167. In like manner, it is evident that the sixth 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.

(31) 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 graveness or acuteness, their duration, and the other qualities intended to be assigned to them, are accurately distinguished.

The characters which denote the graveness or acuteness, 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 graveness or acuteness 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 ascertains 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). 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.

171. As the notes are placed on the lines, and in 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.

172. 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, or double E, EE, &c.

Again, the 'a' above the highest 'g' of the treble staff is placed on a leger line above that staff. The 'b' is placed on the space above the leger line: The next note 'c' 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 recompense 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. Principles equal times, called measures; and each measure is like- of Composi-wise 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 distin- guished; thus, the first note of each time is considered of composition as strong and the others as weak.

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

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

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

(3 n) All the notes, therefore, contained between two bars constitute one measure; although in common lan- guage 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 arrange- ment 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 sort of musical short-hand, 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 r) 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 pro- tracted. See fig. 16.

(3 s) 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 usual- ly 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 \(\frac{2}{4}\), 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 \(\frac{2}{4}\) and \(\frac{3}{4}\) rarely occur.

Three notes are often performed in the time of two of the same name, and are then termed triplets, (fig. 11.) 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 practised 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 X, (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 appoggiature, the shake, the slur, 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 harmonies 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 harmonies may be arranged in different octaves. Their different positions will be most easily seen and best understood from the examples in the plates. Musical Principles: Fundamental Note

For example, if we would mark the perfect major chord DF 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 DF A C, and the chord of the great sixth DF 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 example VIII. IX. shew the chords G Bb D G G Bb D F, G Bb DE (3 §).

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 bass, 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,

(3 §) 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 Bb 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 shew 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 Rules for only capable of carrying a perfect chord, or the chord 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 First Part of this treatise. To be convinced of this we shall find it only necessary to review the articles 34, 91, 122, 124, 126, 127.

Rule I.

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

Rule II.

192. In every chord of the simple dominant, it is necessary 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... 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 it 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 (3 r); 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 X;

(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.

Of the five fundamental rules which have now been stated 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. (3 v); a tonic dominant (124.), see of 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 E G C, F A C E, 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. (3 x).

We must here advertise 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 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 bar by preceding measure. Nevertheless syncopation may be license, sometimes admitted in the fundamental bass, but it is by a license (3 r).

3 Y 2

(3 r) 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 X, in which C would be sharp without the F, is excluded by art. 179.

(3 v) 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.).

(3 x) 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.

(3 r) 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 Bass.

198. The treble is nothing else but a modulation above the fundamental bass, and whose notes are found in the chords of that bass which corresponds with it (189.). Thus in Ex. xvii. the scale C D E F G A B C, is a treble with respect to the fundamental bass 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 bass, when these notes belong to the chord of the same note in the fundamental bass. For example, this modulation C E G E C, may have for its fundamental bass 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 bass. For instance, G alone may answer to these three notes in the bass, C G C (3 z).

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 of composition we call a discord prepared (122). For instance, let us suppose that the note of the fundamental bass 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 bass, C will always be found in the chord of that note in the fundamental bass 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 (4 A).

RULE II.

202. If a note of the fundamental bass 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 bass 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 bass 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.

(3 z) 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 bass 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 proportionably 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.

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

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

and this fundamental bass,

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

(see example L.) the D of the fundamental bass 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 bass 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.

Principles once ought to descend or rise will always be found in the subsequent chord (4 B).

Chap. VIII. Of the Continued Bass and its Rules.

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 sixth, 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 sixth and fourth, composed of a fourth, a sixth, 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 a 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 sixth, 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 singing, and suitable to the voice. See LXVI. in which the fundamental bass which itself is monotonic and little suited for singing, 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 discord, 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 2.

(4 D) This chord is in England called the chord of the second and fourth, and is figured 2.

(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 sixth and of the lesser sixth. (See examples LVII. 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 singing, as the sounds which form it more scrupulously observe 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 4+, 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 LXXV. is much more in the taste of singing, and more agreeable, than the fundamental bass which answers to it.

(41) 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 D F A C, 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.)

(48) When there is a repose in the treble, the note of the continued bass ought to be the same with that of the fundamental bass, (see Example LXVIII.). In the closes 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 LXIX. 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 F, 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 4; 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 4.

We shall repeat here, with respect to the rules of the continued bass, what we have formerly said concerning the rules of the fundamental bass in the note upon the third rule, art. 193. The rules of the continued bass have exceptions, which practice and the perusal of good authors will teach. There are likewise several other rules which might require a considerable detail, and which will be found in the Treatise of Harmony, by M. Rameau, Of the different kinds of Chords by Supposition.

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 a fifth, seventh, ninth, and eleventh. It is figured with a \( \frac{7}{3} \); see LXXVI. (4 N). This chord is not practised 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 \( \frac{3}{5} \).

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 9. 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 resound; 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 resound. 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 \( \frac{7}{3} \): M. Rameau suppresses this 2, and merely marks it to be the seventh redundant by a \( \frac{7}{3} \) or \( \frac{7}{3} \). 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 \( \frac{7}{3} \)? M. Rameau answers, that there is no danger or mistake, because in the seventh major, as the seventh ought to be prepared, it is found in the preceding chord; and thus the sharp subsisting already in the preceding chord, it would be useless to repeat it.

Thus D G, according to M. Rameau, would indicate D F \( \frac{7}{3} \) A C, G B D F \( \frac{7}{3} \). If we would change F \( \frac{7}{3} \) of the second chord into F \( \frac{7}{3} \), it would then be necessary to write D G. In notes such as C, whose natural seventh is major, the figure 7 preceded or followed by a sharp will sufficiently serve to distinguish the chord of the seventh redundant C G B D F, from the simple chord of the seventh C E G B, which is marked with a 7 alone. All this appears just and well founded.

(4 O) Supposition introduces into a chord dissonances 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 dissonance 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 DFAC, we should add the fifth G, we would have the chord GDFAC, called a chord of the eleventh, and which is figured with a \( \frac{2}{4} \) or \( \frac{5}{6} \). (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 EGBD; if there should be added the third C beneath, we shall have this chord of the continued bass CEGBD; 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 LXXIX. (4r).

218. In the chord of the simple dominant, as DFAC, 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 GCD. 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 \( \frac{4}{4} \) (see LXXX.). (4r)

219. Sometimes we only remove the note A, and then the chord ought to be figured with \( \frac{2}{4} \) or \( \frac{5}{6} \) (4r).

220. Finally, in the minor mode, for example, in that of A, where the chord of the tonic dominant (109), is EGGBD; if we add to this chord the third C below, we shall have EGGBD, 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. (4s).

§ 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 EGGBD, in which G is the sensible note. For this chord we sometimes substitute GBGDF, Principles (116), all composed of minor thirds; and which has of composition for its fundamental sound the sensible note G. This chord is called a chord of the flat or diminished seventh, and is figured with a \( \frac{4}{4} \) 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 BDFG, composed of a third, false fifth, and sixth major. They call it the chord of the what, and sixth sensible and false fifth; and it is figured as in how figured. Exam. LXXXV. (Plate CCCLX.).

2. The chord DFGGBD, 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 FGGBD, composed of a second redundant, a tritone, and a sixth. It is called the chord of the second redundant, and figured as in LXXXVII. (4r).

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

1. By adding C, the chord CGGBDF, 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 AGGBDF, composed of a seventh redundant, a ninth, an eleventh, and a thirteenth minor, which is the octave of the sixth minor. It is called the chord of the seventh redundant and sixth minor, and marked as in LXXXIX. It is of all chords the most harsh, and the most rarely practised (4u).

---

(4r) 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 \( \frac{2}{4} \); 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.

(4a) In England it is figured \( \frac{4}{4} \).

(4r) We often remove some dissonances from chords of supposition, either to soften the harshness of the chord, or to remove discords which can neither be prepared nor resolved. For instance, let us suppose, that in the continued 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 CEGBD, 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 GBDF, 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.

(4s) 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 \( \frac{7}{4} \) is a suspension; but in Example LXXXIII, it is a suspension, because it suspends or retards the perfect chord CEGC which the ear expects after the tonic dominant GBDF.

(4t) 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.

(4u) As the chord of the diminished seventh GGBDF, and the chord of the tonic dominant EGBD, 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. No. 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 Principles of Composition.

CHAP. XI. Containing the Method of finding the Fundamental Bass when the continued Bass is figured.

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.

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 IV. We shall distinguish these two cases below. See lvi. figured, and the note (4 y).

3. Every note carrying § gives in the fundamental bass its fifth below not figured. See lxvii.

4. Every note figured with a 7, or a 6, 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 lxiv. (4 z).

6. Every note marked with a 4 gives in the fundamental bass its third above not figured. 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. P'Abbe Roussier, 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 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 The first is that where the note of the continued bass is figured with a 6. We now present the reason of the composition difficulty.

Suppose we should have the dominant D in the fundamental bass, the note which answers to it in the continued bass 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 bass, the subdominant might produce in the continued bass, the same note A figured with a 6. When therefore we find in the continued bass a note marked with a 6, it appears at first uncertain whether we should place in the fundamental bass 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 bass is figured with a 6. For instance, if there should be found F in the continued bass, we may be ignorant whether we ought to insert in the fundamental bass F marked with a 6, or D figured with a 7.

230. This difficulty may be removed by leaving for an instant this uncertain note in suspense, and in examining the succeeding note of the fundamental bass; 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 bass. It is a consequence of this rule, that in the fundamental bass 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 determination of its two-fifths G and F, one in ascending, which is called a tonic dominant, the other in descending which is called a sub-dominant, the scale C D E F A B C may 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 bass of a chord, or with respect to the note which forms its dissonance, or with respect to the nature and species of that discord.

(5 A) We may only add, that here, and in the preceding articles of the text, we suppose, that the continued bass is figured in the manner of M. Rameau. For it is proper to observe, that there are not, perhaps, two musicians who characterize their chords with the same figures; which produce a great inconvenience to the person who plays the accompaniments: but here we do not treat of accompaniments. We prefer the continued basses of M. Rameau to all the others, as by them the fundamental bass 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 bass we ought to choose the third or the fifth below; but it will be easy to see whether the third or the fifth is signified by that figure. This may be distinguished, 1. In observing which of the two notes is excluded by the rules of the fundamental bass. 2. If the two notes may with equal propriety be placed in the fundamental bass, the preference must be determined by the tone or mode of the treble in that particular passage. In the following chapter we shall give rules for determining the mode (note 32).

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 X. It appears difficult to find a fundamental bass for this chord; nor is it indeed much in use amongst us. (See the note upon the art. 115.)

This chord is called in England the chord of the extreme sharp 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. The major mode of F\(\sharp\), of C\(\sharp\), and of G\(\sharp\), are not much practised.

When a piece begins upon C\(\sharp\), 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\(\sharp\). For this reason we substitute here the mode of Db, for that of C\(\sharp\).

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

\[ G\sharp, A\sharp, B\sharp, C\sharp, d\sharp, e\sharp, g\sharp, \]

in which it appears that there are at the same time both a 'g\(\sharp\)' 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 sound 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\(\sharp\), the trouble is eluded.

The double sharp, however, is incidentally used, when in a composition in the key of F\(\sharp\) there is an occasional modulation into the dominant of that key, and it is distinguished by the character X or \(\#\#\). 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 G 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 copious; though it proceeds upon simple principles, and comprehends every possible contingency; yet as the manner of thinking upon which it depends may be less familiar to English readers, if not profoundly skilled in music, it has been thought proper to give a more familiar, though less comprehensive, explanation of the manner in which transposition may be executed.

It will easily occur to every reader, that if each of the intervals through the whole diatonic series were equal, in a mathematical sense, it would be absolutely indifferent upon what note any air were begun, if within the compass of the gammut; because the same equal intervals must always have the same effects. But since, besides the natural semitones, there is another distinction of diatonic intervals into greater and lesser tones; and since these vary their positions in the series of an octave, according as the note from whence you begin is placed, that note is consequently the best key for any tune whose natural series is most exactly correspondent with the intervals which that melody or harmony requires. But in instruments whose scales are fixed, notwithstanding the temperament and other expedients of the same kind, such a series is far from being easily found, and is indeed in common practice almost totally neglected. All that can frequently be done is, to take care that the ear may not be sensibly shocked. This, however, would be the case, if, in transposing any tune, the situation of the semitones, whether natural or artificial, were not exactly correspondent in the series to which your air must be transposed, with their positions in the scale from which you transpose it. Suppose, for instance, your air should begin upon C, requiring the natural diatonic series through the whole gammut, in which the distance between E and E, 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. 34a 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 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), 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 sixth E, and only a semitone from the key note G, according to the laws of the diatonic scale (art. 77.). See Ex. xciv. i. 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 sensible 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, on 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 (5 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. 60.) be made flat (12.).

Proceeding with the series of fifths descending, if Bb, 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 Bb, must, to become the diatonic fourth (art. 60.), also be rendered flat (11.).

If Eb, which is the perfect fifth below Bb, 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 clef 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 clef (9.).

The next scale, that of D flat, is formed by flattening G, and adding its flat to the others at the clef (8.). This is the scale recommended to be used rather than that of C#. (See note 5 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 b, or bb, called a double flat.

We have thus seen, 1st, 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. 2dly, 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. 3dly, 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. 4thly, That in the alteration by sharps, the last sharp is always on the seventh or sensible 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 and flats placed before these notes.

(5 e) Many musicians, and amongst others the ancient musicians of France, as Lulli, Campari, &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 clef; 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 clef, as are contained in the diatonic scale of that mode in ascending; and in the minor mode, as many as are contained in that same scale in descending. Principles bass, 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. vii); 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 (§ F).

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 sounds belong to several different modes, the mode is sometimes considerably undetermined; above all, in the middle of a piece, and during the time of one or two bars. 2. Without giving ourselves much trouble about the mode, it is often sufficient to preserve us from deviating in composition, if we observe in the simplest manner the rules above prescribed (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 indispensable 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 of Composition 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 modes thus, A C E A, it may be almost constantly concluded, be determined that the tone or mode is in A minor, although the notes mixed, 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 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, &c., 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 the original mode be major, or of its third above if it be minor. Thus, for instance, the modes which are most intimately relative to the major mode of 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 (§ 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 closes,

(§ 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.

(§ 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. xciv.) 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. When the mode is ascertained, by the different means which we have pointed out, the fundamental bass 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 bass, under any particular note of the upper part. Yet it sometimes happens that not one of these notes can be used. For example, let it be supposed that we are in the mode of C, and that we find in the melody these two notes A B in succession; if we confine ourselves to place in the fundamental bass one of the three sounds C G F, we shall find nothing for the sounds A and B but this fundamental bass F G; now such a succession as F to G is prohibited by the fifth rule for the fundamental bass, according to which every sub-dominant, as F, shall rise by a fifth; so that F can only be followed by C in the fundamental bass, 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 bass D G; which procedure is agreeable to rules. See Ex. xcvii.

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 bass 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 (51) 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

(51) 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.

(51) 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. Principles other sounds than such as belong to the scale of C. See of Compo- Ex. c.

But if we were to form this fundamental bass,

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

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. ci.

In the same manner, were we to form this fundamen- tal 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 domi- nants, we caused the first D to carry the third major, in this manner, D F A e; this D having become a to- nic dominant, would signify to us the major mode of G, and the G which should follow it, carrying the chord B D G, 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 do- minants, we should cause B to carry F# in this man- ner, B D F# A, this F would show that we had de- parted 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 ma- jor or minor, according as the perfect chord is major or minor.

2. When we find a sub-dominant, we are in the Princi- ple mode of the fifth above that sub-dominant; and the of Compo- 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 to- nic 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 dis- covered 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 presage a change of mode which ought to fol- low 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, especial- ly in the most essential passages, as cadences (5 k).

CHAP. XIV. Of the Chromatic and Enharmonic.

245. We call that melody chromatic which is com- posed 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 To an air most natural and ordinary fundamental bass is a con- catenated series of tonic dominants; all of which fol- low one another in descending by a fifth, or, which is the same thing, in rising by a fourth. See Ex. cv. funda- mental bass. (5 l).

(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 con- tinue 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 dim- inished, and that in this bass there is a false fifth from D to G#, and a fifth redundant from G# to C.

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

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

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

\[ \text{7 7 7 7 7 7 7 7} \]

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

Now this last fundamental bass is formed according to the common rules, unless that there is a broken ca- dence from D to E, and an interrupted cadence from E to C, which are licenses (art. 213. and 214.) When the air is chromatic in ascending, one may form a fundamental bass by a series of tonics and 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.

The enharmonic is very rarely put in practice; and we have explained its formation in the first book, to which we refer our readers.

**CHAP. XV. Of Design, Imitation, and Fugue.**

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

**250. Imitation** consists in causing to be repeated the melody of one or 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 treble, and in the bass, or even in all the parts, if there are more than two.

---

(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,

\[ C, F, G, C, F, G, C, \text{&c.} \]

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 truly 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. cxix. 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 sound F D, and the other 'c a' in rising, C is the fifth of F, and 'a' the twelfth of D. 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 hypothesis 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 sound, considered in the vibrations that cause it, and the dimensions of the vibrating body, came to be reduced to quantity, and as such was the subject of calculation, and expressible by numbers.—For instance, the two sounds that form an octave could be expressed by the numbers 1 and 2, which would represent either the number of vibrations in a given time, or the length of the strings; and would mean that the acuter sound vibrates twice, while the graver vibrates once; or that the string producing the lower sound is twice the length of that which gives the higher. If the vibrations were considered, the higher sound was as 2, the lower as 1; the reverse, if the length was alluded to. In the same manner, in the same sense, the 5th would be expressed by the ratio of 2 to 3, and the 4th by that of 3 to 4.

Aristoxenes, 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 passed from conjecture to certainty.

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

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

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

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

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

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**Musimon**