harmony of the first, and fa to the harmony of the second; and if you invert the first chord, that is to say, if you give to the two chords this form, re, fa, la, ut, sol, fa, re, fa.

104. Besides, the chord fa, la, ut, re, being allowed to succeed the perfect chord ut, mi, sol, ut, it follows for the same reasons, that the chord ut, mi, sol, ut, may be succeeded by re, fa, la, ut; which is not contradictory to what we have above said (art. 37.), that the sounds ut and re cannot succeed one another in the fundamental bass: for in the passage quoted, we had supposed that both ut and re carried with them a perfect chord major; whereas, in the present case, re Theory of Harmony carries the third minor fa, and likewise the found ut, Harmony, by which the chord re fa la ut is connected with that which precedes it ut mi sol ut; and in which the sound ut is found. Besides, this chord, re fa la ut, is properly nothing else but the chord fa la ut re inverted, and, if we may speak so, disguised.

105. This manner of presenting the chord of the Double sub-dominant under two different forms, and of employing it under these two different forms, has been why so called by M. Rameau its double office or employment†. called. This is the source of one of the finest varieties in harmony; and we shall see in the following chapter the advantages which result from it.

We may add, that as this double employment is a kind of licence, it ought not to be practised without some precaution. We have lately seen that the chord re fa la ut, considered as the inverse of fa la ut re, may succeed to ut mi sol ut, but this liberty is not reciprocal: and though the chord fa la ut re may be followed by the chord ut mi sol ut, we have no right to conclude from thence that the chord re fa la ut, considered as the inverse of fa la ut re, may be followed by the chord ut mi sol ut. For this the reason shall be given Chap. XVI.

CHAP. XIII. Concerning the Use of this Double Employment, and its Rules.

106. We have shown (chap. vi.) how the diatonic scale, or ordinary gammut, may be formed from the double use of the fundamental bass fa, ut, sol, re, by twice repeating the word sol in that series; so that this gammut is primitively and originally composed of two similar tetra-chords, the chords, one in the mode of ut, the other in that of sol, impression of the mode Now it is possible, by means of this double employment, to preserve the impression of the mode of ut, through the whole extent of the scale, without twice repeating the note sol, or even without supposing this repetition. For this effect we have nothing to do but form the following fundamental bass,

ut, sol, ut, fa, ut, re, sol, ut;

in which ut is understood to carry with it the perfect chord ut mi sol ut; sol, the chord sol fa re fa; fa, the chord

(AA) "M. Rameau, in several passages of his works (for instance, in p. 110, 111, 112, and 113, of the Generation Harmonique), appears to consider the chord re, fa, la, ut, as the primary chord and generator of the chord fa, la, ut, re, which is nothing but that chord itself reversed; in other passages (particularly in p. 116, of the same performance), he seems to consider the first of these chords as nothing else but the reverse of the second. It would seem that this great artist has neither expressed himself upon this subject with so much uniformity nor with so much precision as is required. For my own part, I think there is some foundation for considering the chord fa, la, ut, re, as primitive: 1. Because in this chord, the fundamental and principal note is the sub-dominant fa, which ought in effect to be the fundamental and principal sound in the chord of the sub-dominant. 2. Because that without having recourse, with M. Rameau, to harmonical and arithmetical progressions, of which the consideration appears to us quite foreign to the question, we have found a probable and even a satisfactory reason for adding the note re to the harmony of the fifth fa (art. 96. and 97.) The origin thus assigned for the chord of the sub-dominant appears to us the most natural, though M. Rameau does not appear to have felt its full value; for scarcely has it been slightly infinuated by him."

Thus far our author. We do not enter with him into the controversy concerning the origin of the chord in question; but only propose to add to his definition of the sub-dominant Rouffleau's idea of the same note. It is a name, says he, given by M. Rameau to the fourth note in any modulation relative to a given key, which of consequence is in the same interval from the key in descending as the dominant in rising; from which circumstance it takes its name. chord fa la ut re; and re, the chord re fa la ut. It is plain from what has been said in the preceding chapter, that in this case ut may ascend to re in the fundamental bass, and re descend to sol, and that the impression of the mode of ut is preserved by the fa natural, which forms the third minor re fa, instead of the third major which re ought naturally to imply.

107. This fundamental bass will give, as it is evident, the ordinary diatonic scale,

ut, re, mi, fa, sol, la, si, UT,

which of consequence will be in the mode of ut alone; and if one should choose to have the second tetrachord in the mode of sol, it will be necessary to substitute fa instead of fa natural in the harmony of re (bb).

108. Thus the generator ut may be followed according to pleasure in ascending diatonically either by a tonic dominant (re fa la ut), or by a simple dominant (re fa la ut).

109. In the minor mode of la, the tonic dominant mi ought always to imply its third major mi sol, when this dominant mi descends to the generator la (art. 83); and the chord of this dominant shall be mi sol fa re, entirely similar to sol fa re fa. With respect to the sub-dominant re, it will immediately imply the third minor fa, to denominate the minor mode; and we may add si above its chord re fa la, in this manner re fa la si, a chord similar to that of fa la ut re; and as we have deduced from the chord fa la ut re that of re fa la ut, we may in the same manner deduce from the chord re fa la si a new chord of the seventh si re fa la, which will exhibit the double employment of dissonances in the minor mode.

110. One may employ this chord si re fa la, to preserve the impression of the mode of la in the diatonic scale of the minor mode, and to prevent the necessity of twice repeating the sound mi; but in this case, the fa must be rendered sharp, and change this chord to si re fa la, the fifth of fa is fa, as we have seen above; this chord is then the inverse of re fa la si, where the subdominant implies the third major, which ought not to surprise us. For in the minor mode of la, the second tetrachord mi fa sol la is exactly the same as it would be in the major mode of la; now, in the major mode of la, the sub-dominant re ought to imply the third major fa.

111. From thence we may see that the minor mode is susceptible of a much greater number of varieties than the major; likewise the major mode is the product of nature alone; whereas the minor is, in some measure, the product of art. But in return, the major mode has received from nature, to which it owes its immediate formation, a force and energy which the minor cannot boast.

CHAP. XIV. Of the Different Kinds of Chords of the Seventh.

112. The dissonance added to the chord of the dominant and of the sub-dominant, though in some measure inferrable by nature (chap. xi.), is nevertheless a work of art; but as it produces great beauties in consequence of harmony by the variety which it introduces into it, let us discover whether, in consequence of this first advancement, art may not still be carried farther.

113. We have already three different kinds of chords of the seventh, viz.

1. The chord sol fa re fa, composed of a third major followed by two thirds minor.

2. The chord re fa la ut, or si re fa la, composed of a third major between two minors.

3. The chord si re fa la, composed of two thirds minor followed by a major.

114. There are still two other kinds of chords of the seventh which are employed in harmony; one is composed of a third minor between two thirds major, ut mi sol fa, or fa la ut mi; the other is wholly composed of thirds minor sol fa re fa. These two chords, which at first appear as if they ought not to enter into harmony if we rigorously keep to the preceding rules, are nevertheless frequently practised with success in the fundamental bass. The reason is this:

115. According to what has been said above, if we would add a seventh to the chord ut mi sol, to make a dominant of ut, one can add nothing but fa; and in this case ut mi sol fa would be the chord of why? the tonic dominant in the mode of fa, as sol fa re fa is the chord of the tonic dominant in the mode of ut; but if you would preserve the impression of the mode of ut in the harmony, you then change this fa into fa natural, and the chord ut mi sol fa becomes ut mi sol fa. It is the same case with the chord fa la ut mi, which is nothing else but the chord fa la ut mi, in which one may substitute for mi, mi natural, to preserve the impression of the mode of ut, or of that of fa.

Besides, in such chords as ut mi sol fa, fa la ut mi, the sounds si and mi, though they form a dissonance with

(b) We need only add, that it is easy to see, that this fundamental bass ut sol, ut fa, ut re, sol ut, which formed the ascending scale ut, re, mi, fa, sol, la, si, UT, cannot by inverting it, and taking it inversely in this manner si, ut, sol, re, mi, fa, sol, la, si, UT, form the diatonic scale UT, si, la, sol, fa, mi, re, ut, in decent. In reality, from the chord sol, fa, re, fa, we cannot pass to the chord re, fa, la, ut, nor from thence to ut, mi, sol, ut. It is for this reason that in order to have the fundamental bass of the scale, UT, si, la, sol, fa, mi, re, ut, in decent, we must either determine to invert the fundamental bass mentioned in art. 55, in this manner, ut, sol, re, sol, ut, fa, ut, sol, ut, in which the second sol and the second ut answer to the sol alone in the scale; or otherwise we must form the fundamental bass ut, sol, re, sol, ut, sol, ut, in which all the notes imply perfect chords major, except the second sol, which implies the chord of the seventh sol, fa, re, fa, and which answers to the two notes of the scale sol, fa, both comprehended in the chord sol, fa, re, fa.

Whichever of these two basses we shall choose, it is obvious that neither the one nor the other shall be wholly in the mode of ut, but in the mode of ut and in that of sol. From whence it follows, that the double employment which gives to the scale a fundamental bass all in the same mode when ascending, cannot do the same in descending; and that the fundamental bass of the scale in descending will be necessarily in two different modes. Theory of Harmony.

116. With respect to the chord of the seventh fol\* si re fa, wholly composed of thirds minor, it may be regarded as formed from the union of the two chords of the dominant and of the sub-dominant in the minor mode. In effect, in the minor mode of la, for instance, these two chords are mi fol\* si re, and re fa la si, whose union produces mi fol\* la fol\* la si, re fol\* (art. 18.) so that, to avoid this inconveniency, the generator la is immediately expunged, which (art. 19.) is as it were understood in re, and the fifth or dominant mi whose place the sensible note fol\* is supposed to hold, thus there remains no more than the chord fol\* si re fa, wholly composed of thirds minor, and in which the dominant mi is considered as understood; in such a manner that the chord fol\* si re fa represents the chord of the tonic dominant mi fol\* si re, to which we have joined the chord of the sub-dominant re fa la si, but in which the dominant mi is always reckoned the principal note (pp).

117. Since, then, from the chord mi fol\* si re, we may pass to the perfect la ut mi la, and vice versa, we may in like manner pass from the chord fol\* si re fa to the chord la ut mi la, and from this last to the chord fol\* si re fa: this remark will be very useful to us in the sequel.

Chap. XV. Of the Preparation of Discords.

118. In every chord of the seventh, the highest note, that is to say, the seventh above the fundamental, is called a dissonance or discord; thus fa is the dissonance of the chord fol\* si re fa, ut in the chord re fa la ut, &c.

119. When the chord fol\* si re fa follows the chord ut mi fol ut, as this may happen, and in reality often happens, it is obvious that we do not find the dissonance fa in the preceding chord ut mi fol ut. Nor ought it indeed to be found in that chord; for this dissonance is nothing else but the sub-dominant added to the harmony of the dominant to determine the mode: now, the sub-dominant is not found in the harmony of the generator.

120. For the same reason, when the chord of the sub-dominant fa la ut re follows the chord ut mi fol ut, the note re, which forms a dissonance with ut, is not found in the preceding chord.

It is not so when the chord re fa la ut follows the chord ut mi fol ut; for ut, which forms a dissonance in the second chord, stands as a consonance in the preceding.

121. In general, dissonance being the production of art (chap. xi.), especially in such chords as are not only to the tonic dominant nor sub-dominant; the only tolerable to means to prevent its displeasing the ear by appearing when found too heterogeneous to the chord, is, that it may be, if in preceding we may speak so, announced to the ear by being found in the preceding chord, and by that means serve chords to connect the two chords. From whence follows this rule:

122. In every chord of the seventh, which is not the chord of the tonic dominant, that is to say, (art. of dissonance 102,) which is not composed of a third major followed by two thirds minor, the dissonance which this chord performed, forms ought to stand as a consonance in the chord which precedes it.

This is what we call a prepared dissonance.

123. From thence it follows, that in order to prepare a dissonance, it is absolutely necessary that the fundamental basfs should ascend by the interval of a second, as

UT mi fol ut, RE fa la ut;

or descend by a third, as

UT mi fol ut, LA ut mi fol;

or descend by a fifth, as

UT mi fol ut, FA la ut mi:

in every other case the dissonance cannot be prepared. This is what may be easily ascertained. If, for instance, the fundamental basfs rises by a third, as ut mi fol ut, mi fol si re, the dissonance re is not found in the chord ut mi fol ut. The same might be said of ut mi fol ut, fol si re fa, and ut mi fol ut, five fa la, in which the fundamental basfs rises by a fifth or descends by a second.

124. It may only be added, that when a tonic, that is to say, a note which carries with it a perfect chord, is followed by a dominant in the interval of a fifth or third, this procedure may be regarded as a process from that same tonic to another, which has been

(cc) On the contrary, a chord such as ut mi fol si, in which mi would be flat, could not be admitted in harmony, because in this chord the si is not included and understood in mi. It is the same case with several other chords, such as si re fa la si, si re fa la, &c. It is true, that in the last of these chords, la is included in fa, but it is not contained in re; and this re likewise forms with fa and with la a double-dissonance, which, joined with the dissonance si fa, would necessarily render this chord not very pleasing to the ear; we shall yet, however, see in the second part, that this chord is sometimes used.

(pp) We have seen (art. 109.) that the chord si re fa la, in the minor mode of la, may be regarded as the inverse of the chord re fa la si; it would likewise seem, that, in certain cases, this chord si re fa la may be considered as composed of the two chords fol si re fa, fa la ut re, of the dominant and of the sub-dominant of the major mode of ut; which chords may be joined together, after having excluded from them, 1. The dominant fol, represented by its third major si, which is presumed to retain its place. 2. The note ut which is understood in fa, which will form this chord si re fa la. The chord si re fa la, considered in this point of view, may be understood as belonging to the major mode of ut upon certain occasions. Part I.

Theory of Harmony.

been rendered a dominant by the addition of the dissonance.

Moreover, we have seen (art. 119, and 120,) that a dissonance does not stand in need of preparation in the chords of the tonic dominant and of the sub-dominant; from whence it follows, that every tonic carrying with it a perfect chord may be changed into a tonic dominant (if the perfect chord be major), or into a subdominant (whether the chord be major or minor) by adding the dissonance all at once.

CHAP. XVI. Of the Rules for resolving Dissonances.

125. We have seen (chap. v. and vi.) how the diatonic scale, so natural to the voice, is formed by the harmonies of fundamental sounds; from whence it follows, that the most natural succession of harmonical sounds is to be diatonic. To give a dissonance then, in some measure, as much the character of an harmonic sound as may be possible, it is necessary that this dissonance, in that part of the modulation where it is found, should descend or rise diatonically upon another note, which may be one of the consonances of the subsequent chord.

126. Now in the chord of the tonic dominant it ought rather to descend than to rise; for this reason. Let us take, for instance, the chord sol si re fa followed by the chord ut mi sol ut; the part which formed the dissonance fa ought to descend to mi rather than rise to sol, though both the sounds mi and sol are more natural and more conformed to the connection which ought to be found in every part of the music, that sol should be found in the same part where sol has already been found, whilst the other part was founding fa, as may be here seen (parts first and fourth.)

First part,

fa mi,

Second,

si ut,

Third,

re ut,

Fourth,

sol sol,

Fundamental bass,

ut mi sol ut.

127. For the same reason, in the chord of the simple dominant re fa la ut, followed by sol si re fa, dissonance ut ought rather to descend to si than rise to re.

128. In short, for the same reason, we shall find, that in the chord of the sub-dominant fa la ut re, the dissonance re ought to rise to mi of the following chord ut mi sol ut, rather than descend to ut; whence may be deduced the following rules.

129. 1°. In every chord of the dominant, whether tonic or simple, the note which constitutes the seventh, that is to say the dissonance, ought diatonically to descend upon one of the notes which form a consonance in the subsequent chord.

2°. In every chord of the subdominant, the dissonance ought to rise diatonically upon the third of the subsequent chord.

3°. A dissonance which descends or rises diatonically according to these two rules, is called a dissonance resolved.

From these rules it is a necessary result, that the chord of the seventh re fa la ut, though one should even consider it as the inverse of fa la ut re, cannot be succeeded by the chord ut mi sol ut, since there is not in this last chord of si any note upon which the dissonance ut of the chord re fa la ut can descend.

One may besides find another reason for this rule, in examining the nature of the double employment of dissonances. In effect, in order to pass from re fa la ut, to ut mi sol ut, it is necessary that re fa la ut should in this case be understood as the inverse of fa la ut re. Now the chord re fa la ut can only be conceived as the inverse of fa la ut re, when this chord re fa la ut precedes or immediately follows the ut mi sol ut; in every other case the chord re fa la ut is a primitive chord, formed from the perfect minor chord re fa la, to which the dissonance ut was added, to take from re the character of a tonic. Thus the chord re fa la ut, could not be followed by the chord ut mi sol ut, but after having been preceded by the same chord. Now, in this case, the double employment would be entirely a futile expedient, without producing any agreeable effect; because, instead of this succession of chords, ut mi sol ut, re fa la ut, ut mi sol ut, it would be much more easy and natural to substitute this other, which furnishes this natural process, ut mi sol ut, fa la ut re, ut mi sol ut. The proper use of the double employment is, that, by means of inverting the chord of the sub-dominant, it may be able to pass from that chord thus inverted to any other chord except that of the tonic, to which it naturally leads.

CHAP. XVII. Of the Broken or Interrupted Cadence.

131. In a fundamental bass which moves by fifths, there is always, as we have formerly observed (chap. perfection viii.), a repose more or less perfect from one found to another; and of consequence there must likewise be another repose more or less perfect from one found to another in the diatonic scale, which results from that bass bass.

It may be demonstrated by a very simple experiment, that the cause of a repose in melody is solely in the fundamental bass expressed or understood. Let anyone person sing these three notes ut re ut, performing on the re a shake, which is commonly called a cadence; the modulation will appear to him to be finished after the second ut, in such a manner that the ear will neither expect nor with anything to follow. The case will be the same if we accompany this modulation with its natural fundamental bass ut sol ut; but if, instead of this bass, we should give it the following, ut sol la; in this case the modulation ut re ut would not appear to be finished, and the ear would still expect and desire something more. This experiment may easily be made.

132. This passage sol la, when the dominant sol cadentially ascends upon the note la, instead of descending by a fifth upon the generator ut, as it ought what, and naturally to do, is called a broken cadence; because the perfect cadence sol ut, which the ear expected after the dominant sol, is, if we may speak so, broken and suspended by the transition from sol to la.

133. From thence it follows, that if the modulation ut re ut appeared finished when we supposed no bass to it at all, it is because its natural fundamental bass ut sol ut is supposed to be implied; because the ear defies something to follow this modulation, as soon as it is reduced to the necessity of hearing another bass.

134. The 134. The interrupted cadence may, as it seems to me, be considered as having its origin in the double employment of dissonances; since this cadence, like the double employment, only consists in a diatonic procedure of the bass ascending (chap. xiii.). In effect, nothing hinders us to descend from the chord sol si re fa to the chord ut mi sol la, by converting the tonic ut into a sub-dominant, that is to say, by passing all at once from the mode of ut to the mode of sol; now to descend from sol si re fa to ut mi sol la is the same thing as to rise from the chord sol si re fa to the chord la ut mi sol, in changing the chord of the subdominant ut mi sol la for the imperfect chord of the dominant, according to the laws of the double employment.

135. In this kind of cadence, the dissonance of the first chord is resolved by descending diatonically upon the fifth of the subsequent chord. For instance, in the broken cadence sol si re fa, la ut mi sol, the dissonance fa is resolved by descending diatonically upon the fifth mi.

136. There is still another kind of cadence called an interrupted cadence, where the dominant descends by a third to another dominant, instead of descending by a fifth upon the tonic, as in this process of the bass, sol si re fa, mi sol si re; in the case of an interrupted cadence, the dissonance of the former chord is resolved by descending diatonically upon the octave of the fundamental note of the subsequent chord, as may be here seen, where fa is resolved upon the octave of mi.

137. This kind of interrupted cadence, as it seems to me, has likewise its origin in the double employment of dissonances. For let us suppose these two chords in succession, sol si re fa, sol si re mi, where the note sol is successively a tonic dominant and sub-dominant; that is to say, in which we pass from the mode of ut to the mode of re; if we should change the second of these chords into the chord of the dominant, according to the laws of the double employment, we shall have the interrupted cadence sol si re fa, mi sol si re.

CHAP. XVIII. Of the Chromatic Species.

138. The series of fundamental basses by fifths produces the diatonic species in common use (chap. vii.): now the third major being one of the harmonies of a fundamental found as well as the fifth, it follows, that we may form fundamental basses by thirds major, as we have already formed fundamental basses by fifths.

139. If then we should form this bass ut, mi, sol, a chromatic the two first sounds carrying each along with them their thirds major and fifths, it is evident that ut will give semitone, sol, and that mi will give sol, now the semitone which was found between this sol and this sol is an interval much less than the semitone which is found in the diatonic scale between mi and fa, or between fa and ut. This may be ascertained by calculation (ee); it is for this reason that the semitone from mi to fa is called major, and the other minor (ff).

140. If the fundamental bass should proceed by thirds minor in this manner, ut, mi, a succession which is allowed when we have investigated the origin of the minor mode (chap. ix.), we shall find this modulation sol, sol, which would likewise give a minor semitone (gg).

141. The minor semitone is hit by young practitioners in intonation with more difficulty than the major semitone. For which this reason may be assigned: The semitone major which is found in the diatonic scale, as from mi to fa, results from a fundamental bass by fifths ut fa, that is to say, by a succession which is most natural, and for this reason the easiest to the ear. On the contrary, the minor semitone arises from a succession by thirds, which is still less natural than the former. Hence, that scholars may truly hit the minor semitone, the following artifice is employed. Let us suppose, for instance, that they intend to rise from sol to sol; they rise at first from sol to la, then descend from la to sol by the interval of a semitone major; for this sol sharp, which is a semitone below la, proves a semitone minor above sol. [See the notes (ee) and (ff).]

142. Every procedure of the fundamental bass by thirds, whether major or minor, rising or descending, gives the minor semitone. This we have already seen be found in the succession of thirds in ascending. The series of thirds minor in descending, ut, la, gives ut, la, the fundamental bass by thirds.

(EE) In reality, ut being supposed 1, as we have always supposed it, mi is \( \frac{4}{7} \), and sol \( \frac{5}{7} \); now sol being \( \frac{5}{7} \), sol\( \frac{5}{7} \) then shall be to sol as \( \frac{5}{7} \) to \( \frac{1}{7} \); that is to say, as 25 times 2 to 3 times 16: the proportion then of sol\( \frac{5}{7} \) to sol is as 25 to 24, an interval much less than that of 16 to 15, which constitutes the semitone from ut to fa, or from fa to mi (note L.)

(ff) It may be observed, that a minor joined to a major semitone will form a minor tone; that is to say, if one rises, for instance, from mi to fa, by the interval of a semitone major, and afterwards from fa to fa\( \frac{5}{7} \) by the interval of a minor semitone, the interval from mi to fa\( \frac{5}{7} \) will be a minor tone. For let us suppose mi to be 1, fa will be \( \frac{5}{7} \), and fa\( \frac{5}{7} \) will be \( \frac{5}{7} \) of \( \frac{5}{7} \); that is to say, 25 times 16 divided by 24 times 15, or \( \frac{5}{7} \); mi then is to fa\( \frac{5}{7} \) as 1 is to \( \frac{5}{7} \), the interval which constitutes the minor tone (note N.)

With respect to the tone major, it cannot be exactly formed by two semitones; for, 1. Two major semitones in immediate succession would produce more than a tone major. In effect, \( \frac{5}{7} \) multiplied by \( \frac{5}{7} \) gives \( \frac{5}{7} \), which is greater than \( \frac{5}{7} \), the interval which constitutes (note N) the major tone. 2. A semitone minor and a semitone major would give less than a major tone, since they amount only to a true minor. 3. And, a fortiori, two minor semitones would give still less.

(gg) In effect, mi\( \frac{5}{7} \) being \( \frac{5}{7} \), sol\( \frac{5}{7} \) will be \( \frac{5}{7} \) of \( \frac{5}{7} \); that is to say, (note C) \( \frac{5}{7} \): now the proportion of \( \frac{5}{7} \) to \( \frac{5}{7} \) (note C) is that of 3 times 25 to 2 times 36; that is to say, as 25 to 24. The minor semitone constitutes the species called chromatic; and with the species which moves by diatonic intervals, resulting from the succession of fifths (chap. v. and vi.), it comprehends the whole of melody.

CHAP. XIX. Of the Enharmonic Species.

144. The two extremes, or highest and lowest notes, ut sol, of the fundamental basis by thirds major, or mi sol, give this modulation ut fa; and these two sounds ut, fa, differ between themselves by a small interval which is called the disjunctive, or enharmonic fourth* of a tone (ll), which is the difference between a semitone major and a semitone minor (mm). This quarter tone is inappreciable by the ear, and impracticable upon several of our instruments. Yet have means been found to put it in practice in the following manner, or rather to perform what will have the same effect upon the ear.

145. We have explained (art. 116.) in what manner the chord sol fa re fa may be introduced into the minor mode, entirely consisting of thirds minor perfectly true, or at least supposed such. This chord supplying the place of the chord of the dominant (art. 116.) from thence we may pass to that of the tonic or generator la (art. 117.). But we must remark,

1. That this chord sol fa re fa, entirely consisting of thirds minor, may be inverted or modified according to the three following arrangements, fa re fa sol, re fa sol fa, fa sol fa re; and that in all these three different states, it will still remain composed of thirds minor; or at least there will only be wanting the enharmonic fourth of a tone to render the third minor between fa and sol entirely just; for a true third minor, as that from mi to sol in the diatonic scale, is composed of a semitone and a tone both major. Now from fa to sol there is a tone major, and from sol to fa there is only a minor semitone. There is then a wanting (art. 144.) the enharmonic fourth of a tone, to render the third fa sol exactly true.

2. But as this division of a tone cannot be found in the gradations of any scale practicable upon most of our instruments, nor be appreciated by the ear, the ear takes the different chords,

\[ \begin{align*} & \text{fa} & \text{re} & \text{fa} & \text{sol} \\ & \text{re} & \text{fa} & \text{sol} & \text{fa} \\ & \text{fa} & \text{sol} & \text{fa} & \text{re} \end{align*} \]

which are absolutely the same, for chords composed every one of thirds minor exactly just.

Now the chord sol fa re fa, belonging to the minor mode of la, where sol is the sensible note; the chord fa re fa sol, or fa re fa la, will, for the same reason, belong to the minor mode of ut, where fa is the sensible note. In like manner, the chord fa sol fa, or fa re fa la ut, will belong to the minor mode of mi, and the chord fa sol fa re, or fa la ut mi, to the minor mode of sol.

After having passed then by the mode of la to the chord sol fa re fa (art. 117.), one may by means of this last chord, and by merely satisfying ourselves to invert it, afterwards pass all at once to the modes of ut minor, of mi minor, or of sol minor; that is to say, into the modes which have nothing, or almost nothing.

(Vol. XII. Part II.)

(LH) La being \( \frac{3}{8} \), ut \( \frac{5}{8} \); that is to say \( \frac{3}{4} \), and ut is 1 : the proportion then between ut and ut \( \frac{5}{8} \) is that of 1 to \( \frac{3}{4} \), or of 24 to 25.

(II) Lab being the third major below ut, will be \( \frac{4}{5} \) (note c): utb, then, is \( \frac{7}{8} \) of \( \frac{4}{5} \); that is to say \( \frac{3}{4} \). The proportion, then, between ut and utb, is as 25 to 24.

(LL) Sol \( \frac{5}{8} \) being \( \frac{3}{8} \) and \( \frac{5}{8} \) being \( \frac{1}{2} \) of \( \frac{5}{8} \), we shall have \( \frac{5}{8} \) equal (note c) to \( \frac{1}{2} \), and its octave below shall be \( \frac{3}{8} \); an interval less than unity by about \( \frac{1}{12} \) or \( \frac{1}{12} \). It is plain then from this fraction, that the \( \frac{5}{8} \) in question must be considerably lower than ut.

This interval has been called the fourth of a tone, and this denomination is founded on reason. In effect, we may distinguish in music four kinds of quarter tones.

1. The fourth of a tone major: now, a tone major being \( \frac{3}{8} \), and its difference from unity being \( \frac{1}{9} \), the difference of this quarter tone from unity will be almost the fourth of \( \frac{1}{9} \); that is to say, \( \frac{1}{12} \).

2. The fourth of a tone minor; and as a tone minor, which is \( \frac{5}{8} \), differs from unity by \( \frac{1}{9} \), the fourth of a minor tone will differ from unity about \( \frac{1}{12} \).

3. One half of a tone major; and as this semitone differs from unity by \( \frac{1}{12} \), one half of it will differ from unity about \( \frac{1}{12} \).

4. Finally, one half of a semitone minor, which differs from unity by \( \frac{1}{12} \); its half then will be \( \frac{1}{24} \).

The interval, then, which forms the enharmonic fourth of a tone, as it does not differ from unity but by \( \frac{1}{12} \), may justly be called the fourth of a tone, since it is less different from unity than the largest interval of a quarter tone, and more than the least.

We shall add, that since the enharmonic fourth of a tone is the difference between a semitone major and a semitone minor; and since the tone minor is formed (note ff) of two semitones, one major and the other minor; it follows, that two semitones major in succession form an interval larger than that of a tone by the enharmonic fourth of a tone; and that two minor semitones in succession form an interval less than a tone by the same fourth of a tone.

(MM) That is to say, that if you rise from mi to fa, for instance, by the interval of a semitone major, and afterwards, returning to mi, you should rise by the interval of a semitone minor to another sound which is not in the scale, and which I shall mark thus, fa +, the two sounds fa + and fa will form the enharmonic fourth of a tone: for mi being \( \frac{1}{2} \), fa will be \( \frac{3}{8} \); and fa + \( \frac{5}{8} \): the proportion then between fa + and fa is that of \( \frac{3}{8} \) to \( \frac{5}{8} \) (note c); that is to say, as 25 times 15 to 16 times 1; or otherwise, as 25 times 15 to 16 times 8, or as 125 to 128. No this proportion is the same which is found, in the beginning of the preceding note, to express the enharmonic fourth of a tone. nothing, in common with the minor mode of la, and which are entirely foreign to it (†).

146. It must, however, be acknowledged, that a transition to abrupt, and to little expected, cannot deceive nor elude the ear; it is struck with a sensation so unlooked-for without being able to account for the passage to itself. And this account has its foundation in the enharmonic fourth of a tone; which is overlooked as nothing, because it is inappreciable by the ear; but of which, though its value is not aforesaid, the whole harshness is sensibly perceived. The instant of surprise, however, immediately vanishes; and that astonishment is turned into admiration, when one feels himself transported as it were all at once, and almost imperceptibly, from one mode to another, which is by no means relative to it, and to which he never could have immediately passed by the ordinary series of fundamental notes.

CHAP. XX. Of the Diatonic Enharmonic Species.

147. If we form a fundamental base, which rises alternately by fifths and thirds, as fa, ut, mi, fa, this base will give the following modulation, fa, mi, mi, re‡; in which the semitones from fa to mi, and from mi to re‡, are equal and major (nn).

This species of modulation or of harmony, in which all the semitones are major, is called the enharmonic diatonic species. The major semitones peculiar to this species give it the name of diatonic, because major semitones belong to the diatonic species; and the tones which are greater than major by the excess of a fourth, resulting from a succession of major semitones, give it the name of enharmonic (note LL).

CHAP. XXI. Of the Chromatic Enharmonic Species.

148. If we pass alternately from a third minor in descending to a third major in rising, as ut, ut, la, ut‡, ut‡, we shall form this modulation mil, mi, mi, mi‡, in which all the semitones are minor (oo).

This species is called the chromatic enharmonical species: the minor semitones peculiar to this kind give it the name of chromatic, because minor semitones belong to the chromatic species; and the semitones which are lesser by the diminution of a fourth result from a succession of minor semitones, give it the name of enharmonic (note LL).

149. These new species confirm what we have already said, that the whole effects of harmony and melody appear along with the fundamental bases.

150. The diatonic species is the most agreeable, because the fundamental base which produces it is formed from a succession of fifths alone, which is the most natural of all others.

151. The chromatic being formed from a succession of thirds, is the most natural after the preceding.

152. Finally, the enharmonic is the least agreeable next of all, because the fundamental base which gives it is not immediately indicated by nature. The fourth of enharmonic tone which constitutes this species, and which is itself inappreciable to the ear, neither produces nor can produce its effect, but in proportion as imagination suggests the fundamental base from whence it results; a base whose procedure is not agreeable to nature, since it is formed of two sounds which are not contiguous one to the other in the series of thirds (art. 144.)

CHAP. XXII. Showing that Melody is the Offspring of Harmony.

153. All that we have hitherto said, as it seems to me, is more than sufficient to convince us, that melody of harmony, expressed or understood, that we ought to have any look for the effects of melody.

154. If this should still appear doubtful, nothing more is necessary than to pay due attention to the first experiment (art. 19.), where it may be seen that the principal sound is always the lowest, and that the sharper sounds which it generates are with relation to it what the treble of an air is to its base.

155. Yet more, we have proved, in treating of broken cadence (chap. xvii.), that the diversification of basses

(†) As this method for obtaining or supplying enharmonic gradations cannot be practised on every occasion when the composer or practitioner would wish to find them, especially upon instruments where the scale is fixed and invariable, except by a total alteration of their economy, and re-tuning the strings, Dr Smith in his Harmonies has proposed an expedient for redressing or qualifying this defect, by the addition of a greater number of keys or strings, which may divide the tone or semitone into as many appreciable or sensible intervals as may be necessary. For this, as well as for the other advantageous improvements which he proposes in the structure of instruments, we cannot with too much warmth recommend the perusal of his learned and ingenious book to such of our readers as aspire to the character of genuine adepts in the theory of music.

(nn) It is obvious, that if fa in the base be supposed, i.e., fa of the scale will be 2, ut of the base 1, and mi of the scale 3 of 4, that is, \( \frac{1}{2} \); the proportion of fa to mi is as 2 to \( \frac{3}{4} \), or as 1 to \( \frac{3}{8} \). Now mi of the base being likewise \( \frac{3}{4} \) of \( \frac{4}{5} \), or \( \frac{3}{5} \) of \( \frac{4}{5} \), of the base is \( \frac{3}{4} \) of \( \frac{4}{5} \), and its third major \( \frac{3}{4} \) of \( \frac{4}{5} \), or \( \frac{3}{5} \) of \( \frac{4}{5} \); this third major, approximated as much as possible to mi in the scale by means of octaves, will be \( \frac{3}{4} \) of \( \frac{4}{5} \); mi then of the scale will be to re‡ which follows it, as \( \frac{3}{4} \) is to \( \frac{4}{5} \) of \( \frac{4}{5} \), that is to say, as 1 to \( \frac{3}{8} \). The semitones then from fa to mi, and from mi to re‡, are both major.

(oo) It is evident that mil is \( \frac{3}{4} \) (note c), and that mi is \( \frac{3}{4} \); these two mi's, then, are between themselves as \( \frac{3}{4} \) to \( \frac{3}{4} \), that is to say, as 6 times 4 to 5 times 5, or as 24 to 25; the interval which constitutes the minor semitone. Moreover, the la of the base is \( \frac{3}{4} \), and ut‡ \( \frac{3}{4} \) of \( \frac{3}{4} \), or \( \frac{3}{4} \); mi‡ then is \( \frac{3}{4} \) of \( \frac{3}{4} \), the mi in the scale is likewise to the mi‡ which follows it, as 24 to 25. All the semitones therefore in this scale are minor. Part I.

Theory of Harmony.

156. Can it be still necessary to adduce more convincing proofs? We have nothing to do but examine the different basses which may be given to this very simple modulation *sol ut*; of which it will be found susceptible of a great many, and each of these basses will give a different character to the modulation *sol ut*, though in itself it remains always the same; in such a manner that we may change the whole nature and effects of a modulation, without any other alteration except that of changing its fundamental bass.

M. Rameau has shown, in his New System of Music, printed at Paris 1726, p. 44, that this modulation *sol ut*, is susceptible of 20 different fundamental basses. Now the same fundamental bass, as may be seen in our second part, will afford several continued or thorough basses. How many means, of consequence, may be practised to vary the expression of the same modulation?

157. From these different observations it may be concluded, 1. That an agreeable melody, naturally implies a bass extremely sweet and adapted for singing; and that reciprocally, as musicians express it, a bass of this kind generally prognosticates an agreeable melody (pp).

2. That the character of a just harmony is only to form in some measure one system with the modulation, so that from the whole taken together the ear may only receive, if we may speak so, one simple and indivisible Theory of impression.

3. That the character of the same modulation may be diversified, according to the character of the bass which is joined with it.

But notwithstanding the dependency of melody upon harmony, and the sensible influence which the latter may exert upon the former; we must not however from thence conclude, with some celebrated musicians, that the effects of harmony are preferable to those of melody. Experience proves the contrary. [See, on this account, what is written on the licence of music, printed in tom. iv. of D'Alembert's Mélanges de Litterature, p. 448.]

General Remark.

The diatonic scale or gammut being composed of twelve semitones, it is clear that each of these semitones taken by itself may be the generator of a mode; and that thus there must be twenty-four modes in all, twelve major and twelve minor. We have assumed the major mode of *ut*, to represent all the major modes in general, and the minor mode of *la* to represent the modes minor, to avoid the difficulties arising from sharps and flats, of which we must have encountered either a greater or lesser number in the other modes. But the rules we have given for each mode are general, whatever note of the gammut be taken for the generator of a mode.

Part II. Principles and Rules of Composition.

158. Composition, which is likewise called counterpoint, is not only the art of composing an agreeable air, but also that of composing a great many airs in such a manner that when heard at the same time, they may unite in producing an effect agreeable and delightful to the ear; this is what we call composing music in several parts.

The highest of these parts is called the treble, the lowest is termed the bass; the other parts, when there are any, are termed middle parts; and each in particular is signified by a different name.

Chap. I. Of the Different Names given to the same Interval.

159. In the introduction (art. 9.), which is at the front of this treatise, we have seen a detail of the most common names which are given to the different intervals. But there are particular intervals which have obtained different names, according to particular circumstances; which it is proper to explain.

160. An interval composed of a tone and a semitone, which is commonly called a third minor, is likewise sometimes called a second redundant; such is the interval from *ut* to *re* in ascending, or that of *la* to *fa* in descending.

This interval is so termed, because one of the sounds which form it is always either sharp or flat, and that, called, if you deduce that sharp or that flat, the interval will be that of a second.

161. An interval composed of two tones and two semitones, as that from *fa* to *fa*, is called a false fifth. This interval is the same with the triton (art. 9.), since two tones and two semitones are equivalent to three tones. There are, however, some reasons for distinguishing them, as will appear below.

162. As the interval from *ut* to *re* in ascending has been called a second redundant, they likewise call the interval from *ut* to *sol* in ascending a fifth redundant, or from *fa* to *mi* in descending, each of which intervals are composed of four tones.

This interval is, in the main, the same with that of the fifth minor (art. 6.); but in the fifth redundant guised there is always a sharp or a flat; insomuch, that if this sharp or flat were deduced, the interval would become a true fifth.

163. For the same reason, an interval composed of three tones and three semitones, as from *sol* to *fa* in ascending,

(pp) There are likewise several eminent musicians, who in their compositions, if we can depend on what has been affirmed, begin with determining and writing the bass. This method, however, appears in general more proper to produce a learned and harmonious music, than a strain prompted by genius and animated by enthusiasm. Musical Principles

**Part II**

**Chap. II. Comparison of the Different Intervals.**

165. If we sing *ut* in descending by a second, and afterwards *ut* in ascending by a seventh, these two *fa*s shall be octaves one to the other; or, as we commonly express it, they will be *replications* one of the other.

166. On account then of the resemblance between every sound and its octave (art. 22.), it follows, that to rise by a seventh, or descend by a second, amount to one and the same thing.

167. In like manner, it is evident that the sixth is nothing but a replication of the third, nor the fourth but a replication of the fifth.

168. The following expressions either are or ought to be regarded as synonymous.

| To rise | To descend | |---------|------------| | by a second. | by a seventh. |

| To descend | To rise | |------------|--------| | by a third. | by a sixth. |

| To rise | To descend | |---------|------------| | by a fourth. | by a fifth. |

| To descend | To rise | |------------|--------| | | |

169. Thus, therefore, we shall employ them indifferently the one for the other; so that when we say, "Principles for instance, to rise by a third, it may be said with equal propriety to descend by a sixth," &c.

**Chap. III. Of the different Clefs; of the Value or Quantity; of the Rithmus; and of Syncopation.**

170. There are three clefs* in music; the clef of *fa* O, or F; the clef ut H; and the clef of sol G. But, in Britain, the following characters are used: The F, or bass-clef E; the C, or tenor clef C; and the G, or treble clef G.

The clef of *fa* is placed on the fourth line, or on the third; and the line upon which this clef is placed gives the name of *fa* or F, to all the notes which are upon that line.

The clef of ut is placed upon the fourth, the third, the second, or the first line; and in these different positions all the notes upon that line where the clef is placed take the name of ut, or C.

Lastly, the clef of sol is placed upon the second or first line; and all the notes upon that line where the clef is placed take the name of sol, or G.

171. As the notes are placed on the lines, and in the spaces between the lines, any one, when he sees the clef, may easily find the name of any note whatever. Thus he may see, that, in the first clef of *fa*, position of the note which is placed on the lowest line ought to be sol; that the note which occupies the space between the two first lines should be la; and that the note which is on the second line is a fa, &c. (RR).

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(*RR*) The chief use of these different denominations is to distinguish chords; for instance, the chord of the redundant fifth and that of the diminished seventh are different from the chord of the fifth; the chord of the seventh redundant, from that of the seventh major. This will be explained in the following chapters.

(*RR*) It is on account of the different compasses of voices and instruments that these clefs have been invented.

The masculine voice, which is the lowest, may at its greatest depth, without straining, descend to sol, which is in the last line of the first clef of *fa*; and the female voice, which is the sharpest, may at its highest pitch rise to a sol, which is a triple octave above the former.

The lowest of masculine voices is adapted to a part which may be called a mean bass, and its clef is that of *fa* on the fourth line; this clef is likewise that of the violoncello and of the deepest instruments. A mean bass extremely deep is called a baritone or counter-bass.

The masculine voice which is next in depth to what we have called the mean bass may be termed the concordant bass. Its clef is that of *fa* on the third line.

The masculine voice which follows the concordant bass may be denominated a tenor; a voice of this pitch is the most common, yet seldom extremely agreeable. Its clef is that of ut on the fourth line. This clef is also that of the bassoon or bass-hautboy.

The highest masculine voice of all may be called counter tenor. Its clef is that of ut on the third line. It is likewise the clef of tenor violins, &c.

The deepest female voice immediately follows the counter tenor, and may be called bass in alt. Its clef is that of ut upon the first line. The clef of ut upon the second line is not in frequent use.

The sharpest female voice is called treble; its clef is that of sol on the second line.

This last clef, as well as that of sol on the first line, is likewise the clef of the sharpest instruments, such as the violin, the flute, the trumpet, the hautboy, the flagelet, &c.

The ut which may be seen in the clefs of *fa* and in the clefs of ut is a fifth above the *fa* which is on the line of the clef of *fa*; and the sol which is on the two clefs of sol is a fifth above ut; insomuch that sol which is Part II.

172. A note before which there is a sharp (marked thus ♯) ought to be raised by a semitone; and if, on the contrary, there is a ♭ before it, it ought to be depressed by a semitone, (♭ being the mark of a flat.)

The natural (marked thus ♮) restores to its natural value a note which had been raised or depressed by a semitone.

173. When you place at the clef a sharp or a flat, all the notes upon the line on which this sharp or flat is marked are sharp or flat. Thus let us take, for instance, the clef of ut upon the first line, and let us place a sharp in the space between the second and third line, which is the place of fa; all the notes which shall be marked in that space will be fa♯; and if you would restore them to their original value of fa natural, you must place a ♭ or a ♭ before them.

In the same manner, if a flat be marked at the clef, and if you would restore the note to its natural state, you must place a ♭ or a ♭ before it.

174. Every piece of music is divided into different equal times, which they call measures or bars; and each bar is likewise divided into different times.

There are properly two kinds of measures or modes of time (See T): the measure of two times, or of common time, which is marked by the figure 2 placed at the beginning of the tune; and the measure of three times, or of triple time, which is marked by the figure 3 placed in the same manner. (See V.)

The different bars are distinguished by perpendicular lines.

In a bar we distinguish between the perfect and imperfect time; the perfect time is that which we beat, the imperfect that in which we lift up the hand or foot. A bar consisting of four times ought to be regarded as compounded of two bars, each consisting of two times: thus there are in this bar two perfect and two imperfect times. In general, by the words perfect and imperfect, even the parts of the same time are distinguished: thus the first note of each time is reckoned as belonging to the perfect part, and the others as belonging to the imperfect.

175. The longest of all notes is a semibreve. A minim is half its value; that is to say, in singing, we only employ the same duration in performing two minims which was occupied in one semibreve. A minim in the same manner is equivalent to two crotchets, the crotchet to two quavers, &c.

176. A note which is divided into two parts by a syncopation, that is to say, which begins at the end of a time, and terminates in the time following, is called (ss) a syncopated note. (See Z; where the notes mi, la, are each of them syncopated.) (+).

177. A

is on the lowest line of the first clef of fa, is lower by two whole octaves than the sol which is on the lowest line of the second clef of sol.

[Thus far the translator has followed his original as accurately as possible; but this was by no means an easy task. Among all the writers on music which he has found in English, there is no such thing as different names for each particular part which is employed to constitute full or complete harmony. He was therefore under a necessity of substituting by analogy such names as appeared most expressive of his author's meaning. To facilitate this attempt, he examined in Rousseau's musical dictionary the terms by which the different parts were denominated in D'Alembert; but even Rousseau, with all his depth of thought and extent of erudition, instead of expressing himself with that precision which the subject required, frequently applies the same names indiscriminately to different parts, without affixing any reason for this promiscuous and licentious use of words. The English reader therefore will be best able to form an accurate idea of the different parts, by the nature and situation of the clefs with which they are marked; and if he should find any impropriety in the names which are given them, he may adopt and associate others more agreeable to his ideas.]

(ss) Syncopation consists in a note which is protracted in two different times belonging partly to the one and partly to the other, or in two different bars; yet not so as entirely to occupy or fill up the two times, or the two bars. A note, for instance, which begins in the imperfect time of a bar, and which ends in the perfect time of the following, or which in the same bar begins in the imperfect part of one time and ends in the perfect of the following, is syncopated. A note which of itself occupies one or two bars, whether the measure consists of two or three times, is not considered as syncopated: this is a consequence of the preceding definition. This note is said to be continued or protracted. In the end of the example Z, the ut of the first bar consisting of three times is not syncopated; because it occupies two whole times. It is the same case with mi of the second bar, and with the ut of the fourth and fifth. These therefore are continued or protracted notes.

(+) Times and bars in music answer the very same end as punctuation in language. They determine the different periods of the movement, or the various degree of completion, which the sentiment, expressed by that movement, has attained. Let us suppose, for instance, a composer in music intending to express grief or joy, in all its various gradations, from its first and faintest sensation, to its acme or highest possible degree. We do not say that such a progress of any passion either has been or can be delineated in practice, yet it may serve to illustrate what we mean to explain. Upon this hypothesis, therefore, the degrees of the sentiment will pass from least to more sensible, as it rises to its most intense degree. The first of these gradations may be called a time, which is likewise the most convenient division of a bar or measure into its elementary or aliquot parts, and may be deemed equivalent to a comma in a sentence; a bar denotes a degree still more sensible, and may be considered as having the force of a semicolon; a strain brings the sentiment to a tolerable degree of perfection, and may be reckoned equal to a colon; the full period is the end of the imitative piece. It must have been remarked by observers of measure in melody or harmony, that the notes of which a bar or measure consists, are not diversified by their different durations alone, but likewise by greater or lesser degrees of emphasis. A note followed by a point or dot is increased half its value. The \( \frac{1}{2} \), for instance, in the fifth bar of the example Y, followed by a point, has the value (*) or duration of a minim and of a crotchet at the same time.

**Chap. IV. Containing a Definition of the principal Chords.**

178. The chord composed of a third, a fifth, and an octave, as ut mi sol ut, is called a perfect chord (art. 32.)

If the third be major, as in ut mi sol ut, the perfect chord is denominated major; if the third be minor, as in la ut mi la, the perfect chord is minor. The perfect chord major constitutes what we call the major mode; and the perfect chord minor, what we term the minor mode (art. 31).

179. A chord composed of a third, a fifth, and a seventh, as sol fa re fa, or re fa la ut, &c. is called a chord of the seventh. It is obvious that such a chord is wholly composed of thirds in ascending.

All chords of the seventh are practised in harmony, save that which might carry the third minor and the seventh major, as ut mi sol fa; and that which might carry a false fifth and a seventh major, as si re fa la, (chap. xiv. Part I.)

180. As thirds are either major or minor, and as they may be differently arranged, it is clear that there are different kinds of chords of the seventh; there is even one, si re fa la, which is composed of a third, a false fifth, and a seventh.

181. A chord composed of a third, a fifth, and a sixth, as fa la ut re, re fa la si, is called a chord of the greater sixth.

182. Every note which carries a perfect chord is called a tonic, and a perfect chord is marked by an 8, by a 3, or by a 5, which is written above the note; but frequently these numbers are suppressed. Thus in the example I. the two ut's equally carry a perfect chord.

183. Every note which carries a chord of the what, and seventh is called a dominant (art. 102); and this chord how figured is marked by a 7 written above the note. Thus in red, the example II. re carries the chord re fa la ut, and sol the chord sol fa re fa.

It is necessary to remark, that among the chords of what, and the seventh we do not reckon the chord of the figured seventh diminished, which is only improperly called a chord of the seventh; and of which we shall say more below.

184. Every note which carries the chord of the Sub-dominant, great fifth, is called a subdominant, (art. 97, and 42.) and how figured is marked with a 6. Thus in the example III. figured, fa carries the chord of fa la ut re. You ought to remark that the sixth should always be major, (art. 97, and 109):

185. In every chord, whether perfect, or a chord fundamental of the seventh, or of the great fifth, the note which carries this chord, and which is the flattest or lowest, is called the fundamental note. Thus ut in the example I. re and sol in the example II., and fa in the example III. are fundamental notes.

186. In every chord of the seventh, and of the Diffidence great fifth, the note which forms the seventh or fifth of a chord, above the fundamental, that is to say, the highest what, note of the chord, is called a diffidence. Thus in the chords of the seventh sol fa re fa, re fa la ut, fa and ut are the diffidences, viz. fa with relation to sol in the first chord, and ut with relation to re in the second. In the chord of the great fifth fa la ut re, re is the diffidence (art. 120.) but that re is only properly

phasis. The most emphatic parts of a bar are called the accented parts; those which require less energy in expression are called the unaccented. The same observation holds with regard to times as bars. The stroke, therefore, of the hand or foot in beating marks the accented part of the bar, the elevation or preparation for the stroke marks the unaccented part. Let us once more resume our composition intended to express the different periods in the progress of grief or joy. There are some revolutions in each of these so rapid as not to be marked by any sensible transition whether diatonic or consonant. In this case, the most expressive tone may be continued from one part of a time or measure to another, and end before the period of that time or measure in which it begins. Here therefore is a natural principle upon which the practice of syncopation may be founded even in melody: but when music is composed in different parts to be simultaneously heard, the continuance of one note not divided by regular times and measures, nor beginning and ending with either of them, whilst the other parts either ascend or descend according to the regular divisions of the movement, has not only a sensible effect in rendering the imitation more perfect, but even gives the happiest opportunities of diversifying the harmony, which of itself is a most delightful perception.

For the various dispositions of accent in times and measures, according to the movement of any piece, see a Treatise on Music by Alexander Malcolm.

For the opportunity of diversifying harmony afforded by syncopation, see Rameau's Principles of Composition.

(*) To prevent ambiguity or confusion of ideas, it is necessary to inform our readers, that we have followed M. D'Alembert in his double sense of the word value, though we could have wished he had distinguished the different meanings by different words. A sound may be either estimated by its different degrees of intensities, or by its different quantities of duration.

To signify both those ideas the word value is employed by D'Alembert. The reader, therefore, will find it of importance to distinguish the value of a note in height from its value in duration. This he may easily do, by considering whether the notes are treated as parts of the diatonic scale, or as continued for a greater or lesser duration. properly speaking, a dissonance with relation to ut from which it is a second, and not with respect to fa from which it is a fifth major (art. 17, and 18).

187. When a chord of the seventh is composed of a third major followed by two thirds minor, the fundamental note of this chord is called the tonic dominant. In every other chord of the seventh the fundamental is called the simple dominant (art. 132.). Thus in the chord sol fa re fa, the fundamental sol is the tonic dominant; but in the other chords of the seventh, as ut mi sol fa, re fa la ut, &c., the fundamentals ut and re are simple dominants.

188. In every chord, whether perfect, or of the seventh, or of the sixth, if you have a mind that the third above the fundamental note should be major, though it is naturally minor, you must place a sharp above the fundamental note. For example, if I would mark the perfect major chord re fa la re, as the third fa above re is naturally minor, I place above re a sharp, as you may see in example IV. In the same manner the chord of the seventh re fa la ut, and the chord of the great sixth re fa la fa, is marked with a ♯ above re, and above the ♯ a 7 or a 6 (see V. and VI.).

On the contrary, when the third is naturally major, and if you should incline to render it minor, you must place above the fundamental note a b. Thus the examples VII. VIII. IX. show the chords sol fa re sol, sol fa re fa, sol fa re mi (tt).

CHAP. V. Of the Fundamental Bafs.

189. Invent a modulation at your pleasure; and under this modulation let there be a baf composed of different notes, of which some may carry a perfect chord, others that of the seventh, and others that of the great fifth, in such a manner that each note of the modulation which answers to each of the bafs, may be one of those which enters into the chord of that note in the baf; this baf being composed according to the rules which shall be immediately given, will be the fundamental baf of the modulation proposed. See Part I. where the nature and principles of the fundamental baf are explained.

Thus (Exam. XVIII.) you will find that this modulation, ut re mi fa sol la fa ut, has or may admit for its fundamental baf, ut sol ut fa ut re fol ut.

In reality, the first note ut in the upper part is found in the chord of the first note ut in the baf, which chord is ut mi sol ut; the second note re in the treble is found in the chord sol fa re fa, which is the chord of the second note in the baf, &c. and the baf is composed only of notes which carry a perfect chord, or that of the seventh, or that of the great fifth. Moreover, it is formed according to the rules which we are now about to give.

CHAP. VI. Rules for the Fundamental Bafs.

190. All the notes of the fundamental baf being only capable of carrying a perfect chord, or the chord formation of the seventh, or that of the great fifth, are either tonics, or dominants, or sub-dominants; and the dominants may be either simple or tonic.

The fundamental baf ought always to begin with a tonic, as much as it is practicable. And now follow the rules for all the succeeding chords; rules which are evidently derived from the principles established in the First Part of this treatise. To be convinced of this, we shall find it only necessary to review the articles 34, 91, 122, 124, 126, 127.

RULE I.

191. In every chord of the tonic, or of the tonic dominant, it is necessary that at least one of the notes which form that chord should be found in the chord that precedes it.

RULE II.

192. In every chord of the simple dominant, it is necessary

(tt) We may only add, that there is no occasion for marking these sharps or flats when they are originally placed at the clef. For instance, if the sharp be upon the clef of fa (see Exam. X.), one may satisfy himself with simply writing re, without a sharp to mark the perfect chord major of re, re fa ♯ la re. In the same manner, in the Example XI. where the flat is at the clef upon fa, one may satisfy himself with simply writing sol, to mark the perfect chord minor of sol fa re fol.

But if a case occurs where there is a sharp or a flat at the clef, if any one should wish to render the chord minor which is major, or vice versa, he must place above the fundamental note a ♭ or natural. Thus the Example XII. marks the minor chord re fa la re, and Example XIII. the major chord sol fa re fol.—Frequently, in lieu of a natural, a flat is used to signify the minor chord, and a sharp to signify the major. Thus Example XIV. marks the minor chord re fa la re, and Example XV. the major chord sol fa re fol.

When in a chord of the great fifth, the dissonance, that is to say, the fifth, ought to be sharp, and when the sharp is not found at the clef, they write before or after the 6 a ♯; and if this fifth should be flat according to the clef, they write a ♭.

In the same manner, if in a chord of the seventh of the tonic dominant, the dissonance, that is to say, the seventh, ought to be flat or natural, they write by the side of the seventh a ♭ or a ♮. Many musicians, when a seventh from the simple dominant ought to be altered by a sharp or a natural, have likewise written by the side of the seventh a ♯ or a ♭; but M. Rameau suppresses these characters. The reason shall be given below, when we speak of chords by supposition.

If there be a sharp on the clef of fa, and if I should incline to mark the chord sol fa re fa, or the chord la ut mi fa, I would place before the seventh or the sixth a ♭ or a ♮.

In the same manner, if there be a flat on the clef at fa, and if I should incline to mark the chord ut mi sol fa, I would place before the seventh a ♯ or a ♭; and so of the rest. Principles necessary that the note which constitutes the seventh, or dissonance, should likewise be found in the preceding chord.

**Rule III.**

193. In every chord of the sub-dominant, at least one of its consonances must be found in the preceding chord. Thus, in the chord of the sub-dominant fa la ut re, it is necessary that fa, la, or ut, which are the consonances of the chord, should be found in the chord preceding. The dissonance re may either be found in it or not.

**Rule IV.**

194. Every simple or tonic dominant ought to descend by a fifth. In the first case, that is to say, when the dominant is simple, the note which follows can only be a dominant; in the second it may be any one you choose; or, in other words, it may either be a tonic, a tonic dominant, a simple dominant, or a sub-dominant. It is necessary, however, that the conditions prescribed in the second rule should be observed, if it be a simple dominant.

This last reflection is necessary, as you will presently see. For let us assume the succession of the two chords fa ut mi sol, re fa la ut (see Exam. XIX.), this succession is by no means legitimate, though in it the first dominant descends by a fifth; because the ut which forms the dissonance in the second chord, and which belongs to a simple dominant, is not in the preceding chord. But the succession will be admissible, if, without meddling with the second chord, one should take away the sharp carried by the ut in the first; or if, without meddling with the first chord, one should render ut or fa sharp in the second (uu); or in short, if one should simply render the re of the second chord a tonic dominant, in causing it to carry fa instead of fa natural (119. & 122.).

It is likewise by the same rule that we ought to reject the succession of the two following chords, re fa la ut, sol fa re fa (see Exam. XX.).

**Rule V.**

195. Every sub-dominant ought to rise by a fifth; and the note which follows it may, at your pleasure, be either a tonic, a tonic dominant, or a sub-dominant.

**Remark.**

Of the five fundamental rules which have now been given, instead of the three first, one may substitute the three following, which are nothing but consequences from them, and which you may pass unnoticed if you think it proper.

**Rule I.**

If a note of the fundamental bass be a tonic, and rise by a fifth or a third to another note, that second note may be either a tonic (34. & 91.), see Examples XXI. and XXII. (xx); a tonic dominant (124.), see XXIII. and XXIV.; or a sub-dominant (124.), see XXV. and XXVI.; or, to express the rule more simply, that second note may be any one you please, except a simple dominant.

**Rule II.**

If a note of the fundamental bass be a tonic, and descend by a fifth or a third upon another note, this second note may be either a tonic (34. & 91.), see Exam. XXVII. and XXVIII.; or a tonic dominant, or a simple dominant, yet in such a manner that the rule of art. 192. may be observed (124.), see XXIX. XXX. XXXI. XXXII.; or a sub-dominant (124.), see XXXIII. and XXXIV.

The procedure of the bass ut mi sol ut, fa la ut mi, from the tonic ut to the dominant fa (Ex. XXXV.), is excluded by art. 192.

**Rule III.**

If a note in the fundamental bass be a tonic, and rise by a second to another note, that note ought to be a tonic dominant, or a simple dominant (101. & 102.). See XXXVI. and XXXVII. (yy).

We must here advertise our readers, that the examples XXXVIII. XXXIX. XL. XLI. belong to the fourth rule above, art. 194.; and the examples XLII. XLIII. XLIV. to the fifth rule above, art. 195. See the articles 34, 35, 121, 123, 124.

**Remark I.**

196. The transition from a tonic dominant to a perfect and tonic is called an absolute repose, or a perfect cadence imperfect (73.); and the transition from a sub-dominant to a tonic is called an imperfect or irregular cadence (73.); how ever the cadences are formed at the distance of four bars played, one from another, whilst the tonic then falls within the first time of the bar. See XLV. and XLVI.

**Remark II.**

197. We must avoid, as much as we can, syncopations in the fundamental bass; that, within the first time of which a bar is constituted, the ear may be entertained with a harmony different from that which it had had by licence.

(uu) In this chord it is necessary that the ut and fa should be sharp at the same time; for the chord re fa la ut, in which ut would be sharp without the fa, is excluded by art. 179.

(xx) When the bass rises or descends from one tonic to another by the interval of a third, the mode is commonly changed; that is to say, from a major it becomes a minor. For instance, if I ascend from the tonic ut to the tonic mi, the major mode of ut, ut mi sol ut, will be changed into the minor mode of mi, mi sol fa mi. For what remains, we must never ascend from one tonic to another, when there is no sound common to both their modes; for example, you cannot rise to the mode of ut, ut mi sol ut, from the minor mode of mi, mi sol fa mi (91.).

(yy) By this we may see, that all the intervals, viz. the third, the fifth, and second, may be admitted in the fundamental bass, except that of a second in descending. For what remains, it is very proper to remark, that the rules immediately given for the fundamental bass are not without exception, as approved compositions in music will certainly discover; but these exceptions being in reality licences, and for the most part in opposition to the great principle of connection, which prescribes that there should be at least one note in common between a preceding and a subsequent chord, it does not seem necessary to entertain initiates with a minute detail of these licences in an elementary work, where the first and most essential rules of the art alone ought to be expected. Part II.

Principles had before perceived in the last time of the preceding of Compo- bar. Nevertheless, syncopation may be sometimes admitted in the fundamental bas, but it is by a li- cence (22).

CHAP. VII. Of the Rules which ought to be ob- served in the Treble with relation to the Funda- mental Bas's.

198. The treble is nothing else but a modulation above the fundamental bas, and whose notes are found in the chords of that bas which corresponds with it (189.). Thus in Ex. XVIII. the scale ut re mi fa sol la si ut, is a treble with respect to the fundamental bas's ut sol fa ut re sol ut.

199. We are just about to give the rules for the treble; but first we think it necessary to make the two following remarks.

1. It is obvious, that many notes of the treble may answer to one and the same note in the fundamental bas, when these notes belong to the chord of the same note in the fundamental bas. For example, this modulation ut mi sol mi ut, may have for its funda- mental bas the note ut alone, because the chord of that note comprehends the sounds ut, mi, sol, which are found in the treble.

2. In like manner, a single note in the treble may,

Vol. XII. Part II.

(22) There are notes which may be found several times in the fundamental bas in succession with a differ- ent harmony. For instance, the tonic ut, after having carried the chord ut mi sol ut, may be followed by another ut which carries the chord of the seventh, provided that this chord be the chord of the tonic dominant ut mi sol si. See LXXII. In the same manner, the tonic ut may be followed by the same tonic ut, which may be rendered a sub-dominant, by causing it to carry the chord ut mi sol la. See LXXIII.

A dominant, whether tonic or simple, sometimes descends or rises upon one another by the interval of a tritone or false fifth. For example, the dominant fa, carrying the chord fa la ut mi, may be followed by another dominant fa, carrying the chord si re fa la. This is a licence in which the musician indulges himself, that he may not be obliged to depart from the scale in which he is; for instance, from the scale of ut to which fa and si belong. If one should descend from fa to si by the interval of a just fifth, he would then depart from that scale, because si is no part of it.

(AAA) There are often in the treble several notes which may, if we choose, carry no chord, and be regarded merely as notes of passage, serving only to connect between themselves the notes that do carry chords, and to form a more agreeable modulation. These notes of passage are commonly quavers. See Exam. XLVII., in which this modulation ut re mi fa sol, may be regarded as equivalent to this other, ut mi sol, as re and fa are no more than notes of passage. So that the bas's of this modulation may be simply ut sol.

When the notes are of equal duration, and arranged in a diatonic order, the notes which occupy the per- fect part of each time, or those which are accented, ought each of them to carry chords. Those which oc- cupy the imperfect part, or which are unaccented, are no more than mere notes of passage. Sometimes, however, the note which occupies the imperfect part may be made to carry harmony; but the value of this note is then commonly increased by a point which is placed after it, which proportionally diminishes the con- tinuance of the note that occupies the perfect time, and makes it pass more swiftly.

When the notes do not move diatonically, they ought generally all of them to enter into the chord which is placed in the lower part correspondent with these notes.

(BBB) There is, however, one case in which the seventh of a simple dominant may be found in a modulation without being prepared. It is when, having already employed that dominant in the fundamental bas, its se- venth is afterwards heard in the modulation, as long as this dominant may be retained. For instance, let us imagine this modulation,

and this fundamental bas,

(see Example LI.) ; the re of the fundamental bas answers to the two notes re ut of the treble. The dif- fidence ut has no need of preparation, because the note re of the fundamental bas having already been em- ployed for the re which precedes ut, the diffusion ut is afterwards presented, below which the chord re may be preferred, or re fa la ut. of the bass be a sub dominant, it ought to rise diatonomically. This dissonance, which rises or descends diatonomically, is what we have called a dissonance saved or resolved (129, 130). See LII. LIII. LIV.

One may likewise observe here, that, according to the rules for the fundamental bass which we have given, the note upon which the dissonance ought to descend or rise will always be found in the subsequent chord (ccc).

**CHAP. VIII. Of the Continued Bass, and its Rules.**

204. A continued or thorough bass, is nothing else but a fundamental bass whose chords are inverted. We invert a chord when we change the order of the notes which compose it. For example, if instead of the chord sol fa re fa, I should say fa re fa sol, or re fa sol fa, &c. the chord is inverted. Let us see then, in the first place, all the possible ways in which a chord may be inverted.

The ways in which a Perfect Chord may be Inverted.

205. The perfect chord ut mi sol ut may be inverted in two different ways.

1. Mi sol ut mi, which we call a chord of the sixth, composed of a third, a fifth, and an octave, and in this case the note mi is marked with a 6. (See LVI.)

2. Sol ut mi sol, which we call a chord of the sixth and fourth, composed of a fourth, a fifth, and an octave; and it is marked with a 4. (See LVII.)

The perfect minor chord is inverted in the same manner.

The ways in which the Chord of the Seventh may be Inverted.

206. In the chord of the tonic dominant, as sol fa re fa, the third major fa above the fundamental note sol is called a sensible note (77); and the inverted chord fa re fa sol, composed of a third, a false fifth, and sixth, is called the chord of the false fifth, and is marked with an 8 or a b5 (see LVIII. and LIX.)

The chord re fa sol fa, composed of a third, a fourth, and a sixth, is called the chord of the sensible sixth, and marked with a 6 or a x6. In this chord thus figured, the third is minor, and the sixth major, as it is easy to be perceived. (See LX.)

The chord fa sol fa re, composed of a second, a tritone, and a fifth, is called the chord of the tritone, and is marked thus 4+, thus x4, or thus x4. (See LXI.)

In the chord of the simple dominant re fa la ut, we find,

1. Fa la ut re, a chord of the great sixth, which is composed of a third, a fifth, and a sixth, and which is figured with a 6. See LXIII. (ddd).

2. La ut re fa, a chord of the lesser sixth, which is figured with a 6. See LXIV. (eee).

3. Ut re fa la, a chord of the second, composed of a second, a fourth, and a fifth, and which is marked with a 2. See LXII. (fff).

The ways in which the Chord of the Sub-Dominant may be Inverted.

208. The chord of the sub-dominant, as fa la ut re, may be inverted in three different manners; but the method of inverting it which is most in practice is the chord of the lesser sixth la ut re fa, which is marked with a 6, and the chord of the seventh re fa la ut. See LXIV.

Rules for the Continued Bass.

209. The continued bass is a fundamental bass, whose chords are only inverted in order to render it more in the taste of singing, and suitable to the voice. See LXV. in which the fundamental bass which in itself is monotonic and little suited for singing, ut sol ut sol ut sol ut, produces, by inverting its chords, this continued

(ccc) When the treble syncopates in descending diatonically, it is common enough to make the second part of the syncopate carry a discord, and the first a concord. See Example LV. where the first part of the syncopated note sol is in concord with the notes ut mi sol ut, which answer to it in the fundamental bass, and where the second part is a dissonance in the subsequent chord la ut mi sol. In the same manner, the first part of the syncopated note fa is in concord with the notes re fa la ut, which answer to it; and the second part is a dissonance in the subsequent chord sol fa re fa, which answer to it, &c.

(ddd) We are obliged to mark likewise, in the continued bass, the chord of the sub-dominant with a 6, which in the fundamental bass is figured with a 6 alone; and this to distinguish it from the chords of the sixth and of the lesser sixth. (See Examples LVI. and LXIV.) For what remains, the chord of the great fifth in the fundamental bass carries always the sixth major, whereas in the continued bass it may carry the sixth minor. For instance, the chord of the seventh ut mi sol fa, gives the chord of the great fifth mi sol fa ut, thus improperly called, since the sixth from mi to ut is minor.

(eee) M. Rameau has justly observed, that we ought rather to figure this lesser sixth with a 4, to distinguish it from the sensible sixth which arises from the chord of the tonic dominant, and from the sixth which arises from the perfect chord. In the mean time he figures in his works with a 6 alone, the lesser sixths which do not arise from the tonic dominant; that is to say, he figures them as those which arise from the perfect chord; and we have followed him in that, though we thought with him, that it would be better to mark this chord by a particular figure.

(fff) The chord of the seventh fa re fa la gives, when inverted, the chord fa la fa re, composed of a third, a tritone, and a fifth. This chord is commonly marked with a 6, as if the tritone were a just fourth. It is his business who performs the accompaniment, to know whether the fourth above fa be a tritone or a fourth redundant. One may, as to what remains, figure this chord thus 6. Part II.

RULE I.

210. Every note which carries the chord of the false fifth, and which of consequence must be what we have called a sensible note, ought (77) to rise diatonically upon the note which follows it. Thus in example L.XV. the note fi, carrying the chord of the false fifth marked with an 8, rises diatonically upon ut (HHH).

(666) The continued bass is proportionably better adapted to fingering, as the sounds which form it more scrupulously observe the diatonic order, because this order is the most agreeable of all. We must therefore endeavour to preserve it as much as possible. It is for this reason that the continued bass in Example L.XV. is much more in the taste of fingering, and more agreeable, than the fundamental bass which answers to it.

(HHH) The continued bass being a kind of treble with relation to the fundamental bass, it ought to observe the same rules with respect to that bass as the treble. Thus a note, for instance re, carrying a chord of the seventh re fa la ut, to which the chord of the sub-dominant fa la ut re corresponds in the fundamental bass, ought to rise diatonically upon mi, (art. 129, no. 2, and art. 322.)

(III) When there is a repose in the treble, the note of the continued bass ought to be the same with that of the fundamental bass, (see example L.XVII.) In the cloes which are found in the treble at fi and ut (bars third and fourth), the notes in the fundamental and continued bass are the same, viz. sol for the first cadence, and ut for the second. This rule ought above all to be observed in final cadences which terminate a piece or a modulation.

It is necessary, as much as possible, to prevent coincidences of the same notes in the treble and continued bass, unless the motion of the continued bass should be contrary to that of the treble. For example, in the second note of the second bar in example L.XVII. mi is found at the same time in the continued bass and in the treble; but the treble descends from fi to mi, whilst the bass rises from re to mi.

Two octaves, or two fifths, in succession, must likewise be shunned. For instance, in the treble sounds sol mi, the bass must be prevented from founding sol mi, ut la, or re fi, because in the first case there are two octaves in succession, sol against sol, and mi against mi; and because in the second case there are two fifths in succession, sol against sol, and la against mi, or re against sol, and fi against mi. This rule, as well as the preceding, is founded upon this principle, that the continued bass ought not to be a copy of the treble, but to form a different melody.

Every time that several notes of the continued bass answer to one note alone of the fundamental, the composer satisfies himself with figuring the first of them. Nay he does not even figure it if it be a tonic; and he draws above the others a line, continued from the note upon which the chord is formed. See example L.XVIII., where the fundamental bass ut gives the continued bass ut mi sol mi; the two mi's ought in this bass to carry the chord 6, and sol the chord 4; but as these chords are comprehended in the perfect chord ut mi sol ut, which is the first of the continued bass, we place nothing above ut, only we draw a line over ut mi sol mi.

In like manner, in the second bar of the same example, the notes fa and re of the continued bass, rising from the note sol alone of the fundamental bass which carries the chord sol fi re fa, we think it sufficient to figure fa with the number of the tritone 4x, and to draw a line above fa and re.

It should be remarked, that this fa ought naturally to descend to mi; but this note is considered as subsisting so long as the chord subsists; and when the chord changes, we ought necessarily to find the mi, as may be seen by that example.

In general, whilst the same chord subsists in passing through different notes, the chord is reckoned the same as if the first note of the chord had subsisted; in such a manner, that, if the first note of the chord is, for instance, the sensible note, we ought to find the tonic when the chord changes. See example L.XIX. or this continued bass, ut fi sol fi re ut, is reckoned the same with this ut fi ut. (Example L.XX.)

If a single note of the continued bass answers to several notes of the fundamental bass, it is figured with the different chords which agree to it. For example, the note sol in a continued bass may answer to this fundamental bass ut sol ut. (see example L.XX.I.) In this case, we may regard the note sol as divided into three parts, of which the first carries the chord 6, the second the chord 7, and the third the chord 4.

We shall repeat here, with respect to the rules of the continued bass, what we have formerly said concerning the rules of the fundamental bass in the note upon the third rule, art. 193. The rules of the continued bass have exceptions, which practice and the perusal of good authors will teach. There are likewise several other rules which might require a considerable detail, and which will be found in the Treatise of Harmony by M. Rameau, and elsewhere. These rules, which are proper for a complete dissertation, did not appear to me indispensably necessary in an elementary essay upon music, such as the present. The books which we have quoted at the end of our preliminary discourse will more particularly instruct the reader concerning this practical detail. § 1. Of Broken and Interrupted Cadences.

213. The broken cadence is executed by means of a dominant which rises diatonically upon another, or upon a tonic by a licence. See, in the example LXXIV. sol la, (142, and 134).

214. The interrupted cadence is formed by a dominant which descends by a third upon another (136). See, in the example LXXV. sol mi (L.L.L.).

These cadences ought to be permitted but rarely and with precaution.

§ 2. Of Supposition.

215. When a dominant is preceded by a tonic in what the fundamental bass, we add sometimes, in the continued bass to the chord of that dominant, a new note which is a third or a fifth below; and the chord which results from it in this continued bass is called a chord by supposition.

For example, let us suppose, that in the fundamental bass we have a dominant sol carrying the chord of the seventh sol fa re fa; let us add to this chord the note ut, which is a fifth below this dominant, and we shall have the total chord ut sol fa re fa, or ut re fa sol fa, which is called a chord by supposition (MMM).

Of the different kinds of chords by supposition.

216. It is easy to perceive, that chords by supposition are of different kinds. For instance, the chord of the tonic sol fa re fa gives,

1. By adding the fifth ut, the chord ut sol fa re fa, called a chord of the seventh redundant, and composed of a fifth, seventh, ninth, and eleventh. It is figured with a \( \frac{7}{9} \); see LXXVI. (NNN). This chord is not practised but upon the tonic. They sometimes have what, and how figured in the note QQQ, upon the art. 219; it is then reduced to ut fa sol re, and marked with \( \frac{7}{9} \) or \( \frac{9}{7} \).

2. By adding the third mi, we shall have the chord mi sol fa re fa, called a chord of the ninth, and composed of a third, fifth, seventh, and ninth. It is figured with a 9. This third may be added to every third of the dominant. See LXXVII. (OOO).

3. If to a chord of the simple dominant, as re fa la ut, we should add the fifth sol, we would have the chord sol re fa la ut, called a chord of the eleventh, and which is figured with a \( \frac{9}{7} \) or \( \frac{7}{9} \). (See LXXVIII.)

Observe.

217. When the dominant is not a tonic dominant, they often take away some notes from the chord. For example, let us suppose that there is in the fundamental bass this simple dominant mi, carrying the chord mi sol fa re; if there should be added the third ut beneath, we shall have this chord of the continued bass ut mi sol fa re, but they suppress the seventh \( \frac{7}{9} \), for reasons which shall be explained in the note QQQ upon art.

(L.L.L.) One may sometimes, but very rarely, cause several tonics in succession to follow one another in ascending or descending diatonically, as ut mi sol ut, re fa la re, sol re fa \( \frac{7}{9} \); but, besides that this succession is harsh, it is necessary, in order to render it practicable, that the fifth below the first tonic should be found in the chord of the tonic following, as here fa, a fifth below the first tonic ut, is found in the chord re fa la re, and in the chord sol re fa \( \frac{7}{9} \) (37 and note G).

(MMM) Though supposition be a kind of licence, yet it is in some measure founded on the experiment related in the note \( \frac{7}{9} \), where you may see that every principal or fundamental sound causes its twelfth and seventeenth major in descending to vibrate, whilst the twelfth and the seventeenth major ascending refund: which seems to authorize us in certain cases to join with the fundamental harmony this twelfth and seventeenth in descending; or, which is the same thing, the fifth or the third beneath the fundamental sound.

Even without having recourse to this experiment, we may remark, that the note added beneath the fundamental sound, causes that very fundamental sound to be heard. For instance, ut added beneath sol, causes sol to refund. Thus sol is found in some measure to be implied in ut.

If the third added beneath the fundamental sound be minor, for example, if to the chord sol fa re fa, we add the third mi, the supposition is then no longer founded on the experiment, which only gives the seventeenth major, or, what is the same thing, the third major beneath the fundamental sound. In this case the addition of the third minor must be considered as an extension of the rule, which in reality has no foundation in the chords emitted by a sonorous body, but is authorized by the sanction of the ear and by practical experiment.

(NNN) Many musicians figure this chord with a \( \frac{7}{9} \); M. Rameau suppresses this 2, and merely marks it to be the seventh redundant by a \( \frac{7}{9} \) or \( \frac{9}{7} \). But it may be said, how shall we distinguish this chord from the seventh major, which, as it would seem, ought to be marked with a \( \frac{7}{9} \)? M. Rameau answers, that there is no danger of mistake, because in the seventh major, as the seventh ought to be prepared, it is found in the preceding chord; and thus the sharp subsisting already in the preceding chord, it would be useless to repeat it.

Thus re sol, according to M. Rameau, would indicate re fa \( \frac{7}{9} \) la ut, sol fa re \( \frac{7}{9} \). If we would change fa \( \frac{7}{9} \) of the second chord into fa, it would then be necessary to write re sol. In notes such as ut, whose natural seventh is major, the figure 7 preceded or followed by a sharp will sufficiently serve to distinguish the chord of the seventh redundant ut sol fa re fa, from the simple chord of the seventh ut mi sol fa, which is marked with a 7 alone. All this appears just and well founded.

(OOO) Supposition introduces into a chord dissonances which were not in it before. For instance, if to the chord mi sol fa re, we should add the note of supposition ut descending by a third, it is plain that, besides the dissonance... Part II.

MUSI C.

Principles art. 219. In this state the chord is simply composed of a third, fifth, and ninth, and is marked with a g. See LXXIX. (ppp)

218. What is more, in the chord of the simple dominant, as re fa la ut, when the fifth sol is added, they frequently obliterate the sounds fa and la, that too great a number of dissonances may be avoided, which reduces the chord to sol ut re. This last is composed only of the fourth and the fifth. It is called a chord of the fourth, and it is figured with a 4. (See LXXX.)

219. Sometimes they only remove the note la, and then the chord ought to be figured with 2 or 4 (qqq).

220. Finally, in the minor mode, for example, in that of la, where the chord of the tonic dominant (109) is mi sol fa re; if we add to this chord the third ut below, we shall have ut mi sol fa re, called the chord of the fifth redundant, and composed of a third, a fifth redundant, a seventh, and a ninth. It is figured with a 5, or +5. See LXXXI. (rrr.)

§ 3. Of the Chord of the Diminished Seventh.

221. In the minor mode, for instance, in that of la, mi a fifth from la is the tonic dominant (109), and carries the chord mi sol fa re, in which sol is the sensible note. For this chord they sometimes substitute that other sol fa re fa, (116), all composed of minor thirds; and which has for its fundamental sound the sensible note sol. This chord is called a chord of the flat, or diminished seventh, and is figured with a 7 in the fundamental bals, (see LXXXII.) but it is always considered as representing the chord of the tonic dominant.

222. This chord in the fundamental bals produces in the continued bals the following chords:

1. The chord fa re fa sol, composed of a third, false fifth, and sixth major. They call it the chord of the what, and sixth sensible and false fifth; and it is figured thus 7, how fig. or +5. (See LXXXIII.)

2. The chord re fa sol fa, composed of a third, a tritone, and a sixth, they call it the chord of the tritone and third minor; and they mark it thus 7. (See LXXXIV.)

3. The chord fa sol fa re, composed of a second redundant, a tritone, and a sixth. They call it the chord of the second redundant, and they figure it thus 7, or +2. See LXXXV. (sss).

223. Besides, since the chord sol fa re fa represents the chord mi sol fa re, it follows, that if we operate by supposition upon the first of these chords, which it must be performed as one would perform it upon mi what, and how figured.

diffonance between mi and re which was in the original chord, we have two new diffonances, ut fi, and ut re; that is to say, the seventh and the ninth. These diffonances, like the others, ought to be prepared and resolved. They are prepared by being syncopated, and resolved by descending diatonically upon one of the confonances of the subsequent chord. The sensible note alone can be resolved in ascending; but it is even necessary that this sensible note should be in the chord of the tonic dominant. As to the diffonances which are found in the primitive chord, they should always follow the common rules. (See art. 202.)

(ppp) Several musicians call this last chord the chord of the ninth; and that which, with M. Rameau, we have simply called a chord of the ninth, they term a chord of the ninth and seventh. This last chord they mark with a 7; but the denomination and figure used by M. Rameau are more simple, and can lead to no error; because the chord of the ninth always includes the seventh, except in the cases of which we have already spoken.

(qqq) They often remove some diffonances from chords of supposition, either to soften the harshness of the chord, or to remove discords which can neither be prepared nor resolved. For instance, let us suppose, that in the continued bals the note ut is preceded by the sensible note fi, carrying the chord of the false fifth, and that we should choose to form upon this note ut the chord ut mi sol fa re, we must obliterate the seventh fi, because in retaining it we should destroy the effect of the sensible note fi, which ought to rise to ut.

In the same manner, if to the harmony of a tonic dominant sol fa re fa, one should add the note by supposition ut, it is usual to retrench from this chord the sensible note fi; because, as the re ought to descend diatonically to ut, and the fi to rise to it, the effect of the one would destroy that of the other. This above all takes place in the supposition, concerning which we shall presently treat.

(rrr) Supposition produces what we call suspension; and which is almost the same thing. Suspension consists in retaining as many as possible of the sounds in a preceding chord, that they may be heard in the chord which succeeds. For instance, if this fundamental bals be given ut sol ut, and this continued bals above it ut ut ut, it is a supposition; but if we have this fundamental bals ut sol sol ut, and this continued bals above it ut sol ut ut, it is a supposition; because the perfect chord of ut, which we naturally expect after sol in the continued bals, is suspended and retarded by the chord ut, which is formed by retaining the sounds sol fi re fa of the preceding chord to join them to the note ut in this manner, ut sol fi re fa; but this chord ut does nothing in this case but suspend for a moment the perfect chord ut mi sol ut, which ought to follow it.

(sss) The chord of the diminished seventh, such as sol fi re fa, and the three derived from it, are termed chords of substitution. They are in general harsh, and proper for imitating melancholy objects. Principles of Composition: To add to the chord *sol* *fa* *re* *la*, the notes *ut* or *la*, which are the third or fifth below *mi*, and which will produce:

1. By adding *ut*, the chord *ut* *sol* *fa* *re* *la*, composed of a fifth redundant, a seventh, a ninth, and eleventh, which is the octave of the fourth. It is called a chord of the fifth redundant and fourth, and thus marked *4*, or *8* (See LXXXVI.)

2. By adding *la*, we shall have the chord *la* *sol* *fa* *re* *la*, composed of a seventh redundant, a ninth, an eleventh, and a thirteenth minor, which is the octave of the sixth minor. It is called the chord of the seventh redundant and sixth minor, and marked *6*, or *8* (See LXXXVII.) It is of all chords the most harsh, and the most rarely practised (***).

In the Treatise of Harmony by M. Rameau, and elsewhere, may be seen a much longer detail of the chords by supposition; but here we delineate the elements alone.

**Chap. X. Of some Licences used in the Treble and Continued Basso.**

Licence 1st.

224. Sometimes in a treble, the dissonance which ought to have been resolved by descending diatonically upon the succeeding note, instead of descending, on the contrary rises diatonically; but in that case, the note upon which it ought to have descended must be found in some of the other parts. This licence ought to be rarely practised.

In like manner, in a continued bass, the dissonance in a chord of the sub-dominant inverted, as *la* in the chord *la* *ut* *mi* *sol*, inverted from *ut* *mi* *sol* *la*, may sometimes descend diatonically instead of rising as it ought to do, art. 129, n° 2; but in that case the note ought to be repeated in another part, that the dissonance may be there resolved in ascending.

Licence 2nd.

225. Sometimes likewise, to render a continued bass more agreeable by causing it to proceed diatonically, they place between two fouads of that bass a note which belongs to the chord of neither. See example principles XCVI, in which the fundamental bass *sol* *ut* produces the continued bass *la* *fa* *sol* *ut*, where *la* is added on account of the diatonic modulation. This *la* has a line drawn above it to show its resolution by passing under the chord *sol* *fa* *re* *la*.

In the same manner, (see XCV), this fundamental bass *ut* *fa* may produce the continued bass *ut* *re* *mi* *ui* *fa*, where the note *re* which is added passes under the chord *ut* *mi* *sol* *ut*.

**Chap. XI. Containing the Method of finding the Fundamental Bass when the Continued Bass is figured.**

226. To exercise yourself with greater ease in finding the fundamental bass, and to render it more familiar to you, it is necessary to observe how eminent mental bass writers, and above all how M. Rameau has put the rules in practice. Now, as they never place anything but figured, the continued bass in their works, it becomes then necessary to know how to find the fundamental bass when the continued bass is figured. This problem may be easily solved by the following rules.

227. 1. Every note which has no figure in the continued bass, ought to be the same, and without a figure in the fundamental bass; it either is a tonic, or reckoned such. (**v**)

2. Every note which in the continued bass carries a 6, ought in the fundamental bass to give its third below not figured *, or its fifth below marked with a 7. *See Figure LXIV, and the note zzz).

3. Every note carrying *gives in the fundamental bass its fifth below not figured. (See LVII.)

4. Every note figured with a 7 or a *g*, is the same in both basses, and with the same figure (**x**).

5. Every note figured with a 2 gives in the fundamental bass the diatonic note above figured with a 7. See LXII. (**v**).

6. Every note marked with a 4 gives in the fundamental bass its third above, figured with a 6. For example, this continued bass *la* *fa* *ut* gives this fundamental bass *ut* *sol* *ut*; but in this case it is necessary that the note figured with a 6 should rise by a fifth, as we see here *ut* rises to *sol*.

(**xxx**) Sometimes a note which carries a 7 in the continued bass, gives in the fundamental bass its third above, figured with a 6. For example, this continued bass *la* *fa* *ut* gives this fundamental bass *ut* *sol* *ut*; but in this case it is necessary that the note figured with a 6 should rise by a fifth, as we see here *ut* rises to *sol*.

(**vvv**) A note figured with a 2, gives likewise sometimes in the fundamental bass its fourth above, figured with Part II.

Principles of Composition.

7. Every note figured with an 8 gives its third below figured with a 7. (See LXXXIII.)

8. Every note marked with a 9 gives the fifth below marked with a 7; (see LX.) and it is plain by art. 187, that in the chord of the seventh, of which we treat in these three last articles, the third ought to be major, and the seventh minor, this chord of the seventh being the chord of the tonic dominant. (See art. 102.)

9. Every note marked with a 9 gives its third above figured with a 7. (See LXXXIV and LXXXV.)

10. Every note marked with a 9 gives the fifth above figured with a 7. (See LXXXVI.)

11. Every note marked with a 9, or with a +5, gives the third above figured with a 7. (See LXXXVII.)

12. Every note marked with a 9 gives a fifth above figured with a 7, or with a 9. (See LXXXVIII.)

It is the same case with the notes marked 4, 5, or 6: which shows a retrenchment, either in the complete chord of the eleventh, or in that of the seventh redundant.

13. Every note marked with a 4 gives a fifth above figured with a 7, or a 9. (See LXXXIX.)

14. Every note marked with a 9 gives the third minor below, figured with a 7. (See LXXXIII.)

15. Every note marked with a 9 gives the tritone above figured with a 7. (See LXXXIV.)

16. Every note marked with a 9 gives the second redundant above, figured with a 7. (See LXXXV.)

17. Every note marked with a 9 gives the fifth redundant above, figured with a 7. (See LXXXVI.)

18. Every note marked with a 9 gives the seventh redundant above, figured with a 7. See LXXXVII.

(zzz).

REMARK.

228. We have omitted two cases, which may cause some uncertainty.

The first is that where the note of the continued bass is figured with a 6. We now present the reason bass.

Suppose we should have the dominant re in the fundamental bass, the note which answers to it in the continued bass may be la carrying the figure 6 (see LXIV.); that is to say, the chord la ut re fa: now if we should have the sub-dominant fa in the fundamental bass, but it is necessary in that case that the note figured with a 6, may even here rise to a fifth. (See note XXX).

These variations in the fundamental bass, as well in the chord concerning which we now treat, as in the chord figured with a 7, and in two others which shall afterwards be mentioned (art. 228 and 229), are caused by a deficiency in the signs proper for the chord of the sub-dominant, and for the different arrangements by which it is inverted.

M. l'Abbe Rouffier, to redress this deficiency, had invented a new manner of figuring the continued bass. His method is most simple for those who know the fundamental bass. It consists in expressing each chord by only signifying the fundamental found with that letter of the scale by which it is denominated, to which is joined a 7 or 6, or a 6, in order to mark all the discords. Thus the fundamental chord of the seventh re fa la ut is expressed by a D; and the same chord, when it is inverted from that of the sub-dominant fa la ut re, is characterized by F; the chord of the second ut re fa la, inverted from the dominant re fa la ut, is likewise represented by D; and the same chord ut re fa la inverted from that of the sub-dominant fa la ut re is signified by F; the case is the same when the chords are differently inverted. By this means it would be impossible to mistake either with respect to the fundamental bass of a chord, or with respect to the note which forms its dissonance, or with respect to the nature and species of that discord.

(zzz) We may only add, that here and in the preceding articles, we suppose, that the continued bass is figured in the manner of M. Rameau. For it is proper to observe, that there are not, perhaps, two musicians who characterize their chords with the same figures; which produces a great inconvenience to the person who plays the accompaniments: but here we do not treat of accompaniments. For every reason, then, we should advise initiates to prefer the continued basses of M. Rameau to all the others, as by them they will most successfully study the fundamental bass.

It is even necessary to advertise the reader, and I have already done it (note XXX), that M. Rameau only marks the lesser sixth by a 6 without a line, when this lesser sixth does not result from the chord of the tonic dominant; in such a manner that the 6 renders it uncertain whether in the fundamental bass we ought to choose the third or the fifth below; but it will be easy to see whether the third or the fifth is signified by that figure. This may be distinguished, 1. In observing which of the two notes is excluded by the rules of the fundamental bass. 2. If the two notes may with equal propriety be placed in the fundamental bass, the preference must be determined by the tone or mode of the treble in that particular passage. In the following chapter we shall give rules for determining the mode.

There is a chord of which we have not spoken in this enumeration, and which is called the chord of the sixth redundant. This chord is composed of a note, of its third major, of its redundant fourth or tritone, and its redundant sixth, as fa la fa re. It is marked with a 6. It appears difficult to find a fundamental bass for this chord; nor is it indeed much in use amongst us. (See the note upon the art. 115.) mental bas, this sub-dominant might produce in the continued bas the same note la figured with a 6. When therefore one finds in the continued bas a note marked with a 6, it appears at first uncertain whether we should place in the fundamental bas the fifth below marked with a 7, or the third below marked with a 6.

229. The second case is that in which the continued bas is figured with a 6. For instance, if there should be found fa in the continued bas, one may be ignorant whether he ought to insert in the fundamental bas fa marked with a 6, or re figured with a 7.

230. You may easily extricate yourself from this little difficulty, in leaving for an instant this uncertain note in suspense, and in examining what is the succeeding note of the fundamental bas; for if that note be in the present case a fifth above fa, that is to say, if it is ut in this case, and in this alone, he may place fa in the fundamental bas. It is a consequence of this rule, that in the fundamental bas every sub-dominant ought to rise by a fifth (195).

CHAP. XII. What is meant by being in a Mode or Tone.

231. In the first part of this treatise (chap. vi), we have explained, how by the means of the note ut, and of its two fifths sol and fa, one in ascending, which is called a tonic dominant, the other in descending, which is called a sub-dominant, the scale ut re mi fa sol la fa ut may be found; the different sounds which form this scale compose what we call the major mode of ut, because the third mi above ut is major. If therefore we would have a modulation in the major mode of ut, no other sounds must enter into it than those which compose this scale; in such a manner that if, for instance, I should find fa in this modulation, this fa discovers to me that I am not in the mode of ut, or at least that, if I have been in it, I am no longer so.

232. In the same manner, if I form this scale in ascending la fa ut re mi fa sol la, which is exactly similar to the scale ut re mi fa sol la fa ut of the major mode of ut, this scale, in which the third from la to la is major, shall be in the major mode of la; and if of composition I incline to be in the minor mode of la, I have nothing to do but to substitute for ut sharp ut natural; so that the major third la ut may become minor la ut; I shall have then

la si ut re mi fa sol la,

which is (85) the scale of the minor mode of la in ascending; and the scale of the minor mode of la in descending shall be (90)

la sol fa mi ut re si la,

in which the sol and fa are no longer sharp. For it is a singularity peculiar to the minor mode, that its scale is not the same in rising as in descending (89).

233. This is the reason why, when we wish to begin a piece in the major mode of la, we place three sharps at the clef upon fa, ut, and sol; and on the contrary, in the minor mode of la, we place none, because the minor mode of la, in descending, has neither sharps nor flats.

234. As the scale contains twelve sounds, each diatonic mode of la differs from the other by the interval of a semitone, it is obvious that each of these sounds can produce both a major and a minor mode, which constitute 24 modes the minor upon the whole. Of these we shall immediately give mode in descending, a table, which may be very useful to discover the mode in which we are.

A TABLE of the Different Modes.

| Maj. Mode | ut re mi fa sol la fa ut | |-----------|-------------------------| | ut | ut re mi fa sol la fa ut | | sol | sol la fa re mi fa sol | | re | re mi fa sol la fa ut re | | la | la si ut re mi fa sol la | | mi | mi fa sol la fa ut re mi | | fa | fa ut re mi fa sol la fa | | sol | sol fa sol la fa sol la | | ut | ut re mi fa sol la fa ut | | sol | sol la fa re mi fa sol | | re | re mi fa sol la fa ut re | | la | la si ut re mi fa sol la | | mi | mi fa sol la fa ut re mi | | fa | fa ut re mi fa sol la fa | | sol | sol fa sol la fa sol la | | ut | ut re mi fa sol la fa ut |

(AAAA) The major mode of fa, of mi or re, and of sol or la, are not much practised. In the opera of Pyramus and Thisbe, p. 267, there is a passage in the scene, of which one part is in the major mode of fa, and the other in the major mode of ut, and there are six sharps at the clef.

When a piece begins upon ut, there ought to be seven sharps placed at the clef; but it is more convenient only to place five flats, and to suppose the key rel, which is almost the same thing with ut. It is for this reason that we substitute here the mode of rel for that of ut.

It is still much more necessary to substitute the mode of la for that of sol; for the scale of the major mode of sol is

sol la fa ut re mi fa sol la,

in which you may see that there are at the same time both a sol natural and a sol; it would then be necessary, even at the same time, that upon sol there should and should not be a sharp at the clef; which is shocking and inconsistent. It is true that this inconvenience may be avoided by placing a sharp upon sol at the clef, and by marking the note sol with a natural through the course of the music wherever it ought to be natural; but this would become troublesome, above all if there should be occasion to transpose. In the article 236, we shall give an account of transposition. One might likewise in this series, instead of sol natural, which is the note immediately before the last, substitute fa, that is to say, fa twice sharp; which, however, is not absolutely the same sound with sol natural, especially upon instruments whose scales are fixed, or whose intervals are invariable. But in that case two sharps may be placed at the clef upon fa, which would produce another inconvenience. But by substituting la for sol, the trouble is eluded. Minor Modes.

Of la

In descending. la sol fa mi re ut si la. In rising. la si ut re mi fa sol la.

Of mi

In descending. mi re ut si la sol fa mi. In rising. mi fa sol la si ut re mi.

Of fa

In descending. fa mi re ut si la sol fa. In rising. fa sol la si ut re mi fa.

Of ut

In descending. ut si la sol fa mi re ut. In rising. ur re mi fa sol la si ut.

Of re

In descending. re ut si la sol fa mi re ut. In rising. re mi fa sol la si ut.

235. These then are all the modes, as well major Modes as minor. Those which are crowded with sharps and flats are little practised, as being extremely difficult in execution.

236. From thence it follows,

1. That when there are neither sharps nor flats at the clef, it is a token that the piece begins in the major mode of ut, or in the minor mode of la.

2. That when there is one single sharp, it will always be placed upon fa, and that the piece begins in the major mode of sol, or the minor of mi, in such a manner that it may be sung as if there were no sharp, by singing fa instead of fa, and in singing the tune as if it had been in another clef. For instance, let there be a sharp upon fa in the clef of sol upon the first line; one may then sing the tune as if there were no sharp: And instead of the clef of sol upon the first line, let there be the clef of ut; for the fa, when changed into fa, will require that the clef of sol should be changed to the clef of ut, as may be easily seen. This is what we call transposition (†).

237. It is evident, that when fa is changed into fa, sol must be changed into ut, and mi into la. Thus by transposition, the air has the same melody as if it were in the major mode of ut, or in the minor mode of la.

(BBBB) We have already seen, that in each mode, the principal note is called a tonic; that the fifth above that note is called a tonic dominant, or the dominant of the mode, or simply a dominant; that the fifth beneath the tonic, or, what is the same thing, the fourth above that tonic, is called a sub-dominant; and in short, that the note which forms a semitone beneath the tonic, and which is a third major from the dominant, is called a sensible note. The other notes have likewise in every mode particular names which it is advantageous to know. Thus a note which is a tone immediately above the tonic, as re in the mode of ut, and si in that of la, is termed a sub-tonic; the following note, which is a third major or minor from the tonic, according as the chord is major or minor, such as mi in the major mode of ut, and ut in the minor mode of la, is called a mediant; in short, the note which is a tone above the dominant, such as la in the mode of ut, and fa in that of la, is called a sub-dominant.

† Though our author's account of this delicate operation in music will be found extremely just and compendious; though it proceeds upon simple principles, and comprehends every possible contingency; yet as the manner of thinking upon which it depends may be less familiar to English readers, if not profoundly skilled in music, it has been thought proper to give a more familiar, though less comprehensive, explanation of the manner in which transposition may be executed.

It will easily occur to every reader, that if each of the intervals through the whole diatonic series were equal, in a mathematical sense, it would be absolutely indifferent upon what note any air were begun, if within the compass of the gammut; because the same equal intervals must always have the same effects. But since, besides the natural semitones, there is another distinction of diatonic intervals into greater and lesser tones; and since these vary their positions in the series of an octave, according as the note from whence you begin is placed, that note is consequently the best key for any tune whose natural series is most exactly correspondent with the intervals which that melody or harmony requires. But in instruments whose scales are fixed, notwithstanding the temperament and other expedients of the same kind, such a series is far from being easily found, and is indeed in common practice almost totally neglected. All that can frequently be done is, to take care that the ear may not be sensibly shocked. This, however, would be the case, if, in transposing any tune, the situation of the semitones, whether natural or artificial, were not exactly correspondent in the series to which your air must be transposed, with their positions in the scale from which you transpose it. Suppose of la. The major mode then of sol, and the minor of mi, are by transposition reduced to those of ut major, and of la minor. It is the same case with all the other modes, as any one may easily be convinced (cccc).

Chap. XIII. To find the Fundamental Bafs of a given Modulation.

238. As we have reduced to a very small number the rules of the fundamental bas, and those which in the treble ought to be observed with relation to this bas, it should no longer be difficult to find the fundamental bas of a given modulation, nay, frequently to find several; for every fundamental bas will be legit. Method of finding a fundamental bas to a timate, when it is formed according to the rules which we have given (Chap. VI.); and that, besides this, the dissonances which the modulation may form with given air this nor difficult, and why.

for instance, your air should begin upon ut or C, requiring the natural diatonic series through the whole gammut, in which the distance between mi and fa, or E and F, as also that between fa and ut, or B and C, is only a semitone. Again, suppose it necessary for your voice, or the instrument on which you play, that the same air should be transposed to sol or G, a fifth above its former key; then because in the first series the intervals between the third and the fourth, seventh and eighth-notes, are no more than semitones, the same intervals must take the same place in the octave to which you transpose. Now, from sol or G, the note with which you propose to begin, the three tones immediately succeeding are full; but the fourth, ut or C, is only a semitone; it may therefore be kept in its place. But from fa or F, the seventh note above, to sol or G, the eighth, the interval is a full tone, which must consequently be redressed by raising your fa a semitone higher. Thus the situations of the semitonic intervals in both octaves will be correspondent; and thus, by conforming the positions of the semitones in the octave to which you transpose, with those in the octave in which the original key of the tune is contained, you will perform your operation with as much success as the nature of fixed scales can admit: But the order in which you must proceed, and the intervals required in every mode, are minutely and ingeniously delineated by our author.

(cccc) Two sharps, fa and ut, indicate the major mode of re, or the minor of fa; and then, by transposition, the ut is changed into fa, and of consequence, re into ut and fa into la.

Three sharps, fa ut sol indicate the major mode of la, or the minor of fa; and it is then sol, which must be changed into fa, and of consequence la into ut, and fa into la.

Four sharps, fa ut sol re, indicate the major mode of mi, or the minor of ut; then the re is changed into fa, and of consequence mi into ut, and ut into la.

Five sharps, fa ut sol re la, indicate the major mode of fa, or the minor of sol; la then is changed into fa, and of consequence fa into ut, and sol into la.

Six sharps, fa ut sol re la mi, indicate the major mode of fa; mi then is changed into fa, and of consequence fa into ut.

Six flats, fa mi la re sol ut, indicate the minor mode of mi; ut is changed into fa, and of consequence mi into la.

Five flats, fa mi la re sol, indicate the major mode of re, or the minor mode of fa; then the fa is changed into fa, and of consequence re into ut, and the fa into la.

Four flats, fa mi la re, indicate the major mode of la, or the minor mode of fa; re then is changed into fa, and of consequence la into ut, and fa into la.

Three flats, fa mi la, indicate the major mode of mi, or the minor of ut; the la then is changed into fa, and of consequence mi into ut, and the sol into la.

Two flats, fa mi, indicate the major mode of fa, or the minor of sol; mi then is changed into fa, and of consequence fa into ut, and the fa into la.

One flat, fa, indicates the major mode of fa, or the minor mode of re, and fa is changed into fa; of consequence the fa is changed into ut, and the re into la.

All the major modes then may be reduced to that of ut, and the modes minor to that of la minor.

It only remains to remark, that many musicians, and amongst others the ancient musicians of France, as Lulli, Campra, &c., place one flat less in the minor mode: so that in the minor mode of re, they place neither sharp nor flat at the clef; in the minor mode of sol, one flat only; in the minor mode of ut, two flats, &c.

This practice in itself is sufficiently indifferent, and scarcely merits the trouble of a dispute. Yet the method which we have here described, according to M. Rameau, has the advantage of reducing all the modes to two; and besides it is founded upon this simple and very general rule, That in the major mode, we must place as many sharps or flats at the clef, as are contained in the diatonic scale of that mode in ascending; and in the minor mode, as many as are contained in that same scale in descending.

However this be, I here present you with a rule for transposition, which appears to me more simple than the rule in common use.

For the Sharps.

Suppose sol, re, la, mi, fa, ut, and change sol into ut if there is one sharp at the clef, re into ut if there are two sharps, la into ut if there are three, &c.

For the Flats.

Suppose fa, fa, mi, la, re, sol, and change fa into ut if there is only one flat at the clef, fa into ut if there are two flats, mi into ut if there are three, &c. Part II.

MUSIC

239. It is of the greatest utility in searching for the fundamental bass, to know what is the tone or mode of the melody to which that bass should correspond.

—But it is difficult in this matter to assign general rules, and such as are absolutely without exception, ascertaining in which nothing may be left that appears indifferent or discretionary; because sometimes we seem to have the free choice of referring a particular melody either to one mode or another. For example, this melody fol ut may belong to all the modes, as well major as minor, in which fol and ut are found together; and each of these two sounds may even be considered as belonging to a different mode.

240. For what remains, one may sometimes, as it should seem, operate without the knowledge of the mode, for two reasons: 1. Because, since the same sounds belong to several different modes, the mode is sometimes considerably undetermined; above all, in the middle of a piece, and during the time of one or two bars. 2. Without giving ourselves much trouble about the mode, it is often sufficient to preserve us from deviating in composition, if we observe in the simplest manner the rules above prescribed (ch. VI.) for the procedure of the fundamental bass.

241. In the mean time, it is above all things necessary to know in what mode we operate at the beginning of the piece, because it is indispensable that the fundamental bass should begin in the same mode, and that the treble and bass should likewise end in it; of composition, that they should even terminate in its fundamental note, which in the mode of ut is ut, and la in that of la, &c. Besides, in those passages of the modulation where there is a cadence, it is generally necessary that the mode of the fundamental bass should be the same with that of the part to which it corresponds.

242. To know upon what mode or in what key a piece commences, our inquiry may be entirely reduced to distinguishing the major mode of ut from the minor of la. For we have already seen (art. 236 and 237.), that all the modes may be reduced to these two, at least in the beginning of the piece. We shall now therefore give a detail of the different means by which these two modes may be distinguished.

1. From the principal and characteristic sounds of the mode, which are ut mi fol in the one, and la ut mi in the other; so that if a piece should, for instance, begin thus, la ut mi la, it may be almost constantly concluded, that the tone or mode is in la minor, although the notes la ut mi belong to the mode of ut.

2. From the sensible note, which is fa in the one, and fol in the other; so that if fol appears in the first bars of a piece, one may be certain that he is in the mode of la.

3. From the adjuncts of the mode; that is to say, the modes of its two fifths, which for ut are fa and fol, and re and mi for la. For example, if after having begun

(DDDD) We often say, that we are upon a particular key, instead of saying that we are in a particular mode. The following expressions therefore are synonymous; such a piece is in ut major, or in the mode of ut major, or in the key of ut major.

We have seen that the diatonic scale or gammut of the Greeks was la si ut re mi fa sol la (art. 49.) A method has likewise been invented of representing each of the sounds in this scale by a letter of the alphabet; la by A, si by B, ut by C, &c. It is from hence that these forms of speaking proceed: Such a piece is upon A, with mi, la, and its third minor; or, simply, it is upon A, with mi, la, and its minor; such another piece upon C, with fol, ut, and its third major; or, simply, upon C, with fol, ut, and its major; to signify that the one is the mode of la minor, or that the other is in that of ut major; this last manner of speaking is more concise, and on this account it begins to become general.

They likewise call the clef of ut faF, the clef of re folG, &c. to denominate the clef of fa, the clef of fol, &c.

They say likewise to take the A mi la; to give the A mi la; that is to say, to take the unison of a certain note called la in the harpsichord, which la is the same that occupies the fifth line, or the highest line in the first clef of fa. This la divides in the middle the two octaves which subsist (note rr) between the fol which occupies the first line in the clef of fol upon that same line, and that fol which occupies the first line in the clef of fa upon the fourth; and as it possesses (if we may speak so) the middle station between the sharpest and lowest sounds, it has been chosen to be the sound with relation to which all the voices and instruments ought to be tuned in a concert (§).

(3) Thus far our author: and though the note is no more than an illustration of the technical phraseology in his native language, we did not think it consistent with the fidelity of a translation to omit it. We have little reason to envy, and still less to follow, the French in their abbreviations of speech; the native energy of our tongue supercedes this necessity in a manner so effectual, that, in proportion as we endeavour to become succinct, our style, without the smallest sacrifice of perspicuity, becomes more agreeable to the genius of our language; whereas, in French, laconic diction is equally ambiguous and disagreeable. Of this we cannot give a more flagrant instance than the note upon which these observations are made, in its original. We must, however, follow the author's example, in reciting a few technical phrases upon the same subject, which occur in our language, and which, if we are not mistaken, will be found equally concise, at the same time that they are more natural and intelligible. When we mean to express the fundamental note of that series within the diatonic octave which any piece of music demands, we call that note the key. When we intend to signify its mode, whether major or minor, we denominate the harmony sharp or flat. When in a concert we mean to try how instruments are in tune by that note upon which, according to the genius of each particular instrument, they may best agree in unison, we desire the musicians who join us to sound A. gun a melody by some of the notes which are common to the modes of ut and of la (as mi re mi fa mi re ut fi ui), I should afterwards find the mode of sol, which I ascertain by the fa, or that of fa which I ascertain by the fa or ut, I may conclude that I have begun in the mode of ut; but if I find the mode of re, or that of mi, which I ascertain by fa, ut, or re, &c. I conclude from thence that I have begun in the mode of la.

4. A mode is not for ordinary deserted, especially in the beginning of a piece, but that we may pass into one or other of these modes which are most relative to it, which are the mode of its fifth above, and that of its third below, if the original mode be major, or of its third above if it be minor. Thus, for instance, the modes which are most intimately relative to the major mode of ut, are the major mode of sol, and that of la minor. From the mode of ut we commonly pass either into the one or the other of these modes; so that we may sometimes judge of the principal mode in which we are, by the relative mode which follows it, or which goes before it, when these relative modes are decisively marked. For what remains, besides these two relative modes, there are likewise two others into which the principal mode may pass, but less frequently, viz. the mode of its fifth below, and that of its third above, as fa and mi for the mode of ut (EEEEE).

5. The modes may still be likewise distinguished by the cadences of the melody. These cadences ought to occur at the end of every two, or at most of every four bars, as in the fundamental bass: now the note of the fundamental bass which is most suitable to these cadences, is always easy to be found. For the sounds which occur in the treble may be consulted M. Rameau, p. 54. of his Nouveau Système de Musique théorique et pratique (FFFF).

When a person is once able to ascertain the mode, and can render himself sure of it by the different means which we have pointed out, the fundamental bass will cause little pains. For in each mode there are three fundamental sounds.

1. The tonic of the mode, or its principal sound, which carries always the perfect chord major or minor, according as the mode itself is major or minor.

Major mode of UT. ut mi sol ut. Minor mode of LA. la ut mi la.

2. The tonic dominant, which is a fifth above the tonic, and which, whether in the major or minor mode, always carries a chord of the seventh, composed of a third major followed by two thirds minor.

Tonic dominant. Major mode of UT. sol fa re fa. Minor mode of LA. mi sol fa re.

3. The sub-dominant, which is a fifth below the tonic, and which carries a chord composed of a third, fifth, and sixth major, the third being either greater or lesser, according as the mode is major or minor.

Sub-dominant. Major mode of UT. fa la ut re. Minor mode of LA. re fa la fi.

These three sounds, the tonic, the tonic dominant, and the sub-dominant, contain in their chords all the notes which enter into the scale of the mode; so that when a melody is given, it may almost always be found which of these three sounds should be placed in the fundamental bass, under any particular note of the upper part. Yet it sometimes happens that not one of these notes can be used. For example, let it be supposed that we are in the mode of ut, and that we find in the melody these two notes la fi in succession; if we confine ourselves to place in the fundamental bass one of the three sounds ut sol fa, we shall find nothing for the sounds la and fi but this fundamental bass fa sol; now such a succession as fa to sol is prohibited by the fifth rule for the fundamental bass, according to which every sub-dominant, as fa, should rise by a fifth; so that fa can only be followed by ut in the fundamental bass, and not by sol.

To remedy this, the chord of the sub-dominant fa la ut re must be inverted into a fundamental chord of the seventh, in this manner, re fa la ut, which has been called the double employment (art. 105.) because it is a secondary manner of employing the chord of the sub-

(EEEEE) It is certain that the minor mode of mi has an extremely natural connection with the mode of ut, as has been proven (art. 92.) both by arguments and by examples. It has likewise appeared in the note upon the art. 93. that the minor mode of re may be joined to the major mode of ut; and thus in a particular sense, this mode may be considered as relative to the mode of ut, but it is still less so than the major modes of sol and fa, or than those of la and mi minor; because we cannot immediately, and without licence, pass in a fundamental bass from the perfect minor chord of ut to the perfect minor chord of re; and if you pass immediately from the major mode of ut to the minor mode of re in a fundamental bass, it is by passing, for instance, from the tonic ut, or from mi sol ut, to the tonic dominant of re, carrying the chord la ut mi sol, in which there are two sounds, mi sol, which are found in the preceding chord; or otherwise from ut mi sol ut to sol fa re mi, a chord of the sub-dominant in the minor mode of re, which chord has likewise two sounds, sol and mi, in common with that which went immediately before it.

(FFFF) All these different manners of distinguishing the modes ought, if we may speak so, to give mutual light and assistance one to the other. But it often happens, that one of these signs alone is not sufficient to determine the mode, and may even lead to error. For example, if a piece of music begins with these three notes, mi ut sol, we must not with too much precipitation conclude from thence that we are in the major mode of ut; although these three sounds, mi ut sol, be the principal and characteristical sounds in the major mode of ut; we may be in the minor mode of mi, especially if the note mi should be long. You may see an example in the fourth act of Zoroaster, where the first air sung by the priests of Arimanus begins thus with two times sol mi fa, each of these notes being a crotchet. The air is in the minor mode of sol, and not in the major mode of mi, as one would at first be tempted to believe it. Now we may be sensible that it is in sol minor, by the relative modes which follow, and by the notes where the cadences fall. Part II.

Principles sub-dominant. By these means we give to the modulation la fa, this fundamental bafis re sol, which procedure is agreeable to rules.

Here then are four chords, ut mi sol ut, sol fa re fa, fa la ut re, re fa la ut, which may be employed in the major mode of ut. We shall find in like manner, for the minor mode of la, four chords,

la ut mi la, mi sol fa re, re fa la fa, fa re fa la.

And in this mode we sometimes change the last of these chords into fa re fa la, substituting the fa for jah. For instance, if we have this melody in the minor mode of la mi fa sol la, we would cause the first note mi to carry the perfect chord la ut mi la, the second note fa to carry the chord of the seventh, fa re fa la, the third note sol fa, the chord of the tonic dominant mi sol fa re, and in short, the last the perfect chord la ut mi la.

On the contrary, if this melody is given always in the minor mode la la sol la, the second la being syncopated, it might have the same bafis as the modulation mi fa sol la, with this difference alone, that jah might be substituted for fa in the chord fa re a la, the better to mark out the minor mode.

Besides these chords which we have just mentioned, and which may be regarded as the principal chords of the mode, there are still a great many others; for example, the series of dominants,

ut la re sol ut ja, ja mi la re sol ut,

which are terminated equally in the tonic ut, either entirely belong, or at least may be reckoned as belonging (GGGG) to the mode of ut; because none of these dominants are tonic dominants, except sol, which is the tonic dominant of the mode of ut; and besides, because the chord of each of these dominants forms no other sounds than such as belong to the scale of ut.

But if I were to form this fundamental bafis,

7 7 7 7

ut la re sol ut,

considering the last ut as a tonic dominant in this manner, ut mi sol fa; the mode would then be changed at the second ut, and we should enter into the mode of ja, because the chord ut mi sol fa indicates the tonic dominant of the mode of la; besides, it is evident that the mode is changed, because fa does not belong to the scale of ut.

In the same manner, were I to form this fundamental bafis

7 7 7 6

ut la re sol ut,

considering the last ut as a tonic dominant in this manner, ut mi sol la; this last ut would indicate the mode of sol, of which ut is the sub-dominant.

In like manner, still, if in the first series of dominants, I caused the first re to carry the third major, in this manner, re ja la ut; this re having become a tonic dominant, would signify to me the major mode of sol, and the sol which should follow it, carrying the chord fa re fa, would relapse into the mode of ut, from whence we had departed.

Finally, in the same manner, if in this series of dominants, one should cause fa to carry fa in this manner, fa re fa la, this fa would show that we had departed from the mode ut, to enter into that of sol.

From hence it is easy to form this rule for discovering the changes of mode in the fundamental bafis.

1. When we find a tonic in the fundamental bafis, we are in the mode of that tonic; and the mode is major or minor, according as the perfect chord is major or minor.

2. When we find a sub-dominant, we are in the mode of the fifth above that sub-dominant; and the mode is major or minor, according as the third in the chord of the sub-dominant is major or minor.

3. When we find a tonic dominant, we are in the mode of the fifth below that tonic dominant. As the tonic dominant carries always the third major, one cannot be secure by the affluence of this dominant alone, whether the mode be major or minor; but it is only necessary for the composer to cast his eye upon the following note, which must be the tonic of the mode in which he is; by the third of this tonic he will discover whether the mode be major or minor.

243. Every change of the mode supposes a cadence; and when the mode changes in the fundamental bafis, it is almost always either after the tonic of the mode in which we have been, or after the tonic dominant of that mode, considered then as a tonic by favour of a close which ought necessarily to be found in that place; Whence it happens that cadences in a melody for the most part preface a change of mode which ought to follow them.

244. All these rules, joined with the table of modes which we have given (art. 234.), will serve to discover in what mode we are in the middle of a piece, especially in the most essential passages, as cadences (HHHH).

I here subjoin the folloquy of Armida, with the continued and fundamental bafis. The changes of the mode will be easily distinguished in the fundamental bafis,

(GGGG) I have said, that they may be reckoned as belonging to this mode, for two reasons: 1. Because, properly speaking, there are only three chords which essentially and primitively belong to the mode of ut, viz. ut carrying the perfect chord, fa carrying that of the sub-dominant, and sol that of the tonic dominant, to which we may join the chord of the seventh, re fa la ut (art 105.) but we here regard as extended the series of dominants in question, as belonging to the mode of ut, because it preserves in the ear the impression of that mode. 2. In a series of dominants, there are a great many of them which likewise belong to other modes; for instance, the simple dominant la belongs naturally to the mode of sol, the simple dominant fa to that of la, &c. Thus it is only improperly, and by way of extension, as I have already said, that we regard here these dominants as belonging to the mode of ut.

(HHHH) Two modes are so much more intimately relative as they contain a greater number of sounds common to both; for example, the minor mode of ut and the major of sol, or the major mode of ut and the minor Musical Principles

Chap. XIV. Of the Chromatic and Enharmonic.

245. We call that melody chromatic which is composed of several notes in succession, whether rising or descending by semitones. (See LXXXVIII. and LXXXIX.)

246. When an air is chromatic in descending, the most natural and ordinary fundamental bass is a concatenated series of tonic dominants; all of which follow one another in descending by a fifth, or which is the same thing, in rising by a fourth. See LXXXVIII (llll).

247. When the air is chromatic in ascending, one may form a fundamental bass by a series of tonics and of tonic dominants, which succeed one another alternately by the interval of a third in descending, and of a fourth in ascending, (see LXXXIX). There are many other ways of forming a chromatic air, whether in rising or descending; but these details in an elementary essay are by no means necessary.

248. With respect to the enharmonic, it is very rarely put in practice; and we have explained its formation in the first book, to which we refer our readers. We shall content ourselves with saying, that, in the beautiful soliloquy of the fourth act of Dardanus, at the words "lieux funestes," &c., "fatal places, &c." we find an example of the enharmonic; an example of the diatonic enharmonic in the trio of the Fatal Sisters, in Hippolitus and Aricia, at the words, "Ou cours-tu malheureux," "Whither, unhappy, dost thou run?" and that there are no examples of the chromatic enharmonic, at least in our French operas. M. Rameau had imitated an earthquake by this species of music, in the second act of the Gallant Indians; but he informs us, that in 1735 he could not cause it to be executed by the band. The trio of the Fatal Sisters in Hippolitus has never been sung in the opera as it is composed. But M. Rameau affirms, (and we have heard it elsewhere by people of taste, before whom the piece was performed), that the trial had succeeded when made by able hands that were not mercenary, and that its effect was astonishing.

Chap. XV. Of Design, Imitation, and Fugue.

249. In music, the name of design, or subject, is generally given to a particular air or melody, which the composer intends should prevail through the piece; whether it is intended to express the meaning of words to which it may be set, or merely inspired by the impulse of taste and fancy. In this last case, design is distinguished into imitation and fugue.

250. Imitation consists in causing to be repeated the melody of one or of several bars in one single part, or in the whole harmony, and in any of the various modes that what.

of la: on the contrary, two modes are less intimately relative as the number of sounds which they contain as common to both is smaller; for instance, the major mode of u and the minor of f, &c.

When you find yourself led away by the current of the modulation, that is to say, by the manner in which the fundamental bass is constituted, into a mode remote from that in which the piece was begun, you must continue in it but for a short time, because the ear is always impatient to return to the former mode.

(llll) It is extremely proper to remark, that we have given the fundamental, the continued bass, and in general the modulation of this soliloquy, merely as a lesson in composition extremely suitable to beginners; not that we recommend the soliloquy in itself as a model of expression. Upon this last object what we have said may be seen in what we have written concerning the liberties to be taken in music, Vol. IV. p. 435, of our Literary Miscellany. It is precisely because this soliloquy is a proper lesson for initiates, that it would be a bad one for the mature and ingenious artist. The novice should learn tenaciously to observe his rules; the man of art and genius ought to know on what occasions and in what manner they may be violated when this expedient becomes necessary.

(llll) We may likewise give to a chromatic melody in descending, a fundamental bass, into which may enter chords of the seventh and of the diminished seventh, which may succeed one another by the intervals of a false fifth and a fifth redundant: thus in the Example XC, where the continued bass descends chromatically, it may easily be seen that the fundamental bass carries successively the chords of the seventh and of the seventh diminished, and that in this bass there is a false fifth from re to sol, and a fifth redundant from sol to u.

The reason of this licence is, as it appears to me, because the chord of the diminished seventh may be considered as representing (art. 221.) the chord of the tonic dominant; in such a manner that this fundamental bass

\[ \begin{array}{cccc} 7 & \sharp & 7 & 7 \\ la & re & sol & ut \\ \end{array} \]

(see Example XCI.) may be considered as representing (art. 116.) that which is written below,

\[ \begin{array}{cccc} 7 & \sharp & 7 & 7 \\ la & re & mi & ut \\ \end{array} \]

Now this last fundamental bass is formed according to the common rules, unless that there is a broken cadence from re to mi, and an interrupted cadence from mi to u, which are licenses (art. 213 and 214.) Part II.

MUSIC

Courant is very slow sarabande; this last is no longer in use. The paffpied is properly a very brisk minuet, which does not begin like the common minuet, with a stroke of the foot or hand; but in which each strain begins in the last of the three times of which the bar consists.

The loure is an air whose movement is slow, whose time is marked with $\frac{4}{4}$, and where two of the times in which the bar consists are beaten; it generally begins with that in which the foot is raised. For ordinary the note in the middle of each time is shortened, and the first note of the same time pointed.

The jig is properly nothing else but a loure very brisk, and whose movement is extremely quick.

The forlana is a moderate movement, and in a mediocrity between the loure and the jig.

The rigadoon has two times in a bar, is composed of two strains, each to be repeated, and each consisting of 4, 8, or 12 bars; its movement is lively; each strain begins, not with a stroke of the foot, but at the last note of the second time.

The bourée is almost the same thing with the rigadoon.

The gavotte has two times in each bar, is composed of two strains, each to be repeated, and each consisting of 4, 8, or 12 bars; the movement is sometimes slow, sometimes brisk; but never extremely quick, nor very slow.

The tambourin has two strains, each to be repeated, and each consisting of 4, 8, or 12 bars, &c. Two of the times that make up each bar are beaten, and are very lively; and each strain generally begins in the second time.

The mufette consists of two or three times in each bar; its movement is neither very quick nor very slow; and for its bas it has often no more than a single note, which may be continued through the whole piece.

APPENDIX.

The treatise of D'Alembert, of which we have given a translation, is well entitled to the merit of accuracy; but perhaps a person who has not particularly studied the subject, may find difficulty in following the scientific deductions of that author.—We join, therefore, a few general observations on the philosophy of musical sound, commonly called harmonics, which may perhaps convey the full portion of knowledge of the theory of music, with which one in search only of general information, and not a professed student of this particular science, would choose to rest satisfied.

The theory of musical sound, which only in the beginning of the present century was ultimately established by mathematical demonstration, is no other than that which distinguished the ancient musical sect who

(MMMM) Yet there may be two fifths in succession, provided the parts move in contrary directions, or, in other words, if the progress of one part be ascending, and the other descending; but in this case they are not properly two fifths, they are a fifth and a twelfth; for example, if one of the parts in descending should found $a$ re, and the other $la$ in rising, $a$ is the fifth of $fa$, and $la$ the twelfth of $re$. who followed the opinions of Pythagoras on that subject.

No part of natural philosophy has been more fruitful of hypothesis than that of which musical sound is the object. The musical speculators of Greece arranged themselves into a great number of different sects, the chief of whom were the Pythagoreans and the Aristoxenians.

Pythagoras supposed the air to be the vehicle of sound; and the agitation of that element, occasioned by a similar agitation in the parts of the sounding body, to be the cause of it. The vibrations of a string or other sonorous body, being communicated to the air, affected the auditory nerves with the sensation of sound; and this sound, he argued, was acute or grave in proportion as the vibrations were quick or slow.—He discovered by experiment, that of two strings equal in every thing but length, the shorter made the quicker vibrations, and emitted the acuter sound:—in other words, that the number of vibrations made in the same time, by two strings of different lengths, was inversely as those lengths; that is, the greater the length the smaller the number of vibrations in any given time. Thus found, considered in the vibrations that cause it, and the dimensions of the vibrating body, came to be reduced to quantity, and as such was the subject of calculation, and expressible by numbers.—For instance, the two sounds that form an octave could be expressed by the numbers 1 and 2, which would represent either the number of vibrations in a given time, or the length of the strings; and would mean, that the acuter sound vibrates twice, while the graver vibrates once; or that the string producing the lower sound is twice the length of that which gives the higher. If the vibrations were considered, the higher sound was as 2, the lower as 1; the reverse, if the length was alluded to. In the same manner, in the same sense, the 5th would be expressed by the ratio of 2 to 3, and the 4th by that of 3 to 4.

Aristoxenus, in opposition to the calculations of Pythagoras, held the ear to be the sole standard of musical proportions. That sense he accounted sufficiently accurate for musical, though not for mathematical purposes; and it was in his opinion absurd to aim at an artificial accuracy in gratifying the ear beyond its own power of distinction. He, therefore, rejected the velocities, vibrations, and proportions of Pythagoras as foreign to the subject, in so far as they substituted arbitrary causes in the room of experience, and made music the object of intellect rather than of sense.

Of late, however, as has been already mentioned, the opinions of Pythagoras have been confirmed by absolute demonstration; and the following propositions, in relation to musical sound, have passed from conjecture to certainty.

Sound is generated by the vibrations of elastic bodies, which communicate the like vibrations to the air, and these again the like to our organs of hearing. This is evident, because sounding bodies communicate tremors to other bodies at a distance from them. The vibrating motion, for instance, of a musical string, excites motion in others, whose tension and quantity of matter dispose their vibrations to keep time with the undulations of air propagated from it (the string first set in motion.)

If the vibrations be isochronous, and the sound musical, continuing at the same pitch, it is said to be acuter, sharper, or higher, than any other sound whose vibrations are slower; and graver, flatter, or lower, than any other whose vibrations are quicker.—For while a musical string vibrates, its vibrations become quicker by increasing its tension or diminishing its length; its sound at the same time will be more acute; and, on the contrary, by diminishing its tension or increasing its length, the vibrations will become slower and the sound graver. The like alteration of the pitch of the sound will follow, by applying, by means of a weight, an equal degree of tension to a thicker or heavier and to a smaller or lighter string, both of the same length, as in the smaller string the mass of matter to be moved by the same force is less.

If several strings, however, different in length, density, and tension, vibrate altogether in equal times, their sounds will have all one and the same pitch, however they may differ in loudness or other qualities.—They are called unisons. The vibrations of unisons are isochronous.

The vibrations of a musical string, whether wider or narrower, are nearly isochronous. Otherwise, while the vibrations decrease in breadth till they cease, the pitch of the sound could not continue the same (which we perceive by experience it does), unless where the first vibrations are made very violently; in which case, the sound is a little acuter at the beginning than afterwards.

Lastly, the word vibration is understood to mean the time which passes between the departure of the vibrating body from any assigned place and its return to the same.

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**MUS**

**Glass-Music.** See Harmonica.

**Musimon,** in natural history, the name of an animal esteemed a species of sheep, described by the ancients as common in Corsica, Sardinia, Barbary, and the north-east parts of Asia. It has been doubted whether the animal described under this name is now anywhere to be found in the world; and whether it was not, probably, a spurious breed between two animals of different species, perhaps the sheep and goat.

N° 234.

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**MUS**

which, like the mule, not being able to propagate its species, the production of them may have been discontinued.

Buffon supposes it to be the sheep in a wild state; and it is described as such by Mr Pennant. These animals live in the mountains, and run with great swiftness among the rocks. Those of Kamtschatka are so strong, that 10 men can scarcely hold one; and the horns are so large as sometimes to weigh 30 pounds, and

Plate CCCXXIII.

A ut re mi fa sol la si vt

B ut re mi fa sol la si vt re mi fa sol &c

Scale first. Scale Second.

The diatonic Scale of the Greeks.

Si vt re mi fa sol la Sol vt sol vt fa vt fa

The Fundamental bass.

C K H G C V T E S R F L N O P Q M

The Chromatic Species.

Scale.

Sol Sol &c vt mi sol *

The Fundamental bass.

D

E Ut Re Mi Fa Sol Sol La Si Ut Ut Sol Ut Fa Ut Sol Re Sol Ut

The Fundamental bass.

F Scale first. Scale Second.

The first Scale of the minor mode.

Sol La Si Ut Re Mi Fa Mi La Mi La Re La Re

The Fundamental bass.

G

N

Scale.

Mi Mi Mi Mi Mi Ut Ut La Ut Ut

The Fundamental bass.

The second Scale of the minor mode.

La Si Ut Re Mi mi fa* Solx La La Mi La Re La Mi Si Mi La

The Fundamental bass.

Scale.

I Ut Re Mi Fa Sol La Si Ut Ut Sol Ut Fa Ut Re Sol Ut

M

Scale.

Fa Mi Mi Re Fa Ut Mi Si

The Fundamental bass.

W.B. Smith, London, fecit. En fin, il est en ma puissance, Ce fatal ennemi, Ce superbe vainqueur, Le charme du sommeil le livre à ma vengeance, Je vais percer son invincible cœur; Par luy, tous mes Captifs sont sortis d'esclavage. dois me venger anjour d'huy! Ma colere se teint Quand j'approche de luy

Plus je le vois! plus ma vengeance est vaine; Mon bras tremblant se re-

suse à ma haine: Ah! quelle cruauté de luy ravir le jour! A ce jeune He- Translation. Intended to give such Readers as do not understand French, an idea of the Song.

At length the victim in my power I see, This fatal year resigns him to my rage; Subdued by sleep he lies, and leaves me free, With chastning hand my fury to asswage. That mighty heart invincible and fierce, Which all my captives freed from servile chains; That mighty heart, my vengeful hand shall pierce; My rage inventive wanton in his pains. Ha! in my soul what perturbation reigns! What would compassion in his favour plead? Strike, hand, O heaven! what charm thy force restrains? Obey my wrath, I sigh; yet let it bleed. And is it thus my just revenge improves The fair occasion to chastise my foe? As I approach, a softer passion moves, And all my boasting fury melts in wo. Trembling, relax'd, and faithless to my hate, The dreadful task this coward arm declines.

How cruel thus to urge his instant fate, Depriv'd of life amid his great designs! In youth how blooming! what a heavenly grace, Thro' all his form, resistless power displays! How sweet the smile that dwells upon his face, Relentless rage disarming whilst I gare! Tho' to the prowess of his conquering arms Earth stood with all her hosts opposed in vain; Yet is he form'd to spread more mild alarms, And bind all nature in a softer chain. Can then his blood, his precious blood, alone Extinguish all the vengeance in my heart? Tho' still surviving, might he not stone. For all the wrongs I feel, by gentler smart? Since all my charms, unfeeling, he defies, Let Magic force his stubborn soul subdue; Whilst I, inflexible to tears and sighs, With hate (if I can hate) his peace pursue. and so capacious, that young foxes often shelter themselves in the hollow of such as by accident fall off in the deserts. See Ovis.