s one of the seven sciences commonly called liberal; and is comprehended also among the mathematical, as having for its object discrete quantity or number, but not considering it in the abstract like arithmetick, but in relation to time and sound, in order to make a delightful harmony: Or it is the art of disposing. disposing and conducting sounds considered as acute and grave; and proportioning them among themselves, and separating them by just intervals pleasing to the sense.

Mr Malcome defines it a science that teaches how sound, under certain measures of time and tune, may be produced; and so ordered and disposed, as either in consonance (i.e., joint-founding,) or succession, or both, they may raise agreeable sensations.

From this definition, the science naturally divides into two general parts, viz. theoretical and practical.

PART I. THE THEORY OF MUSICK.

CHAP. I. OF MUSICAL SOUNDS.

AXIOM I. The ear is the sole judge of sound. Every sound is not a musical sound. For to this two things are required: first, That the sound please the ear; secondly, That it be within a certain compass. A musical sound is clear, uninterrupted, and uniform; and ought not to exceed the power of the ear to judge of it.

For sounds, very deep or very high, are not easily distinguished, but by an ear very conversant in music.

Sound being a simple idea, cannot be defined but by an imperfect description of its cause; which is an undulatory motion of the air, communicated by the vibration of the parts of bodies to the organ of hearing.

The diversities of sounds, and their proportions, are perfectly discerned by the ear, are the object of the theory of music, the grounds and principles of the practice, as well as the causes of pleasure in the sense and imagination.

These diversities of sounds are expressed by the terms high and low, acute and grave, or sharp and flat. Hence, from any given sound, we can conceive a succession of them; wherein the last in order is more acute than the foregoing; and this series is called notes ascending.

Or, on the contrary, when, in a succession of sounds, the last in order is more grave than the former; this series is called notes descending.

AXIOM II. From this order of notes ascending and descending are deduced all the proportions of sounds; the properties of the same proportions; and the relations which arise out of their various combinations and successions. These include the whole business of harmony and melody; and the knowledge of them is the groundwork or basis of the science of music.

COROLLARY I. Hence nothing can be admitted in musical composition which doth not immediately depend on the foregoing axiom, and which cannot be demonstrated from it.

Sect. 2. Of the Diversity of Sounds.

The diversity of sounds succeeding in the natural order is not however extended through any number of sounds which may be expressed by a musical instrument, or even by the human voice. For universal experience, conducted by the judgment of the ear, hath demonstrated and ascertained the number seven to comprehend all the variety that music is capable of affording. Therefore, the number eight is the bound or limit of the materials of musical composition; and this eighth note or sound is called an octave.

This octave may be conceived as unity, or the first note of another series ascending or descending.

This series, though it may be repeated at pleasure, must still come under the restriction above mentioned; which is, that it must not exceed the power of the ear to judge of it. The greatest compass of the human voice will scarcely reach above two octaves, or fifteen notes. Instruments are framed with more, to answer the most interesting purpose of music, which is variety.

The seven sounds in music are named from the first seven letters of the alphabet, viz. A, B, C, D, E, F, G.

The distance between any two of those, whether immediate or remote, is called an interval. And every interval is named from the natural numbers; beginning at unity.

In naming of an interval, it is always understood of the ascending notes; and both terms are inclusive. Thus AB is called a 2d, AE a 5th, BE a 4th, EG a 3d, AA an 8th or octave; and so of the rest.

We proceed now to lay down an exact description of all the intervals in music. For from the knowledge of these are discovered all the proportions which constitute harmony; and upon which the whole superstructure of music is raised.

First, of the intervals of sounds lying in their natural order. Of these there are seven intervals, named either the greater tone, the lesser tone, and the half-tone or semitone. See the Music Plates, No. 1, 2.

| Interval of a | |---------------| | No. 1 | | Major | | Minor | | Semitone | | Major | | Minor | | Minor | | Semitone |

| Interval of a | |---------------| | No. 2 | | Major | | Semitone | | Minor | | Major | | Semitone | | Minor | | Major |

In this series of 8 notes are contained 5 whole tones; three greater, and two lesser; and 2 semitones. Reducing them therefore to the lowest denomination, they will be found to contain 12 half-tones; and inclusively 13. Every octave then contains 13 half-tones; out of the various combinations of which arise the several concords and discords, as will be shown in its proper place. The lesser tones are alike divided into half tones, as are the greater. We shall therefore, for brevity sake, hereafter use the distinction only of whole tones and half tones: the reason for which shall be assigned below.

From the inequality in the order of these intervals we draw the following corollary. Cor. II. Harmonical proportion is of a species different from all other proportions, and can be demonstrated only from principles peculiar to itself. This will be seen when we come to show the method of dividing a line harmonically; as well as from the proportion stated in numbers.

The first of the notes in the examples above is called the key-note, or key. Notwithstanding the intervals may be reckoned from any given note; yet it will answer our purpose better to begin with the key.

In the first example,

The first interval, or distance between the key and second, contains 2

Between the 2d and 3d 2 3d and 4th 1 4th and 5th 2 5th and 6th 2 Semitones, 6th and 7th 2 7th and 8th 1

In the second example,

The first interval, or distance between the key and second, contains 2 semitones.

Between the 2d and 3d 1 3d and 4th 2 4th and 5th 2 Semitones. 5th and 6th 1 6th and 7th 2 7th and 8th 2

From this comparison of the two series, it is evident there is but one difference, and this arising from the order of the notes, or place of the semitone. For if you begin to read the second series at the interval between the 3d and 4th, the semitones will be found exactly in the same order as in the first example.

In the first example, the first semitone falls on the 4th note, or that which is next above the 3d to the key; which 3d is 5 half tones above the key inclusively. In the 2d example, the half-tone falls on the 3d note; and is therefore itself the third to the key, and is four half-tones above the key inclusively.

This distinction of the place of the semitone is most worthy of observation, it being the only essential difference of tune, the groundwork of all that beautiful variety which may be introduced in the air or melody, as well as it is the principle or hinge on which turns the resolution of every discord. The key-note of every tune is that whereon the tune ends; which though it may be altered for variety in the upper part, yet the last note of the bass is ever the key.

When the 3d to the key is 5 semitones to the key, as in the first example, that tune is said to be composed in a sharp key. When the 3d to the key is 4 semitones to the key, as in the second example, the tune is in a flat key. And this, as was said before, is the only difference in tune.

This distinction of flat and sharp third holds good, not only in relation to the key, but likewise to every note in the scale of music. And in this light it is the foundation of composing in different keys; of changing the key in the same tune which introduces the so much desired variety in music; and of writing the same tune in divers keys, which is called transposition. Hence we establish the following axioms.

Axiom III. As the difference of the flat and sharp third to the key constitutes the key, and is essential to the tune: so no tune composed in a sharp key can be composed into a flat one, nor a flat into a sharp; for that would be altering the permanent nature of things.

The truth of this axiom will most evidently appear, when we shall, in the second part, or practice, have learned the art of transposition.

Axiom IV. The great and constant object which must be sought after in music, whether in composition, or performance of thorough bass, is variety with uniformity. For the proportions already laid down, and the prodigious variety emerging from them, as they lie in the order of nature, before they are modified, divided or combined by art, do not only point out this variety to us; but the concords and discords likewise made out of these, and arranged by art, will not only not suffer us to recede from the established precept, but by a kind of sweet violence constrain us to pursue this darling object.

On the truth of this axiom is grounded the reason for the mixture of discord with harmony, and the occasion of this precept in playing thorough bass, namely, that the hands should as much as possible move in a contrary direction.

As to the place of the other semitone, which in the flat key is on the 6th, the reason shall be told in its proper place. And moreover, it must be observed, that the greater 7th in the sharp key, which causes the second semitone to fall on the 8th in that key, is also common to the flat key in many passages, but unexceptionably at the end of the tune, or close.

Sect. 3. Of the Concords and Discords.

Of the intervals standing in their natural order are compounded the greater intervals, namely, the concords and discords.

These are the next things to be considered. Now, to investigate the order of these, and their proportions to each other, we must have recourse to the original cause of sound; that is, to the tremulous motion of the air, excited by the percussion of some solid body, as a bell, string, or pipe.

This trembling of the air is in proportion quick or slow as the impression given it by the voice or an instrument. The quicker the trembling is, the more acute the sound; the slower, the more grave or flat. The same sound is the effect of the same degree of quickness of the air's motion continued. Hence a string founding the same note to the end of its vibration, proves, that the vibrations are in equal times, from the greatest to the least ranges of its motion.

The shorter a musical string is, the quicker are its vibrations, and therefore the more acute the sound.

The longer a string is, the more slow are its vibrations, and so the more grave the sound.

Therefore, from the division of a musical string, the proportions of acuteness or gravity are computed. Hence we raise the following axiom.

Axiom V. The quickness of the vibrations is reciprocal. Proportions of the lesser or flat 6th.

Let \( AB \), a musical string, be divided equally in \( C \), and stop there; \( CB \) will found an octave to the whole or open string \( AB \). Now, \( CB, AB \), are as 1 to 2; therefore, the vibrations are as 2 to 1; that is, the proportion of the octave or diapason is double, or 2 to 1. Q.E.D.

Proportion of the 5th.

Let \( AB \) be divided into three equal parts, and stop in \( C \); \( CB \) will found a 5th to the whole or open string. Now, \( CB \) is to \( AB \) as 2 to 3; therefore the vibrations are as 3 to 2; that is, the proportion of the 5th, or diapente, is sesquialteral, or 3 to 2.

Proportion of the 4th.

Let the string be stop in \( C \), which is a 4th part of the whole; \( CB \) will found a 4th to the whole \( AB \) or open string. Now, \( CB \) is to \( AB \) as 3 to 4; therefore the vibrations are as 4 to 3; or, the proportion of the 4th, or diatessaron, is 4 to 3.

Proportion of the sharp 3d.

Stop the string in \( C \), the 5th part; \( CB \) will found a greater 3d to \( AB \). But \( CB \) is to \( AB \) as 4 to 5. Therefore the vibrations are as 5 to 4; or, the proportion of the sharp 3d is as 5 to 4.

Proportion of the flat 3d.

Stop in \( C \) the 6th part; \( CB \) will found the lesser or flat third. But, &c. Therefore the proportion of the flat third is as 6 to 5.

Proportion of the greater or sharp 6th.

\( CB \frac{3}{8} \text{ths of } AB \) will found the greater 6th. Therefore the proportion of the sharp 6th, is as 5 to 3.

The harmonic proportion of three numbers in this natural succession of fractions, extends as far as the chord of the flat 3d. Which third, being \(\frac{1}{3}\) of the whole number, limits this equality of proportion, seeing that the number 7 is no aliquot part. But as to the fourth proportional, it cannot be found even from that number which expresses the sharp 3d, which is still of shorter extent. This limitation of proportion then explains the extent of harmony, and likewise becomes the principle of the same; as will be seen in the definition of harmony.

Hence it is evident, the remaining concords of the sharp 6th, which is \(\frac{4}{3}\), and of the flat 6th, or \(\frac{8}{5}\), are not included in this equality of proportion.

There are the concords, their order and proportions; any one of which founded together with the open strings, is concordant with it, and produces harmony.

Example of the names and order of the intervals in concord with the open string or bass, and the semitones contained in each, Music Plates, No. 3.

Again, two of these concordant intervals, namely, the 5th and — 8th

Sharp 3d and — 8th

Flat Flat 3d and —— 8th Sharp 3d and —— 5th Flat 3d and —— 5th Sharp 3d and Sharp 6th Flat 3d and Flat 6th 6th and —— 4th Flat 6th and —— 4th

founded with the open string, or bass note, are concordant all together; and therefore produce harmony.

Example of two concording with the open string or bass. No 4.

Next follows an example of three concording with the bass. No. 5.

Having thus discovered the concords, their order and proportions; it is worth remarking, that the first concord, or 8th, which arises from the most simple division of a line, is the most perfect concord; the 5th is the next perfect concord; and so of the rest, in the order they have been found by the division of the string. For the nature and perfection of the 4th, accounted by some a very imperfect concord, shall be explained in the corollaries of the demonstration of the harmony, in Part II., on practice.

The 8ths and 5ths then are called the perfect concords. The 3ds and 6ths imperfect concords. The 4th, of a middle nature between the others, may be called an improper concord; for this reason, that with the 6th, with which it is always accompanied in harmony, though it make perfect harmony with the given note, yet they change the chord into that of the 4th to that note.

Likewise the 6th, whether joined with the 3d or 4th to the given note, tho' it make perfect harmony with either, yet they change the chord into that of the 6th or 4th to the same note.

Hence the reason why the 6ths are more imperfect concords than the 3ds.

From the order and perfection of the concords thus discovered; we deduce the following corollary.

Cor. III. The most perfect harmony is that which will be produced by the perfect concords, namely, the 3d, 5th, and 8th. Thus No. 6.

From the foregoing corollary, we are able to give a just definition of harmony. Harmony consists in one certain invariable proportion of distance of four sounds performed at the same instant of time, and most pleasing to the ear.

These proportions of the first series are called simple concords. If the notes of a second series be added to the first octave, the proportion of any two concording notes compounded with the octave retains the name and nature of the simple concord; as a tenth, compounded of an octave and third, is called a third; a twelfth, compounded of an octave and fifth, is called a fifth; a fifteenth, compounded of two octaves, is called an octave, or double octave. And so on to a third series.

These are the compound concords.

All other proportions founded together are harsh and disagreeable to the ear; and are for this reason called discords.

From the compounding and dividing the proportions delivered, not only the harmonical intervals are computed, but the discords likewise.

And this the following calculations demonstrate.

The proportion of the octave is the proportion of the 4th and 5th: for, by compounding $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$, or $\frac{5}{3}$ the proportion of the octave.

Again, it is the proportion of the sharp 3d and flat 6th: for, $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$ in its lowest terms.

Again, the flat 3d and sharp 6th: for, $\frac{5}{3} \cdot \frac{6}{5} = \frac{30}{15}$ or $\frac{2}{1}$ the proportion of the octave.

Now, since the 4th and 5th, the 3d and 6th, as also the 2d and 7th, compounded, make the octave; that is, any two numbers making 9, the middle term or note being repeated, or common to both; it follows, that to fall a 4th or rise a 5th, as also to fall a 3d or rise a 6th, and to fall a 2d or rise a 7th, and the contrary, answers the same purpose of harmony; for they meet in the octave.

This observation will be of great use in setting the bass, and figuring the same, by producing that variety and contrary motion demonstrated necessary in the 4th axiom.

Proportion of the 5th.

The proportion of the 5th is the proportion of the sharp 3d and flat 3d; for by compounding $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$ or $\frac{5}{3}$ the equilateral and known proportion of the 5th.

Proportion of the Sharp 6th.

The proportion of the sharp 6th is the compound proportions of the fourth and sharp 3d; for $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$, or $\frac{5}{3}$.

Of the flat 6th, the proportion is of the 4th and flat 3d; for $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$ or $\frac{5}{3}$.

By the same manner of compounding are found the proportions of the concords of the 3ds; which shall be shewn when we shall have got the tones and semitones; which, as being discords, arise by dividing the harmonic proportions as follows.

Proportions of the Discords proved.

Proportion of the Greater Tone.

The proportion of the greater tone is the difference of the 4th and 5th; for $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$ the proportion of the greater tone.

Proportion of the Lesser Tone.

The proportion of the lesser tone is the difference of the 5th and sharp 6th; for $\frac{5}{3} \cdot \frac{6}{5} = \frac{30}{15}$ the proportion of the lesser tone.

Proportion of the Semitone.

The proportion of the semitone is the difference of the sharp 2d and 4th; for $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$ the proportion of the semitone.

Having now the proportions of the tones and semitones, we are enabled to prove the proportion of the semitone, or flat 2d and sharp 7th to the 8th; as likewise all the remaining proportions, whether discord or concord: For, the 5th and sharp 3d, $\frac{4}{3} \cdot \frac{5}{4}$, give $\frac{20}{12}$ the greater 7th; and the sharp 9th and semitone $\frac{4}{3} \cdot \frac{5}{4} = \frac{20}{12}$ in its lowest terms $\frac{5}{3}$ the proportion of the octave.

To go on; The proportion of the sharp 3d is that of the greater greater and lesser tones; for, $\frac{3}{4} \div \frac{1}{2} = \frac{3}{2}$ in its lowest terms, the greater 3d.

And the proportion of the flat 3d is compounded of the greater tone and semitone; for, $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ in its lowest terms $\frac{3}{2}$ the proportion of the flat 3d.

The proportion of the 4th is that of the sharp 3d and semitone; for, $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ in its lowest terms the proportion $\frac{3}{2}$ of the 4th. Or it is the proportions of the flat 3d and lesser tone; for, $\frac{3}{4} \div \frac{1}{2} = \frac{3}{2}$ in its lowest terms $\frac{3}{4}$.

The proportion of the discord of the sharp 4th is found by compounding its constituent intervals, the 4th and semitone; for, $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the proportion of the sharp 4th.

And lastly, the proportion of the flat 7th is compounded of two 4ths, or is $\frac{1}{2}$; for, $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the proportion of the flat 7th.

These proportions, in the natural order of the first series, or sharp key, stand thus:

K 2d Sharp 3d 4th 5th Sharp 6th Sharp 7th 8th

$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$

In the second series, or flat key, thus:

K 2d Flat 3d 4th 5th Flat 6th Flat 7th 8th

$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$

Hence we can demonstrate (what before was taken for granted) the places of the greater and lesser tones, and semitone.

Now, the relative proportion or difference is found by division of the two next proportions in the natural order as above.

The places of the greater and lesser tones and semitones in the sharp key.

1) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the greater tone, or 2d.

2) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ in its lowest terms $\frac{3}{2}$ the lesser tone, or sharp 3d.

3) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the semitone, or 4th.

4) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the greater tone, or 5th.

5) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the lesser tone, or sharp 6th.

6) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ or $\frac{3}{2}$ the greater tone, or sharp 7th.

7) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the semitone, or 8th.

The places of the greater and lesser tones and semitones in the flat key.

1) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the greater tone, or 2d.

2) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the semitone, or flat 3d.

3) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the lesser tone, or 4th.

4) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the greater tone, or 5th.

5) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the semitone, or flat 6th.

6) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the lesser tone, or flat 7th.

7) $\frac{3}{4} \div \frac{1}{2} = \frac{3}{4}$ the greater tone, or 8th.

The use of this theory is chiefly on account of ascertaining the places of the semitone; the difference of the major and minor tones, which is as $\frac{3}{2}$, having not been hitherto reduced to practice.

We shall therefore hereafter admit no other distinction than that of whole and half tones.

The intervals then contained in the octave, in both keys, excluding the first term, will be more easily described thus:

The —— 2d is one tone. Flat 3d a tone and a half. Sharp 3d two tones. 4th two tones and a half. Sharp 4th three tones. 5th three tones and a half. Flat 6th four tones. Sharp 6th four tones and a half. Flat 7th five tones. Sharp 7th five tones and a half. 8th six tones.

Having found all the intervals, their order and proportions; it will be necessary to take in one view the semitones of the octave, marked by their different names and intervals. For every semitone hath two names in respect to the preceding and following note in the natural order. As in the following example. The knowledge of this is most necessary to learning the art of composition. No 7.

The discords being, as hath been shewn, the lesser 2d, or semitone, and greater 2d, the sharp 4th or false 5th, the lesser and greater 7ths, it is to be understood, that not any two or three of these are to be founded together, to frame the discord; as the members of any concord are, to make the harmony; but each discordant note hath its discordant and concordant notes proper to itself, which fill up the discord; and which are called the accompaniments.

The five discords, then, being distinct and unlike each other; the definition of discord must be this:

Discord consists in certain variable proportions of the distance of sounds, performed at the same instant of time, and disagreeable to the ear.

Chap. II. Of the Scale of Musick.

Having found that the larger combinations of the 13 semitones in the octave constitute the concords and discords; for the better application of them to our purpose, we shall next consider them singly and distinctly.

Diapason — — — — — — — — Octave — 8th. Semidiapason Defective 8th, or — — — — greater sharp 7th. Sept. major — — — — — — lesser flat 7th. Sept. minor — — — — — — greater sharp 6th. Hexachordon major — — — — — — lesser flat 6th. Hexachordon minor — — — — — — — — Diapente — — — — — — — — 5th. Semidiapente, or — — — — — — — — sharp 4th, Tri tone — — — — — — — — or false 5th. Diatessaron — — — — — — — — 4th. Ditone — — — — — — — — greater, or sharp 3d. Semiditone — — — — — — — — lesser, or flat 3d. Tone — — — — — — — — greater — 2d. Semitone — — — — — — — — lesser — 2d. Unison — — — — — — — — one sound.

As they succeed each other in the natural order of both keys, as above demonstrated; this is called the scale of musick.

In this scale we shall likewise take a view of the concords of the same denomination, as they arise in succession from the same natural order of the simple tones, and also of the discords as oft as they occur.

There are two scales in use: the diatonic scale, and the chromatic.

In the diatonic scale, the notes arise by two tones, a semitone, semitone, and one tone to the 5th; and thence by two tones and a semitone to the 8th. This is the order in a sharp key; where note, that the semitones are in the 4th and 8th places. No. 8.

In a flat key, the notes ascend according to the following example. No. 9.

One tone, a semitone, and two tones to the 5th; a semitone and two tones to the 8th.

In the next example, the same proportion of the flat third is illustrated by comparison with the instance in the sharp key. No. 10.

The number of the tones and semitones in both flat and sharp key are equal. The difference arises from the places of the semitone; which, in the flat key, are the 3d and 6th. This is that essential difference of tune already mentioned, which creates such variety in musical strains, as well as in the harmony. These are some effects of the semitone; others we shall see in its proper place.

The Chromatic Scale.

The chromatic rises by a tone and 5 semitones to the 8th, and thence by 5 semitones more to the 8th. Thus No. 11.

The chromatic scale, which is no other than the natural semitones in their order, except the first tone, is only used when mixed with the diatonic. That is to say, when a semitone, not belonging to the harmony of the key, is introduced in the middle of a tune. And this may be done by the note ascending by a semitone, or descending: in either of which cases, the key is changed in that part of the strain. This is the cause of great variety in the air; as well as it new-modulates the harmony. This is another effect of the semitone, on which turns so much variety and elegance. It must be executed by the composer with all the address and art imaginable. For this we must refer to the second part or practice; where will be given the rules for the mixture of the chromatic. Pieces of music where it is frequently used, are now commonly called chromatic music.

The diatonic scale being that which we are chiefly concerned to understand, as well as the first in order, and before any use of the chromatic can take place; we shall proceed to view it in another light, whereby we shall discover such properties of it as will be useful to the composer. Sharp key. No. 12.

From the key the thirds ascend, as in the above example, by one sharp 3d, two flat 3ds, two sharp 3ds, a flat 3d, and lastly another flat 3d on the 2d to the key.

All the 4ths being perfect, are like; except that one which falls on the sharp 7th; this is called a sharp 4th, or false 5th. No. 13.

All the 5ths are perfect, and therefore like; except that formed by the sharp 7th and 4th, which likewise is a flat 5th or sharp 4th. No. 14.

The 6ths stand thus: two sharp 6ths, one flat 6th; two sharp 6ths, two flat 6ths. No. 15.

There are but two greater 7ths which are the sharp 3d and sharp 7th to the key: they stand under the two semitones.

All the 8ths are perfect and alike.

Vol. III. No. 82.

From these theorems, and axioms, the 2d and 4th, we deduce this practical corollary.

Cor. IV. The 3ds and 6ths are the intervals most frequently to be used in composition; the 8ths, 5ths, and 4ths most rarely.

The 4th being an improper concord, and the 7th a discord; we cannot ascertain their use till we come to the demonstration of the harmony and accompaniments of the discords in the 2d part.

On this 4th corollary is grounded the reason of forbidding two 8ths and two 5ths in consecution, either in composition, or performing thorough bass.

Next follow examples of the same in a flat key., No. 16, 17, 18, 19.

There are but two sharp 7ths which are the 2d and 5th to the key. They stand under the semitones.

All the octaves are perfect and alike.

It is evident to sight, that the intervals in the flat and sharp key do not in the least vary; except in the order they succeed each other, beginning from the key. And it is equally evident, that this variation is owing to the different places of the semitone.

This demonstrates what has been said in page 319, col. 1. concerning the semitone; and illustrates what is asserted in the 4th axiom, that variety, amidst uniformity, must be the great object attended to in music, since that uniformity and variety both subsist in the very principles.

Let it be observed, that this uniformity is preserved by bringing in the flat 6th and flat 7th to the key. And this must of necessity be so, since they are the places of the semitone and lesser tone in the natural flat key wherein the example is set, according to the demonstration of the same.

For, by corollary 1. nothing can be admitted in composition which doth not immediately depend on the 2d axiom, and which cannot be demonstrated from it; namely, that the proportions of sounds, and their relations, must be deduced from the natural order of the notes. This is the true reason for introducing the flat 6th and flat 7th in every flat key.

The harmonic proportions and discords having been demonstrated from the division of a line in arithmetical progression; we shall, in the next place, try what are the effects of a musical string divided in geometrical proportion. No. 20.

Let A B, a musical string, be divided equally in C; C B, the half next the bridge, will sound an octave to the whole or open string, as we have shewn in the harmonic proportions.

Again, let C B be equally divided in D; D B will sound the octave to C B, or double octave to the open string, A B.

And thus, by an equal division of a string between either the nut and bridge, or stop and bridge; the half next the bridge will give the octave above continually.

But the same proportion is not preserved in the equal division between the nut and stop, or between any two stops. For the length of the octave to the open string, which is between the nut and stop, being equally divided; the half next the nut gives the sound of the 4th to the open string; and the half next the stop, or bridge, found the 5th, which two are the constituent intervals of the octave.

And the same division of the sounds is constantly preserved, if the length of an octave be equally divided between any two stops.

Again, the length of a 5th between two stops, or nut and stop, equally divided; the half next the bridge gives the greater 3d, the other half the lesser 3d.

And again, the length of the greater 3d, thus divided, gives the greater and lesser tone. And the greater tone's length, equally divided, gives the greater and lesser semitone. And the length of the greater semitone, equally divided, gives the sounds in proportion as 5 and 4. The greater interval being next the bridge, and so continually.

Hence the necessity of the greater and lesser tones and semitones in music is evident; and the truth confirmed, which is asserted in the 2d corollary.

Now, in the diatonic scale, wheresoever the semitones lie, that is, whether the air be in a flat or sharp key, the graver part of the tone will be the lesser semitone, and the acuter the great semitone; and in the chromatic, which ascends by semitones, the greater and lesser semitones will, for the same reason, succeed each other alternately. Wherefore, if any series of chromatic notes be removed a semitone higher or lower; it must happen, that the lesser semitone will succeed into the place of the greater, and the greater into the place of the lesser. Hence dissonances will happen in the diatonic scale, as being composed of the same materials with the chromatic, if the key be injudiciously changed by transposition. For, as the dissonance will be evident, if the transposition be by one semitone; so the disproportion will still appear, if the removal be by any odd number of semitones within the compass of the 4th.

As the proportions of the concords have been demonstrated from the division of a line; so are they likewise to be found in the geometrical proportions of solid bodies, and therefore may be illustrated by the same.

We shall begin with the proportion of the 8th.

The proportion of the 8th being the compound proportions of the 5th and 4th, is, by corollary of the 34th proposition of Archimedes, as the whole superficies of a right cylinder described about a sphere, to the whole superficies of an equilateral cylinder inscribed as 2 to 1.

For, the circumscribed is to the spheric superficies as 12 is to 8 (by 32 of this,) but the spheric is to the inscribed as 8 is to 6. By this present proposition: therefore the circumscribed is to the inscribed as 12 is to 6, or 2 to 1.

In harmonic terms thus expressed: the 5th is to a given note or key as 12 is to 8; but the proportion of the 4th is as 8 to 6. Therefore, the proportion of the 8th is \( \frac{12}{8} = \frac{3}{2} \) as 2 to 1.

Again, the proportion of the 5th, and the next harmonical proportion arising out of the 5th, is beautifully illustrated in the admirable proportion of the sphere, right cylinder, and equilateral cone circumscribed about each other. The last proportion being invented by Andrew Faguet; and that of the two first by Archimedes, as demonstrated in his 45th proposition in Tacquet's Euclid.

We cannot forbear transcribing at length this wonderful proposition, and demonstration of the same: for that on this proportion is erected the whole superstructure of harmonic chords.

**ARCHIMEDES'S PROPOSITION 45.**

**THEOREM.** An equilateral cone circumscribed about a sphere, and a right cylinder in like manner circumscribed about the same sphere, and the same sphere itself, continue the same proportion, to wit, the equilateral, as well in respect of the solidity, as of the whole superficies. For, by 32d of this book, the right cylinder () encompassing the sphere, is to the sphere, as well in respect of solidity, as of the whole superficies, as 3 is to 2, or as 6 to 4. But by the foregoing, the equilateral cone circumscribed about the sphere, is to the sphere, in both the said respects, as 9 to 4. Therefore, the same cone is to the cylinder, both in respect of solidity and surface, as 9 is to 6. Wherefore, these three bodies, a cone, cylinder, and sphere, are, betwixt themselves, as the numbers 9, 6, and 4; and consequently continue the sesquialteral proportion, Q, E, D.

In harmonic terms expressed thus: the 5th is to the key as 3 is to 2, or as 6 to 4; but the 9th is to the key, (that is, the 5th to the 5th) as 9 is to 4; (for the 2d and its 8th \( \frac{9}{8} = \frac{3}{2} \)) therefore, the same 9th is to the 5th as 9 is to 6. Wherefore these three tones, the 9th, the 5th, and the key, are, betwixt themselves, as the numbers 9, 6, and 4; and consequently continue the sesquialteral proportion, Q, E, D.

Therefore, the proportions of the key, the 5th and its 9th, being the same sesquialteral proportion continued, are the same proportions as that of the equilateral cone, right cylinder and sphere; the two first described about the sphere, Q, E, D.

On the proportion of these three is erected every other proportion of harmony; which we shall pursue one step further, forasmuch as these truths will be most manifest and established in the practice when we shall have delivered the rules of harmony.

The 5th divided arithmetically, or equally, gives, as hath been shewn, the greater 3d or next perfect concord; the sesquialteral proportion to which gives the greater 7th; for \( \frac{5}{4} \times \frac{3}{2} \) gives \( \frac{15}{8} \) the greater 7th, as demonstrated above.

Hence, from the sesquialteral proportion thrice repeated, namely, to the key, its 5th and 3d, we are furnished with the perfect harmony, or concords of the key and 5th; to which every harmonic proportion, where ever found, is analogous; that is, partaking of the nature, proportion, and relation of the key and its 5th. It is worth remarking in this place, that the members of these chords arise out of the proportions, as above demonstrated, by turns. The key being first supplied with one proportion; and then in its turn the 5th with the same. Wherefore, the mixture of the harmony of the key and 5th is scarcely separable: A truth which will abundantly discover itself in the practice, both in the rules of harmony, and in every other part of composition. On this is founded the following axiom.

Ax1012. AXIOM VI. These 5 tones therefore, namely, the key, the 3rd, 5th, sharp 7th, and 9th, are the foundation of the whole superstructure of musick.

We shall conclude this theory with the harmonical division of a line.

To divide a Line harmonically. No. 21.

A right line, AD is said to be divided harmonically, if being cut into three parts, AB, BC, CD, the case be so, that, as the whole AD (or Z) is to either extreme a or c; so shall the other extreme be to the intermediate part m; that is,

\[ Z : a :: c : m \] \[ Z : c :: a : m \]

Wherefore \( Z = a + c \).

And, to divide any given right line thus harmonically, suppose AD: From either end of it draw a right line, as DG; make an angle with it, and of any length; connect the end of this line with the other end A, by drawing AG; and then taking any point, as B, at pleasure, in the given line, there draw EF parallel to DG, and in it take BE equal to BF; then draw EG, and that shall find the point C required; and then calling, as above, the whole line Z, \( A'B = a, B'C = m, \) and \( D'C = c \), I say \( Z : a :: c : m \).

For the triangles ADG, ABF, and BEC, are all similar; and consequently \( AD : AB :: DG : BF :: \) or as \( DG : BF = BE \). But as \( DG : BE : CD : BC \); (by working about the equal angles D and E, BC) wherefore, by equality, \( AD : A'B :: CD : CB \); that is, \( Z : a :: c : m \).

Q.E.D.

And from hence it is plain, that the ratio of the whole line AD, to the segment AB, may be taken at pleasure; but that the intermediate part BC must be less than either AB or CD.

PART II. THE PRACTICE OF MUSICK.

The practice of musick is founded on the principles delivered in the theory. Its several parts are, composition, figuring the bass, melody, transposition, and singing by note.

Of these we shall treat separately in the above order.

Composition is the setting together two or more notes in harmony, to be sounded at the same time.

When in the succession of concords, in the parts, the notes of each part are of the same length, or time of sounding, the composition is called counterpoint.

When the succession of concords is by notes of different lengths in the several parts, it is called plain descant.

The mixture of discord and concord, by notes of the same or different lengths or time in the parts, is called figurate descant. Of these in their order: and first of counterpoint.

In order to attain more easily the art of composition, it is necessary to premise a few things concerning other affections of sounds; as the time or lengths of sounding the notes; the time of musick, or movement of the air; and the different cliffs wherein the parts of musick are usually written.

The following account of the proportions of the lengths of notes, the time and cliffs, being well understood by every one acquainted ever so little with musick, might well have been omitted in an essay of this kind; (where, instead of using repetitions, it is hoped we have offered to the public something new; at least in the manner of demonstrating the rules of composition, both in discord and harmony;) but that we would leave nothing in our power untold, which may contribute to form a compleat musician.

The longest note, now generally in use in instrumental musick, is called a semibreve. Its time is as long as you can distinctly count for.

Out of the division and subdivision of the semibreve are formed the lengths of all other notes; according to the following proportions No. 22.

A semibreve, whose time is as one, two, three, four, is as long.

\[ \begin{align*} \text{2 Minims,} \\ \text{4 Crochets} \\ \text{8 Quavers,} \\ \text{16 Semiquavers,} \\ \text{32 Demi-semiquavers.} \end{align*} \]

A dot after any note, signifies the time of such note must be lengthened to one-half of the plain note. No. 23.

The proportions are thus;

A dotted semibreve is equal to 3 minims. A dotted minim to 3 crochets. A dotted crotchet to 3 quavers. And so of the rest.

Thus we are furnished with notes according to the odd and even numbers. And this naturally divides the time of any long or music into odd and even time.

COMMON TIME.

When the air moves according to the even numbers; and every bar is measured by beating the time into two equal and even parts, the music is composed in common time: known by one of the following marks prefixed to the tune; as the letter C, having 4 crochets in a bar; or \( \frac{3}{4} \), denoting two crochets in a bar.

TRIPLE TIME.

But when the music moves according to the odd numbers, and every bar is measured by beating the time into two unequal parts, as two and one, the song is composed in triple time; which is known by one of these signs prefixed to the tune.

\[ \begin{align*} \text{3, or } \frac{3}{4}, \text{ for 3 minims in a bar,} \\ \text{3 Crochets in a bar.} \\ \text{3 Quavers in a bar.} \\ \text{9 Crotchets in a bar.} \\ \text{9 Quavers in a bar.} \end{align*} \]

The uppermost number being the numerator of a vulgar fraction; and the lower, or denominator, the aliquot part of the semibreve.

There is also another proportion of the length of notes in use. And this is, when three quavers are, by diminution, nution of their lengths, contracted into the time of 2 quavers, or one crotchet, constantly noted by the figure (3) over them.

And lastly, the most common movement of jiggs, which is by six or twelve quavers in a bar, have their bals, for the smoothness of the movement, often written in plain crotchets; 2 in a bar for the treble $\frac{6}{8}$; and four, marked thus C, for the treble $\frac{4}{8}$. It is plain, therefore, that all tunes in these movements truly belong to common time, since every bar is measured by the beating, or dividing it into even parts, as expressed in the bals.

A pause or rest in music, is a cessation of the sound, in one or more of the parts; or of all the parts together. Nothing hath a finer effect in music than a pause of all the parts judiciously made; or of one, or more of the parts, for the sake of imitation. The rests therefore are written down in the place of notes, and each note hath its own rest, which is of the same length with the note whose name it bears. Thus,

A semibreve rest is as long as a semibreve. A minum rest as long as a minum. And so of the rest.

The next thing to be considered is the cliff in which any part of the music is said to be written; according as the cliff is prefixed to each stave of the writing.

The use of the cliff is to ascertain the names of the notes; and to denominate that part of the music to which it is prefixed.

There are three cliffs, to answer and distinguish the three parts in music: The bals, or F cliff; the tenor, or C cliff; and the treble, or G cliff. No. 24.

The bals is so called, from its being the lowest part, or that wherein are set the graver tones.

The tenor, or middle part, hath its name from holding the bals and upper parts together. This will be clearly understood, when we shall have learned to compose in four parts.

The uppermost part is called the counter-tenor in vocal music; and, in instrumental, the first treble.

The bals and treble cliffs are now constantly written in the same places as in the examples. The tenor cliff is often removed, according to the fancy of the composer or writer of music; to answer the convenience of the notes standing, as much as may be, within the compass of the five lines, or stave. Which convenience is the reason for the invention of the diversity of cliffs, as well as the uses already named. For it is easy to apprehend, that the natural tones, and their proportions, are invariably the same, whether expressed by the voice, or an instrument, however they may be distinguished by artificial signs. Observe, that the cliffs, according to their names, rise above each other by the interval of a fifth: thus the tenor is equally distant from each other part. For C is a fifth to F, as it is also a fifth below G.

**CHAP. I. OF COMPOSITION IN COUNTERPOINT.**

Composition in counterpoint is when, in the succession of concords in the parts, the notes of each part are of the same length, or time of sounding.

According to the 2d axiom, we shall begin with the harmony of the key note; and proceed to demonstrate the harmony of the remaining notes of the octave in their natural order.

**Demonstration of the harmony of the key.**

The harmony of the key is the concord of itself.

The harmony of the key must be perfect harmony. Now, the notes concording in perfect harmony, are, by corollary 3d, the 3d, 5th, and 8th: But these, with the key, are the concord of itself: Therefore, the harmony of the key is the concord of itself.

This demonstration is grounded on this evident truth; namely, that any other concord would, by the term, or name of it, in effect change the key; whereby the unity of the tune would be destroyed, and by this contradiction the author's meaning rendered unintelligible. The necessity of perfect harmony in the key being evident, no other form of demonstration is required, nor indeed can be admitted.

**Prob.** Let it be required to set a bals to the notes of an octave ascending in G sharp. No 25.

Any one of the three notes in the bals is concording, by corollary 3; but the 8th is preferable when it is the first or last note of the tune; for thus it best ascertains the key. The preference of either of the other two depends on the following rules.

First, The 5th cannot take place when the concord immediately preceding shall happen to be a 5th, the forbidding the consecution of 5ths being asserted in corollary the 4th.

Again, the movement of the bals ought generally to be by descending a 5th, or rising a 4th, 6th, or 8th, or any other great interval; thereby meeting the treble, and effecting variety and contrary motion of the parts; the established rules of harmony by the 4th axiom.

Lastly, The air of the bals must be consulted; and, if possible, an imitation of some foregoing passage in the upper part.

The application of these rules will decide which of the two or three notes is preferable in this or any other concord.

**Demonstration of the harmony of the 2d.**

The harmony of the 2d is the concord of the 5th.

The harmony of the key having been shewn, we must consider it as an immovable point, in relation to which we are to order the rest of our computations, consistent with the established principle of uniformity.

The 2d to the key immediately descending into the key, will have, for its next concording note, the greater 7th; which at the same time ascends by a semitone into the key; to which 7th the 2d is a 3d.

For, by axiom 2, the combination of sounds are deduced from the natural order of notes ascending and descending. But the 2d and 7th can admit no other concordant note but the 5th to the key. For the 3d is discord with the 2d; and the 4th, 6th, and 8th discord with the sharp 7th.

Now, the 2d, 7th, and 5th are the concord of the 5th; therefore, the harmony of the 2d is the concord of the 5th. No. 26. The 5th must always be taken at the close; or when the treble is descending into the key; for then the bass will fall a 5th into the key; which movement is called the great cadence. Otherwise the 7th or 5th may be taken indiscriminately; yet, under the restriction of the rules, (p. 326. col. 2.) for setting the harmony of the key.

Demonstration of the harmony of the 3d.

The harmony of the 3d is the concord of the key.

From the demonstration of the harmony of the key, the key will have its 3d; and, by inverting, the 3d will have the key. Now, the key and its 3d will admit no other discordant note but its 5th: For the 2d is discord to both, the 4th is discord to the 3d, and the 7th discord to the key; but the key, its 3d and 5th, are the concord of the key. Therefore, the harmony of the 3d is the concord of the key.

The 6th indeed, which is a 4th to the 3d, which is an improper concord, will, with the key, form the concord of the 6th; but the demonstration of the concord of the 6th in a sharp key, depending on another principle, as will be shewn in its place, can, for the same reason, bear no relation to the harmony of the 3d, which is a member of the key: No. 27.

The two notes in the bass may be taken indiscriminately; yet complying with the rules, (p. 326. col. 2.) But, if the 3d in the treble be prepared to descend into the key, by its passage into the 2d, then the 5th is more eligible; which falling an 8th for the next note, thence descends by a 5th into the key. This is the most striking movement of the bass; and, at the same time, the most common, at a final close in either flat or sharp key.

Demonstration of the harmony of the 4th.

The harmony of the 4th is the concord of itself.

In a sharp key, the places of the two greater 7ths are the sharp 3d and sharp 7th to the key: and, of the semitones, the 4th and 8th, or key; Therefore, the 3d is to the 4th as the sharp 7th to the key. Now, since by axiom 24, the combinations are deduced from the natural order of the notes ascending and descending; the harmony of the 4th will be as the harmony of the key: But the harmony of the key is (by demonstration 1.) the concord of itself: therefore, the harmony of the 4th is the concord of itself: No. 28.

The notes in the bass may be taken indiscriminately: only observing the foregoing rules. If a close on the 4th be prepared from the 4th itself, either note will do: yet the key is preferable, in order to prepare for the great cadence.

Demonstration of the harmony of the 5th.

The harmony of the 5th is the concord of the key.

From the demonstration of the harmony of the key, the key will have its 5th; and, by inverting, the 5th will have the key. Now, the key and its 5th will admit no other discordant note than the 3d. For, the 2d and 4th are discord with the key and 5th; the 6th discord with the 5th; and the 7th discord with the key. But the key, its 5th and 3d, are the concord of the key:

Therefore, the harmony of the 5th is the concord of the key. No. 29.

There is no exception in the choice of the bass notes; but the disallowance of the consecution of 5ths. But if there be a preparation for a close on the key, and the 5th stand in the bass; in order to make the great cadence, the 5th will have its own concord. This depends on the demonstration of the 2d.

Demonstration of the harmony of the 6th.

The harmony of the 6th is the concord of the 4th.

From the demonstration of the 4th, its harmony is its own concord. The 4th, then, will have its 3d; and, by inverting its 3d (that is, the 6th) will have the 4th. Now, the 4th and 6th will admit no other discordant note than the 8th: For the 2d is, (with the 4th and 6th,) a discord, as will be shewn in the demonstration of the discords. The 3d and 5th are discord to the 4th, and the 7th to the 6th. But the 4th, 6th, and 8th, are the concord of the 4th: therefore, the harmony of the 6th is the concord of the 4th. No. 30.

Either note in the bass may be taken at will. But if there be a preparation for a close on the 4th, the second note, or key, is preferable, for the reasons assigned in the demonstration of the 4th, which is, to make the great cadence, prepared by the bass, first descending an 8th, and thence a 5th, into the 4th, or close.

Demonstration of the harmony of the 7th.

The harmony of the 7th is the concord of the 5th.

The harmony of the 7th is part of the harmony of the 2d, (by demonstration 2.) but the harmony of the 2d is the concord of the 5th: therefore the harmony of the 7th is the concord of the 5th. No. 31.

If the 7th, or treble note, precede a close on the key, the first note in the example must be the bass note, in order to make the great cadence.

The 8th being the key, hath for harmony its own concord; as by demonstration 1.

From the foregoing demonstrations, the bass notes, set to the 8 ascending notes in the treble, will stand thus.

General Rule. The consecution of 8ths, 5ths, and 4ths, is not allowed, (as by corollary 4.) except by contrary motion of the parts, or in the passage of very quick notes in composition of many parts. No. 32.

From taking in one view the harmony of the seven notes, we shall deduce some useful corollaries. No. 33.

Key 2d, 3d, 4th, 5th, 6th, 7th, hath the harmony of the Key 5th, Key 4th, Key and 5th, 4th, 5th, Semitone.

Cor. I. Every note in the octave (except the 2d to the key) admits in its harmony a 3d.

Cor. II. The key, the 2d and 5th, admit in their harmony a 4th.

Scholia.

When the key admits a 4th, the concord is of the 4th. When the 2d admits a 4th, the concord is of the 5th. When the 5th admits a 4th, the concord is of the key. Hence the interval of that note which admits a 4th, is in fact a 5th: therefore two 4ths are no more allowed in consecution than two 5ths. And hence likewise the interval of the 4th, which we have called an improper concord, appears to be of a middle nature between concord and discord; being a fourth in name and appearance in the natural order of sounds; yet a 5th in name and effect in composition, as member of that chord wherein it makes a part of harmony.

Cor. III. The key, the 4th and 5th, admit in their harmony a 5th.

Scholia.

When the key admits a 5th, the concord is that of the key.

When the 4th admits a 5th, the chord is of the 4th.

When the 5th admits a 5th, the concord is of the 5th.

Hence, when a note admits a 5th, the harmony is the concord of the same note.

Cor. IV. Every note but the 4th admits a 6th; for, the 4th having its concord for harmony, will have only its 5th.

Every note admits its 8th; for any note may be substituted for its octave. But 8ths are (by corollary 4. of the theory) to be the most sparingly used, as not producing that variety or mixture of sounds requisite to bind the harmony, especially where it can be best avoided, in the composition of two parts.

From the foregoing demonstrations and corollaries, arise the following observations.

The 3ds and 6ths most frequently occur in composition. This then demonstrates what was asserted, by way of precept, in the 4th corollary of the theory as well as part of the 4th axiom; namely, that the proportions of musical sounds, and the variety emerging from them, point out to us this variety, and will not suffer us to depart from the established precept.

It will be necessary to see the same truths confirmed in the descending notes. We shall therefore set down instances of composition in the descending notes of the octave upon the same principles, and wherein the same demonstrations and corollaries do take place.

Example of composition in the descending notes of the octave. No. 34.

In the ascending notes, when the upper part rises by a semitone, the bass generally falls a 5th; when the upper part falls by a whole tone to a close, the bass also falls a 5th. This fall of the bass, or great cadence, must be effected when chromatic notes are introduced ascending; it being the property of the new semitone, thus formed by the note rising a half tone, to imitate the key or close. By axiom 2, the proportions of sounds, and properties of the same, are deduced from the natural order of the notes. Now, by the new semitone introduced, the note below imitates the greater 7th to the key: therefore, in this case, as in a close on the key, the bass must fall a 5th.

Notwithstanding, this must be understood not of the passage of quick notes; and chiefly at a close.

Sect. 2. Of Composition in a Flat Key.

From the difference between the flat and the sharp key which lies in the different places of the semitone, there will arise a variety in the composition in a flat key, yet resting on the principles and demonstrations delivered in the last section.

The places of the semitone in the sharp key are the 4th and 8th. In a flat key, the semitone stands in the 3d and 6th places. The variety in the composition will happen where the semitones are concerned. For, as the middle close is made in the sharp key on the 4th, which is the semitone; or, as the 4th in the sharp key hath (by demonstration of the harmony of the 4th) its own concord for harmony: so the middle close in the flat key is made on the 3d, which is the semitone; or the flat 3d will have for harmony its own concord. Now, as the 4th hath its 3d and 5th for harmony, (which are the 6th and 8th of the key;) so the flat 3d will have its 3d and 5th, which are the 5th and 7th of the key.

Again, the flat 7th of the key being the 5th to the 3d, will, like the 5th of the sharp key, have for harmony its own concord. This will cause the 2d of the key to appear as the sharp 7th to the 3d, and the 4th of the key as a 2d (which it really is) to the 3d. Thus the whole harmony will be new modulated by the power of the semitone. Again, the flat 6th being the semitone, a middle close may be made on that note; and then the same proportional variety succeeds, and new harmony, as in the former case.

Lastly, at the end of the music, where there must of necessity be a close, the flat key will have the greater 7th, like the sharp one. Of so great consequence is the semitone. Nor indeed can a close be made at all, without the passage of a semitone in one or other of the parts. No. 35.

Differences in the flat key noted.

In the first example the harmony of the 2d is the concord of the flat 7th, as being 5th to the third.

The close is made on the 3d, the bass falling a 5th.

The 4th hath its own concord, as in the sharp key.

The 5th standing in an octave, may be understood as part of the harmony of the 3d, as the 3d to the key, in a sharp key.

The 6th is part of the 4th's concord, as in the sharp key; as above in the remark on the 4th.

At the close, there is the sharp 7th, from which the bass makes the great cadence.

In this example there happen four 8ths: the first and last are absolutely necessary to ascertain the key; by the second there is a close made on the 3d; and that on the 5th, is for the sake of the air in the bass.

In the second example, the harmony of the 4th is the concord of the flat 7th, as 5th to the 3d.

The harmony of the flat 6th is it own concord, being the place of the semitone; where the bass rises a 4th (the same as falling a 5th) as on a close in the treble ascending by the semitone.

In the 3d example, these differences of the flat key are left out; and the notes set as if they were part of a sharp key: that is to say, there is no close made on the 3d; the 4th hath its chord for harmony; and the 6th is likewise part of the harmony of the 4th.

For, notwithstanding the propriety of making a close on the 3d and 6th, which are semitones; yet the composer is not under the necessity of making a close in these places in every passage; and then he is at liberty of setting the notes as in the example. This observation clearly points out the difference of composition in a flat key, and where it is to be practised.

And indeed an author, whose sole end is to please the ear, will designedly introduce a close on the flat 3d, and in as many other passages as he can, to create the variety so much desired. In these cases, the rules delivered for composition in the flat key must undoubtedly take place.

The fourth example is set to show the movement of the bass to the descending notes. The composition is the same as in the other examples.

Let us now take, in one view, the full harmony of every note in the flat key, and where the difference between it and the sharp key lies: from which we may derive some useful corollaries. No. 36.

The harmony of the

Flat 7th, 3d, Flat 6th, Flat 7th.

Cor. I. The 2d admits a 3d; then the concord is of the flat 7th.

Cor. II. The 3d admits a 5th; the concord is of the 2d.

Cor. III. Again, the 3d admits a 4th; then the concord is of the flat 6th.

Cor. IV. The 4th admits a 4th and 6th; the concord is the flat 7th.

By comparing these with the corollaries on the sharp key, it will be evident, that each note in the flat key admits for its harmony that note which was excluded in the sharp key. And therefore, that all harmony is divided between the flat and sharp keys; and wonderfully diversified by changing the places of the semitone.

From the demonstration of the harmony of the 5th, with corollary 3. on the sharp key, and the scholium 3. on the same, we gather how great a share of the harmony belongs to the 5th. For it is part of the harmony of the key, and of the 2d (which chord is its own, or that of the 5th) in both flat and sharp key: and, in the flat key, it is likewise in the harmony of the 3d.

The nature and properties of the semitone being the same in both keys, we can now more clearly demonstrate the harmony of it in the following manner.

The harmony of every semitone is the concord of the same.

The key always stands between the greater 7th below, and the whole tone, or 2d, above. Now, by axiom 2. of the theory, the proportions, properties and relations of sounds are deduced from the natural order of the notes ascending and descending: The 4th (in a sharp key,) the flat 3d, and flat 6th, being semitones, are distant by a half tone below, and a whole tone above, as is the key; therefore they have the same properties with the key. But the harmony of the key is the chord of the same: therefore the harmony of the semitone, or 4th, flat 3d, and flat 6th, is the chord of the same. Hence we raise the following axiom.

Axiom I. The harmony of every member of the concord of the key, is the concord of the key. And the harmony of every note in the compass of music, proved by the rules of harmony, is part either of the concord of the key, or of its 5th, or of a semitone. Hence variety in music is introduced by the contrary motion of the parts, and by changing the key, by bringing in new semitones. The better to illustrate this axiom, we shall hereafter in the examples set harmonical figures over every note, expressive of the chord.

First Example.

Let it be required to set a bass to this treble in G sharp. No. 37.

Harmony of the

Second Example. No. 38.

Harmony of the

Third Example. In A flat, No. 39.

Fourth Example. In G flat, No. 40.

Fifth Example. In A flat, No. 41.

Sixth Example. In D sharp, No. 42.

In these examples every passage occurs which hath been delivered in the precepts of composition.

Take notice, that in the last example, the four passages, where the harmonic figures are not set over the notes, are part of a discord; which would take place, if the composition were in three parts; and which we cannot explain till we come to figurate discant.

Sect. 3. Of Composition in Three Parts.

The harmony, or full concord, of every note being well understood, both by reading the foregoing examples, as well as making application of the rules of composition on which the examples are framed, by trial of setting basses to other airs; the next step will be to proceed to composition of three parts.

This requireth no other precept than those already delivered, touching the harmony of each note. For the third part consists of the remaining notes of each concord which have not been made use of in the composition of two parts. Yet this caution must be used, that the two upper parts stand in the nearest concord to each other: that is to say, in 3ds as much as may be, and is consistent with variety and contrary motion of them. For hereby two points will be gained: first, it will bind the harmony; and secondly, the bass, being more at liberty to rise and fall by greater intervals, will meet the upper parts at every point, and produce variety by his contrary motion. The following examples are the same set in two parts above.

Prob. Let it be required to set a bass and second part to this treble. No. 43.

Prob. Let it be required to set the ascending notes of the octave in three parts, in a flat key.

In G flat, No. 44. In A flat, No. 45. In G flat, No. 46. The harmony of the seventh bar in the last example, is altered in the repetition; though the notes of the first treble be the same.

In the first instance, the concords are of the key and 5th. In the repetition, the concords are of the 3d and flat 7th.

In the first instance, the passage from the 5th into the key, (in the 8th bar,) being the great cadence, is just.

But if otherwise a close had been made on the 3d, (in the 8th bar,) the harmony in the second instance must, for the same reason, be preferred.

Hence it will be easy to decide in all flat keys, (to which only this case belongs) when the harmony of the 3d is to be part of the concord of the key, and that of the 2d the concord of the 5th; or, when the harmony of the 3d is to be its own concord, and that of the 2d or 4th part of the concord of the flat 7th.

Hence, and from corollary 1. of the theory, and from the demonstration of the harmony of the semitone, we deduce this general theorem.

The truest harmony is produced by the whole concords taken together falling in succession, as frequent as is consistent with the approved rules of harmony, by a 5th.

We shall put an end to composition in three parts, with the following example in a flat key, being one of these above in two parts. No. 47.

The use we shall make of this example is to remark, that although the bass be altered from that which is set in the same example in two parts: yet the harmony is the same, as is evident from the harmonic figures set over each.

Secondly, The 4th having its own concord, passes into the key, or 3d, in the passage of quick notes, and where there is not a close. But where, on the 6th bar, a close is made on the 3d, the bass making the great cadence, the 4th in the preceding bar is part of the concord of the flat 7th. And thus the whole harmony falls a 5th.

We have altered the bass also to answer the purpose of the movement of the upper parts in the closest harmony. And likewise to prove, that composition of many parts differs from that of two only. A truth which every composer should always have in view. For it will be found, upon trial, that, when the music is set in two parts, if it be required to add a third, it will not be in the power of the composer to give that third part an air. A matter which ought to be studied by all means; and which, it is evident from the example, can be executed, without injuring the harmony in the least, by composing the three parts together.

We therefore recommend it to the practitioner to make himself perfect in the composition of two parts, before he engages in three; as he will thereby not only sooner become master of the harmony; but also, by discovering more clearly the difference we are pointing out, will execute the composition of three parts with more ease and propriety.

Sect. 4. Composition of Four Parts.

In composition of 4 parts, every note in the concord is taken; or to every note there is full harmony.

The fourth part, or tenor, now to be added, consists of the remaining note of the concord, which was not used in composition of three. The octave therefore will take place in the concord of every note. The consecution of which, as well as of 5ths and 4ths, is to be avoided between the same parts. The rules already delivered in the composition of three parts must be attended to in this.

Example of the ascending notes of the octave in composition of 4 parts. No. 48.

Second Example.

In the descending notes of the octave in composition of 4 parts. No. 49

Third Example in 4 Parts.

In A flat, No. 50.

Fourth Example in 4 Parts.

In G flat, No. 51.

In composition of 4 parts, it was said, that to every note there is full harmony. Notwithstanding, in the first example, the sixth notes of the first treble and tenor are in unison; each being a 6th to the bass; so that the octave hath no part in that concord. This is done to avoid the consecution of 8ths, by the succeeding note of the tenor, whose place must be, for the air's sake, the 8th to the bass, as well as to bind the harmony.

In the second example, the seventh notes in the first and second trebles are in unison; both being a 5th to the bass. Let not this be understood to be a consecution of 5ths, as they are members of the same chord; but is done for the sake of the air of the second treble. Let this remark serve for every like instance which may happen hereafter.

In the fifth bar of the third example, the second note, the second treble and tenor are in unison. This is done to avoid the consecution of 4ths, which, had the tenor kept his place, would have happened from the foregoing note between the first treble and tenor.

In the same example, there is a consecution of 5ths in the two next bars; by the tenor falling a 4th, and the bass rising a 5th. This seeming error is tolerated, since it is effected by contrary motion of these parts. For as well as it is by the contrary motion of the parts, that the consecution of perfect concords is avoided; so for the same reason, the sameness of the harmony disappears, or escapes the ear; especially in composition of many parts.

By this reason, the consecution of 4ths is prevented by the bass rising a 3d, according to the first observation on this example, at the 5th bar. For that would have happened by all the parts descending; that is, not having contrary motion. The second note then of the bass in that bar is changed from that which is set in the same example in three parts.

A few general remarks occur in this place, from comparing the composition of four parts with that of three.

First, Whereas the perfect concords have place in some part of the harmony of every note in music of 4 parts; so the chances of the consecution of 8ths and 5ths being more frequent, the more skill and attention will be required to avoid them.

Secondly, Secondly, Composition of 4 parts differs in many particular passages from that of three, though the general precepts of harmony belong to both. For, by comparing the same example in both cases, there will be seen a variation of some passages in the lower parts. The necessity of complying with the established precept of variety, by preventing the succession of perfect concords, hath caused this alteration. See the 5th bar of the example in A flat in the three different compositions.

Hence arises a new reason for saying the composition of many parts differs from that of few, or two only. Therefore, in whatever number of parts the music is to be composed, one design must be first laid down; and to adjust and perfect the harmony, and to create as much variety as possible, the whole work must be planned at once, and executed agreeable to that design.

Lastly, Of the tenor in particular, we have this to remark, That, whereas in composition of three parts, there is often a liberty left of taking any one of the concordant notes to the bass; in four parts, the fourth or tenor coming in leaves no room for that liberty; but obliges us to a certain disposition of each member of the harmony, and by this means holds together the parts, the octave everywhere sounding and binding the inner notes together.

This remark on the tenor is more particularly true at almost every close, where the tenor note is the 8th to the bass on the last note but one of the close; and, by keeping its place, while the bass making the great cadence falls a 5th, the same tenor note becomes its 5th. Thus the two concords are held together and entire by the tenor's not removing.

Sect. 5. Composition of Five Parts.

The four concordant notes answering exactly to four parts in composition; when a fifth part is to be added, it is evident one note of the harmony must be repeated in every concord. The fifth part therefore consists of the notes which are by turns repeated in each of the former, in which the avoiding the consecution of the perfect concords is to be observed as before; and the air of this part attended to as far as may be consistent with the rules delivered.

Example of composition of 5 parts in the ascending notes of the octave. N. 52.

Second Example in 5 parts. No. 53.

The two octaves between the tenor and bass on the sixth and seventh notes of this example, are allowed; as the parts do not move into other notes, or make a new concord.

Third Example in 5 parts. No. 54.

The consecution of 5ths between the tenor and bass is admitted, as they meet by contrary motion of the parts.

Fourth Example in 5 parts. No. 55.

It is observable from these examples, that the most difficult composition is that of 4 parts. The other 4 parts, consisting of a repetition of one or more of the concordant notes of the first four parts, are more easily contrived, nothing more being required than to avoid the consecution of the perfect concords between any two parts.

Vol. III. No. 82.

Sect. 6. Composition of Six Parts.

In music of six parts there is a repetition of two concordant notes. The sixth part therefore consists of the notes which take place in each of the five former, by turns.

An example or two will sufficiently illustrate this,

Example of composition of 6 parts. No. 56.

There is a consecution of 8ths between the fourth line and bass, on the 3rd and 4th notes; but it being the effect of contrary motion is admitted.

Second Example in 6 parts. No. 57.

The two 8ths between the tenor and bass are allowed; for, as they do not move, they are in effect but one.

Third Example in 6 parts. No. 58.

The consecution of 8ths is by repetition of the same note, and therefore reckoned as one.

Fourth Example in 6 parts. No. 59.

In the 8th bar there is a consecution of 8ths between the third part and tenor effected by contrary motion of these parts.

Sect. 7. Composition of Seven Parts.

In composition of seven parts, three notes of the harmony are repeated in each concord. The seventh part therefore consists of the notes which are taken by turns from each of the six former; or, which is the same thing, from the first four; under the restriction of the rules concerning the consecution of 8ths, 5ths and 4ths between any two of the parts, unless produced by contrary motion of the same, or repetition of the notes, or in the octave, as said above. The seventh part is written in the tenor cliff, and is a second tenor to the first; so that, like the upper parts, it must stand in the nearest concord to the first tenor, or next part.

First example of composition of 7 parts. No. 60.

Notwithstanding it hath been said, that three notes of the harmony are repeated in each concord in seven parts; yet it doth not appear in every instance in the example. The reasons are, that in every close, whether middle or final, it is preferable that most of the parts should end in the concord note of the close, and especially of the last close, that the harmony of the key may make the deeper impression on the sense.

Secondly, The air of each part should be consulted; for this will not only justify, but demand the changing of one note of the harmony for another.

Again, as the parts next each other should stand in the closest concord; so, in order to effect this sometime by contrary motion, they will meet in unison; and therefore the repetition of the three notes will not take place in every chord.

These rules will be sufficient to answer any doubt, or determine any choice to be made of any note of the concord, as well as justify the meeting of the parts in unison, in music of any number of parts whatever.

Second example in 7 parts. No. 61.

Third example in 7 parts. No. 62.

Sect. 8. Composition of Eight Parts.

The eighth and last part is the second bass; concerning ing which the following rules and observations must be premised.

If the musick be composed for voices and instruments in full choir, it will be elegant and proper that the second bass stand in the nearest concord with the first, after the example of the trebles and tenors.

The reason is, that the two choirs singing either together, or in responses, will thus exhibit greater variety.

After this manner we shall set the two following examples.

If the musick be for instruments only, the difference of the basses consists in two things: First, The organ hath the figures of the thorough bass written. Secondly, The bass viol performs the solo parts, while the organ rests. And in full concert the two basses move in unison. This is the manner in which the basses of instrumental musick are set by the most approved masters.

We shall in this place offer our opinion on the subject of two basses in instrumental musick, relating to some alteration from the usual method of practice described above; which, as being perhaps new, will be received according to the notice it deserves.

We would have the part for the organ move in long notes, and by the least intervals; the figures filling up the harmony and discord; while the part for the violoncello moving in quicker notes, and greater intervals, becomes descant to the other bass. Of this an example shall be given when we come, in the next place, to treat of plain descant.

To return: In musick of 8 parts, the four notes of every chord are repeated, (allowing the exceptions remarked above.) Therefore, the full harmony of every note is double. The due mixture of which, according to the rules delivered, and the contrary motion of the parts, produce all the variety which harmony without discord is capable of affording.

First example of composition of 8 parts. No. 63. Second example in 8 parts. No. 64.

Having given examples sufficient for instruction in composition of harmony in the several parts of musick, and having illustrated in the same examples what hath been said concerning the harmony proper both to flat and sharp keys; we shall proceed to make such observations on composition in general, as may assist the practitioner in the application of the rules at the beginning, or first attempts.

And first, concerning the consecution of perfects; which must be avoided, except in contrary motion of the parts, or repetition of the concord in the same notes in each part, or in the octave.

Let the intervals which constitute the octave be remembered, as hath been said above (in p. 321.) namely, a 5th and 4th, a 6th and 3rd, a 7th and 2nd; taking care, that when the note of one part rises or falls by one of these intervals, the note of the other part should not fall or rise by the interval which is the complement of the octave.

Thus the consecution of 8ths will be easily avoided, and much trouble thereby saved in setting the parts.

Again, by the same caution, we avoid the consecution of 5ths. For the notes set in the concord of a 5th, rising and falling together in the different parts, by the same constituent intervals of the octave, will likewise meet in a 5th; and, by avoiding such movement of the notes of a 4th, the consecution of the same is in the same manner prevented.

One instance of each will show this evidently. No. 65.

The consecution of perfects, it is true, is tolerated, when effected by contrary motion of the parts. But these observations are made for the sake of a beginner, that he may not too often incur the abuse of this liberty.

The next observation is concerning the harmony of the 4th in a flat key.

The 4th in a flat key is either part of the concord of the flat 7th, or hath for harmony its own chord.

The harmonical figures over the examples point out this to sight. Notwithstanding, it may be asked in what case either harmony is to be preferred.

We shall endeavour to ascertain this matter upon the principles on which what hath been already taught is demonstrated.

In page 430, col. 1. we deduced this general theorem; That the truest harmony is produced by the whole concords taken together, falling in succession, as frequent as is consistent with the approved rules of harmony, by a 5th.

Therefore, when a close is made on the semitone, or flat 3d, the harmony of the 4th or immediately preceding note, must be the concord of the flat 7th. For thus the whole chord, or harmony, according to the foregoing theorem, falls a 5th.

And this theorem extends to the harmony of every note whose interval is a semitone, or which stands a half tone above and a whole tone below its contiguous notes, whose movement into the next chord must be falling a 5th.

The places of the semitone in the harmony of the key, are the key, the flat 3d, the 4th (in a sharp key,) and the flat 6th, and not in the harmony of the key, wherefore a semitone is introduced by the addition of a sharp or flat, whereby a close may be made on the semitone above.

Thus the truth of the first axiom of the practice is established; where it is said, That the harmony of every note in the compass of musick, proved by the rules of harmony, is part of the concord of the key, or its 5th, or a semitone. For the flat 7th is to the flat 3d a 5th.

In the other case, when a close is not made on the flat 3d, or when the harmony need not fall a 5th, then the 4th may have for harmony its own chord. And here the 4th stands generally in the bass. Thus No. 66.

The proof of this depends on the relative proportion of the flat and sharp keys; and will be given, by that analogy, under the article of transposition.

The last observation is in respect of the practice of authors of instrumental musick, in composition of many parts.

Whereas in the composition of 7 and 8 parts, the harmony of the notes are doubled: this is effected by the common practice, after the most easy manner, by doubling whole parts; that is to say, by two alternate parts moving through every note in unison, when in full concert; and likewise other two alternate parts. Thus the first and third violins play in unison; and the second and fourth. And, when these parts are not thus doubled, the third and fourth parts rest. Or otherwise, in some passages they take part of the harmony from the other parts, as in the examples above of 7 and 8 parts; excepting only in longer notes than the upper parts. The concertos of Corelli, Geminiani, and the overtures of Handel, are instances of this.

In a word, whatever form the parts of music may be disposed in, the principles of harmony are the same.

And when the rules of composition in counterpoint, which is the ground-work, are well understood, and confirmed by practice, the remaining part will become easy in proportion as the composer will find himself more at liberty to dispose of the parts to such advantage as he will judge most suitable to the genius of the music he is about to compose.

**CHAP. II. Of PLAIN DESCANT.**

The second manner of composing is when the succession of concords is by notes of different lengths in the several parts. It differs from counterpoint, not in the principles of harmony, but only in the form.

The effect of descant is variety; which is produced, either when two or more notes of one part are set against one note in another, or when a long passage in one part is set against a single note in the other. This last manner is properly called descanting on that note. Or lastly, When a subject is set in the bass, and constantly repeated; while at every repetition of the same, there is a variation in the treble, which diversifies the harmony, but doth not deviate from the rules of art.

This bass, which is the first written part, is called a ground bass; and the piece of music is called a ground.

Again, in descanting, it is usual for the parts to relieve each other; the bass sometimes holding the note, while the descant is in the treble; and again, the note is held in the treble, while the descant is in the bass.

The finest descant is where the discords are introduced in the passage of the notes. For here the air is less constrained; and the variety, in respect of the harmony, greater. This is figurate descant, which shall be treated of in the next chapter.

To return to plain descant: The movement of every part is more free than in counterpoint; and not so easy and unconstrained as in figurate descant. From this then arise its chief uses.

In the first place, it is the best introduction to the practitioner, to give an air to every part of his music: It is also the ground-work of inventing variations in the treble upon a plain subject in the same part. The rules for plain descant are these.

Every note in the descant must be one of the harmony of each note in the bass; as demonstrated in composition in counterpoint.

Secondly, If the descant be variation on a given subject, in the treble, the original air must be preserved as much as possible in imitation of the same.

The harmony then being the same as in counterpoint, the difference being only in the form or length of the notes, one example, after so many given in counter-

point before, will be sufficient to illustrate this part. No. 67.

Whatever the subject of descant may be, whether a ground bass, or air in the treble, (the descant on which is called variations,) the practice is the same; as in the example, where the bass is the ground to the four trebles, the uppermost line of which may be called the air; and the other three descant on the bass; as well as variations on the subject, or first line.

For the more easy execution of both kinds, take these following rules of practice.

If you are to raise descant on a ground bass, then on this supposition the bass is first framed. Next let plain notes in the treble be set, as in the example, though no other use were to be made of them than to guide the composer's eye, and thereby furnish matter more readily for a better air and for the descant.

If the subject be an air in the treble, on which you are to make variations; as, in this case, the air is the first part written, so it is the object on the book to which you are to attend constantly as a pattern for the variations or descant.

It would be advisable also to set a plain bass to the treble or plain song, before you begin the variations. For, as the bass, or second note, in many cases determines the concord of the note; it thereby afflicts and rules the descant to be raised.

In general, when the two parts are set in plain harmony, the descant ought to imitate, and not depart from that design. If otherwise a discord be introduced in the composition of the two plain parts, or a discordant note be brought in the treble or air, the descant must take part of the discord.

This properly belongs to figurate descant. Notwithstanding, it is an elegance common in practice, to throw in a discordant note in the variation, which is not in the plain song.

But these rules are addressed only to beginners.

Having done with plain descant, we shall here give an example of what hath been offered (p. 332, col. i.) relating to the manner in which we would have the two basses let in composition of many parts; which is, That the part for the organ should move in long notes, and by the least intervals; the figures filling up the harmony and discord; while the part for the violoncello, moving by quicker notes, and greater intervals, becomes descant to the other bass.

The manner of setting the two basses depending on the principles of plain descant, and implying nothing more than what is contained in the last example, one instance of this will sufficiently answer our intention here.

Example of two basses in composition of many parts. No. 68.

The variety will be still greater if this manner be pursued in figurate descant. For as undoubtedly that is the best and most perfect composition where discord is intermixed; so there is no variety, which music is capable of, produced from the form or disposition of the parts, that will not receive improvement from the more perfect composition. The last example therefore, and what hath been said of the two basses in plain descant, is meant as an introduction to a trial of the same in the more perfect, or figurate descant, both in the composition and performance.

Chap. III. Of FIGURATE DESCANT.

Figurate descant is the mixture of discord and concord, by notes of the same or different lengths or time, in the several parts.

Every interval in musick which is not harmony, must be discord.

The discords therefore are six: namely, the lesser and greater 2d; the sharp 4th, or flat 5th; the lesser and greater 7th; and the 9th. The reason for repeating the 9th, which is the 8th to the 2d, shall be shewn in its place.

The use of discord is twofold: To give a better air to every part of the musick; and to create variety. For the discords standing in the natural order of the notes, between the concords, afford an easy passage of the same; and, at the same time, mix with, and bind the harmony.

Discords are introduced in composition several ways.

First, When the notes passing in the natural order, two, three, or more of one part are set against one of another part. This passage of the notes is said to be by diminution; as in the following example.

Example of discords in passage by diminution. No. 69.

When the treble descends, the discords descend likewise into the concords; that is, the 9ths pass into 8ths, the 7ths into 6ths, and so forth. The same thing happens when the bass ascends.

When the treble ascends, or the bass descends, the contrary happens; that is, the 2ds pass into 3ds, and the 7ths into 8ths, or the discords into concords, according to the natural numbers.

When a single discordant note is set, the change of that note into a concord is properly called the passage of the discord.

When the accompaniments are set along with the discordant note; that is, when the whole discord, either in composition of many parts, or in figures in the bass, is expressed, the change of the same into a concord is justly called the resolution of the discord. The passage of the discord in single notes moving according to the natural order, as in the last example, is evident.

The accompaniments and resolution of the whole discord depend on certain principles; on which we shall, in its place, demonstrate the same.

To proceed, then, on single discords.

The second way in which discords are used in composition is, when the notes of each part move alternately, a long note between two short ones, so that the note of one part breaks off and ends in the middle of the note of the other part. This is called syncopation or binding; for the frequent mixture of the discord here supports and binds the harmony;

As in this example. No. 70.

In this manner the air of either part is less constrained, by the constant return of the discord, and passage into the concord. In some places this happens by the natural succession of the notes by diminution, as in the first example, though not so frequently as in the last manner by syncopation. But there is a passage of the discord into the concord, formed by the notes moving by greater intervals.

This musick is preferable on account of the great variety produced by this unexpected, and, we may say, surprising mixture of the discord and harmony. For variety itself causes new pleasure, when it is least expected, or when attended by novelty.

This moving of the notes by greater intervals, is the third way of introducing discord; and is the effect of the discords and concords constantly meeting by contrary motion of the parts.

In this manner the variety arises from the passage into concords, different from those which must succeed, either in the natural order of the notes, or by syncopation.

The variety also is greater by the constant succession of discord and harmony almost through every note.

For here the composer is at liberty to pass into any concord he pleases; and to resume any discord. For as the passage into the perfect concords, between two parts, cannot be effected, but from the nearest discord, when the notes move in the natural order; so, when the notes move by greater intervals, there is opportunity for many passages, which could not take place in any other way.

All this will be evident, when we come to understand the resolution of the discords.

Example of the more perfect mixture of discord and harmony. No. 71.

In this example are set forth the two first ways of using discords; namely, by diminution and syncopation, as well as passing by greater intervals; being set according to the rules relative to each manner. Where the discord and harmony move by greater intervals, there the passage is from discords new and unpractised in the other two.

The composition, where this liberty is taken, does most justly challenge the name of ornamental or figurate descant.

And the musick, wherein the discord is used these three several ways in their turn, must be esteemed the best, as exhibiting greater variety than could be expressed the other ways only. Let it be remarked in this place, that this new passage of the discord is effected by both parts generally moving by a semitone; the power of which will be seen when we shall have demonstrated the accompaniments and resolutions of the whole discord.

There is a fourth way wherein discords are admitted in composition. This is when discords succeed each other; or, where there is no passage into a concord. This is setting discords note against note. It is to be done two ways. First, when the discord passes into one of another denomination, in the natural order of the notes, by contrary passage of the notes of each part, of the same or nearly equal quantity.

This passage of the discord is necessary, as we cannot ascend or descend by the degrees of a great interval; but the intermediate discords will take place, and thence oftentimes two will succeed each other. This liberty is to be used chiefly in short notes, and by diminution.

Example of discords succeeding each other, or set note against note. No. 72.

Secondly, Discords are admitted, note against note, when the same discord is often repeated. This liberty is taken with the discord of the flat 7th above all others; and is most justly practised in musick of three parts. When this discord is brought in successively in two parts, the complement of the chord ought to be written in the treble.

Example of the discord of the flat 7th successively. No. 73.

This example is taken out of the eleventh sonata of Corelli's fourth opera.

In this example, it is remarkable, that the first and second trebles furnish by turns the discord to the bass; which constantly descends by a 5th, while the intervals of the upper parts are 5ths and 4ths to the bass alternately. Observe, when the note in the bass is flat, the discord will be the sharp 7th.

The imitation of this passage may be learned by inspection of the example. The demonstration depends on the demonstration of the third resolution of the discords following; which we must therefore reserve for that place.

An example of this passage shall be given when we come to reach the use of discords in musick of three parts.

Having done with the single discords and their passages, we proceed, in the next place, to the complex ones; by which are meant the same discords with their accompaniments and resolutions.

Now, as, by axiom the second of the theory, from the natural order of notes, the properties, proportions and relations of sounds, which arise out of their various combinations and successions, are deduced; we shall demonstrate the properties of the discords upon the same principle.

The first property of the discord is the notes which are to be played in the thorough bass, in concert with the discord. These notes are called the accompaniments.

On the exact knowledge of these depends the second property of the discord; namely, its passage into a succeeding concord.

This passage is called the resolution of the discord, as mentioned above.

Each discord hath its own distinct properties. Therefore the definition of discord already given is just; where it is said, that discord consists in certain variable proportions of the distance of sounds.

As two single notes standing at a certain interval, form the discord, so they may easily be resolved into the proper succeeding concord, as we have already shewn.

And on instruments which have not keys, no more than the two notes can well be performed. Yet, as the resolutions of the discords cannot be demonstrated without the knowledge of the accompaniments, we shall consider the whole discord together; and demonstrate the accompaniments of each particular discord, after the same method we have proved the harmony of each note of the octave in counterpoint.

Vol. III. No. 83.

THEOREM. Since every interval in musick is discord or harmony, the accompaniments of most discords will be harmony in themselves; for thus they will be discord to the given note. But it will also happen, that some note of the accompaniment in other cases will also be harmony to the given note, yet the whole accompaniment discord in itself. For the soul so accords with harmony, as not to bear an entire perfect discord.

Now, as more or less of discord with the given note prevails; so the discords are naturally divided into proper and inharmonic.

A proper discord is the concord of some member of itself, and only discord in part with the bass or given note.

An inharmonic discord is an absolute discord in itself, and partly concord to the bass or given note.

There are five proper discords; namely, the lesser and greater 2d, the sharp 4th (or flat 5th,) the sharp 7th, and the 9th.

There is one inharmonic discord; which is, the flat 7th. It hath three places in the compass of the octave; where it appears in three different forms.

It is called inharmonic; not only because it is an absolute discord in itself, but also because it is not the accompaniment to the bass note, from whence the order of the discords is traced in the natural series; except in one place or form, which is the second; wherein the flat 7th is the uppermost note of the chord. This will be seen most clearly, when we shall have gone through the discords of each kind in their natural order, in the table of discord and harmony. No. 82.

We proceed therefore to the demonstration of the discords. And, according to the 2d axiom of the theory, shall begin with the demonstration of the accompaniments of the 2d.

As in the demonstration of the concords we begin with the key-note, which we considered as an immoveable point, from whence our calculations were to proceed; so we shall here consider the bass, or lower note of the discord, that immoveable point; and the upper discordant note the interval in question, whose properties are to be found.

Demonstration of the accompaniments of the 2d.

The accompaniments of the 2d are the 4th and 6th to the bass or given note, or the discord of the 2d is the concord of the same.

The 2d is a proper discord: Therefore the accompaniments of the 2d are its 3d and 5th. But the 3d and 5th to the 2d, or discordant note, are to the given note the 4th and 6th; therefore, the accompaniments of the 2d are the 4th and 6th.

Example of the first discord, or discord of the 2d. No. 74.

Proper discord.

The discord of the 2d must be a proper discord: for the 3d and 5th to the bass with the 2d would be intolerable discord, seeing they are three notes in the natural order, and the 5th and 7th is the harmony of the 2d; therefore they must be the 6th and 8th, which is the gi- ven note: But the 6th will have the 4th; therefore the discord of the 2d is a proper discord.

By the 2d axiom of the theory, the properties, proportions and relation of sounds are deduced from the natural order of the same. Which axiom is extended to the discords, as they are combined of the natural notes, and differ from the concords only in form.

On this axiom, then, we are to investigate the next succeeding discord. The 2d discord is the 2d, 4th, and sharp 7th to the given note. For these are the next succeeding discordant notes.

Demonstration of the second discord.

The 2d and 4th cannot have the flat 7th; for they are harmony, or concord of the flat 7th; and the 8th is the given note: therefore it remains, that the second discord is the 2d, 4th, and sharp 7th to the given note.

Example of the second discord. No. 75.

Inharmonic discord.

This is an inharmonic discord; being an absolute discord in itself. It hath but one concording note with the bass; which is the 4th. This 4th is the flat 7th to the given note's 5th: which 5th is the bass to this discord; the given note in this place being considered only as a point, or unity, from which we are to investigate the next discordant notes, according to the 2d axiom.

The property of this inharmonic discord, or flat 7th, is, that its own discordant interval, or that which is formed by the accompaniment, is always a sharp 4th, or flat 5th, which distinguishes it at first from every other discord. And every inharmonic, where-ever found, hath the same property... The resolution also of every inharmonic is the same; as we shall see, when we come, in the next place, to shew the resolutions of the discords.

The next discord, according to the 2d axiom, is the sharp 3d, 5th, and flat 7th to the given note. This is also an inharmonic, or flat 7th; and having the same property with the former, namely, the flat 5th, must not be accounted a new discord. No. 76.

Inharmonic discord.

This is the inharmonic discord in that form, whose accompaniments are relative to the bass, or given note.

The third discord is the 3d, 5th, and sharp 7th to the bass, or given note.

Demonstration of the third discord.

The sharp 3d, 5th, and sharp 7th, must constitute the next discord. For the flat 3d, 5th, and flat 7th, are harmony, or concord of the flat 3d; and the 8th with the 3d and 5th, are the chord of the bass note; and the flat 7th, with the sharp 3d and 5th, are the inharmonic last mentioned; therefore, the sharp 3d, 5th, and sharp 7th, are the 3d discord.

Example of the third discord. No. 77.

Proper discord.

This is a proper discord, being the concord of the 3d to the bass; and the sharp 7th the discordant note.

To proceed then according to our 2d axiom, the next discordant notes in order, are the 4th, 6th, and 9th: But these being the notes which constitute the first discord, varying only in place and name of the 9th, for the 2d, are in effect the same discord.

The next succeeding discordant notes are, according to our well known axiom, the 4th, sharp 7th, and 9th. But these likewise constitute the 2d discord in like manner, as was said in the former case; and therefore cannot be reckoned a new discord.

To proceed then by our axiom: The next ascending notes, by the smallest intervals, are the sharp 4th, 6th, and 8th. This is an inharmonic, or flat 7th; its flat 5th being formed by the sharp 4th and 8th; therefore no new discord. No. 78.

To go on, the next discordant notes will be found the sharp 4th, 6th, and 9th.

Demonstration of the fourth discord.

From the proof of the last inharmonic discord, the sharp 4th and 6th can form a proper discord with no other interval but the 9th; for the 7th would produce three notes in the natural order, and intolerable discord. Therefore the fourth discord is the sharp 4th, 6th, and 9th.

Example of the fourth discord. No. 79.

This is a proper discord, being a concord in itself, and only discordant to the bass note. The discordant notes of it are the sharp 4th and 9th.

The next which presents itself, is the 5th, sharp 7th, and 9th, by the same axiom.

Demonstration of the fifth discord.

The 5th will admit no other discordant notes but the sharp 7th and 9th. For the 8th and 10th make the concord of the bass note; and the sharp 7th and 10th is, with the 5th, the third discord already proved; and any other note would be double discord, and intolerable: therefore, the fifth discord is the 5th, sharp 7th, and 9th.

Example of the fifth discord. No. 80.

Proper discord.

This is a proper discord, being a concord in itself; and discordant only with the given note. Its discordant notes are the 7th and 9th.

We have purposely referred the discord of the lesser 2d to the sixth and last place, 1st, Because, as the interval next above the key is always a whole tone, we cannot, according to our 2d axiom, erect this discord as relative to the given note, or key; as we have done the other five. 2dly, The resolution of this discord will be found different from that of the greater second; for reasons which will abundantly appear, when we speak of the resolutions. This discord may properly be called the discord of the semitone.

Demonstration of the discord of the semitone.

The discord of the semitone, or lesser 2d, is, like that of the greater 2d, or whole tone, the 2d, 4th, and 6th. The demonstration is the same as that of the greater 2d, and therefore need not be repeated here.

Example of the sixth discord. No. 81.

This is a proper discord, like that of the greater 2d, being being a concord itself. Its note discordant with the bass is the 2d.

It hath been said, that all harmony is divided between the flat and sharp keys.

The mixture of discord and harmony enables us to extend the like observation in this place much further.

Hence the following corollary.

The composition of all music, of any number of parts whatever, is divided between the harmony of the flat and sharp keys, and the just mixture of discord with it.

To illustrate these truths, we shall set in one view every concord and discord, in the whole compass of music, in their natural order. No. 82.

Hence we shall derive some useful corollaries, which will lead us to discover what is next to be considered, the second property of the discords, or their resolutions into the concords.

The manner of reading this is as follows:

This concord is the concord of the key.

This concord is the concord of the 2d to the key, or given note.

This concord is the concord of the flat 7th.

This discord is inharmonic, and so forth: descending still from the uppermost lines of harmony, or discord, to the lowest line, or bass.

In this view is seen the mixture of discord with harmony, each in the natural order. Wherein, indeed, nothing regular or proportioned appears to fight. The reason of this is evident from the demonstration of the harmonical proportions. For, if they be of a species different from all other proportions, as by corollary 2d of the theory, and must be demonstrated on principles peculiar to them; then the succession of the discords, constantly taking place between the intervals of harmony, must be disproportioned too. This appears to fight in the next example, or view of harmony and discord in the natural order. No. 82.

However irregular this may seem, an uniformity prevails through the whole, which supports that variety in music so desirable: Without which variety, there could have been but one concord among sounds; a sameness prevailing through the whole; without semitones, and consequently without discord. In this case, music never could have existed as an object of pleasure to the sense; much less of science.

This admirable structure is raised on the power and property of the semitone, which shall be the subject of the following corollaries.

Cor. I. Every semitone in the octave hath either a concord or discord proper and peculiar to itself. Yet, the natural succession of the concords and discords is not according to the ascending and descending semitones. For, it is evident, in the annexed table, that the corresponding bass notes constantly descend by 3ds, the variety, at the same time, shining throughout the harmony and discord in the upper parts, ascending by semitones. Yet the bass expresses every semitone in its passage by 3ds, uniformly to its period.

This most strongly illustrates the truth of the 4th axiom of the theory; namely, that the concords and discords, either in their natural order, or arranged by art, will not suffer us to depart from the established precept of variety amidst uniformity.

The same uniformity, or rather unity, is exhibited still more plainly in the 5th discord, in the coincidence of discord and harmony in the same individual sounds.

For this discord, which is the discord of the 9th, is also the harmony of the 5th.

This is truly admirable, and furnishes us with the most interesting remark in the compass of music, as in the following corollary.

Cor. II. The scope of music, and motion of the parts, must at length terminate, and meet in one invariable thing Harmony.

Thus are we arrived at the full extent, or bounds of music. It is fit we now return to make such further observations as will lead us to the knowledge of the resolutions of the discords, which is the next thing to be spoken to.

The division of discords into proper and inharmonic, we have made for the sake of clearness and method. The difference already pointed out between the discords must be remembered; which is, that the property of the inharmonic, or flat 7th (which note does ever, with another note of the chord, frame the sharp 4th, or flat 5th) is the same, in whatever place or form it is met with; whereas the proper discords essentially differ from each other, and in every particular.

The three inharmonic therefore, in the natural order of the discords, are not so properly three, as the same discord in different light; where it is a preparation for a close on the key, and on the 4th and 5th to the key.

The bass to the discords moves by 3ds descending in a sharp key.

The notes of the bass, corresponding to the proportions of the flat key, have no relation to the discords in the line next above; but are the bass to the concords in the flat key, as demonstrated in the rules of harmony.

The two discords, which are a repetition of the first and second, are set down in compliance with the 2d axiom, to pursue the natural order. And hence they serve to demonstrate there can be no other discord than those exemplified in the scheme. For there is no semitone in the octave which doth not appear there to have its discord or harmony connected with it.

In this scheme then is comprised every interval of music, with the members of each chord respectively, both discord and harmony, in the natural order.

From the same order, we shall demonstrate the passage of the discords, into the concords or resolutions of the same.

In the theory, it hath been said, that the semitone is the principle or hinge, on which turns the resolution of every discord.

On this principle, then, we shall now demonstrate the same.

The discords stand in the natural order between the concords; but every note of the chord is not equally near respectively.

From the idea of harmony, which is fitness or propor- The passage of the discord must be to the nearest concord; therefore, the resolution will be by the smallest interval, that is, by the semitone.

This is the general theorem for the resolution of every discord. We will now apply it,

The resolution of the discord of the 2d is into the concord of the given note.

The 4th, or semitone, will move into the 3d, but the 3d will have for harmony its 3d; therefore, the 6th must descend into the 5th, and the 2d's passage by the nearest interval will be into the given note. By these passages is formed the chord of the same; therefore, the resolution of the discord of the 2d is into the concord of the given note.

In a flat key, there are two passages by semitones; that of the flat 6th into the 5th, and of the 2d ascending, by contrary motion, into the 3d.

Example of the resolution of the discord of the 2d, or first discord. No. 83.

Proper discord.

The resolution of this discord being into the given note, the bass does not move.

The resolution of the second discord is into the concord of the given note.

The second discord is the 2d, 4th, and sharp 7th. The sharp 7th and 4th move by the semitones and contrary motion into the 8th and 3d, while the 2d falls a 5th into the 5th. These are the concord of the given note; therefore the resolution of this discord is into the concord of the given note.

Example of the resolution of the second discord being inharmonic. No. 84.

Inharmonic discord.

The resolution of this discord being into the concord of the given note, the bass ascends by a semitone.

The passage of this inharmonic by contrary motion of two semitones, and the other note falling a 5th, is the resolution of every inharmonic, wherever introduced. This therefore needs no repetition. But if the succeeding concord has a flat 3d, the passage is by two ascending semitones; the 2d rising into the 3d, the 7th into the 8th, and the 4th by a whole tone into the 5th. This movement can only happen in the resolution into the key.

The other, of much more extensive use, is the true resolution of the inharmonic discord; and more interesting, as, by its contrary motion of the semitones, it better binds the harmony.

It is necessary here to explain further the nature of the inharmonic discord.

The inharmonic discord, then, is always the chord in a sharp 3d, with a flat 7th; which two notes frame the interval which characterizes this chord, namely, the sharp 4th, or flat 5th; when the flat 7th is the upper note of the two, the interval is the flat 5th; when the sharp 3d is the upper note, the same interval is called the sharp 4th. These notes being relative to the fundamental note either of them determine the chord.

As the inharmonic is found in different places of the octave, so consequently the note of the chord must vary accordingly; the second inharmonic therefore only, in the natural order, hath reference to the given note in the table, as that happens to be the note of the chord. For the given note there is to be accounted only a point or unity, from whence we proceeded to trace the discords in their natural order, as they lie between the concords. The chord note, therefore, of the first inharmonic is the 5th to the given note; and of the last inharmonic, it is the 2d to the same. Now, as the whole chord falls a 5th in the resolution, so the first is a preparation for a close on the key, the second for a close on the 4th, and the last for a close on the 5th; now, as any note of the chord may stand in the bass, so the third is often preferred before the chord note, for the sake of the movement of the bass by a semitone, as well as because falling a 5th in the bass is more properly the part of harmony.

The flat 7th is likewise chosen for the bass note; for the same reason, the movement of the bass by a semitone descending; which is no inconsiderable use of discords. For in figurate descant, as we have said, all the parts move more freely.

The next discord is likewise inharmonic. It is the sharp 3d, 5th, and flat 7th to the given note; which note is likewise that of the chord. No. 85.

Note, that the resolution of every inharmonic being into its 5th below the chord, the resolution of this will be into the 4th of the given note; as rising a 4th, and falling a 5th, answers the same thing in estimating the intervals of harmony.

Inharmonic discord.

Example of the second inharmonic discord and its resolution: here the bass note is the note of the chord; therefore, in the resolution it falls a 5th, which is the 4th to the given note.

The flat 7th descends by a semitone into the 3d, the 3d rises by a semitone into the 8th, and the 5th falls a 5th into the 5th of the concord. The resolution therefore is into the 4th of the key.

This is that form of the inharmonic discord on which the composition of the passage taken out of Corelli, (No. 73.) is grounded. Observe, that in the cited passage, and also in every like passage, the two notes of the bass move to only one note of the second part, which becomes the flat 7th by this movement of the bass.

Thus the flat 7th is given, in the upper parts by turns, to every note in the bass, as hath been before remarked.

Resolution of the third discord.

The third discord is the 3d, 5th, and sharp 7th, to the given note; it is resolved into the 6th to the same. For the 7th ascends into the 8th or 6th's 3d, the 5th rises a whole tone into the 6th, and the 3d not moving becomes the 5th. The resolution of this discord therefore is into the chord of the 6th.

Example of the resolution of the third discord. No. 86.

Proper discord.

This is a proper discord; in the resolution of which the bass falls a 3d, while the whole discord falls a fifth. The next discord is the sharp 4th, 6th, and 8th. It is inharmonic. Its resolution is into the chord of the given note's 5th. No. 87.

Resolution of the third inharmonic discord.

Of this discord the bass note is the flat 7th; it descends by a semitone, while the whole chord falls a 5th.

Inharmonic discord.

Its resolution is the same as that of every inharmonic, in what form soever, by the contrary motion of the two semitones, while the third note falls a 5th.

Resolution of the fourth discord.

The fourth discord is the sharp 4th, 6th, and 9th. Its resolution is likewise into the chord of the note's 5th.

For the sharp 4th ascends into the succeeding concord's 8th, the 6th passes into the 3d, and the 9th not moving becomes the 5th. Those are the chord of the note's 5th.

Example of the resolution of the fourth discord. No. 88.

Proper discord.

This discord is a mixture of the proper and inharmonic. It is a proper discord, for that the notes of the treble are concord; and inharmonic, in respect of the bass, with which it makes the discord of the sharp 4th.

It differs from the foregoing, where the flat 7th is expressed in both treble and bass; whereas, in this, it is only in the bass.

The bass here also descends by a semitone, while the chord falls a 5th.

Resolution of the fifth discord.

The fifth discord is the 5th, sharp 7th, and 9th. It is resolved into the concord of the bass note.

For the 5th is that note's 5th; the sharp 7th ascends by a semitone into the 8th; and the 9th (or 2d) passes into the 3d. Thus it is resolved into the chord of the given, or bass note.

Example of the resolution of the fifth discord. No. 89.

Proper discord.

In this resolution the whole chord falls a 5th, while the bass stands still, or descends into the octave.

This is plainly the last discord in the order of sounds. Its resolution is into the given note or key, by the passage of the great cadence, or descent by a 5th. It is a concord in itself; and is in harmony the concord of the 5th.

In this chord discord and harmony are united. When it stands in discord with the bass, the bass doth not move in the resolution; when it sounds perfect harmony with the bass, then the bass descends a 5th.

Therefore we conclude, Harmony and discord are like two finite lines, whose beginnings are at a certain distance; and in the natural progression converge constantly, until they meet in a point.

The discord, which we have reserved to the sixth place, is that of the lesser 2d, or semitone.

Its places in a flat key are the 3d and 6th; and in a sharp key the 4th and 8th, or wherever a new semitone is introduced.

It is a proper discord, being a concord in itself, whose chord hath always a sharp 3d.

Its properties are everywhere alike; but its resolution differs from that of the greater 2d, for the reason assigned in the resolution of every discord; that is, the passage by the semitone.

Resolution of the lesser 2d, or semitone.

The discord of the lesser 2d is the 2d, 4th, and 6th to the bass; or, the concord of itself.

Its resolution is into the concord of its own 3d or 5th. It rises into the concord of its 3d by the single passage of the semitone descending. And into the concord of its 5th by the 4th descending along with the semitone.

Example of the resolution of the discord of the semitone. No. 90.

In the first resolution, the chord rises a 3d, and the bass falls a 5th. In the second, the chord rises a 5th, and the bass falls a 3d, the reverse of the former.

From the resolutions of the discords we derive the following corollaries.

Cor. I. There is no interval of harmony that is performed by the bass in the resolution of one discord or another.

Hence we may conceive that harmony regulates even the discords, and presides in every part of music.

Cor. II. The inharmonic discord, or flat 7th, is a preparation to a close on a key, the 4th and 5th, flat 3d and flat 6th; for into the harmony of these it is resolved; they being the intervals on which closes may be made according to the established rules of melody. And universally, wheresoever a close may be made by introducing a new semitone, the preparation may be made by the flat 7th, or inharmonic discord.

This discord, being of such extensive use, will deserve some further remarks, which may render the setting of the same more easy, and assist the performer in the taking and resolution of it.

In the inharmonic discord, then, are three notes chiefly concerned, which are the note of the concord; its 3d, (which is always sharp;) and the flat 7th: either of these may be set in the bass. Hence there will arise three varieties.

If the note of the chord be the bass note, the figure is the flat 7th; the chord is that of the same note; and the bass falls a 5th.

Secondly, When the 3d is the bass note, the figures are flat 5th and 6th; the chord is that of the 6th to the same 3d; and the bass ascends by a semitone.

Thirdly, If the flat 7th stand in the bass, the figures are the sharp 4th, 6th, and 9th; the chord is that of the 2d to the bass note; and the bass descends by a semitone.

Example. No. 91.

The 5th of the chord may likewise stand in the bass; but as the movement of the same is by a whole tone descending, it is very seldom used.

The figures are sharp 6th.

Flat 3d. Cor. III. Hence the bass ascending or descending by a semitone, furnishes an opportunity of introducing notes in the upper parts, which will constitute the inharmonic discord.

And again, the sharp 3d of any chord in the treble, or any note having the addition of a sharp, and thereby becoming the greater 7th, may be the 3d of an inharmonic; the bass taking the flat 7th. For the sharp 3d of the chord, (which is the sharp 4th to the flat 7th in the bass, or elsewhere,) and the flat 7th, in whatsoever part they are set, in bass or treble, or both in the treble, constantly move each his own way; the first ascending, and the latter descending by a semitone.

These are the simple discords as they are found to lie in the natural order of sounds between the concords; whose accompaniments are, for the most part, harmony among themselves. It is evident, from the method in which we traced them, that there is no other discord among sounds. Nowwithstanding, from the combination of two simple discords, a new form of discord may be framed, which taketh part of the inharmonic and discord of the semitone; which, therefore, we call the compound discord.

This discord is the sharp 3d, flat 7th, and flat 2d, or semitone to any note whose chord hath a sharp 3d.

It is resolved, by the passage of three semitones, two descending, and one ascending, into any concord with a sharp 3d; and therefore may be introduced as a preparation to any concord, in either flat or sharp key, where the greater 3d is.

Example of the compound discord, and its resolution. No. 92.

The resolution of this discord, as it is compounded of the discords of the flat 7th, and semitone, will partake of the resolution of the same. Thus the upper note, or semitone, descends into the 5th of the concord; and the flat 7th and 3d meet by contrary passage of a semitone each into the 3d and 8th part of the resolution of every inharmonic, while the bass descends a 5th. Thus the passage into every note of the concord is by a semitone; so great a favourite of nature is the semitone.

By changing the form of this discord, it will be resolved into a chord with a flat 3d, by one semitone descending, and two ascending; the two extreme notes of which are the same as in the example above; but the middle note is the 5th ascending into the 3d, instead of the flat 7th descending. The upper notes therefore form the flat 5th, or inharmonic interval. No. 93.

The properties of those discords, and of the inharmonic, furnish us with some practical observations; which are, that the two discordant notes of these discords are, in music of two parts, a preparation to, or pass by contrary motion of the semitones into the concords of the sharp 3d, flat 6th, and 5th. When the flat 7th is the uppermost note, and the lower the sharp 3d, they pass into a sharp 3d.

When the upper note is the sharp 3d, and the flat 7th is below, they pass into the flat 6th; so do likewise the flat 7th above, and the chord note below.

Lastly, the two extreme notes of the compound discord pass into the concord of the 5th.

Example of the passage of discords in music of two parts. No. 94.

In like manner there is a passage into the octave from a discordant interval; the upper note of which is part of a concord, and the lower the semitone of the compound discord. No. 95. Or the contrary.

There is no passage by two semitones into the flat 3d and sharp 6th from any discordant interval, except the semitone. For, in the passage of quick notes encountering each other by contrary motion, this, or any other discordant interval, may fall into the concords. But such, being tolerated only for their quickness, need not be reduced, as indeed they cannot, to any rules of art.

Lastly, the bass will admit two notes together, each concording with it, namely, the 5th and 6th, and making a discord between themselves.

This discord, which differs from the proper and inharmonic, is rightly called the mixed discord; each of the two notes being in harmony with the bass, and discordant to each other.

The framing of this discord depends upon the rules of harmony, and may be let to any note of the bass which hath for harmony its own concord, and is likewise the member of another.

Therefore, in the sharp key, the key and 5th, and in the flat key, the key 3d, 5th, and flat 7th, admit a 5th and 6th.

When we shall have proved, under the article of transposition, the 4th in a flat key, and 6th in a sharp key, to have their own concord; they will be found, no doubt, to have the privilege of admitting a 5th and 6th.

For thus it is understood. The key hath a 5th in its own right, and a 6th as member of the 4th.

The 5th hath a 5th in its own right, and a 6th as member of the key.

The flat 3d hath a 5th in his own right, and a 6th as member of the 3d.

Thus the 5th and 6th will stand together to the bass.

The properties of sounds in the natural order may be transferred by art, and improved into all the variety possible; as this is no other than an imitation of nature.

Hence we infer, that every note, which assumes the nature of the key, by the addition of the greater 7th, will admit a 5th and 6th.

This 5th is easily distinguished from the 5th of the inharmonic, which is always an imperfect one, and ought constantly to have a flat prefixed.

Now the chord of the inharmonic with a flat 5th is that of the 6th to the bass note, as hath been said. But the chord of the mixt discord may be better understood to be the chord of the bass note, with a 6th added.

Sect. 2. Of Figuring the Bass.

Having delivered all that hath fallen under our observation concerning the nature, proportion, and use of discord; we shall now make an application of the same, in order to explain the figuring of the bass; the next article proposed to be spoken to.

First, Of figuring the concords. That the notes whose harmony is their own chords, need no figures, is evident from the definition of harmony; which consists of one certain, invariable proportion of sounds.

These are the key, flat 3d, 4th of a sharp key, 5th, flat 6th, flat 7th.

They which, as members of other chords, require the figures of harmony set over them, are these following; and are reduced to this general rule:

The 3d of every concord hath a $\frac{6}{7}$; and the 5th of every concord hath a $\frac{5}{4}$.

Therefore,

| Key | Key 2d | |-----|--------| | Flat 3d and sharp | Flat 5d and sharp | | 5th of a flat key | have a 4th of a flat key | | have a 5th of a sharp key | have a 5th of a sharp key |

Lastly, The 4th in a flat key, when it has its own chord, must have a 5th set over it, to distinguish the chord from that of the flat 7th. And the 6th in the sharp key, when it hath its own chord, must have a 5th likewise set over it, to distinguish the chord from that of the 4th.

The proof of the 6th in a sharp key, having for harmony its own concord, depends on the relative proportion of the flat and sharp keys; as will be shewn in the chapter on transposition. So much for figuring the bas in concords.

Let us now inquire into the shortest and clearest method of figuring the discords. This will be no difficult matter, when we consider well the whole discords, as they are full figured in every example.

There the figures set over the bas express the intervals which the notes in the upper parts form with them. These taken together make the whole discord. And as these, with the harmonic chords in succession, express the whole composition, they are therefore called the thorough-bas.

To render the performance of the thorough-bas easy and expeditious being the chief intention of figuring the bas; this will be best answered by distinguishing the discords, which have some figures in common with each other, by such figures only as will strongly mark each discord. For though all the figures set down in the examples be necessary to demonstrate the properties of the discords, and truth of the composition; the case is quite otherwise in respect to the sight; many marks causing perplexity and confusion; when one single mark in this, as in all other cases, best discovers the difference. The proper discords then being concords in themselves, the figure, or figures, discording with the bas note, will distinguish each of these.

The inharmonic discords being the same in different form, will be distinguished by the discording figures peculiar to each form.

Of the properties of this discord, and manner of taking the same, we have spoken sufficiently. We shall therefore only set down the discording figures of each form, in the following example.

Example of the proper discords figured for taking the same at sight. No. 96.

The inharmonic discords figured for taking the same at sight. No. 97.

Moreover, the proper discords being concords, each in itself; every discord will be concord to some note different from the bas, or discordant note.

To remember this note will render the taking of this discord ready at sight, it appearing in this light a chord of harmony.

The same is in some measure true of the inharmonic discord.

Therefore the figures in the examples, under the bas lines, express the names of these concords relative to each.

It remains to be observed, that the accompaniments of the 2d are the $\frac{6}{7}$, and of the 9th the sharp $\frac{7}{8}$, as they appear in the natural order of the proportion of the same, (No. 82.) It will be necessary to demonstrate here the truth of this.

The 4th and 6th being the accompaniments of the 2d, the fifth and sharp 7th are the accompaniments of the 9th.

For the proportion of the 9th to the 2d (whose 8th it is) is $2:1$. And the proportion of the 4th and 6th $\frac{7}{8}:\frac{5}{4}$.

Therefore, by the rule of proportion, say,

$$\text{as } \frac{1}{\frac{5}{4}} : \frac{2}{\frac{7}{8}} : : \frac{3}{\frac{5}{4}} : \frac{3}{\frac{7}{8}};$$

but the 4th number, or $\frac{3}{\frac{5}{4}}$, is the proportion of the 5th and sharp 7th; (for $\frac{1}{\frac{5}{4}}$, give $\frac{3}{\frac{5}{4}}$;) therefore the 5th and sharp 7th, are the accompaniments of the 9th. Q.E.D.

The last use we shall mention of the discord is the furnishing the inward parts of musick, in composition of many parts.

As, in harmony, each part of the four takes one of the concording notes, and hence, by the continual mixture of these by their contrary motion, the composition is framed; so, in figurate descant, the notes of the discord furnish the parts in their turn.

And the same notes, passing by semitones chiefly, form among themselves that resolution which the bas performs alone.

The composer therefore, when, setting a bas, he introduces a discord, is as well prepared for the notes of the other parts in this case as in setting a concord; and if he be well skilled in the resolutions, will with great ease compose the succeeding concord.

For the thorough bas, being the whole composition in one view, the knowledge of the one and of the other must be the same. As, then, he who understands the rules of harmony and the discords, and their resolutions, will succeed with ease in composition; so, on the other hand, he cannot be a skilful composer who is ignorant of the properties of harmonical chords and discords.

One example in three parts, being the same set above in two, will sufficiently illustrate this.

Example of the use of discord in the composition of many parts. No. 98.

**Chap. IV. Of MELODY.**

HITHERTO we have considered musick in its several parts taken together, or the art of composition. Our next business will be inquire into the method of framing a single part, or making the melody.

MELODY. Melody is the air of the uppermost or first part in musick, commonly called the tune.

In a plain song, the air is formed without considering the relation which the other parts, which may be set in composition with it, may bear. For, being first framed, and for the sole end of pleasing the ear and fancy; it must, it is evident, be independent of them.

For as to framing the bass first, and setting the treble to it, there appears no necessity either in reason or the rules of composition; they equally serving the purpose of beginning with any part, no part being privileged with any particular member of discord or harmony; as is abundantly manifest from the various positions which the discordant notes have been shewn to stand in; as well as from the 4th axiom of the theory, which establishes variety for conducting and rendering even harmony acceptable; a sameness in the successive concords being the only thing exceptionable in that part of composition.

All the parts of musick then being equally concerned in the composition; to prefer any one part, as a basis, or unerring guide, on which to erect the musick, or bring in the parts, is doing injury to that liberty which nature and the rules of art put us in possession of.

But the air of the first part so essential to the tune, or rather the tune itself, compels us to decide in favour of framing the treble first. In which it will be found impossible to succeed, when it is confined to what the bass, if it be first framed, must of necessity prescribe.

This preference in framing the treble first, chiefly respects a plain song, or air. For, in more elaborate pieces, where the design of the author is imitation of passages in the several parts by turns, according to his choice or fancy making use of the same liberty, he will take any for the leading part, and accordingly write the passage in that part, and finish the composition in the rest.

The air or first part in instrumental musick is called the first treble; the air for a single voice is called the voice part, or song; and in musick for many voices, the upper part is called the counter-tenor.

In this musick, the air of the tenor, and of every part performed by the voice, is studied with more exactness than the inward parts of instrumental musick.

The reason for this difference is, that in instrumental musick, the first violin generally presides, or leads the musick by its air: as this is the composer's design, the other parts must of necessity be accommodated to it.

Whereas in musick for voices, every voice repeating the same words, that is, expressing the same sense, at the same time, or immediately succeeding; nothing can defeat the end of the musick so much, which is the setting of words, or rather sentiments, to notes as expressive of the sense as inarticulate sounds can possibly do, as for one part to excel the others so much in this necessary point, as by comparison to depreciate, weaken, or alter the sense in the others.

The air, therefore, of every part in vocal musick must be consulted; not only for the sake of harmony, (for a good air in each part improves even the harmony;) but also for the sentiment sake, without which the musick must be absurd and dissonant.

Notwithstanding the liberty which every one may justly challenge of framing an air agreeable to his own fancy; yet it cannot be said, that this liberty is uncontrollable, or beyond the power of art to prescribe bounds to: For then indeed every strain composed by even a bad and injudicious ear might stand in competition with the most finished pieces. But as this will not be allowed on any hand, even an undistinguishing ear conceiving a degree of pleasure in hearing good musick; so there is no doubt but that there must be some precept or manner found out by experience, to ascertain and conduct the air or strain, and which will render it to a good and judicious ear plainly preferable.

The rules therefore which we shall lay down for melody, are such only as are founded in truth and reason; the result of experience, joined to skill; and which are admitted in every liberal art: These are unity, imitation, and order. If it shall be said, that persons unskilled in musick, but otherwise very capable from a natural good ear, will sing an air which an artist cannot find fault with, we confess it may be in some sort true.

But the strains of such composers are always very short; and as they seldom or never depart from the key, so they afford not that variety so desirable in musick: Nay, what is this but saying that the rules of art are conclusions taken from nature, as in truth they are; so then they must be assuredly right? This must be so, when the appeal is made from art to nature.

As to those essays called voluntaries, there was never a good one performed but by a good master. The musick was always good in proportion to the master's skill in the art; in proportion to the variety he introduced according to the rules of art. Therefore even voluntaries are the effects of knowledge and deliberation.

But to return. The first rule of melody is unity.

The unity of tune is said to be in respect of the key, and of the subject.

Every tune must be written in some key, in which it must begin and end.

As every air is said to be in such a key as is the last note, especially the last note of the bass; so there is the same necessity for the first and last note of every air to be some member of the concord of the key. This discovers the design of the author. Having thus fixed the attention of the hearer to this particular, the ear and imagination can no other way be satisfied than by holding to and executing the same design.

The unity of tune is as necessary in this respect, as consistency in the words and sentiments of an orator is requisite to discover the scope and meaning of his discourse.

Secondly, the unity of tune, in respect of the subject, signifies, that there should be one subject of every piece of musick repeated and insisted on, as often as conveniently can be, throughout the whole piece.

And this repetition will be in proportion to the length of the tune, and design of the composer. Even in a minuet, or any other exact piece confined to a certain number of bars, the repetition of the subject may be effected. Now, the subject of every air, or piece of musick, is the first passage of the same, for any number of bars, bars, be they more or less, as it shall happen; every tune being stamped with some prevailing idea or fancy peculiar to itself, and therefore distinguishing it from every other.

The subjects of grand pieces of instrumental music are contrived with care and study; and invented with design to enlarge or descant upon at will; not being confined to any length, or certain number of bars. Such pieces, being the efforts of great and masterly genius, afford all the pleasure that design and invention, carried on by every masterly stroke of art, can give.

The second rule of melody is imitation. As in the executing other arts, a similitude and proportion of the members ought to be preserved; so imitation, or a repetition of the most striking passages, answers to this in music.

Imitation may be performed many ways. First, when the repetition of the passage is made, beginning on the note above the leading note of the passage; or on the third, fifth, eighth, or any other interval.

A passage also may be imitated in any of the descending notes. A repetition on the octave below is frequent in every good author.

In the repetition of passages, there are two varieties.

The first is, when the passage is repeated in notes belonging to the harmony of the key. It will seldom happen in this case, that the passage will in the repetition be precisely the same, in respect of the intervals of the notes, though the movement be an exact imitation.

The reason of this will be evident, if we consider that the intervals in both flat and sharp keys respectively ascend by different degrees; the semitone changing the intervals almost continually.

See example, No. 12. in the theory.

In these examples no more than two flat or sharp thirds succeed each other. And where they do succeed, the semitone is in a different place in the two like intervals of flat thirds; it being the third of one interval, and 2d of the next ascending 3ds, or the contrary.

In the sharp key, two sharp 3ds ascend from the 4th and 5th, and is the flat key from the 6th and 7th. The inequality of the flat 3ds, and of the few instances of their succession, is owing to the places of the semitone.

To the inequality of the 3ds is owing the inequality of the 4ths, 5ths, and every other unequal interval in the course of the notes; the greater necessarily partaking of the inequality of the lesser, which is included in it. All this is evident.

Therefore, the repetition of a passage will not be precisely as the passage, except in the places abovementioned; that is to say, a repetition of sharp thirds from the 4th and 5th of the sharp key; and of the same, on the 6th and 7th of the flat key. And in the sharp key, there may be an imitation in the compass of six notes ascending; namely, from the key and its 5th. We have been particular in remarking the want of exactness in imitation on notes belonging to the key. Not that we mean to mark it as a defect; for it is beyond doubt, that every passage in the harmony of the key must be pleasing whether it be a perfect imitation or not.

Besides, this dissimilitude, arising from the place of the semitone being changed, is so far from being chargeable with a defect, that, as hath been often said, it produces that sweet variety which is founded in the principles, and which every artful pursuing will succeed in; as in this he doth no other than copy after nature.

These remarks on the imitation of a passage in the notes of the harmony of the key, will lead us to the second manner of imitation; which is such, as that every note in the repetition stands exactly in the same interval respectively as the notes of the first passage.

This then is a perfect imitation. Which, as it cannot take place in the harmony of the key, except in the few cases abovementioned, it must be effected by art; that is, by altering the places of the semitones in the key, so as to correspond with those in the original passage, by marking a sharp for the semitone ascending, if the repetition be in notes above the passage; and a flat for the semitone removed lower, if the repetition be in the descending notes.

In this manner there can be a perfect imitation of any passage of any length whatsoever, and of any compass: in every instance of which, the key is changed, by introducing notes not belonging to the harmony of the same.

As every interval of the first passage must be preserved in the repetition: it will sometimes happen, that many flats, or sharps, must be added to the notes in the repetition. The rule of this practice will be well understood, when we shall have learned the art of transposition; the repetition of any passage in this manner being no other than a transposition of the same into another key. In regard to this perfect imitation, we have one remark; which is, that if a repetition be made on the note next above, and repeated again the note still higher, it will have a good effect; for this will create such a novelty in the strain as is surprising; besides that it affords the author an opportunity both of making new descant or enlarging on the subject in this new key, as well as of shewing the greatest skill by returning from that digression into the original key with art and propriety.

This will be no difficult matter to one who understands well the art of transposition. Now, the repetition of the subject transposed into a key different from the original belongs to this second rule of melody; as the repeating the subject in its own key respects the rule of unity.

The repetition of the subject after these two manners, and throughout the several parts, as treble, bass, tenor, and so on successively and constantly, each part taking it up immediately, or as soon as the repetition is finished in another, whereby the several parts seem to move in pursuit of each other, is called a fugue.

Music composed on this design is justly esteemed above all other, not only on account of its excellent contrivance, but for the sake of the pleasure also which it affords.

In a just fugue is represented all the variety possible; at the same time that an uniform progression of the parts is preserved throughout the whole, without the least discovery of the signs of art.

The reason for this may be, that the repetition of so interesting a passage as the subject is, is so natural to the imagination imagination and ear, as not to be easily distinguished as the effect of art. The construction of a fugue will be understood, from this description, to be in the following manner.

The subject being first written in that part which the composer intends to be the leading part, the same must be set down again in the next part wherein the repetition is appointed to be made, either in unison, or on the 4th or 5th to the key or subject, the 2d, or any other interval; in which matter the composer is at liberty.

Yet the repetition on the 4th seems more natural to the flat key; as, on the 5th, it is to the sharp key.

If the music be in two parts only, the subject being written in each part in succession; the next step will be to frame descant to that part where the repetition is, and which therefore will be written in the leading part. Henceforward the parts move on at liberty, that is, nowhere repeating the subject, but expressing all the variety in descant each to the other, which the fancy, invention, and skill of the author suggest, until the subject is again repeated, either in the key, or some interval of the harmony of the key, or perhaps in a new key.

The imitation of this must immediately follow in the other part, in union, or otherwise.

If the music be of three or four parts, let the subject be first written in every part, in succession; and in the order you intend. Then fill up with descant the second bar, or more, of the leading part; that is, as far as the subject reaches in the other part; and proceed likewise on the next repeating part with other new descant; and so on through every part, until all the staves are equally full.

After which the parts move at liberty, as before in a two-part fugue, until another repetition of the subject.

But where, or how frequent, the repetition of the subject may be made; or on what interval, whether above or below; or by what succession of the parts, (for they need not preserve the order they began in;) is neither the business, nor in the compass of the rules of art, to prescribe.

In these matters, the composer is as much at liberty as his genius and invention can furnish matter and variety. So that in some places the subject may be repeated continually, in the different parts, on intervals and in a key different from the original key or order.

Sometimes, the movement of the subject being the same, the notes are changed from ascending to descending, or the contrary. Sometimes even the movement to the contrary. At other times, a new subject is introduced; and then it is called a double fugue. And lastly, for the sake of variety, the subject is repeated backwards, or inverted; so that the parts seem to pass each other by contrary motion, instead of pursuing.

In a word, there is no passage which expresses variety, which may not be introduced in a just fugue; while the uniformity is preserved in the imitation of the same, and returning the original subject, and key, towards the conclusion or close of the piece.

We shall only add, that if the descant which fills up the bars be constantly written in all the parts successively and in order throughout the whole piece, the fugue is, from the exactness of this repetition, called a canon. In framing of which, observe, that if the canon consist of three or more parts; when the third part takes up the subject, the descant in the leading part must be part of the harmony of the other two.

What remains to be spoken to on this second rule of melody, or imitation, is the method of returning into the original key, after a passage in a remote one. This will lead us to consider the half-tones not belonging to the harmony of the key, or chromatic notes.

For if a passage, or repetition of a passage, be in a new key, which is the imitation we are now speaking of, the returning the key immediately will be by chromatic notes; descending if the repetition were above; and if the repetition was in notes below the passage, we may ascend into the key by chromatics likewise, or half-tones ascending.

This is evident. For every key, flat or sharp, having its semitones in their proper places, a passage is not in the key, when the semitones are out of their places.

Therefore the returning into the key, from a passage not in the harmony of the same, must be by removing the new half-tones.

This depends on the knowledge of transposition.

The new semitones introduced in a passage, or imitation of one, we have called chromatic notes; because every semitone not belonging to the harmony of the key are to be found only in the chromatic scale.

Yet this is but improperly. For two, three, or more semitones succeeding each other are properly called chromatic notes; which, to music wherein the frequent use of these is made, gives the name of chromatic music.

The use of chromatic notes is to raise the attention by the uncommon and unexpected variety they produce.

For every new half-tone ascending, being understood by the ear as the greater 7th, implies a new key. Three or more semitones ascending after each other, do therefore raise the expectation of so many new keys: whereby the curiosity is greatly excited; and the expectation of the ear being gratified, in the imitation of a close, by every new semitone, the music becomes, as in all other cases where novelty takes place, truly the object of admiration.

In a series of semitones ascending, the last is in the place of the key. When chromatic notes descend, the last but one is the key, for the same reason; namely, the semitone below sounding as the greater 7th, every key being a semitone to its greater 7th or half-tone below relatively.

Chromatic notes ascending, by alarming the ear and imagination, elevate the soul, thereby imitating the sublime.

Chromatic notes descending, express the pathetic, which is free from any alarm or terror. The performance of these notes should always be with softness, which naturally removes the apprehension of terror. Ascending and descending semitones partake of the nature of the sharp and flat keys; as hath been said, concerning the power of musical sounds to touch the passions.

Chromatic notes may be introduced in many places. If, in a passage, the semitones of the key occur among others, they are to be accounted as chromatic. Therefore fore flat and sharp keys are equally capable of improvement by them.

Notwithstanding chromatic notes create so much variety and elegance, it must not be understood that they are to be introduced injudiciously, or without any address; for then they would not only be useless, but injure the music. Chromatic notes being so affecting and expressive, as we have shown, their place in vocal music will easily be determined by the sentiment.

As, on the other hand, to introduce lively or pathetic sounds, where the sense is dissonant from either, is introducing contradiction and confusion.

Neither is it natural in instrumental music to break in upon a lively strain by slowly-moving chromatic notes. Though at the end of a brisk movement, the transition is good. For music which moves in semitones, though quick notes, must appear slow to the ear, which expects the greater intervals of the diatonic scale, or whole tones.

Yet instrumental music does properly admit the mixture of chromatic notes, when they are accommodated to the genius of the strain or subject. Neither will it be difficult to judge of this propriety. For musical sounds having a natural tendency to express our ideas, the place of chromatics will readily be found by this mark; it being in the power of the composer to imagine ideas without the help or intervention of words, and to substitute these ideas in the place of words, and make them the subject of his strain. Thus he may fill his mind with the imaginary passions of love, sorrow, anger, dejection, pity, and the like; the expressions of which will be most easily distinguished by musical sounds, and varied as the subject requires. Thus, if two or more of the passions, especially contrary ones, be represented by turns, it will form in the imagination a kind of conversation between persons, which never fails to strike the attention stronger, and make a deep impression on the hearer.

Besides this, the imagination of the composer will be assisted in the invention of variety; and the different passages of the piece will be furnished with notes proper and natural to each; for the same reason that choice and expressive words flow in upon a good writer who is master of his subject. We shall only add, that if sometimes the different passages be allotted to the bass and treble by turns, it will greatly diversify the subject, mark the sentiment stronger, and thereby cause new pleasure. So much for chromatic notes. Notwithstanding, what hath been said in this place doth not respect chromatics only; but in general the whole process of an elaborate piece, in every form and transition of the melody; wherein only there is opportunity for application of what we have here suggested.

The reason why we have given place to chromatics under this second rule of melody, or imitation, we have already assigned; namely, that the use of half-tones is necessary where there is a repetition of a passage in another key. For, whatever proportion of sounds is found in the natural order, the same may be transferred by art, and improved upon every occasion, as thereby imitating nature. And this, by the way, is likewise the true reason for the resolution of discords by semitones; being taken from the original pattern, or reso-

lution by the two semitones in the natural series of sounds in the octave.

The third rule of melody is order.

Order in music is the conducting the melody or air, according to a certain rule, through several intermediate cloises, from the beginning to the final close or end of the tune.

A close is the termination of a passage in a concord; which, like a period in sense, is framed with design, and from the preparation from the notes immediately preceding, which are the whole tone above, the half tone below, and the 5th (generally the bass note) is expected by the ear. This is the description of a full close, which is ever the final close, especially of quick movements. This preparation is the concord of the 5th to the note on which the close is made; the bass making the great cadence, as we have taught in the rules of composition.

This preparation is in full harmony. But it must be remembered, that there is a preparation also from discord; chiefly the discord of the flat 7th, of which we have said enough in the chapter of discord or figurate descant. In slow movements there is a preparation to a full close; which shall be described presently.

Every intermediate close hath its preparation in imitation of the final close, more or less. For there is no necessity for the parts taking invariably the same member of the chord; so that the treble oft times makes the cadence from the 5th, particularly in quick passages; in which likewise, in the middle of a strain, many imperfect closes may occur, the parts taking the notes indiscriminately as they happen, without any preparation designed. It is enough to mention these.

But in order to conduct the air in each strain, if there be more strains than one, with propriety and method, there must be a full close in several places in the harmony of the key. These are called proper closes.

These are also closes made by the introduction of a new semitone, by the addition of a sharp, making the greater 7th not belonging to the harmony of the key. These are rightly called improper closes.

The places and order of both these we shall now assign.

The first proper close falls naturally on the key. This is not meant of the final close; for a close may be made on the key, within a few bars of the beginning; yet this close is seldom made, as not affording variety. Again, the first strain sometimes closes on the key; yet the close of that strain is more properly on the 5th; and the close next after that on the 5th falls naturally on the key.

When a close is made on the 5th, the 4th of the key being removed a semitone higher, becomes the greater 7th by the addition of a sharp. For, in imitation of the final close on the key, there must be a semitone ascending.

Notwithstanding, a close made on the 5th, with a sharp 3d always, from the chord of the 4th in a flat key, without altering the 4th, or bringing in the greater 7th, is accounted an elegance; in which passage the bass takes the flat 6th or 3d to the 4th, and thence descends by the semitone into the 5th.

This is never practised but in very slow movements. From the necessity of a semitone ascending to every full close, except this last instance, we draw this inference, That a close may be made on every semitone in the key. This ascends the places in both flat and sharp keys: in the closes of both which the key is a semitone; in a sharp key, the 4th also; and in the flat key, the 3rd and 6th.

These are the places of proper closes, or such as are made in the harmony of the key.

The order of closes is now to be considered, which, in a sharp key, may be thus. The key, the 5th; the key, the 4th; the 5th, the key.

Notwithstanding, every composer, being at liberty to pursue his own design, will prefer that order which will suit best with the manner of the air, or answer his intention. Our business is only to point out the places where closes may be made, and give a general idea of the order. For we have observed much difference in practice among the best authors; and indeed it cannot be otherwise in long and finished pieces, considering that liberty inseparable from every composer who invents; and therefore every manner which may increase variety, is to be recommended.

The order of closes, then, in the flat key may be, The key, the 3rd the 5th, the key.

Or, the key, the 5th the 3rd the 5th, the key.

Or again, the 5th the key, the 3rd the 5th, the 3rd the 6th, the 5th the key.

In pieces of a considerable length, closes may be repeated in these several places, and the succession of them altered from that wherein we have set them down above.

Improper closes, or such as are made on any other note than the key, flat 3rd, 4th, in a sharp key, 5th or 6th in a flat key, are made by bringing in a semitone not in the harmony of the key, by the addition of a flat or sharp, the note below the new-made semitone being always the greater 7th, and which thereby determines the note on which the close is to be made.

By this art a close may be made on any note; as on the 2nd, 3rd, 6th, or 7th of a sharp key; on the 2nd, 4th, and flat 7th of a flat key. A close on the 6th of a sharp key is much in use, though no semitone: this depends on the 6th having for harmony its own chord, which will be proved in the next chapter.

It remains to be remarked, that as every new key is formed by the addition of a sharp or flat; so the return into the former key, whether the original, or otherwise, is effected by taking off the flat or sharp from that note, the next time it occurs in the course of the strain.

In music of two parts, the greater 7th, or that which makes the new semitone, will not in every passage be expressed; the treble sometimes descending from the 3rd above into the close. Yet, if the bass falls the 5th, these two members of the chord do properly lead to the close.

However, it must not be understood that in every close the bass must move the same way, by descending from the 5th, as was said before; no part having any member of a chord proper to it by any necessary or natural dependance; for otherwise the second treble, or tenor, could never ascend above the first, nor the bass above either; than which nothing is more common. Custom indeed has appointed to the bass this movement at a close, for the most part, and especially at a full or final close: and justly; the descending from the 5th, being so interesting a movement, is better expressed, as well as more suitable to the grave notes of an instrument performing the bass.

There is another passage in practice, which, though not a close, yet comes properly in this place to be spoken of. It is a sudden or unexpected stop of all the parts made on a discord.

As this is generally practised in quick movements; so it is often, though not always, succeeded by a slow movement: during this stop, the ear is held in suspense by the discord, and waits for the resolution into the concord.

The suddenness and novelty of this passage recommend it. It seems contrary to a close: for as, being a discord, it hath no preparation, and, not being resolved as soon as the ear expects, seems to lose its connection with the following chord; its meaning therefore is undetermined, and the sense confused: yet it hath a good effect by alarming the imagination, resembling an affected perturbation in the order of the words and sentences of an oration.

**CHAP. V. OF TRANSPOSITION.**

Transposition is the removing a tune from one key into another. The use of transposing is to bring a tune within the compass of some instrument, or for the more easy performance on an instrument; some keys being more difficult to perform in than others; especially in wind instruments, as the German flute, &c. For as to instruments that are stopped, as the violin and bass-viol; and instruments with keys, as the organ and harpsichord; all keys are easy to a good performer, who is said to be master of the scale of the instrument.

Secondly, Transposition is absolutely necessary in music for voices and instruments, when it happens that the key in which the music is written is too high or too low for one or more of the voices. In this case, the music must be transposed for the instruments into the key which is nearest to and will best suit the pitch and compass of the voice. For as to the vocal performer, it matters not in what key the music be written for his part, provided he can sing in the cliff, (or indeed, if he can transpose, as shall be taught in the last chapter, Of singing by note, whether he can sing in the cliff before him, or not,) if the instrument be accommodated to his voice.

There are two ways of transposing. The first is, when the tune is written in another key, at any distance above or below the original, with the proper flats or sharps prefixed. When this is done, the performance on the instrument is easy; the half-tones of the key, and every other, keeping their due places expressed in the writing. This may be called transposition by writing.

The other method of transposing is by the cliff; that is to say, when the cliff is removed, or supposed to be removed, from the place wherein it stands prefixed to the tune.

This removal of the cliff at once transposes the whole, without alteration of the writing; the use of the cliff being, as hath been said, to ascertain the names of the notes. We shall shew both these methods of transposition; and first of transposition by writing. Transposition, in general, is writing or playing a tune in a different key from that wherein it is written, preserving the places of all the semitones.

If the notes in the scale of music expressed no other than whole tones, transposition would be evident to sight. For a series of tones at equal intervals would, when removed to any distance or interval, preserve their places of themselves (to speak,) or without the help of art. And the performance of the same on an instrument would be equally easy in all keys; no alteration happening thereby, but changing the names of the notes. And even this would be unnecessary.

The semitones therefore are the causes of any obliquity in the scale of transposition; and therefore the keeping them in their due places is the art we are now speaking of. As it is said in the theory, they are the foundation of writing the same tune in divers keys, which is transposition.

We must have recourse therefore to what hath been said in the theory concerning the essential difference of tune, or the different places of the semitone.

When the first semitone above the key is on the 4th, or 6 semitones to the key; the 3d, which is 5 semitones, is the greater, or sharp 3d; and the tune is from hence said to be in a sharp key.

Again, when the first semitone above the key is on the 3d, or is 4 semitones to the key, that being the lesser or flat 3d, the tune is in a flat key. Again, the next semitone in the sharp key is the 8th; and in the flat key on the 6th.

In every sharp key, the semitone must stand in the same places, that is, the 4th and 8th; and in the flat key, in the 3d and 6th.

The name of the sharp key in the scale, whose semitones are in their places without the addition of flat or sharp, is C. Hence C is called the naturally sharp key. Its semitones are F and C.

This is the pattern for transposing in all sharp keys: chiefly by remembering the letters or names of the semitones.

The key in the scale, whose semitones are in their places without the addition of flat or sharp, is A. Hence this is called the naturally flat key. Its semitones are C and F. This is the pattern for transposing in all flat keys; remembering the names of the semitones.

It may be required to transpose from any key with a sharp 3d, into any other of the same; or likewise from any flat key respectively. Notwithstanding, we shall proceed, according to our method, to show what are the keys in the natural succession into which a tune will be transposed, beginning at C the naturally sharp key. As the properties of every key will be discovered by this method; so it will answer everything that can be required in transposition; or show how a tune may immediately be transposed from any key into any other interval or key that may be required.

For instance, let it be required to transpose from the key of C into that which is next in the natural order of transposition. F is the first semitone in the key of C; which, by the addition of a sharp, becomes F sharp, and consequently the greater 7th to the semitone above it, which is G. Therefore G is the new key into which the tune is transposed. No. 99. For,

Again, let it be required to find the next key to be transposed into, from the former, or G.

The first semitone in the key of G is C; which by the addition of a sharp, which is removing the semitone, becomes C sharp, or the greater 7th; therefore the new key is D. No. 100. For,

Observe that every former sharp which is set down for the tune transposed, is included in every succeeding transposition. That is to say, C, for instance, cannot be marked sharp, unless also F be marked.

The reason of proportion demonstrates this to sight. These are the two semitones of the open keys, as they are termed; the observation will hold true of all others, as will be seen in the following instances.

To transpose out of D into the next key; the first semitone in the key of D is G. But G sharp is the greater 7th to A. Therefore A is the new key. No. 101.

Take notice, that the first semitone, or 4th in the sharp key, being removed by the addition of a sharp; the tune is hereby transposed into the 5th to the key.

This removal into the 5th, by changing the first semitone, being the same in all sharp keys, it need not be repeated in more examples. For the next to A will consequently be E, with the addition of A sharp to D.

The next B with A sharp.

The next will be F sharp with E sharp; never used.

The next would be C sharp with B sharp; never used.

The next would be G sharp with F having two sharps; never used.

The next is D sharp, with C having two sharps; which last is D natural. This key is in frequent use; but the name of the key is changed, and marked with its equivalent E flat; and the name of its greater 7th is likewise changed; and, instead of C having two sharps, is left, as in truth it is, D natural.

The next is B flat; its greater 7th A natural.

The next is F natural; its greater 7th E natural.

From this we ascend, according to the rule, by a 5th into C where we began.

Hence it is evident, that of the twelve semitones in the octave, nine are in use as keys with a sharp 3d.

Of which the following are examples in their order. No. 102.

The last three examples marked with flats may be demonstrated in another manner; which is, by changing the place of the other semitone, or 8th, by removing the sharp 7th, by prefixing to it a flat. The transposition will, by this alteration, happen in an inverted order from that in the foregoing examples. For, as the 4th or first semitone in the former manner, being removed by the addition of a sharp, became the greater 7th; so in this case, the greater 7th being removed by prefixing a flat to it, becomes the 4th of the next key. No. 103.

This manner of demonstrating the last three keys is preferable. For, by changing the first semitone, as in the former manner, it doth not so plainly appear, as that depends upon the foregoing key, which never was in use. Besides, the number of sharps to be prefixed to these keys might perplex a beginner in transposing; whereas the flats, as prefixed in the example, do more methodically follow the removing the greater 7th by a flat.

The demonstration is equally certain and clear in both manners. Let it be remembered, that in transposing, by altering the greater 7th, the key is removed to the 4th above.

Let us proceed to transposition in a flat key; the general rule of which is the same as for the sharp, to remove the semitones, and thereby preserve the proportion of the key.

The different places of the semitone will cause some variation in the effect, or interval of transposition.

Let it be required, according to our method, to transpose out of the open or naturally flat key A, into that which is next by natural succession.

The first semitone in the key of A is the 3d, or C; which, by the addition of a sharp, becomes the greater 7th; wherefore, the new key, or that into which the tune will be transposed, is D. No. 104. For,

Let it be remarked, that the greater 7th which determines the new key in this as well as in the sharp key, is never prefixed to the tune; which to the sharp key always is.

The reason is, that the flat 7th is the property of the flat key. To prefix therefore a sharp to the place of this note on the stave, or beginning of the tune, would be a constant contradiction to all the flat 7ths that may occur throughout the whole air. But the sharp 7th, when brought in at a close, middle or final, or elsewhere where there is no close, is marked particularly as occasion requires.

Another reason for not prefixing a sharp on the stave to the place of the 7th of a flat key is, that, in some keys, the 6th must have its flat set on the stave, or beginning of the writing: a sharp, then, on the place of the 7th would appear a contradiction to sight, and ought therefore to be avoided.

The removal by transposition in the flat key being always to the 4th above, as in the last example, there needs no other example at length of transposition in this key, the same proportion obtaining throughout every key with a flat 3d respectively.

According, then, to this proposition, the next key after D will be G; the 3d of the former, or F natural, becoming F sharp the greater 7th, E flat its 6th.

The next will be C. Its greater 7th is B natural, with the addition of a flat to A its 6th.

The next will be F, with the addition of a flat to D its 6th.

The next would be B flat, with the addition of a flat to G its 6th, never used.

The next would be E flat, with the addition of C flat its 6th, never used.

The next would be A flat, with the addition of a flat to F its 6th, never used.

The next would be D flat, with the addition of two flats to B, never used.

The next would be F sharp, its 6th D, never used.

The next is B, its greater 7th A sharp.

The next is E, from whence we ascend by a 4th into the first key A.

Hence it is plain there are seven flat keys in use, out of the twelve semitones in the octave. Of which the following are examples, with their proper signs prefixed.

No. 104.

It may be observed, that by changing the place of the other semitone or 6th, by adding a sharp, the transposition is by two degrees of the former at once; as from A into G, and so on. But this is no more than what hath been done in the second step of this example; yet by putting this sharp to the 6th, it is a short way of transposing into the whole tone below.

It appears from hence, that there are in all fifteen keys in use, a fund for great variety: among which you will observe, that some sharp keys have flats prefixed, and some flat keys have sharps; which cannot by this time appear strange to one who perceives the necessity of preserving the proportion of each, and who must now understand the truth of the 3d axiom of the practice, That no tune composed in a sharp key can be transposed into a flat one, nor a flat one into a sharp; for that would be altering the permanent nature of things.

From comparison of the examples, will be seen what is most worthy of remarking; which is, That whatever sharps or flats belong to any flat key, the same are likewise the property of that sharp key, which is on the same flat key's 3d: for instance, A with a flat 3d, and C with a sharp 3d; so D and F, G and B; and so of all others respectively having the same signs belonging to and prefixed to each.

Hence we collect, that the essential difference of tune consists in the form, that is, the position of the semitone, and not in the materials of music.

A truth which appeared before in the comparison of discord with harmony; and which will be of great service hereafter, in the art of learning to sing by note.

This inference furnishes us with a proof of the 4th of a flat key, and 6th of a sharp key, having for harmony each his own concord.

A proof which was wanting; as it could not be had from the rules of harmony, neither of these intervals being in the place of a semitone. The general theorem is this: As by axiom 2d the proportion of one single sound is to another according to the natural order of sounds; so the proportion of one chord to another will be according to the natural succession of chords. For a chord is no other than an unity of sounds. But it appears that the succession of chords by transposition in the flat key, is by 4ths, that is, the chord of a note, with a flat 3d; and that of the note's 4th is the same. Now, the harmony of the key or given note is the concord of itself; therefore the harmony of the 4th in a flat key is the concord of itself.

In like manner we demonstrate the harmony of the 6th of a sharp key to be the chord of the same.

And in this the proof lies nearer the truth than in the former case.

For the chord or proportion here is not only the same, but the individual sounds.

For as by comparison, as above, of any flat key with the key of its 3d (which must ever be a sharp key, the flat and sharp 3ds being the compound intervals of the 5th) the properties and proportions are not only like, but the same, the difference consisting in the form or place of the semitone; so the chord of one key will be to the chord of the other, not only like, but the same. Now the harmony of every flat key is, by the rules of harmony, the concord of itself: but the concord of the flat key is the relative sharp key's 6th; therefore the harmony or chord of the sharp key's 6th is the concord of itself.

Let us now apply these rules of transposition to the second rule of melody, or perfect imitation, which is the repetition of a passage in notes not belonging to the harmony of the key; by which notes we understand all that have a sharp or flat added, which was not prefixed to the beginning of the flave or tune.

The general rule of which is, First name the key in which the passage is written, whether the same be the original key, or that of the tune, or some other; then name the interval of the first note of the passage to the same key.

Whatever interval the imitation begins on, whether a 2d, 3d, 4th, 5th, or any other above or below the passage, it bears the same proportion to the key of the imitation as the first note of the passage to its key. Thus

G is to C as A is to D, or G : C :: B : E, or again D : A :: B : F, or D : B :: E : C.

The leading note and key being thus expressed, both of the passage and imitation, shews the proportion of the imitation above or below passage.

If the passage be in a sharp 3d, as the imitation must be too, the signs, as prefixed to one of the examples of the sharp keys, will be required to be added to the notes of the imitation; and if the passage move with a flat 3d, the examples of the flat key discover the marks wanting in the repetition.

Thus G, the leading note of a passage in C, with a sharp 3d, repeated in the note above; as A and D require two sharps, namely on F and C, the property of D with a sharp 3d.

And the same passage in C with a flat 3d, will in the note above require B marked flat, the property of D with a flat 3d:

And G C sharp 3d, transposed into a 3d, or B E, requires four sharps, on F, C, G, and D, the properties of E with a sharp 3d.

The same passage with a flat 3d, or any other passage, will, when transposed into E, require but one sharp on F:

And D B, with a sharp 3d repeated in E C the natural sharp key, require neither sharp nor flat.

But the same, or any other passage with a flat 3d, will, when transposed into C, have three flats on B, E, and A, the property of C with a flat 3d: and if there be no leading note, as it may often happen, nothing more is to be considered than the key.

Thus the use of transposition in perfect imitation is evident.

The second method of transposition is by the cliff. The use of the cliff is to ascertain the names of the notes. Therefore, the names of the notes will be changed by removal of the cliff.

Now as, in transposing, it is necessary that every interval be preserved, or that the semitones keep their due places; so by altering the name of the first note of the tune, by removing the cliff, all the other notes are altered in proportion.

Thus the removal of the cliff effects at once what was done in the other method by the transposition of every single note of the tune by writing.

This way is easier to the writer, but much more difficult to the performer.

Inasmuch as a confirmed habit in anything is harder to be changed for a new method, than it is to learn by a certain rule at first.

Therefore the performance in transposition by the cliff can no otherwise be attained, than by constant and repeated practice in all the cliffs, and in all such places as they are used to be set for convenience.

Every one therefore who desires to become a master in performance, after he is well acquainted with the three cliffs in their usual places, ought to accustom himself to perform in every cliff, in whatever place it may be set. This knowledge will not only render the performance convenient to his private amusement, by the variety with which he can furnish himself, by playing the same air in whatever key he pleases; but will also make him an useful member in a concert, by transposing at sight, whenever it may be required to accommodate the instrument to the voice. For let it be understood, that to him who is so well acquainted with the places of the cliffs, as to perform in any of them at sight, nothing more is wanting to his transposing by the cliff at sight than to imagine the cliff is prefixed to such or such a place, and commit to his memory the name of the key in which he is to perform, by transposing according to the removal of the cliff which may be most convenient.

This will fix the places of the semitones, or assign the sharps or flats belonging to the new key, as they are set in the example in the first method of transposing by the writing.

Any of the cliffs may be removed; yet the C cliff or tenor is most commonly in use for this purpose.

The general rule for transposing by the cliff is this.

To transpose into any interval above the key, remove the cliff by the same interval descending. And if the instrument be too high for the voice, to transpose into a lower key, remove the cliff to a convenient interval higher.

For, raising the cliff depresses the notes; and, contrary, setting the cliff lower raises the notes, or transposes them into a higher key, in proportion.

CHAP. VI. OF SINGING BY NOTE.

The art of singing by note is founded on the principles and practice of music. Therefore we have reserved this subject to the last.

To sing by note seems in some respects more difficult to attain than performance on some instruments. In other respects, it is easier and sooner acquired. The more time is laid out on the practice on some instru- ments, the more difficult the execution grows, in some sense, that is, according to the construction and compass of the instrument. On the contrary, all the difficulties in learning to sing by note present themselves in the be- ginning, in appearance greater than they really are; and which a knowledge of the principles of music, and a little of the practice, with a tolerable good ear, will with ease overcome.

Besides, a little time and experience will convince any one of what compass his voice is, and what degree of performance he is capable of attaining. The principles of singing therefore being well understood, there re- mains no further difficulty; no one having a right to expect he can execute more than what is within his na- tural powers.

If this art is not so commonly understood, or the knowledge of it sought, it may be owing to this, that the precepts for learning, or the manner invented, and constantly used, are more perplexed than the subject de- mands.

How far this may be true, will appear from an obser- vation or two which we shall make on the method now in use.

The art of singing by note relies on these two principles; the finding the places of the semitones, and tuning them and the whole tones of the octave aright.

The first of these has been delivered in the theory; and also in the practice, under the last article Transposi- tion. The tuning the notes is the subject we are now engaged in.

Let us first examine how far these have been prosecuted in the present method.

The notes of the octave, besides their names in the scale, have been used to be distinguished by these four syllables, Sol, la, mi, fa; accommodated to the purpose of singing by note, in the following order.

Fa, sol, la, mi, fa, sol, la.

Whereof Sol being thrice repeated in the octave, La twice, Fa twice, and Mi once; four syllables express the 8 notes.

The art of tuning by these, or assigning the places of the semitones, is by appointing to Mi the place of the greater 7th; and then Fa immediately following expres- ses the semitone or key, and the other Fa the 4th. How well forever this may answer the purpose of tuning the half-tones in a sharp key; yet in a flat key, the places of the half-tones being the 3rd and 6th, will, according to this order, be expressed by La, the semitone syllable Fa consequently expressing whole tones.

To obviate this difficulty, and reduce things to order, another place of Mi must be assigned, which is the 2nd of a flat key: for this Fa will express the semitones on the 3rd and 6th.

It is evident then, that before any half-tone or whole- tone can be tuned, the first business must be to find out the place of this mi: now how this can be done by vir- tue of the sound, or name, or order of these syllables, is not so easy to comprehend.

But admitting the place of mi, or the key, to be known by some previous precept, as indeed ought to be; yet tuning the key as the first semitone in a sharp 3rd, and the 3rd as the first semitone in a flat key, is beginning at the wrong end in the first case, and thereby not marking the essential difference of tune, which consists in the flat and sharp 3ds, the order of which is disturbed by this varia- tion of the place of mi.

From this want of marking the essential difference in tuning by these syllables, and wherein the beginning and ending is not on the key, some confusion and much trouble and untunableness must arise. And indeed it cannot be imagined, that this or any other essential dif- ference of things can be marked by the same invariable artificial signs, if they be not exactly accommodated to the nature of things. An invention that fails in this, however ingenious it may be in speculation, not being a just representation of nature, doth not merit the name of art.

For instance, if you tune eight notes, whose key hath a sharp 3rd beginning on the 5th, your seventh note, which is the 4th of the key, and therefore a whole tone from the succeeding note, will sound like the flat 7th. Again, if you begin to tune on the 2nd, your 3rd, which is the 4th of the key, is flat; and the sound in this suc- cession will appear as if you were tuning in a flat key.

And again, if you tune from the 6th, the deception of a flat 3rd is the same as in the last case.

Secondly, if you tune 8 notes whose key hath a flat 3rd, and begin on the 7th, your 3rd, which is the 2nd of the key, is sharp, and your tuning will be as if in a sharp key.

The same deception will appear if you begin to tune on the 3rd or 6th.

In a word, whatever other interval you begin on, to tune either with flat or sharp 3rd except the key, some semitones will be out of their places: This is rendering what is at first sight attended with some difficulty, more perplexed and obscure.

The ear, the judge of sounds, is deceived, and the judgment misled.

But on the other hand, the ear will naturally and easily distinguish the flat and sharp key, when the key and its 3rd are ascertained by beginning and ending on the key.

But otherwise, and where these marks are promiscu- ously used, the difference of tune, or infallible sign, will appear neither to the ear nor understanding.

We shall end these remarks with one general observa- tion; which is, that by assigning the place of mi to the greater 7th or 2nd, in order to find out the key, is resolv- ing one difficulty by a greater, and requiring to do a thing without any means of information offered to com- pass it.

For as it is true that when the greater 7th or 2nd is known, the key is known also, and again, the key being given, you have consequently the 7th or 2nd; yet to do either of these, without some intermediate helps, is taking for known the thing sought, which is directly contrary to reason.

Proceed we now to our method of singing by note.

The first principle of singing, is the finding the places of the two semitones in the octave, in any given key. This hath been pointed out in general, in the theory, where are shewn the places of the semitones in the sharp key to be the 4th and 8th, and in the flat key the 3rd and 6th.

But the particular names of the notes, on which the semitones fall in any key whatsoever, and which it is evident must depend on the name of the key, are demonstrated, and examples given, in the practice, under the article of Transposition.

We shall therefore transfer only the examples into this place, in a concise order, which will fully answer our inquiry into the names and places of the semitones.

The nine sharp keys. No. 106. The seven flat keys. No. 107.

By these examples the particular names of all the semitones are known at sight; as they depend on the name of the key.

Therefore in the example of the sharp keys; the first key being C, the semitones are F and C. The second G; the semitones C and G. The third D; the semitones G and D, the 4th and 8th of each respectively.

In the example of the flat keys; the first being A, the semitones are C and F, The second D; the semitones are F and B flat. The third G; the semitones B flat and E flat. And so on, the 3ds and 6ths respectively.

To apply this to the purpose of tuning the notes by the voice: At sight of the sharps or flats prefixed to the tune to be sung, and looking at the key-note, you have of course the places of the semitones, by referring these to the original in the examples set above.

Having thus discovered the difference of tune, you are at the same time determined whether you are to tune the notes of the octave with a flat or sharp 3d.

This tuning of the eight notes, tones and semitones, in their due order, is the first step or principle of tuning all other intervals, or of fingering by note.

It will most readily be learned by imitating another voice, or following the notes of an instrument; this is the only case wherein there is need of any foreign affluence to fingering by note.

The instrument we would recommend for this purpose is the organ or harpsichord; as the 4 or 5 semitones, which ascertain the flat or sharp 3d, succeeding each other, being visible on the keys of that instrument in any part, the learner can in this case assist himself, by striking the notes of the octave in either flat or sharp key, on any part of the instrument which will best suit the pitch of his voice, and distinctly repeating them by turns, until his ear becomes a perfect judge of the difference of the flat and sharp 3d, as well descending as ascending, and his voice perfect in tuning both.

As musical sounds will be best expressed in tuning by articulate ones; we shall, to answer this convenience, take the four syllables already in use.

As we shall apply them to another purpose than they serve at present; so the order or manner we shall dispose them in, will be altogether different from that.

In tuning, then, the notes of the octave with the instrument, let the syllables be expressed with the notes, in the order of the following examples.

In G sharp. No. 108.

Now, since the flats or sharps adjusting the semitones of any sharp key are exactly the same which belong to the flat key respectively on the 6th, as we have said before in comparing the examples of flat and sharp keys in transposition; therefore the eight notes ascending in a flat key will have the syllables annexed to each, as in the following example on the 6th, without disturbing or departing from the order of the sharp.

In E flat. No. 109.

The semitones and the tones below them being distinguished by the syllables fa and mi, in their respective places in both keys, for descending as well as ascending notes, is the sole use we intend by these syllables; the tuning of the notes, which is to be learned by the instrument, being entirely independent of them.

For tuning the descending, notes then, there need no other examples than the two above written; for reading the same backwards will serve this purpose.

When the ear becomes well acquainted with tuning the notes of the octave by the instrument, it will then be proper to sing the same looking on the notes written on the book; and this should be done in every example of both keys. And let it be remembered, that tuning the notes thus in the natural order, should be to a beginner the prelude to singing any song proposed.

The next step will be, before the learner attempts to sing any part of a song, to tune by the notes the greater intervals, both concord and discord.

The general rule for which is, Tune all the notes of the interval in the natural order, ascending if the interval ascend, and descending if the interval descend. Then immediately tune both notes of the interval, beginning with the concords.

Thus. No. 110.

Concords in succession.

The discords are. No. 111.

The semitone being the distance between the 3rd and 4th, is already known by tuning the notes of this interval.

Note, The name of every greater 7th introduced by a sharp prefixed, is mi.

Next tune the concords of the thirds in succession. In this manner. No. 112.

The 4ths and 5ths being all perfect and like, except one of each, need no repetition.

The 6ths in succession are tuned thus. No. 113.

Lastly, mix the discords and concords as they stand in the natural order; than which nothing will better confirm the just tuning of the intervals, when these rules are to be applied to future practice.

In this manner. No. 114.

This line may be tuned various ways; as, secondly, beginning still on the left hand, tune the 3d and 2d notes, reading backwards; and so on, each two under the slur.

Again, beginning on the right hand, tune the uppermost note and second downwards; then the first and third; and so on, still missing one, and omitting constantly the G, or key not below.

And lastly, beginning still on the right hand, tune the second and third, the third and first, the fourth and first, and so on, omitting the G or key note constantly.

The practice of tuning the notes descending of all these examples, is by reading the same backwards.

The tuning the greater intervals in the flat key depending in like manner on tuning the eight notes in succession, according to that series; it is unnecessary to set examples of the same.

The same method of practice equally serving this key, except that the syllables annexed to this key must be repeated, as in the proper example 109.

As in this example of the intervals of the 3d, 6th, and 7th, wherein this key differs from the sharp. No. 115.

The general rule of tuning the intermediate notes of each interval first likewise taking place here.

In order to establish these rules in the memory, and render them of immediate service to the practitioner, especial notice must be taken of the flat and sharp thirds, as also of the flat and sharp sixths, in what places they stand, or how they succeed each other in the order of the key.

The not attending to these differences being the only obstacle that can stand in the way of singing at sight, see them set down at large in the theory, and in the examples of this chapter, No. 112, 113.

When these are well recorded in the memory, together with the sharp 4th or flat 5th, the art of singing by note will not appear so mysterious. This knowledge of the intervals at sight will render the syllables of little or no use, as hath been observed, and especially if words be set to the airs you intend to practice; which we would advise.

When the interval of each note is known at sight by constant practice, and the sound of every interval become familiar to the ear, and thereby distinguished immediately upon hearing the same, the learner may make an essay to sing by note some plain song; which is no more than tuning the same intervals, with which he is supposed to be well acquainted in the foregoing lessons.

For as to any other article of knowledge requisite to the performance of the song; as the time of the movement, and lengths of the notes, and the like; if the practitioner hath not been acquainted with them by practice on some instrument beforehand, the principles of them have been delivered in few words in the introduction to this essay.

But besides that this is not the place for speaking of these matters, so neither is there occasion for this knowledge in the very beginning, in strictness of speaking; it being advisable for a beginner to study the tuning the intervals of the song, without respect to any other affection of the sounds; and when he is master of this, to add the practice of the lengths of the notes, as a second consideration.

We shall here set the notes of a plain song, in order to make such application of the rules as may be an introduction towards the further execution of them. No. 116.

First, find out the key, by looking at the last or keynote; then see whether it hath a flat or sharp 3d.

The key of this example being G, with one sharp prefixed, is a sharp key; being the second instance in the examples of the nine sharp keys.

Therefore tune the notes of the octave ascending, and descending, in a sharp 3d. Immediately after tune the concords in succession. After that, tune the concords of the 3ds, ascending and descending. This will be prelude enough for fixing your attention and ear to the 3ds of the key, and for pitching your voice. Having repeated this two or three times, begin the song in the same pitch or key wherein you sung the prelude. For nothing contributes more to singing in tune, than frequent repetition of notes in one key. Therefore, if your voice be rightly pitched in the prelude, seek not to change it in the song.

The two first notes in the example are Fa, Mi; which interval, it is presumed, you can tune at sight. If otherwise, you must have recourse to the general rule, and tune the intermediate note of that interval ascending, thus, Fa, sol, mi; immediately repeating the interval you want to found, thus, Fa, Mi.

The next note is the 2d, or Sol, which may be tuned from Mi, the last note, by descending; or from the key, as it hath an equal reference to both. We have laid before you this choice in consideration of your first attempt. But when from experience you are become more perfect in tuning the intervals, the most approved way will be; to make the last note you sung relative to the succeeding one, whose interval you are to tune. Whereby your singing an air will be no other than tuning the intervals as they succeed each other in the movement of the song; which you practised often before in the natural order, and with which you are supposed to be well acquainted.

The fourth note in the example being Sol, and a 4th to the last note you sung, you will now tune a perfect 4th; not considering this note in relation to the key, to which it is a 5th; but in relation to the last founded note Sol, to which it is a 4th.

This is the method you will pursue in every interval after some improvement gained by practice. Notwithstanding, it will be convenient sometimes to have recourse to the key, by founding it, and taking the interval from the same; whereby you will sing better in tune by keeping to the pitch or key you began in; particularly if the interval from the last note be a great one, or discord, or lie near the key, above or below.

To sum up all; every new note introduced; or not belonging to the harmony of the key, bearing the proportion of some concord or discord, to the preceding note or to the key, will come within the rules laid down, and therefore needs no repetition.

On the principles of the theory and practice of musick, we shall now demonstrate the art of transposing with the voice, or singing in any cliff at sight, wherewith it may happen a person is not acquainted; and this from the knowledge of singing in any other cliff. Plate C XVIII.

A Semibreve whose Time is as long as 1, 2, 3, 4, or 8 Minims

1 Crotchets

8 Quavers

16 Semiquavers

32 Demisemiquavers Theorem. The intervals of the notes of all sharp keys and flat keys respectively, are proportional. Therefore, the fingering at sight in an unknown cliff will be by transposing out of the given cliff, into that you are acquainted with.

This is done by naming the key in the cliff you are to transpose into, and distinguishing whether the song hath a flat or sharp 3d, compared to the examples of the nine sharp and seven flat keys in use, knowing the name of each cliff. Now, the name of the bass cliff is F, of the tenor C, and of the treble G.

Let it be required to sing the notes in the following example in the bass cliff unknown; transposed into the treble with which you are acquainted. No. 117.

The notes in the uppermost line, in the bass cliff, are in G.

In the second line and treble cliff, they are in E.

Demonstration. By the rules of transposition they are the second and fifth instances in the sharp key; then they are proportional: if proportional, the semitones are preserved in their proper places: but keeping or singing the semitones in their places, is tuning the notes of the octave right; therefore this transposition from the bass cliff into the treble is fingering by note right.

The same example in the tenor. No. 118.

The notes of the tenor are in F; those of the treble in E. But they are the ninth and fifth instances of sharp keys, therefore proportional; and if proportional, &c.

Again, let the treble be the unknown cliff. No. 119.

Musik, a dry, light, and friable substance, of a dark blackish colour, tinged with purple; it is a kind of perfume of a very strong scent, and only agreeable when in a very small quantity, or moderated by the mixture of some other perfume. It is found in a kind of bag or tumour which grows under the belly of the muskus moschiferus. See Moschus.

Musk is brought to us sewed up in a kind of bladders or cases of skin of a pigeon's egg, or larger, each containing from two or three drams to an ounce of musk. These are covered with a brownish hair, and are the real capsules in which the musk is lodged while on the animal. That which is unadulterated appears in masses, of loose and friable granules, which are soft to the touch, and easily crumble between the fingers, feeling somewhat smooth and unctuous.

Musk taken inwardly produces ease from pain, quiet sleep, and a copious diaphoresis: hence it has been found of great use in spasmodic disorders, petechial, malignant, putrid fevers, the jail distemper, hiccoughs, &c. Dr Wall observes, that it has been found useful in spasmodic disorders, given by way of clyster. The operation of musk in some respects resembles that of opium; but it does not leave behind it any stupor or languidness, which the latter often does. Musk likewise seems likely to answer in those low cases where sleep is much wanted, and opiates are improper. It is said to be best given in a bolus, in which form those who are most averse to perfumes may take it without inconvenience. Fifteen grains or more are now given in a dose with great success.