Theory of Harmony.

Minor, of its fifth, and of its octave. In effect, these two kinds of chords are exhibited by nature; but the first more immediately than the second. The first are called perfect chords major, the second perfect chords minor.

CHAP. III. Of the Succession by Fifths, and of the Laws which it observes.

Fundamental bas, what.

33. Since the sound C causes the sound G to be heard, and is itself heard in the sound F, which sounds G and F are its two twelfths, we may imagine a modulation composed of that sound C and its two twelfths, or, which is the same thing (art. 22.), of its two fifths, F and G, the one below, the other above; which gives the modulation or series of fifths F, C, G, which we call the fundamental bas of C by fifths.

We shall find in the sequel (Chap. XVIII.), that there may be some fundamental bases by thirds, deduced from the two seventeenth, of which the one is an attendant of the principal sound, and of which the other includes that sound. But we must advance step by step, and satisfy ourselves at present to consider immediately the fundamental bases by fifths.

34. Thus, from the sound C, one may make a transition indifferently to the sound G, or to the sound F.

35. One may, for the same reason, continue this kind of fifths in ascending, and in descending, from C, in this manner:

E♭, B♭, F, C, G, D, A, &c.

And from this series of fifths one may pass to any sound which immediately precedes or follows it.

36. But it is not allowed in the same manner to pass from one sound to another which is not immediately contiguous to it; for instance, from C to D, or from D to C: for this very simple reason, that the sound D is not contained in the sound C, nor the sound C in that of D; and thus these sounds have not any alliance the one with the other, which may authorize the transition from one to the other.

37. And as these sounds C and D, by the first experiment, naturally bring along with them the perfect chords consisting of greater intervals C, E, G, &c., and D, F♯, A, d'; hence may be deduced this rule, That two perfect chords, especially if they are major (T), cannot succeed one another diatonically in a fundamental base; we mean, that in a fundamental base two sounds 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.

Mode in general, what.

38. A mode, in music, is, the order of sounds prescribed, as well in harmony as melody, by the series of fifths. Thus the three sounds F, C, G, and the harmonies of each of these three sounds, that is to say, their thirds major and their fifths, compose all the major modes which are proper to C.

39. The series of fifths then, or the fundamental bases Modes, F, C, G, of which C holds the middle space, may be represented by regarding as representing the mode of C. One may take the series of fifths, or fundamental bases, fifths, C, G, D, as representing the mode of G; in the same manner B♭, F, C, will represent the mode of F.

Thus the mode of G, or rather the fundamental base of that mode, has two sounds in common with the fundamental bases of the mode of C. It is the same with the fundamental bases of the mode F.

40. The mode of C (F, C, G) is called the principal mode with respect to the modes of these two fifths, which mode, and are called its two adjuncts.

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

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

43. As in the series of fifths, we may indifferently pass from one sound to that which is contiguous; for, having passed from the mode of C to that of G, one how to be may from thence proceed to the mode of D. And on managed, the other hand, having passed from the mode of C to that of F we may then pass to the mode of B♭. 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 (U) 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 sounds which are contiguous may be placed in immediate succession in the series of fifths, F, C, G, it follows, that one may form

(t) We say especially if they are major; for in the major chord D, F♯, A, d', besides that the sounds C and D have no common harmonical relation, and are even dissonant between themselves (art. 13.), it will likewise be found, that F♯ forms a dissonance with C. The minor chord D, F, A, d', would be more tolerable, because the natural F, which occurs in this chord carries along with it its fifth C, 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.

(u) As the mere English reader, unacquainted with the technical phraseology of music, may be surprised at the form this modulation, or this fundamental bals, by fifths,

G, C, G, C, F, C, F.

45. Each of the sounds which forms this modulation brings necessarily along with itself its third major, its fifth, and its octave; inasmuch that he who, for instance, sings the note G, may be reckoned to sing at the same time the notes G, B, 'd', g': in the same manner the sound C in the fundamental bals brings along with it this modulation, C, E, G, C; and, in short, the sound F brings along with it F, A, C, 'f'. This modulation then, or this fundamental bals,

G, C, G, C, F, C, F,

gives the following diatonic series,

B, 'c', d, e, f, g, a';

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 bals G, C, G, C, F, C, F; and that of consequence this bals is justly called fundamental, as being the real primitive modulation, that which conduces the ear, and which it feels to be implied in the diatonic modulation, B, 'c', d, e, f, g, a' (x).

46. We shall be still more convinced of this truth by the following remarks.

the use of the word phrase when transferred from language to that art, we have though proper to infer the definition of Rousseau.

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.

(x) Nothing is easier than to find in this scale the value or proportions of each sound with relation to the sound C, which we call 1; for the two sounds G and F in the bals are \(\frac{3}{4}\) and \(\frac{2}{3}\); from whence it follows,

1. That 'c' in the scale is the octave of C in the bals; that is to say, 2. 2. That 'b' is the third major of G; that is to say \(\frac{3}{4}\) of \(\frac{3}{4}\) (note Q), and of consequence \(\frac{9}{8}\). 3. That 'd' is the fifth of G; that is to say \(\frac{5}{4}\) of \(\frac{3}{4}\), and of consequence \(\frac{15}{8}\). 4. That 'e' is the third major of the octave of C, and of consequence the double of \(\frac{3}{4}\); that is to say, \(\frac{3}{2}\). 5. That 'f' is the double octave of F of the bals, and consequently \(\frac{3}{2}\). 6. That 'g' of the scale is the octave of G of the bals, and consequently 3. 7. That 'a' in the scale is the third major of 'f' of the scale; that is to say, \(\frac{3}{2}\) of \(\frac{3}{2}\), or \(\frac{9}{4}\).

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

| Diatonic Scale | Fundamental Bals | |----------------|-----------------| | B, c, d, e, f, g, a' | G, C, G, C, F, C, F |

And if, for the convenience of calculation, we choose to call the sound C of the scale 1; in this case we have only to divide each of the numbers by 2, which represent the diatonic scale, and we shall have

\[ \begin{array}{cccccc} \frac{1}{2} & \frac{3}{4} & \frac{9}{8} & \frac{15}{8} & \frac{3}{2} & \frac{9}{4} \\ B, c, d, e, f, g, a' \end{array} \]

(y) In order to compare 'd' with 'f', we need only compare \(\frac{3}{4}\) with \(\frac{5}{4}\); 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 'd' to 'f', 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 'd' and 'f', 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 'e' and 'g', in the continued tone of those honourable bodies of which 'e' and 'g' 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. In reality, in order that the sound 'b' may succeed immediately in the scale to the found 'a', it is necessary that the note 'g', which is the only one from whence 'b' as a harmonic may be deduced, should immediately succeed to the found 'f', in the fundamental baf, which is the only one from whence 'a' can be harmonically deduced. Now, the diatonic succession from F to G cannot be admitted in the fundamental baf, according to what we have remarked (art. 36.). The founds 'a' and 'b', then, cannot immediately succeed one another in the scale: we shall see in the sequel why this is not the case in the series 'c, d, e, f, g, a, b', c, which begins upon C; whereas the scale in question here begins upon B.

49. The Greeks likewise, to form an entire octave, added below the first B the note A, which they distinguished and separated from the rest of the scale, which for that reason they called proflambanomenes, that is to say, a string or note subadded to the scale, and put before B to form the entire octave.

50. The diatonic scale B, 'c, d, e, f, g, a', is composed of two tetrachords, that is to say, of two diatonic scales, each consisting of four founds, B, 'c, d, e, f', g, a'. These two tetrachords are exactly similar; for from 'c' to 'd' there is the same interval as from B to 'c'; from 'd' to 'g' the same as from 'c' to 'd'; from 'g' to 'a' the same as from 'd' to 'e' (z): this is the reason why the Greeks distinguished these two tetrachords; yet they joined them by the note 'a' which is common to both, and which gave them the name of conjunctive tetrachords.

51. Moreover, the intervals between any two founds, taken in each tetrachord in particular, are precisely true: thus, in the first tetrachord, the intervals of C 'e', and B 'd', are thirds, the one major and the other minor, exactly true, as well as the fourth B 'e' (AA); it is the same thing with the tetrachord 'e, f, g, a', since this tetrachord is exactly like the former.

52. But the case is not the same when we compare two founds taken each from a different tetrachord; for we have already seen, that the note 'd' in the first tetrachord forms with the note 'f' in the second a third minor, which is not true. In like manner it will be found, that the fifth from 'd' to 'a' is not exactly true, which is evident; for the third major from 'f' to 'a' is true, and the third minor from 'd' to 'f' 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 another 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 re-scale into four, for distinguishing the scale into these two tetra-chords.

54. It may be ascertained by calculation, that in the The source tetrachord B, 'c, d, e', the interval, or the tone from 'c' to 'e', is a little less than the interval or tone from 'c' to 'd' (BB). In the same manner, in the second tetrachord figured, 'e, f, g, a', which is, as we have proved, perfectly similar to the first, the note from 'g' to 'a' is a little less than the note from 'f' to 'g'. It is for this reason that they distinguished two kinds of tones; the greater tone *, as Greater tone. * See from 'c' to 'd', from 'f' to 'g', &c.; and the lesser tone. Lesser tone. † See Inter- from 'd' to 'e', from 'g' to 'a', &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, B, 'c, d, e, g, a', how formed only, F, C, G; but to form the scale 'c, d, e, f, g, a, b', c, which we use at present, we must necessarily add to the fundamental baf the note D, and form, with these four founds F, C, G, D, the following fundamental baf:

C, G, C, F, C, G, D, G, C;

from whence we deduce the modulation or scale

See fig. 5.

'c, d, e, f, g, a, b', c.

In effect (cc), 'c' in the scale belongs to the harmony. Of C which corresponds with it in the baf; 'd', which is the second note in the gammut, is included in the harmony of G, the second note of the baf; 'e', the third note of the gammut, is a natural harmonic of C, which is the third found in the baf, &c.

56. From

(z) The proportion of B to 'c' is as 14 to 1, that is to say as 15 to 16; that between 'e' and 'f' is as 4 to 5, that is to say (note Q), as 5 times 3 to 4 times 4, or as 15 to 16; these two proportions are equal. In the same manner, the proportion of 'c' to 'd' is as 1 to 8, or as 8 to 9; that between 'f' and 'g' is as 4 to 5; that is to say (note Q), as 8 to 9. The proportion of 'e' to 'c' is as 4 to 5, or as 5 to 4; that between 'f' and 'a' is as 4 to 5, or as 5 to 4; the proportions here then are likewise equal.

(AA) The proportion of 'e' to 'c' is as 4 to 5, or as 5 to 4, which is a true third major; that from 'd' to 'b' is as 5 to 4; 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 'c' to 'b' is as 4 to 5; that is to say, as 5 times 16 to 15 times 4, or as 4 to 3, which is a true fourth.

(BB) The proportion of 'd' to 'c' is as 8 to 9, or as 9 to 8; that of 'e' to 'd' is as 4 to 5, that is to say, as 40 to 36, or as 10 to 9; now 9 is less removed from unity than 8; the interval then from 'd' to 'c' is a little less than that from 'c' to 'd'.

If any one would wish to know the proportion which 9 bear to 8, he will find (note Q) 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 'd', 'f', of which we have taken notice in the scale (note X); for we have seen, that this third minor thus altered is in the proportion of 80 to 81 with the true third minor.

(cc) The values or estimates of the notes shall be the same in this as in the former scale, excepting only the tone 56. Hence 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 C; whereas ours is originally and primitively formed, not only from the mode of (F, C, G), but likewise from the mode of G, (C, G, D).

It will likewise appear, that this last scale consists of two parts; of which the one, 'c, d, e, f, g,' is in the mode of C; and the other, 'g, a, b, c,' in that of G.

57. For this reason the note 'g' is twice repeated in the diatonic scale from its harmonic relations to the fundamental basis.

The modern scale, composed of two disjunctive tetrachords of different modes.

The mode of G introduced in the fundamental basis productive of conveniences.

58. The scale of the moderns, then, may be considered as consisting of two tetrachords, disjunctive indeed, but perfectly similar one to the other, 'c, d, e, f,' and 'g, a, b, c,' one in the mode of C, the other in that of G. We shall see in the sequel, by what artifice one may cause the scale 'c, d, e, f, g, a, b, c,' to be regarded as belonging to the mode of C alone. For this purpose it is necessary to make some changes in the fundamental basis, which we have already assigned; but this shall be explained at large in chap. xiii.

59. The introduction of the mode proper to G in the fundamental basis has this happy effect, that the notes 'f, g, a, b,' 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 C alone. Whence it follows:

tone 'a'; for 'd' being represented by \( \frac{2}{3} \), its fifth will be expressed by \( \frac{3}{5} \); so that the scale will be numerically figured thus:

\[ \begin{array}{cccccc} 1 & \frac{2}{3} & \frac{3}{5} & \frac{4}{5} & \frac{5}{8} & \frac{6}{5} \\ c, d, e, f, g, a, b, c, \end{array} \]

Where you may see, that the note 'a' of this scale is different from that in the scale of the Greeks; and that the 'a' in the modern series stands in proportion to that of the Greeks as \( \frac{3}{5} \) to \( \frac{5}{8} \), that is to say, as 81 to 80; these two 'a's then likewise differ by a comma.

(DD) In the scale of the Greeks, the note 'a' being a third from 'f', there is an altered fifth between 'a' and 'd'; but in ours, 'a' being a fifth to 'd', produces two altered thirds, 'f'a' and 'a'c'; and likewise a fifth altered, 'a'e', 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.

(EE) But here it may be with some colour objected: The scale of the Greeks, it may be said, has a fundamental basis 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 basis, but the arrangement of ours is more suitable to natural intonation. Our scale commences by the fundamental sound c, and it is in reality from that sound that we ought to begin; it is from this that all the others naturally arise, and upon this they depend; nay, if we may speak so, in this they are included: on the contrary, neither the scale of the Greeks, nor its fundamental basis, commences with C; but it is from this C that we depart, in order to regulate our intonation, whether in rising or descending; now, in ascending from 'c', the intonation, even of the Greek scale, gives the series 'c, d, e, f, g, a'; and to be sure it is that the fundamental sound C is here the genuine guide of the ear, that if, before we modulate the sound 'c', we should... Part I.

Theory of that it must be different according as the note 'a' has Harmony. 'f' or 'd' for its base. See the note (cc).

CHAP. VII. Of Temperament.

64. The alterations which we have observed in the intervals between particular sounds of the diatonic scale, naturally lead us to speak of temperament. To give a clear idea of this, and to render the necessity of it 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 us choose in that harpsichord one of the strings which will found the note C, and let us tune the string G to a perfect fifth with C in ascending; let us afterwards tune to a perfect fifth with this G the d' which is above it; we shall evidently perceive that this d' will be in the scale above that from which we set out: but it is also evident that this d' must have in the scale a D which corresponds with it, and which must be tuned a true octave below d'; and between d' and G there should be the interval of a fifth; so that the D in the Theory of first scale will be a true fourth below the G of the same scale. We may afterwards tune the note A of the first scale to a just fifth with this last D; then the note e' in the highest scale to a true fifth with this new A, and of consequence the E in the first scale to a true fourth beneath this same A: Having finished this operation, it will be found that the last E, thus tuned, will by no means form a just third major from the found C (ff): that is to say, that it is impossible for E to constitute at the same time the third major of C and the true fifth of A; or, what is the same thing, the true fourth of A in descending.

65. If, after having successively and alternately tuned the strings C, G, d', A, E, in perfect fifths and fourths one from the other, we continue to tune successively by true fifths and fourths the strings E, B, F, C, G, d', e', E, B; we shall find, that though B, being a semitone higher than the natural note, should be equivalent to c' natural, it will by no means form a just octave to the first C in the scale, but be considerably higher (gc); yet this B upon the harpsichord ought not

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 B, and by the semitone from B to c'. Now to make a transition from B to c', by this semitone, the ear must of necessity be predisposed for that modulation, and consequently preoccupied with the mode of C: if this were not the case, we should naturally rise from B to c', and by this operation pass into another mode.

(ff) The A considered as the fifth of D is \( \frac{3}{2} \), and the fourth beneath this A will constitute \( \frac{4}{3} \) of \( \frac{3}{2} \), that is to say, \( \frac{8}{5} \); \( \frac{3}{2} \) then shall be the value of E, considered as a true fourth from A in descending: now E, considered as the third major of the found C, is \( \frac{5}{4} \), or \( \frac{9}{8} \); these two E's then are between themselves in the proportion of 81 to 80; thus it is impossible that E should be at the same time a perfect third major from C, and a true fourth beneath A.

(gc) 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.

C, G, a fifth; A a fourth; A a fifth; E a fourth; F a fourth; C a fifth; G a fourth; d' a fifth; A a fourth; e' or f' a fifth; B a fourth: now it will be found, by a very easy computation, that the first C being represented by 1, D \( \frac{3}{2} \), A \( \frac{5}{4} \), E \( \frac{9}{8} \), &c. and so of the rest, till you arrive at B\( \frac{12}{11} \), which will be found \( \frac{12}{11} \times \frac{12}{11} \). This fraction is evidently greater than the number 2, which expresses the perfect octave c to its correspondent C; and the octave below B\( \frac{12}{11} \) would be one half of the same fraction, that is to say \( \frac{12}{11} \times \frac{12}{11} \), which is evidently greater than C represented by unity. This last fraction \( \frac{12}{11} \times \frac{12}{11} \) 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 B\( \frac{12}{11} \), is not equal to the unity which expresses the value of the found C, though, upon the harpsichord, B\( \frac{12}{11} \) and C are identical. This fraction rises above unity by \( \frac{12}{11} - 1 = \frac{1}{11} \), that is to say, by about \( \frac{1}{7} \); 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 bb), and which is only \( \frac{1}{8} \).

We have already proved that the series of fifths produces a c' different from B\( \frac{12}{11} \), the series of thirds major gives another still more different. For, let us suppose this series of thirds, C, E, G\( \frac{12}{11} \), B\( \frac{12}{11} \), we shall have E equal to \( \frac{5}{4} \), G\( \frac{12}{11} \) to \( \frac{12}{11} \), and B to \( \frac{12}{11} \), whose octave below is \( \frac{12}{11} \times \frac{12}{11} \); from whence it appears, that this last B is less than unity (that is to say than C), by \( \frac{1}{8} \), or by \( \frac{1}{8} \), 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 B\( \frac{12}{11} \), deduced from the series of thirds, is to the B\( \frac{12}{11} \) deduced from the series of fifths, as \( \frac{12}{11} \) is to \( \frac{12}{11} \times \frac{12}{11} \); that is to say, in multiplying by \( \frac{12}{11} \times \frac{12}{11} \), as 125 multiplied by 4096 is to 331441, or as 51200 to 331441, that is to say, nearly as 26 is to 27: from whence it may be seen, that these two B\( \frac{12}{11} \) 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 \( \frac{1}{8} \).

Moreover, if, after having found the G\( \frac{12}{11} \) equal to \( \frac{12}{11} \), we then tune by fifths and by fourths, G\( \frac{12}{11} \), d' \( \frac{12}{11} \), A\( \frac{12}{11} \), C\( \frac{12}{11} \), B\( \frac{12}{11} \), as we have done with respect to the first series of fifths, we find that the B\( \frac{12}{11} \) must be \( \frac{12}{11} \times \frac{12}{11} \); its difference, then, from unity, or, in other words, from C, is \( \frac{1}{8} \), that is to say, about \( \frac{1}{8} \); a comma still less than any of the preceding, and which the Greeks have called apotome minor; not to be different from the octave above C; for every B♭ and every c' is the same found, since the octave or the scale only consists of twelve semitones.

66. From thence it necessarily follows, 1. That it is impossible that all the octaves and all the fifths should be just at the same time, particularly in instruments which have keys, where no intervals less than a semitone are admitted. 2. That, of consequence, if the fifths are justly tuned, some alteration must be made in the octaves; now the sympathy or sound which subsists between any note and its octave, does not permit us to make such an alteration: this perfect coalescence of sound is the cause why the octave should serve as limits to the other intervals, and that 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 c' or B♭ 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, C, G, D, A, E, B, F♯, C♯, G♯, d♯, A♯, e♯, B♯, 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 this question.—The alternate succession of fifths and fourths having been given, (art. 66.), in which B♭ or C is not the true octave of the first C; it is proposed to alter all the fifths equally, in such a manner that the two C's may be in a perfect octave the one to the other.

70. For a solution of this question, we must begin by tuning the two C's in a perfect octave the one to the other; in consequence of which, we will render all the semitones which compose the octave as equal as possible. By this means (HH) the alteration made in each

In a word, if, after having found E equal to § in the progression of thirds, we then tune by fifths and fourths E, B, F♯, C♯, &c. we shall arrive at a new B♭, which shall be §, and which will not differ from unity but by about §, which is the last and smallest of all the commas; but it must be observed, that, in this case, the thirds major from E to G♯, from G♯ to B♭ or C, &c. are extremely false, and greatly altered.

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

These twelve semitones are formed by a series of thirteen sounds, of which C and its octave c' 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.

| C | C♯ | D | D♯ | E | F | F♯ | G | G♯ | |---|----|---|----|---|---|----|---|----| | 1 | √2 | √3 | √4 | √5 | √6 | √7 | √8 | √9 |

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 C to G, which should be §, ought to be diminished by about § of §; that is to say, by §, a quantity almost inconceivably small.

It is true, that the thirds major will be a little more altered; for the third major from C to E, for instance, shall be increased in its interval by about §; 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 C, E, G♯, B♭, that this last B♭ is very different from c' (note GG); from whence it follows, that if we would tune this B♭ in unison with the octave of C, 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 each fifth will be very considerable, but equal in all of them.

71. In this, then, the theory of temperament consists; but as it would be difficult in practice to tune a harpsichord or organ by thus considering all the semitones 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, C; then tune the note G 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 in ascent; and as the ear does not appreciate so 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 E to B, which should of themselves be in tune; that is to say, they ought to be in such a state, that B, the highest note of the two which compose the fifth, may be identical with the found C, with which you began, or at least the octave of that found perfectly just: it will be necessary then to try if this C, or its octave, forms a just fifth with the last found E or F, 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 intervention (11).

(11) We have only to acknowledge, with M. Rameau, that this temperament is far remote from that which is now in practice: it may here be seen in what this last temperament consists as applied to the organ or harpsichord. They begin with C in the middle of the keys, and they flatten the four first fifths G, D, A, E, till they form a true third major from E to C; afterwards, setting out from this E, they tune the fifths B, F, C, G, but flattening them still less than the former, so that G may almost form a true third major with E. When they have arrived at G, they stop; they resume the first C, and tune to it the fifth F in descending, then the fifth B, &c., and they heighten a little all the fifths till they have arrived at A, which ought to be the same with the G 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 so; insomuch, 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 anything. 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 C to that of G, from the mode of G to that of D, &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 C,

C, D, E, F, G, A, B, C,

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

E, F, G, A, B, c, d, e,

which, in their judgement, renders the modes of C and E 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 assigned 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 with, or at least very little different from that which is practised 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 M. Rameau, in his *New System of Music*, printed in 1726, 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, 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 alteration 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. Let the three keys E, G, B be struck upon an organ, and the minor perfect chord only will be heard; though E, by the construction of that instrument, must cause G likewise to be heard; though G should have the same effect upon D, and B upon F; insomuch that the ear is at once affected with all these sounds, D, E, F, G, G, B: 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!

In what is different from what we name cadence in harmony. In the first case, this word only signifies an agreeable and rapid alteration between two contiguous sounds, called likewise a trill or shake; in the second, it signifies a repose or clofe. It is however true, that this shake implies, or at least frequently enough prefaces, a repose, either present or impending, in the fundamental base (ll.).

76. Since there is a repose in passing from one sound to another in the fundamental base, there is also a repose in passing from one note to another in the diatonic scale, which is formed from it, and which this base represents; and as the absolute repose G C is of all others the most perfect in the fundamental base, the repose from B to c', 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.

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 or leading 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. 20., 30., 31., and 32.) by what means, and upon what principle, the minor chord C, Eb, G, 'c', may be formed, which is the characteristic chord of the minor mode. Now what we have there said, taking C 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, C, Eb, G, 'c', there occurs an Eb 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. D, F, A, 'd'; A, 'c', e, a'; and E, G, B, 'e'. Among these three we shall choose A, 'c', e, a'; because this chord, without including any sharp or flat, has two sounds in common with the major chord C, E, G, 'c'; and besides, one of these two sounds is the very same 'c': so that this chord appears to have the most immediate, and at the same time the most simple, relation with the chord C, E, G, 'c'. Concerning this we need only add, that this preference of the chord A, 'c', e, a', 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. In every mode, whether major or minor, the principal sound which implies the perfect chord, whether major or minor, is called the tonic note or key; thus what C is the key in its proper mode, A in the mode of A, See Principal &c. Having laid down this principle,

81. We have shown how the three sounds, F, C, G, which constitute (art. 58.) the mode of C, of which the first, F, and the last, G, are the two fifths of C, one scale purifying, the other rising, produce the scale, B, 'c', d, f, e, f, g, a', of the major mode, by means of the fundamental base G, C, G, C, F, C, F; let us in the same manner take the three sounds D, A, E, which constitute the mode of A, for the same reason that the sounds F, C, G, constitute the mode of C; and of them let us form this fundamental base, perfectly like the preceding E, A, F, A, D, A, D; let us afterwards place See fig. 7, 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 D and A as implying their thirds minor in the fundamental base to characterize the minor mode; and we shall have the diatonic scale of that mode,

G, A, B, 'c', d, e, f'.

82. The G, which corresponds with E in the fundamental base, forms a third major with that E, 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 A.

83. It is true, that, in causing E to imply its third See Imply minor G, one might also rise to A, by a diatonic progression. But that manner of rising to A would be less perfect than the preceding; for this reason (art. 76.), that the absolute repose or perfect cadence E, A, 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 A, the key itself upon which the repose is made. From whence it follows, that the preceding note G ought rather to be sharp than natural; because G, being included in E (art. 19.), much more perfectly represents the note E in the base, than the natural G could do, which is not included in E.

84. We may remark this first difference between Diversities in the scales of the major and minor mode.

G, A, B, 'c', d, e, f', and the scale which corresponds with it in the major mode

B, 'c', d, e, f, g, a', that from 'e' to 'f', which are the two last notes of the former scale, there is only a semitone; whereas from 'g' to 'a', 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.

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(ll) 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. 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,

\[ c, d, e, f, g, g, a, b', c. \]

That last series, as we have seen, was formed by means of the fundamental bas \( F, C, G, D, G, C \), disposed in this manner,

\[ C, G, C, F, C, G, D, G, C. \]

Let us take in the same manner the fundamental bas \( D, A, E, B \), and arrange it in the following order,

\[ A, E, A, D, A, E, B, E, A, \]

and it will produce the scale immediately subjoined,

\[ A, B, c', d, e, f, g, g, a, \]

in which \( c' \) forms a third minor with \( A \), which in the fundamental bas corresponds with it, which designates the minor mode; and, on the contrary, \( g \) forms a third major with \( E \) in the fundamental bas, because \( g \) rises towards \( a \) (art. 82, 83.)

86. We see besides an \( f \), which does not occur in the former,

\[ G, A, B, c', d, e, f, \]

where \( f \) is natural. It is because, in the first scale, \( f \) is a third minor from \( D \) in the bas; and in the second, \( f \) is the fifth from \( B \) in the bas (NN).

87. Thus the two scales of the minor mode are still 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 \( f \) and \( g \) are sharp in the sharp when ascending in the minor mode; besides the minor mode, and why.

The case different in the fifth of the generator, ought not to imply the third descending, major \( g \); but in the case when that \( E \) descends to the generator \( A \) to form a perfect repose (art. 77. and 83.) and in this case the third major \( g \) rises to the generator \( a \); but the fundamental bas \( AE \) may, in descending, give the scale \( a, g \); natural, provided \( g \) does not rise again to \( a \).

92. It is much more difficult to explain how the \( f \) which ought to follow this \( g \) in descending, is natural and not sharp; for the fundamental bas

\[ A, E, B, E, A, D, A, E, A, \]

produces in descending,

\[ a, g, f, e, d, c', B, A. \]

And it is plain that the \( f \) cannot be otherwise than

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(MM) Besides, without appealing to the proof of the fundamental bas, \( f \) obviously presents itself as the sixth note of this scale; because the seventh note being necessarily \( g \) (art. 77.) if the sixth were not \( f \), but \( f' \), there would be an interval of three semitones between the sixth and the seventh, consequently the scale would not be diatonic, (art. 8.)

(NN) When \( g \) is said to be natural in descending the diatonic scale of the minor mode of \( A \), it is only meant that this \( g \) is not necessarily sharp in descending as it is in rising; for it may be sharp, as may be proved by numerical examples, of which all musical compositions are full. It is true, that when \( g \) is found sharp in descending to the minor mode of \( A \), we are not sure that the mode is minor till the \( f \) or \( c' \) natural is found; both of which imparts a peculiar character on the minor mode, viz. \( c' \) natural, in rising and in descending, and the \( f \) natural in descending.

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Chap. X. Of Relative Modes.

91. Two modes of such a nature that we can pass from the one to the other, are called relative modes. Modes relative to the major mode of \( C \) is relative to the major mode of \( F \) and to that of \( G \). It has also been seen, what how many intimate connexions there are between the major mode of \( C \), and the minor mode of \( A \). For,

1. The perfect chords, one major, \( C, E, G, c' \), the other minor, \( A, c', e, a' \), which characterize each of those two kinds of modulation * or harmony, have two sounds in common, \( c' \) and \( e' \).

2. The scale of the minor mode of \( A \) in descent, absolutely contains the same sounds with the scale of the major mode of \( C \).

Hence the transition is so natural and easy from the major mode of \( C \) to the minor mode of \( A \), or from the minor mode of \( A \) to the major mode of \( C \), as experience proves.

92. In the minor mode of \( E \), the minor perfect chord \( E, G, B, c' \), which characterizes it, has likewise two sounds, \( E, G \), in common with the perfect chord major \( C, E, G, c' \), which characterizes the major mode of \( C \). Theory of Harmony. But the minor mode of E is not so closely related nor allied to the major mode of C as the minor mode of A; because the diatonic scale of the minor mode of E in descent, has not, like the series of the minor mode of A, all their sounds in common with the scale of C. In reality, this scale is e, d, c', B, A, G, F, E, where there occurs an f' (sharp which is not in the scale of C. Though the minor mode of E is thus less relative to the major mode of C than that of A; yet the artist does not hesitate sometimes to pass immediately from the one to the other.

When we pass from one mode to another by the interval of a third, whether in descending or rising, as from C to A, or from A to C, from C to E, or from E to C, 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 C. It is the minor mode of C itself in which the perfect minor chord C, Eb, G, c', has two sounds, C and G, in common with the perfect major chord C, E, G, c'. Nor is there anything more common than a transition from the major mode of C to the minor mode, or from the minor to the major (oo).

CHAP. XI. Of Dissonance.

94. We have already observed, that the mode of C (F, C, G,) has two sounds in common with the mode of G (C, G, D); and two sounds in common with the mode of F (Bb, F, C); of consequence, this procedure of the bass C G may belong to the mode of C, or to the mode of G, as the procedure of the bass F C, or C F, may belong to the mode of C or the mode of F. When one therefore passes from C to F or to G in a fundamental bass, he is still ignorant 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 sounds G and F in the same harmony, that is to say, by joining to the harmony G, B, d' of the fifth G, the other fifth F in this manner, G, B, d', f'; this f' which is added, forms a dissonance with G (art. 18.) Hence the chord G, B, d', f' is called a dissonant chord, or a chord of the seventh. It serves to distinguish the fifth G from the generator C, which always implies, without mixture or alteration, the perfect chord C, E, G, c', resulting from nature itself (art. 32.) By this we may see, that when we pass from C to G, one passes at the same time from C to F, because f' is found to be comprehended in the chord of G; and the mode of C by these means plainly appears to be determined, because there is none but that mode to which the sounds F and G at once belong.

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

97. For this reason, instead of G, we shall take its chord of fifth f', the sound that approaches it nearest; and the great we shall have, instead of the fifth F, the chord F, A, c', d', which is called a chord of the great fifth.

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

98. The fifth G, in rising above the generator, gives a chord entirely consisting of thirds ascending from G, ces continuer C, B, d', f'; now the fifth F being below the generator C in descending, we shall find, as we go lower by thirds from c' towards E, the same sounds c', A, F, D, which form the chord F, A, c', d', given to the fifth F.

99. It appears besides, that the alteration of the harmony in the two fifths consists only in the third minor D, F, 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 sounds to Account of their octaves, that the chord F, A, c', d', is in effect the double of the same as the chord D, F, A, c', taken inversely f', employment that the inverse of the chord C, A, F, D, has been § See In found (art. 8.) in descending by thirds, from the generator C (pp.).

(oo) There are likewise other minor modes, into which we may pass in our progress from the mode major of C; as that of F minor, in which the perfect minor chord F, A, c', includes the sound c', and whole scale in ascent F, G, Ab, Bb, c', d, e, f', only includes the two sounds Ab, Bb, which do not occur in the scale of C. This transition, however, is not frequent.

The minor mode of D has only in its scale ascending D, E, F; G, A, B, c', d', one c' sharp which is not found in the scale of C. For this reason a transition may likewise be made, without grating the ear, from the mode of C major to the mode of D minor; but this passage is less immediate than the former, because the chords C, E, G, c', and D, F, A, d', not having a single sound in common, one cannot (art. 37.) pass immediately from the one to the other.

(pp) 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 D, F, A, C, as the primary chord and generator of the chord E, A, c', d', which is that chord 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... 101. The chord D, F, A, 'c', is a chord of the seventh like the chord G, B, 'd', f'; with this only difference, that the latter in the third G, B, is major; whereas in the former, the third D, F, is minor. If the F were sharp, the chord D, F, A, 'c', would be a genuine chord of the dominant, like the chord G, B, D, 'd'; and as the dominant G may descend to C in the fundamental bass, the dominant D implying or carrying with it the third major F might in the same manner descend to G.

102. Now if the F should be changed into F natural, D, the fundamental tone of this chord D, F, A, 'c', might still descend to G; for the change from F to F natural will have no other effect, than to preserve the impression of the mode of C, instead of that of the mode of G, which the F would have here introduced. The note D will, however, preserve its character as a dominant, on account of the mode of C, which forms a seventh. Thus in the chord of which we treat, (D, F, A, 'c'), D may be considered as an imperfect dominant: we call it imperfect, because it carries with it the third minor F, instead of the third major F. It is for this reason that in the sequel we shall call it simply the dominant, to distinguish it from the dominant G, which shall be named the tonic dominant.

103. Thus the sounds F and G, which cannot succeed each other (art. 36.) in a diatonic bass, when they only carry with them the perfect chords F A C, G B d, may succeed one another, if 'd' be added to the harmony of the first, and 'c' to the harmony of the second; and if the first chord be inverted, that is to say, if the two chords take this form, D, F, A, C, G, B, d, a.

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

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

We may add, that as this double employment is a kind of license, it ought not to be practised without some precaution. We have lately seen that the chords D, F, A, 'c', considered as the inverse of F, A, 'c', may succeed to C, E, G, 'c', but this liberty is not reciprocal: and though the chord F, A, 'c', may be followed by the chord C, E, G, 'c', we have no right to conclude from thence that the chord D, F, A, 'c', considered as the inverse of F, A, 'c', may be followed by the chord C, E, G, 'c'. For this reason shall be given in chap. xvi.

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

106. We have shown (chap. xvi.) how the diatonic scale, or ordinary gammut, may be formed from the fundamental bass F, C, G, D, by twice repeating the note G in that series; so that this gammut is primitive-ly composed of two similar tetrachords, one in the mode of C, the other in that of G. Now it is possible, by means of this double employment, to preserve the impression of the mode of C through the whole extent preferred, of the scale, without twice repeating the note C, or even without supposing this repetition. For this effect we form the following fundamental bass,

C, G, C, F, C, D, G, C:

in which C is understood to carry with it the perfect chord C, E, G, 'c'; G, the chord G, B, 'd', f'; F the chord F, A, 'c', d'; and D, the chord D, F, A, 'c'. It is plain from what has been said in the preceding chapter, that in this case C may ascend to D in the fundamental bass, and D descend to G, and that the impression of the mode of C is preserved by the 'd' natural, which forms the third minor 'd', instead of the third major which D ought naturally to imply.

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

'tc, d, e, f, g, a, b', c,

which of consequence will be in the mode of C alone; and if one should choose to have the second tetrachord in the mode of G, it will be necessary to substitute 'f', instead of 'f', in the harmony of D (QQ.).

108. Thus the generator C may be followed according... Part I.

Theory of ing to pleasure in ascending diatonically either by a Harmony, tonic dominant (DF A C), or by a simple dominant (DF A C).

109. In the minor mode of A, the tonic dominant E ought always to imply its third major EG X, when this dominant E descends to the generator A (art. 83.), and the chord of this dominant shall be EG X B d f, entirely similar to G B d f. With respect to the sub-dominant D, it will immediately imply the third minor F, to denominate the minor mode; and we may add B above its chord DF A, in this manner DF A B, a chord similar to that of FA c d; and as we have deduced from the chord FA c d that of DF A c, we may in the same manner deduce from the chord DF A B a new chord of the seventh B d f a', which will exhibit the double employment of diffiances in the minor mode.

110. One may employ this chord B d f a', to preserve the impression of the mode of A in the diatonic scale of the minor mode, and to prevent the necessity of twice repeating the sound E; but in this case, the F must be rendered sharp, and the chord changed to B d f a', the fifth of B being f' X, as we have seen above. This chord is then the inverse of DF X A B, the sub-dominant implying the third major, which ought not to surprise us; for in the minor mode of A, the second tetrachord E F X G X A is exactly the same as it would be in the major mode of A. Now, in the major mode of A the subdominant D ought to imply the third major F X.

111. Hence the minor mode is susceptible of a much greater number of varieties than the major: the major mode is founded in 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 subdominant, though in some measure suggested by nature (chap. xi.), is nevertheless a work of art; but as it produces great beauties in harmony by the variety which it introduces into it, let us discover whether, in consequence of this first advance, Theory of art may not still be carried farther.

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

1. The chord G B d f', composed of a third major followed by two thirds minor. 2. The chord D F A c', or B d f X a', a third major between two minors. 3. The chord B d f a', 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, C E G B, or FA c e'; the other is wholly composed of thirds minor G X B d f'. 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 C E G, to make it admit a dominant of C, one can add nothing but B b; and if, in this case C E G B b would be the chord of the tonic why dominant in the mode of F, as G B d f' is the chord of the tonic dominant in the mode of C; but if we would preserve the impression of the mode of C in the harmony, we change this B b into B natural, and the chord C E G B becomes C E G B. It is the same case with the chord FA c e', which is nothing else but the chord FA c e' b; in which one may substitute for 'e' b, e' natural, to preserve the impression of the mode of C, or of that of F.

Besides, in such chords as C E G B, FA c e', the sounds B and e', though they form a dissonance with C in the first case, and with F in the second, are nevertheless supportable to the ear, because these sounds B and e' (art. 19.) are already contained and underlaid, the first in the note E of the chord C E G B, as likewise in the note G of the same chord; the second in the note A of the chord FA c e', as likewise in the note c' of the same chord. All together they seem to allow the artist to introduce the note B and e' into these two chords (RR).

116. With respect to the chord of the seventh G X B d f', wholly composed of thirds minor, it may be regarded as formed from the union of the two chords of and extended. Theory of the Dominant and of the Sub-Dominant in the Minor Mode.

In effect, in the minor mode of A, for instance, these two chords are E G B d f, and D E A B, whose union produces E G B d f, f a. Now, if we should suffer this chord to remain thus, it would be disagreeable to the ear, by its multiplicity of dissonances, D E, F F, F G B d f, A B, D G B d f (art. 18); so that, to avoid this inconvenience, the generator A is immediately expunged, which, (art. 19) is as it were understood in D, and the fifth or dominant E, whose place the sensible note G B d f is supposed to hold; thus there remains only the chord G B d f, wholly composed of thirds minor, and in which the dominant E is considered as understood; in such a manner that the chord G B d f represents the chord of the tonic dominant E G B d f, to which we have joined the chord of the sub-dominant D E A B, but in which the dominant E is always reckoned the principal note (ss).

117. Since, then, from the chord E G B d f, we may pass to the perfect A C e a, and vice versa, we may in like manner pass from the chord G B d f to the chord A C e a, and from this last to the chord G B d f; this remark will be very useful to us in the sequel.

Chap. XV. Of the Preparation of Dissonances.

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 d' is the dissonance of the chord G B d f; e' in the chord D F, A c', &c.

Manner of preparing dissonances investigated.

119. When the chord G B d f follows the chord C E G c', as often happens, it is obvious that we do not find the dissonance d' in the preceding chord C E G c'. 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 F A c d follows the chord C E G c', the note d', which forms a dissonance with c', is not found in the preceding chord.

It is not so when the chord D F A c' follows the chord C E G c'; for c', 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 of the tonic dominant nor sub-dominant, the only means to prevent its displeasing the ear by appearing too heterogeneous to the chord, is, that it may be, if we may speak so, announced to the ear by being found in the preceding chord, and by that means connect the two chords. Hence 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. Hence, in order to prepare a dissonance, the fundamental bas must necessarily ascend by the interval of a second, as

C E G c', D F A c';

or descend by a third, as

C E G c', A C E G;

or descend by a fifth, as

C E G c', F A C E;

in every other case the dissonance cannot be prepared. This may be easily ascertained. If, for instance, the fundamental bas rises by a third, as C E G c', E G B d f, the dissonance d' is not found in the chord C E G c'. The same might be said of C E G c', G B d f, and C E G c', B D f a, in which the fundamental bas rises by a fifth or descends by a second.

124. 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 succession may be regarded as a process from that same tonic to another, which has been rendered a dominant by the addition of the dissonance.

Moreover, we have seen (art. 119. and 120.) that a dissonance does not require preparation in the chords of the tonic dominant and of the sub-dominant: 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 sub-dominant (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 ceaseless 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, and make in some measure, as much the character of an harmonic as 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 to be the chord of the tonic dominant, to which the dissonance may be joined together after having excluded from them, i. The dominant G, represented by its friend, and third major B, which is presumed to retain its place. 2. The note C which is underlaid in F, which will form why this chord B d f a. The chord B d f a, considered in this point of view, may be understood as belonging to the major mode of C upon certain occasions.

(ss) We have seen (art. 109.) that the chord B d f a, in the minor mode of A, may be regarded as the inversion of the chord D F A B; it would likewise seem, that, in certain cases, this chord B d f a may be considered as should be recomposed of the two chords G B d f, F A c d of the dominant and of the sub-dominant of the major mode of C, than descend, and third major B, which is presumed to retain its place. ought rather to descend than to rise; for this reason.

Let us take, for instance, the chord G B d f, followed by the chord C E G c; the part which formed the dissonance f ought to descend to e rather than rise to g, though both the sounds E and G are found in the subsequent chord C E G c; because it is more natural and more conformed to the connexion which ought to be found in every part of the music, that G should be found in the same part where G has already been founded, whilst the other part was founding f, as may be here seen (Parts First and Fourth).

First part, - f, e Second, - d, c Third, - B Fourth, - G G Fundamental bass, - G C

127. So, in the chord of the simple dominant D F A c, followed by G B d f, the dissonance e ought rather to descend to B than rise to d.

128. And, for the same reason, in the chord of the sub-dominant F A c d, the dissonance d ought to rise to e of the following chord C E G c, rather than descend to c; 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 sub-dominant, the dissonance ought to rise diatonically upon the third of the subsequent chord.

130. 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 D F A c, though it should even be considered as the inverse of F A c d, cannot be succeeded by the chord C E G c, since there is not in this last chord the note B, upon which the dissonance e of the chord D F A c 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 D F A c, to C E G c, it is necessary that D F A c should in this case be understood as the inverse of F A c d. Now the chord D F A c can only be conceived as the inverse of F A c d, when this chord D F A c precedes or immediately follows the C E G c; in every other case the chord D F A c is a primitive chord, formed from the perfect minor chord D F A, to which the dissonance e was added, to take from D the character of a tonic. Thus the chord D F A c could not be followed by the chord C E G c, 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, C E G c, D F A c, C E G c, it would be much more easy and natural to substitute this other, which furnishes this natural succession C E G c, F A c d, C E G c. 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. viii.), a repose more or less perfect from one found to another; and of consequence there must likewise be a repose more or less perfect from one found to another in the fundamental bass, which results from that 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 any person sing these three notes c d g, performing on the id a shake, which is commonly called a cadence; the modulation will appear to him to be finished after the second c, 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 C G C: but if, instead of this bass, we should give it the following, C G A: in this case the modulation c d c 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 G A, when the dominant G diatonically ascends upon the note A instead of descending by a fifth upon the generator C, as it ought naturally to do, is called a broken cadence; because the perfect cadence G C, which the ear expects after the dominant G, is, if we may speak so, broken and suspended by the transition from G to A.

133. Hence it follows, that if the modulation c d c appeared finished when we supposed no bass to it at all, it is because its natural fundamental bass C G C is implied; for the ear desires something to follow this modulation, as soon as it is reduced to the necessity of hearing another bass.

134. The broken cadence may be considered as having its origin in the double employment of dissonances; broken since this cadence, like the double employment, only consists in a diatonic procedure of the bass ascending (chap. xii.) In effect, nothing hinders us to descend from the chord G B d f to the chord C E G A by converting the tonic C into a sub-dominant, that is to say, by passing all at once from the mode of C to the mode of G: now to descend from G B d f to C E G A is the same thing as to rise from the chord G B d f to the chord A c e g, in changing the chord of the sub-dominant C E G A 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 performing the fifth of the subsequent chord. For instance, in this case the broken cadence G B d f, A c e g, the dissonance f is resolved by descending diatonically upon the fifth e.

136. There is another kind of cadence, called an interrupted cadence, where the dominant descends by a cadence, third to another dominant, instead of descending by a what fifth upon the tonic, as in this succession of the bass G B d f.

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 C, to represent all the major modes in general, and the minor mode of A 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, called also counterpoint, is not only the art of composing an agreeable air, but also that of composing several 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.), we have seen a detail of the most common names given to the different intervals. But particular intervals have obtained different names, according to 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 C to D in ascending, or that of A to Gb descending.

This interval is so termed, because one of the sounds which form it is always either sharp or flat, and that, if that sharp or flat be taken away, the interval will be that of a second (3 c).

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

162. As the interval from C to D in ascending fifth has been called a second redundant, we likewise call redundant, or from B to Eb in descending, each of which intervals are composed of four tones (3 h).

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

163. For the same reason, an interval composed of three tones and three semitones, as from G to f' in ascending, is called a seventh diminished; because, if we remove the sharp from G, the interval from G to f' will become that of an ordinary seventh. The interval of a seventh diminished is in other respects the same with that of the sixth major (art. 9.) (3 i).

164. The major seventh is likewise sometimes called a seventh redundant (3 k).

Chap. II. Comparison of the Different Intervals.

165. If we sing 'c' B in descending by a second, and afterwards C B in ascending by a seventh, these different two B'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 others every

general more proper to produce a learned and harmonious music, than a strain prompted by genius and animated by enthusiasm.

(3 c) For the same reason, this interval is frequently termed by English musicians an extreme sharp second.

(3 h) This interval is usually termed by English theorists a sharp fifth.

(3 i) The material difference between the diminished seventh and the major sixth is, that the former always implies a division of the interval into three minor thirds, whereas a division into a fourth and third major, or into a second and major and minor third, is usually supposed in the latter.

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

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

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

To rise by a second.—To descend by a seventh. To descend by a second.—To rise by a seventh. To rise by a third.—To descend by a sixth. To descend by a third.—To rise by a sixth. To rise by a fourth.—To descend by a fifth. To descend by a fourth.—To rise by a fifth.

(3 L) Our author has treated this part of his subject with somewhat less perspicuity than usual. He has neither described the staffs or systems of lines on which the clefs are placed, nor explained their relation to each other. We have therefore attempted to supply the deficiency.

Musical sounds, like language, are represented by written characters, by which their gravenefs or acutenefs, their duration, and the other qualities intended to be assigned to them, are accurately distinguished.

The characters which denote the gravenefs or acutenefs, or, as it is termed, the pitch of sounds, are intended to represent the ordinary limits of the human voice, in the exercise of which, or the employment of instruments of nearly the same compass with it, all practical music consists.

From the lowest distinct note, without straining, of the masculine voice, to the highest note generally produced by the female voice, there is an interval of three octaves, or twenty-two diatonic notes.

These notes are represented by characters described alternately on eleven parallel lines, and the spaces between them, forming what we shall here term the general system.

The characters representing the notes are differently formed according to their duration, but with this we have at present no concern. We shall employ the simplest, a small circle or ellipse.

The whole extent of the human voice, then, if described upon the general system, would be represented as at Plate CCCLV, fig 1.

The masculine voice, rising from the lowest note of the general system, will, generally speaking, reach the note on the central line; and an ordinary female voice will reach the same note, descending from the highest. Male voices more acute, and female voices graver than usual, will consequently execute this note with greater facility.

This central note then, being producible by every species of voice, has been assumed as a fundamental or key note, by which all the others are regulated (art. 4.). And to it is assigned the name of C, by which, in the theory of harmony, (as we have seen), the fundamental sound of the diatonic scale is distinguished.

The other notes take their denominations accordingly. The note below it is B, that above it 'd', &c.; and to distinguish this central C from its octaves, it is called the middle or tenor C.

As no human voice can execute the whole twenty-two notes, the general system is divided into portions of five lines, each portion representing the compass of an ordinary voice; and different portions are made use of, according to the gravenefs or acutenefs of different voices.

The five lines in this state form what is called a staff. Each staff is subdivided into lines and spaces. On the lines, and in the spaces, the heads of the notes are placed. The lines and spaces are counted upwards, from the lowest to the highest; the lowest line is termed the first line; the space between it and the second line is denominated the first space, and so on. Both lines and spaces have the common name of degrees; the staff thus contains nine degrees, viz. five lines and four spaces.

To ascertain what part of the general system is formed by a staff, one of the clefs mentioned in the text is placed at the beginning of the staff, on one or other of the lines of it.

The C or tenor clef always denotes the line on which it is placed to be that which carries the tenor C. The G or treble clef distinguishes the line carrying 'g', the perfect fifth above the tenor C. And the F or bass clef affords the line which represents F the perfect fifth below the tenor C.

The figures of the clefs, (which are characters gradually corrupted from the Gothic C, G, and F), and their places in the general system, appear on Plate CCCLV, Fig. 2.

By this disposition of the clefs, we see that the staff, which includes the line bearing the treble clef, is formed by the five highest lines of the general system; and that the staff which comprehends the bass clef consists of the five lowest.

The central line, which carries the tenor C, belongs neither to the treble nor the bass staves. But as that note frequently occurs in composition written on these staves, a small portion of the tenor line is occasionally introduced below the treble clef and above that of the bass (fig. 3.) Part II.

Principles: all the notes on the same line with the clef take the name of C.

The G clef is placed on the second or first line; and all the notes on the line of the clef take the name of G.

Names of the notes to the spaces between the lines, the name of any note may be discovered from the position of the clef. Thus, in the F clef, the note on the lowest line is G; the note on the space between the two first lines A; the note on second line B, &c.

Marks and power of sharps, flats, and naturals.

A note before which there is a sharp (marked thus ♯) must be raised by a semitone; and if there be a flat (marked ♭) before it, it must be depressed by a semitone.

As notes still more remote from the staff in use are sometimes introduced, small portions of the lines to which these lines belong are employed in the same manner. Thus, if in writing in the bass staff we want the note properly placed on the lowest line of the treble staff, we draw two short lines above the bass staff, one representing the tenor line, and the other the lowest line of the treble staff, and on this last short line we place the note in question, (fig. 4.)

On the other hand, if, in writing on the treble staff, we would employ a note properly belonging to the bass staff, we place it below the treble staff, and insert the requisite short lines, representing the corresponding lines of the general system (fig. 5.)

The occasional short lines thus employed are termed leger lines.

The same expedient is used to represent notes beyond the limits of the general system. Thus, we write the F which is one degree lower than the lowest G of the bass staff, on the space below that G; the E immediately lower, or on a leger line below the bass staff, and so on. Notes in this position are termed double; thus, the F just mentioned is double F, or FF; the E, double E, or EE, &c.

Again, the ♯ above the highest G of the treble staff is placed on a leger line above that staff. The ♭ is placed on the space above the leger line: The next note ♭ is set on a second leger line, and so on. These high notes are, in compositions for some instruments, carried more than an octave above the general system. Those in the first octave are said to be in alt; those beyond it, to be in altissimo.

The tenor or C clef is employed to form different intermediate staves between the treble and bass, according to the compass of the voice or instrument for which the staff is wanted.

Compositions for the gravest masculine voices and instruments are written on the bass clef, and those for female voices and instruments highest in tone, on the treble staff.

For masculine voices next in depth to the bass and for the higher octave of the violoncello and bassoon, a staff, called the tenor staff, is formed by adding to the tenor line the three highest lines of the bass staff and the lowest line of the treble (fig. 6. 1.)

For the highest masculine voices, which are called counter tenor, and for the tenor violin, a staff is formed by the tenor line, the two highest lines of the bass, and the two lowest of the treble staff (fig. 6. 2.)

For the gravest female voices, which are called mezzo soprano, the tenor line and four lowest lines of the treble form a staff (fig. 6. 3.)

The relation of all the staves to the general system, and to each other, will appear from fig. 6.

The bass clef on the third line, the tenor clef on the second, and the treble clef on the first, rarely occur, except in old French music.

The tenor clef, and the staves distinguished by it, are now less frequently used than the treble and bass clefs. Those who cultivate music only as an amusement find it irksome to learn so many modes of notation. The tenor staves are accordingly banished from compositions for keyed instruments. Secular compositions for voices are likewise now written in the treble and bass staves only; although in this there is some inaccuracy, as the tenor parts now written in the treble staff, must often be sung an octave below that in which they appear. The chief use of the tenor clef is in choral music and compositions for the bassoon and tenor violin; and its principal advantage, the facility of reading ancient music, which is almost exclusively written in this clef, has seldom been deemed an insufficient compensation for the labour of acquiring it.

(3 M) The disposition of sharps or flats at the clef, which is termed the signature, depends upon the mode, or tone assumed in the composition as a fundamental or key note, and will be afterwards explained.

The sharps or flats of the signature affect not only the notes placed on the same degree with themselves, as mentioned in the text, but also all the notes of the same letter, in every octave throughout the movement.

The sharps or flats of the signature determine the scale in which the movement is composed, and are therefore said to be essential; those which occur in the course of the piece on an occasional change of the scale, are termed accidental.

* Compositions for French horns are written in the treble staff, although the tone of the instrument be very grave; but this is because the horn is borrowed from and has the same natural intervals with the Trumpet, which is an acute instrument. equal times, called measures; and each measure is likewise divided into different times.

There are properly two kinds of measures or modes of time; the measure of two times, or common time, marked by the figure 2 at the beginning of the time (fig. 10); and the measure of three times, or triple time, marked by the figure 3 placed in the same manner (fig. 11).

The different measures are distinguished by perpendicular lines (3 n), called bars.

In a measure, we distinguish between the strong and the weak time: the strong time is that which is beat; the weak, that in which the hand or foot is raised. A measure consisting of four times ought to be considered as compounded of two measures, each consisting of two times; thus there are in this measure two strong and two weak times. In general by the words strong and weak even the parts of the same time are distinguished; thus, the first note of each time is considered as strong and the others as weak.

175. The longest of all notes is a semibreve. A minim is half its value; that is to say, two minims are to be performed in the time occupied by one semibreve duration. A minim in the same manner is equivalent to two crotchets, the crotchet to two quavers (3 o).

176. A note which is divided into two parts by a bar, that is, which begins at the end of a measure, and terminates in the measure following, is called a syncopated note (3 p).

179. A note followed by a point or dot is increased by half its value. Thus a dotted semibreve is equivalent to a semibreve and a minim, a dotted minim, to a minim and a crotchet, &c. (Fig. 17.) (3 q).

(3 n) All the notes, therefore, contained between two bars constitute one measure; although in common language the word bar is improperly used for measure.

(3 o) The notes, in their figure, consist of a head and a stem, except the semibreve, which has a head only.

The place of the note in the staff is determined by the head, which must be placed on the line, or in the space, assigned to the note. The stem may be turned either up or down.

The quaver is equivalent to two semiquavers, and the semiquaver to two demi-semiquavers. In modern music the demi-semiquaver is also subdivided.

The quaver and the notes of shorter duration may be grouped together, by two, three, or four, &c., and joined by as many black lines across the ends of the stem as there are hooks in the single note (fig. 12). This arrangement is convenient in writing, and assists the eye in performance.

When quavers, or the shorter notes, are to be repeated in the same degree for a time equal to the duration of a longer note, the iterations are, by a fort of musical shorthand, represented by writing the long note only, and placing over or under it, as many short lines as the short note has hooks (fig. 13). And the repetition of a series of short notes is represented by merely writing for each repetition as many short lines as there are hooks to the short notes of which the series is composed (fig. 14).

(3 p) A note in the middle of a measure is also said to be syncopated when it begins on a strong, and ends on a weak part of the measure, (see fig. 15,) where D, C, and B are each of them syncopated.

A note which of itself occupies one, two, or more measures, is not said to be syncopated, but continued or prolonged. See fig. 16.

(3 q) Notes have sometimes in modern music a double dot after them, which makes them longer by three-fourths. Thus a minim twice dotted is equal to three crotchets and a half, or seven quavers, &c.

Our author, in this chapter, has omitted the explanation of rests, and of the particular modifications of time.

Rests are characters indicating the temporary suspension of musical sounds. There are as many different rests as there are notes. Thus the semibreve rest indicates a pause of the duration of a semibreve; the minim rest, of a minim, &c. (fig. 18.)

The semibreve rest also denotes the silence of one entire measure, in triple as well as common time. The silence of several measures is marked as in fig. 18.; but where the silence exceeds three bars, the number is usually marked over the rests.

Common time is either of a semibreve, or of a minim to the measure.

Common time of a semibreve is indicated by the letter C at the clef, fig. 1. of Plate CCCLVI. When it is meant to be somewhat quicker than usual, a perpendicular line is drawn through the C, (fig. 2.)

Common time of a minim to the measure, which is called half time, is indicated by the fraction \(\frac{2}{3}\), that is, two-fourths of a semibreve, or two crotchets equal to a minim, (fig. 3.)

In triple time the measure consists of three minims, three crotchets or three quavers, six crotchets or six quavers, nine quavers or twelve quavers.

Triple time of three minims is marked at the clef \(\frac{3}{4}\), that is, three halves of a semibreve, (fig. 4.)

Triple time of three crotchets is indicated by the fraction \(\frac{3}{4}\), (three-fourths of a semibreve) (fig. 5.) and that of three quavers by \(\frac{3}{4}\) (three-eighths of a semibreve,) (fig. 6.)

In the last three examples the measure is divided into three times, of which the first is strong, and the two others weak.

The measure of six crotchets is marked \(\frac{6}{4}\), (fig. 7.) and that of six quavers, \(\frac{6}{4}\), (fig. 8.) In both there are two times, of which the first is strong, and the second weak.

The measure of nine quavers is marked \(\frac{9}{4}\), (fig. 9.) and is divided into one strong and two weak times. That of twelve quavers is marked \(\frac{12}{4}\), (fig. 10.) and is accented as if it were two measures of six quavers.

The measures of \(\frac{3}{4}\) and \(\frac{9}{4}\) rarely occur.

Three notes are often performed in the time of two of the same name, and are then termed triplets, (fig. 11.) Part II.

Chap. IV. Definition of the principal Chords.

178. (3 r) The chord composed of a third, a fifth, and an octave, as C, E, G, C, is called a perfect chord (art. 32.).

If the third be major, as in C, E, G, C, the perfect chord is denominated major; if the third be minor, as in A, C, E, A, the perfect chord is minor. The perfect chord major constitutes the major mode; and the perfect chord minor, the minor mode (art. 31.).

179. A chord composed of a third, a fifth, and a seventh, as G, B, D, F, or D, F, A, C, &c., is called a chord of the seventh. Such a chord is wholly composed of thirds in ascending.

All chords of the seventh are specified in harmony, save that which might carry the third minor and the seventh major, as C Eb G B; and that which might carry a false fifth and a seventh major, B D F A, (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, B D F A, 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 F A C D, D F A B, 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 C's equally carry a perfect chord.

183. Every note which carries a chord of the seventh is called a dominant (art. 102.); and this chord is marked by a 7 written above the note. Thus in the example ii. D carries the chord D F A C, and G the chord G B D F.

It is necessary to remark, that among the chords

Vol. XIV. Part II.

where the groups of quavers in the second measure are triplets, and each triplet occupies the time of two quavers only. Triplets also occur in triple time, fig. 12.

Certain other characters will be with propriety explained here.

The Pause signifies that the regular time is to be delayed, and the note marked with the pause protracted. See fig. 13, where the pause is on the last note of the second measure.

The Repeat, a character resembling an S, denotes, that the following part of the movement must be repeated. See fig. 14.

The Direct (fig. 15.) is placed at the end of the staff, to shew upon what degree the first note of the following staff is placed.

When the inner sides of two bars are dotted, the measures between them are to be repeated (fig. 16.) The word bis is sometimes placed over such passages.

The double bar distinguishes the end of a movement or strain, (fig. 17.) If the double bar be dotted on one or both sides, the strain is to be repeated, (fig. 18.) The double bar does not affect the time; so that when the strain terminates before the end of a measure, as is often the case, the double bar only marks the conclusion of the strain, but the time is kept exactly as if it were not inserted. See fig. 19.

The graces of exertion and expression, such as the appogiature, the shake, the flur, the crescendo, the diminuendo, &c. are not necessary to the consideration of the theory of music or principles of composition, but belong to the performer only. See Shake, &c.

(3 r) In this part of our subject, we shall, in mentioning the harmonics of the chords, make use of the capital letters only, as the general names of the notes, without distinguishing octaves by minuscular or Italic letters. The harmonics may be arranged in different octaves. Their different positions will be most easily seen and best understood from the examples in the plates. fundamental note. For example, if we would mark the perfect major chord D F A D, as the third F above D is naturally minor, we place above D a sharp, as in Example iv. In the same manner, the chord of the seventh D F A C, and the chord of the great sixth D F A B, is marked with a ♯ above D, and above the ♭ a 7 or a 6 (see v. and vi.).

On the contrary, when the third is naturally major, and if we would render it minor, we place above the fundamental note a ♭. Thus the examples vii. viii. ix. show the chords G B♭ D G G B♭ D F, G B♭ D E (35).

**Chap. V. Of the Fundamental Bass.**

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

Thus (Exam. xvi.) it will be found that this modulation, C D E F G A B C, has or may admit for its fundamental bass, C G C F C D G C.

In reality, the first note C in the upper part is found in the chord of the first note C in the bass, which chord is G E G C; the second note D in the treble is found in the chord G B D G, which is the chord of the second note in the bass, &c., and the bass is composed only of notes which carry a perfect chord,

(35) 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 F which indicates the key of G (see Exam. x.) it is sufficient to write D, without a sharp, to mark the perfect chord major of D, D F A D. In the same manner, in the Example xi. where the flat is at the clef upon B, which denotes the key of F, it is sufficient to write G, to mark the perfect chord minor of G B♭ D G.

But where there is a sharp or a flat at the clef, if we would render the chord minor which is major, or vice versa, we must place above the fundamental note a ♭ or natural. Thus the Example xii. marks the minor chord D F A D, and Example xiii. the major chord G B D G.—Sometimes, in lieu of a natural, a flat is used to signify the minor chord, and a sharp to signify the major. Thus Example xiv. in the key of G, marks the minor chord D F A D, and Example xv. in F, the major chord G B D G.

When in a chord of the great sixth, the dissonance, that is to say, the sixth, ought to be sharp, and when the sharp is not found at the clef, we write before or after the 6 a ♯; and if this sixth should be flat according to the clef, we 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, we 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 one sharp at the clef, and if we would mark the chord G B D F♯, or the chord A C E F♯, we ought to place before the seventh or the sixth a ♬ or a ♭.

In the same manner, if there be one flat at the clef, and if we would mark the chord C E G B♭, we ought to place before the seventh a ♬ or a ♭; and so of the rest.

All these intricate combinations of figuring show the superior convenience of the modern method of writing the notes themselves instead of the figures, which has the farther advantage of exhibiting the proper arrangement of the chord, see Example ii.

**Chap. VI. Rules for the Fundamental Bass.**

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

The fundamental bass 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 Fifth 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 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 F A C D, it is necessary that F, A, or C, which are the consonances... Part II.

Principles consonances of the chord, should be found in the chord preceding. The dissonance D 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; 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 will presently be seen. For, let us assume the succession of the two chords A C E G, D F A C (see Exam. xvii.), this succession is by no means legitimate, though in the first dominant descends by a fifth; because the C 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, we take away the sharp carried by the C in the first; or if, without meddling with the first chord, we render C and F sharp in the second (3t); or, if we simply render the D of the second chord a tonic dominant, in causing it to carry F instead of F# (119, and 122).

It is likewise by the same rule that we ought to reject the succession of the two following chords,

D F A C, G B D F#;

(see Exam. xviii.).

RULE V.

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

REMARK.

Other rules Of the five fundamental rules which have now been substituted, given, instead of the three first, one may substitute the three following, which are consequences from them.

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 Principles xix. and xx., (3u); a tonic dominant (124.), see Composition xxI. and xxII.; or a sub-dominant (124.), see xxIII. and xxIV.; or, to express the rule more simply, that second note may be any one, 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. xxv. and xxvi.; or a tonic dominant, or a simple dominant, yet in such a manner that the rule of art. 192. may be observed (124.), see xxvii. xxviii. xxix. and xxx.; or a sub-dominant (124.), see xxxi. and xxxii.

The succession of the bass C Eb G C, FACE, 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 xxxiv. and xxxv. (3x).

We must here advertize our readers, that the examples xxxvi. xxxvii. xxxviii. xxxix. belong to the fourth rule above, art. 194.; and the examples xl. xli. xlii. 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 cadences, tonic is called an imperfect or irregular cadence (73.) how ever the tonic falls upon the accented part of the bar. See employed.

XLIII. XLIV. XLV. XLVI.

REMARK II.

197. We must avoid, as much as we can, syncopations in the fundamental bass; that the ear may accurately distinguish the primarily accented part of a measure, by means of a harmony different from that which it had before perceived in the last unaccented part of the bass by the preceding measure. Nevertheless, syncopation may be sometimes admitted in the fundamental bass, but it is by a license (3y).

(3t) In this chord it is necessary that the C and F should be sharp at the same time; for the chord D F A C#, in which C would be sharp without the F, is excluded by art. 179.

(3u) 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 we ascend from the tonic C to the tonic E, the major mode of C, C E G C, will be changed into the minor mode of E, E G B E. We must never ascend from one tonic to another, when there is no sound common to both their modes: for example, we cannot rise from the mode of C, C E G C, to the minor mode of Eb, Eb Gb Bb Eb (91.).

(3x) Thus all the intervals, viz. the third, the fifth, and second, may be admitted in the fundamental bass, except that of a second in descending. The rules now given for the fundamental bass are not, however, 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 enter into 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.

(3y) There are notes which may be found several times in the fundamental bass in succession with a different CHAP. VII. Of the Rules which ought to be observed in the Treble with relation to the Fundamental Bafs.

198. The treble is nothing else but a modulation above the fundamental bafs, and whole notes are found in the chords of that bafs which corresponds with it (189.) Thus in Ex. xvi. the scale C D E F G A B C, is a treble with respect to the fundamental bafs C G C F C D G C.

199. We are 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 bafs, when these notes belong to the chord of the same note in the fundamental bafs. For example, this modulation C E G E C, may have for its fundamental bafs the note C alone, because the chord of that note comprehends the sounds C, E, G, which are found in the treble.

2. In like manner, a single note in the treble may, for the same reason, answer to several notes in the bafs. For instance, G alone may answer to these three notes in the bafs, C G C (32).

RULE I. For the TREBLE.

200. If the note which forms the seventh in a chord of the simple dominant, is found in the treble, the note which precedes it must be the very same. This is what we call a disjunct prepared (122). For instance, let us suppose that the note of the fundamental bafs shall be D, bearing the chord of the simple dominant D F A C; and that this C, which (art. 18. and 118.) is the dissonance, should be found in the treble; it is necessary that the note which goes before it in the treble should likewise be a C.

201. According to the rules which we have given for the fundamental bafs, C will always be found in the chord of that note in the fundamental bafs which precedes the simple dominant D. See XLVIII., XLIX., L. In the first example the dissonance is C, in the second G, and in the third E; and these notes are already in the preceding chord (4A).

RULE II.

202. If a note of the fundamental bafs be a tonic dominant, or a simple dominant, and if the dissonance be found in the treble, this dissonance in the same treble ought to descend diatonically. But if the note of the bafs be a sub-dominant, it ought to rise diatonically. This dissonance, which rises or descends diatonically, is what we have called a dissonance saved or resolved (129, 130.) See LII., LIII., LIV.

203. According to the rules for the fundamental bafs which we have given, the note upon which the dissonance

ferent harmony. For instance, the tonic C, after having carried the chord C E G C, may be followed by another C which carries the chord of the seventh, provided that this chord be the chord of the tonic dominant C E G Bb. In the same manner, the tonic C may be followed by the same tonic C, which may be rendered a sub-dominant, by causing it to carry the chord C E G A.

A dominant, whether tonic or simple, sometimes descends or rises to another by the interval of a tritone or false fifth. For example, the dominant F carrying the chord F A C E, may be followed by another dominant B carrying the chord B D F A. 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 C to which F and B belong. If one should descend from F to Bb by the interval of a just fifth, he would then depart from that scale, because Bb is no part of it.

(32) 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 Example XLVII. (Plate CCCLVIII.), in which this modulation C D E F G, may be regarded as equivalent to this other, C E G, as D and F are no more than notes of passage. So that the bafs of this modulation may be simply C G.

When the notes are of equal duration, and arranged in a diatonic order, the notes which are accented ought each of them to carry chords. Those which are unaccented, are mere notes of passage. Sometimes, however, the unaccented note may be made to carry harmony; but the duration of this note is then commonly increased by a point placed after it, which proportionally diminishes the continuance of the accented note, 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.

(4A) 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 bafs, its seventh is afterwards heard in the modulation, while the dominant is still retained. For instance, let us imagine this modulation,

\[ \begin{array}{c|c} C & D C B C \\ \hline D & GC G \end{array} \]

and this fundamental bafs,

\[ \begin{array}{c|c} C & D C B C \\ \hline D & GC G \end{array} \]

(see example LI.); the D of the fundamental bafs answers to the two notes D C of the treble. The dissonance C has no need of preparation, because the note D of the fundamental bafs having already been employed for the D which precedes C, the dissonance C is afterwards presented, below which the chord D may be preserved, or D F A C. Part II.

204. The continued bass, is 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 G B D F, we should say B D F G or D F G B, &c., the chord is inverted.

The ways in which a Perfect Chord may be Inverted.

205. The perfect chord C E G C may be inverted in two different ways.

1. E G C E, which we call a chord of the sixth, composed of a third, a fifth, and an octave; and in this case the bass note E is marked with a 6. (See LVI.)

2. G C E G, which we call a chord of the fifth and fourth, composed of a fourth, a fifth, and an octave; and it is marked with a 6. (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 G B D F, the third major B above the fundamental note G is called a sensible note (77); and the inverted chord B D F G composed of a third, a false fifth and sixth, is called the chord of the false fifth, and is marked as in examples LVIII. and LIX.

The chord D F G B, composed of a third, a fourth, and a sixth, is called the chord of the sensible fifth, and marked as in Example LX. (4 c). In this chord, the third is minor, and the sixth major.

The chord F G B D, composed of a second, a tritone, and a sixth, is called the chord of the tritone, and is marked as in Example LXI. (4 d).

207. In the chord of the simple dominant D F A C, we find,

1. F A C D, 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 LXII. (4 E).

2. A C D F, a chord of the lesser sixth, which is figured with a 6. See LXIII. (4 F).

3. C D F A, a chord of the second, composed of a second, a fourth, and a sixth, and which is marked with a 2. See LXIV. (4 G).

The ways in which the Chord of the sub-dominant may be Inverted.

208. The chord of the sub-dominant, as F A C D, 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 A C D F (LXIII.), and the chord of the seventh D F A C. See LXV.

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 fingering, and suitable to the voice. See LXVI. in which the fundamental bass which in itself is monotonic and little fitted for fingering, C G C G C G C, produces, by inverting its chords, this continued bass highly proper to be sung, C B C D E F E, &c. (4 H.)

The continued bass then is properly a treble with respect to the fundamental bass. Its rules immediately follow, which are properly those already given for the treble.

RULE I.

210. Every note which carries the chord of the false fifth,

(4 b) When the treble syncopates in descending diatonically, it is common enough to make the second part of the syncopate carry a dissonance, and the first a concord. See Example LV. where the first part of the syncopated note G, is in concord with the notes C E G C, which answers to it in the fundamental bass, and where the second part is a dissonance in the subsequent chord A C E G. In the same manner, the first part of the syncopated note F is in concord with the notes D F A C, which answer to it; and the second part is a dissonance in the subsequent chord G B D F, which answers to it, &c.

(4 c) This chord is called by English musicians, the chord of the third and fourth, and generally figured 4.

(4 d) This chord is in England called the chord of the second and fourth, and is figured 4.

(4 e) 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 fifth and of the lesser sixth. (See examples LVI. and LXIII.) The chord of the great sixth 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 C E G B, gives the chord of the great sixth E G B C, thus improperly called, since the sixth from E to C is minor.

(4 f) M. Rameau has justly observed, that we ought rather to figure this lesser sixth with a 6, 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 notation, though we thought with him, that it would be better to mark this chord by a particular figure.

(4 g) The chord of the seventh B D F A gives, when inverted, the chord F A B D, composed of a third, a tritone, and a sixth. The 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 F be a tritone or a fourth redundant.

One may figure this chord thus, 4.

(4 h) The continued bass is proportionably adapted to fingering, as the sounds which form it more scrupulously observe. Musical Principles

RULE II.

211. Every note carrying the chord of the tritone should descend diatonically upon the subsequent note. Thus in the same example LXVI. F, which carries the chord of the tritone figured with a 4t, descends diatonically upon E (art. 202.)

RULE III.

212. The chord of the second is commonly put in practice upon notes which are syncopated in descending 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 LXV. is much more in the taste of singing, and more agreeable, than the fundamental bass which answers to it.

(4 t) 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 D, carrying a chord of the seventh DFAC, to which the chord of the sub-dominant F A C D corresponds in the fundamental bass, ought to rise diatonically upon E, (art. 129. No. 1. and art. 202.)

(4 k) When there is a rebele in the treble, the note of the continued bass ought to be the same with that of the fundamental bass, (see Example LXXIII.) In the places which are found in the treble at D and C (measures second and fourth), the notes in the fundamental and continued bass are the same, viz. G for the first cadence, and C for the second. This rule ought above all to be observed in 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 first note of the second measure in Example LXXIX. D is found at the same time in the continued bass and in the treble; but the treble rises from C to D, and from D to E, whilst the bass descends from E to D, and from D to C.

Two octaves, or two fifths, in succession, must likewise be avoided. For instance, in the treble sounds G E, the bass must be prevented from sounding G E, C A, or D B; because in the first case there are two octaves in succession, E against E, and G against G; and because in the second case there are two fifths in succession, C against E, and A against G, or D against G, and B against E. 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 LXX. (Plate CCCLIX.), where the fundamental bass C gives the continued bass C E G E; the two E's ought in this bass to carry the chord 6, and G the chord 3; but as these chords are comprehended in the perfect chord C E G C, which is the first of the continued bass, we place nothing above C, only we draw a line over C E G E.

In like manner, in the second measure of the same example, the notes F and D of the continued bass, arising from the note G alone of the fundamental bass which carries the chord G B D F, we think it sufficient to figure F only, and to draw a line above F and D because the same harmony is used with both.

It should be remarked, that this F ought naturally to descend to E; but this note is considered as subsisting so long as the chord subsists; and when the chord changes, we ought necessarily to find the E, 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 LXXI. where this continued bass, C B D B G C, is reckoned the same with this C, B C. (Example LXXII.)

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 G in a continued bass may answer to this fundamental bass C G C, (see Example LXXIII.) in this case, we may regard the note G as divided into three parts, of which the first carries the chord 6, the second the chord 7, and the third the chord 3.

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 Of Supposition.

215. When a dominant is preceded by a tonic in 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 G carrying the chord of the seventh G B D F; let us add to this chord the note C, which is a fifth below this dominant, and we shall have the total chord C G B D F, or C D F G, which is called a chord by supposition (4 m).

216. Chords by supposition are of different kinds. For instance, the chord of the tonic dominant G B D F gives,

1. By adding the fifth C, the chord C G B D F, called a chord of the seventh redundant, and composed of chords of a fifth, seventh, ninth, and eleventh. It is figured what, and with a \( \frac{7}{8} \); see LXXVI. (4 n). This chord is not how figured, but upon the tonic. They sometimes leave out the sensible note, for reasons which we shall give in the note (4 o), upon the art. 219; it is then reduced to C F G D, and marked with \( \frac{3}{4} \) or \( \frac{5}{8} \).

2. By adding the third E, we shall have the chord E G B D F, called a chord of the ninth, and composed of a third, fifth, seventh, and ninth. And it is figured with a \( \frac{9}{8} \). This third may be added to every third of the dominant. See LXXVII.

3. If

and elsewhere. These rules, which are proper for a complete dissertation, did not appear indispensably necessary in an elementary essay on 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.

(4 l) One may sometimes, but very rarely, cause several tonics in succession to follow one another in ascending or descending diatonically, as C E G C, D F A D, Bb D F Bb; 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 F, a fifth below the first tonic C, is found in the chord D F A D, and in the chord Bb D F Bb (37, and note T).

(4 m) Though supposition be a kind of license, yet it is in some measure founded on the experiment related in the note (s), 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, C added beneath G, causes G to refund. Thus G is found in some measure to be implied at C.

If the third added beneath the fundamental sound be minor, for example, if to the chord G B D F, we add the third E, 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.

(4 n) Many musicians figure this chord with a \( \frac{7}{8} \); M. Rameau supposes this \( \frac{7}{8} \), and merely marks it to be the seventh redundant by a \( \frac{7}{8} \) or \( \frac{7}{8} \). 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 \( \frac{7}{8} \)? 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 subsiding already in the preceding chord, it would be useless to repeat it.

Thus D G, according to M. Rameau, would indicate D F \( \frac{7}{8} \) A C, G B D F \( \frac{7}{8} \). If we would change F \( \frac{7}{8} \) of the second chord into F \( \frac{7}{8} \), it would then be necessary to write D G. In notes such as C, whose natural seventh is major, the figure \( \frac{7}{8} \) preceded or followed by a sharp will sufficiently serve to distinguish the chord of the seventh redundant C G B D F, from the simple chord of the seventh C E G B, which is marked with a \( \frac{7}{8} \) alone.

All this appears just and well founded.

(4 o) Supposition introduces into a chord differences which were not in it before. For instance, if to the chord E G B D, we should add the note of supposition C descending by a third, it is plain that, besides the difference between E and D which was in the original chord, we have two new dissonances, C B, and C D; that is to say, the seventh and the ninth. These dissonances, like the others, ought to be prepared and resolved. They are prepared by being syncopated, and resolved by descending diatonically upon one of the consonances 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 dissonances which are found in the primitive chord, they should always follow the common rules. (See art. 202.) 3. If to a chord of the simple dominant, as D F A C, we should add the fifth G, we would have the chord G D F A C, called a chord of the eleventh, and which is figured with a \( \frac{2}{4} \) or \( \frac{4}{5} \). (See LXXXVIII.)

**Observe.**

217. When the dominant is not a tonic dominant, we often take away some notes from the chord. For example, let us suppose that there is in the fundamental bass this simple dominant E, carrying the chord E G B D; if there should be added the third C beneath, we shall have this chord of the continued bass C E G B D; but we suppress the seventh B, for reasons which shall be explained in the note upon art. 210. In this state the chord is simply composed of a third, fifth, and ninth, and is marked with a 9. See LXXXIX. (4 p).

218. In the chord of the simple dominant, as D F A C, when the fifth G is added, we frequently obliterate the sounds F and A, that too great a number of dissonances may be avoided, which reduces the chord to G C D. 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 (4 Q). (See LXXX.)

219. Sometimes we only remove the note A, and then the chord ought to be figured with \( \frac{2}{4} \) or \( \frac{4}{5} \) (4 R).

220. Finally, in the minor mode, for example, in that of A, where the chord of the tonic dominant (109), is E G B D; if we add to this chord the third C below, we shall have E G B D, called the chord of the fifth redundant, and composed of a third, a fifth redundant, a seventh, and a ninth. It is figured as in LXXXI. (4 S)

§ 3. Of the Chord of the Diminished Seventh.

221. In the minor mode, for instance, in that of A, E a fifth from A is the tonic dominant (109), and carries the chord E G B D, in which G is the sensible note. For this chord we sometimes substitute G B D F, Principles (116), all composed of minor thirds; and which has for its fundamental found the sensible note G. This chord is called a chord of the flat or diminished seventh, and is figured with a \( \frac{2}{4} \) in the fundamental bass, (see LXXXIV.); but it is always considered as representing the chord of the tonic dominant.

222. This chord by inversion produces in the continued bass the following chords:

1. The chord B D F G, composed of a third, false bass by this fifth, and sixth major. They call it the chord of the what, and sixth sensible and false fifth; and it is figured as in Exam. LXXXV. (Plate CCCLX).

2. The chord D F G B, composed of a third, a tritone, and a sixth. It is called the chord of the tritone and third minor; and marked as in LXXXVI.

3. The chord F G B D, composed of a second redundant, a tritone, and a sixth. It is called the chord of the second redundant, and figured as in LXXXVII. (4 T).

223. Besides, since the chord G B D F represents the chord E G B D, it follows, that if we operate by supposition upon the first of these chords, it must be which they performed as one would perform it upon E G B D produce, that is to say, that it will be necessary to add to the what, and chord G B D F, the notes C or A, which are the third or fifth below E, and which will produce,

1. By adding C, the chord C G B D F, 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 marked as in LXXXVIII.

2. By adding A, we shall have the chord A G B D F, 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 fifth minor, and marked as in LXXXIX. It is of all chords the most harsh, and the most rarely practised (4 U).

(4 P) 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 \( \frac{2}{3} \); 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.

(4 Q) In England it is figured \( \frac{2}{3} \).

(4 R) We often remove some dissonances from chords of supposition, either to soften the harshness of the chord, or to remove discords which cannot be prepared nor resolved. For instance, let us suppose, that in the continued bass the note C is preceded by the sensible note B carrying the chord of the false fifth, and that we should choose to form upon this note C the chord C E G B D, we must obliterate the seventh B, because in retaining it we should destroy the effect of the sensible note B, which ought to rise to C.

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

(4 S) 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, in Example LXXXII. the C bearing \( \frac{2}{3} \) is a supposition; but in Example LXXXIII. it is a suspension, because it suspends or retards the perfect chord C E G C which the ear expects after the tonic dominant G B D F.

(4 T) The chord of the diminished seventh, and the three derived from it, are termed chords of substitution. They are in general harsh, and proper for imitating melancholy objects.

(4 U) As the chord of the diminished seventh G B D F, and the chord of the tonic dominant E G B D, only differ... 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 license ought to be rarely practised.

In like manner, in a continued bass, the dissonance in a chord of the sub-dominant inverted, as A in the chord A C E G, inverted from C E G A, 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.

225. Sometimes likewise, to render a continued bass more agreeable by causing it to proceed diatonically, we place between two sounds of that bass a note which belongs to the chord of neither. See Example xcii., in which the fundamental bass G C produces the continued bass G A B G C, where A is added on account of the diatonic modulation. This A has a line drawn above it, to show its resolution by passing under the chord G B D F.

In the same manner, (see xciii.) this fundamental bass C F may produce the continued bass C D E C F,

Vol. XIV. Part II.

where the note D, which is added, passes under the chord C E G C.

226. As the continued bass alone appears in practical compositions, it becomes 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 is either a tonic, or reckoned such (4 x).

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 Figured. We shall distinguish these two cases below. See l.vi. and the note (4 y).

3. Every note carrying a 2 gives in the fundamental bass its fifth below not figured. See l.vii.

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

5. Every note figured with a 2 gives in the fundamental bass the diatonic note above figured with a 7. See l.xiv. (4 z).

6. Every note marked with a 4 gives in the fundamental bass its third above, figured with a 6. For example, this continued bass A B C gives this fundamental bass C G C; but in this case it is necessary that the note figured with a 6 should rise by a fifth, as we see here C rise to G.

(4 z) A note figured with a 2, gives likewise sometimes in the fundamental bass its fourth above, figured with a 6; but it is necessary in that case that the note figured with a 6, may even here rise to a fifth. (See note 4 y.)

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'Abbé Ronsier, 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 sound with that letter of the scale by which it is denominated, to which is joined a 7 or 4, or a 6, in order to mark all the discords. Thus the fundamental chord of the seventh D F A C is expressed by a D; and the same chord, when it is inverted from that of the sub-dominant F A C D, is characterized by F; the chord of the second C D F A, inverted from the dominant D F A C, is likewise represented by D; and the same chord C D F A, inverted from that of the sub-dominant F A C D, is signified by F; the case is mental bas the diatonic note above, figured with a 7. (See LVI.)

7. Every note figured with a g gives its third below figured with a 7. (See LVIII.)

8. Every note marked with a 6 gives the fifth below marked with a 7; (see IX.) 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 LXXVII. and LXXXIX.)

10. Every note marked with a 5 gives the fifth above figured with a 7. (See LXXXVIII.)

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

12. Every note marked with a 5 gives a fifth above figured with a 7, or with a 5. (See LXXXVI.) It is the same case with the notes marked 4, 3, or 2: 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 5. (See LXXX.)

14. Every note marked with a 5 gives the third minor below, figured with a 7. (See LXXXV.)

15. Every note marked with a 5 gives the tritone above figured with a 7. (See LXXXVI.)

16. Every note marked with a 5 gives the second redundant above, figured with a 7. (See LXXXVII.)

17. Every note marked with a 5 gives the fifth redundant above, figured with a 7. (See LXXXVIII.)

18. Every note marked with a 5 gives the seventh redundant above, figured with a 7. (See LXXXIX.) (5 A).

REMARK.

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

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

Suppose we should have the dominant D in the fundamental bas, the note which answers to it in the continued bas may be A carrying the figure 6 (see LXIV.); that is to say, the chord A C D F: now if we should have the subdominant F in the fundamental bas, this subdominant might produce in the continued bas the same note A figured with a 6. When therefore we find 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 F in the continued bas, we may be ignorant whether we ought to infer in the fundamental bas F marked with a 6, or D figured with a 7.

230. This difficulty may be removed by leaving for solution an instant this uncertain note in suspense, and in examining the succeeding note of the fundamental bas; for if that note be in the present case a fifth above F, that is to say, if it be C, in this case, and in this alone, we may place F 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 C, and determining of its two-fifths G and F, one in ascending, which is mode in called a tonic dominant, the other in descending, which we is called a sub-dominant, the scale C D E F A B C may are be found: the different sounds which form this scale compose

the same when the chords are differently inverted. By this means it would be impossible to mistake either with respect to the fundamental bas of a chord, or with respect to the note which forms its dissonance, or with respect to the nature and species of that discord.

(5 A) We may only add, that here, and in the preceding articles of the text, we suppose, that the continued bas 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. We prefer the continued bases of M. Rameau to all the others, as by them the fundamental bas will be most easily discovered.

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 bas 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 bas. 2. If the two notes may with equal propriety be placed in the fundamental bas, 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 (note 3 z).

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 F A B D X. It is marked with a 6. It appears difficult to find a fundamental bas for this chord; nor is it indeed much in use amongst us. (See the note upon the art. 115.)

This chord is called in England the chord of the extreme flat sixth. When accompanied by the third only, it is called the Italian sixth. When the fifth is substituted for the tritone, it has been called the German sixth. Part II.

MUSIC

Principles of Composition.

compose the major mode of C, because the third E above C is major. If therefore we would have a modulation in the major mode of C, no other sounds must enter into it than those which compose this scale; in such a manner that if, for instance, we should find F in this modulation, this F discloses to us that we are not in the mode of C, or at least that, if we have been in it, we are no longer so.

232. In the same manner, if we form this scale in ascending A B C D E F G A, which is exactly similar to the scale C D E F G A B C of the major mode of C, this scale, in which the third from A to C is major, shall be in the major mode of A; and if we incline to be in the minor mode of A, we have only to substitute for C sharp C natural; so that the major third A C may become minor A C: we shall have then

A B C D E F G A,

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

A G F E C D B A,

in which the G and F 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.).

Hence it appears what sharps and flats should be placed at the clef upon F, C, and G; and on the contrary, in the minor mode of A, we place none, because at the clef the minor mode of A, in descending, has neither sharps nor flats.

234. As the scale contains twelve sounds, each distant 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 in all.

Of A.

In descending, A G F E D C B A. In rising, A B C D E F G A.

Of E.

In descending, E D C B A G F E. In rising, E F G A B C D E.

Of B.

In descending, B A G F E D C B. In rising, B C D E F G A B C.

Of F.

In descending, F E D C B A G E X. In rising, F X G A B C D E F.

Of C.

In descending, C X B A G F E D C X. In rising, C X D E F G A B C X.

Of G or A.

In descending, G X F E D C X B A X G X. In rising, A X B X C X D X E X F X G X.

Of D or E.

In descending, E D C B A G F E B. In rising, E B F G A B C D E.

Of A or B.

In descending, B A G F E D C B. In rising, B D C E F G A B.

Of

(5 b) The major mode of F, of C, and of G, are not much practised.

When a piece begins upon C, 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 Db, which is almost the same thing with C. For this reason we substitute here the mode of Db, for that of C.

It is still much more necessary to substitute the mode of Ab for that of G; for the scale of the major mode of G is,

G, A, B, C, D, E, F, G, A, B, C,

in which it appears that there are at the same time both a 'gh' and a 'g': it would then be necessary, even at the same time, that upon G there should and should not be a sharp at the clef; which is inconsistent. It is true that this inconvenience may be avoided by placing a sharp upon G at the clef; and by marking the note G 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.

We might likewise in this series, instead of G natural, which is the note immediately before the last, substitute F, that is to say, F twice sharp: which, however, is not absolutely the same found with G natural, especially upon instruments whose scales are fixed, or whose intervals are invariable. But in that case two sharps must be placed at the clef upon F, which would produce another inconvenience. But by substituting Ab for G, the trouble is eluded.

The double sharp, however, is incidentally used, when in a composition in the key of F there is an occasional modulation into the dominant of that key, and it is distinguished by the character X or XX. 235. These then are all the modes, as well major as minor. Those which are crowded with sharps and flats are little practised, as being extremely difficult in execution.

(5c) 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 below 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 below 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 D in the mode of C, and B in that of A is termed a super-tonic; the following note, which is a third major or minor from the tonic, according as the chord is major or minor, such as E in the major mode of C, and C in the minor mode of A, is called a mediant; and the note which is a tone above the dominant, such as A, in the mode of C, and F in that of A, is called a super-dominant.

(5d) 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 gamut; 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, for instance, your air should begin upon C, requiring the natural diatonic series through the whole gamut, in which the distance between E and F, as also that between 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 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 G, the note with which you propose to begin, the three tones immediately succeeding are full; but the fourth C is only a semitone; it may therefore be kept in its place. But from F, the seventh note above, to G, the eighth, the interval is a full tone, which must consequently be redressed by raising the F 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.

The order to be observed in these alterations of the intervals, is deduced from the relation which the fifth ascending and descending bear to the fundamental (art. 34, 35); and therefore the farther we depart from the natural fundamental C by a series of fifths ascending or descending, the alterations, and consequently the number of sharps or flats indicating them, will be the greater.

Thus if G, which is the perfect fifth ascending from C, therefore the note most nearly allied to C (art. 39, 40), Part II.

MUSIC

B, G must be changed into C, and E into A. Thus, by transposition, the air has the same melody as if it were in the major mode of C, or in the minor mode of A. The major mode then of G, and the minor of E, are by transposition reduced to those of C major, and of A minor. It is the same case with all the other modes (§ e).

All the modes reducible to the major of C and the minor of A.

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

238. As we have reduced to a very small number the rules of the fundamental basis, and those which in the treble ought to be observed with relation to this basis, given air not difficult, and why.

be taken for a fundamental, F, which is the seventh of the scale of G, must be made sharp, that it may be a whole tone from the fifth E, and only a semitone from the key note G, according to the laws of the diatonic scale (art. 77.). See Ex. xciv. 1. 2.

Again, if D, the perfect fifth ascending from G, and the second in the series of progressive fifths ascending from C, be used as a fundamental, C, which is the seventh of the scale of D, must, to render it the feasible or leading note (art. 77.), be made sharp in addition to F; so that in the scale of D, there are two sharps, F and C. See Ex. xciv. (3.).

If A, the perfect fifth above D, and the third in the series of fifths ascending from C, be the fundamental, the seventh G must, in addition to F and C, be made sharp, for the same reason (4.); and so on, in the scale of E, which is next in order, F, C, G, and D, must be sharp (5.): in that of B, the sharps must be F, C, G, D and A (6.).

The perfect fifth above B is F♯, and in that scale F, C, G, D, A, and E, must be sharp (7.). And in the next scale C♭ all the notes of the system are sharp (8.).

This, for the reasons mentioned in the note (§ b), is the last scale to which we can properly go by the progressions of fifths ascending.

Returning to the natural scale of C, if, instead of assuming G, the perfect fifth above, for a fundamental, we take F, the perfect fifth below; B, which is the fourth note above F, and forms a tritone or sharp fourth to it, must, to become a perfect fourth, according to the laws of the diatonic scale, (art. 6c.) be made flat (12.).

Proceeding with the series of fifths descending, if Eb, which is the perfect fifth below F, be taken for a fundamental; E, which, in its natural state, is the tritone or sharp fourth to B♭, must, to become the diatonic fourth (art. 6c.), also be rendered flat (11.).

If Eb, which is the perfect fifth below B♭, and the third in the series of fifths descending from C, be made the fundamental, A, the sharp fourth, must, to become the diatonic fourth, be made flat, and the flats marked at the cleft are B, E and A (10.).

To form the next scale in the series of fifths descending, which is that of A flat, D must be flattened; and B, E, A, and D, are marked flat at the cleft (9.).

The next scale, that of D flat, is formed by flattening G, and adding its flat to the others at the cleft (8.).

This is the scale recommended to be used rather than that of C♯. (See note § b.)

We do not proceed farther with the series of fifths descending, since the next scale, that of G♭, would just or very nearly exhibit the sounds already represented by the scale of F♯ (7.). This scale is, however, sometimes written in the key of G flat, and we even meet with the scale of its fifth below, C flat, and, with an occasional modulation from that key into its fifth below, F flat, where B being necessarily twice flattened, is distinguished by this character ♭, or ♯♭, called a double flat.

We have thus seen, first, that each of the notes of the diatonic scale of C, and each of the semitones into which the whole tones of that scale are divided, may be taken for the fundamental note of a diatonic scale, called the scale of that note. Secondly, that the notes of the natural scale are more or less altered, as the note assumed for a fundamental is more or less distant from C, in a progression of fifths ascending or descending. Thirdly, that in the progression by fifths ascending, the notes are altered by sharps, and in the progression by fifths descending, the alterations are by flats. Fourthly, that in the alteration by sharps, the last sharp is always on the seventh or feasible note of the scale; and where there are more than one, is always on the fifth above the sharp immediately preceding; and in the alteration by flats, the last flat is always on the fourth of the scale; and where there are more than one, is always on the fifth below the flat immediately preceding.

The signatures of sharps and flats at the clefs, belonging to the twelve major scales, are also used for their relative minor scales. The occasional elevation and depression of the sixths and sevenths of the minor scales, are denoted by occasional sharps or flats placed before these notes.

(§ e) 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 D, they place neither sharp nor flat at the cleft; in the minor mode of G, one flat only; in the minor mode of C, 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 cleft, 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. Musical Composition

It should no longer be difficult to find the fundamental bass of a given modulation, nay, frequently to find several; for every fundamental bass will be legitimate, 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 this bass, will both be prepared, if it is necessary that they should be so, and always resolved.

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, 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 G C may belong to all the modes, as well major as minor, in which G and C are found together; and each of these two sounds may even be considered as belonging to a different mode.

240. We may sometimes, as it should seem, operate without the knowledge of the mode, for two reasons:

1. Because, since the same sound belongs 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 (chap. 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 indissoluble that the fundamental bass should begin in the same mode, and that the treble and bass should likewise end in it; nay, that they should even terminate in its fundamental note, which in the mode of C is C, and A in that of A, &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 distinguish the major mode of C from the minor of A. 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 C E G in the one, and A C E in the other; so that if a piece should, for instance, begin thus, A C E A, it may be almost constantly concluded, that the tone or mode is in A minor, although the notes A C E belong to the mode of C.

2. From the sensible note, which is B in the one, and G in the other; so that if G appears in the first bars of a piece, we may be certain that we are in the mode of A.

3. From the adjuncts of the mode, that is to say, the modes of its two-fifths, which for C are F and G, and D and E for A. For example, if after having begun a melody by some of the notes which are common to the modes of C and of A (as E D E F E D C B C), we should afterwards find the mode of G, which we ascertain by the F, or that of F which we ascertain by the Bb or C#, we may conclude that we have begun in the mode of C; but if we find the mode of D, or that of E, which we ascertain by Bb, C#, or D#, we conclude from thence that we have begun in the mode of A.

4. A mode is not usually changed, especially in the beginning of a piece, unless in order to pass into one or other of the modes most relative to it, which are the mode of its fifth above, and that of its third below, if it be minor. Thus, for instance, the modes which are most intimately relative to the major mode of C, are the major mode of G, and that of A minor. From the mode of C 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. 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 F and E for the mode of C (5 g).

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,

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

(5 g) It is certain that the minor mode of E has an extremely natural connection with the mode of C, 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 D may be joined to the major mode of C: and thus in a particular sense, this mode may be considered as relative to the mode of C, but it is still less so than the major modes of G and F, or than those of A and E minor; because we cannot immediately, and without licence, pass in a fundamental bass from the perfect minor chord of C to the perfect minor chord of D; and if you pass immediately from the major mode of C to the minor mode of D in a fundamental bass, it is by passing, for instance, from the tonic C, or from E G C, to the tonic dominant of D, carrying the chord A C E G, in which there are two sounds, E G, which are found in the preceding chord, (Ex. xcvi.) or otherwise from C E G C to G Bb D E, a chord of the sub-dominant in the minor mode of D, which chord has likewise two sounds, G and E, in common with that which went immediately before it. See Ex. xcvi. cloves*, is always easy to be found. For the sounds which occur in the treble, M. Rameau may be consulted, p. 54. of his Nouveau Système de Musique théorique et pratique (5H).

When the mode is ascertained, by the different means which we have pointed out, the fundamental bas will cost 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 C, C E G 'c'.

Minor mode of A, A C E A.

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 C, G B D 'f'.

Minor mode of A, E G B 'd'.

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 C, F A C 'd'.

Minor mode of A, D F A B.

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 bas, 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 C, and that we find in the melody these two notes A B in succession; if we confine ourselves to place in the fundamental bas one of the three sounds C G F, we shall find nothing for the sounds A and B but this fundamental bas F G; now such a succession as F to G is prohibited by the fifth rule for the fundamental bas according to which every sub-dominant, as F, should rise by a fifth; so that F can only be followed by C in the fundamental bas, and not by G.

To remedy this, the chord of the sub-dominant F A C 'd' must be inverted into a fundamental chord of the seventh, in this manner, D F A 'c', which has been called the double employment (art. 105.) because it is a secondary manner of employing the chord of the sub-dominant. By these means we give to the modulation A B this fundamental bas D G; which procedure is agreeable to rules. See Ex. xcvi.

Here then are four chords, C E G 'c', G B D 'f', F A C 'd', D F A 'c', which may be employed in the major mode of C. We shall find in like manner, for the minor mode of A, four chords.

A C 'e' a', E G B 'd', D F A B, B D 'f' a'.

And in this mode we sometimes change the last of these chords into B D 'f' a', substituting the 'f' for 'f'. For instance, if we have this melody in the minor mode of A, E F G A, we would cause the first note E to carry the perfect chord A C E A; the second note F to carry the chord of the seventh B D F A; the third note G, the chord of the tonic dominant E G B D, and the last the perfect chord A C E A. See Ex. xcviii.

On the contrary, if this melody is given always in the minor mode, A A G A, the second A being syncopated, it might have the same bas as the modulation E F G A, with this difference alone, that F might be substituted for F in the chord B D F A, the better to mark out the minor mode. See Exam. xcix.

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,

C A D G C F B E A D G C,

which are terminated equally in the tonic C, either entirely belong, or at least may be reckoned as belonging (5I) to the mode of C; because none of these dominants are tonic dominants, except G, which is the tonic dominant of the mode of C; and besides, because the chord of each of these dominants forms no other

(5H) 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, E C G, we must not with too much precipitation conclude from thence that we are in the major mode of C, although these three sounds, E C G, be the principal and characteristical sounds in the major mode of C: we may be in the minor mode of E, especially if the note E should be long.

(5I) 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 C, viz. C carrying the perfect chord, F carrying that of the sub-dominant, and G that of the tonic dominant, to which we may join the chord of the seventh, D F A C (art. 105.): but we here regard as extended the series of dominants in question, as belonging to the mode of C, 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 A belongs naturally to the mode of G, the simple dominant B to that of A, &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 C. other sounds than such as belong to the scale of C. See Ex. c.

But if we were to form this fundamental bass,

\[ \text{C A D G Bb C} \]

considering the last C as a tonic dominant in this manner, C E G Bb; the mode would then be changed at the second C, and we should enter into the mode of F, because the chord C E G Bb indicates the tonic dominant of the mode of F; besides, it is evident that the mode is changed, because Bb does not belong to the scale of C. See Ex. c1.

In the same manner, were we to form this fundamental bass

\[ \text{C A D G C} \]

considering the last C as a sub-dominant in this manner, C E G A; this last C would indicate the mode of G, of which C is the sub-dominant. See Ex. cii.

In like manner, still, if in the first series of dominants, we caused the first D to carry the third major, in this manner, D F X A c', this D having become a tonic dominant, would signify to us the major mode of G, and the G which should follow it, carrying the chord B D f', would relapse into the mode of C, from whence we had departed. See Ex. ciii.

Finally, in the same manner, if in this series of dominants, we should cause B to carry F X in this manner, B D F X A, this F would show that we had departed from the mode C, to enter into that of G. See Ex. civ.

Hence it is easy to form this rule for discovering the changes of mode in the fundamental bass.

1. When we find a tonic in the fundamental bass, 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, it cannot be ascertained from this dominant alone, whether the mode be major or minor; but it is only necessary to examine the following note, which must be the tonic of the mode in which he is; by the third of this tonic it will be discovered whether the mode be major or minor.

243. Every change of the mode supposes a cadence; and when the mode changes in the fundamental bass, 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 (5 k).

CHAP. XIV. Of the Chromatic and Enharmonic.

245. We call that melody chromatic which is composed of several notes in succession, whether rising or what descending by semitones. See cv. and cvi.

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 Ex. cv. fundamental basses (5 l).

247. What.

(5 k) 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 C and the major of G, or the major mode of C and the minor of A; 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 C and the minor of B, &c.

When the composer, 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, he ought to continue in it but for a short time, because the ear is always impatient to return to the former mode.

(5 l) 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 cvii., 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 D to G X, and a fifth redundant from G X to C.

The reason of this licence is, as it appears to us, 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

\[ \text{A D G X C F X B E A} \]

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

\[ \text{A D E C F X B E A} \]

Now this last fundamental bass is formed according to the common rules, unless that there is a broken cadence from D to E, and an interrupted cadence from E to C, which are licenses (art. 213 and 214.). Music.

Plate CCCLIV.

Fig. 1. C D E F G A B c

Fig. 2. C D E F G A B c d e f g a b c d e f g a b

Scale First

Scale Second

Scale Third

H I

Fig. 3. K L M N R S T U V W

The Diatonic Scale of the Greeks

B c d e f g a

G C G C F C F

The Fundamental Bass

Fig. 4.

Fig. 5. C G C F C G D G C

The Fundamental Bass

Fig. 6. C, C#, D, D#, E, E#, F, F#, G, G#, A, A#, B, B#, c, c#, d, d#, e, e#

Scale First

Scale Second

The first Scale of the Minor Mode

G A B c d e f

E A E A D A D

The Fundamental Bass

Fig. 7.

Fig. 8. A E A D A E B E A

The Fundamental Bass

Fig. 9. C G C F C D G C

Scale

c d e f g a b c

Fig. 10. C E G#

The Fund. Bass

Fig. 11. C E G#

The Chromatic Species

g g# &c

Fig. 12. F C E B

The Fund. Bass

Fig. 13. C C A C# C#

The Fund. Bass 247. When the air is chromatic in ascending, one may form a fundamental bass by a series of tones and of tonic dominants, which succeed one another alternately by the interval of a third in descending, and of a fourth in ascending. (See Ex. cvi.) 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. The enharmonic is very rarely put in practice; and we have explained its formation in the first book, to which we refer our readers.

See Design.

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

Design, what.

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.

See Imitation.

Imitation, what.

250. Imitation consists in causing to be repeated the melody of one or of several measures in one single part, or in the whole harmony, and in any of the various modes that may be chosen. When all the parts absolutely repeat the same air or melody, and beginning one after the other, this is called a canon (5 m).

Fugue consists in alternately repeating that air in the treble, and in the bass, or even in all the parts, if there are more than two.

Vol. XIV. Part II.

(5 m) Compositions in strict canon, where one part begins with a certain subject, and the other parts are bound to repeat the very same subject, or the reply, as it is called, in the unison, fifth, fourth, or octave, depend on the following rules, which are nothing more than a summary of the system explained by our author:

1. The chords to be employed are the tonic, and its two adjuncts; the subdominant, susceptible of an added sixth, and the dominant, susceptible of an added seventh.

2. The subject must begin in the harmony of the tonic, and as the fundamental progression from the dominant to the subdominant is not permitted (art. 33-36.), the subdominant must follow the tonic, and the dominant the subdominant, thus,

\[ C, \frac{6}{5} F, \frac{7}{6} G, C, \frac{6}{5} F, \frac{7}{6} G, C, \&c. \]

3. As the diatonic scale consists of two tetrachords, of which the first is also the second tetrachord of the mode of the sub-dominant, and the second the first tetrachord of the dominant; so, in canon, when the reply is meant to be in the mode of the dominant, the subject must be in the first tetrachord of the tonic, by which means the corresponding first tetrachord of the dominant being the second tetrachord of the tonic, the whole piece is true in that mode. On the other hand, if the reply is to be in the mode of the sub-dominant, the subject must be in the second tetrachord of the tonic, the corresponding tetrachord of the sub-dominant being the first tetrachord of the tonic, and the mode of the tonic being thus preserved.

4. For the same reason, where the reply is in the dominant, the subject is only allowed to modulate into the mode of the sub-dominant, and the reply of course into that of the tonic. And where the reply is in the dominant, the subject is to modulate only into the mode of the sub-dominant, the reply following of course into that of the tonic. Were the contrary modulation permitted, the reply would depart too far from the mode of the tonic.

Lastly, when the reply is to be in the mode of the dominant it must commence in the measure bearing that harmony; and in the same way, the reply in the sub-dominant must begin in the measure which bears the harmony of the sub-dominant.

If these rules be observed, and due attention paid to the preparation and resolution of dissonances, composition in strict canon, in any number of parts, will be found to be by no means difficult. See Ex. cix. and cx.

(5 n) 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 F D, and the other 'c a' in rising, C is the fifth of F, and 'a' the twelfth of D.

APPENDIX.

The treatise of D'Alembert 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 subjoin, 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 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 followed the opinions of Pythagoras on that subject.

No part of natural philosophy has been more fruitful of hypotheses than that of which musical sound is the object. The musical speculators of Greece arranged themselves into a great number of 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 caused 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 fifth would be expressed by the ratio of 2 to 3, and the fourth 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 abstract 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 failed 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 uniform. The vibrations of uniforms 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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**Music**

**Musimon.** See Harmonica.