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deduced them from one simple experiment; and to have Preliminary established upon this foundation the most common and Difficultie essential rules of the musical art. But if the intimate and unalterable conviction which can only be produced by the strongest evidence is not here to be required, we must also doubt whether a clearer elucidation of our subject be possible.

After this declaration, it will not excite surprise, that, amongst the facts deduced from our fundamental experiment, some should immediately appear to depend upon that experiment, and others to result from it in a way more remote and less direct. In disquisitions of natural philosophy, where we are scarcely allowed to use any other arguments than those which arise from analogy or congruity, it is natural that the analogy should be sometimes more and sometimes less sensible; and we will venture to pronounce that mind very unphilosophical, which cannot recognize and distinguish this gradation and the different circumstances on which it proceeds. It is not even surprising, that, in a subject where analogy alone can take place, this conductors should defeat us all at once in our attempts to account for certain phenomena. This likewise happens in the subject which we now treat; nor do we conceal the fact, however mortifying, that there are certain points (though their number be but small) which appear still in some degree unaccountable from our principle. Such, for instance, is the procedure of the diatonic scale of the minor mode in descending, the formation of the chord commonly termed the sixth redundant † or superfluous, and some other facts of less importance, for which as yet we can scarcely offer any satisfactory account except from experience alone.

Thus, though the greatest number of the phenomena of music appear to be deducible in a simple and easy manner from the protracted tone of sonorous bodies, it ought not perhaps with too much temerity to be affirmed as yet that this mixed and protracted tone is demonstratively the only original principle of harmony. But in the mean time it would not be less unjust Rameau's to reject this principle, because certain phenomena appear to be deduced from it with less success than others, has not as yet been found by future experiments means may be found for reducing counted for these phenomena to this principle; or that harmony all the phenomena perhaps some other unknown principle, more general than that which results from the protracted and perhaps compounded tone of sonorous bodies, and of which this some other is only a branch; or, lastly, that we ought not perhaps may be need to attempt the reduction of the whole science of music to one and the same principle; which, however, is the natural effect of an impatience so frequent even among philosophers themselves, which induces them to take a part for the whole, and to judge of objects in their full extent by the greatest number of their appearances.

In those sciences which are called phisico mathematical (and amongst this number perhaps the science of sounds may be placed), there are some phenomena which depend only upon one single principle and one single experiment; there are others which necessarily suppose a greater number both of experiments and principles, whose combination is indispensable in forming an exact and complete system; and music perhaps is in this last case. It is for this reason, that whilst Preliminary we bestow on M. Rameau all due praise, we should not at the same time neglect to stimulate the learned in their endeavours to carry them still to higher degrees of perfection, by adding, if it is possible, such improvements as may be wanting to consummate the science.

Whatever the result of their efforts may be, the reputation of this intelligent artist has nothing to fear; he will still have the advantage of being the first who rendered music a science worthy of philosophical attention; of having made the practice of it more simple and easy; and of having taught musicians to employ in this subject the light of reason and analogy.

We would the more willingly persuade those who are skilled in theory and eminent in practice to extend and improve the views of him who before them pursued and pointed out the career, because many amongst them have already made laudable attempts, and have even been in some measure successful in diffusing new light through the theory of this enchanting art. It was with this view that the celebrated Tartini has presented us in 1734 with a treatise of harmony, founded on a principle different from that of M. Rameau. This principle is the result of a most beautiful experiment (c). If at once two different sounds are produced from two instruments of the same kind, these two founds generate a third different from both the others. We have inserted in the Encyclopédie, under the article Fundamental, a detail of this experiment according to M. Martini; and we owe to the public an information, of which in composing this article we were ignorant: M. Romieu, a member of the Royal Society at Montpelier, had presented to that society in the year 1753, before the work of M. Tartini had appeared, a memorial printed the same year, and where may be found the same experiment displayed at full length. In relating this fact, which it was necessary for us to do, it is by no means our intention to detract in any degree from the reputation of M. Tartini; we are persuaded that he owes this discovery to his own researches alone: but we think ourselves obliged in honour to give public testimony in favour of him who was the first in exhibiting this discovery.

But whatever be the case, it is in this experiment that M. Tartini attempts to find the origin of harmony: his book, however, is written in a manner so obscure, that it is impossible for us to form any judgment of it; and we are told that others distinguished for their knowledge of the science are of the same opinion. It were to be wished that the author would engage some man of letters, equally practised in music and skilled in the art of writing, to unfold these ideas which

(c) Had the utility of the preliminary discourse in which we are now engaged been less important and obvious than it really is, we should not have given ourselves the trouble of translating, or our readers that of perusing it. But it must be evident to every one, that the cautions here given, and the advices offered, are no less applicable to students than to authors. The first question here decided is, Whether pure mathematics can be successfully applied to the theory of music? The author is justly of a contrary opinion. It may certainly be doubted with great justice, whether the solid contents of sonorous bodies, and their degrees of cohesion or elasticity, can be ascertained with sufficient accuracy to render them the subjects of musical speculation, and to determine their effects with such precision as may render the conclusions deduced from them geometrically true. It is admitted, that sound is a secondary quality of matter, and that secondary qualities have no obvious connexion which we can trace with the sensations produced by them. Experience, therefore, and not speculation, is the grand criterion of musical phenomena. For the effects of geometry in illustrating the theory of music (if any will still be so credulous as to pay them much attention), the English reader may consult Smith's Harmonics, Malcolm's Dissertation on Music, and Pleydel's Treatise on the same subject inserted in a former edition of this work. Our author next treats of the famous discovery made by Signor Tartini, of which the reader may accept the following compendious account.

If two sounds be produced at the same time properly tuned and with due force, from their conjunction a third sound is generated, so much more distinctly to be perceived by delicate ears as the relation between the generating sounds is more simple; yet from this rule we must except the unison and octave. From the fifth is produced a sound unison with its lowest generator; from the fourth, one which is an octave lower than the highest of its generators; from the third major, one which is an octave lower than its lowest; and from the sixth minor (whose highest note forms an octave with the lowest in the third formerly mentioned) will be produced a sound lower by a double octave than the highest of the lesser sixth; from the third minor, one which is double the distance of a greater third from its lowest; but from the sixth major (whose highest note makes an octave to the lowest in the third minor) will be produced a sound only lower by double the quantity of a greater third than the highest; from the second major, a sound lower by a double octave than the lowest; from the second minor, a sound lower by triple the quantity of a third major than the highest; from the interval of a diatonic or greater semitone, a sound lower by a triple octave than the highest; from that of a minor or chromatic semitone, a sound lower by the quantity of a fifth four times multiplied than the lowest, &c. &c. But that these musical phenomena may be tried by experiments proper to ascertain them, two hautboys tuned with scrupulous exactness must be procured, whilst the musicians are placed at the distance of some paces one from the other, and the hearers in the middle. The violin will likewise give the same chords, but they will be less distinctly perceived, and the experiment more fallacious, because the vibrations of other strings may be supposed to enter into it.

If our English reader should be curious to examine these experiments and the deductions made from them in the theory of music, he will find them clearly explained and illustrated in a treatise called Principles and Power of Harmony, printed at London in the year 1771. If philosophical musicians ought not to lose their time in searching for mechanical explications of the phenomena in music, explications which will always be found vague and unsatisfactory; much less is it their province to exhaust their powers in vain attempts to rise above their sphere into a region still more remote from the prospect of their faculties, and to lose themselves in a labyrinth of metaphysical speculations upon the causes of that pleasure which we feel from harmony. In vain would they accumulate hypotheses on hypotheses, to find a reason why some chords should please us more than others. The futility of these supposititious accounts must be obvious to every one who has the least penetration. Let us judge of the rest by the most probable which has till now been invented for that purpose. Some ascribe the different degrees of pleasure which we feel from chords, to the more or less frequent coincidence of vibrations; others to the relations which these vibrations have among themselves, as they are more or less simple. But why should this coincidence of vibrations, that is to say, their simultaneous impulse on the same organs of sensation, and the accident of beginning frequently at the same time, prove so great a source of pleasure? Upon what is this gratuitous supposition founded? And though it should be granted, would it not follow, that the same chord should successively and rapidly affect us with contrary sensations, since the vibrations are alternately coincident and different? On the other hand, how should the ear be so sensible to the simplicity of relations, whilst for the most part these relations are entirely unknown to him whose organs are notwithstanding sensibly affected with the charms of agreeable music? We may conceive without difficulty how the eye judges of relations, but how does the ear form similar judgments? Besides, why should certain chords which are extremely pleasing in themselves, such as the fifth, lose almost nothing of the pleasure which they give us, when they are altered, and of consequence when the simplicity of their relations are destroyed; whilst other chords, which are likewise extremely agreeable, such as the third, become harsh almost by the smallest alteration; nay, whilst the most perfect and the most agreeable of all chords, the octave, cannot suffer the most inconsiderable change? Let us in sincerity confess our ignorance concerning the genuine causes of these effects (ii). The meta-

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(ii) We have as great an aversion as our author to the explication of musical phenomena from mechanical principles; yet we fear the following observations, deduced from irresistible and universal experience, evidently show that the latter necessarily depend on the former. It is, for instance, universally allowed, that consonances grate, and concords please a musical ear: It is likewise no less unanimously agreed, that in proportion as a chord is perfect, the pleasure is increased; now the perfection of a chord consists in the regularity and frequency of coincident oscillations between two sonorous bodies impelled to vibrate: thus the third is a chord less perfect than the fifth, and the fifth than the octave. Of all these consonances, therefore, the octave is most pleasing to the ear; the fifth next, and the third last. In absolute discords, the vibrations are never coincident, and of consequence a perpetual pulsation or jarring is recognized between the protracted sounds, which exceedingly hurts the ear; but in proportion as the vibrations coincide, those pulsations are superceded, and a kindred formed betwixt the two continued sounds, which delights even the corporeal senses: that relation, therefore, without recognizing the aptitudes which produce it, must be the obvious cause of the pleasure which chords give to the ear. What we mean by coincident vibrations is, that while one sonorous body performs a given number of vibrations, another performs a different number in the same time: so that the vibrations of the quickest must sometimes be simultaneous with those of the slowest, as will plainly appear from the following Preliminary physical conjectures concerning the acoustic organs are probably in the same predicament with those which are formed concerning the organs of vision, if one may speak so, in which philosophers have even till now made such inconsiderable progress, and in all likelihood will not be surpassed by their successors.

Since the theory of music, even to those who confine themselves within its limits, implies questions from which every wise musician will abstain; with much greater reason should they avoid idle excursions beyond the boundaries of that theory, and endeavours to investigate between music and the other sciences chimerical relations which have no foundation in nature. The singular opinions advanced upon this subject by some even of the most celebrated musicians, deserve not to be refused from oblivion, nor refuted; and ought only to be regarded as a new proof how far men of genius may err, when they engage in subjects of which they are ignorant.

The rules which we have attempted to establish concerning the track to be followed in the theory of the musical art, may suffice to show our readers the end which we have proposed, and which we have endeavoured to attain in this Work. We have here (we repeat it), nothing to do with the mechanical principles of protracted and harmonic tones produced by sonorous bodies; principles which have hitherto been and perhaps may yet be long explored in vain: we have less to do with the metaphysical causes of the sensations impressed on the mind by harmony; causes which are still less discovered, and which, according to all appearances, will remain latent in perpetual obscurity. We are alone concerned to show how the principal laws of harmony may be deduced from one single experiment; preliminary for which, if we may speak so, preceding artists have been under a necessity of groping in the dark.

With an intention to render this work as generally useful as possible, we have endeavoured to adapt it to the capacity even of those who are absolutely uninstructed in music. To accomplish this design, it appeared necessary to pursue the following plan.

To begin with a short introduction, in which are defined the technical terms most frequently used in this treatise, such as chord, harmony, key, third, fifth, octave, &c.

Afterwards to enter into the theory of harmony, which is explained according to M. Rameau, with all possible perspicuity. This is the subject of the First Part; which, as well as the introduction, presupposes no other knowledge of music than that of the names of the notes, C, D, E, F, G, A, B, which all the world knows (1).

The theory of harmony requires some arithmetical calculations, necessary for comparing sounds one with another. These calculations are short, simple, and may be comprehended by every one; they demand no operation but what is explained, and which every schoolboy may perform. Yet, that even the trouble of this may be spared to such as are not disposed to take it, these calculations are not inserted in the text, but in the notes, which the reader may omit, if he can take for granted the propositions contained in the text which will be found proved in the notes.

These calculations we have not endeavoured to multiply; we could even have wished to suppress them, if it had been possible: so much did it appear to us to be apprehended that our readers might be misled upon this subject, and might either believe, or suspect us of believing,

following deduction: Between the extremes of a third, the vibrations of the highest are as 5 to 4 of the lowest; those of the fifth as 3 to 2; those of the octave as 2 to 1. Thus it is obvious, that in proportion to the frequent coincidence of periodical vibrations, the compound sensation is more agreeable to the ear. Now, to inquire why that organ should be rather pleased with these than with the pulsation and tremulous motion of encountering vibrations which can never coalesce, would be to ask why the touch is rather pleased with polished than rough surfaces; or, why the eye is rather pleased with the waving line of Hogarth than with sharp angles and abrupt or irregular prominences? No alteration of which any chord is susceptible will hurt the ear unless it should violate or destroy the regular and periodical coincidence of vibrations. When alterations can be made without this disagreeable effect, they form a pleasing diversity; but still this fact corroborates our argument, that in proportion as any chord is perfect, it is impatient of the smallest alteration; for this reason, even in temperament, the octave endures no alteration at all, and the fifth as little as possible.

(1) In our former editions, the French syllabic names of the notes ut, re, mi, fa, sol, la, si, were retained, as being thought to convey the idea of the relative sounds more distinctly than the seven letters used in Britain. It is no doubt true, that by constantly using the syllables, and considering each as representing one certain sound in the scale, a finger will in time associate the idea of each sound with its proper syllable, so that he will habitually give ut the sound of the first or fundamental note, re that of a second, mi of a third, &c. but this requires a long time, and much application: and is, besides, useless in modulation or changes of the key, and in all instrumental music. Teachers of sol-fa'ing as it is called, or fingering by the syllables, in Britain, have long discarded, (if they ever used), the syllables ut, re, and si: and the prevalent, and we think, the founder opinion is now, that a scholar will, by attending to the sounds themselves rather than to their names, soon learn their distinct characters and relations to the key, and to each other, and be able of course to assign to each its proper degree in the scale which he employs for the time, by whatever name the note representing that degree may be generally known. See Holden's Essay towards a Rational System of Music, Part I, chap. i. § 32, 33.

We have therefore, in our present edition, preferred to the French syllables the British nomenclature by the letters C, D, E, F, G, A, B, as being more simple, more familiar to British musicians, and equally applicable to instrumental as to vocal music. Preliminary believing, all this arithmetic necessary to form an artist's course. Calculations may indeed facilitate the understanding of certain points in the theory, as of the relations between the different notes in the gamut and of the temperament; but the calculations necessary for treating of these points are so simple, and of so little importance, that nothing can require a less ostentatious display. Let us not imitate those musicians who, believing themselves 'geometers,' or those geometers who, believing themselves musicians, fill their writings with figures upon figures; imagining, perhaps, that this apparatus is necessary to the art. The propensity of adorning their works with a false air of science, can only impose upon ignorance, and render their treatises more obscure and less instructive.

This abuse of geometry in music may be condemned with much more reason, that in this subject the foundations of those calculations are in some manner hypothetical, and can never arise to a degree of certainty above hypothesis. The relation of the octave as 1 to 2, that of the fifth as 2 to 3, that of the third major as 4 to 5, &c., are not perhaps the genuine relations established in nature; but only relations which approach them, and such as experience can discover. For are the results of experience anything more but mere approaches to truth?

But happily these approximated relations are sufficient, though they should not be exactly agreeable to truth, for giving a satisfactory account of those phenomena which depend on the relations of sound; as in the difference between the notes in the gamut, of the alterations necessary in the fifth and third, of the different manner in which instruments are tuned, and other facts of the same kind. If the relations of the octave, of the fifth, and of the third, are not exactly such as we have supposed them, at least no experiments can prove that they are not so; and since these relations are signified by a simple expression, since they are besides sufficient for all the purposes of theory, it would be useless, and contrary to sound philosophy, to invent other relations in order to form the basis of any system of music less easy and simple than that which we have delineated in this treatise.

The second part contains the most essential rules of composition*, or in other words the practice of harmony. These rules are founded on the principles laid down in the first part; yet those who wish to understand no more than is necessary for practice, without exploring the reasons why such practical rules are necessary, may limit the objects of their study to the introduction and the second part. They who have read the first part, will find at every rule contained in the second, a reference to that passage in the first where the reasons for establishing that rule are given.

That we may not present at once too great a number of objects and precepts, we have transferred to the Preliminary notes in the second part several rules and observations, which are less frequently put in practice, which perhaps it may be proper to omit till the treatise is read a second time, when the reader is well acquainted with the essential and fundamental rules explained in it.

This second part presupposes, no more than the first, any habit of singing, nor even any knowledge of music; it only requires that one should know, not even the intonation, but merely the position of the notes in the clef F on the fourth line, and that of G upon the second; and even this knowledge may be acquired from the work itself; for in the beginning of the second part we explain the position of the clefs and of the notes. Nothing is necessary but to render it a little familiar, and any difficulty in it will disappear.

It would be wrong to expect here all the rules of composition, and especially those which direct the rules of composition of music in several parts, and which, being less severe and indispensible, may be chiefly acquired by practice, by studying the most approved models, in the assistance of a proper master, but above all by the cultivation of the ear and of the taste. This treatise is properly nothing else, if the expression may be allowed, but the rudiments of music, intended for explaining to beginners the fundamental principles, not the practical detail of composition. Those who wish to enter more deeply into this detail, will either find it in M. Rameau's treatise of harmony, or in the code of music which he published more lately (k), or lastly in the explication of the theory and practice of music by M. Bethizy (l); this last book appears to us clear and methodical (m).

Is it necessary to add, that in order to compose music in a proper taste, it is by no means enough to have familiarized with much application the principles of the mechanism of the art; it is the province of nature alone to accomplish the rest. Without her assistance, it is no more possible to compose agreeable music by having read these elements, than to write verses in a proper manner with the Dictionary of Richelet. In one word, it is the elements of music alone, and not the principles of genius, that the reader may expect to find in this treatise.

DEFINITIONS.

I. What is meant by Melody, by Chord, by Harmony, by Interval.

1. Melody is a series of sounds which succeed one to another in a manner agreeable to the ear.

2. A Chord is a combination of several sounds heard together; and Harmony is properly a series of chords of which the succession pleases the ear. A single chord is what.

(k) From my general recommendation of this code, I except the reflections on the principle of sound which are at the end, and which I should not advise anyone to read.

(l) Printed at Paris by Lambert in the year 1754.

(m) In addition to the works mentioned in the text, we recommend to our readers, Holden's Essay, Glasgow 1770, Edin. 1805; Kollmann's Essay on Musical Harmony, 1796; his Essay on Musical Composition, fol. 1799; Shield's Introduction, 1800; and Dr Callcott's Musical Grammar, 1806. Definitions. is likewise sometimes called harmony, to signify the coalescence of the sounds which form the chord, and the sensation produced in the ear by that coalescence. We shall occasionally use the word harmony in this last sense, but in such a manner as never to leave our meaning ambiguous.

3. An Interval, in melody and harmony, is the distance, or difference in pitch, between one sound, and another higher or lower than it.

4. That we may learn to distinguish the intervals, and the manner of perceiving them, let us take the ordinary scale C, D, E, F, G, A, B, c, which every person whose ear or voice is not extremely false naturally modulates. The following observations will occur to us in singing this scale:

The sound D is higher or sharper than the sound C, the sound E higher than the sound D, the sound F higher than the sound E, &c., and so through the whole octave; so that the interval, or the distance from the sound C to the sound D, is less than the interval or distance between the sound C and the sound E, the interval from C to F is less than that between C and F, &c., and in short that the interval from the first to the second C is the greatest of all.

To distinguish the first from the second C, we have marked the last with a small letter (n).

5. In general, the interval between two sounds is proportionally greater, as one of these sounds is higher or lower with relation to the other: but it is necessary to observe, that two sounds may be equally high or low, though unequal in their force. The stringing of a violin touched with a bow produces always a sound grave, equally high, whether strongly or faintly struck; the sound will only have a greater or lesser degree of strength. It is the same with vocal modulation; let any one form a sound by gradually swelling the voice, the sound may be perceived to increase in force, whilst it continues always equally low or equally high.

6. We must likewise observe concerning the scale, between that the intervals between C and D, between D and E, between F and G, between G and A, between A and B, are equal, or at least nearly equal; and that the intervals between E and F, and between B and C, are likewise equal among themselves, but consist almost only of half the former. This fact is known and recognized by every one: the reason for it shall be given in the sequel; in the mean time every one may ascertain its reality by the affiance of an experiment (o).

7. It

(n) We shall afterwards find that three different series of the seven letters are used, which we have distinguished by capitals, small Roman, and Italic characters. When the notes represented by small Roman characters occur in this treatise we shall, merely to distinguish them from the typography of the text, place them in inverted commas, thus 'c', 'd', &c.

(o) This experiment may be easily tried. Let any one sing the scale C, D, E, F, G, A, B 'c', it will be immediately observed without difficulty, that the last four notes of the octave G, A, B, 'c', are quite similar to the first C, D, E, F; infomuch, that if, after having sung this scale, one would choose to repeat it, beginning with C in the same tone which was occupied by G in the former scale, the note D of the last scale would have the same sound with the note A in the first, the E with the B, and the F with the 'c'.

Whence it follows, that the interval between C and D, is the same as between G and A; between D and F, as between A and B, and E and F, as between B and 'c'.

From D to E, from F to G, there is the same interval as from C to D. To be convinced of this, we need only sing the scale once more; then sing it again, beginning with C, in this last scale, in the same tone which was given to D in the first; and it will be perceived, that the D in the second scale will have the same sound, at least as far as the ear can discover, with the E in the former scale; whence it follows, that the difference between D and E is, at least as far as the ear can perceive, equal to that between C and D. It will also be found, that the interval between F and G is, so far as our senses can determine, the same with that between C and D.

This experiment may perhaps be tried with some difficulty by those who are not inured to form the notes and change the key; but such may very easily perform it by the assistance of a harpsichord, by means of which the performer will be saved the trouble of retaining the sounds in one intonation whilst he performs another. In touching upon this harpsichord the keys G, A, B, 'f', and in performing with the voice at the same time C, D, E, F, in such a manner that the same sound may be given to C in the voice with that of the key G in the harpsichord, it will be found that D in the vocal intonation shall be the same with A upon the harpsichord, &c.

It will be found likewise by the same harpsichord, that if one should sing the scale beginning with C in the same tone with E on the instrument, the D, which ought to have followed C, will be higher by an extremely perceptible degree than the F which follows E: thus it may be concluded, that the interval between E and F is less than between C and D; and if one would rise from F to another sound which is at the same distance from F, as F from E, he would find, in the same manner, that the interval from E to this new sound is almost the same as that between C and D. The interval then from E to F is nearly half of that between C and D.

Since then, in the scale thus divided, C, D, E, F, G, A, B, 'c', the first division is perfectly like the last; and since the intervals between C and D, between D and E, and between F and G, are equal; it follows, that the intervals between G and A, and between A and B, are likewise equal to every one of the three intervals between C and D, between D and E, and between F and G; and that the intervals between E and F and between B and 'c' are also equal, but that they only constitute one half of the others. 7. It is for this reason that they have called the interval from E to F, and from B to C, a semitone; whereas those between C and D, D and E, F and G, G and A, and A and B, are tones.

The tone is likewise called a second major *, and the semitone a second minor +.

8. To descend or rise diatonically, is to descend or rise from one sound to another by the interval of a tone or a semitone, or in general by seconds, whether major or minor; as from D to C, or from C to D, from F to E, or from E to F.

II. The terms by which the different Intervals of the Scale are denominated.

9. An interval composed of a tone and a semitone, as from E to G, from A to C, or from D to F, is called a third minor.

An interval composed of two full tones, as from C to E, and from F to A, or from G to B, is called a third major.

An interval consisting of two tones and a semitone, as from C to F, or from G to C, is called a fourth.

An interval consisting of three full tones, as from F to B, is called a tritone or fourth redundant.

An interval consisting of three tones and a semitone, as from C to G, from F to C, from D to A, or from E to B, &c., is called a fifth.

An interval composed of three tones and two semitones, as from E to C, is called a sixth minor.

An interval composed of four tones and a semitone, as from C to A, is called a sixth major.

An interval consisting of four tones and two semitones, as from D to C, is called a seventh minor.

An interval composed of five tones and a semitone, as from C to B, is called a seventh major.

And in short, an interval consisting of five tones and two semitones, as from C to 'c' is called an octave.

Several of the intervals now mentioned, are distinguished by other names, as may be seen in the beginning of the second part; but those now given are the most common, and the only terms which our present purpose demands.

10. Two sounds equally high, or equally low, however unequal in their force, are said to be in unison one Definitions, with the other.

11. If two sounds form between them any interval, whatever it be, we say, that the highest when ascending is in that interval with relation to the lowest; and when descending, we pronounce the lowest in the same interval with relation to the highest. Thus in the third minor, E, G, where E is the lowest and G the highest, found, G is a third minor from E ascending, and E is a third minor from G in descending.

12. In the same manner, if speaking of two sonorous bodies, we should say, that the one is a fifth above the other in ascending; this infers that the sound given by the one is at the distance of a fifth ascending from the sound given by the other.

III. Of Intervals greater than the Octave.

13. If, after having sung the scale C, D, E, F, G, Fig. 2. A, B, c, one would carry this scale still farther in ascent, it would be discovered without difficulty that a new scale would be formed, 'c, d, e, f', &c., entirely similar to the former, and of which the sounds will be an octave ascending, each to its correspondent note in the former scale; thus 'd', the second note of the second scale, will be an octave in ascent to the D of the first scale; in the same manner 'e' shall be the octave to E, &c., and so of the rest.

14. As there are nine notes from the first C to the Ninth, second 'd', the interval between these two sounds is called what is called a ninth, and this ninth is composed of fix full tones and two semitones. For the same reason the interval from C to 'f' is called an eleventh, and the interval between C and 'g' a twelfth, &c.

It is plain that the ninth is the octave of the second, Eleventh the eleventh of the fourth, and the twelfth of the fifth, &c.

The octave above the octave of any sound is called a double octave *; the octave of the double octave is called a triple octave, and so of the rest.

The double octave is likewise called a fifteenth; and for the same reason the double octave of the third is called a seventeenth, the double octave of the fifth a nineteenth, &c. (p).

IV.

(p) Let us suppose two vocal strings formed of the same matter, of the same thickness, and equal in their tension, but unequal in their length; it will be found by experience,

1stly. That if the shortest is equal to half the longest, the sound which it will produce must be an octave above the sound produced by the longest.

2ndly. That if the shortest constitutes a third part of the longest, the sound which it produces must be a twelfth above the sound produced by the longest,

3rdly. That if it constitutes the fifth part, its sound will be a seventeenth above.

Besides, it is a truth demonstrated and generally admitted, that in proportion as one musical string is less than another, the vibrations of the least will be more frequent (that is to say, its departures and returns through the same space) in the same time; for instance, in an hour, a minute, a second, &c., in such a manner that one string which constitutes a third part of another, forms three vibrations, whilst the largest has only accomplished one. In the same manner, a string which is one half less than another, performs two vibrations, whilst the other only completes one; and a string which is only the fifth part of another, will perform five vibrations in the same time which is occupied by the other in one.

From thence it follows, that the sound of a string is proportionally higher or lower, as the number of its vibrations is greater or smaller in a given time; for instance, in a second.

It is for that reason, that if we represent any sound whatever by 1, one may represent the octave above by 2, that is to say, by the number of vibrations formed by the string which produces the octave, whilst the longest string only vibrates once; in the same manner we may represent the twelfth above the sound 1 by 3, the seventeenth IV. What is meant by Sharps and Flats.

15. It is plain that one may imagine the five tones which enter into the scale, as divided each into two semitones; thus one may advance from C to D, forming in his progress an intermediate sound, which shall be higher by a semitone than C, and lower in the same degree than D. A sound in the scale is called sharp, when it is raised by a semitone; and it is marked with this character ♯: thus C ♯ signifies C sharp, that is to say, C raised by a semitone above its pitch in the natural scale. A sound in the scale depressed by a semitone is called flat, and is marked thus ♭: thus A ♭ signifies A flat, or A depressed by a semitone.

V. What is meant by Consonances and Dissonances.

16. A chord composed of sounds whose union or coalescence pleases the ear is called a consonance; and the sounds which form this chord are said to be consonant one with relation to the other. The reason of this denomination is, that a chord is found more perfect, as the sounds which form it coalesce more closely among themselves.

17. The octave of a sound is the most perfect of consonances of which that sound is susceptible; then the fifth, afterwards the third, &c. This is a fact founded on experiment.

18. A number of sounds simultaneously produced Diffonance, whose union is displeasing to the ear is called a dissonance, and the sounds which form it are said to be dissonant one with relation to the other. The second, the tritone, and the seventh of a sound, are dissonants See Diffonance, and the seventh of a sound, are dissonants See Diffonance, with relation to it. Thus the sounds C D, C B, or cord, F B, &c. simultaneously heard, form a dissonance. The reason which renders dissonance disagreeable, is, that the sounds which compose it, seem by no means coalescent to the ear, and are heard each of them by itself as distinct sounds, though produced at the same time.

PART I. THEORY OF HARMONY.

CHAP. I. Preliminary and Fundamental Experiments.

EXPERIMENT I.

19. WHEN a sonorous body is struck till it gives a sound, the ear, besides the principal sound and its octave, perceives two other sounds very high, of which one is the twelfth above the principal sound, that is to say, the octave to the fifth of that sound; and the other is the seventeenth major about the same sound, that is to say, the double octave of its third major.

20. This experiment is peculiarly sensible upon the thick strings of the violoncello, of which the sound being extremely low, gives to an ear, though not very much practised, an opportunity of distinguishing with sufficient ease and clearness the twelfth and seventeenth now in question (Q).

teeth major above 5, &c. But it is very necessary to remark, that by these numerical expressions we do not pretend to compare sounds as such; for sounds in themselves are nothing but mere sensations, and it cannot be said of any sensation that it is double or triple to another: thus the expressions 1, 2, 3, &c. employed to denote a sound, its octave above, its twelfth above, &c. signify only, that if a string performs a certain number of vibrations, for instance, in a second, the string which is in the octave above shall double the number in the same time, the string which is in the twelfth above shall triple it, &c.

Thus to compare sounds among themselves is nothing else than to compare among themselves the numbers of vibrations which are formed in a given time by the strings that produce these sounds.

(Q.) Since the octave above the sound 1 is 2, the octave below that same sound shall be ½; that is to say, that the string which produces this octave shall have performed half its vibration, whilst the string which produces the sound 1 shall have completed one. To obtain therefore the octave above any sound, the operator must multiply the quantity which expresses the sound by 2; and to obtain the octave below, he must on the contrary divide the same quantity by 2.

It is for that reason that if any sound whatever, for instance C, is denominated

| Its octave above | 2 | |-----------------|---| | Its double octave above | 4 | | Its triple octave above | 8 | | In the same manner its octave below will be | ½ | | Its double octave below | ¼ | | Its triple octave below | ⅛ |

And so of the rest.

| Its twelfth above | 3 | |------------------|---| | Its twelfth below | ⅓ | | Its 17th major above | 5 | | Its 17th major below | ⅕ |

The fifth then above the sound 1 being the octave beneath the twelfth, shall be, as we have immediately observed, 21. The principal sound is called the generator*; and the two other sounds which it produces, and with which it is accompanied, are, inclusive of its octave, called its harmonics†.

**Experiment II.**

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

Let it be supposed that any person has an inclination to sing a tune, and having at first begun this air

\[ \text{Vol. XIV. Part II.} \]

served, \( \frac{1}{2} \); which signifies that this string performs \( \frac{3}{4} \) vibrations; that is to say, one vibration and a half during a single vibration of the string which gives the sound \( \frac{1}{2} \).

To obtain the fourth above the sound \( \frac{1}{2} \), we must take the twelfth below that sound, and the double octave above that twelfth. In effect, the twelfth below \( C \), for instance, is \( F \), of which the double octave \( f \) is the fourth above \( c \). Since then the twelfth below \( c \) is \( \frac{1}{2} \), it follows that the double octave above this twelfth, that is to say, the fourth from the sound \( c \) in ascending, will be \( \frac{1}{2} \) multiplied by 4, or \( \frac{4}{5} \).

In short, the third major being nothing else but the double octave beneath the seventeenth, it follows, that the third major above the sound \( \frac{1}{2} \) will be \( \frac{5}{6} \) divided by 4, or in other words \( \frac{5}{8} \).

The third major of a sound, for instance the third major \( E \), from the sound \( C \), and its fifth \( G \), form between them a third minor \( E, G \); now \( E \) is \( \frac{3}{4} \), and \( G \) \( \frac{5}{8} \), by what has been immediately demonstrated: from whence it follows, that the third minor, or the interval between \( E \) and \( G \), shall be expressed by the relation of the fraction \( \frac{3}{4} \) to the fraction \( \frac{5}{8} \).

To determine this relation, it is necessary to remark, that \( \frac{3}{4} \) are the same thing with \( \frac{1}{2} \), and that \( \frac{5}{8} \) are the same thing with \( \frac{5}{6} \): so that \( \frac{3}{4} \) shall be to \( \frac{5}{8} \) in the same relation as \( \frac{1}{2} \) to \( \frac{5}{6} \); that is to say, in the same relation as 10 to 12, or as 5 to 6. If, then, two sounds form between themselves a third minor, and that the first is represented by \( \frac{5}{6} \), the second shall be expressed by \( \frac{5}{6} \); or, what is the same thing, if the first is represented by \( \frac{5}{6} \), the second shall be expressed by \( \frac{5}{6} \).

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

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

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

Let us suppose, for example, that the third major of the fifth \( \frac{1}{2} \) is sought. That third major ought to be, by what has been shown above, the \( \frac{3}{4} \) of the fifth; for the third major of any sound whatever is the \( \frac{3}{4} \) of that sound. We must then look for a fraction which expresses the \( \frac{3}{4} \) of \( \frac{1}{2} \); which is done by multiplying the numerators and denominators of both fractions one by the other, from whence results the new fraction \( \frac{3}{8} \). It will likewise be found that the fifth of the fifth is \( \frac{5}{6} \), because the fifth of the fifth is the \( \frac{5}{6} \) of \( \frac{5}{6} \).

Thus far we have only treated of fifths, fourths, thirds major and minor, in ascending; now it is extremely easy to find by the same rules the fifths, fourths, thirds major and minor in descending. For suppose \( C \) equal to 1, we have seen that its fifth, its fourth, its third, its major and minor in ascending, are, \( \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10} \). To find its fifth, its fourth, its third, its major and minor in descending, nothing more is necessary than to reverse these fractions, which will give \( \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10} \).

(Q*) It is not then imagined that we change the value of a sound in multiplying or dividing it by 2, by 4, or by 8, &c., the number which expresses these sounds, since by these operations we do nothing but take the simple double, or triple octave, &c., of the sound in question, and that a sound coalesces with its octave. 23. To render our ideas still more precise and permanent, we shall call the tone produced by the sonorous body C: it is evident, by the first experiment, that this sound is always attended by its 12th and 17th major; that is to say, with the octave of G, and the double octave of E.

24. This octave of G then, and this double octave of E, produce the most perfect chord which can be joined with C, since that chord is the work and choice of nature (r).

25. For the same reason, the modulation formed by C with the octave of G, and the double octave of E, sung one after the other, would likewise be the most simple and natural of all modulations which do not descend or ascend directly in the diatonic order, if our voices had sufficient compass to form intervals so great without difficulty: but the ease and freedom with which we can substitute its octave to any sound, when it is more convenient for the voice, afford us the means of representing this modulation.

26. It is on this account that, after having sung the tone C, we naturally modulate the third E, and the fifth G, instead of the double octave of E, and the octave of G; from whence we form, by joining the octave of the found G, this modulation, C, E, G, 'c', which in effect is the simplest and easiest of all; and which likewise has its origin even in the prolonged and compounded tones produced by a sonorous body.

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

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

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

30. This new arrangement, C, E b, G, in which the sounds C and E b have both the power of causing G to resound, though C does not cause E b to resound, is not indeed equally perfect with the first arrangement C, E, G; because in this the two sounds E and G are both the one and the other generated by the principal sound C; whereas, in the other, the sound E b, is not generated by the sound C; but this arrangement C, E b, G, is likewise dictated by nature (art. 19.), though less immediately than the former; and accordingly experience evinces that the ear accommodates itself almost as well to the latter as to the former.

31. In this modulation or chord, C, E b, G, C, Origin of it is evident that the third from C to E b is minor; mode minor (s).

32. The most perfect chords then are, 1. All chords related one to another, as C, E, G, 'c', consisting of any found, of its third major, of its fifth, and of its chords, octave. 2. All chords related one to another, as C what. E b, G, 'c', consisting of any found, of its third minor,

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

(s) The origin which we have here given of the mode minor, is the most simple and natural that can possibly be given. M. Rameau deduces it, more artificially, from the following experiment.—If you put in vibration a musical string HI, and if there are at the same time contiguous to this two other strings KN, RW, of which the first shall be a twelfth, and the second a seventeenth major below the string HI, the strings KN, RW will vibrate without being struck as soon as the string HI shall give a sound, and divide themselves by a kind of undulation, the first into three, the last into five equal parts; in such a manner, that, in the vibration of the string KN, you may easily distinguish two points at rest LM, and in the tremulous motion of the string RW, four quiescent points S, T, U, V, all placed at equal distances from each other, and dividing the strings into three or five equal parts. In this experiment, says M. Rameau, if we represent by the note C the tone of the string HI, the two other strings will represent the sounds F and A b; and from thence M. Rameau deduces the modulation F, A b, C, and of consequence the mode minor. The origin which we have assigned to the minor mode, appears more direct and more simple, because it presupposes no other experiment than that of art. 19., and because also the fundamental sound C is still retained in both the modes, without being obliged, as M. Rameau found himself, to change it into F. theory of minor, of its fifth, and of its octave. In effect, these harmony two kinds of chords are exhibited by nature; but the first more immediately than the second. The first are called perfect chords major, the second perfect chords minor.

**Chap. III. Of the Succession by Fifths, and of the Laws which it observeth.**

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

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

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

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

E♭, B♭, F, C, G, D, A, &c.

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

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

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

**Chap. IV. Of Modes in General.**

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

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

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

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

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

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

43. As in the series of fifths, we may indifferently pass from one sound to that which is contiguous: to transition having passed from the mode of C to that of G, one may from thence proceed to the mode of D. And on how to be the other hand, having passed from the mode of C to managed, that of F we may then pass to the mode of B♭. But it is necessary, however, to observe, that the ear, which has been immediately affected with the principal mode, feels always a strong propensity to return to it. Thus the further the mode to which we make a transition is removed from the principal mode, the less time we ought to dwell upon it; or rather, to speak in the terms of the art, the less ought the phrase (u) of that mode to be protracted.

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

44. From this rule, that two sounds which are contiguous may be placed in immediate succession in the series of fifths, F, C, G, it follows, that one may form

(t) We say especially if they are major; for in the major chord D, F♯, A, d', besides that the sounds C and D have no common harmonical relation, and are even dissonant between themselves (art. 13.), it will likewise be found, that F♯ forms a dissonance with C. The minor chord D, F, A, d', would be more tolerable, because the natural F, which occurs in this chord carries along with it its fifth C, or rather the octave of that fifth: It has likewise been sometimes the practice of composers, though rather by a licence indulged them than strictly agreeable to their art, to place a minor in diatonic succession to a major chord.

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

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

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

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

gives the following diatonic series,

B, c, d, e, f, g, a?

which is precisely the diatonic scale of the Greeks. We are ignorant upon what principles they had formed this scale; but it may be sensibly perceived, that that series arises from the bafs G, C, G, C, F, C, F; and that of consequence this baf is justly called fundamental, as being the real primitive modulation, that which conduces the ear, and which it feels to be implied in the diatonic modulation, B, c, d, e, f, g, a? (x).

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

In the modulation B, c, d, e, f, g, a?, the founds c, d, and f? form between themselves a third minor, which is not so perfectly true as that between c? and g? (y). Nevertheless, this alteration in the third minor between d? and f? gives the ear no pain, because that d? and that f? which do not form between themselves a true third minor, form, each in particular, consonances perfectly just with the founds in the fundamental baf which correspond with them: for d? in the scale is the true fifth of G, which answers to it in the fundamental baf; and f? in the scale is the true octave of F, which answers to it in the same baf.

47. If, therefore, these founds in the scale form consonances perfectly true with the notes which correspond therewith, no to them in the fundamental baf, the ear gives itself objection little trouble to investigate the alterations which there may be in the intervals which these founds in the scale form between themselves. This is a new proof that the fundamental baf is the genuine guide of the ear, and the true origin of the diatonic scale.

48. Moreover, this diatonic scale includes only seven founds, and goes no higher than b?, which would why this be the octave of the first: a new singularity, for which reason may be given by the principles above established.

---

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

A phrase, according to him, is in melody a series of modulations, or in harmony a succession of chords, which form without interruption a sense more or less complete, and which terminates in a repose by a cadence more or less perfect.

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

1. That c? in the scale is the octave of C in the baf; that is to say, 2.

2. That d? is the third major of G; that is to say \( \frac{3}{4} \) of \( \frac{3}{4} \) (note Q), and of consequence \( \frac{3}{4} \).

3. That e? is the fifth of G; that is to say \( \frac{3}{4} \) of \( \frac{3}{4} \), and of consequence \( \frac{3}{4} \).

4. That f? is the double octave of F of the baf, and consequently \( \frac{3}{4} \).

5. That g? of the scale is the octave of G of the baf, and consequently 3.

6. That a? in the scale is the third major of f? of the scale; that is to say, \( \frac{3}{4} \) of \( \frac{3}{4} \), or \( \frac{3}{4} \).

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

| Diatonic | Scale | Fundamental | Bafs | |----------|-------|-------------|------| | \( \frac{3}{4} \) | 2 | \( \frac{3}{4} \) | \( \frac{3}{4} \) | | \( \frac{3}{4} \) | 3 | \( \frac{3}{4} \) | \( \frac{3}{4} \) |

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

\( \frac{3}{4} \) I \( \frac{3}{4} \) \( \frac{3}{4} \) \( \frac{3}{4} \)

B, c, d, e, f, g, a?

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

M. Rameau, when he published, in 1726, his New theoretical and practical System of Music, had not as yet found the true reason of the alteration in the consonance which is between d? and f?, and of the little attention which the ear pays to it. For he pretends, in the work now quoted, that there are two thirds minor, one in the proportion of 5 to 6, the other in the proportion of 27 to 32. But the opinion which he has afterwards adopted, seems much preferable. In reality, the genuine third minor, is that which is produced by nature between c? and g?, in the continued tone of those sonorous bodies of which c? and g? are the two harmonics: and that third minor, which is in the proportion of 5 to 6, is likewise that which takes place in the minor mode, and not that third minor which is false and different, being in the proportion of 27 to 32. In reality, in order that the found 'b' may succeed immediately in the scale to the found 'a', it is necessary that the note 'g', which is the only one from whence 'b' as a harmonic may be deduced, should immediately succeed to the found 'f', in the fundamental baf's, which is the only one from whence 'a' can be harmonically deduced. Now, the diatonic succession from F to G cannot be admitted in the fundamental baf's, according to what we have remarked (art. 36.). The founds 'a' and 'b', then, cannot immediately succeed one another in the scale: we shall see in the sequel why this is not the case in the series 'c, d, e, f, g, a, b', c, which begins upon C; whereas the scale in question here begins upon B.

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

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

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

52. But the scale is not the same when we compare two founds taken each from a different tetrachord; for we have already seen, that the note 'd' in the first tetrachord forms with the note 'f' in the second a third minor, which is not true. In like manner it will be found, that the fifth from 'd' to 'a' is not exactly true, which is evident; for the third major from 'f' to 'a' is true, and the third minor from 'd' to 'f' is not so; now, in order to form a true fifth, a third major and a third minor, which are both exactly true, are necessary.

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

54. It may be ascertained by calculation, that in the four tetrachords B, 'c, d, e', the interval, or the tone from 'd' to 'e', is a little less than the interval or tone from 'c' to 'd' (BB). In the same manner, in the second tetrachord 'e, f, g, a', which is, as we have proved, perfectly similar to the first, the note from 'g' to 'a' is a little less than the note from 'f' to 'g'. It is for this reason that they distinguish two kinds of tones; the greater tone *, as Greater from 'c' to 'd', from 'f' to 'g', &c.; and the lesser †, tone ‡, from 'd' to 'e', from 'g' to 'a', &c.

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

55. We have just shown in the preceding chapter, how the scale of the Greeks is formed, B, 'c, d, e, g, a', determined scale, by means of a fundamental baf composed of three founds only, F, C, G; but to form the scale 'c, d, e, f, g, a, b', c, which we use at present, we must necessarily add to the fundamental baf's the note D, and form, with these four founds F, C, G, D, the following fundamental baf's:

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

from whence we deduce the modulation or scale 'c, d, e, f, g, a, b', c.

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

56. Hence

(z) The proportion of B to 'c' is as \( \frac{1}{2} \) to 1, that is to say as 15 to 16; that between 'e' and 'f' is as \( \frac{3}{4} \) to \( \frac{4}{5} \), that is to say (note Q), as 5 times 3 to 4 times 4, or as 15 to 16; these two proportions then are equal. In the same manner, the proportion of 'e' to 'd' is as 1 to \( \frac{3}{8} \), or as 8 to 9; that between 'f' and 'g' is as \( \frac{4}{5} \) to \( \frac{3}{4} \); that is to say (note Q), as 8 to 9. The proportion of 'e' to 'c' is as \( \frac{3}{4} \) to 1, or as 5 to 4; that between 'f' and 'a' is as \( \frac{4}{5} \) to \( \frac{3}{4} \), or as 5 to 4: the proportions here then are likewise equal.

(AA) The proportion of 'e' to 'c' is as \( \frac{3}{4} \) to 1, or as 5 to 4, which is a true third major; that from 'd' to 'b' is as \( \frac{3}{4} \) to \( \frac{5}{6} \); that is to say, as 9 times 16 to 15 times 8, or as 9 times 2 to 15, or as 6 to 5. In like manner we shall find, that the proportion of 'c' to 'b' is as \( \frac{3}{4} \) to \( \frac{5}{6} \); that is to say, as 5 times 16 to 15 times 4, or as 4 to 3, which is a true fourth.

(BB) The proportion of 'd' to 'c' is as \( \frac{3}{4} \) to 1, or as 9 to 8; that of 'e' to 'd' is as \( \frac{3}{4} \) to \( \frac{5}{6} \), that is to say, as 40 to 36, or as 10 to 9; now \( \frac{1}{9} \) is less removed from unity than \( \frac{1}{9} \); the interval then from 'd' to 'e' is a little less than that from 'c' to 'd'.

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

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

(CC) The values or estimates of the notes shall be the same in this as in the former scale, excepting only the tone Hence it follows, that the diatonic scale of the Greeks is, at least in some respects, more simple than ours; since the scale of the Greeks (chap. v.) may be formed alone from the mode proper to C; whereas ours is originally and primitively formed, not only from the mode of C (F, C, G), but likewise from the mode of G, (C, G, D).

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

For this reason the note 'g' is twice repeated in immediate succession in this scale; once as the fifth of C, which corresponds with it in the fundamental bass; and again as the octave of G, which immediately follows G in the same bass. These two consecutive 'g's are otherwise in perfect union. For this reason we find only one of them when we modulate the scale 'c, d, e, f, g, a, b, c'; but this does not prevent us from employing a pause or repose, expressed or understood, after the found 'f'.

There is no person who does not perceive this whilst he himself sings the scale.

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

The introduction of the mode proper to G in the fundamental bass has this happy effect, that the notes 'f, g, a, b,' may immediately succeed each other in ascending the scale, which cannot take place (art. 48) in the diatonic series of the Greeks, because that series is formed from the mode of C alone. Whence it follows:

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

2. That if these three tones are sung in succession in the scale 'c, d, e, f, g, a, b, c,' this cannot be done but by the affluence of a pause expressed or understood after the note 'f'; insomuch, that the three tones 'f, g, a, b,' are supposed to belong to two different tetra-chords.

60. It ought not then any longer to surprise us, Change of that we feel some difficulty whilst we ascend the scale mode the in singing three tones in succession, because this is cause of the impracticable without changing the mode; and if one ringing pauses in the same mode, the fourth sound above the three con-firming first note will never be higher than a semitone above (ective that which immediately precedes it; as may be seen by tones al-tering, and by 'g, a, b, c,' where there is no more than a semitone between 'e' and 'f,' and between 'b' and 'c.'

61. We may likewise observe in the scale 'c, d, e, f,' Intervals, that the third minor from 'd' to 'f,' is not true, for though all-reasons which have been already given (art. 49). It is the same case with the third minor from 'a' to 'c,' and form true with the third major from 'f' to 'a'; but each of these consonances forms otherwise consonances perfectly true, with the with their correspondent sounds in the fundamental bass.

62. The thirds 'a' c, 'f'a,' which were true in the former scale, are false in this; because in the former scale 'a' was the third of 'f,' and here it is the fifth of D, which corresponds with it in the fundamental bass.

63. Thus it appears, that the scale of the Greeks contains fewer consonances that are altered than ours; and this likewise happens from the introduction of the mode of G into the fundamental bass.

We see likewise that the value of 'a' in the diatonic scale, a value which authors have been divided in after-taining, solely depends upon the fundamental bass, and that

tone 'a'; for 'd' being represented by §, its fifth will be expressed by §; so that the scale will be numerically signified thus:

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

Where you may see, that the note 'a' of this scale is different from that in the scale of the Greeks; and that the 'a' in the modern series stands in proportion to that of the Greeks as § to §, that is to say, as 81 to 80; these two 'a's then likewise differ by a comma.

(DD) In the scale of the Greeks, the note 'a' being a third from 'f,' there is an altered fifth between 'a' and 'd'; but in ours, 'a' being a fifth to 'd,' produces two altered thirds, 'f'a' and 'a' c; and likewise a fifth altered, 'a' e, as we shall see in the following chapter. Thus there are in our scale two intervals more than in the scale of the Greeks which suffer alteration.

(EE) But here it may be with some colour objected: The scale of the Greeks, it may be said, has a fundamental bass more simple than ours; and besides, in it there are fewer chords which will not be found exactly true; why then, notwithstanding this, does ours appear more easy to be sung than that of the Greeks? The Grecian scale begins with a semitone, whereas the intonation prompted by nature seems to impel us to rise by a full tone at once. This objection may be thus answered. The scale of the Greeks is indeed better disposed than ours for the simplicity of the bass, but the arrangement of ours is more suitable to natural intonation. Our scale commences by the fundamental sound c, and it is in reality from that sound that we ought to begin; it is from this that all the others naturally arise, and upon this that they depend; nay, if we speak so, in this they are included: on the contrary, neither the scale of the Greeks, nor its fundamental bass, commences with C; but it is from this C that we must depart, in order to regulate our intonation, whether in rising or descending; now, in ascending from 'c,' the intonation, even of the Greek scale, gives the series 'c, d, e, f, g, a': and so true is it that the fundamental sound C is here the genuine guide of the ear, that if, before we modulate the found 'c,' we should that it must be different according as the note 'a' ha 'f' or 'd' for its bas. See the note (cc).

CHAP. VII. Of Temperament.

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

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

65. If, after having successively and alternately tuned the strings C, G, 'd', A, E, in perfect fifths and fourths one from the other, we continue to tune successively by true fifths and fourths the strings E, B, FXX, CXX, GXX, 'dXX', EXX, BXX; we shall find, that though BXX, being a semitone higher than the natural note, should be equi- valent to c natural, it will by no means form a just octa- tive to the first C in the scale, but be considerably higher (gg); yet this BXX upon the harpsichord ought not

should attempt to rise to it by that note in the scale which is most immediately contiguous, we cannot reach it but by the note B, and by the semitone from B to c'. Now to make a transition from B to c', by this fe- mitone, the ear must of necessity be predisposed for that modulation, and consequently preoccupied with the mode of C: if this were not the case, we should naturally rise from B to c', and by this operation pass into an- other mode.

(ff) The A considered as the fifth of D is $\frac{7}{12}$, and the fourth beneath this A will constitute $\frac{3}{4}$ of $\frac{7}{12}$, that is to say, $\frac{8}{12}$; $\frac{9}{12}$ then shall be the value of E, considered as a true fourth from A in descending: now E, considered as the third major of the found C, is $\frac{5}{12}$, or $\frac{8}{12}$: these two E's then are between themselves in the proportion of $\frac{8}{12}$ to $\frac{5}{12}$; thus it is impossible that E should be at the same time a perfect third major from C, and a true fourth beneath B.

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

C, G, a fifth; D a fourth; A a fifth; E a fourth; B a fifth; FXX a fifth; CXX a fifth; GXX a fourth; 'dXX' a fifth; AXX a fourth; 'eXX' or 'fXX' a fifth; BXX a fourth: now it will be found, by a very easy computation, that the first C being represented by 1, G shall be $\frac{3}{4}$, D $\frac{5}{12}$, A $\frac{7}{12}$, E $\frac{9}{12}$, &c. and so of the rest till you arrive at BXX, which will be found $\frac{11}{12}$. This fraction is evidently greater than the number 2, which expresses the perfect octave c to its correspondent C: and the octave below BXX would be one half of the same fraction, that is to say $\frac{11}{24} = \frac{1}{2}$, which is evidently greater than C represented by unity. This last fraction $\frac{11}{24}$ is compo- sed of two numbers; the numerator of the fraction is nothing else but the number 3 multiplied 11 times in succes- sion by itself, and the denominator is the number 2 multiplied 18 times in succession by itself. Now it is evi- dent, that this fraction which expresses the value of BXX, is not equal to the unity which expresses the value of the found C, though upon the harpsichord, BXX and C are identical. This fraction rises above unity by $\frac{1}{24}$, that is to say, by about $\frac{1}{24}$; and this difference was called the comma of Pythagoras. It is palpable that this comma is much more considerable than that which we have already mentioned (note bb), and which is only $\frac{1}{80}$.

We have already proved that the series of fifths produces a 'c' different from BXX, the series of thirds major gives another still more different. For, let us suppose this series of thirds, C, E, GXX, BXX, we shall have E equal to $\frac{5}{12}$, GXX to $\frac{7}{12}$, and B to $\frac{9}{12}$, whose octave below is $\frac{11}{12}$; from whence it appears, that this last B is less than unity (that is to say than C), by $\frac{1}{24}$, or by $\frac{1}{24}$, or near it: A new comma, much greater than the preceding, and which the Greeks have called apotome major.

It may be observed, that this BXX, deduced from the series of thirds, is to the BXX deduced from the series of fifths, as $\frac{11}{24}$ is to $\frac{1}{24}$: that is to say, in multiplying by 524288, as 125 multiplied by 4996 is to 531441, or as 51200 to 531441, that is to say, nearly as 26 is to 27: from whence it may be seen, that these two B'sXX are very considerably different one from the other, and even sufficiently different to make the ear sensible of it; because the difference consists almost of a minor semitone, whose value, as will afterwards be seen (art. 139.), is $\frac{1}{24}$.

Moreover, if, after having found the GXX equal to $\frac{7}{12}$, we then tune by fifths and by fourths, GXX, 'dXX', AXX, CXX, BXX, as we have done with respect to the first series of fifths, we find that the BXX must be $\frac{11}{24}$; its differ- ence, then, from unity, or, in other words, from C, is $\frac{1}{24}$, that is to say, about $\frac{1}{24}$; a comma still less than any of the preceding, and which the Greeks have called apotome minor. not to be different from the octave above C; for every B♭ and every c' is the same sound, since the octave or the scale only consists of twelve semitones.

66. From thence it necessarily follows, i. That it is impossible that all the octaves and all the fifths should be just at the same time, particularly in instruments which have keys, where no intervals less than a semitone are admitted. 2. That, of consequence, if the fifths are justly tuned, some alteration must be made in the octaves; now the sympathy or sound which subsists between any note and its octave, does not permit us to make such an alteration: this perfect coalescence of sound is the cause why the octave should serve as limits to the other intervals, and that all the notes which rise above or fall below the ordinary scale, are no more than replications, i.e., repetitions, of all that have gone before them. For this reason, if the octave were altered, there could be no longer any fixed point either in harmony or melody. It is then absolutely necessary to tune the c' or B♭ in a just octave with the first; from whence it follows, that, in the progression of fifths, or, what is the same thing, in the alternate series of fifths and fourths, C, G, D, A, E, B, F♯, C♯, G♯, d♭, A♭, e♭, B♭, it is necessary that all the fifths should be altered, or at least some of them. Now, since there is no reason why one should rather be altered than another, it follows, that we ought to alter them all equally. By these means, as the alteration is made to influence all the fifths, it will be in each of them almost imperceptible; and thus the fifth, which, after the octave, is the most perfect of all consonances, and which we are under the necessity of altering, must only be altered in the least degree possible.

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

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

69. It results then from all that we have now said, Principle that the theory of temperament may be reduced to this question.—The alternate succession of fifths and fourths having been given, (art. 66.), in which B♭ or C is not the true octave of the first C; it is proposed to alter all the fifths equally, in such a manner that the two C's may be in a perfect octave the one to the other.

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

In a word, if, after having found E equal to \( \frac{4}{3} \) in the progression of thirds, we then tune by fifths and fourths E, B, F♯, C♯, &c. we shall arrive at a new B♭, which shall be \( \frac{15}{16} \), and which will not differ from unity but by about \( \frac{1}{16} \), which is the least and smallest of all the commas; but it must be observed, that, in this case, the thirds major from E to G♯, from G♯ to B♭ or C, &c. are extremely false, and greatly altered.

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

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

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

\[ \begin{array}{cccccccc} C & C^{\#} & D & D^{\#} & E & F & F^{\#} & G & G^{\#} \\ \sqrt{2} & \sqrt{2^{\frac{1}{2}}} & \sqrt{2^{\frac{1}{3}}} & \sqrt{2^{\frac{1}{4}}} & \sqrt{2^{\frac{1}{5}}} & \sqrt{2^{\frac{1}{6}}} & \sqrt{2^{\frac{1}{7}}} & \sqrt{2^{\frac{1}{8}}} & \sqrt{2^{\frac{1}{9}}} \\ A & A^{\#} & B & c' & \sqrt{2^{\frac{1}{10}}} & \sqrt{2^{\frac{1}{11}}} & \sqrt{2^{\frac{1}{12}}} & \sqrt{2^{\frac{1}{13}}} & \sqrt{2^{\frac{1}{14}}} \\ \end{array} \]

It is obvious, that in this temperament all the fifths are equally altered. One may likewise prove, that the alteration of each in particular is very inconsiderable; for it will be found, for instance, that the fifth from C to G, which should be \( \frac{3}{2} \), ought to be diminished by about \( \frac{1}{7} \) of \( \frac{3}{2} \); that is to say, by \( \frac{1}{7} \), a quantity almost inconceivably small.

It is true, that the thirds major will be a little more altered; for the third major from C to E, for instance, shall be increased in its interval by about \( \frac{1}{7} \); but it is better, according to M. Rameau, that the alteration should fall upon the third than upon the fifth, which after the octave is the most perfect chord, and from the perfection of which we ought never to degenerate but as little as possible.

Besides, it has appeared from the series of thirds major C, E, C♯, B♭, that this last B♭ is very different from c' (note GG); from whence it follows, that if we would tune this B♭ in unison with the octave of C, and alter at the same time each of the thirds major by a degree as small as possible, they must all be equally altered. This is what occurred in the temperament which we propose; and if in it the third be more altered than the fifth, it is a consequence of the difference which we find between the degrees of perfection in these intervals; a difference with which, if we may speak so, the temperament proposed conforms itself. Thus this diversity of alteration is rather advantageous than inconvenient. each fifth will be very considerable, but equal in all of them.

71. In this, then, the theory of temperament consists: but as it would be difficult in practice to tune a harpsichord or organ by thus rendering all the semitones equal, M. Rameau, in his *Generation Harmonique*, has furnished us with the following method, to alter all the fifths as equally as possible.

72. Take any key of the harpsichord which you please; but let it be towards the middle of the instrument; for instance, C: then tune the note G a fifth above it, at first with as much accuracy as possible; this you may imperceptibly diminish: tune afterwards the fifth to this with equal accuracy, and diminish it in the same manner; and thus proceed from one fifth to another in ascent: and as the ear does not appreciate so exactly sounds that are extremely sharp, it is necessary, when by fifths you have risen to notes extremely high, that you should tune in the most perfect manner the octave below the last fifth which you had immediately formed; then you may continue always in the same manner; till in this process you arrive at the last fifth from E to B, which should of themselves be in tune; that is to say, they ought to be in such a state, that B, the highest note of the two which compose the fifth, may be identical with the sound C, with which you began, or at least the octave of that sound perfectly just: it will be necessary then to try if this C, or its octave, forms a just fifth with the last found E or F, which has been already tuned. If this be the case, we may be certain that the harpsichord is properly tuned. But if this last fifth be not true, in this case it will be too sharp, and it is an indication that the other fifths have been too much diminished, or at least some of them; or it will be too flat, and consequently discover that they have not been sufficiently diminished. We must then begin and proceed as formerly, till we find the last fifth in tune of itself, and without our immediate interpolation (ii).

Vol. XIV. Part II.

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

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

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

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

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

which, in their judgment, renders the modes of C and E proper for different manners of expression. But after all that we have laid in this treatise on the formation of diatonic intervals, every one should be convinced, that, according to the intention of nature, the diatonic scale ought to be perfectly the same in all its modes: The contrary opinion, says M. Rameau, is a mere prejudice of musicians. The character of an air arises chiefly from the intermixture of the modes; from the greater or lesser degrees of vivacity in the movement; from the tones, more or less grave, or more or less acute, which are assigned to the generator of the mode; and from the chords more or less beautiful, as they are more or less deep, more or less flat, more or less sharp, which are found in it.

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

Yet M. Rameau, in his *New System of Music*, printed in 1726, adopted the ordinary temperament. In that work, (as may be seen chap. xxiv.), he pretends that the alteration of the fifths is much more supportable than that of the thirds major; and that this last interval can hardly suffer a greater alteration than the octave, which, as we know, cannot suffer the slightest alteration. He says, that if three strings are tuned, one by an octave, the other by a fifth, and the next by a third major to a fourth string, and if a sound be produced from the last, the strings tuned by a fifth will vibrate, though a little less true than it ought to have been; but that the octave and the third major, if altered in the least degree, will not vibrate: and he adds, that the temperament which is now practised, is founded upon that principle. M. Rameau goes still farther; and as, in the ordinary temperament, By this method all the twelve sounds which compose the scales shall be tuned; nothing is necessary but to tune with the greatest possible exactness their octaves in the other scales, and the harpsichord shall be well tuned.

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

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

We may give another experiment. Let the three keys E, G, B be struck upon an organ, and the minor perfect chord only will be heard; though E, by the construction of that instrument, must cause G likewise to be heard; though G should have the same effect upon D, and B upon F; insomuch that the ear is at once affected with all these sounds, D, E, F, G, G, B: how many dissonances perceived at the same time, and what a jarring multitude of discordant sensations, would result from thence to the ear, if the perfect chord with which it is preoccupied had not power entirely to abstract its attention from such sound as might offend!

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

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

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

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

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

That the reader may have a clear idea of the term before he enters upon the subject of this chapter, it may be necessary to caution him against a mistake into which he may be too easily led by the ordinary signification of the word repose. In music, therefore, it is far from being synonymous with the word rest. It is, on the contrary, the termination of a musical phrase which ends in a cadence more or less emphatic, as the sentiment implied in the phrase is more or less complete. Thus a repose in music answers the same purpose as punctuation in language. See Repose, in Rousseau's Musical Dictionary. In the first case, this word only signifies an agreeable and rapid alteration between two contiguous sounds, called likewise a trill or shake; in the second, it signifies a repose or close. It is however true, that this shake implies, or at least frequently enough prefaces, a repose, either present or impending, in the fundamental base (ll.).

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

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

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

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

78. In the second chapter, we have explained (art. 20, 30, 31, and 32) by what means, and upon what principle, the minor chord C, Eb, G, 'c', may be formed, which is the characteristic chord of the minor mode. Now what we have there said, taking C for the principal and fundamental sound, we might likewise have said of any other note in the scale, assumed in the same manner as the principal and fundamental sound: but as in the minor chord, C, Eb, G, 'c', there occurs an Eb which is not found in the ordinary diatonic scale, we shall immediately substitute, for greater ease and convenience, another chord, which is likewise minor and exactly similar to the former, of which all the notes are found in the scale.

79. The scale affords us three chords of this kind, viz. D, F, A, 'd'; A, 'c', e, a'; and E, G, B, 'c'. Among these three we shall choose A, 'c', e, a'; because this chord, without including any sharp or flat, has two sounds in common with the major chord C, E, G, 'c'; and besides, one of these two sounds is the very same 'c': so that this chord appears to have the most immediate, and at the same time the most simple, relation with the chord C, E, G, 'c'. Concerning this we need only add, that this preference of the chord A, 'c', e, a' to every other minor chord, is by no means in itself necessary for what we have to say in this chapter upon the diatonic scale of the minor mode. We might in the same manner have chosen any other minor chord; and it is only, as we have said, for greater ease and convenience that we fix upon this.

80. In every mode, whether major or minor, the Tonic or principal sound which implies the perfect chord, whose key in harsher major or minor, is called the tonic note or key; thus, if C is the key in its proper mode, A in the mode of A, See Principal &c. Having laid down this principle,

81. We have shown how the three sounds, F, C, See Tonic, G, which constitute (art. 38.) the mode of C, of which the formation of the first, F, and the last, G, are the two fifths of C, one descending, the other rising, produce the scale, B, 'c', d, fixed e, f, g, a'; of the major mode, by means of the fundamental bases G, C, G, C, F, C, F; let us in the same manner take the three sounds D, A, E, which constitute the mode of A, for the same reason that the sounds F, C, G, constitute the mode of C; and of them let us form this fundamental base, perfectly like the preceding E, A, F, A, D, A, D; let us afterwards place See fig. 7, below each of these sounds one of their harmonies, as we have done (chap. v.), for the first scale of the major mode; with this difference, that we must suppose D and A as implying their thirds minor in the fundamental base to characterize the minor mode; and we shall have the diatonic scale of that mode,

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

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

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

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

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

B, 'c', d, e, f, g, a', that from 'c' to 'f' which are the two last notes of the former scale, there is only a semitone; whereas from 'g' to 'a', which are the two last sounds of the latter series, there is the interval of a complete tone; but this is not the only discrimination which may be found between the scales of the two modes.

---

(ll.) M. Rousseau, in his letter on French music, has called this alternate undulation of different sounds a trill, from the Italian word trillo, which signifies the same thing; and some French musicians already appear to have adopted this expression. 85. To investigate these differences, and to discover the reason for which they happen, we shall begin by forming a new diatonic scale of the minor mode, similar to the second scale of the major mode,

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

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

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

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

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

and it will produce the scale immediately subjacent,

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

in which \( c' \) forms a third minor with \( A \), which in the fundamental base corresponds with it, which designates the minor mode; and, on the contrary, \( g\# \) forms a third major with \( E \) in the fundamental base, because \( g\# \) rises towards \( a' \) (art. 82, §3.).

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

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

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

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

88. From thence it may be seen why \( f' \) and \( g' \) are sharp when ascending in the minor mode; besides the \( f' \) is only natural in the first scale \( G\# , A, B, c', d, e, f' \), because this \( f' \) cannot rise to \( g\# \), (art. 48.).

89. It is not the same case in descending. For \( E \), the fifth of the generator, ought not to imply the third major \( g\# \), but in the case when \( E \) descends to the generator \( A \) to form a perfect repose (art. 77. and 83.) and in this case the third major \( g\# \) rises to the generator \( a' \); but the fundamental base \( AE \) may, in descending, give the scale \( a', g' \), natural, provided \( g' \) does not rise again to \( a' \).

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

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

produces in descending,

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

And it is plain that the \( f' \) cannot be otherwise than sharp, since \( f\# \) is the fifth of the note \( B \) of the fundamental base. Experience, however, evinces that the \( f' \) is natural in descending in the diatonic scale of the major mode of \( A \), especially when the preceding \( g' \) is natural; and it must be acknowledged, that here the fundamental base appears defective.

M. Rameau has attempted the following solution of this difficulty. In the diatonic scale of the minor mode in descending, \( (a, g, f, e, d, c', B, A,) \) \( g' \) may be regarded simply as a note of passage, merely added to give yet unsweetness to the modulation, and as a diatonic gradation factory, by which we may descend to \( f' \) natural. This is easily perceived, according to M. Rameau, by the fundamental base,

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

which produces

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

which may be regarded, as he says, as the real scale of the minor mode in descending; to which is added \( g' \) natural between \( a' \) and \( f' \), to preserve the diatonic order.

This appears the only possible answer to the difficulty above proposed; but we know not whether it will fully satisfy the reader; whether he will not feel with regret, that the fundamental base does not produce, to speak properly, the diatonic scale of the minor mode in descent, when at the same time this same base so happily produces the diatonic scale of that identical mode in ascending, and the diatonic scale of the major mode whether in rising or descending (\( NN \)).

CHAP. X. Of Relative Modes.

91. Two modes of such a nature that we can pass from the one to the other, are called relative modes. Modes re. Thus the major mode of \( C \) is relative to the major mode of \( F \) and to that of \( G \). It has also been seen what how many intimate connexions there are between the major mode of \( C \), and the minor mode of \( A \). For, 1. The perfect chords, one major, \( C, E, G, c' \), the other minor, \( A, c', e, a' \), which characterize each of those two kinds of modulation * or harmony, have two sounds in common, \( c' \) and \( e' \). 2. The scale of the minor mode of \( A \) in descent, absolutely contains the same sounds with the scale of the major mode of \( C \).

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

92. In the minor mode of \( E \), the minor perfect chord \( E, G, B, c' \), which characterizes it, has likewise two sounds, \( E, G \), in common with the perfect chord major \( C, E, G, c' \), which characterizes the major mode of

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

(\( NN \)) When \( g' \) is said to be natural in descending the diatonic scale of the minor mode of \( A \), it is only meant that this \( g' \) is not necessarily sharp in descending as it is in rising; for it may be sharp, as may be proved by numberless examples, of which all musical compositions are full. It is true, that when \( g' \) is found sharp in descending to the minor mode of \( A \), we are not sure that the mode is minor till the \( f' \) or \( c' \) natural is found; both of which impress a peculiar character on the minor mode, viz. \( c' \) natural, in rising and in descending, and the \( f' \) natural in descending. But the minor mode of E is not so closely related nor allied to the major mode of C as the minor mode of A; because the diatonic scale of the minor mode of E in descent, has not, like the series of the minor mode of A, all these founds in common with the scale of C. In reality, this scale is e, d, c', B, A, G, Fx, E, where there occurs an f sharp which is not in the scale of C. Though the minor mode of E is thus less relative to the major mode of C than that of A; yet the artist does not hesitate sometimes to pass immediately from the one to the other.

When we pass from one mode to another by the interval of a third, whether in descending or rising, as from C to A, or from A to C, from C to E, or from E to C, the major mode becomes minor, or the minor mode becomes major.

93. There is still another minor mode, into which an immediate transition may be made in issuing from the major mode of C. It is the minor mode of C itself in which the perfect minor chord C, Eb, G, c', has two founds, C and G, in common with the perfect major chord C, E, G, c'. Nor is there anything more common than a transition from the major mode of C to the minor mode, or from the minor to the major (oo).

**Chap. XI. Of Diffonance.**

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

95. This advantage may be obtained by uniting at the same time the founds G and F in the same harmony, that is to say, by joining to the harmony G, B, d', of the fifth G, the other fifth F in this manner, G, B, d', f'; this f' which is added, forms a diffonance with G (art. 18.). Hence the chord G, B, d', f', is called a diffonant chord, or a chord of the seventh. It serves to distinguish the fifth G from the generator C, which always implies, without mixture or alteration, the perfect chord C, E, G, c', resulting from nature itself (art. 32.). By this we may see, that when we pass from C to G, one passes at the same time from C to F, because f' is found to be comprehended in the chord of G; and the mode of C by these means plainly appears to be determined, because there is none but that mode to which the founds F and G at once belong.

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

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

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

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

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

**Chap. XII. Of the Double Use or Employment of Diffonance.**

100. It is evident by the resemblance of sounds to account of their octaves, that the chord F, A, c', d', is in effect the double employment of the same as the chord D, F, A, c', taken inversely f', that the inverse of the chord C, A, F, D, has been found (art. 98.) in descending by thirds, from the generator C (pp).

101. The

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

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

(pp) M. Rameau, in several passages of his works (for instance, in p. 110, 111, 112, and 113, of the Generation Harmonique), appears to consider the chord D, F, A, C, as the primary chord and generator of the chord E, A, c', d', which is that chord reversed; in other passages (particularly in p. 116, of the same performance), he seems to consider the first of these chords as nothing else but the reverse of the second. It would seem that this 101. The chord D, F, A, 'c', is a chord of the seventh like the chord G, B, 'd, f'; with this only difference, that the latter in the third G, B, is major; whereas in the former, the third D, F, is minor. If the F were sharp, the chord D, F, A, 'c', would be a genuine chord of the dominant, like the chord G, B, D, 'f'; and as the dominant G may descend to C in the fundamental bass, the dominant D implying or carrying with it the third major F might in the same manner descend to G.

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

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

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

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

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

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

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

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

in which C is understood to carry with it the perfect chord C E G 'c'; G, the chord G B 'd f'; F the chord F A 'c d'; and D, the chord D F A 'c'.

It is plain from what has been said in the preceding chapter, that in this case C may ascend to D in the fundamental bass, and D descend to G, and that the impression of the mode of C is preserved by the 'f' natural, which forms the third minor 'd f', instead of the third major which D ought naturally to imply.

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

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

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

108. Thus the generator C may be followed according... ing to pleasure in ascending diatonically either by a tonic dominant (D F A C), or by a simple dominant (D F A C).

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

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

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

**Chap. XIV. Of the different Kinds of Chords of the Seventh.**

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

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

1. The chord G B 'd f', composed of a third major followed by two thirds minor. 2. The chord D F A 'c', or B 'd f X a', a third major between two minors. 3. The chord B 'd f a', two thirds minor followed by a major.

114. There are still two other kinds of chords of the seventh which are employed in harmony; one is composed of a third minor between two thirds major, C E G B, or F A 'c e'; the other is wholly composed of thirds minor G X B 'd f'. These two chords, which at first appear as if they ought not to enter into harmony if we rigorously keep to the preceding rules, are nevertheless frequently practised with success in the fundamental basses.

The reason is this:

115. According to what has been said above, if we would add a seventh to the chord C E G, to make last described, a dominant of C, one can add nothing but Bp; and ed admitted in this case C E G Bp would be the chord of the tonic dominant in the mode of F, as G B 'd f' is the chord of the tonic dominant in the mode of C; but if we would preserve the impression of the mode of C in the harmony, we change this Bp into B natural, and the chord C E G Bp becomes C E G B. It is the same case with the chord F A 'c e', which is nothing else but the chord F A 'c eb'; in which one may substitute for 'eb', 'e' natural, to preserve the impression of the mode of C, or that of F.

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

116. With respect to the chord of the seventh G X B 'd f', wholly composed of thirds minor, it may be regarded as formed from the union of the two chords of the seventh of the scale, c, 'b', a, g, f, e, d, c', in descent, we must either determine to invert the fundamental bass mentioned in art. 55. In this manner, C, G, D, G, C, F, C, G, C, in which the second G and the second C answer to the G alone in the scale; or otherwise we must form the fundamental bass C, G, D, G, C, G, C, in which all the notes imply perfect chords major, except the second G, which implies the chord of the seventh G, B, 'd, f', and which answers to the two notes of the scale G, F, both comprehended in the chord G, B, 'd, f'.

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

( rr ) On the contrary, a chord such as C Ep G B, in which E would be flat, could not be admitted in harmony, because in this chord the B is not included and underlined in Ep. It is the same case with several other chords, such as B D F A X, B D X F A, &c. If it true, that in the last of these chords, A is included in F, but it is not contained in D X; and this D X likewise forms with F and with A a double dissonance, which, joined with the dissonance B F, would necessarily render this chord not very pleasing to the ear; we shall yet, however, see in the second part, that this chord is sometimes used. Theory of the dominant and of the sub-dominant in the minor mode. In effect, in the minor mode of A, for instance, these two chords are E G B, d', and D E A B, whose union produces E G B, d', f, a. Now, if we should suffer this chord to remain thus, it would be disagreeable to the ear, by its multiplicity of dissonances, D E, E F, F G, A B, D G, (art. 18.) so that, to avoid this inconveniency, the generator A is immediately expunged, which, (art. 19.) is as it were undertood in D, and the fifth or dominant E, whose place the sensible note G is supposed to hold; thus there remains only the chord G B d'f', wholly composed of thirds minor, and in which the dominant E is considered as understood: in such a manner that the chord G B d'f' represents the chord of the tonic dominant E G B d', to which we have joined the chord of the sub-dominant D F A B, but in which the dominant E is always reckoned the principal note (ss).

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

**CHAP. XV. Of the Preparation of Discords.**

Diffidence, what.

118. In every chord of the seventh, the highest note, that is to say, the seventh above the fundamental, is called a diffidence or discord; thus f' is the diffidence of the chord G B d'f'; c' in the chord D F, A c', &c.

Manner of preparing discidences investigated.

119. When the chord G B d'f' follows the chord C E G c', as often happens, it is obvious that we do not find the diffidence f' in the preceding chord C E G c'. Nor ought it indeed to be found in that chord; for this diffidence is nothing else but the sub-dominant added to the harmony of the dominant to determine the mode: now, the sub-dominant is not found in the harmony of the generator.

120. For the same reason, when the chord of the sub-dominant F A c'd' follows the chord C E G c', the note d', which forms a diffidence with c', is not found in the preceding chord.

It is not so when the chord D F A c' follows the chord C E G c'; for c' which forms a diffidence in the second chord, stands as a consonance in the preceding.

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

122. In every chord of the seventh, which is not preparation the chord of the tonic dominant, that is to say, (art. 120.) which is not composed of a third major followed by two thirds minor, the diffidence which this chord performed, forms ought to stand as a consonance in the chord which precedes it.

This is what we call a prepared diffidence. See Pre-

123. Hence, in order to prepare a diffidence, the preparation, fundamental bas must necessarily ascend by the interval of a second, as

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

or descend by a third, as

C E G c', A C E G;

or descend by a fifth, as

C E G c', F A C E;

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

124. When a tonic, that is to say, a note which carries with it a perfect chord, is followed by a dominant in the interval of a fifth or third, this succession may be regarded as a process from that same tonic to another, which has been rendered a dominant by the addition of the diffidence.

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

**CHAP. XVI. Of the Rules for resolving Diffidences.**

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

126. Now in the chord of the tonic dominant it ought chord of the tonic dominant, the diffidence should rather be than descend, and why.

(ss) We have seen (art. 109.) that the chord B d'f'a', in the minor mode of A, may be regarded as the inverse of the chord D F A B; it would likewise seem, that, in certain cases, this chord B d'f'a' may be considered as composed of the two chords G B d'f', F A c'd' of the dominant and of the sub-dominant of the major mode of C, which chords may be joined together after having excluded from them, 1. The dominant G, represented by its why third major B, which is presumed to retain its place. 2. The note C which is understood in F, which will form this chord B d'f'a'. The chord B d'f'a', considered in this point of view, may be understood as belonging to the major mode of C upon certain occasions. ought rather to descend than to rise; for this reason.

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

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

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

128. And, for the same reason, in the chord of the sub-dominant F A 'c', the dissonance 'd' ought to rise to 'e' of the following chord C E G 'c', rather than descend to 'c'; whence may be deduced the following rules.

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

20. In every chord of the sub-dominant, the dissonance ought to rise diatonically upon the third of the subsequent chord.

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

From these rules it is a necessary result, that the chord of the seventh D F A 'c', though it should even be considered as the inverse of F A 'c', cannot be succeeded by the chord C E G 'c', since there is not in this last chord the note B, upon which the dissonance 'c' of the chord D F A 'c' can descend.

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

CHAP. XVII. Of the Broken or Interrupted Cadence.

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

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

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

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

134. The broken cadence may be considered as having its origin in the double employment of dissonances, because this cadence, like the double employment, only the double confits in a diatonic procedure of the bass ascending employed (chap. xii.). In effect, nothing hinders us to descend from the chord G B 'd f' to the chord C E G A by converting the tonic C into a sub-dominant, that is to say, by passing all at once from the mode of C to the mode of G: now to descend from G B 'd f' to C E G A is the same thing as to rise from the chord G B 'd f' to the chord A 'c e g', in changing the chord of the sub-dominant C E G A for the imperfect chord of the dominant, according to the laws of the double employment.

135. In this kind of cadence, the dissonance of the first chord is resolved by descending diatonically upon the fifth of the subsequent chord. For instance, in performing the broken cadence G B 'd f', A 'c e g', the dissonance 'f' is resolved by descending diatonically upon the fifth 'e'.

136. There is another kind of cadence, called an interrupted cadence, where the dominant descends by a third to another dominant, instead of descending by a fifth upon the tonic, as in this succession of the basses G B 'd f', 'cadence'. Theory of G B d f', E G B d f'; in the case of an interrupted cadence, the difference of the former chord is resolved by descending diatonically upon the octave of the fundamental note of the subsequent chord, as may be here seen, where d f' is resolved upon the octave of E.

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

Chap. XVIII. Of the Chromatic Species.

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

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

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

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

142. Every procedure of the fundamental bases by thirds, whether major or minor, rising or descending, gives the minor semitone. This we have already seen from the succession of thirds in ascending. The series of thirds minor in descending, CA, gives, C, CX the fundamental base by thirds.

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

Chap. XIX. Of the Enharmonic Species.

144. The two extremes, or highest and lowest notes, Diéis or CG X, of the fundamental bases by thirds major CEG X, enharmonically give this modulation c'e' B X; and these two sounds c'e' nic inter-vene what, B X, differ between themselves by a small interval which is called the diéis, or enharmonic fourth* of a tone (3A), formed.

* See Fourth of a Tone.

Fig. 11.

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

(uu) A minor joined to a major semitone will form a minor tone; that is to say, if one rises, for instance, from E to F, by the interval of a semitone major, and afterwards from F to F X by the interval of a minor semitone, the interval from E to F X will be a minor tone. For let us suppose E to be 1, F will be 4/3, and F X will be 5/4 of 4/3; that is to say, 25 times 16 divided by 24 times 15, or 10/9; E then is to F X as one is to 10/9, the interval which constitutes the minor tone (note BB).

With respect to the tone major, it cannot be exactly formed by two semitones; for, 1. Two major semitones in immediate succession would produce more than a tone major. In effect, 10/9 multiplied by 4/3 gives 5/4, which is greater than 10/9, the interval which constitutes (note BB) the major tone. 2. A semitone minor and a semitone major would give less than a major tone, since they amount only to a true minor. 3. And, à fortiori, two minor semitones would still give less.

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

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

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

(3A) G X being 5/4 and B X being 4/3 of 5/4, we shall have B X equal (note Q) to 5/4, and its octave below shall be 5/4; an interval less than unity by about 1/28 or 1/27. It is plain then from this fraction, that the B X in question must be considerably lower than C. This interval has been called the fourth of a tone, and this denomination is founded on reason. In effect, we may distinguish in music four kinds of quarter tones.

1. The fourth of a tone major; now, a tone major being \( \frac{3}{4} \), and its difference from unity being \( \frac{1}{8} \), the difference of this quarter tone from unity will be almost the fourth of \( \frac{1}{2} \); that is to say, \( \frac{1}{2} \).

2. The fourth of a tone minor; and as a tone minor, which is \( \frac{5}{8} \), differs from unity by \( \frac{1}{9} \), the fourth of a minor tone will differ from unity about \( \frac{1}{7} \).

3. One half of a semitone major; and as this semitone differs from unity by \( \frac{1}{8} \), one half of it will differ from unity about \( \frac{1}{16} \).

4. Finally, one half of a semitone minor, which differs from unity by \( \frac{1}{16} \): its half then will be \( \frac{1}{32} \).

The interval, then, which forms the enharmonic fourth of a tone, as it does not differ from unity but by \( \frac{1}{16} \), may justly be called the fourth of a tone, since it is less different from unity than the largest interval of a quarter tone, and more than the least.

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

(3 b) That is to say, that if you rise from E to F, for instance, by the interval of a semitone major, and afterwards, returning to E, you should rise by the interval of a semitone minor to another sound which is not in the scale, and which I shall mark thus, \( F_{+} \), the two sounds \( F_{+} \) and F will form the enharmonic fourth of a tone: for E being \( \frac{1}{2} \), F will be \( \frac{1}{2} \); and \( F_{+} \) is that of \( \frac{1}{2} \) to \( \frac{1}{2} \) (note q); that is to say, as 25 times 15 to 16 times 24; or otherwise, as 25 times 5 to 16 times 8, or as 125 to 128.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(3 D) It is obvious, that if F in the bas be supposed 1, 'f' of the scale will be 2, C of the bas 3 and 'e' of the scale 4 of 4, that is, 1/5; the proportion of 'f' to 'e' is as 2 to 1/5, or as 1 to 1/5. Now E of the bas being likewise 4 of 4, or 1/5; B of the bas is 3 of 4, and its third major D of 4 of 3, or 1/5 of 1/5; this third major, approximated as much as possible to 'e' in the scale by means of octaves, will be 1/5 of 1/5; 'e' then of the scale will be to 'd' which follows it, as 1/5 is to 1/5 of 1/5, that is to say, as 1 to 1/5. The semitones then from 'f' to 'e', and from 'e' to 'd', are both major.

(3 E) It is evident that 'e' is 6 (note Q), and that 'e' is 6: these two 'e's, then, are between themselves as 6 to 3, that is to say, as 6 times 4 to 5 times 5, or as 24 to 25, the interval which constitutes the minor semitone. Moreover, the A of the bas is 5, and C of 5 of 5, or 2/5; 'e' then is 5 of 2/5, the 'e' in the scale is likewise to the 'e' which follows it, as 24 to 25. All the semitones therefore in this scale are minor.

(3 F) Many composers begin with determining and writing the bas; a method, however, which appears in general General Remark.

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

PART II. PRINCIPLES AND RULES OF COMPOSITION.

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

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

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

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

160. An interval composed of a tone and a semitone, which is commonly called a third minor, is likewise sometimes called a second redundant; such is the interval from C to D in ascending, or that of A to Gb descending.

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

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

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

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

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

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

CHAP. II. Comparison of the Different Intervals.

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

166. On account then of the resemblance between every

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

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

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

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

(3 K) The chief use of these different denominations is therefore to distinguish chords: for instance, the chord of the redundant fifth and that of the diminished seventh are different from the chord of the sixth; the chord of the seventh redundant, from that of the seventh major. This will be explained in the following chapters. Musical sounds, like language, are represented by written characters, by which their graveness or acuteness, their duration, and the other qualities intended to be assigned to them, are accurately distinguished.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

171. As the notes are placed on the lines, and in the spaces between the lines, the name of any note may be discovered from the position of the cleff. Thus, in the F cleff, the note on the lowest line is G; the note on the space between the two first lines A; the note on second line B, &c.

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

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

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

The occasional short lines thus employed are termed leger lines.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

175. The longest of all notes is a semibreve. A minim is half its value; that is to say, two minims are to be performed in the time occupied by one semibreve.

A minim in the same manner is equivalent to two crotchets, the crotchet to two quavers (3 O).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The measure of nine quavers is marked \( \frac{9}{8} \), (fig. 9.) and is divided into one strong and two weak times. That of twelve quavers is marked \( \frac{12}{8} \), (fig. 10.) and is accented as if it were two measures of six quavers.

The measures of \( \frac{9}{8} \) and \( \frac{12}{8} \) rarely occur.

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

Chap. IV. Definition of the principal Chords.

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

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

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

All chords of the seventh are practised in harmony, save that which might carry the third minor and the seventh major, as C E G B; and that which might carry a false fifth and a seventh major, B D F A, (chap. xiv. Part I.).

180. All thirds are either major or minor, and as they may be differently arranged, it is clear that there are different kinds of chords of the seventh; there is even one, B D F A, which is composed of a third, a false fifth, and a seventh.

181. A chord composed of a third, a fifth, and a sixth, as F A C D, D F A B, is called a chord of the greater sixth.

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

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

It is necessary to remark, that among the chords

Vol. XIV. Part II.

of the seventh we do not reckon the chord of the seventh diminished, which is only improperly called a chord of the seventh; and of which we shall say more below.

184. Every note which carries the chord of the sub-dominant, great fifth, is called a subdominant, (art. 97. and 42.) and what, and is marked with a 6. Thus in the example iii. how F carries the chord of F A C D. The fifth should always be major, (art. 97. and 109.).

185. In every chord, whether perfect, or a chord fundamental of the seventh, or of the great fifth, the note which carries this chord, and which is the flattest or lowest, what, is called the fundamental note. Thus C in the ex-Sample 1. D and C in the example ii. and F in the ex-mental. Example iii. are fundamental notes.

186. In every chord of the seventh, and of the great difference sixth, the note which forms the seventh or fifth above of a chord, the fundamental, that is to say, the highest note of the chord, is called a dissonance. Thus in the chords of the seventh G B D F, D F A C, F and C are the dissonances, viz. F with relation to G in the first chord, and C with relation to D in the second. In the chord of the great fifth F A C D, D is the dissonance (art. 120.); but that D is only, properly speaking, a dissonance with relation to C from which it is a second, and not with respect to F from which it is a sixth major (art. 17. and 18.).

187. When a chord of the seventh is composed of Tonie and a third major followed by two thirds minor, the fundamental note of this chord is called the tonic dominant. In every other chord of the seventh the fundamental is called the simple dominant (art. 102.). Thus in the chord G B D F, the fundamental G is the tonic dominant; but in the other chords of the seventh, as C E G B, D F A C, &c. the fundamentals C and D are simple dominants.

188. In every chord, whether perfect, or of the major seventh, or of the sixth, if it is meant that the third chords, above the fundamental note should be major though how rendered minor, a sharp must be placed above the fundamental, vice versa.

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

Certain other characters will be with propriety explained here.

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

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

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

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

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

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

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

On the contrary, when the third is naturally major, and if we would render it minor, we place above the fundamental note a ♭. Thus the example vii. viii. ix. show the chords G B♭ D G G B♭ D F, G B♭ D E (3 s).

**Chap. V. Of the Fundamental Bas.**

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

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

In reality, the first note C in the upper part is found in the chord of the first note C in the bas, which chord is G E G C; the second note D in the treble is found in the chord G B D G; which is the chord of the second note in the bas, &c. and the bas is composed only of notes which carry a perfect chord, or that of the seventh, or that of the great sixth. Moreover it is formed according to the rules which we of Composition are now about to give.

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

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

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

**Rule I.**

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

**Rule II.**

192. In every chord of the simple dominant, it is necessary that the note which constitutes the seventh, or dissonance, should likewise be found in the preceding chord.

**Rule III.**

193. In every chord of the sub-dominant, at least one of its consonances must be found in the preceding chord. Thus, in the chord of the sub-dominant F A C D, it is necessary that F, A, or C, which are the consonances

(3 s) We may only add, that there is no occasion for marking these sharps or flats when they are originally placed at the clef. For instance, if the sharp be upon F which indicates the key of G (see Exam. x.) it is sufficient to write D, without a sharp, to mark the perfect chord major of D, D F A D. In the same manner, in the Example xi. where the flat is at the clef upon B, which denotes the key of F, it is sufficient to write G, to mark the perfect chord minor of G B♭ D G.

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

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

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

If there be one sharp at the clef, and if we would mark the chord G B D F♯ or the chord A C E F♯, we ought to place before the seventh or the fifth a ♯ or a ♭.

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

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

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

This last reflection is necessary, as will presently be seen. For, let us assume the succession of the two chords A C E G, D F A C (see Exam. xvii.), this succession is by no means legitimate, though in the first dominant descends by a fifth; because the C which forms the dissonance in the second chord, and which belongs to a simple dominant, is not in the preceding chord. But the succession will be admissible, if, without meddling with the second chord, we take away the sharp carried by the C in the first; or if, without meddling with the first chord, we render C and F sharp in the second (3 T); or, if we simply render the D of the second chord a tonic dominant, in causing it to carry F instead of F (119, and 122.).

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

D F A C, G B D F;

(see Exam. xviii.).

RULE V.

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

REMARK.

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

RULE I.

If a note of the fundamental bass be a tonic, and rise by a fifth or a third to another note, that second

note may be either a tonic (34, & 91.), see Examples Principes xix. and xx. (3 U); a tonic dominant (124.), see Composition, xxI. and xxII.; or a sub-dominant (124.), see xxIII., and xxIV.; or, to express the rule more simply, that second note may be any one, except a simple dominant.

RULE II.

If a note of the fundamental bass be a tonic, and descend by a fifth or a third upon another note, this second note may be either a tonic (34, & 91.) see Exam. xxv. and xxvi.; or a tonic dominant, or a simple dominant, yet in such a manner that the rule of art. 192. may be observed (124.), see xxvii. xxviii., xxix. and xxx.; or a sub-dominant (124.), see xxxi. and xxxii.

The succession of the bass C Eb G C, F A C E, is excluded by art. 192.

RULE III.

If a note in the fundamental bass be a tonic, and rise by a second to another note, that note ought to be a tonic dominant, or a simple dominant (101. & 102.). See xxxiv. and xxxv. (3 X).

We must here advertize our readers, that the examples xxxvi. xxxvii. xxxviii. xxxix. belong to the fourth rule above, art. 194.; and the examples xl. xli. xlii. to the fifth rule above, art. 195. See the articles 34, 35, 121, 123, 124.

REMARK I.

196. The transition from a tonic dominant to a perfect and tonic is called an absolute repose, or a perfect cadence (73.); and the transition from a sub-dominant to a what, and tonic is called an imperfect or irregular cadence (73.); where the tonic falls upon the accented part of the bar. See played.

XLIII. XLIV. XLV. XLVI.

REMARK II.

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

3 Y 2

CHAP.

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

(3 U) When the bass rises or descends from one tonic to another by the interval of a third, the mode is commonly changed; that is to say, from a major it becomes a minor. For instance, if we ascend from the tonic C to the tonic E, the major mode of C, C E G C, will be changed into the minor mode of E, E G B E. We must never ascend from one tonic to another, when there is no found common to both their modes: for example, we cannot rise from the mode of C, C E G C, to the minor mode of Eb, Eb Gb Bb Eb (91.).

(3 X) Thus all the intervals, viz. the third, the fifth, and second, may be admitted in the fundamental bass, except that of a second in descending. The rules now given for the fundamental bass, are not, however, without exception, as approved compositions in music will certainly discover; but these exceptions being in reality licences, and for the most part in opposition to the great principle of connection, which prefers that there should be at least one note in common between a preceding and a subsequent chord, it does not seem necessary to enter into a minute detail of these licences in an elementary work, where the first and most essential rules of the art alone ought to be expected.

(3 Y) There are notes which may be found several times in the fundamental bass in succession with a different 198. The treble is nothing else but a modulation above the fundamental bass, and whose notes are found in the chords of that bass which corresponds with it (189.). Thus in Ex. xvi. the scale C D E F G A B C, is a treble with respect to the fundamental bass C G C F C D G C.

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

1. It is obvious, that many notes of the treble may answer to one and the same note in the fundamental bass, when these notes belong to the chord of the same note in the fundamental bass. For example, this modulation C E G E C, may have for its fundamental bass the note C alone, because the chord of that note comprehends the sounds C, E, G, which are found in the treble.

2. In like manner, a single note in the treble may, for the same reason, answer to several notes in the bass. For instance, G alone may answer to these three notes in the bass, C G C (3 z).

Rule I. For the TREBLE.

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

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

Rule II.

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

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

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

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

(3 z) There are often in the treble several notes which may, if we choose, carry no chord, and be regarded merely as notes of passage, serving only to connect between themselves the notes that do carry chords, and to form a more agreeable modulation. These notes of passage are commonly quavers. See Example XLVII. (Plate CCCLVIII). In which this modulation C D E F G, may be regarded as equivalent to this other, C E G, as D and F are no more than notes of passage. So that the bass of this modulation may be simply C G.

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

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

(4 A) There is, however, one case in which the seventh of a simple dominant may be found in a modulation without being prepared. It is when, having already employed that dominant in the fundamental bass, its seventh is afterwards heard in the modulation, while the dominant is still retained. For instance, let us imagine this modulation,

\[ \begin{array}{ccc} C & D & C \\ \end{array} \]

and this fundamental bass,

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

(see example LI.) the D of the fundamental bass answers to the two notes D C of the treble. The dissonance C has no need of preparation, because the note D of the fundamental bass having already been employed for the D which precedes C, the dissonance C is afterwards presented, below which the chord D may be preserved, or DFAC. Chap. VIII. Of the Continued Bass and its Rules.

284. The continued bass, is a fundamental bass whose chords are inverted. We invert a chord when we change the order of the notes which compose it. For example, if instead of the chord G B D F, we should say B D F G or D F G B, &c. the chord is inverted.

The ways in which a PERFECT CHORD may be INVERTED.

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

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

2. G C E G, which we call a chord of the sixth and fourth, composed of a fourth, a fifth, and an octave; and it is marked with a 6. (See LVII.)

The perfect minor chord is inverted in the same manner.

The ways in which the CHORD of the SEVENTH may be INVERTED.

286. In the chord of the tonic dominant, as G B D F, the third major B above the fundamental note G is called a sensible note (77.) and the inverted chord B D F G composed of a third, a false fifth and a sixth, is called the chord of the false fifth, and is marked as in examples LVIII. and LIX.

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

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

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

1. F A C D, a chord of the great sixth, which is composed of a third, a fifth, and a sixth, and which is figured with a 6. See LXII. (4 e).

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

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

The ways in which the CHORD of the sub-DOMINANT may be Inverted.

288. The chord of the sub-dominant, as F A C D, may be inverted in three different manners; but the method of inverting it which is most in practice is the chord of the lesser sixth A C D F (LXIII.), and the chord of the seventh D F A C. See LXV.

RULES for the CONTINUED BASS.

289. The continued bass is a fundamental bass, whose chords are only inverted in order to render it more in the taste of singing, and suitable to the voice. See LXVI. in which the fundamental bass, which in itself is monotonic and little suited for singing, C G C G C G C, produces, by inverting its chords, this continued bass highly proper to be sung, C B C D E F E, &c. (4 h).

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

RULE I:

290. Every note which carries the chord of the false fifth, Principles fifth, and which of consequence must be what we have of Composition called a sensible note, ought (77.) to rise diatonically upon the note which follows it. Thus in example LXIV. the note B, carrying the chord of the false fifth, rises diatonically upon C (4 r).

RULE II.

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

RULE III.

212. The chord of the second is commonly put in practice upon notes which are syncopated in descending, because these notes are dissonances which ought to be prepared and resolved (200, 302.). See the example LXVII. where the second C, which is syncopated, and which descends afterwards upon B, carries the chord of the second (4 k).

CHAP. IX. Of some Licenses assumed in the Fundamental Bass.

§ 1. Of Broken and INTERRUPTED CADENCES.

213. The broken cadence is executed by means of a dominant which rises diatonically upon another, or upon a tonic by a license. See, in the example LXXIV. G A, executed (132. and 134.).

214. The interrupted cadence is formed by a do-Interrupted cadence, how formed.

observe the diatonic order, because this order is the most agreeable of all. We must therefore endeavour to preserve it as much as possible. It is for this reason that the continued bass in Example LXXV. is much more in the taste of singing, and more agreeable, than the fundamental bass which answers to it.

(4 r) The continued bass being a kind of treble with relation to the fundamental bass, it ought to observe the same rules with respect to that bass as the treble. Thus a note, for instance D, carrying a chord of the seventh D F A C, to which the chord of the sub-dominant F A C D corresponds in the fundamental bass, ought to rise diatonically upon E, (art. 129. No. 1. and art. 202.).

(4 k) When there is a repose in the treble, the note of the continued bass ought to be the same with that of the fundamental bass, (see Example LXXXIII.). In the closings which are found in the treble at D and C (measures second and fourth), the notes in the fundamental and continued bass are the same, viz. G for the first cadence, and C for the second. This rule ought above all to be observed in cadences which terminate a piece or a modulation.

It is necessary, as much as possible, to prevent coincidences of the same notes in the treble and continued bass, unless the motion of the continued bass should be contrary to that of the treble. For example, in the first note of the second measure in Example LXXXIX. D is found at the same time in the continued bass and in the treble; but the treble rises from C to D, and from D to E, whilst the bass descends from E to D, and from D to C.

Two octaves, or two fifths, in succession, must likewise be avoided. For instance, in the treble sounds G E, the bass must be prevented from sounding G E, C A, or D B; because in the first case there are two octaves in succession, E against E, and G against G; and because in the second case there are two fifths in succession, C against E, and A against G, or D against G, and B against E. This rule, as well as the preceding, is founded upon this principle, that the continued bass ought not to be a copy of the treble, but to form a different melody.

Every time that several notes of the continued bass answer to one note alone of the fundamental, the composer satisfies himself with figuring the first of them. Nay he does not even figure it if it be a tonic; and he draws above the others a line, continued from the note upon which the chord is formed. See Example LXXX. (Plate CCCLIX.) where the fundamental bass C gives the continued bass C E G E; the two E's ought in this bass to carry the chord 6, and G the chord 5; but as these chords are comprehended in the perfect chord C E G C, which is the first of the continued bass, we place nothing above C, only we draw a line over C E G E.

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

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

In general, whilst the same chord subsists in passing through different notes, the chord is reckoned the same as if the first note of the chord had subsisted; in such a manner, that, if the first note of the chord is, for instance, the tenible note, we ought to find the tonic when the chord changes. See Example LXXI. where this continued bass, C B D B C C, is reckoned the same with this, C, B C. (Example LXXXII.).

If a single note of the continued bass answers to several notes of the fundamental bass, it is figured with the different chords which agree to it. For example, the note G in a continued bass may answer to this fundamental bass C G C, (see example LXXXIII.) in this case, we may regard the note G as divided into three parts, of which the first carries the chord 6, the second the chord 7, and the third the chord 8.

We shall repeat here, with respect to the rules of the continued bass, what we have formerly said concerning the rules of the fundamental bass in the note upon the third rule, art. 193. The rules of the continued bass have exceptions, which practice and the perusal of good authors will teach. There are likewise several other rules which might require a considerable detail, and which will be found in the Treatise of Harmony, by M. Rameau, and 215. When a dominant is preceded by a tonic in the fundamental bass, we add sometimes, in the continued bass to the chord of that dominant, a new note which is a third or a fifth below; and the chord which results from it in this continued bass is called a chord by supposition.

For example, let us suppose, that in the fundamental bass we have a dominant G carrying the chord of the seventh G B D F; let us add to this chord the note C, which is a fifth below this dominant, and we shall have the total chord C G B D F, or C D F G, which is called a chord by supposition (4 m).

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

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

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

3. If

and elsewhere. These rules, which are proper for a complete dissertation, did not appear indispensably necessary in an elementary essay on music, such as the present. The books which we have quoted at the end of our preliminary discourse will more particularly instruct the reader concerning this practical detail.

(4 l) One may sometimes, but very rarely, cause several tonics in succession to follow one another in ascending or descending diatonically, as C E G C, D F A D, Bb D F Bb; but, besides that this succession is harsh, it is necessary, in order to render it practicable, that the fifth below the first tonic should be found in the chord of the tonic following, as here E, a fifth below the first tonic C, is found in the chord D F A D, and in the chord Bb D F Bb (37. and note T).

(4 m) Though supposition be a kind of license, yet it is in some measure founded on the experiment related in the note (8), where you may see that every principal or fundamental sound causes its twelfth and seventeenth major in descending to vibrate, whilst the twelfth and the seventeenth major ascending resound: which seems to authorize us in certain cases to join with the fundamental harmony this twelfth and seventeenth in descending; or, which is the same thing, the fifth or the third beneath the fundamental sound.

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

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

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

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

(4 o) Supposition introduces into a chord dissonances which were not in it before. For instance, if to the chord E G B D, we should add the note of supposition C defending by a third, it is plain that, besides the dissonance between E and D which was in the original chord, we have two new dissonances, C B, and C D; that is to say, the seventh and the ninth. These dissonances, like the others, ought to be prepared and resolved. They are prepared by being syncopated, and resolved by descending diatonically upon one of the consonances of the subsequent chord. The sensible note alone can be resolved in ascending; but it is even necessary that this sensible note should be in the chord of the tonic dominant. As to the dissonances which are found in the primitive chord, they should always follow the common rules. (See art. 202.). 3. If to a chord of the simple dominant, as D F A C, we should add the fifth G, we would have the chord G D F A C, called a chord of the eleventh, and which is figured with a \( \frac{2}{4} \) or \( \frac{5}{6} \). (See LXXXVIII.)

**Observe.**

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

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

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

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

§ 3. Of the Chord of the Diminished Seventh.

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

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

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

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

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

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

1. By adding C, the chord C G B D F, composed of a fifth redundant, a seventh, a ninth, and eleventh, which is the octave of the fourth. It is called a chord of the fifth redundant and fourth, and marked as in LXXXVIII.

2. By adding A, we shall have the chord A G B D F, composed of a seventh redundant, a ninth, an eleventh, and a thirteenth minor, which is the octave of the fifth minor. It is called the chord of the seventh redundant and fifth minor, and marked as in LXXXIX. It is of all chords the most harsh, and the most rarely practised (4 U).

(4 P) Several musicians call this last chord the chord of the ninth; and that which, with M. Rameau, we have simply called a chord of the ninth, they term a chord of the ninth and seventh. This last chord they mark with a \( \frac{2}{4} \); but the denomination and figure used by M. Rameau are more simple and can lead to no error; because the chord of the ninth always includes the seventh, except in the cases, of which we have already spoken.

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

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

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

(4 S) Supposition produces what we call suspension; and which is almost the same thing. Suspension consists in retaining as many as possible of the sounds in a preceding chord, that they may be heard in the chord which succeeds. For instance, in Example LXXXII., the C bearing \( \frac{2}{4} \) is a supposition; but in Example LXXXIII., it is a suspension, because it suspends or retards the perfect chord C E G C which the ear expects after the tonic dominant G B D F.

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

(4 U) As the chord of the diminished seventh G B D F, and the chord of the tonic dominant E G B D, only differ 224. Sometimes in a treble, the dissonance which ought to have been resolved by descending diatonically upon the succeeding note, instead of descending, on the contrary rises diatonically; but in that case, the note upon which it ought to have descended must be found in some of the other parts. This license ought to be rarely practised.

In like manner, in a continued bass, the dissonance in a chord of the sub-dominant inverted, as A in the chord A C E G, inverted from C E G A, may sometimes descend diatonically instead of rising as it ought to do, art. 129, No. 2; but in that case the note ought to be repeated in another part, that the dissonance may be there resolved in ascending.

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

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

Vol. XIV. Part II.

where the note D, which is added, passes under the Principles of Composition.

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

226. As the continued bass alone appears in practical compositions, it becomes necessary to know how to find the fundamental bass when the continued bass is figured. This problem may be easily solved by the following rules.

1. Every note which has no figure in the continued bass, ought to be the same, and without a figure in the fundamental bass; it is either a tonic, or reckoned such (4 x).

2. Every note which in the continued bass carries a 6, ought in the fundamental bass to give its third below not figured *, or its fifth below marked with a 7. * See Fi. We shall distinguish these two cases below. See LVI. figured, and the note (4 y).

3. Every note carrying 6 gives in the fundamental bass its fifth below not figured. See LVII.

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

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

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

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

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

M. l'Abbé Roussier, to redress this deficiency, had invented a new manner of figuring the continued bass. His method is most simple for those who know the fundamental bass. It consists in expressing each chord by only signifying the fundamental sound with that letter of the scale by which it is denominated, to which is joined a 7 or 7, or a 6, in order to mark all the discords. Thus the fundamental chord of the seventh D F A C is expressed by a D; and the same chord, when it is inverted from that of the sub-dominant F A C D, is characterized by F; the chord of the second C D F A, inverted from the dominant D F A C, is likewise represented by D; and the same chord C D F A, inverted from that of the sub-dominant F A C D, is signified by F; the case is mental bass the diatonic note above, figured with a 7. (See LVI.)

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

8. Every note marked with a 6 gives the fifth below marked with a 7; (see LX.) and it is plain by art. 187, that in the chord of the seventh, of which we treat in these three last articles, the third ought to be major, and the seventh minor, this chord of the seventh being the chord of the tonic dominant. (See art. 102.)

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

10. Every note marked with a 4 gives the fifth above figured with a 7. (See LXXVIII.)

11. Every note marked with a 5, or with a +5, gives the third above figured with a 8. (See LXXXI.)

12. Every note marked with a 7 gives a fifth above figured with a 7, or with a 8. (See LXXXVI.) It is the same case with the notes marked 7, 3, or 5; which shows a retrenchment, either in the complete chord of the eleventh, or in that of the seventh redundant.

13. Every note marked with a 4 gives a fifth above figured with a 7, or a 8. (See LXXX.)

14. Every note marked with a 6 gives the third minor below, figured with a 7. (See LXXXV.)

15. Every note marked with a 6 gives the tritone above figured with a 7. (See LXXXVI.)

16. Every note marked with a 2 gives the second redundant above, figured with a 7. (See LXXXVII.)

17. Every note marked with a 4 gives the fifth redundant above, figured with a 7. (See LXXXVIII.)

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

Remark.

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

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

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

229. The second case is that in which the continued bass is figured with a 5. For instance, if there should be found F in the continued bass, we may be ignorant whether we ought to insert in the fundamental bass F marked with a 6, or D figured with a 7.

230. This difficulty may be removed by leaving for solution an instant this uncertain note in suspense, and in examining the succeeding note of the fundamental bass; for if that note be in the present case a fifth above F, that is to say, if it be C, in this case, and in this alone, we may place F in the fundamental bass. It is a consequence of this rule, that in the fundamental bass every sub-dominant ought to rise by a fifth (195).

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

231. In the first part of this treatise (chap. vi.) we have explained, how by the means of the note C, and determining the mode of its two-fifths G and F, one in ascending, which is called a tonic dominant, the other in descending, which which we is called a sub-dominant, the scale C D E F A B C may be found: the different sounds which form this scale compose the same when the chords are differently inverted. By this means it would be impossible to mistake either with respect to the fundamental bass of a chord, or with respect to the note which forms its dissonance, or with respect to the nature and species of that discord.

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

M. Rameau only marks the lesser sixth by a 6 without a line, when this lesser sixth does not result from the chord of the tonic dominant; in such a manner that the 6 renders it uncertain whether in the fundamental bass we ought to choose the third or the fifth below; but it will be easy to see whether the third or the fifth is signified by that figure. This may be distinguished, i. In observing which of the two notes is excluded by the rules of the fundamental bass. 2. If the two notes may with equal propriety be placed in the fundamental bass, the preference must be determined by the tone or mode of the treble in that particular passage. In the following chapter we shall give rules for determining the mode (note 3 z).

There is a chord of which we have not spoken in this enumeration, and which is called the chord of the sixth redundant. This chord is composed of a note, of its third major, of its redundant fourth or tritone, and its redundant fifth, as F A B D X. It is marked with a 6. It appears difficult to find a fundamental bass for this chord; nor is it indeed much in use amongst us. (See the note upon the art. 115.)

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

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

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

A B C D E F G A,

which is (85.) the scale of the minor mode of A in ascending; and the scale of the minor mode of A in descending shall be (90.), A G F E C D B A,

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

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

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

A TABLE of the Different Modes.

Major Modes.

| Maj. Mode | of C; | C, D, E, F, G, A, B, c. | |-----------|------|------------------------| | | of G; | G, A, B, c, d, e, f, g. |

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

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

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

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

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

The double sharp, however, is incidentally used, when in a composition in the key of F there is an occasional modulation into the dominant of that key, and it is distinguished by the character X or XX.

Minor Modes.

Of A.

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

Of E.

In descending. e d c B A G F E. In rising. E F G A B C d e.

Of B.

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

Of F.

In descending. f e d c B A G E. In rising. F G A B C d e f.

Of C.

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

Of G or Ab.

In descending. g f e d c B A G. In rising. A B C d e f g a.

Of D or Eb.

In descending. c b d c B A G F Eb. In rising. Eb F G A B c d e.

Of A or Bb.

In descending. Bb A Bb Gb F Eb Db C Bb. In rising. Db C Db Eb F G A Bb.

3 Z 2 Of E♭ or F♯.

In descending. f F eb db c Bb Ab GF. In rising. F G Ab Bb c d e f.

Of C.

In descending. c Bb Ab G F Eb DC. In rising. CD Eb FG A B c.

Of G.

In descending. g feb d C Bb AG. In rising. GA Bb cd ef G g.

Of D.

In descending. dc Bb AG F ED. In rising. DE FG AB c G d (5c).

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

(5c) We have already seen, that in each mode, the principal note is called a tonic; that the fifth above that note is called a tonic dominant, or the dominant of the mode, or simply a dominant; that the fifth below the tonic, or, what is the same thing, the fourth above that tonic, is called a sub-dominant; and in short, that the note which forms a semitone below the tonic, and which is a third major from the dominant, is called a sensible note. The other notes have likewise in every mode particular names which it is advantageous to know. Thus a note which is a tone immediately above the tonic, as D in the mode of C, and B in that of A is termed a super-tonic; the following note, which is a third major or minor from the tonic, according as the chord is major or minor, such as E in the major mode of C, and C in the minor mode of A, is called a mediant; and the note which is a tone above the dominant, such as A, in the mode of C, and F♯ in that of A, is called a super-dominant.

(5d) Though our author's account of this delicate operation in music will be found extremely just and copious; though it proceeds upon simple principles, and comprehends every possible contingency; yet as the manner of thinking upon which it depends may be less familiar to English readers, if not profoundly skilled in music, it has been thought proper to give a more familiar, though less comprehensive, explanation of the manner in which transposition may be executed.

It will easily occur to every reader, that if each of the intervals through the whole diatonic series were equal, in a mathematical sense, it would be absolutely indifferent upon what note any air were begun, if within the compass of the gammut; because the same equal intervals must always have the same effects. But since, besides the natural semitones, there is another distinction of diatonic intervals into greater and lesser tones; and since these vary their positions in the series of an octave, according as the note from whence you begin is placed, that note is consequently the best key for any tune whose natural series is most exactly correspondent with the intervals which that melody or harmony requires. But in instruments whose scales are fixed, notwithstanding the temperaments and other expedients of the same kind, such a series is far from being easily found, and is indeed in common practice almost totally neglected. All that can frequently be done is, to take care that the ear may not be sensibly shocked. This, however, would be the case, if, in transposing any tune, the situation of the semitones, whether natural or artificial, were not exactly correspondent in the series to which your air must be transposed, with their positions in the scale from which you transpose it. Suppose, for instance, your air should begin upon C, requiring the natural diatonic series through the whole gammut, in which the distance between E and F, as also that between B and C, is only a semitone. Again, suppose it necessary for your voice, or the instrument on which you play, that the same air should be transposed to G, a fifth above its former key; then because in the first series the intervals between the third and the fourth, seventh and eighth notes, are no more than semitones, the same intervals must take the same place in the octave to which you transpose. Now, from G, the note with which you propose to begin, the three tones immediately succeeding are full; but the fourth C is only a semitone; it may therefore be kept in its place. But from F, the seventh note above, to G, the eighth, the interval is a full tone, which must consequently be redressed by raising the F a semitone higher. Thus the situations of the semitone intervals in both octaves will be correspondent; and thus, by conforming the positions of the semitones in the octave to which you transpose, with those in the octave in which the original key of the tune is contained, you will perform your operation with as much success as the nature of fixed scales can admit.

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

Thus if G, which is the perfect fifth ascending from , therefore the note most nearly allied to C (art. 39, 40), B, G must be changed into C, and E into A. Thus, by transposition, the air has the same melody as if it were in the major mode of C, or in the minor mode of A. The major mode then of G, and the minor of E, are by transposition reduced to those of C major, and of A minor. It is the same case with all the other modes (5 E).

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

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

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

This, for the reasons mentioned in the note (5 B), is the last scale to which we can properly go by the progressions of fifths ascending.

Returning to the natural scale of C, if, instead of assuming G, the perfect fifth above, for a fundamental, we take F, the perfect fifth below; B, which is the fourth note above F, and forms a tritone or sharp fourth to it, must, to become a perfect fourth, according to the laws of the diatonic scale, (art. 60.) be made flat (12.).

Proceeding with the series of fifths descending, if B, which is the perfect fifth below F, be taken for a fundamental; E, which, in its natural state, is the tritone or sharp fourth to B; must, to become the diatonic fourth (art. 60.), also be rendered flat (11.).

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

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

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

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

We have thus seen, 1stly, That each of the notes of the diatonic scale of C, and each of the semitones into which the whole tones of that scale are divided, may be taken for the fundamental note of a diatonic scale, called the scale of that note. 2dly, That the notes of the natural scale are more or less altered, as the note assumed for a fundamental is more or less distant from C, in a progression of fifths ascending or descending. 3dly, That in the progression by fifths ascending, the notes are altered by sharps, and in the progression by fifths descending, the alterations are by flats. 4thly, That in the alteration by sharps, the last sharp is always on the seventh or sensible note of the scale; and where there are more than one, is always on the fifth above the sharp immediately preceding; and in the alteration by flats, the last flat is always on the fourth of the scale; and where there are more than one, is always on the fifth below the flat immediately preceding.

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

(5 E) Many musicians, and amongst others the ancient musicians of France, as Lulli, Campra, &c. place one flat less in the minor mode: so that in the minor mode of D, they place neither sharp nor flat at the clef; in the minor mode of G, one flat only; in the minor mode of C, two flats, &c.

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

It should no longer be difficult to find the fundamental bas of a given modulation, nay, frequently to find several; for every fundamental bas will be legitimate, when it is formed according to the rules which we have given (chap. vi.); and that, besides this, the dissonances which the modulation may form with this bas, will both be prepared, if it is necessary that they should be so, and always resolved.

239. It is of the greatest utility in searching for the fundamental bas, to know what is the tone or mode of the melody to which that bas should correspond. But it is difficult in this matter to assign general rules, and such as are absolutely without exception, in which nothing may be left that appears indifferent or discretionary; because sometimes we seem to have the free choice of referring a particular melody either to one mode or another. For example, this melody G C may belong to all the modes, as well major as minor, in which G and C are found together; and each of these two sounds may even be considered as belonging to a different mode.

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

1. Because, since the same sounds belong to several different modes, the mode is sometimes considerably undetermined; above all, in the middle of a piece, and during the time of one or two bars.

2. Without giving ourselves much trouble about the mode, it is often sufficient to preserve us from deviating in composition, if we observe in the simplest manner the rules above prescribed (chap. vi.) for the procedure of the fundamental bas.

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

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

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

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

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

4. A mode is not usually changed, especially in the beginning of a piece, unless in order to pass into one or other of the modes most relative to it, which are the mode of its fifth above, and that of its third below, if the original mode of major, or of its third above if it be minor. Thus, for instance, the modes which are most intimately relative to the major mode of C, are the major mode of G, and that of A minor. From the mode of C we commonly pass either into the one or the other of these modes; so that we may sometimes judge of the principal mode in which we are, by the relative mode which follows it, or which goes before it, when these relative modes are decisively marked. Besides these two relative modes, there are likewise two others into which the principal mode may pass, but less frequently, viz. the mode of its fifth below, and that of its third above, as F and E for the mode of C (5 c).

5. The modes may still be likewise distinguished by the cadences of the melody. These cadences ought to occur at the end of every two, or at most of every four bars, as in the fundamental bas; now the note of the fundamental bas which is most suitable to these cloics,

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

(5 c) It is certain that the minor mode of E has an extremely natural connection with the mode of C, as has been proven (art. 92.) both by arguments and by examples. It has likewise appeared in the note upon the art. 93., that the minor mode of D may be joined to the major mode of C; and thus in a particular sense, this mode may be considered as relative to the mode of C, but it is still less so than the major modes of G and F, or than those of A and E minor; because we cannot immediately, and without licence, pass in a fundamental bas from the perfect minor chord of C to the perfect minor chord of D; and if you pass immediately from the major mode of C to the minor mode of D in a fundamental bas, it is by passing, for instance, from the tonic C, or from E G C, to the tonic dominant of D, carrying the chord A C E G, in which there are two sounds, E G, which are found in the preceding chord, (Ex. xciv.) or otherwise from C E G C to G B D E, a chord of the subdominant in the minor mode of D, which chord has likewise two sounds, G and E, in common with that which went immediately before it. See Ex. xcvi. closer, is always easy to be found. For the sounds which occur in the treble, M. Rameau may be consulted, p. 54, of his Nouveau Systeme de Musique theorique et pratique (§ H).

When the mode is ascertained, by the different means which we have pointed out, the fundamental bass will cost little pains. For in each mode there are three fundamental basses.

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

Major mode of C, C E G c'.

Minor mode of A, A C E A.

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

Tonic dominant. Major mode of C, G B D f'. Minor mode of A, E G X B d'.

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

Sub-dominant. Major mode of C, F A C d'. Minor mode of A, D F A B.

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

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

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

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

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

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

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

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

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

(§ H) All these different manners of distinguishing the modes ought, if we may speak so, to give mutual light and assistance one to the other. But it often happens, that one of these signs alone is not sufficient to determine the mode, and may even lead to error. For example, if a piece of music begins with these three notes, E C G, we must not with too much precipitation conclude from thence that we are in the major mode of C, although these three sounds, E C G, be the principal and characteristical sounds in the major mode of C: we may be in the minor mode of E, especially if the note E should be long.

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

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

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

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

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

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

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

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

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

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

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

3. When we find a tonic dominant, we are in the mode of the fifth below that tonic dominant. As the tonic dominant carries always the third major, it cannot be ascertained from this dominant alone, whether the mode be major or minor; but it is only necessary to examine the following note, which must be the tonic of the mode in which he is; by the third of this tonic it will be discovered whether the mode be major or minor.

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

244. All these rules, joined with the table of modes which we have given (art. 234.), will serve to discover in what mode we are in the middle of a piece, especially in the most essential passages, as cadences (5K).

**CHAP. XIV. Of the Chromatic and Enharmonic.**

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

246. When an air is chromatic in descending, the most natural and ordinary fundamental bas is a concatenated series of tonic dominants; all of which follow one another in descending by a fifth, or, which is the same thing, in rising by a fourth. See Ex. cv. fundamental bas, (5L).

(5K) Two modes are so much more intimately relative, as they contain a greater number of sounds common to both; for example, the minor mode of C and the major of G, or the major mode of C and the minor of A: on the contrary, two modes are less intimately relative as the number of sounds which they contain as common to both is smaller; for instance, the major mode of C and the minor of B, &c.

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

(5L) We may likewise give to a chromatic melody in descending, a fundamental bas, into which may enter chords of the seventh and of the diminished seventh, which may succeed one another by the intervals of a false fifth and a fifth redundant: thus in the Example cvII., where the continued bas descends chromatically, it may easily be seen that the fundamental bas carries successively the chords of the seventh and of the seventh diminished, and that in this bas there is a false fifth from D to G X, and a fifth redundant from G X to C.

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

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

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

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

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

PLATE CCCLIV.

Scale

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

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

Scale First Scale Second Scale Third

H I

Fig. 3. K L M N

R S T U V W

The Diatonic Scale of the Greeks

B c d e f g a

G C G C F C F

The Fundamental Bass

Fig. 4.

The Chromatic Species

g g# &c

C E G#

The Fund. Bass

Fig. 5.

c d e f g g a b c

C G C F C G D G C

The Fundamental Bass

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

Scale First Scale Second

The first Scale of the Minor Mode

G A B c d e f

E A E A D A D

The Fundamental Bass

Fig. 7.

f e e d#

F C E B

The Fund. Bass

Fig. 12.

The Second Scale of the Minor Mode

A B c d e e f # g # a

A E A D A E B E A

The Fundamental Bass

Fig. 8.

e b e e e e #

C C A C # C

The Fund. Bass

Fig. 13. CV. Chromatic Modulation descending.

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

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

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

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

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

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

**Vol. XIV. Part II.**

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

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

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

\[ C, F, G, C, F, G, C, \&c. \]

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

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

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

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

(5 N) Yet there may be two fifths in succession, provided the parts move in contrary directions, or, in other words, if the progress of one part be ascending, and the other descending; but in this case they are not properly two fifths, they are a fifth and a twelfth: for example, if one of the parts in descending should found F D, and the other 'c a' in rising, C is the fifth of F, and 'a' the twelfth of D. search only of general information, and not a professed student of this particular science, would choose to rest satisfied.

The theory of musical sound, which only in the beginning of the present century was ultimately established by mathematical demonstration, is no other than that which distinguished the ancient musical sects who followed the opinions of Pythagoras on that subject.

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

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

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

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

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

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

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

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

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

---

**Mus**

**Glass-Music.** See Harmonica.

**Musimon,** in Natural History, the name of an animal esteemed a species of sheep, described by the ancients as common in Corsica, Sardinia, Barbary, and the north-east parts of Asia. It has been doubted whether the animal described under this name is now anywhere to be found in the world, and whether it was not, probably, a spurious breed between two animals of different species, perhaps the sheep and goat, which, like the mule, not being able to propagate its species, the production of them may have been discontinued.

Buffon supposes it to be the sheep in a wild state; and it is described as such by Mr Pennant. These animals live in the mountains, and run with great swiftness among the rocks. Those of Kamtschatka are so strong, that 10 men can scarcely hold one; and the horns are so large as sometimes to weigh 30 pounds, and MUSKIVUM aurum. See Chemistry, No 1806.