Part I.

Theory of Harmony.

G B d f', E G B d'; in the case of an interrupted cadence, the dissonance of the former chord is resolved by descending diatonically upon the octave of the fundamental note of the subsequent chord, as may be here seen, where d' is resolved upon the octave of E.

137. This kind of interrupted cadence has likewise its origin in the double employment of dissonances. For let us suppose these two chords in succession, G B d f', G B d e', where G is successively a tonic dominant and sub-dominant; that is to say, in which we pass from the mode of C to the mode of D; if we should change the second of these chords into the chord of the dominant, according to the laws of the double employment, we shall have the interrupted cadence G B d f', E G B d'.

CHAP. XVIII. Of the Chromatic Species.

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

A chromatic interval or minor semitone, how found. See fig. 10.

139. If then we should form this bass C, E, G, the two first sounds carrying each along with it their thirds major and fifths, it is evident that C will give G, and that E will give G; now the semitone which is between this G and this G is an interval much less than the semitone which is found in the diatonic scale between E and F, or between B and c'. This may be ascertained by calculation (tt): and for this reason the semitone from E to F is called major, and the other minor (uu).

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

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

142. Every procedure of the fundamental bass by thirds, whether major or minor, rising or descending, gives the minor semitone. This we have already seen be found from the succession of thirds in ascending. The series of thirds minor in descending, C A, gives, C, C, Ab, gives C, Cb, (zz).

143. The minor semitone constitutes the species called chromatic; and with the species which moves by diatonic intervals, resulting from the succession of fifths (chap. v. and vi.), it comprehends the whole melody.

CHAP. XIX. Of the Enharmonic Species.

144. The two extremes, or highest and lowest notes, C G, of the fundamental bass by thirds major CEG, give this modulation c'B; and these two sounds c'nic intervals differ between themselves by a small interval which is called the dieisis, or enharmonic fourth of a tone (3A), and how formed.

(ttt) In reality, C being supposed 1, as we have always supposed it, E is 4/3, and G is 5/4; now G being 5/4, G then shall be to G as 4/5 to 3/4; that is to say, as 25 times 2 to 3 times 16: the proportion then of G to G is as 25 to 24, an interval much less than that of 16 to 15, which constitutes the semitone from c' to B, or from F to E (note z).

(uuu) A minor joined to a major semitone will form a minor tone; that is to say, if one rises, for instance, from E to F, by the interval of a semitone major, and afterwards from F to F by the interval of a minor semitone, the interval from E to F will be a minor tone. For let us suppose E to be 1, F will be 4/3, and F will be 5/4 of 4/3; that is to say, 25 times 16 divided by 24 times 15, or 16/15; E then is to F as 1 is to 9, the interval which constitutes the minor tone (note BB).

With respect to the tone major, it cannot be exactly formed by two semitones; for, 1. Two major semitones in immediate succession would produce more than a tone major. In effect, 4/3 multiplied by 4/3 gives 16/9, which is greater than 4/3, the interval which constitutes (note BB) the major tone. 2. A semitone minor and a semitone major would give less than a major tone, since they amount only to a true minor. 3. And, a fortiori, two minor semitones would still give less.

(xx) In effect, Eb being 4/3, Gb will be 5/4 of 4/3; that is to say, (note Q) 4/3: now the proportion of 1 to 4/3 (note Q) is that of 3 times 25 to 2 times 36; that is to say, as 25 to 24.

(yyy) A being 4/3, C is 5/4 of 4/3; that is to say 4/3, and C is 1: the proportion then between C and C is that of 1 to 4/3, or of 24 to 25.

(zzz) Ab being the third major below C, will be 4/3 (note Q): Cb, then, is 5/4 of 4/3; that is to say 4/3. The proportion, then, between C and Cb, is as 25 to 24.

(3A) G being 4/3 and B being 4/3 of 4/3, we shall have B equal (note Q) to 4/3, and its octave below shall be 4/3; an interval less than unity by about 1/3 or 1/4. It is plain then from this fraction, that the B in question must be considerably lower than C. which is the difference between a semitone major and a semitone minor (3 b). This quarter tone is inappre- ciable by the ear, and impracticable upon several of our instruments. Yet have means been found to put it in practice in the following manner, or rather to perform what will have the same effect upon the ear.

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

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

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

| B | d | f | g | |---|---|---|---| | D | F | G | B | | F | G | B | d |

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

Now the chord G B d f, belonging to the minor mode of A, where G is the sensible note; the chord B d f g, or B d f a b, will, for the same reason, belong to the minor mode of C, where B is the sensible note. In like manner, the chord D F G B, or D F A b c b, will belong to the minor mode of Eb, and the chord F G B d, or F A b c b e b, to the minor mode of Gb.

After having passed then by the mode of A to the chord G B d f (art. 117.), one may by means of this last chord, and by merely satisfying ourselves to invert it, afterwards pass all at once to the modes of C minor, of Eb minor, or of Gb minor; that is to say, into the modes which have nothing, or almost nothing, in common with the minor mode of A, and which are entirely foreign to it (3 c).

146. It must, however, be acknowledged, that a The alternation transition so abrupt, and so little expected, cannot deceive nor elude the ear; it is struck with a sensation which it is so unlooked-for, without being able to account for the effectuated passage to itself. And this account has its foundation abrupt and in the enharmonic fourth of a tone; which is overlooked.

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

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

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

3. One half of a semitone major; and as this semitone differs from unity by \( \frac{1}{8} \), one half of it will differ from unity about \( \frac{1}{8} \).

4. Finally, one half of a semitone minor, which differs from unity by \( \frac{1}{8} \): its half then will be \( \frac{1}{8} \).

The interval, then, which forms the enharmonic fourth of a tone, as it does not differ from unity but by \( \frac{1}{8} \), may justly be called the fourth of a tone, since it is less different from unity than the largest interval of a quarter tone, and more than the least.

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

(3 b) That is to say, that if you rise from E to F, for instance, by the interval of a semitone major, and afterwards, returning to E, you should rise by the interval of a semitone minor to another sound which is not in the scale, and which I shall mark thus, F +, the two sounds F + and F will form the enharmonic fourth of a tone: for E being 1, F will be \( \frac{9}{8} \); and \( \frac{9}{8} + \frac{1}{8} = \frac{10}{8} \): the proportion then between F + and F is that of \( \frac{9}{8} \) to \( \frac{10}{8} \) (note Q); that is to say, as 25 times 15 to 16 times 24; or otherwise, as 25 times 5 to 16 times 8, or as 125 to 128.

Now this proportion is the same which is found, in the beginning of the preceding note, to express the enharmonic fourth of a tone.

(3 c) As this method for obtaining or supplying enharmonic gradations cannot be practised on every occasion when the composer or practitioner would wish to find them, especially upon instruments where the scale is fixed and invariable, except by a total alteration of their economy, and re-tuning the strings, Dr Smith in his Harmonics has proposed an expedient for redressing or qualifying this defect, by the addition of a greater number of keys or strings, which may divide the tone or semitone into as many appreciable or sensible intervals as may be necessary. For this, as well as for the other advantageous improvements which he proposes in the structure of instruments, we cannot with too much warmth recommend the perusal of his learned and ingenious book to such of our readers as aspire to the character of genuine adepts in the theory of music. ed as nothing, because it is inappreciable by the ear; but of which, though its value is not ascertained, the whole harmony is sensibly perceived. The infant of surprise, however, immediately vanishes; and that astonishment is turned into admiration, when one feels himself transported as if he were all at once, and almost imperceptibly, from one mode to another, which is by no means relative to it, and to which he never could have immediately passed by the ordinary series of fundamental notes.

**Chap. XX. Of the Diatonic Enharmonic Species.**

147. If we form a fundamental bass, which rises alternately by fifths and thirds, as F, C, E, B, this bass will give the following modulation, 'f', e, e, d'f'; in which the semitones from 'f' to 'e', and from 'e' to 'd'f', are equal and major (3 D).

This species of modulation or of harmony, in which all the semitones are major, is called the enharmonic diatonic species. The major semitones peculiar to this species give it the name of diatonic, because major semitones belong to the diatonic species; and the tones which are greater than major by the excess of a fourth, resulting from a succession of major semitones, give it the name of enharmonic (note 3 A).

**Chap. XXI. Of the Chromatic Enharmonic Species.**

148. If we pass alternately from a third minor in descending to a third major in rising, as C, C, A, C', C', we shall form this modulation 'c', b, e, e, e c'; in which all the semitones are minor (3 E).

This species is called the chromatic enharmonic species: the minor semitones peculiar to this kind give it the name of chromatic, because minor semitones belong to the chromatic species; and the semitones which are lesser by the diminution of a fourth resulting from a succession of minor semitones, give it the name of enharmonic (note 3 F).

149. These new species confirm what we have all along said, that the whole effects of harmony and melody reside in the fundamental bass.

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

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

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

**Chap. XXII. Showing that Melody is the Offspring of Harmony.**

153. All that we have hitherto said, as it seems to me, is more than sufficient to convince us, that melody has its original principle in harmony; and that it is investigated in harmony, expressed or underfed, that we ought to pay attention to the effects of melody.

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

155. Yet more, we have proved, in treating of the broken cadence (chap. xvii.), that the diversification of basses produces effects totally different in a modulation which, in other respects, remains the same.

156. Can it be still necessary to adduce more convincing proofs? We have but to examine the different basses which may be given to this very simple modulation G C. It will be found susceptible of many, and each will give a different character to the modulation G C, though in itself it remains always the same. We may thus change the whole nature and effects of a modulation, without any other alteration than that of its fundamental bass.

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

157. From these different observations it may be concluded, 1. That an agreeable melody, naturally implies a bass extremely sweet and adapted for fingering; this principle and that reciprocally, as musicians express it, a bass of this kind generally prognosticates an agreeable melody (3 F).

2. That the character of a just harmony is only to form in some measure one system with the modulation, fo

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(3 D) It is obvious, that if F in the bass be supposed 1, 'f' of the scale will be 2, C of the bass 3 and 'e' of the scale 4 of 3, that is, 1/8; the proportion of 'f' to 'e' is as 2 to 1/8, or as 1 to 1/4. Now E of the bass being likewise 1/4 of 3, or 1/8; B of the bass is 1/4 of 1/8, and its third major D'f' 1/8 of 1/8, or 1/8 of 1/8; this third major, approximated as much as possible to 'e' in the scale by means of octaves, will be 1/8 of 1/8; 'e' then of the scale will be to D'f' which follows it, as 1/8 is to 1/8 of 1/8, that is to say, as 1 to 1/8. The semitones then from 'f' to 'e', and from 'e' to D'f', are both major.

(3 E) It is evident that 'c' is 1/8 (note Q), and that 'e' is 1/8; these two 'e's, then, are between themselves as 1/8 to 1/8, that is to say, as 6 times 4 to 5 times 5, or as 24 to 25, the interval which constitutes the minor semitone. Moreover, the Δ of the bass is 1/8, and C'f' 1/8 of 1/8, or 1/8 of 1/8; 'e' then is 1/8 of 1/8, the 'e' in the scale is likewise to the 'e' which follows it, as 24 to 25. All the semitones therefore in this scale are minor.

(3 F) Many composers begin with determining and writing the bass; a method, however, which appears in general Part II.