ments, which correspond to the two preceding:

That which answers to harmonical music, and which the ancients called melopee, teaches the rules for combining and varying the intervals, whether consonant or dissonant, in an agreeable and harmonious manner.

The second, which answers to the rhythmical music, and which they called rhythmopoea, contains the rules for applying the different modes of time, for understanding the feet by which verses were scanned, and the diversities of measure; in a word, for the practice of the rhythmus.

Music is at present divided more simply into melody and harmony; for since the introduction of harmony the proportion between the length and shortness of sounds, or even that between the distance of returning cadences, are of less consequence amongst us. For it often happens in modern languages, that the verses assume their measures from the musical air, and almost entirely lose the small share of proportion and quantity which in themselves they possess.

By melody the successions of sound are regulated in such a manner as to produce pleasing airs.*

Harmony consists in uniting to each of the sounds, in a regular succession, two or more different sounds, which simultaneously striking the ear soothe it by their concurrence. See Harmony.

Music, according to Rousseau, may be, and perhaps likewise ought to be, divided into the physical and the imitative. The first is limited to the mere mechanism of sounds, and reaches no further than the external senses, without carrying its impressions to the heart, and can produce nothing but corporeal sensations more or less agreeable. Such is the music of songs, of hymns, of all the airs which only consist in combinations of melodious sounds, and in general all music which is merely harmonious.

It may, however, be questioned whether every sound, even to the most simple, is not either by nature, or by early and confirmed association, imitative. If we may trust our own feelings, there is no such thing in nature as music which gives mechanical pleasure alone. For if so, it must give such pleasure as we receive from taffes, from odours, or from other grateful titillations; but we absolutely deny that there are any musical sensations or pleasures in the smallest degree analogous to those. Let any piece of music be reduced into its elementary parts and their proportions, it will then easily appear from this analysis, that sense is no more than the vehicle of such perceptions, and that mind alone can be susceptible of them. It may indeed happen, from the number of the performers and the complication of the harmony, that meaning and sentiment may be lost in the multiplicity of sounds; but this, though it may be harmony, loses the name of music.

The second department of his division, by lively and accentuated inflections, and by sounds which may be said to speak, expresses all the passions, paints every possible picture, reflects every object, subjects the whole of nature to its skillful imitations, and impresses even on the heart and soul of man sentiments proper to affect them in the most sensible manner. This, continues he, which is the genuine lyric and theatrical music, was what gave double charms and energy to ancient poetry; this is what, in our days, we exert ourselves in applying to the drama, and what our singers execute on the stage. It is in this music alone, and not in harmonics or the resonance of nature, that we must expect to find accounts of those prodigious effects which it formerly produced.

But, with Mr Rousseau's permission, all music which is not in some degree characterized by these pathetic and imitative powers, deserves no better name than that of a musical jargon, and can only be effectuated by such a complication and intricacy of harmony, as may confound, but cannot entertain the audience. This character, therefore, ought to be added as essential to the definition of music; and it must be attributed to our neglect of this alone, whilst our whole attention is bestowed on harmony and execution, that the best performances of our artists and composers are heard with listless indifference and officiation, nor ever can conciliate any admirers, but such as are induced, by pedantry and affectation, to pretend what they do not feel. Still may the curse of indifference and inattention pursue and harrow up the souls of every composer or performer, who pretends to regale our ears with this musical legerdemain, till the grin of scorn, or the hiss of infamy, teach them to correct this depravity of taste, and entertain us with the voice of nature!

Whilst moral effects are sought in the natural effects of sound alone, the scrutiny will be vain, and disputes will be maintained without being understood: but sounds, as representatives of objects, whether by nature or association, introduce new scenes to the fancy and new feelings to the heart; not from their mechanical powers, but from the connection established by the Author of our frame between sounds and the objects which either by natural resemblance or unavoidable association they are made to represent.

It would seem that music was one of those arts which were first discovered; and that vocal was prior to instrumental music, if in the earliest ages there was any music which could be said to be purely instrumental. For it is more than probable, that music was originally formed to be the vehicle of poetry; and of consequence, though the voice might be supported and accompanied by instruments, yet music was never intended for instruments alone.

We are told by ancient authors, that all the laws whether human or divine, exhortations to virtue, the knowledge of the characters and actions of gods and heroes, the lives and achievements of illustrious men, were written in verse, and sung publicly by a quire to the sound of instruments; and it appears from the Scriptures, that such from the earliest times was the custom among the Israelites. Nor was it possible to find means more efficacious for impressing on the mind of man the principles of morals, and inspiring the love of virtue. Perhaps, however, this was not the result of a premeditated plan; but inspired by sublime sentiments and elevation of thought, which in accents that were suited and proportioned to their celestial nature endeavoured to find a language worthy of themselves and expressive of their grandeur.

It merits attention, that the ancients were so sensible of the value and importance of this divine art, not only as a symbol of that universal order and symmetry which prevails through the whole frame of material and intelligent nature, but as productive of the most momentous effects both in moral and political life. Plato and Aristotle, who disagreed almost in every other maxim of politics, are unanimous in their approbation of music, as an efficacious instrument in the formation of the public character and in conducing the state; and it was the general opinion, that whilst the gymnastic exercises rendered the constitution robust and hardy, music humanized the character, and softened those habits of roughnesses and ferocity by which men might otherwise have degenerated into savages. The gradations by which voices were exerted and tuned, by which the invention of one instrument succeeded to another, or by which the principles of music were collected and methodized in such a manner as to give it the form of an art and the dignity of a science, are topics so fruitful of conjecture and so void of certainty, that we must leave them to employ minds more speculative and inventions more prolific than ours, or transfer them to the History of Music as a more proper place for such disquisitions. For the amusement of the curious, Rousseau in his Musical Dictionary, Plates C and N, has transcribed some fragments of Grecian, Persian, American, Chinese, and Swiss music, with which performers may entertain themselves at leisure. When they have tried the pieces, it is imagined they will be less languidly fond than that author of ascribing the power of music to its affinity with the national accents where it is composed. This may doubtless have its influence; but there are other causes more permanent and less arbitrary to which it owes its most powerful and universal charms.

The music now most generally celebrated and practised is that of the Italians, or their successful imitations. The English, from the invasion of the Saxons, to that more late tho' lucid era in which they imbibed the art and copied the manner of the Italians, had a music which neither pleased the soul nor charmed the ear. The primitive music of the French deserves no higher panegyric. Of all the barbarous nations, the Scots and Irish seem to have possessed the most affecting original music. The first consists of a melody characterized by tenderness; it melts the soul to a pleasing pensive languor. The other is the native expression of grief and melancholy. Taffoni informs us, that in his time a prince from Scotland had imported into Italy a lamentable kind of music from his own country; and that he himself had composed pieces in the same spirit. From this expressive, though laconic description, we learn, that the character of our national music was even then established; yet so gross is our ignorance and credulity, that we ascribe the best and most impassioned airs which are extant among us to David Rizzio; as if an Italian Lutanist, who had lived for a short time in Scotland, could at once, as it were by inspiration, have imbibed a spirit and composed in a manner so different from his own. It is yet more surprising that Geminiani should have entertained and published the same prejudice, upon the miserable authority of popular tradition alone; for the fact is authenticated by no better credentials. The primitive music of the Scots may be divided into the marchal, the pastoral, and the festive. The first consists either in marches, which were played before the chieftains, in imitation of the battles which they fought, or in lamentations for the catastrophes of war and the extinction of families. These wild effusions of natural melody preserve several of the rules prescribed for composition. The strains, though rude and untutored, are frequently terrible or mournful in a very high degree. The part or march is sometimes in common, sometimes in treble time; regular in its measures, and exact in the distance between its returning cadences; most frequently, though not always, loud and brisk. The pibroch, or imitation of battles, is wild, and abrupt in its transitions from interval to interval and from key to key; various and deftly in its movements; frequently irregular in the return of its cadences; and in short, through the whole, seems inspired with such fury and enthusiasm, that the hearer is irresistibly infected with all the rage of precipitate courage, notwithstanding the rudeness of the accents by which it is kindled. To this the pastoral forms a striking contrast. Its accents are plaintive, yet soothing; its harmony generally flat; its modulations natural and agreeable; its rhythm simple and regular; its returning cadences at equal distance; its transitions from one concinnous interval to another, at least for the most part; its movements slow, and may be either in common or treble time. It scarcely admits of any other harmony than that of a simple bass. A greater number of parts would cover the air, and destroy the melody. To this we shall add what has been said upon the same subject by Dr Franklin. Writing to Lord K———, he proceeds thus:

"Give me leave on this occasion, to extend a little the sense of your position, 'That melody and harmony are separately agreeable, and in union delightful;' and to give it as my opinion, that the reason why the Scotch tunes have lived so long, and will probably live for ever (if they escape being stifled in modern affected ornament) is merely this, that they are really compositions of melody and harmony united, or rather that their melody is harmony. I mean, the simple tunes sung by a single voice. As this will appear paradoxical, I must explain my meaning. In common acceptation, indeed, only an agreeable succession of sounds is called melody; and only the coexistence of agreeable sounds, harmony. But since the memory is capable of retaining for some moments a perfect idea of the pitch of a past sound, so as to compare it with the pitch of a succeeding sound, and judge truly of their agreement or disagreement, there may and does arise from thence a sense of harmony between the present and past sounds, equally pleasing with that between two present sounds. Now the construction of the old Scotch tunes is this, that almost every succeeding emphatical note is a third, a fifth, an octave, or in short some note that is in concord with the preceding note. Thirds are chiefly used, which are very pleasing concords. I use the word emphatical, to distinguish those notes which have a stress laid on them in fingering the tune, from the lighter connecting notes that serve merely, like grammar-articles in common speech, to tack the whole together.

"That we have a most perfect idea of a sound just past, I might appeal to all acquainted with music, who knows how easy it is to repeat a sound in the same pitch with one just heard. In tuning an instrument, a good ear can as easily determine that two strings are in unison by bounding them separately, as by bounding them together; their disagreement is also as easily, I believe I may say more easily and better distinguished when founded separately; for when founded together, though you know by the beating that one is higher than the other, you cannot tell which it is. I have ascribed to memory the ability of comparing the pitch of a present tone with that of one past. But if there should be, as possibly there may be, something in the ear similar to what we find in the eye, that ability would not be entirely owing to memory. Possibly the vibrations given to the auditory nerves by a particular sound may actually continue some time after the cause of these vibrations is past, and the agreement or disagreement of a subsequent sound become by comparison with them more discernible. For the impression made on the visual nerves by a luminous object will continue for 20 or 30 seconds."

After some experiments to prove the permanency of visible impressions, he continues thus:

"Farther, when we consider by whom these ancient tunes were composed, and how they were first performed, we shall see that such harmonical successions of sounds was natural and even necessary in their construction. They were composed by the minstrels of those days, to be played on the harp accompanied by the voice. The harp was strung with wire, which gives a sound of long continuance; and had no contrivance like that of the modern harpsichord, by which the sound of the preceding note could be stopt the moment a succeeding note begin. To avoid actual discord, it was therefore necessary that the succeeding emphatic note should be a chord with the preceding, as their sounds must exist at the same time. Hence arose that beauty in those tunes that has so long pleased, and will please forever, though men scarce know why. That they were originally composed for the harp, and of the most simple kind, I mean a harp without any half-notes but those in the natural scale, and with no more than two octaves of strings, from C to C, I conjecture from another circumstance; which is, that not one of these tunes, really ancient, has a single artificial half-note in it; and that in tunes where it is most convenient for the voice to use the middle notes of the harp, and place the key in F, there the B, which if used should be a B flat, is always omitted, by passing over it with a third. The connoisseurs in modern music will say I have no taste; but I cannot help adding, that I believe our ancestors, in having a good song, distinctly articulated, sung to one of those tunes, and accompanied by the harp, felt more real pleasure than is communicated by the generality of modern operas, exclusive of that arising from the scenery and dancing. Most tunes of late composition, not having this natural harmony united with their melody, have recourse to the artificial harmony of a bass, and other accompanying parts. This support, in my opinion, the old tunes do not need, and are rather confused than aided by it. Whoever has heard James Oswald play them on his violincello, will be less inclined to dispute this with me. I have more than once seen tears of pleasure in the eyes of his auditors; and yet, I think, even his playing those tunes would please more if he gave them less modern ornament."

As these observations are for the most part true and always ingenious, we need no other apology for quoting them at length. It is only proper to remark, that the transitions in Scots music by consonant intervals, does not seem, as Dr Franklin imagines, to arise from the nature of the instruments upon which they played. It is more than probable, that the ancient British harp was not strung with wire, but with the same materials as the Welsh harps at present. These strings have not the same permanency of tone as metal, so that the sound of a preceding emphatic note must have expired before the subsequent accented note could be introduced. Besides, they who are acquainted with the manoeuvre of the Irish harp, know well that there is a method of discontinuing sounds less easy and effectual than upon the harpsichord. When the performer finds it proper to interrupt a note, he has no more to do but return his finger gently upon the string immediately struck, which effectually stops its vibration.

That species of Scotch music which we have distinguished by the name of fiddle seems now limited to reels and country-dances. These may be either in common or treble time. They most frequently consist of two strains: each of these contains eight or twelve bars. They are truly rhythmical; but the mirth which they excite seems rather to be inspired by the vivacity of the movement, than either by the force or variety of the melody. They have a manoeuvre and expression peculiar to themselves, which it is impossible to describe, and which can only be exhibited by good performers.

Thus far we have pursued the general idea of music. We shall, after the history, give a more particular detail of the science from Monsieur D'Alembert.

HISTORY OF MUSIC.

The uncertainty of most cultivated nations, is now either so entirely facts in musick, or so unhappily obscured, that we can make but ficial history few certain, and perhaps no satisfactory discoveries whether ancient or modern.

THE ancient history of music, even among the people as are called barbarous, our accounts of it must be still less authentic and satisfactory, than those of the former. Even at periods which are more recent, and may for that reason be thought more within the sphere of our investigation, we are equally at a loss both for the causes and the authors of some essential improvements in music. Yet those parts of its history, which are either already known, or may be discovered, if related at full length with proper illustrations, would produce a work little inferior in size to the whole extent of that Encyclopedia of which it only constitutes a part. All, therefore, which can be expected from this preliminary account, is to give a short and cursory detail of its primary state, and its most important revolutions, so far as history will enable us, by enlightening our researches, to accomplish this design. But if our accounts are thought concise and imperfect, we shall all along direct direct the views of our readers to sources which may prove more copious and more adequate to their curiosity.

It has been pretended by Father Kircher and others, that music prevailed in Egypt before it was known in Greece. These authors derive its name from a word which is primitive in the Egyptian language, and attribute the invention of the art to the iridulous murmur of the winds whistling through reeds, or other vegetable tubes, which grew upon the banks of the river Nile. But if this idle and legendary account of the discovery merits any attention at all, it must relate to instrumental music alone: for it cannot be imagined that mankind, if in the least degree attentive to the natural modulation of their own voices, and to such transitions of sound as were agreeable or disagreeable, would have recourse for their ideas of melody to objects so extrinsic and so contingent as the whistling of winds through a reed. Man is certainly as much a musical as he is a vocal animal; nor is the act of singing in him less instinctive than in birds, though his powers are more extensive and more susceptible of culture than theirs. If we believe the accounts of such as have been attentive to the music of the groves, they will tell us, that though the feathered warblers have a musical instinct, yet the modes of its exertion are as really acquired by birds from their parents or tutors as by men*. Nor is it easy to conceive a human creature, endowed with the natural powers of musical formation, and advanced to any degree of maturity, without supposing at the same time that he has tried several musical experiments, and that in some degree he has formed and cultivated his natural organs. At the same time, it cannot be denied, that the degrees of sound, passing through tubes of different textures, lengths, and diameters, or of strings whose magnitudes and degrees of cohesion were different, must be ascertained by experiment alone. But whether these experiments were the result of contingency or design, whether observation took the hint from nature, or began of itself to make trials and preserve their results, it seems now too late to determine.

The origin of instrumental music appears to have been at a period much prior to the date of authentic history; and when we look for its epoch or its discoverer, we are carried at once into the wild regions of fable and mythology. The god Mercury, or Hermes, is said to be the inventor of the lyre*, by distending strings of different tensions and diameters upon the shell of a tortoise which he found upon the shore. The first exhibition of the fistula, or shepherd's pipe, is ascribed to Pan. But of these beings and their actions, little or nothing can be ascertained with proper evidence†. We must therefore content ourselves with such later accounts as merit any degree of confidence.

The Grecian lyre, in its original state, seems to have been an instrument of the utmost simplicity: for, according to some, the Mercurian lyre consisted only of three, and according to others only of four, strings. These being touched open, could only produce the same number of sounds: from whence we may easily conclude, that the powers of this instrument could not be very extensive. This tetrachord, as some say, was conjoined; others maintain that it was disjoined, and that its intervals were not even diatonic. It is, however, allowed, that its two extremes produced an octave; and that the two intermediate strings divided it by a fourth on each side, with a tone in the midst, in the following manner:

\[ \begin{align*} U_t & \quad \text{Trite diezeugménon.} \\ S_o & \quad \text{Lichanos mélon.} \\ F_a & \quad \text{Parhypate méfon.} \\ U_t & \quad \text{Parhypate hypaton.} \end{align*} \]

This is what Boëtius calls the tetrachord of Mercury; though Diodorus affirms, that the lyre of Mercury had only three strings. This system did not long remain confined to so small a number of sounds. Cho-rebus, the son of Athis king of Lydia, added to it a fifth string; Hyagnis, a sixth; Terpander, a seventh, to equal the number of the planets; and at last, Lycaon of Samos the eighth.

This is the account of Boëtius. But Pliny says, that Terpander having added three strings to the four which were original, first played upon the cithara with seven strings: that Simonides joined to them an eighth, and Timotheus a ninth. Nicomachus the Gerætanian attributes this eighth chord to Pythagoras, the ninth to Theophrastus of Pierceus, afterwards the tenth to Hyfeus of Colophon. Pherecratus, in the dialogue of Plutarch, makes the system advance with a more rapid progress: he gives twelve strings to the cithara of Menalippides, and as many to that of Timotheus. And as Pherecratus was contemporary with these musicians, if we suppose that he really said what Plutarch attributes to him, his testimony will have considerable importance in a fact which was obvious to his own immediate observation.

But how shall we obtain any certainty among such a number of contradictions as are found not only in the authors on doctrines of the authors, but in the order of the events which they relate? For instance, the tetrachord of Mercury evidently gives the octave or diapason. How then could it happen, that, after the addition of three strings, the whole scale was found to be diminished by one degree, and reduced to the interval of a seventh? This is, however, what the greatest number of authors leave us to understand; and among others Nicomachus, who tells us, that Pythagoras, finding the whole system composed only of two conjoined tetrachords, which between their extremes formed a dissonant interval, rendered it a consonance, by dividing these two tetrachords by the interval of a tone, which produced the octave.

Whatever be the case, there is at least one thing certain, that the system of the Greeks was infensibly extended as well above as below, till it reached, and even surpassed, the compass of a disdiapason or double octave; a series which they call a perfect system, and which was likewise termed the greatest and the most unchangeable; because, between its two extremes, which betwixt themselves formed a perfect consonance, were contained all the simple, the double, the direct, or the inverted chords, every particular system, and according to them the greatest intervals, which can take place in melody.

This whole system consisted of four tetrachords, three conjoined and one disjoined; and of a single of the pentatone redundant, which was added below the whole to complete the double octave; from whence the string... which formed it took the name of proflambanomene, or the additional string. This, one would imagine, could only form fifteen notes in the diatonic genus; there were, however, sixteen. This was because the disjunction being sometimes perceived between the second and third tetrachord, and at other times between the third and fourth, it happened, in the first case, that the found la or A, the highest in the second tetrachord, the fi or B natural, with which the third tetrachord began, immediately followed in ascending; or otherwise, in the second case, that the same found la, with which note itself the third tetrachord begun, was immediately followed by fi or B flat; for the first gradation of every tetrachord, in the diatonic species, consisted always of a semitone. This difference then produced a sixteenth found, on account of the fi or B, which was natural or flat according to its various positions in the different tetrachords. The sixteen founds were expressed by eighteen different names; that is to say, that ut or C, and re or D, being either the sharpest or the middle founds of the third tetrachord, according to the two manners of disjoining the tetrachords, they gave to each of these two founds a name which determined its position.

But as the fundamental found was varied according to the mode, from the situation occupied by each mode in the general system arose a difference of acuteness and gravity, which very much multiplied the founds; for though the different modes had many founds in common, there were likewise some peculiar to each mode, or to some of them alone. Thus, in the diatonic genus alone, the extent of all the founds admitted in the fifteen modes enumerated by Aliphius amounted to three octaves; and as the difference between the fundamental found of each mode and that of its contiguous found was a semitone only, it is evident, that all that space divided by semitones produced, in the general scale, the quantity of thirty-four founds practised in ancient music; which, if we deduct all the replicates of the same found, and confine ourselves to the limits of an octave, it will be found to be chromatically divisible into twelve different founds, as in modern music. This is obvious from the table placed by Meibomius at the front of Aliphius's work. These remarks are necessary to refute the error of those who believe, upon the credit of some moderns, that the whole of ancient music was limited to sixteen founds.

In Rousseau's Musical Dictionary, Plate H, fig. 12, will be found a table of the general system amongst the Greeks, taken in one mode only, and according to the diatonic genus. With respect to the enharmonic and chromatic genera, the tetrachords were divided by very different proportions; but as they always contained four founds and three consecutive intervals, in the same manner as the diatonic genus, each of these founds, in its particular genus, bore the same names which corresponded with them in the diatonic. For this reason Rousseau, whom we follow, has not given particular tables for each of these genera. The curious may consult those of Meibomius, placed at the front of the work of Arithoxenus. They will there find five; one for the enharmonic genus, three for the chromatic, and two for the diatonic, according to the situations of each of these genera in the system of Aristoxenus.

Such, in its perfection, was the general system of the Greeks; which remained almost in the same state till the eleventh century, the time when Guy d'Arezzo made considerable changes in it. He added below a new string, which he called hypoproflambanomene, or "sub," now in added," and above a fifth tetrachord. Besides this, he invented, as they say, a flat, to distinguish the second found of a conjunctive tetrachord from the first of the same tetrachord when disjunctive; that is to say, he fixed the double signification of the letter B, which St Gregory before him had already given to the note fi or B. For since it is certain that the Greeks had for a long time these very conjunctions and disjunctions of the tetrachord, and of consequence signs for expressing each degree in these different cases, it follows, that this was not a new found introduced into the system of Guido, but merely a new name which he gave to that found; thus reducing to one degree what, among the Greeks, had constituted two. It must likewise be observed concerning his hexachords, which were substituted for their tetrachords, that it was less a change of system than of method; and that all which resulted from it was another manner of following the same founds. But the character of Guido, and the alterations which he made in the ancient scale, may be more properly resumed when we reach the period in which he lived. We have already seen from Rousseau, that the different accounts of the system and its improvements, of the different kinds of music, and of the modes to be met with among ancient harmonists, are so various and so obscure, that, in these disquisitions, little or no satisfaction can be obtained. For ascertaining with accuracy the diversity of intervals, Pythagoras, the philosopher of Samos, invented the monochord, or the division of the string by which the consonances were produced, and found the same ratios which are given in the subsequent elements of music, in Malcolm's account of the scale, and in several other authors unnecessary to be enumerated. For a fuller and more exact account of this monochord, and its use, see the History of Music by Sir John Hawkins, Vol. I. p. 449, where the necessity of applying it to practice is inculcated by Guido.

Had succeeding writers upon the science been more attentive to the real constitution of the scale, and the principles derived from a monochord properly divided, we might have expected their account of the other phenomena in music to have been more precise and more perspicuous; but for a considerable time after that philosopher, the accounts of ancient music transmitted to us are either superficial and cursory, or unintelligible. The modes, of which Aliphins reckoned fifteen, are by Ptolemy limited to seven. Even of the seven Ptolemaic modes, it would seem that five opinions must be merely possible and nominal; two only real, concerning and practical. These appear to coincide with the major and minor mode of the moderns, by which effects similar to those ascribed to the ancient modes are produced. Still, however, this hypothesis is attended with some difficulty: The effects attributed to the modes of the moderns seem to be no more than cheerfulness and melancholy; whereas it would appear that different sentiments were thought to be naturally excited by all the different modes of the ancients, such as courage and terror, fury and complacency, &c. Yet Yet if by ancient modes we are to understand any given intervals which predominate in a piece of music, it is far from being easy to conceive any other explication which will so rationally account for the modes of Ptolemy, as that which we have immediately before recited. A more particular detail of this author, of Boetius, and of Arithides Quintilianus, than it is in our power to give, circumscribed as we are by limits much too narrow for such an undertaking, will be found in Sir John Hawkins's History of Music, Vol. I.

These are some of the chief writers whose works remain to us, and have escaped the depredations of time. Most of the other ancient writers upon music either appear to have been lost, or only to have treated the subject occasionally. Among these may be reckoned Vitruvius, author of a treatise on architecture, who, in his description of theatres, takes the opportunity of proposing some musical improvements, of making some casual observations upon the art, and of describing an hydraulic organ. But as a more particular account of these would throw no additional light upon the theory of ancient music, for this we must once more remit the curious to Melibonius de re Musica, and to the history by Sir John Hawkins above quoted.

The province to which our efforts are necessarily confined, directs our attention not so much to the history of those who cultivated the art, as to the art itself, and its various revolutions.

The discovery of the monochord and its divisions, was not the only speculation in music peculiar to Pythagoras. He likewise thought the earth and seven planets, or solar system, resembled a musical diapason; and from thence formed the romantic idea of the music of the spheres. For a more satisfactory account of this celestial concert, the curious reader may peruse the Somnium Scipionis, a fragment of Cicero, and the Observations upon numbers by his commentator Macrobius.

Pythagoras, the philosopher of Samos, as we have said above, who taught in Italy, was the first who investigated the relations of sound by measuring a musical string, and observing the tones produced by the vibrations of its different parts, whilst the others were at rest. These he expressed by numbers, and thus ascertained the ratio which one found bears to another. This investigation was afterwards carried farther, and delineated more distinctly, by Euclid; and gave rise to a controversy which divided the theoretical writers on ancient music into two principal sects, viz. the followers of Pythagoras, who maintained that intervals could only be ascertained by the vibrations of sonorous bodies compared one with another; and those of Aristoxenus, who asserted the judgment of the ear to be the ultimate criterion of intervals. Perhaps neither were absolutely right, nor entirely wrong. Without ascertaining by experiments and calculations the distances of tones, or quantities of intervals, we can by no means obtain the same certainty of their exactitude, whether in tuning instruments of fixed scales, or in performing upon those whose notes admit of variation, and where the temperament is immediate and occasional. So far the Pythagoreans are right. Yet the Aristoxenians might likewise urge, that, though we could suppose a being acquainted with all the properties, relations, and modes of quantity, in their full extent; if such a one, with all this knowledge, should attempt from mere theory to compose a piece of good music, he might be eternally engaged in the same employment to no purpose, and have the mortification to see himself every instant outdone by a mere mechanical performer, who had been long inured to judge of intervals, and practised in the laws of harmony. In short, the whole powers of geometry and algebra may be exhausted, without producing a musical strain which will give real pleasure to the ear. An adept, therefore, in this delightful art, will regulate his practice by his theory, and confirm his theory by his practice. He will not imagine the necessity of experiment and calculation superceded by the decision of his ear; nor will he endeavour to extort from the abstract nature of numbers (which are equally applicable to all subjects that contain quantity) those rules which taste and sensation alone can suggest, and of which they are the ultimate standard.

Nicomachus the Gerasenian lived A.C. 60, and wrote a book called Introduction to harmony, which seems to be one of the clearest and most intelligible of the Greeks.

In the Symposiaca of Plutarch is a dialogue on music, containing many anecdotes with respect to the invention of several different species of music and poetry. There Pliny and Timotheus are recorded to have been stigmatized for adding what were esteemed superumerary strings to the lyre, which at that time had only seven, to mark the different degrees of the diapason. But the additional strings were tuned by intervals less than diatonic. This dialogue, however, is acknowledged to be obscure, and its authenticity questioned.

After exploring what can be known concerning the ancient music, from the theories and writings of those whose works have been transmitted to us, the forms and powers of their instruments occur next to be examined. These can only be collected from verbal descriptions, or from designs either expressed in colours or by sculpture. From these, modern musicians have not of ancient scruple to form a most contemptible idea of practical music among the ancients. But are we sure, that the descriptions are perfectly complete and thoroughly understood? If they were, does there not still remain a possibility, that they might be tuned and handled in a manner productive of effects to which we are strangers? Of our instruments now in use, the difference between one manner of performing and another is so astonishing, that one should imagine it might render us cautious in forming any conclusions concerning instruments, which are perhaps neither perfectly described nor exactly delineated, described by authors of a period sufficiently distant to render the idioms of the language in which they wrote obscure. And tho' the forms exhibited in colours or by sculpture may be thought more permanent and more universally intelligible, they are yet sufficiently subjected to the injuries of time to render their representations suspicious. The power it cannot be doubted, but that the accounts of the music of ancient, of the power and efficacy of their music, were exaggerated frequently fabulous and hyperbolical; but still they are in fable, such as, when divested of these accidental circumstances, must have been great must convince any man of common sense, who admits to give the fables credit. the evidence of history, that they are superior to what we at present experience in music with all its boasted improvements. It may well be admitted, that the miracles ascribed to Orpheus and Amphion are false in their literal sense; but no person will imagine, that, even among the superstitious and illiterate vulgar, fables of this kind could have obtained any degree of attention, or been entertained with any other sentiments than those of ridicule, if the truths which they adumbrated had not been uncommonly striking. Nor would it have been relished as a tolerable legend, that music had the power of animating stones and trees, if its visible effects upon sensitive beings at that period had not been wonderfully transporting. It is therefore a degree of incredulity which does no great honour to the authority of modern testimony, to doubt the assertion of Horace, when he tells us, that, by the force of music, the human savage was allured from his acorns, his brutal pastimes, and his sanguine broils, to the more decent habits and amiable employments of social life. It has been formerly observed, that among such nations as were esteemed barbarous, we meet with no accounts either of music or its instruments which either deserve credit or attention. It is not easy to conceive how the Jews, who had made such a great progress in arts and civilization, should still have remained so backward in their musical acquisitions, as they must have been if we take for granted the figures and powers of their instruments, as delineated by Kircher, and transcribed by Sir John Hawkins. Nor will the advantages which are generally allowed to the instruments of other barbarous nations, afford a satisfactory account how they were able either to compose or perform such pieces of music as we know them to have possessed. We must therefore with good reason suspect, that the authors of such descriptions have either been grossly ignorant of the subject, or shamefully careless and remiss in the performance of their task.

Almost in every period since the restitution of literature, an important controversy has been agitated by virtuosos of different opinions in the theory of music. Some have maintained that harmony was, and others that it was not, known to the ancients. By some of these it was contended, that the knowledge of harmony naturally results from the knowledge of consonances; that by tuning their instruments the ancients must have been familiar to the various coalescences of sound, and that of consequence they could not be ignorant of the pleasure which they produce. Several passages likewise from such dissertations or fragments as have escaped the rage of time, are collected to prove that the ancients must have been acquainted with harmony or symphonical music.

The opponents of this hypothesis have alleged, that from the fancies or ideas of simple chords no conception could be formed of the effects produced by their conjunction or succession. It is on all hands agreed, that several voices and instruments were used by the Greeks and Romans in performing the same piece of music; but the antiharmonists, as we may term them, will not admit that the intervals of these voices or instruments were varied: nay, it is affirmed that they performed always in octave or unison one to the other; and from thence it is pretended, that all the passages which seem to import the acquaintance of the ancients with practical harmony may be rationally and confidently explained. This, however, notwithstanding the labours of French critics, will still remain extremely doubtful to any person who has either perused the dialogue above mentioned as ascribed to Plutarch, or other passages to the same purpose.

Nor can it be reasonably thought, that, at a period so barbarous as the 12th century, harmony, though rude and simple, should have been the creature of naked invention in places where every other branch of literature and degree of culture were unknown. Yet it is clear from the monkish historians of that era, that harmony was even then in practice, where it could hardly be supposed to be immediately transmitted by a progress so rapid from other parts of the world, where the finer organization of the natives, the more propitious aspect of nature, and the more obvious vestiges of ancient improvement, might be thought favourable to the invention, culture, and propagation of the fine arts. Nor is it a weak presumption, in favour of the knowledge of antiquity in harmony, that the adherence of a contrary opinion can neither ascertain the epoch nor the parent of symphonic music. Yet had it been, as they pretend, a modern invention, barbarous and ignorant as the general character of human nature was during that gloomy interval from the decline of the Roman empire to the resuscitation of letters, the author of an improvement to new and extraordinary could not have escaped the public notice. His name, his character, and his discoveries, must have been recorded by the cloistered authors of his time with panegyric and admiration. We cannot therefore cease to think with the author of The principles and power of harmony, p. 133, that the ancients were better acquainted with this species of music than the moderns are willing to allow; though perhaps it may be admitted, that its powers were neither so thoroughly known, nor so generally and successfully practised, as afterwards.

After the long and cruel devastation of the Goths and Vandals, music seems first to have been revived for the service of the church. It was then of two different kinds, one of which was called the Ambrosian and the other the Gregorian chant. Of these, the last prevailed, and became universal, till corrupted by the ignorance or false taste of its teachers and performers. This degeneracy became at last the subject of high remonstrance and complaint. It seems to have consisted in a total negligence of rhythm, and in a perversion of that licence of gracing the notes, which is so essential to all emphatic and animated music. It became, however, so contagious and diffusive, that monarchs thought the rescue of the Cantus Gregorianus an object worthy of their interposition. They accordingly authorized more profound adepts and more accurate performers to teach and practise it in its purity through their several dominions. The antiphonaries, or books of ecclesiastical music, were rectified, and a more correct and legitimate taste re-established. Thus the Cantus Gregorianus once more triumphed over ignorance and barbarity, and obtained a reception worthy of its original sublimity. It is denominated among the French, and by Rousseau in particular, plain chant. That author scruples not to reckon it History.

it a precious remain of antiquity. For a short account of the nature and revolutions of this music, may be consulted the article Plain Chant in his Musical Dictionary. From whence it appears, that the Gregorian music was not originally different from the Ambrosian, but the latter only an improvement upon the former. One would be tempted to suspect, that the first gradation of this music towards its decline was occasioned by transferring it from verse to prose. In consequence of which, that strict and inviolable regard to measured sounds, so conspicuous in ancient music, and so effectually preserved by the aptitude of measured notes to measured syllables, was lost. There is, we know, even in profaic compositions, a rhythmus. The Roman orators were accustomed to scan their sentences in prose. But though even periods of this kind were by no means emancipated from the laws of rhythmus, yet were they much more loose and indefinite than poetical numbers, which were constituted by feet and syllables whose quantities were determined. From thence, and from the cadences by which they are marked, alone, can result that regularity and satisfaction in which the musical ear acquiesces, and without which every thing is unintelligible. It was this religious observation of determined and regular quantities in ancient poetry, which preserved and regulated the due proportion of sounds, and which, when abandoned, left the value of notes, with respect to their duration, impossible to be determined, till other characters and signs were superadded, which discovered the real estimate of every note, and showed to what degree it should be protracted, or by what quantity of duration limited. This seems to have been the next advance in musical improvement; but it had one pernicious effect, which was, to render music independent of poetry. Yet these latter-arts seem to be twin-born from heaven; and perhaps, in no case could the laws of nature have suffered a more cruel and impious violation than in separating the one from the other. Modulated sound is a more genuine, powerful, and universal vehicle of sentiment, than any articulate or arbitrary signs can possibly be. But articulate signs may be so happily adjusted by convention, as to express degrees, varieties, and modes of sentiment or emotion, which in modulated sounds are less definitely signified, if signified at all. Thus sounds give energy and sweetness to words, words variety and definiteness to sounds.

We have already observed, that Guy d’Arezzio, otherwise named Guido Aretinus†, was the inventor of that disposition of the musical scale which is now in use. He could not, therefore, be the author of harmony, which we know to have been practised some centuries before his time, but only of a new set of characters by which it was expressed. This musician, by changing the tetrachords into hexachords, highly improved the scale, discovered more accurately the position of semitones, and rendered its intonation much more practicable. He likewise adapted the syllables ut, re, mi, fa, sol, la, to the various sounds which compose it, from the following Sapphic verses in a hymn to St John.

UT queant laxis RESonare fibris Mira gestorum FAmuli tuorum SOLve polluti LABii reatum.

SANCTE JOANNES.

The rhythmus in music, or the regular division and measures of sound, had formerly been determined by the quantities of the feet in poetry; and, independent of these, seems to have been entirely indefinite. The invention of a rhythmus capable of subsisting by itself, is ascribed to one Johannes de Muris. Yet there is considerable reason to believe that it had been invented by one Franco, who lived a number of years before him.

In these times there was a secular as well as sacred Division of music. The Troubadours, or Provençal poets, composed songs of different kinds, which they sung to call and their harps or violins for public entertainment. Hence secular. It happened, that harmony, melody, and rhythmus, admitted of immensely greater varieties than they had hitherto done. We have formerly said, that in ancient music, the quantities or values of every note were determined by those of the syllables to which they answered. It is, however, by no means improbable, that at a very early period, in their private rehearsals, or practice for improvement, whether in taste or execution, the musicians frequently played the instrumental parts without being accompanied either by the voice or the words to which they had been set. The impressions of those poetical measures to which the parts corresponded, were abundantly sufficient to preserve in the memory of the performer the idea of the rhythmus, and of course to determine the value of each particular note. But when airs were either set to pieces in prose, or composed without any regard to syllabic duration, the quantity of each note was absolutely indefinite. When therefore music begun to be set Origin of in parts, it was indispensably necessary that the points the term which mark the notes intended to correspond one with counter another, should be set in direct opposition. Hence the point denomination of counterpoint. But when characters, or different forms of characters, were invented for expressing the different durations of sounds, or their relative proportions one to another, the same precision in opposing note to note became less necessary, and was on that account less scrupulously observed. It might, perhaps, be neither an unpleasing nor uninstructive deduction, after having delineated the nature of simple counterpoint, to trace it thro’ all its different species or divisions; but the contracted sphere in which we are at present constrained to move, obliges us to confine these excursions. Such readers as may wish more profoundly and minutely to examine this matter, will find it more perspicuously and fully explained in Sir John Hawkins’s History. To this they may likewise recur for an idea of the characters or methods by which the precise duration of particular notes might be ascertained. For us, it suffices to add, that the method now in practice, which are explained in the following elements, will be found more simple, whilst at the same time, it is equally expressive and intelligible.

The airs into which secular music was originally di- Division of stinguished seem to have been the madrigal, the song, secular mu- the cantata, the canon. These were vocal, or at least fic common to voices and instruments; but the solo, the phantasia, the concerto, were progressive changes in instrumental music. By what gradations they proceeded, and who were the inventors of each particular species, we cannot attempt to show, not only because such a disquisition would be incompatible with the limits of our plan, but because we should find it frequently impracticable either to investigate the hints from which such innovations arose, or the persons by whom they were made.

If music be allowed to possess imitative powers, it will follow, that in proportion as the objects are interesting, the imitation must likewise engage and command attention. From this, it will be acknowledged, that as imitation is the chief purpose of dramatic music; as the actions, characters, and situations exhibited in the drama, are the most interesting that can possibly be displayed; and as the dramatic is allowed to be the most perfect of all possible imitations; so of all music, the dramatic, in its perfection, ought to be the most powerful and enchanting. It is therefore a research of no small importance, to discover when this kind of music was first revived, and by what degrees it arrived at its present state.

It is generally agreed, that the Greeks and Romans sang their tragedies and comedies from beginning to end; but no monument of these compositions remains to us; so that the music of the drama is as really a modern invention as if no such thing had subsisted among the ancients, since the mere knowledge of a fact could by no means throw any light upon the manner in which it was produced. All that has been transmitted to us concerning the ancient theatrical music, can only inform us, that it was pathetic and imitative to a high degree. But upon these hints few composers will think themselves sufficiently instructed to proceed. This arduous enterprise, however, was nobly begun and successfully prosecuted by one Jacopo Peri. A poet, whose name was Ottavio Rinuccino in the city of Florence, having composed a dramatic pastoral upon the story of Apollo and Daphne, engaged this excellent musician to set it. Both being warmed with the same ideas, and animated by the same design, so happily succeeded, that other poets and musicians were generally approved and admired in proportion as they pursued the vestiges of these great matters. A second performance of the same kind, called Euridice, composed by the authors of the former pastoral, was represented in Florence in the year 1600, upon occasion of the marriage of Mary de Medici with Henry IV of France. But a detail of the gradations by which theatrical music rose to its present perfection, would be a task too extensive for the limits by which we are circumscribed. Nor is it in our power, for the same reason, to enter more minutely and critically into the nature of those compositions called operas. Let it suffice to add, that, in common with tragedy and comedy, they are representations of action. In consequence of this, they require the same unity of design, the same diversity of characters and passions, with the former. Hence it follows, that some parts of them will be simply narrative, some pathetic, and others more emphatically descriptive. Music suited to the first of these is called recitative. Its distinguishing characteristics are, to express the nature and degree of sentiment exhibited by the speaker, to be scrupulously adapted to the peculiar genius of that language which it is designed to accompany; and to be exactly modelled according to the accents of that nation, for which it was formed. Some authors have pretended that the irresistible efficacy of melody was founded upon this principle alone. But if that position be true, in what manner shall we account for the wonderful influence of an Italian recitative upon a British audience, and for other phenomena of the same kind too numerous to be mentioned? Such parts of the music as are intended for more pathetic declamations may be called airs. In these words, both with respect to their quantity and order, may be treated with greater freedom. The melody is less in the tone of conversation, and the harmony more complex. In this, however, there is no small hazard lest sentiment should be lost in sound; and it requires no small degree of judgment, delicacy, and taste in the composer, at once to fill the harmony and preserve the sentiment. But of this the reader will find a more complete account under the article Air in this Dictionary. The chorus is intended to express some emphatic event, to celebrate some distinguished hero, or to praise some beneficent god. It is properly the voice of triumph and exultation. The harmony should therefore be as full and expressive as possible. But for the rules of such compositions, one must refer the reader to such theoretical and practical musicians as have been most successful in describing and cultivating dramatic music. What remains for us is to subjoin a list of those who have been most remarkable for their accuracy in the theory, or for their excellence in the practice, of music in general.

Of John de Muris we have already spoken, who lived in the year 1330, and to whom, by mistake, has been attributed the invention of those characters by practical which, in modern times, the value of notes, and their musicians' relative proportions one to another, have been ascertained. But this expedient for making visible the different durations of notes as constituent of one rhythmus or particular movement, we have found to be first introduced by one Franco, who lived prior to John de Muris.

Latus was the first who wrote on music; but his work is lost, as well as several other books of the Greeks and Romans upon the same subject. Aristoxenus, the disciple of Aristotle, and leader of a sect in music, is the most ancient author who remains to us upon this science. After him came Euclid of Alexandria. Arithidius Quinilianus wrote after Cicero. Alphus afterwards succeeded; then Gaudentius, Nicomachus, and Bacchius.

Marcus Meibomius has favoured us with a beautiful edition of these seven Greek authors, with a Latin translation and notes.

Plutarch, as has already been said, wrote a dialogue upon music. Ptolemy, a celebrated mathematician, wrote in Greek a treatise intitled The Principles of Harmony, about the time of the emperor Antoninus. This author endeavoured to preserve a medium between the Pythagoreans and the Aristoxenians. A long time afterwards, Manuel Pryennius wrote likewise upon the same subject.

Among the Latins, Boetius wrote in the times of Theodoric; and not distant from the same period Martianus, Caffiodorus, and St Augustine.

The number of the moderns is almost indefinite. The most distinguished are, Zarino, Salinas*, Valguero, Galileo, Doni, Kircher, Merfome, Parran, Pearticleault, Wallis, Delcartes, Holden, Mengoli, Malcolm, Blind, Baretti, Vallotti, Marcus Meibomius, Christopher Simpson; Tartini, whose book is full of deep researches. History.

MUSIC

and of genius, but tedious from its prodigious length, and perplexed with obscurity; and M. Rameau, whose writings have had this singular good luck, to have produced a great fortune without being read almost by any one. Besides, the world may now be spared the pains of perusing them, since M. d'Alembert has taken the trouble of explaining to the public the system of the fundamental bas, the only useful and intelligible discovery which we find in Rameau's writings. To these we may add Dr Smith, author of a learned and mathematical treatise, intitled, Harmonics, or The Philosophy of musical Sounds; Mr Stillingfleet, author of the Principles and the Power of Harmony, or An explication of Tartini's system; Dr Pepusch, and his noble pupil the Lord Abercorn; Mr Avison, late organist at Newcastle, who wrote a treatise on Musical Expression with the politeness and elegance of a gentleman, the depth and precision of a scholar, the spirit and energy of a genius. The names of Rousseau and d'Alembert have been so often repeated during the course of these musical lucubrations, that it would be superfluous to resume their characters in this place. Amongst the authors already mentioned, it would be unpardonable to omit the names of Sir John Hawkins and Dr Burney, each of whom has favoured the world with a history of music: The first protracted to five volumes in quarto, replete with musical erudition, but seldom original; frequently careless, and sometimes too circumstantial and inelegant to be entertaining. Of the last the world has only as yet seen one volume. This abounds with descriptions, events, and disquisitions, highly worthy of attention: but, on account of the limits which the author has prescribed to himself, many things have been omitted which would have been equally acceptable to literary curiosity, and explicative of musical science.

ELEMENTS OF MUSIC,

Theoretical and Practical (†).

PRELIMINARY DISCOURSE.

MUSIC may be considered, either as an art, which has for its object one of the greatest pleasures of which our senses (‡) are susceptible; or as a science, by which that art is reduced to principles. This is the double view in which we mean to treat of music in this work.

It has been the case with music as with all the other arts invented by men: some facts were at first discovered by accident; soon afterwards reflection and observation investigated others; and from these facts, properly disposed and united, philosophers were not slow in forming a body of science, which afterwards increased by degrees.

The first theories of music were perhaps as ancient as the earliest age which we know to have been distinguished by philosophy, even as the age of Pythagoras; nor does history leave us any room to doubt, that from the period when that philosopher taught, the ancients cultivated music, both as an art and as a science, with great assiduity. But there remains to us much uncertainty concerning the degree of perfection to which they brought it. Almost every question which has been proposed with respect to the music of the ancients has divided the learned; and may probably still continue to divide them, for want of monuments sufficient in their number, and incontestable in their nature, from whence we might be enabled to exhibit testimonies and discoveries instead of suppositions and conjectures. As we cannot throw any new light upon this subject, all that can be done is to refer our readers to the different authors who have treated of ancient music. It were fitter to wish, that, in order to elucidate as much as possible, we should have recourse to the works of those authors who have treated of ancient music. It were fitter to wish, that, in order to elucidate as much as possible, we should have recourse to the works of those authors who have treated of ancient music.

(†) To deliver the elementary principles of music, theoretical and practical, in a manner which may prove at once entertaining and instructive, without protracting this article much beyond the limits prescribed in our plan, appears to us no easy task. We therefore hesitated for some time, whether to try our own strength, or to follow some eminent author on the same subject. Of these the last seemed preferable. Amongst these authors, none appeared to us so eligible as the treatise of M. D'Alembert, being the most methodical, perspicuous, concise, and elegant dissertation which we have met with. As this work is hitherto unknown to English readers, it ought to have all the merit of an original. We have given a faithful translation of it; but in the notes, several remarks are added, and many authors quoted, which will not be found in the original. It is a work so systematically composed, that all attempts to abridge it, without rendering it obscure and imperfect, would be impracticable. It is perhaps impossible to render the system of music intelligible in a work of less compass than that with which our readers are now presented; and, in our judgment, a performance of this kind, which is written in such a manner as not to be generally understood, were much better suppressed.

(‡) In this passage, and in the definitions of melody and harmony, our author seems to have adopted the vulgar error, that the pleasure of music terminates in corporeal sense. He would have pronounced it absurd to affect the same thing of painting. Yet if the former be no more than a mere pleasure of corporeal sense, the latter must likewise be ranked in the same predicament. We acknowledge that corporeal sense is the vehicle of sound; but it is plain from our immediate feelings, that the results of sound arranged according to the principles of melody, or combined and disposed according to the laws of harmony, are the objects of a reflex or internal sense.

For a more satisfactory discussion of this matter, the reader may consult that elegant and judicious treatise on Musical Expression by Mr Avison. In the meantime it may be necessary to add, that, in order to shun the appearance of affectation, we shall use the ordinary terms by which musical sensations, or the mediums by which they are conveyed, are generally denominated. as possible, a point so momentous in the history of the sciences, some person of learning, equally skilled in the Greek language and in music, should exert himself to unite and discuss in the same work the most probable opinions established or proposed by the learned upon a subject so difficult and curious. This philosophical history of ancient music is a work which might highly embellish the literature of our times.

In the meantime, till an author can be found, sufficiently instructed in the arts and in history, to undertake such a labour with success, we shall content ourselves with considering the present state of music, and limit our endeavours to the explication of those acquisitions which have accrued to the theory of music in these latter times.

There are two departments in music, melody* and harmony†. Melody is the art of arranging several sounds in succession one to another in a manner agreeable to the ear; harmony is the art of pleasing that organ by the union of several sounds which are heard at one and the same time. Melody has been known and felt through all ages; perhaps the same cannot be affirmed of harmony (§), we know not whether the ancients made any use of it or not, nor at what period it began to be practised.

Not but that the ancients certainly employed in their music those chords which were most perfect and simple; such as the octave, the fifth, and the third: but it seems doubtful, whether they knew any of the other consonances or not, or even whether in practice they could deduce the same advantages from the simple chords which were known to them, that have afterwards accrued from experience and combinations.

If that harmony which we now practise owes its origin to the experience and reflection of the moderns, there is the highest probability, that the first essays of this art, as of all the others, were feeble, and the progress of its efforts almost imperceptible; and that, in the course of time, improving by small gradations, the successive labours of several geniuses have elevated it to that degree of perfection in which at present we find it.

The first inventor of harmony escapes our investigation, from the same causes which leave us ignorant of those who first invented each particular science; because the original inventors could only advance one step, a succeeding discoverer afterwards made a more sensible improvement, and the first imperfect essays in every kind were lost in the more extensive and striking views to which they led. Thus the arts which we now enjoy, are for the most part far from being due to any particular man, or to any nation exclusively: they are produced by the united and successive endeavours of mankind; they are the results of such continued and united reflections, as have been formed by all men at all periods and in all nations.

It might, however, be wished, that after having ascertained, with as much accuracy as possible, the state of ancient music by the small number of Greek authors which remain to us, the same application were immediately directed to investigate the first incontestable traces of harmony which appear in the succeeding ages, and to pursue these traces from period to period. The products of these researches would doubtless be very imperfect, because the books and monuments of the middle ages are by far too few to enlighten that gloomy and barbarous era; yet these discoveries would still be precious to a philosopher, who delights to observe the human mind in the gradual evolutions of its powers, and the progress of its attainments.

The first compositions upon the laws of harmony which we know, are of no higher antiquity than two ages prior to our own; and they were followed by many others. But none of these essays was capable of satisfying the mind concerning the principles of harmony; imperfectly confined themselves almost entirely to the single occupation of collecting rules, without endeavouring to account for them; neither had their analogies one with another, nor their common source, been perceived; a blind and unenlightened experience was the only compass by which the artist could direct and regulate his course.

M. Rameau was the first who began to transmute its precepts light and order through this chaos. In the different notes produced by the same sonorous body, he found the most probable origin of harmony, and the cause of all till by that pleasure which we receive from it. His principle M. Rameau unfolded, and shewed how the different phenomena of music were produced by it: he reduced all the consonances to a small number of simple and fundamental chords, of which the others are only combinations or various arrangements. He has, in short, been able to discover, and render sensible to others, the mutual dependence between melody and harmony.

Though these different topics may be contained in the writings of this celebrated artist, and in these writings may be understood by philosophers who are like-wise for wife adepts in the art of music; still, however, such writing musicians as were not philosophers, and such philosophers as were not musicians, have long desired to see these objects brought more within the reach of their capacity: such is the intention of the treatise I now present to the public. I had formerly composed it for the use of some friends. As the work appeared to them clear and methodical, they have engaged me to publish it, persuaded (though perhaps with too much credulity), that it might be useful to facilitate the progress of initiates in the study of harmony.

This was the only motive which could have determined me to publish a book of which I might without

(§) Though no certainty can be obtained what the ancients understood of harmony, nor in what manner and in what period they practised it; yet it is not without probability, that both in speculation and practice, they were in possession of what we denominate counterpoint. Without supposing this, there are some passages in the Greek authors which can admit of no satisfactory interpretation. See the Origin and Progress of Language, Vol. II. Besides, we can discover some vestiges of harmony, however rude and imperfect, in the history of the Gothic ages, and amongst the most barbarous people. This they could not have derived from more cultivated countries, because it appears to be incorporated with their national music. The most rational account, therefore, which can be given, seems to be, that it was conveyed in a mechanical or traditional manner through the Roman provinces from a more remote period of antiquity. out hesitation assume the honour, if its materials had been the fruits of my own invention, but in which I can now boast no other merit than that of having developed, elucidated, and perhaps in some respects improved, the ideas of another (c).

The first edition of this essay, published 1752, having been favourably received by the world, and copies no longer to be found in the hands of booksellers, I have endeavoured to render this more perfect. The detail which I mean to give of my labours will present the reader with a general idea of the principle of M. Rameau, of the consequences deduced from it, of the manner in which I have disposed this principle and its consequences; in short, of what is still awaiting, and might be advantageous to the theory of this amiable art; of what still remains for the learned to contribute towards the perfection of this theory; of the rocks and quicksands which they ought to avoid in this research, and which could serve no other purpose than to retard their progress.

Every sonorous body, besides its principal sound, likewise exhibits to the ear the 12th and 17th major of that sound. This multiplicity of different, yet concordant sounds, known for a considerable time, constitutes the basis of the whole theory of M. Rameau, and the foundation upon which he builds the whole superstructure of a musical system*. In these our elements may be seen, how from this experiment one may deduce, by an easy operation of reason, the chief points of melody and harmony; the perfect chord, as well major as minor; the two tetrachords employed in ancient music; the formation of our diatonic scale; the different values which the same sound may have in that scale, according to the turn which is given to the bass; the alterations which we observe in that scale, and the reason why they are totally imperceptible to the ear; the rules peculiar to the mode major; the difficulty in intonation of forming three tones in succession; the reason why two perfect chords are proscribed in immediate succession in the diatonic order; the origin of the minor mode, its subordination to the mode major, and its variations; the use of discord; the causes of such effects as are produced by different kinds of music, whether diatonic, chromatic*, or enharmonic†; the principles and laws of temperament‡. In this discourse we can only point out those different objects, the subsequent essay being designed to explain them with the minuteness and precision which they require.

One end which we have proposed in this treatise, was not only to place the discoveries of M. Rameau in their most conspicuous and advantageous light, but even in particular respects to render them more simple. For instance, besides the fundamental experiment which we have mentioned above, that celebrated musician, to render the explication of some particular phenomena in music more accessible, had recourse to another experiment; I mean that which shows that a sonorous body struck and put in vibration, forces its 12th and 17th major in descending to divide themselves and produce a tremulous sound. The chief use which M. Rameau made of this second experiment was to investigate the origin of the minor mode, and to give a satisfactory account of some other rules established in harmony; and with respect to this in our first edition we have implicitly followed him: in this we have found means to deduce from the first experiment alone the formation of the minor mode, and besides to disengage that formation from all the questions which were foreign to it.

It is the same case with some other points (as the origin of the chord of the sub-dominant §, and the See Sub-explication of the seventh in some peculiar respects), upon which it is imagined that we have simplified, and perhaps in some measure extended, the principles of the celebrated artist.

We have likewise banished from this edition, as from the former, every consideration of geometrical, arithmetical, and harmonical proportions and progressions, which authors have endeavoured to find in the mixture and protraction of tones produced by a sonorous body; persuaded as we are, that M. Rameau was under no necessity of paying the least regard to these proportions, which we believe to be not only useless, but even, if we may venture to say so, fallacious when applied to the theory of music. In short, though the relations produced by the octave, the fifth, and the third, &c., were quite different from what they are; though in these chords we should neither remark any progression nor any law; though they should be incommensurable one with another; the protracted tone of a sonorous body, and the multiplied sounds which result from it, are a sufficient foundation for the whole harmonic system.

But though this work is intended to explain the theoretical theory of music, and to reduce it to a system more complete and more luminous than has hitherto been done, we ought to caution those who shall read this to the advantage of themselves, either by misapprehending the nature of mathematical objects, or the end which our endeavours pursue.

We must not here look for that striking evidence which is peculiar to geometrical discoveries alone, and which can be so rarely obtained in these mixed disquisitions, where natural philosophy is likewise concerned: into the theory of musical phenomena there must always enter a particular kind of metaphysics, which these phenomena implicitly take for granted, and which brings along with it its natural obscurity. In this subject, therefore, it would be absurd to expect what is called demonstration: it is an achievement of no small importance, to have reduced the principal facts to a system consistent with itself, and firmly connected in its parts; to have deduced them from one simple experiment; and to have established upon this foundation the most common and essential rules of the musical art. But in another view, if here it be improper to require that intimate and unalterable conviction which can only be produced by the strongest evidence, we remain in the mean time doubtful whether it is possible to elucidate this subject more strongly.

After this declaration, one should not be astonished, that, amongst the facts which are deduced from our fundamental experiment, there should be some which appear immediately to depend upon that experiment, and others which are deduced from it in a way more remote

(c) See M. Rameau's letter upon this subject, Merc. de Mai 1752. remote and less direct. In disquisitions of natural philosophy, where we are scarcely allowed to use any other arguments, except such as arise from analogy or congruity, it is natural that the analogy should be sometimes more sometimes less sensible; and we will venture to assert, that such a mind must be very improper for philosophy, 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 conductress should desert 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 there number be but small) which appear still in some degree unaccountable from our principle. Such, for instance, is the procedure of the diatonic scale 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 in the art of music appear to be deducible in a simple and easy manner from the protracted tone of sonorous bodies, one ought not perhaps with too much temerity to affirm as yet, that this mixed and protracted tone is demonstratively the only original principle of harmony (b). But in the mean time it would not be less unjust to reject this principle, because certain phenomena appear to be deduced from it with less success than others. It is only necessary to conclude from this, either that by future scrutinies means may be found for reducing these phenomena to this principle; or that harmony has perhaps some other unknown principle, more general than that which results from the protracted and compounded tone of sonorous bodies, and of which this is only a branch; or lastly, that we ought not perhaps to attempt the reduction of the whole science of music to one and the same principle;

(d) The demonstration of the principles of harmony by M. Rameau was not thus entitled in the exposition which he presented in the year 1749 to the Academy of Sciences, and which that society besides approved with all the eulogiums which the author deserved; the title, as inserted in the register of the academy, was, "A memorial, in which are explained the foundations of a system of music theoretical and practical." It is likewise under this title that it was announced and approved of by the Commissioners, who in their printed report, which the public may read along with M. Rameau's memorial, have never dignified his theory with any other name than that of a *system*, the only name in reality which is expressive of its nature. M. Rameau, who, after the approbation of the academy, has thought himself at liberty to adorn his system with the name of a *demonstration*, did not certainly recollect what the academy has frequently declared; that, in approving any work, it was by no means implied that the principles of that work appeared to them demonstrated. In short, M. Rameau himself, in some writings posterior to what he calls his *demonstration*, acknowledges, that upon particular points in the theory of the musical art, he is under a necessity of having recourse to analogy and aptitude; this excludes every idea of demonstration, and restores the theory of the musical art, exhibited by M. Rameau, to the class in which it can only be ranked with propriety, I mean the class of probabilities.

(t) 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, nor 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. found sounds generate a third different from both the others. They have inserted in the Encyclopédie, under the article Fundamental, a detail of this experiment according to M. Tartini; and we owe to the public an information of which in composing this article we were ignorant: M. Rameau, a member of the Royal Society at Montpellier, 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 a 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 he has not discovered with sufficient perspicuity, and from whence the art might perhaps derive considerable advantage if they were placed in a proper light. Of this I am so much the more persuaded, that even though this experiment should not be regarded by others in the same view with M. Tartini as the foundation of the musical art, it is nevertheless extremely probable that one might use it with the greatest advantage to enlighten and facilitate the practice of harmony.

In exhorting philosophers and artists to make new attempts for the advancement of the theory of music, we ought at the same time to let them know the danger of mistaking what is the real end of their researches. Experience is the only foundation upon which they can proceed; it is alone by the observation of facts, by bringing them together in one view, by shewing their dependency upon one, if possible, or at least upon a very small number of primary facts, that they can reach the end to which they so ardently aspire, the important end of establishing an exact theory of music, where nothing is wanting, nothing obscure, but everything discovered in its full extent, and in its proper light. The philosopher who is properly enlightened, will not give himself the trouble to explain such facts as are less essential to his art, because he can discern those on which he ought to expatiate for its proper illustration. If one would estimate them according to their proper value, he will only find it necessary to cast his eyes upon the attempts of natural philosophers who have discovered the greatest skill in their science; to explain, for instance, the multiplicity of tones produced by sonorous bodies. These sages, after having remarked (what is by no means difficult to conclude) that the universal vibration of a musical string is a mixture of several partial vibrations, from thence infer, that a sonorous body ought to produce a multiplicity of tones, as it really does. But why should this multiplied sound only appear to contain three, and why these three preferable to others? Some pretend that there are particles in the air, which, by their different degrees of magnitude and texture, being naturally susceptible of different oscillations, produce the multiplicity of sound in cally true. It is admitted, that sound is a secondary quality of matter, and that secondary qualities have no obvious connection 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, Malcom'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 Sig. 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 a 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 powers of harmony, printed at London in the year 1771.

(e) See the article Fundamental in the French Encyclopédie, vol. vii. p. 63. in question. But what do we know of all this hypothetical doctrine? And though it should even be granted, that there is such a diversity of tension in these aerial particles, how should this diversity prevent them from being all of them confounded in their vibrations by the motions of a sonorous body? What then should be the result, when the vibrations arrive at our ears, but a confused and inappreciable noise, where one could not distinguish any particular sound? (f)

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 hypothesis on hypothesis, 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 one should grant it, would it not follow from thence, that the same chord should successively and rapidly affect us with contrary sensations, since the vibrations are alternately coincident and discrepant? 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, I mean the octave, cannot suffer the most inconsiderable change? Let us in sincerity confess our ignorance concerning the genuine causes of these effects.

The metaphysical 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 (g).

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 rescued from oblivion, nor refuted; and ought only to be regarded as a new proof how far men of genius may deviate from truth and taste, when they engage in subjects of which they are ignorant.

The rules which we have attempted to establish concerning the track which every one ought to pursue 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 nothing to do here (for it is proper that we repeat it), we have nothing to do with the mechanical principles of protracted and harmonic tones produced by sonorous bodies; principles which, till now, have been explored in vain, and which perhaps may be long explored with the same success: we have still left to do with the metaphysical causes of those pleasing sensations which are impressed on the mind by harmony; causes which are still left discovered, and which, according to all appearances, will remain latent in perpetual obscurity. We are alone concerned to show how the chief and most essential laws of harmony may be deduced from one single experiment; and 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, I 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 treatises of this art; such as chord, harmony, tone, 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 and powers of the syllables ut, re, mi, fa, sol, la, si, which

(f) One may see this subject treated at greater length in the Encyclopédie, at the word Fundamental.

(g) To these arguments others may still be added, which may be found under the article Consonance in the Encyclopédie, where this question has been very successfully treated by M. Rousseau.—Thus far the author; but with respect to his strictures concerning the metaphysics of vision, the little progress which philosophers have made in it, and the little probability of their being surpassed by their successors, we cannot forbear to remark, that M. D'Alembert would have been less precipitate and sanguine in his decisions had he read Dr Reid's Inquiry into the human mind on the principles of common sense. The theory of harmony requires some arithmetical calculations, which are necessary for comparing sounds one with another. These calculations are very short, extremely simple, and conducted in such a manner as to be sensibly comprehended by every one; they demand no operation but what is clearly explained, and which every school-boy with the slightest attention may perform. Yet, that even the trouble of this may be spared to such as are not disposed to take it, I have not inserted these calculations in the body of the treatise, but transferred them to the notes, which the reader may omit, if he can satisfy himself by taking for granted the propositions contained in the work, which will be found proved in the notes.

These calculations I have not endeavoured to multiply; I could even have wished to suppress them, if it had been possible: so much did it appear to me to be apprehended that my readers might be misled upon this subject, and might either believe themselves, or at least suspect me of believing, all this arithmetic necessary to form an artist. Calculations may indeed facilitate the understanding of certain points in the theory, as of the relations between the different notes in the gammut and of the temperament; but the calculations necessary for treating of these points are so simple, and, to speak more properly, of so little importance, that nothing can require a less minute or ostentatious display. Do not let us 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 credulity and ignorance, and serve no other purpose but to render their treatises more obscure and less instructive. In the character of a geometer, I think I have some right to protest here (if I may be permitted to express myself in this manner) against such ridiculous abuse of geometry in music.

This I may do with so 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 gammut, 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 not only be useless, but even contrary to sound philosophy, should any one incline to invent other relations, 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 the 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, I have transferred to the 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 to the notes a second time, when the reader is well acquainted with the essential and fundamental rules explained in it.

This second part, strictly speaking, 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 rules and manner of intonation, but merely the position of the notes in the clef f on the fourth line, and of that of f on the second; and even this knowledge may be acquired from the work itself; for in the beginning of the second part I explain the positions of the clefs and of the notes. Nothing else is necessary but to render it a little familiar to our memory, and we shall have no more difficulty in it.

It would be wrong to expect here all the rules of composition, and especially those which direct the composition of music in several parts, and which, being less severe and indispensable, may be chiefly acquired by practice, by studying the most approved models, in an elementary way.

* See Compositions.

The names of the seven notes used by the French are here retained, and will indeed be continued throughout the whole ensuing work; as we imagine, that, if properly associated with the sounds which they designate, they will tend to impress these sounds more distinctly on the memory of the scholar than the letters C, D, E, F, G, A, B, from which characters, except in folk-singing the notes in the diatonic series are generally named in Britain. Amongst us, in the progress of intonation, the syllables ut, re, and fa, have been omitted, by which means the teachers of church-music have rendered it still more difficult to express by the four remaining denominations the various changes of the semitones in the octave. As these artificially change their places, the seven syllables above-mentioned also diversify their powers, and are variously arranged according to the intervals in which the notes they are intended to signify may be placed.

For an account of these variations, see Rouiléau's musical dictionary, article GAMME. See also the Essay towards a rational system of music, by John Holden, part i. chap. i. by 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 I may be allowed the expression; 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 Mr Rameau's treatise of harmony, or in the code of music which he published more lately (1), or lastly in the explication of the theory and practice of music by M. Bethizy (k): this last book appears to me clear and methodical.

One may look upon it (with respect to a practical detail) as a supplement to my own performance; I do this justice to the author with so much more cheerfulness, as he is entirely unknown to me, and as his animadversions upon my work appear to me less severe than it deserved (L).

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 explained in this treatise? Here can only be learned 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.

Such was the aim I pursued in its composition, and such should be the ideas of the reader in its perusal. Once more let me add, that to the discovery of its fundamental principles I have not the remotest claim. The sole end which I proposed was to be useful; to reach that end, I have omitted nothing which appeared necessary, and I should be sorry to find my endeavours unsuccessful.

DEFINITIONS OF SEVERAL TECHNICAL TERMS.

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

1. Melody is nothing else but a series of sounds which succeed one to another in a manner agreeable to the ear.

2. That is called a chord which arises from the mixture of several sounds heard at the same time; and harmony is properly a series of chords which in their succession one to another delights the ear. A single chord is likewise sometimes called harmony, to signify

(1) From my general recommendation of this code, I except the reflections on the principles of sound which are at the end, and which I should not advise any one to read.

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

(L) That criticism and my answers may be seen in the Journeaux Economiques of 1752.

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

From whence it follows, that the interval between ut and re, is the same as between sol and la; between re and Elements.

Definitions. 7. It is for this reason that they have called the interval from mi to fa, and from fa to ut, a semitone; whereas those between ut and re, re and mi, and fa and sol, la, la, and fa, 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 of a semitone, or in general by seconds, whether major or minor; as from re to ut, or from ut to re; from fa to mi, or from mi to fa.

II. The Terms by which the different Intervals of the Gammut are denominated.

9. An interval composed of a tone and a semitone, as from mi to sol, from la to ut, or from re to fa, is called a third minor.

An interval composed of two full tones, as from ut to mi, from fa to la, or from sol to fa, is called a third major.

An interval composed of two tones and a semitone, as from ut to fa, or from sol to ut, is called a fourth.

An interval consisting of three full tones, as from fa to fa, is called a triton or fourth redundant.

An interval consisting of three tones and a semitone, as from ut to sol, from fa to ut, from re to la, or from mi to fa, &c., is called a fifth.

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

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

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

An interval composed of five tones and two semitones, as from ut to fa, is called a seventh major.

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

A great many of the intervals which have now been mentioned, are still signified by other names, as may be seen in the beginning of the second part; but those which we have now given are the most common, and octave, 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 whatever 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 mi, sol, where mi is the lowest and sol the highest found, sol is a third minor from mi ascending, and mi is third minor from sol 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 ut, re, mi, fa, sol, la, fa, UT, one would carry this scale still farther in ascent, it would be discovered without difficulty that a new scale would be formed, UT, RE, MI, FA, &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 RE, the second note of the second scale, will be an octave in ascent to the re of

and mi, as between la and fa; and mi and fa, as between fa and ut.

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

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 sol, la, fa, ut, and in performing with the voice at the same time ut, re, mi, fa, in such a manner that the same sound may be given to ut in the voice with that of the key sol in the harpsichord, it will be found that re in the vocal intonation shall be the same with la upon the harpsichord, &c.

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

Since then, in the scale thus divided,

ut, re, mi, fa, sol, la, fa, UT,

the first division is perfectly like the last; and since the intervals between ut and re, between re and mi, and between fa and sol, are equal; it follows, that the intervals between sol and la, and between la and fa, are likewise equal to every one of the three intervals between ut and re, between re and mi, and between fa and sol; and that the intervals between mi and fa and between fa and ut are also equal, but that they only constitute one half of the others. Definitions of the first scale; in the same manner MI shall be the octave to MI, &c., and so of the rest.

14. As there are nine notes from the first ut to the second RE, the interval between these two sounds is called a ninth, and this ninth is composed of six full tones and two semitones. For the same reason the interval from ut to FA is called an eleventh, and the interval between ut and SOL, a twelfth, &c.

It is plain that the ninth is the octave of the second, 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. (n)

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 ut to re, forming in his progress an intermediate sound, which shall be higher by a semitone than ut, and lower in the same degree than re. A sound in the scale is called sharp, when it is raised by a semitone; and it is marked with this character ♯: thus ut ♯ signifies ut sharp; that it is to say, ut 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 la♭ signifies la flat, or la depressed by a semitone.

V. What is meant by Consonances and Dissonances.

16. A chord composed of sounds whose union or concordance 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 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 triton, and the seventh of a sound, are dissonants with relation to it. Thus the sounds ut re, ut fa, or fa ut, &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, tho' produced at the same time.

PART

(b) 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 major above by 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. Musick.

Tone tone semitone tone tone semitone

A ut re mi fa sol la si vt B ut re mi fa sol la si vt re mi fa sol &c

Scale first. Scale Second.

The diatonic Scale of the Greeks.

D Si Ut Re Mi Fa Sol La Sol Ut Sol Ut Fa Ut Fa

The Fundamental bass.

The Chromatic Species.

C K H G C X V T E R F L N O P Q M

Scale.

K Sol Sol &c. Ut Mi Sol*

The Fundamental bass.

E Ut Re Mi Fa Sol Sol La Si Ut Ut Sol Ut Fa Ut Sol Re Sol Ut

The Fundamental bass.

L Ut Mi Si* Ut Mi Sol*

The Fundamental bass.

F vt, vt*, re, re*, mi, mi*, fa*, sol, sol*, la*, si si*, ut, ut*, re, re*, mi, mi*,

Scale first. Scale Second.

The first Scale of the minor mode.

G Sol La Si Ut Re Mi Fa Mi La Mi La Re La Re

The fundamental bass.

N Mi Mi Mi Mi Mi Ut Ut La Ut Ut

The Fundamental bass.

The second scale of the minor mode.

H La Si Ut Re Mi mi fa* Sol* La La Mi La Re La Mi Si Mi La

The Fundamental bass.

M Fa Mi Mi Re Fa Ut Mi Si

The Fundamental bass.

I Ut Re Mi Fa Sol La Si Ut Ut Sol Ut Fa Ut Re Sol Ut

A Bell Sculp! Enfin, il est en ma puissance, Ce fatal ennemi, Ce superbe vainqueur. Le charme du sommeil le livre à ma vengeance, Je vais percer son invincible cœur; Par lui, tous mes captifs sont sortis d'esclavage. dois me venger aujour d'huy! Ma colere se teint Quand j'approche de luy

Plus je le vois! plus ma vengeance est vaine; Mon bras tremblant se re-

fusé à ma haine: Ah! quelle cruauté de luy ravir le jour! A ce jeune He- Translation. Intended to give such Readers as do not understand French, an idea of the Song.

At length the victim in my power I see, This fatal year resigns him to my rage; Subdued by sleep he lies, and leaves me free, With chastening hand my fury to affrave. That mighty heart invincible and fierce, Which all my captives freed from servile chains; That mighty heart, my vengeful hand shall pierce, My rage inventive wanton in his pains. Ha! in my soul what perturbation reigns! What would compassion in his favour plead? Strike, hand. O heaven! what charm thy force restrains? Obey my wrath. I fight; yet let it bleed. And is it thus my just revenge improves The fair occasion to chastize my foe? As I approach, a softer passion moves, And all my boasting fury melts in wo. Trembling, relaxed, and faithless to my hate, The dreadful task this coward arm declines.

How cruel thus to urge his instant fate, Depriv'd of life amid his great designs! In youth how blooming, what a heavenly grace, Thro' all his form, resistless power displays! How sweet the smile that dwells upon his face, Relentless rage disarming whilst I gaze! Tho' to the prowess of his conquering arms Earth stood with all her hosts opposed in vain; Yet is he form'd to spread more mild alarms, And bind all nature in a softer chain. Can then his blood, his precious blood, alone Extinguish all the vengeance in my heart? Tho' still surviving, might he not atone For all the wrongs I feel, by gentler smart? Since all my charms, unfeeling, he defies, Let Magic force his stubborn soul subdue; Whilst I, inflexible to tears and sighs, With hate (if I can hate) his peace pursue. 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 above the same sound, that is to say, the double octave of its third major.

19. 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 (c).

21. The principal sound is called the generator *; * See Generator.

(c) Since the octave above the sound 1 is 2, the octave below that same sound shall be $\frac{1}{2}$; 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 ut, is denominated

| Its octave above will be | 2 | |-------------------------|---| | Its double octave above | 4 | | Its triple octave above | 8 |

In the same manner its octave below will be

| Its double octave below | $\frac{1}{2}$ | |-------------------------|------------| | Its triple octave below | $\frac{1}{4}$ |

And so of the rest.

| Its twelfth above | 3 | |------------------|---| | Its twelfth below | $\frac{1}{3}$ |

Its seventeenth major above

| Its seventeenth major below | $\frac{1}{5}$ |

The fifth third above the sound 1 being the octave beneath the twelfth, shall be, as we have immediately observed, $\frac{1}{2}$; which signifies that this string performs $\frac{1}{2}$ vibrations, that is to say one vibration and a half, during a single vibration of the string which gives the sound 1.

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

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

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

To determine this relation, it is necessary to remark, that $\frac{5}{4}$ are the same thing with $\frac{1}{\frac{4}{5}}$, and that $\frac{3}{4}$ are the same thing with $\frac{1}{\frac{4}{3}}$: so that $\frac{5}{4}$ shall be to $\frac{3}{4}$ in the same relation as $\frac{1}{\frac{4}{5}}$ to $\frac{1}{\frac{4}{3}}$; 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 5, the second shall be expressed by 6; or what is the same thing, if the first is represented by 1, the second shall be expressed by $\frac{6}{5}$.

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

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 $\frac{5}{4}$, multiplied by 2 of the fraction $\frac{3}{4}$ 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. Theory of Harmony.

22. There is no person insensible of the resemblance which subsists 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 be further 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 upon a pitch too high or too low for his voice, so that he is obliged, lest he should strain himself too much, to sing the tune in question on a key higher or lower than the first; I affirm, that, without being initiated in the art of music, he will naturally take his new key in the octave below or the octave above the first; and that in order to take this key in any other interval except the octave, he will find it necessary to exert a sensible degree of attention. This is a fact of which we may easily be persuaded by experience.

Another fact. Let any person sing a tune in our presence, and let it be sung in a tone too high or too low for our voice; if we wish to join in singing this air, we naturally take the octave below or above, and frequently, in taking this octave, we imagine it to be the unison (E).

Chap. II. The Origin of the Modes Major and Minor; of the most natural Modulation, and the most perfect Harmony.

23. To render our ideas still more precise and permanent, we shall call the tone produced by the sonorities, or sound body ut; 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 sol, and the double octave of mi.

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

25. For the same reason, the modulation formed by ut with the octave of sol and the double octave of mi, sung one after the other, would likewise be the most simple and natural of all modulations which do not reduce to descend or ascend directly in the diatonic order, if chords, our voices had sufficient compass to form intervals of fifths, and octaves, 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 Mode ma the tone ut, we naturally modulate the third mi, and jor, what, the fifth sol, instead of the double octave of mi, and the octave of sol; from whence we form, by joining the octave of the found ut, this modulation, ut, mi, sol, ut, which in effect is the simplest and easiest of them all; and which likewise has its origin even in the protracted and compounded tones produced by a sonorous body.

27. The modulation ut, mi, sol, ut, in which the chord ut, mi, is a third major, constitutes that kind Mode, See of harmony or melody which we call the mode major; likewise from whence it follows, that this mode results from the immediate operation of nature.

28. In the modulation ut, mi, sol, of which we have now been treating, the sounds mi and sol are so nor, what proportioned one to the other, that the principal found ut (art. 19.) causes both of them to resound; but the second tone mi does not cause sol to resound, which only forms the interval of a third minor.

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

Let us suppose, for example, that the third major of the fifth § is sought. That third major ought to be, by what has been shown above, the § of the fifth; for the third major of any found whatever is the § of that found. We must then look for a fraction which expresses the § of §; which is done by multiplying the numerators and denominators of both fractions one by the other, from whence results the new fraction §. It will likewise be found that the fifth of the fifth is §, because the fifth of the fifth is the § of §.

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 ut equal to 1, we have seen that its fifth, its fourth, its third, its major and minor in ascending, are §, §, §, §. To find its fifth, its fourth, its third major and minor in descending, nothing more is necessary than to reverse these fractions, which will give §, §, §, §.

(D) 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.

(E) 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 triomphé, &c.: in this passage of the chorus, the two parts of the vocal and instrumental bass gives 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 in short, the second part, or tenor above, gives the twelfth. Part I.

Theory of arc in search shall be a semitone beneath the natural Harmony. mi, and of consequence mi b.

30. This new arrangement, ut, mi b, sol, in which the founds ut and mi b have both the power of causing sol to refund, though ut does not cause mi b to refund, is not indeed equally perfect with the first arrangement ut, mi, sol; because in this the two founds mi and sol are both the one and the other generated by the principal found ut; whereas, in the other, the found mi b is not generated by the found ut; but this arrangement ut, mi b, sol, 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 ut, mi b, sol, ut, it is evident that the third from ut to mi b is minor; and such is the origin of that mode which we call minor (f).

32. The most perfect chords then are, 1. All chords related one to another, as ut, mi, sol, ut, consisting of any sound of its third major, of its fifth, and of its octave. 2. All chords related one to another, as ut, mi b, sol, ut, consisting of any sound, of its third minor, of its fifth, and of its octave. In effect, these two kinds of chords are exhibited by nature; but the first more immediately than the second. The first are called perfect chords major, the second perfect chords minor.

CHAP. III. Of the Series which the Fifth requires, and of the Laws which it observes.

33. Since the found ut causes the found sol to be heard, and is itself heard in the found fa, which founds sol and fa are its two-twelfths, we may imagine a modulation composed of that found ut and its two-twelfths, or, which is the same thing (art. 22.), of its two-fifths, fa and sol, the one below, the other above; which gives the modulation or series of fifths fa, ut, sol, which I call the fundamental base of ut by fifths.

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

34. Thus, from the found ut, one may make a transition indifferently to the found sol, or to the found fa.

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

mi b, fa, sol, ut, sol, re, la, &c.

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

36. But it is not allowed in the same manner to pass from one found to another which is not immediately contiguous to it; for instance, from ut to re, or from re to ut: for this very simple reason, that the found re is not contained in the found ut; nor the found ut in that of re; 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 founds ut and re, by the first experiment, naturally bring along with them the perfect chords consisting of greater intervals ut, mi, sol, ut, re, fa, la, re; hence may be deduced this rule, That two perfect chords, especially if they are major (g), cannot succeed one another diatonically in a fundamental base; we mean, that in a fundamental base two sounds cannot be diatonically placed in succession, each of which, with its harmonics, forms a perfect chord, especially if this perfect chord be major in both.

CHAP. IV. Of Modes in general.

38. A mode, in music, is nothing else but the order of sounds prescribed, as well in harmony as melody, general, by the series of fifths. Thus the three sounds fa, ut, what, sol, 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 ut...

39. The series of fifths then, or the fundamental Modes, bas fa, ut, sol, of which ut holds the middle space, how represented by the series of fifths.

(f) The origin which we have here given of the mode minor, is the most simple and natural that can possibly be given. In the first edition of this treatise, I had followed M. Rameau in deducing it from the following experiment.—If you put in vibration a musical string AB, and if there are at the same time contiguous to this two other strings CF, LM, of which the first shall be a twelfth below the string AB, and the second LM a seventeenth major below the same AB, the strings CF, LM, will vibrate without being struck as soon as the string AB 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 CF, you may easily distinguish two points at rest D, E, and in the tremulous motion of the string LM four acoustical points N, O, P, Q, 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 ut the tone of the string AB, the two other strings will represent the sounds fa and la b; and from thence M. Rameau deduces the modulation fa, la b, ut, and of consequence the mode minor. The origin which we have assigned to the minor mode in this new edition, appears to me more direct and more simple, because it presupposes no other experiment than that of art. 19. and because also the fundamental found ut is still retained in both the modes, without being obliged, as M. Rameau found himself, to change it into fa.

(g) I say especially if they are major; for in the major chord re, fa, la, re, besides that the founds ut and re have no common harmonical relation, and are even dissonant between themselves (Art. 18.), it will likewise be found, that fa forms a dissonance with ut. The minor chord re, fa, la, re, would be more tolerable, because the natural fa which occurs in this chord carries along with it its fifth ut, 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. Theory of Harmony may be regarded as representing the mode of ut. One may likewise take the series of fifths, or fundamental base, ut, sol, re, as representing the mode of sol; in the same manner fa, fa, ut, will represent the mode of fa.

By this we may see, that the mode of sol, or rather the fundamental base of that mode, has two founds in common with the fundamental base of the mode of ut. It is the same with the fundamental base of the mode of fa.

40. The mode of ut (fa, ut, sol,) is called the principal mode with respect to the modes of these two fifths, which 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 founds in common with the principal mode. Yet the mode of sol seems a little more eligible: for sol is heard amongst the harmonies of ut, and of consequence is implied and signified by ut; whereas ut does not cause fa to be heard, though ut is included in the same sound fa. It is hence that the ear, affected by the mode of ut, is a little more prepossessed for the mode of sol than for that of fa. Nothing likewise is more frequent nor more natural, than to pass from the mode of ut to that of sol.

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

43. It remains to add, as we have seen in the preceding chapter, that, in the series of fifths, we may indifferently pass from one sound to that which is contiguous: In the same manner, and for the same reason, one may pass from the mode of sol to the mode of re, after having made a transition from the mode of ut to the mode of sol, as from the mode of fa to the mode of fi. 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 (‡aa) 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 founds which are contiguous may be placed in immediate succession in the series of fifths, fa, ut, sol, it follows, that one may form this modulation, or this fundamental base, by fifths,

sol, ut, sol, ut, fa, ut, fa.

45. Each of the founds which forms this modulation brings necessarily along with itself its third major, its fifth, and its octave; insomuch that he who, for the instance, sings the note sol may be reckoned to sing at Greek diatonic scale the notes sol, fa, re, sol; in the same tonic scale manner the sound ut in the fundamental base brings along with it this modulation, ut, mi, sol, ut; and, in basi, short, the same sound fa brings along with it fa, la, ut, fa. This modulation then, or this fundamental base,

sol, ut, sol, ut, fa, ut, fa,

gives the following diatonic series,

fa, ut, re, mi, fa, sol, la,

which is precisely the diatonic scale of the Greeks. We see D. are ignorant upon what principles they had formed this scale; but it may be sufficiently perceived, that that series arises from the base sol, ut, sol, ut, fa, ut, fa; and that of consequence this base is justly called fundamental, as being the real primitive modulation, that which conducts the ear, and which it feels to be implied in the diatonic modulation fa, ut, re, mi, fa, sol, la, (‡).

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

In the modulation fa, ut, re, mi, fa, sol, la, the founds

(‡aa) As the mere English reader, unacquainted with the technical phraseology of music, may be surprised at the use of the word phrase when transferred from language to that art, we have thought proper to infer the definition of Rousseau.

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

(‡) Nothing is easier than to find in this scale the value or proportions of each sound with relation to the found ut, which we call i; for the two founds sol and fa in the base are \( \frac{1}{2} \) and \( \frac{3}{4} \); from whence it follows,

1. That ut in the scale is the octave of ut in the base; that is to say, 2.

2. That fa is the third major of sol; that is to say \( \frac{3}{4} \) of \( \frac{3}{4} \) (note c), and of consequence \( \frac{9}{8} \).

3. That re is the fifth of sol; that is to say \( \frac{5}{4} \) of \( \frac{3}{4} \), and of consequence \( \frac{15}{8} \).

4. That mi is the third major of the octave of ut, and of consequence the double of \( \frac{3}{4} \); that is to say, \( \frac{9}{4} \).

5. That fa is the double octave of fa of the base, and consequently \( \frac{9}{4} \).

6. That sol of the scale is the octave of sol of the base, and consequently \( \frac{9}{4} \).

7. In short, that la in the scale is the third major of fa of the scale; that is to say, \( \frac{9}{4} \) of \( \frac{9}{4} \), or \( \frac{27}{16} \).

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

| Diatonic Scale | Fundamental Base | |----------------|------------------| | fa, ut, re, mi, fa, sol, la. | sol, ut, sol, ut, fa, ut, fa. |

And if, for the convenience of calculation, we choose to call the found ut, of the scale 1; in this case there is nothing to do but to divide each of the numbers by 2, which represent the diatonic scale, and we shall have

\( \frac{9}{8}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4} \)

\( \frac{9}{8}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4}, \frac{9}{4} \) Theory of sounds re and fa form between themselves a third minor harmony, which is not so perfectly true as that between mi and sol (1). Nevertheless this alteration in the third minor between re and fa gives the ear no pain, because that re and that fa, which do not form between themselves a true third minor, form, each in particular, consonances perfectly just with the sounds in the fundamental bas which correspond with them: for re in the scale is the true fifth of sol, which answers to it in the fundamental bas; and fa in the scale is the true octave of fa, which answers to it in the same bas.

47. If, therefore, these sounds in the scale form consonances perfectly true with the notes which correspond to them in the fundamental bas, the ear gives itself little trouble to investigate the alterations which there may be in the intervals which these sounds in the scale form between themselves. This is a new proof that the fundamental bas is the genuine guide of the ear, and the true origin of the diatonic scale.

48. Moreover, this diatonic scale includes only seven sounds, and goes no higher than fa, which would be the octave of the first: a new singularity, for which a reason may be given by the principles above established. In reality, in order that the sound fa may succeed immediately in the scale to the sound la, it is necessary that the note sol, which is the only one from whence fa as a harmonic may be deduced, should immediately succeed to the sound fa, in the fundamental bas, which is the only one from whence la can be harmonically deduced. Now, the diatonic succession from fa to sol cannot be admitted in the fundamental bas, according to what we have remarked (art. 36.) The sounds la and fa, then, cannot immediately succeed one another in the scale: we shall see in the sequel why this is not the case in the series ut, re, mi, fa, sol, la, si, ut, which begins upon ut; whereas the scale in question here begins upon si.

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

50. The diatonic scale si, ut, re, mi, fa, sol, la, is composed of two tetrachords, that is to say, of two diatonic scales, each consisting of four sounds, si, ut, re, mi, and mi, fa, sol, la. These two tetrachords are exactly similar; for from mi to fa there is the same interval as from si to ut; from fa to sol the same interval as from ut to re, from sol to la the same as from re to mi; this is the reason why the Greeks distinguished these two tetrachords; yet they joined them by the note mi, which is common to both, and which gave them the name of conjunctive tetrachords.

51. Moreover, the intervals between any two sounds taken in each tetrachord in particular, are precisely true: thus, in the first tetrachord, the intervals of ut qual. mi, and si re, are thirds, the one major and the other minor, exactly true, as well as the fourth si mi (m); it is the same thing with the tetrachord mi, fa, sol, la, since this tetrachord is exactly like the former.

52. But the case is not the same when we compare two sounds taken each from a different tetrachord; for we have already seen, that the note re in the first tetrachord forms with the note fa in the second a third minor, which is not true. In like manner it will be found, that the fifth from re to la is not exactly true, which is evident; for the third major from fa to la is true, and the third minor from re to fa 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 tetrachords.

54. It may be ascertained by calculation, that in tones the tetrachord si, ut, re, mi, the interval, or the tone major and minor investigated.

(1) In order to compare re with fa, we need only compare 8 with 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 re to fa, 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 re and fa, 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 mi and sol, in the continued tone of those sonorous bodies, of which mi and sol 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.

(1) The proportion of si to ut is as 15 to 16; that is to say, as 5 to 4; that between mi and fa is as 8 to 9; that is to say, as 5 to 4; that between fa and la is as 8 to 9. The proportion of mi to ut is as 5 to 4; that between fa and la is as 8 to 9; the proportions hereafter are likewise equal.

(m) The proportion of mi to ut is as 5 to 4, which is a true third major; that from re to si is as 8 to 9; 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 mi to si is as 5 to 4; that is to say, as 5 times 16 to 15 times 4, or as 4 to 3, which is a true fourth. Theory of Harmony.

Theory of from re to mi, is a little less than the interval or tone Harmony from ut to re (n). In the same manner, in the second tetrachord mi, fa, sol, la, which is, as we have proved, perfectly similar to the first, the note from sol to la is a little less than the note from fa to sol. It is for this reason that they distinguish two kinds of tones; the greater tone *, as from ut to re, from fa to sol, &c.; and the lesser †, as from re to mi, from sol to la, &c.

Interval.

Greater tone. See Interval. Lesser tone. See Interval.

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, fa, ut, re, mi, fa, sol, la, by means of a fundamental bass composed of three founds only, fa, ut, sol: but to form the scale ut, re, mi, fa, sol, la, fi, UT, which we use at present, we must necessarily add to the fundamental bass the note re, and form, with these four founds fa, ut, sol, re, the following fundamental bass:

ut, sol, ut, fa, ut, sol, re, sol, ut,

from whence we deduce the modulation or scale:

ut, re, mi, fa, sol, la, fi, UT.

In effect (o), ut in the scale belongs to the harmony of ut which corresponds with it in the bass; re, which is the second note in the gammut, is included in the harmony of sol, the second note of the bass; mi, the third note of the gammut, is a natural harmonic of ut, which is the third found in the bass, &c.

56. From thence 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 ut; whereas ours is originally and primitively formed, not only from the mode of ut (fa, ut, sol), but likewise from the mode of sol, (ut, sol, re.)

It will likewise appear, that this last scale consists of two parts; of which the one, ut, re, mi, fa, sol, is in the mode of ut; and the other, sol, la, fi, ut, is in that of sol.

57. It is for this reason that the note sol is found to be twice repeated in immediate succession in this scale; once as the fifth of ut, which corresponds with it in the fundamental bass; and again, as the octave of sol, which immediately follows ut in the same bass. As to what remains, these two consecutive sol's are otherwise in perfect unison. It is for this reason that we are satisfied with fingering only one of them when one modulates the scale ut, re, mi, fa, sol, la, fi, UT: but this does not prevent us from employing a pause or repose, expressed or understood, after the sound fa. There is no person who does not perceive this whilst he himself fingers the scale.

58. The scale of the moderns, then, may be considered as consisting of two tetrachords, disjunctive indeed, but perfectly similar one to the other, ut, re, mi, fa, and sol, la, fi, ut, one in the mode of ut, the other in that of sol. For what remains, we shall see in the sequel by what artifice one may cause the scale ut, re, mi, fa, sol, la, fi, UT, to be regarded as belonging to the mode of ut 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.

59. The introduction of the mode proper to sol in the mode of sol introduces this happy effect, that the notes fa, sol, la, fi, may immediately succeed each other in ascending the scale, which cannot take place in the diatonic series of the Greeks, because that series is formed from the mode of ut alone. From convenience it follows:

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

2. That if these three notes are sung in succession in the scale ut, re, mi, fa, sol, la, fi, UT, this cannot be done but by the assistance of a pause expressed or understood after the note fa; insomuch, that the three tones fa, sol, la, fi, (three only because the note sol is repeated) are supposed to belong to two different tetrachords.

60. It ought not then any longer to surprise us, Change of that we feel some difficulty whilst we ascend the scale in fingering three tones in succession, because this is impracticable without changing the mode; and if one pausing pauses in the same mode, the fourth found above the third note will never be higher than a semitone above that which immediately precedes it; as may be seen by ending ut, re, mi, fa, and by sol, la, fi, ut, where there is no more than a semitone between mi and fa, and between fi and ut.

61. We

(n) The proportion of re to ut is as \( \frac{3}{4} \) to 1, or as 9 to 8; that of mi to re is as \( \frac{3}{4} \) to \( \frac{9}{8} \), that is to say, as 40 to 36, or as 10 to 9: now \( \frac{9}{8} \) is less removed from unity than \( \frac{3}{4} \); the interval then from re to mi is a little less than that from ut to re.

If any one would wish to know the proportion which \( \frac{9}{8} \) bear to \( \frac{3}{4} \), he will find (note c), that it is as 8 times 10 to 9 times 9, that is to say, as 80 to 81. Thus the proportion of a minor to a major tone is as 80 to 81; this difference between the major and minor tone, is what the Greeks called a comma. Though real, it is imperceptible to the ear.

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 re fa, of which we have taken notice in the scale (note i); for we have seen, that this third minor thus altered is in the proportion of 80 to 81 with the true third minor.

(o) The values or estimates of the notes shall be the same in this as in the former scale, excepting only the tone la; for re being represented by \( \frac{9}{8} \), its fifth shall be expressed by \( \frac{3}{4} \); so that the scale will be numerically signified thus:

\[ \begin{array}{cccccc} 1 & \frac{3}{4} & \frac{3}{4} & \frac{3}{4} & \frac{3}{4} & \frac{3}{4} \\ ut, re, mi, fa, sol, la, fi, UT. \end{array} \]

Where you may see, that the note la of this scale is different from that in the scale of the Greeks; and that the la in the modern series stands in proportion to that of the Greeks as \( \frac{3}{4} \) to \( \frac{9}{8} \), that is to say, as 81 to 80; these two la's then likewise differ by a comma. Part I.

Theory of Harmony.

61. We may likewise observe in the scale ut, re, mi, fa, that the third minor from re to fa is not true, for the reasons which have been already given (art. 49.) It is the same case with the third minor from la to ut, and with the third major from fa to la; but each of these forms sound otherwise consonances perfectly true, with their correspondent sounds in the fundamental base.

62. The thirds la ut, fa la, which were true in the former scale, are false in this; because in the former scale la was the third of fa, and here it is the fifth of re, which corresponds with it in the fundamental base.

63. Thus it appears, that the scale of the Greeks contains fewer consonances that are altered than ours (p); and this likewise happens from the introduction of the mode of sol into the fundamental base (q).

We see likewise that the value of la in the diatonic scale, a value which authors have been divided in ascertaining, solely depends upon the fundamental base, and that it must be different according as the note la has fa or re for its base. See the note (o).

Chap. VII. Of Temperament.

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

(p) In the scale of the Greeks, the note la being a third from fa, there is an altered fifth between la and re; but in ours, la being a fifth to re, produces two altered thirds, fa la, and la ut; and likewise a fifth altered, la mi, 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.

(q) But here it may be with some colour objected. The scale of the Greeks, it may be said, has a fundamental base 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 base, but the arrangement of ours is more suitable to natural intonation. Our scale commences by the fundamental sound ut, and it is in reality from that found that we ought to begin; it is from this that all the others naturally arise, and upon this that they depend; nay, if I may speak so, in this they are included: on the contrary, neither the scale of the Greeks, nor its fundamental base, commences with ut; but it is from this ut that we must depart, in order to regulate our intonation, whether in rising or descending: now, in ascending from ut, the intonation, even of the Greek scale, gives the series ut, re, mi, fa, sol, la; and so true is it that the fundamental found ut is here the genuine guide of the ear, that if, before we modulate the found ut, we 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 fa, and by the semitone from fa to ut. Now to make a transition from fa to ut by this semitone, the ear must of necessity be predisposed for that modulation, and consequently preoccupied with the mode of ut: if this were not the case, we should naturally rise from fa to ut, and by this operation pass into another mode.

(r) The LA considered as the fifth of re is $\frac{7}{12}$, and the fourth beneath this LA will constitute $\frac{4}{12}$ of $\frac{7}{12}$; that is to say, $\frac{8}{12} - \frac{4}{12} = \frac{4}{12}$; then shall be the value of mi, considered as a true fourth from LA in descending: now mi, considered as the third major of the found UT, is $\frac{4}{12}$, or $\frac{8}{12}$; these two mi's then are between themselves in the proportion of 8:1 to 8:0; thus it is impossible that mi should be at the same time a perfect third major from UT, and a true fourth beneath LA.

(s) In effect, if you tune 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.

UT, SOL, a fifth; re a fourth; LA a fifth; mi a fourth; fa a fifth; fa a fourth; ut a fifth; sol a fourth; RE a fifth; la a fourth; MI or FA a fifth; fa a fourth: now it will be found, by a very easy computation, that the first UT being represented by 1, SOL shall be $\frac{4}{12}$, LA $\frac{8}{12}$, mi $\frac{8}{12}$, &c. and so on. Theory of Harmony.

66. From thence it necessarily follows, 1. That it is impossible that all the octaves and all the fifths should be just at the same time, particularly in instruments which have keys, where no intervals less than a semitone are admitted. 2. That, of consequence, if the fifths are justly tuned, some alteration must be made in the octaves; now the sympathy of 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 ut or fis 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, UT, SOL, re, LA, mi, fis, fis, ut, sol, re, la, mi, fis, 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 Theory of Harmony under a necessity of altering, must only be altered in the least degree possible.

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

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

69. It results then from all that we have now said, that the theory of temperament may be reduced to this question.—The alternate succession of fifths and fourths having been given, UT, SOL, re, LA, mi, fis, fis, ut, sol, re, la, mi, fis, it is proposed to alter all the fifths equally, in such a manner that the two ut'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 ut'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 (τ) the alteration made in each fifth will be very inconsiderable, but equal in all directions for temperament.

This fraction is evidently greater than the number 2, which expresses the perfect octave ut to its correspondent UT; and the octave below fis would be one half of the same fraction, that is to say, \( \frac{1}{2} \times \frac{144}{85} \), which is evidently greater than UT represented by unity. This last fraction \( \frac{144}{85} \) is composed of two numbers; the numerator of the fraction is nothing else but the number 3 multiplied 11 times in succession by itself, and the denominator is the number 2 multiplied 18 times in succession by itself. Now it is evident, that this fraction, which expresses the value of the sound UT; though, upon the harpsichord, fis and UT are identical. This fraction rises above unity by \( \frac{1}{2} \times \frac{144}{85} \), that is to say, by about \( \frac{1}{144} \); 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 (N), and which is only \( \frac{1}{85} \).

We have already proved that the series of fifths produces an ut different from fis, the series of thirds major gives another still more different. For, let us suppose this series of thirds, ut, mi, sol, fis, we shall have mi equal to \( \frac{3}{4} \), sol to \( \frac{3}{5} \), and fis to \( \frac{3}{6} \), whose octave below is \( \frac{3}{7} \); from whence it appears that this last fis is less than unity (that is to say, than ut), by \( \frac{1}{144} \), or by \( \frac{1}{85} \), 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 fis, deduced from the series of thirds, is to the fis deduced from the series of fifths, as \( \frac{3}{4} \) is to \( \frac{3}{5} \times \frac{144}{85} \); that is to say, in multiplying by 524288, as 125 multiplied by 4096 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 fis 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 \( \frac{1}{144} \).

Moreover, if, after having found the fis equal to \( \frac{3}{5} \), we then tune by fifths and by fourths, sol, re, la, mi, fis, as we have done with respect to the first series of fifths, we find that the fis must be \( \frac{3}{5} \times \frac{144}{85} \); its difference, then, from unity, or, in other words, from UT, is \( \frac{1}{144} \), that is to say, about \( \frac{1}{85} \); a comma still less than any of the preceding, and which the Greeks have called apotome minor.

In a word, if, after having found mi equal to \( \frac{3}{4} \) in the progression of thirds, we then tune by fifths and fourths mi, fis, fis, ut, &c., we shall arrive at a new fis, which shall be \( \frac{3}{5} \times \frac{144}{85} \), and which will not differ from unity but by about \( \frac{1}{144} \), which is the last and smallest of all the commas; but it must be observed, that, in this case, the thirds major from mi to sol, from sol to fis or ut, &c., are extremely false, and greatly altered.

(τ) All the semitones being equal in the temperament proposed by M. Rameau, it follows, that the twelve semitones ut, ut, re, re, mi, mi, &c., shall form a continued geometrical progression; that is to say, a series in which ut shall be to ut in the same proportion as ut to re, as re to re, &c., and so of the rest. Part I.

Theory of Harmony.

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, UT; then tune the note SOL 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 mi to fa, which should of themselves be in tune; that is to say, they ought to be in such a state, that fa, the highest note of the two which compose the fifth, may be identical with the found UT, with which you began, or at least the octave of that found perfectly just: it will be necessary then to try if this UT, or its octave, forms a just fifth with the last found mi or fa 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 interposition (v).

These twelve semitones are formed by a series of thirteen founds, of which UT and its octave ut are the first and last. Thus to find by computation the value of each found 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 practiced 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.

\[ \begin{array}{ccccccc} UT & ut & re & re & mi & fa & fa \\ 1 & \sqrt{2} & \sqrt{2^2} & \sqrt{2^3} & \sqrt{2^4} & \sqrt{2^5} & \sqrt{2^6} \\ la & la & la & la & la & la & la \\ \sqrt{2^9} & \sqrt{2^{10}} & \sqrt{2^{11}} & \sqrt{2^{12}} & \sqrt{2^{13}} & \sqrt{2^{14}} & \sqrt{2^{15}} \end{array} \]

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 ut to fa, which should be \( \frac{3}{4} \), ought to be diminished by about \( \frac{1}{12} \) of \( \frac{3}{4} \); that is to say, by \( \frac{1}{8} \), a quantity almost inconceivably small.

It is true, that the thirds major will be a little more altered; for the third major from ut to mi, for instance, shall be increased in its interval by about \( \frac{1}{12} \): 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 ut, mi, sol, fa, fa, that this last fa is very different from ut (note s); from whence it follows, that if we would tune this fa in unison with the octave of ut, 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.

(v) All that remains, is to acknowledge, with M. Rameau, that this temperament is far remote from that which is now in practice: you may here see in what this last temperament consists as applied to the organ or harpsichord. They begin with UT in the middle of the keys, and they flatten the four first fifths sol, re, la, mi, till they form a true third major from mi to ut; afterwards, setting out from this mi, they tune the fifths fa, fa, ut, sol, but flattening them still less than the former, so that sol may almost form a true third major with mi. When they have arrived at sol, they stop; they resume the first ut, and tune to it the fifth fa in descending, then the fifth fa, &c. and they heighten a little all the fifths till they have arrived at la, which ought to be the same with the sol already tuned.

If, in the temperament commonly practiced, some thirds are found to be less altered than in that prescribed by M. Rameau, in return, the fifths in the first temperament are much more false, and many thirds are likewise so; insomuch, that upon a harpsichord tuned according to the temperament in common use, there are five or six modes which the ear cannot endure, and in which it is impossible to execute anything. On the contrary, in the temperament suggested by M. Rameau, all the modes are equally perfect; which is a new argument in its favour, since the temperament is peculiarly necessary in passing from one mode to another, without shocking the ear; for instance, from the mode of ut to that of sol, from the mode of sol to that of re, &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; By this method all the twelve sounds which compose one of the scales shall be tuned: nothing is necessary but to tune with the greatest possible exactness their octaves in the other scales, and the harpsichord shall be well tuned.

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

Simple and daily experiments confirm what we now advance. Listen to a voice which is accompanied, in singing, by different instruments; though the temperament of the voice, and the temperaments 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 the differences.

We may give another experiment. Strike upon an organ the three keys mi, sol, fa, you will hear nothing but the minor perfect chord; tho' mi, by the construction of that instrument, must cause sol like wise to be heard; though sol should have the same effect upon re, and fa upon sol; infomuch, that the ear is at once affected with all these sounds, re, mi, sol, fa, sol, sol, fa; 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 attract its attention from such sounds as might offend!

So that, according to them, the scale of ut,

ut, re, mi, fa, sol, la, si, UT,

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

mi, fa, sol, la, si, ut, re, mi,

which, in their judgment, renders the modes of ut and mi proper for different manners of expression. But after all that we have said in this treatise on the formation of diatonic intervals, every one should be convinced, that, according to the intention of nature, the diatonic scale ought to be perfectly the same in all its modes:

The contrary opinion, says M. Rameau, is a mere prejudice of musicians. The character of an air arises chiefly from the intermixture of the modes; from the greater or lesser degrees of vivacity in the movement; from the tones, more or less grave, or more or less acute, which are assigned to the generator of the mode; and from the chords more or less beautiful, as they are more or less deep, more or less flat, more or less sharp, which are found in it.

In short, the last advantage of this temperament is, that it will be found conformed, or at least very little different from that which they practice 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 we must not suffer our readers to be ignorant, that M. Rameau, in his New System of Music, printed in 1726, had 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 string 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, there is a necessity for altering the last thirds major, and to make them a little more sharp, that they may naturally return to the octave of the principal sound, he pretends that this alteration is tolerable, not only because it is almost insensible, but because it is found in modulations not much in use, unless the composer should choose it on purpose to render the expression stronger. "For it is proper to remark, (says he), that we receive different impressions from the intervals in proportion to their different alterations: for instance, the third major, which naturally elevates us to joy, in proportion as we feel it, heightens our feelings even to a kind of fury, when it is tuned too sharp; and the third minor, which naturally inspires us with tenderness and serenity, depresses us to melancholy when it is too flat." All this strain, as you may see, is immensely different from that which this celebrated musician afterwards exhibited in his Generation Harmonique, and in the performances which followed it. From this we can only conclude, that the reasons, which, after him, we have urged for the new temperament, must without doubt have appeared to him very strong, because in his mind they have superceded those which he had formerly adduced in favour of the ordinary temperament.

We do not pretend to give any decision for either the one or the other of these methods of temperament, each of which appears to us to have its particular advantages. We shall only remark, that the choice of the one or the other must be left absolutely to the taste and inclination of the reader; without, however, admitting this choice to have any influence upon the principles of the system of music, which we have followed even till this period, and which must always submit, whatever temperament we adopt. Part I.

CHAP. VIII. Of Reposes or Cadences (†).

73. In a fundamental baf whose procedure is by fifths, there always is, or always may be, a repose, or crisis, in which the mind acquiesces in its transition from one sound 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 ut to sol; 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 sound 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.)

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

ut, sol, ut, fa, ut, sol, re, sol, ut,

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

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

In the first case, this word only signifies an agreeable and rapid alternation 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 prefigures, a repose, either present or impending, in the fundamental baf (x).

76. Since there is a repose in passing from one sound to another in the fundamental baf, there is also a repose in passing from one note to another in the diatonic scale, which is formed from it, and which this baf represents: and as the absolute repose sol, ut, is of all others the most perfect in the fundamental baf, the repose from fa to ut, which answers to it in the scale, and which is likewise terminated by the generator, is for that reason the most perfect of all others in the diatonic scale ascending.

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

We have already proved, that the fundamental baf 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 baf.

CHAP. IX. Of the Minor Mode and its Diatonic Series.

78. In the second chapter, we have explained (art. 29, 30, 31, and 32.) by what means, and in what series of upon what principle the minor chord ut, mi, sol, mode altered, may be formed, which is the characteristic chord gained by the minor mode. Now what we have there said, different taking ut for the principal and fundamental sound, we examples, 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 ut, mi, sol, ut, there occurs a mi 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. re, fa, la, re, la, ut, mi, la, and mi, sol, fa, mi. Amongst these three we shall choose la, ut, mi, la; because this chord, without including any sharp or flat, has two sounds in common with the major chord ut, mi, sol, ut; and besides, one of these two sounds is the very same ut: so that this chord appears to have the most immediate, and at the same time the most simple, relation with the chord ut, mi, sol, ut. Concerning this we need only add, that this preference of the chord la, ut, mi, la, 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. Let us now remark, that in every mode, whether major or minor, the principal sound which implies the perfect chord, whether major or minor, may be called the tonic note or key; thus ut is the key in its proper mode, la in the mode of la, &c. Having laid down this principle,

81. We have shown how the three sounds fa, ut, sol, which constitutes (art. 38.) the mode of ut, of which the first fa and the last sol are the two fifths of ut, scale purfing one descending the other rising, produce the scale ed.

(†) 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 Repos in Rouffleau's Musical Dictionary.

(x) M. Rouffleau, 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. Theory of \( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \) of the major mode, by means of the fundamental bals \( \text{sol}, \text{ut}, \text{sol}, \text{ut}, \text{fa}, \text{ut} \).

See fig. D.

\( \text{fa} \): let us in the same manner take the three sounds \( \text{re}, \text{la}, \text{mi} \), which constitute the mode of \( \text{la} \); for the same reason that the sounds \( \text{fa}, \text{ut}, \text{sol} \), constitute the mode of \( \text{ut} \); and of them let us form this fundamental bals, perfectly like the preceding, \( \text{mi}, \text{la}, \text{mi}, \text{la}, \text{re}, \text{la} \); \( \text{re} \): let us afterwards place below each of these sounds one of their harmonics, as we have done (chap. v.) for the first scale of the major mode; with this difference, that we must suppose \( \text{re} \) and \( \text{la} \) as implying their thirds minor in the fundamental bals to characterize the minor mode; and we shall have the diatonic scale of that mode,

\( \text{sol} \), \( \text{la}, \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa} \).

82. The \( \text{sol} \), which corresponds with \( \text{mi} \) in the fundamental bals, forms a third major with that \( \text{mi} \), 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 \( \text{la} \).

83. It is true, that, in causing \( \text{mi} \) to imply its third major \( \text{sol} \), one might also rise to \( \text{la} \) by a diatonic progress. But that manner of rising to \( \text{la} \) would be less perfect than the preceding; for this reason (art. 76.), that the absolute repose or perfect cadence \( \text{mi}, \text{la} \), which is found in the fundamental bals, 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 \( \text{la} \), the key itself upon which the repose is made. From whence it follows, that the preceding note \( \text{sol} \) ought rather to be sharp than natural; because \( \text{sol} \), being included in \( \text{mi} \) (art. 19.), much more perfectly represents the note \( \text{mi} \) in the bals, than the natural note \( \text{sol} \) could do, which is not included in \( \text{mi} \).

84. We may remark this first difference between the scale

\( \text{sol}, \text{la}, \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa} \),

and the scale which corresponds with it in the major mode

\( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{la} \),

that from \( \text{mi} \) to \( \text{fa} \), which are the two last notes of the former scale, there is only a semitone; whereas from \( \text{sol} \) to \( \text{la} \), 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.

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,

\( \text{ut}, \text{re}, \text{mi}, \text{fa}, \text{sol}, \text{sol}, \text{la}, \text{fa}, \text{ut} \).

That last series, as we have seen, was formed by means of the fundamental bals \( \text{fa}, \text{ut}, \text{sol}, \text{re} \), disposed in this manner,

\( \text{ut}, \text{sol}, \text{ut}, \text{fa}, \text{ut}, \text{sol}, \text{re}, \text{sol}, \text{ut} \).

Let us take in the same manner the fundamental bals \( \text{re}, \text{la}, \text{mi}, \text{fa} \), and arrange it in the following order,

\( \text{la}, \text{mi}, \text{la}, \text{re}, \text{la}, \text{mi}, \text{fa}, \text{mi}, \text{la} \),

and it will produce the scale immediately subjoined,

\( \text{la}, \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{mi}, \text{fa}, \text{sol}, \text{la} \),

in which \( \text{ut} \) forms a third minor with \( \text{la} \), which in the fundamental bals corresponds with it, which denominates the minor mode; and on the contrary \( \text{sol} \) forms a third major with \( \text{mi} \) in the fundamental bals, because \( \text{sol} \) rises towards \( \text{la} \), (art. 82. and 83.)

86. We see besides a \( \text{fa} \), which does not occur in the former,

\( \text{sol}, \text{la}, \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa} \),

where \( \text{fa} \) is natural. It is because, in the first scale, \( \text{fa} \) is a third minor from \( \text{re} \) in the bals; and in the second, \( \text{fa} \) is the fifth from \( \text{fa} \) in the bals.

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

88. From thence it may be seen why \( \text{fa} \) and \( \text{sol} \) are Fa and Sol sharp when ascending in the minor mode; nay, be-tharp in the sides, the \( \text{fa} \) is only natural in the first scale \( \text{sol}, \text{la}, \text{minor} \), \( \text{fa}, \text{ut}, \text{re}, \text{mi}, \text{fa} \), because this \( \text{fa} \) cannot rise to \( \text{sol} \), why.

89. It is not the same case in descending. For \( \text{mi} \), the fifth of the generator, ought not to imply the third different in major \( \text{sol} \), but in the case when that \( \text{mi} \) descends to descending, the generator \( \text{la} \) to form a perfect repose, (art. 77. and why. 83.) and in this case the third major \( \text{sol} \) rises to the generator \( \text{la} \); but the fundamental bals \( \text{la} \text{mi} \text{may} \), in descending, give the scale \( \text{la} \text{sol} \text{natural} \), provided \( \text{sol} \) does not rise towards \( \text{la} \).

90. It is much more difficult to explain how the \( \text{fa} \), which ought to follow this \( \text{sol} \) in descending, is natural and not sharp; for the fundamental bals \( \text{la}, \text{mi}, \text{fa}, \text{mi}, \text{la}, \text{re}, \text{la}, \text{mi}, \text{la} \), produces in descending,

\( \text{la}, \text{sol}, \text{fa}, \text{mi}, \text{mi}, \text{mi}, \text{re}, \text{ut}, \text{fa}, \text{la} \).

And it is plain that the \( \text{fa} \) cannot be otherwise than difficult, sharp, since \( \text{fa} \) is the fifth of the note \( \text{fa} \) of the fundamental bals. In the mean time, experience evinces that the \( \text{fa} \) is natural in descending in the diatonic scale of the major mode of \( \text{la} \), especially when the preceding \( \text{sol} \) is natural; and it must be acknowledged, that here the fundamental bals appears in some measure defective.

M. Rameau has invented the following means for Rameau's obtaining a solution of this difficulty. According to solution, him, in the diatonic scale of the minor mode in descending, \( \text{la}, \text{sol}, \text{fa}, \text{mi}, \text{re}, \text{ut}, \text{fa}, \text{la} \), may be yet unfitted regarded simply as a note of passage, merely added to satisfy, give sweetness to the modulation, and as a diatonic gradation by which we may descend to \( \text{fa} \) natural. It is easily perceived, according to M. Rameau, by this fundamental bals,

\( \text{la}, \text{re}, \text{la}, \text{re}, \text{la}, \text{mi}, \text{la} \),

which produces

\( \text{la}, \text{fa}, \text{mi}, \text{re}, \text{ut}, \text{fa}, \text{la} \);

which may be regarded, as he says, as the real scale of the minor mode in descending; to which is added \( \text{sol} \) natural between \( \text{la} \) and \( \text{fa} \), to preserve the diatonic order.

This answer appears the only one which can be given to the difficulty above proposed; but I know not whether it will fully satisfy the reader; whether he will not see with regret, that the fundamental bals does not produce, to speak properly, the diatonic scale of the minor Part I.

Theory of minor mode in descent, when at the same time this harmony same bass 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 (v).

Chap. X. Of relative Modes.

91. Two modes which are of such a nature that we can pass from the one to the other, are called relative modes. Thus we have already seen, that the major mode of ut is relative to the major mode of fa and to that of sol. It may likewise appear from what goes before, how many intimate connections there are between the species (†) or minor mode of ut, and the species or minor mode of la. For, 1. The perfect chords, one major ut mi sol ut, the other minor la ut mi la, which characterize each of those two kinds of modulation * or harmony, have two sounds in common, ut and mi. 2. The diatonic scale of the minor mode of la in descent, absolutely contains the same sounds with the gamut or diatonic scale of the major mode of ut.

It is for this reason that the transition is so natural and easy from the major mode of ut to the minor mode of la, or from the minor mode of la to the major mode of ut, as experience proves.

92. In the minor mode of mi, the minor perfect chord mi sol fa mi, which characterizes it, has likewise two sounds, mi sol, in common with the perfect chord major ut mi sol ut, which characterizes the major mode of ut. But the minor mode of mi is not so closely related nor allied to the major mode of ut as to the minor mode of la; because the diatonic scale of the minor mode of mi in descent has not, like the series of the minor mode of la, all these sounds in common with the scale of ut. In reality, this scale is mi re ut fa la sol fa mi, where there occurs a fa sharp which is not in the scale of ut. We may add, that though the minor mode of mi is less relative to the major mode of ut than that of la; yet the artist does not hesitate some-

(v) For what remains when sol is said to be natural in descending the diatonic scale of the minor mode of la, this only signifies, that this sol is not necessarily sharp in descending as it is in rising; for this sol, besides, may be sharp in descending to the minor mode of la, as may be proved by numberless examples, of which all musical composers are full. It is true, that when the sound sol is found sharp in descending to the minor mode of la, still we are not sure that the mode is minor till the fa or ut natural is found; both of which impress a peculiar character on the minor mode, viz. ut natural, in rising and in descending, and the fa natural in descending.

(†) Species was the only word which occurred to the translator in English by which he could render the French word genre. It is, according to Rousseau, intended to express the different divisions and dispositions of the intervals which formed the two tetrachords in the ancient diatonic scale; and as the gamut of the moderns consists likewise of two tetrachords, though diversified from the former, as our author has shown at large, the genre, or species as the translator has been obliged to express it, must consist in the various dispositions and divisions of the different intervals between the notes or semitones which compose the modern scale.

(z) There are likewise other minor modes, into which we may pass in our egrets from the mode major of ut; as that of fa minor, in which the perfect minor chord fa la sol ut, includes the sound ut, and whose scale in ascent fa sol la sol fa la sol ut re mi fa, only includes the two sounds la sol, which do not occur in the scale of ut. We find an example of this transition from the mode major of ut to that of fa minor, in the opera of Pygmalion by M. Rameau, where the sarabande is in the minor mode of fa, and the rigadoon in the mode major of ut. This kind of transition, however, is not frequent.

The minor mode of re has only in its scale ascending re mi fa sol la fa ut re, one ut sharp which is not found in the scale of ut. For this reason a transition may likewise be made, without grating the ear, from the mode of ut major to the mode of re minor; but this passage is less immediate than the former, because the chords ut mi sol ut re fa la re, not having a single sound in common, one cannot (art. 37.) pass immediately from the one to the other.

Of this may be seen one instance (among many others) in the prologue des Amours des Dieux, at this passage, Onide est l'objet de la fête, which is in the minor mode of mi, though what immediately precedes it is in the major mode of ut.

We may see besides, that when we pass from one mode to another by the interval of a third, whether in descending or rising, as from ut to la, or from la to ut, from ut to mi, or from mi to ut, 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 ut. It is the minor mode of ut itself in which the perfect minor chord ut mi sol ut has two sounds, ut and sol, in common with the perfect major chord ut mi sol ut. Nor is there anything more common than a transition from the major mode of ut to the minor mode, or from the minor to the major (z).

Chap. XI. Of Dissonance.

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

95. This advantage may be obtained by uniting at the same time the sounds sol and fa in the same harmony, that is to say, by joining to the harmony sol fa generator re of the fifth sol, the other fifth fa in this manner, and its sol fifths, and by that means determine the mode un-

[ e ] Theory of Harmony

Mus. re, fa; this fa which is added, forms a dissonance with sol (art. 18.) It is for that reason that the chord sol, re, fa, is called a dissonant chord, or a chord of the seventh. It serves to distinguish the fifth sol from the generator ut, which always implies, without mixture or alteration, the perfect chord ut, mi, sol, ut, resulting from nature itself (art. 32.) By this we may see, that when we pass from ut to sol, one passes at the same time from ut to fa, because fa is found to be comprehended in the chord of sol; and the mode of ut by these means plainly appears to be determined, because there is none but that mode to which the sounds fa and sol at once belong.

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

97. For this reason, instead of sol, we shall take its fifth re, which is the sound that approaches it nearest; and we shall have, instead of the fifth fa, the chord fa, la, ut, re, which is called a chord of the great sixth.

One may here remark the analogy there is observed between the harmony of the fifth sol, and that of the fifth fa.

98. The fifth sol, in rising above the generator, gives a chord entirely consisting of thirds ascending from sol, sol, fa, re, fa; now the fifth fa being below the generator ut in descending, we shall find, as we go lower by thirds from ut towards fa, the same sounds ut, la, fa, re, which form the chord fa, la, ut, re, given to the fifth fa.

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

(αα) "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 fa, la, ut, re, as the primary chord and generator of the chord fa, la, ut, re, which is nothing but that chord itself 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 great artist has neither expressed himself upon this subject with too much uniformity nor with too much precision as is required. For my own part, I think there is some foundation for considering the chord fa, la, ut, re, as primitive; 1. Because in this chord, the fundamental and principal note is the sub-dominant fa, which ought in effect to be the fundamental and principal sound in the chord of the sub-dominant. 2. Because that without having recourse, with M. Rameau, to harmonical and arithmetical progressions, of which the consideration appears to us quite foreign to the question, we have found a probable and even a satisfactory reason for adding the note re to the harmony of the fifth fa, (art. 96 and 97.) The origin thus assigned for the chord of the sub-dominant appears to us the most natural, though M. Rameau does not appear to have felt its full value; for scarcely has it been slightly intimated by him."

Thus far our author. We do not enter with him into the controversy concerning the origin of the chord in question; but only propose to add to his definition of the sub-dominant, Rousseau's idea of the same note. It is a name, says he, given by M. Rameau to the fourth note in any modulation relative to a given key, which of consequence is in the same interval from the key in descending as the dominant in rising; from which circumstance it takes its name.

Chap. XII. Of the Double Use, or Employment of Harmony, Dissonance.

100. It is evident by the resemblance of sounds to Account of their octaves, that the chord fa, la, ut, re, is in effect the double the same as the chord re, fa, la, ut, taken inversely *, employment that the inverse of the chord ut, la, fa, re, has been *See Inversion found (art. 98.) in descending by thirds from the generator ut (αα).

101. The chord re, fa, la, ut, is a chord of the seventh like the chord sol, si, re, fa: with this only difference, that in this the third sol, si, is major; whereas dominant in the second, the third re, fa, is minor. If the fa, dominant, were sharp, the chord re, fa*, la, ut, would be a genuine chord of the dominant, like the chord sol, si, re, fa; and as the dominant sol may descend to ut in the fundamental bass, the dominant re implying or carrying with the third major fa* might in the same manner descend to sol.

102. Now I say, that if the fa* should be changed into fa natural, re, the fundamental tone of this chord re, fa, la, ut, might still descend to sol; for the change from fa* to fa natural, will have no other effect, than to preserve the impression of the mode of ut, instead of that of the mode of fa, which the fa* would have here introduced. For what remains, the note re will always preserve its character as the dominant, on account of the mode of ut, which forms a seventh. Thus in the chord of which we treat, re, fa, la, ut, re, may be considered as an imperfect dominant; I call it imperfect, because it carries with it the third minor fa*, instead of the third major fa*. It is for this reason that in the sequel I shall call it simply the dominant, to distinguish it from the dominant sol, which shall be named the tonic dominant.

103. Thus the sounds fa and sol, which cannot succeed each other (art. 36.) in a diatonic bass, when they only carry with them the perfect chords fa, la, ut, sol, si, re, may succeed one another if you join re to the harmony of the first, and fa to the harmony of the second; and if you invert the first chord, that is to say, if you give to the two chords this form, re, fa, la, ut, sol, si, re, fa.

104. Besides, the chord fa, la, ut, re, being allowed to succeed the perfect chord ut, mi, sol, ut, it follows recon- Theory of follows for the same reasons, that the chord ut, mi, sol, Harmony, ut, may be succeeded by re, fa, la, ut; which is not contradictory to what we have above said (art. 37.), that the founds ut and re cannot succeed one another in the fundamental bas; for in the passage quoted, we had supposed that both ut and re carried with them a perfect chord major; whereas, in the present case, re carries the third minor fa and likewise the found ut, by which the chord re fa la ut is connected with that which precedes it ut mi sol ut, and in which the found ut is found. Besides, this chord, re fa la ut, is properly nothing else but the chord fa la ut re inverted, and, if we may speak so, disguised.

105. This manner of presenting the chord of the sub-dominant 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 is the source of one of the finest varieties in harmony; and we shall see in the following chapter the advantages which result from it.

We may add, that as this double employment is a kind of licence, it ought not to be practised without some precaution. We have lately seen that the chord re fa la ut, considered as the inverse of fa la ut re, may succeed to ut mi sol ut; but this liberty is not reciprocal: and though the chord fa la ut re, may be followed by the chord ut mi sol ut, we have no right to conclude from thence that the chord re fa la ut, considered as the inverse of fa la ut re, may be followed by the chord ut mi sol ut. For this reason shall be given Chap. XVI.

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

106. We have shown (Chap. vi.) how the diatonic scale, or ordinary gammut, may be formed from the fundamental bas fa, ut, sol, re, by twice repeating the word sol in that series; so that this gammut is progressively and originally composed of two similar tetrachords, one in the mode of ut, the other in that of sol. Now it is possible, by means of this double employment, to preserve the impression of the mode of ut through the whole extent of the scale, without twice repeating the note sol, or even without supposing this repetition. For this effect we have nothing to do but form the following fundamental bas,

ut, sol, ut, fa, ut, re, sol, ut;

in which ut is understood to carry with it the perfect chord ut mi sol ut; sol, the chord sol fa re fa; fa, the chord fa la ut re; and re, the chord re fa la ut. It is plain from what has been said in the preceding chapter, that in this case ut may ascend to re in the fundamental bas, and re descend to sol; and that the impression of the mode of ut is preserved by the fa natural which forms the third minor re fa, instead of the third major which re ought naturally to imply.

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

ut, re, mi, fa, sol, la, si, UT,

which of consequence will be in the mode of ut alone; and if one should choose to have the second tetrachord in the mode of sol, it will be necessary to substitute fa instead of fa natural in the harmony of re (ss).

108. Thus the generator ut may be followed according to pleasure in ascending diatonically either by a tonic dominant (re fa la ut), or by a simple dominant (re fa la ut).

109. In the minor mode of la, the tonic dominant mi ought always to imply its third major mi sol *, when this dominant mi descends to the generator la (art. 83.); and the chord of this dominant shall be mi sol * si re, entirely similar to sol si re fa. With respect to the sub-dominant re, it will immediately imply the third minor fa, to denominate the minor mode; and we may add si above its chord re fa la, in this manner re fa la si, a chord similar to that of fa la ut re; and as we have deduced from the chord fa la ut re, that of re fa la ut, we may in the same manner deduce from the chord re fa la si, a new chord of the seventh si re fa la, which will exhibit the double employment of dissonances in the minor mode.

110. One may employ this chord si re fa la, to preserve the impression of the mode of la in the diatonic scale of the minor mode, and to prevent the necessity of twice repeating the found mi: but in this case, the fa must be rendered sharp, and change this chord to si re fa * la, the fifth of si is fa *, as we have seen above; this chord is then the inverse of re fa * la si, where the sub-dominant implies the third major; which ought not to surprise us. For in the minor mode of la, the second tetrachord mi fa * sol * la is exactly the same as it would be in the major mode of la; now, in the major mode of la, the sub-dominant re ought to imply the third major fa *.

111. From thence we may see that the minor mode is susceptible of a much greater number of varieties in the minor than the major: likewise the major mode is the product of nature alone; whereas the minor is, in some measure, than in the major.

*BB We need only add, that it is easy to see, that this fundamental bas ut sol, ut fa, ut re, sol ut, which formed the ascending scale ut, re, mi, fa, sol, la, si, UT, cannot by inverting it, and taking it inversely in this manner si, ut, sol, re, ut, fa, ut, sol, UT, form the diatonic scale UT, si, la, sol, fa, mi, re, ut, in deficient. In reality, from the chord sol, si, re, fa, we cannot pass to the chord re, fa, la, ut, nor from thence to ut, mi, sol, ut. It is for this reason that in order to have the fundamental bas of the scale UT, si, la, sol, fa, mi, re, ut, in deficient, we must either determine to invert the fundamental bas mentioned in art. 55. in this manner, ut, sol, re, sol, ut, fa, ut, sol, ut, in which the second sol and the second ut answer to the sol alone in the scale; or otherwise we must form the fundamental bas ut, sol, re, sol, ut, sol, ut, in which all the notes imply perfect chords major, except the second sol, which implies the chord of the seventh sol, si, re, fa, and which answers to the two notes of the scale sol, fa, both comprehended in the chord sol, si, re, fa.

Which ever of these two basse we shall choose, it is obvious that neither the one nor the other shall be wholly in the mode of ut, but in the mode of ut and in that of sol. From whence it follows, that the double employment which gives to the scale a fundamental bas all in the same mode when ascending, cannot do the same in descending; and that the fundamental bas of the scale in descending will be necessarily in two different modes. Theory of 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 sub-dominant, though in some measure intimated 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 *sol fa re fa*, composed of a third major followed by two thirds minor.

2. The chord *re fa la ut*, or *fa re fa la*, composed of a third major between two minors.

3. The chord *si re fa la*, composed of 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, *ut mi sol fa*, or *fa la ut mi*; the other is wholly composed of thirds minor *sol fa re fa*. These two chords, which at first appear as if they ought not to enter into harmony if we rigorously keep to the preceding rules, are nevertheless frequently practised with success in the fundamental bass. The reason is this:

115. According to what has been said above, if we would add a seventh to the chord *ut mi sol*, to make a dominant of *ut*, one can add nothing but *fa*; and in this case *ut mi sol fa* would be the chord of the tonic dominant in the mode of *fa*, as *sol fa re fa* is the chord of the tonic dominant in the mode of *ut*: but if you would preserve the impression of the mode of *ut* in the harmony, you then change this *fa* into *si* natural, and the chord *ut mi sol si* becomes *ut mi sol si*. It is the same case with the chord *fa la ut mi*, which is nothing else but the chord *fa la ut mi b*; in which one may substitute for *mi b*, *mi* natural, to preserve the impression of the mode of *ut*, or of that of *fa*.

Besides, in such chords as *ut mi sol si*, *fa la ut mi*, the sounds *si* and *mi*, though they form a dissonance with *ut* in the first case, and with *fa* in the second, are nevertheless supportable to the ear, because these sounds *si* and *mi* (art. 19.) are already contained and understood, the first in the note *mi* of the chord *ut*

(cc) On the contrary, a chord such as *ut mi b sol si*, in which *mi* would be flat, could not be admitted in harmony, because in this chord the *si* is not included and understood in *mi b*. It is the same case with several other chords, such as *si re fa la*, *fa re fa la*, &c. It is true, that in the last of these chords, *la* is included in *fa*, but it is not contained in *re*; and this *re* likewise forms with *fa* and with *la* a double dissonance, which, joined with the dissonance *si fa*, 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.

(dd) We have seen above (art. 109.) that the chord *si re fa la*, in the minor mode of *la*, may be regarded as the inverse of the chord *re fa la si*: it would likewise seem, that, in certain cases, this chord *si re fa la* may be considered as composed of the two chords *sol si re fa*, *fa la ut re*, of the dominant and of the sub-dominant of the major mode of *ut*; which chords may be joined together, after having excluded from them,

1. The dominant *sol*, represented by its third major *fa*, which is presumed to retain its place.

2. The note *ut* which is understood in *fa*; which will form this chord *si re fa la*. The chord *si re fa la*, considered in this point of view, may be understood as belonging to the major mode of *ut* upon certain occasions. Part I.

Theory of sub-dominant fa la ut re follows the chord ut mi sol ut, the note re, which forms a dissonance with ut, is not found in the preceding chord.

It is not so when the chord re fa la ut follows the chord ut mi sol ut; for ut, which forms a dissonance in the second chord, stands as a consonance in the preceding.

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

122. In every chord of the seventh, which is not the chord of the tonic dominant, that is to say, (art. 102.) which is not composed of a third major followed by two thirds minor, the dissonance which this chord forms ought to stand as a consonance in the chord which precedes it.

This is what we call a prepared dissonance.

123. From thence it follows, that in order to prepare a dissonance, it is absolutely necessary that the fundamental bass should ascend by the interval of a second, as

UT mi sol ut, RE fa la ut;

or descend by a third, as

UT mi sol ut, LA ut mi sol;

or descend by a fifth, as

UT mi sol ut, FA la ut mi;

in every other case the dissonance cannot be be prepared. This is what may be easily ascertained. If, for instance, the fundamental bass rises by a third, as ut mi sol ut, mi sol fa re, the dissonance re is not found in the chord ut mi sol ut. The same might be said of ut mi sol ut, sol fa re fa, and ut mi sol ut, fa re fa la, in which the fundamental bass rises by a fifth or descends by a second.

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

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

Chap. XVI. Of the Rule for resolving Dissonances.

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

126. Now in the chord of the tonic dominant it ought rather to descend than to rise; for this reason, let us take, for instance, the chord sol fa re fa followed by the chord ut mi sol ut; the part which formed the dissonance fa ought to descend to mi rather than rise to sol, though both the sounds mi and sol are found should raise in the subsequent chord ut mi sol ut; because it is more natural and more conformed to the connection, and which ought to be found in every part of the music, why, that sol should be found in the same part where sol has already been sounded, whilst the other part was sounding fa, as may be here seen (parts first and fourth)

First part,

Second,

Third,

Fourth,

Fundamental bass,

fa mi,

si ut,

re ut,

sol sol,

sol ut.

127. For the same reason, in the chord of the consequent simple dominant re fa la ut, followed by sol fa re fa, the dissonance ut ought rather to descend to si than former rise to re.

128. In short, for the same reason, we shall find, that in the chord of the sub-dominant fa la ut re, the dissonance re ought to rise to mi of the following chord ut mi sol ut, rather than descend to ut; whence may be deduced the following rules.

129. 1°. In every chord of the dominant, whether tonic or simple, the note which constitutes the seventh, is that to say the dissonance, ought diatonically to descend upon one of the notes which form a consonance in the subsequent chord.

2°. In every chord of the sub-dominant, the dissonance ought to rise diatonically upon the third of the subsequent chord.

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

From these rules it is a necessary result, that the chord of the seventh re fa la ut, though one should even consider it as the inverse of fa la ut re, cannot be succeeded by the chord ut mi sol ut, since there is not in this last chord of si any note upon which the dissonance ut of the chord re fa la ut 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 re fa la ut, to ut mi sol ut, it is necessary that re fa la ut, should in this case be understood as the inverse of fa la ut re. Now the chord re fa la ut, can only be conceived as the inverse of fa la ut re, when this chord re fa la ut precedes or immediately follows the ut mi sol ut; in every other case the chord re fa la ut is a primitive chord, formed from the perfect minor chord re fa la, to which the dissonance ut was added, to take from re the character of a tonic. Thus the chord re fa la ut, could not be followed by the chord ut mi sol ut, 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... Theory of cession of chords, ut mi sol ut, re fa la ut, ut mi Harmony sol ut, it would be much more easy and natural to substitute this other, which furnishes this natural process, ut mi sol ut, fa la ut re, ut mi sol ut. 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 base which moves by fifths, there is always, as we have formerly observed (Chap. vii.), a repose more or less perfect from one found to another; and of consequence there must likewise be a repose more or less perfect from one found to another in the diatonic scale, which results from that base. It may be demonstrated by a very simple experiment, that the cause of a repose in melody is solely in the fundamental base expressed or understood. Let any person sing these three notes ut re ut, performing on the re a shake, which is commonly called a cadence; the modulation will appear to him to be finished after the second ut, 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 base ut sol ut; but if, instead of that base, we should give it the following, ut sol la; in this case the modulation ut re ut 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 sol la, when the dominant sol diatonically ascends upon the note la, instead of descending by a fifth upon the generator ut, as it ought naturally to do, is called a broken cadence; because the perfect cadence sol ut, which the ear expected after the dominant sol, is, if we may speak so, broken and suspended by the transition from sol to la.

133. From thence it follows, that if the modulation ut re ut appeared finished when we supposed no base to it at all, it is because its natural fundamental base ut sol ut is supposed to be implied; because the ear desires something to follow this modulation, as soon as it is reduced to the necessity of hearing another base.

134. The interrupted cadence may, as it seems to me, be considered as having its origin in the double employment of dissonances; since this cadence, like the double employment, only consists in a diatonic procedure of the base ascending (chap. xii.) In effect, nothing hinders us to descend from the chord sol fi re fa to the chord ut mi sol la, by converting the tonic ut into a sub-dominant, that is to say, by passing all at once from the mode of ut to the mode of sol; now to descend from sol fi re fa to ut mi sol la is the same thing as to rise from the chord sol fi re fa to the chord la ut mi sol, in changing the chord of the subdominant ut mi sol la for the imperfect chord of the dominant, according to the laws of the double employment.

135. In this kind of cadence, the dissonance of the first chord is resolved by descending diatonically upon performing the fifth of the subsequent chord. For instance, in the broken cadence sol fi re fa, la ut mi sol, the dissonance fa is resolved by descending diatonically upon the fifth mi.

136. There is still another kind of cadence called Interrupted an interrupted cadence, where the dominant descends what, by a third to another dominant, instead of descending See Cadence, by a fifth upon the tonic, as in this process of the base, sol fi re fa, mi sol fi re; in the case of an interrupted cadence, the dissonance of the former chord is resolved by descending diatonically upon the octave of the fundamental note of the subsequent chord, as may be here seen, where fa is resolved upon the octave of mi.

137. This kind of interrupted cadence, as it seems Origin of to me, has likewise its origin in the double employment of dissonances. For let us suppose these two cadences, chords in succession, sol fi re fa, sol fi re mi, where the double note sol is successively a tonic dominant and sub-do-employment; that is to say, in which we pass from the mode of ut to the mode of re; 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 sol fi re fa, mi sol fi re.

CHAP. XVIII. Of the Chromatic Species.

138. The series or fundamental bases by fifths produces the diatonic species in common use (chap. vi.) tal bases now the third major being one of the harmonics of a formed by fundamental sound as well as the fifth, it follows thirds major, 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 ut, mi, sol, a chroma- the two first sounds carrying each along with it their interval, thirds major and fifths, it is evident that ut will give semitone, sol, and that mi will give sol; now the semitone which how found, is between this sol and this sol, is an interval much See fig. k, lefs than the semitone which is found in the diatonic scale between mi and fa, or between fi and ut. This may be ascertained by calculation (EE); it is for this reason that the semitone from mi to fa is called major, and the other minor (FF).

140. If

(EE) In reality, ut being supposed 1 as we have always supposed it, mi is \( \frac{4}{5} \), and sol \( \frac{5}{6} \); now sol being \( \frac{5}{6} \), sol then shall be to sol as \( \frac{5}{6} \) to \( \frac{4}{5} \); that is to say, as 25 times 2 to 3 times 16: the proportion then of sol \( \frac{5}{6} \) to sol is as 25 to 24, an interval much lefs than that of 16 to 15, which constitutes the semitone from ut to fi, or from fa to mi (note L).

(FF) It may be observed, that a minor joined to a major semitone, will form a minor tone; that is to say, if one rises, for instance, from mi to fa, by the interval of a semitone major, and afterwards from fa to fa\( \frac{5}{6} \) by the interval of a minor semitone, the interval from mi to fa\( \frac{5}{6} \) will be a minor tone. For let us suppose mi to be 1, fa will \( \frac{5}{6} \), and fa\( \frac{5}{6} \) will be \( \frac{5}{6} \) of \( \frac{5}{6} \); that is to say, 25 times 16 divided 24 times 15, or \( \frac{5}{6} \); mi then is to fa\( \frac{5}{6} \) as 1 is to \( \frac{5}{6} \), the interval which constitutes the minor tone (note N.)

With respect to the tone major, it cannot be exactly formed by two semitones; for, 1. Two major semitones in 140. If the fundamental bas should proceed by thirds minor in this manner, ut, mi, a succession which is allowed when we have investigated the origin of the minor mode (chap. ix.), we shall find this modulation sol, sol, which would likewise give a minor semitone (ee).

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 mi to fa; results from a fundamental bas by fifths ut fa, 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 sol to sol; they rise at first from sol to la, then descend from la to sol by the interval of a semitone major; for this sol sharp, which is a semitone major below la, proves a semitone minor above sol. [See the notes (ee) and (ff).]

142. Every procedure of the fundamental bas 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, ut, la, gives ut, ut, (HH) and the series of thirds major in descending, ut, la, gives ut, ut (ii).

143. The minor semitone constitutes the species called chromatic; and with the species which moves by semitone diatonic intervals, resulting from the succession of when prevalent, it comprehends the whole of chromatic melody.

CHAP. XIX. Of the Enharmonic Species.

144. The two extremes, or highest and lowest notes, Dleis or ut sol, of the fundamental bas by thirds major, ut mi sol, give this modulation ut fi; and these two founds ut, fi, differ between themselves by a small interval which is called the diekty, or enharmonic fourth formed af a tone (ll), which is the difference between a semitone major and a semitone minor (mm). This quarter tone is inappreciable by the ear, and impracticable upon several of our instruments. Yet have means been found to put it in practice in the following manner, or rather to perform what will have the same effect upon the ear.

145. We have explained (art. 116.) in what manner the chord sol fa re fa may be introduced into the minor mode, entirely consisting of thirds minor perfectly true, or at least supposed such. This chord supposing the place of the chord of the dominant (art. instruments 116.) of fixed scales.

In immediate succession would produce more than a tone major. In effect, \(\frac{1}{2}\) multiplied by \(\frac{1}{2}\) gives \(\frac{1}{4}\), which is greater than \(\frac{1}{2}\), the interval which constitutes (note n), 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 give still less.

(EE) In effect, mi being \(\frac{1}{2}\), sol will be \(\frac{1}{2}\) of \(\frac{1}{2}\); that is to say, (note c) \(\frac{1}{2}\): now the proportion of \(\frac{1}{2}\) to \(\frac{1}{2}\) (note c) is that of 3 times 25 to 2 times 36; that is to say, as 25 to 24.

(HH) La being \(\frac{1}{2}\), ut is \(\frac{1}{2}\) of \(\frac{1}{2}\); that is to say \(\frac{1}{2}\), and ut is 1: the proportion then between ut and ut is that of 1 to \(\frac{1}{2}\), or of 24 to 25.

(ll) La being the third major below ut, will be \(\frac{1}{2}\) (note c): utb, then, is \(\frac{1}{2}\) of \(\frac{1}{2}\); that is to say \(\frac{1}{2}\). The proportion, then, between ut and utb is as 25 to 24.

(lll) Sol is \(\frac{1}{2}\), and fi being \(\frac{1}{2}\) of \(\frac{1}{2}\), we shall have fi equal (note c) to \(\frac{1}{2}\), and its octave below shall be \(\frac{1}{2}\); an interval less than unity by about \(\frac{1}{2}\) or \(\frac{1}{2}\). It is plain then from this fraction, that the fi in question must be considerably lower than ut.

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 \(\frac{1}{2}\), and its difference from unity being \(\frac{1}{2}\), 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 \(\frac{1}{2}\), differs from unity by \(\frac{1}{2}\), the fourth of a minor tone will differ from unity about \(\frac{1}{2}\).

3. One half of a tone major; and as this semitone differs from unity by \(\frac{1}{2}\), one half of it will differ from unity about \(\frac{1}{2}\).

4. Finally, one half of a semitone minor, which differs from unity by \(\frac{1}{2}\): its half then will be \(\frac{1}{2}\).

The interval, then, which forms the enharmonic fourth of a tone, as it does not differ from unity but by \(\frac{1}{2}\), 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 ff) 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.

(MM) That is to say, that if you rise from mi to fa, for instance, by the interval of a semitone major, and afterwards, returning to mi, 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, fa+, the two sounds fa+ and fa will form the enharmonic fourth of a tone: for mi being 1, fa will be \(\frac{1}{2}\); and fa+ \(\frac{1}{2}\): the proportion then between fa+ and fa is that of \(\frac{1}{2}\) to \(\frac{1}{2}\) (note c); 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. Theory of 116.) from thence we may pass to that of the tonic Harmony, or generator la (art. 117.). But we must remark,

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

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

| fa | re | fa | sol | |----|----|----|----| | re | fa | sol | fa | | fa | sol | fa | re |

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

Now the chord sol fa re fa, belonging to the minor mode of la, where sol is the sensible note; the chord fa re fa sol, or fa re fa la, will, for the same reason, belong to the minor mode of ut, where fa is the sensible note. In like manner, the chord re fa sol fa, or fa re fa la ut, will belong to the minor mode of mi, and the chord fa sol fa re, or fa la ut mi, to the minor mode of sol.

After having passed then by the mode of la to the chord sol fa re fa (art. 117.), one may by means of this last chord, and by merely satisfying ourselves to invert it, afterwards pass all at once to the modes of ut minor, of mi minor, or of sol minor; that is to say, into the modes which have nothing, or almost nothing, in common with the minor mode of la, and which are entirely foreign to it (†).

146. It must, however, be acknowledged, that a transition so abrupt, and so little expected, cannot deceive nor elude the ear; it is struck with a sensation so unlooked-for without being able to account for the abrupt and passage to itself. And this account has its foundation in the enharmonic fourth of a tone; which is overlooked as nothing, because it is inappreciable by the ear; but of which, tho' its value is not ascertained, the whole harshness is sensibly perceived. The instant of surprise, 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 passed by the ordinary series of fundamental notes.

Chap. XX. Of the Diatonic Enharmonic Species.

147. If we form a fundamental bass, which rises alternately by fifths and thirds, as fa, ut, mi, fa, this bass will give the following modulation, fa, mi, mi, re; in which the semitones from fa to mi, and from mi to re, are equal and major (nn).

This species of modulation or of harmony, in which all the semitones are major, is called the enharmonic diatonical 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 LL).

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 ut, ut, la, ni inter ut, ut, in which all the semitones are minor (oo), formed.

This species is called the chromatic enharmonical 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 LL).

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

150. The diatonic species is the most agreeable, because species most agreeable, and why.

(†) As this method for obtaining or supplying enharmonic gradations cannot be practised on every occasion when the composer or practitioner would wish to find them, especially upon instruments where the scale is fixed and invariable, except by a total alteration of their economy, and re-tuning the strings, Dr Smith in his Harmonics has proposed an expedient for redressing or qualifying this defect, by the addition of a greater number of keys or strings, which may divide the tone or semitone into as many appreciable or sensible intervals as may be necessary. For this, as well as for the other advantageous improvements which he proposes in the structure of instruments, we cannot with too much warmth recommend the perusal of his learned and ingenious book to such of our readers as aspire to the character of genuine adepts in the theory of music.

(nn) It is obvious, that if fa in the bass be supposed 1, fa of the scale will be 2, ut of the bass 3, and mi of the scale 4 of 5, that is, 4 of 5; the proportion of fa to mi is as 2 to 4 of 5, or as 1 to 2 of 5. Now mi of the bass being likewise 4 of 5, or 4 of 5, fa of the bass is 4 of 5 of 4 of 5, and its third major re of 4 of 5 of 4 of 5, or 4 of 5; this third major, approximated as much as possible to mi in the scale by means of octaves, will be 4 of 5 of 4 of 5: mi then of the scale will be to re which follows it, as 4 of 5 is to 4 of 5, that is to say, as 1 to 4 of 5. The semitones then from fa to mi, and from mi to re, are both major.

(oo) It is evident that mi is 4 of 5 (note c), and that mi is 4 of 5: these two mi's, then, are between themselves as 4 of 5 to 4 of 5, 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 la of the bass is 4 of 5, and ut 4 of 5, or 4 of 5: mi then is 4 of 5 of 4 of 5, the mi in the scale is likewise to the mi which follows it, as 24 to 25. All the semitones therefore in this scale are minor. Part I.

Theory of Harmony.

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

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

CHAP. XXII. Showing that Melody is the Offspring of Harmony.

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

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

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

156. Can it be still necessary to adduce more convincing proofs? We have nothing to do but examine the different basses which may be given to this very simple modulation sol ut; of which it will be found susceptible of a great many, and each of these basses will give a different character to the modulation sol ut, though in itself it remains always the same; in such a manner that we may change the whole nature and effects of a modulation, without any other alteration except that of changing its fundamental bass.

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

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

2. That the character of a just harmony is only to form in some measure one system with the modulation, so that from the whole taken together the ear may only receive, if we may speak so, one simple and indivisible impression.

3. That the character of the same modulation may be diversified, according to the character of the bass which is joined with it.

But notwithstanding the dependency of melody upon harmony, and the sensible influence which the latter may exert upon the former; we must not however from thence conclude, with some celebrated musicians, that the effects of harmony are preferable to those of melody. Experience proves the contrary. [See, on this account, what is written on the licence of music, printed in tom. iv. of D'Alembert's Mélanges de Littérature, p. 448.]

GENERAL REMARK.

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

Part II. PRINCIPLES and RULES of COMPOSITION.

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

158. Composition, which is likewise called counterpoint, is not only the art of composing an agreeable air, but also that of composing a great many 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.

There are likewise several eminent musicians, who in their compositions, if we can depend on what has been affirmed, begin with determining and writing the bass. See l'Encyclopédie, tom. 7, p. 61. This method, however, appears in general more proper to produce a learned and harmonious music, than a strain prompted by genius and animated by enthusiasm. Principles sometimes called a second redundant; such is the Compo interval from ut to re in ascending, or that of la to sol descending.

This interval is so termed, because one of the sounds which form it is always either sharp or flat, and that, if you deduce that sharp or that flat, the interval will be that of a second.

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

As the interval from ut to re in ascending, has been called a second redundant, they likewise call the interval from ut to sol in ascending a fifth redundant, or from si to mi in descending, each of which intervals are composed of four tones.

This interval is, in the main, the same with that of the sixth minor (art. 9.); but in the fifth redundant there is always a sharp or a flat; insomuch, that if this sharp or flat were deduced, the interval would become a true fifth.

For the same reason, an interval composed of three tones and three semitones, as from sol to fa in ascending, is called a seventh diminished; because, if you deduced the sharp from sol, the interval from sol to fa will become that of an ordinary seventh. The interval of a seventh diminished is in other respects the same with that of the sixth major (art. 9.)

The major seventh is likewise sometimes called a seventh redundant (QQ.)

CHAP. II. Comparison of the Different Intervals.

If we sing ut si in descending by a second, and afterwards ut si in ascending by a seventh, these scales representing two fifths shall be octaves one to the other; or, as we commonly express it, they will be replications one of the other.

On account then of the resemblance between every sound and its octave (art. 22.), it follows, that to rise by a seventh, or descend by a second, amount to one and the same thing.

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

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

To rise by a second. To descend by a seventh.

To descend by a third. To rise by a sixth.

To descend by a fourth. To rise by a fifth.

Thus, therefore, we shall employ them indifferently the one for the other; so that when we say, for instance, to rise by a third, it may be said with equal propriety to descend by a sixth, &c.

There are three clefs * in music; the clef of fa, or sol; the clef of ut; and the clef of sol.

But, in Britain, the following characters are used: The F, or bass-clef; the C, or tenor clef; and the G, or treble clef.

The clef of fa is placed on the fourth line, or on placed the third; and the line upon which this clef is placed gives the name of fa, or F, to all the notes which are upon that line.

The clef of ut is placed upon the fourth, the third, the second, or the first line; and in these different positions all the notes upon that line where the clef is placed take the name of ut, or C.

Lastly, the clef of sol is placed upon the second or first line; and all the notes upon that line where the clef is placed take the name of sol, or G.

As the notes are placed on the lines, and in notes to be the spaces between the lines, any one, when he sees investigated the clef, may easily find the name of any note whatever.

Thus he may see, that, in the first clef of fa, the clef, the note which is placed on the lowest line ought to be sol; that the note which occupies the space between the two first lines should be la; and that the note which is on the second line is a si, &c. (RR).

The chief use of these different denominations is 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.

It is on account of the different compasses of voices and instruments that these clefs have been invented.

The masculine voice, which is the lowest, may at its greatest depth, without straining, descend to sol, which is in the last line of the first clef of fa; and the female voice, which is the sharpest, may at its highest pitch rise to a sol, which is a triple octave above the former.

The lowest of masculine voices is adapted to a part which may be called a mean bass, and its clef is that of fa on the fourth line; this clef is likewise that of the violoncello and of the deepest instruments. A mean bass extremely deep is called a baritone or counter-bass.

The masculine voice, which is next in depth to what we have called the mean bass, may be termed the concordant bass. Its clef is that of fa on the third line.

The masculine voice which follows the concordant bass may be denominated a tenor; a voice of this pitch is the most common, yet seldom extremely agreeable. Its clef is that of ut on the fourth line. This clef is also that of the bassoon or bass hautboy.

The highest masculine voice of all may be called counter tenor. Its clef is that of ut on the third line. It is likewise the clef of tenor violins, &c. A note before which there is a sharp (marked thus ♯) ought to be raised by a semitone; and if, on the contrary, there is a ♭ before it, it ought to be depressed by a semitone, (♭ being the mark of a flat).

The natural (marked thus ♮) restores to its natural value a note which had been raised or depressed by a semitone.

When you place at the clef a sharp or a flat, all the notes upon the line on which this sharp or flat is marked are sharp or flat. Thus let us take, for instance, the clef of ut upon the first line, and let us place a sharp in the space between the second and third line, which is the place of fa; all the notes which shall be marked in that space will be fa♯; and if you would restore them to their original value of fa natural, you must place a ♭ or a ♭ before them.

In the same manner, if a flat be marked at the clef, and if you would restore the note to its natural state, you must place a ♭ or a ♭ before it.

Every piece of music is divided into different equal times, which they call measures or bars; and each bar is likewise divided into different times.

There are properly two kinds of measures or modes of time (See T): the measure of two times, or of common time, which is marked by the figure 2 placed at the beginning of the tune; and the measure of three times, or of triple time, which is marked by the figure 3 placed in the same manner. (See V).

The different bars are distinguished by perpendicular lines.

In a bar we distinguish between the perfect and imperfect time; the perfect time is that which we beat, the imperfect that in which we lift up the hand or feet. A bar consisting of four times ought to be regarded as compounded of two bars, each consisting of two times; thus there are in this bar two perfect and two imperfect times. In general, by the words perfect and imperfect, even the parts of the same time are distinguished: thus the first note of each time is reckoned as belonging to the perfect part, and the others as belonging to the imperfect.

The longest of all notes is a semibreve. The value minim is half its value; that is to say, in singing, we of notes in only employ the same duration in performing two minims which was occupied in one semibreve. A minim in the same manner is equivalent to two crotchets, the crotchet to two quavers, &c.

A note which is divided into two parts by a Syncopation-time, that is to say, which begins at the end of a time, and terminates in the time following, is called (ss) a syncopated note. (See Z; where the notes ut, fa, and la, are each of them syncopated.) (+).

The deepest female voice immediately follows the counter tenor, and may be called bass in alt. Its clef is that of ut upon the first line. The clef of ut upon the second line is not in frequent use.

The sharpest female voice is called treble; its clef is that of sol on the second line.

This last clef, as well as that of sol on the first line, is likewise the clef of the sharpest instruments, such as the violin, the flute, the trumpet, the hautboy, the flagelet, &c.

The ut which may be seen in the clefs of fa and in the clefs of ut is a fifth above the fa which is on the line of the clef of fa; and the sol which is on the two clefs of sol is a fifth above ut: insomuch that sol which is on the lowest line of the first clef of fa, is lower by two whole octaves than the sol which is on the lowest line of the second clef of sol.

[Thus far the translator has followed his original as accurately as possible; but this was by no means an easy task. Among all the writers on music which he has found in English, there is no such thing as different names for each particular part which is employed to constitute full or complete harmony. He was therefore under a necessity of substituting by analogy such names as appeared most expressive of his author's meaning. To facilitate this attempt, he examined in Rousseau's musical dictionary the terms by which the different parts were denominated in D'Alembert; but even Rousseau, with all his depth of thought and extent of erudition, instead of expressing himself with that precision which the subject required, frequently applies the same names indiscriminately to different parts, without assigning any reason for this promiscuous and licentious use of words. The English reader therefore will be best able to form an accurate idea of the different parts, by the nature and situation of the clefs with which they are marked; and if he should find any impropriety in the names which are given them, he may adopt and associate others more agreeable to his ideas.]

(ss) Syncopation consists in a note which is protracted in two different times belonging partly to the one and partly to the other, or in two different bars; yet not so as entirely to occupy or fill up the two times, or the two bars. A note, for instance, which begins in the imperfect time of a bar, and which ends in the perfect time of the following, or which in the same bar begins in the imperfect part of one time and ends in the perfect of the following, is syncopated. A note which of itself occupies one or two bars, whether the measure consists of two or three times, is not considered as syncopated; this is a consequence of the preceding definition. This note is said to be continued or protracted. In the end of the example Z, the ut of the first bar consisting of three times is not syncopated, because it occupies two whole times. It is the same case with mi of the second bar, and with the ut of the fourth and fifth. These therefore are continued or protracted notes.

(+) Times and bars in music answer the very same end as punctuation in language. They determine the different periods of the movement, or the various degree of completion, which the sentiment, expressed by that movement, has attained. Let us suppose, for instance, a composer in music intending to express grief or joy, in all its various gradations, from its first and faintest sensation, to its acme or highest possible degree. We do not say that such a progress of any passion either has been or can be delineated in practice, yet it may serve to illustrate what we mean to explain. Upon this hypothesis, therefore, the degrees of the sentiment will pass from less to more sensible, as it rises to its most intense degree. The first of these gradations may be called a A note followed by a point or dot is increased half its value. The \( f \), for instance, in the fifth bar of the example Y, followed by a point, has the value (*) or duration of a minim and of a crotchet at the same time.

**CHAP. IV. Containing a Definition of the principal Chords.**

178. The chord composed of a third, a fifth, and an octave, as ut mi sol ut, is called a perfect chord (art. 32).

If the third be major, as in ut mi sol ut, the perfect chord is denominated major; if the third be minor, as in la ut mi la, the perfect chord is minor. The perfect chord major constitutes what we call the major mode; and the perfect chord minor, what we term the minor mode (art. 31).

179. A chord composed of a third, a fifth, and a seventh, as sol fa re fa, or re fa la ut, &c., is called a chord of the seventh. It is obvious that 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 ut mi sol fa; and that which might carry a false fifth and a seventh major, as fa re fa la. (chap. xiv. Part I.).

180. As thirds are either major or minor, and as they may be differently arranged, it is clear that there are different kinds of chords of the seventh; there is even one, fa re fa la, 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 fa la ut re, re fa la fa, is called a chord of the greater sixth, which is likewise the most convenient division of a bar or measure into its elementary or aliquot parts, and may be deemed equivalent to a comma in a sentence; a bar denotes a degree still more sensible, and may be considered as having the force of a semicolon; a strain brings the sentiment to a tolerable degree of perfection, and may be reckoned equal to a colon: the full period is the end of the imitative piece. It must have been remarked by observers of measure in melody or harmony, that the notes of which a bar or measure consists, are not diversified by their different durations alone, but likewise by greater or lesser degrees of emphasis. The most emphatic parts of a bar are called the accented parts; those which require less energy in expression are called the unaccented. The same observation holds with regard to times as bars. The stroke, therefore, of the hand or foot in beating marks the accented part of the bar, the elevation or preparation for the stroke marks the unaccented part. Let us once more resume our composition intended to express the different periods in the progress of grief or joy. There are some revolutions in each of these so rapid as not to be marked by any sensible transition whether diatonic or consonant. In this case, the most expressive tone may be continued from one part of a time or measure to another, and end before the period of that time or measure in which it begins. Here therefore is a natural principle upon which the practice of syncopation may be founded even in melody: but when music is composed in different parts to be simultaneously heard, the continuance of one note not divided by regular times and measures, nor beginning and ending with either of them, whilst the other parts either ascend or descend according to the regular divisions of the movement, has not only a sensible effect in rendering the imitation more perfect, but even gives the happiest opportunities of diversifying the harmony, which of itself is a most delightful perception.

For the various dispositions of accent in times and measures, according to the movement of any piece, see a Treatise on Music by Alexander Malcolm.

For the opportunities of diversifying harmony afforded by syncopation, see Rameau's Principles of Composition.

(*) To prevent ambiguity or confusion of ideas, it is necessary to inform our readers, that we have followed M. D'Alembert in his double sense of the word value, though we could have wished he had distinguished the different meanings by different words. A sound may be either estimated by its different degrees of intensity, or by its different quantities of duration.

To dignify both those ideas the word value is employed by D'Alembert. The reader, therefore, will find it of importance to distinguish the value of a note in height from its value in duration. This he may easily do, by considering whether the notes are treated as parts of the diatonic scale, or as continued for a greater or lesser duration. Part II.

Principles chords of the seventh sol fa re fa, re fa la ut, fa and Compo ut are the dissonances, viz. fa with relation to sol in the first chord, and ut with relation to re in the second. In the chord of the great fifth fa la ut re, re is the dissonance (art. 120); but that re is only, properly speaking, a dissonance with relation to ut from which it is a second, and not with respect to fa from which it is a sixth major (art. 17, and 18).

187. When a chord of the seventh is composed of 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 sol fa re fa, the fundamental sol is the tonic dominant; but in the other chords of the seventh, as ut mi fa si, re fa la ut, &c. the fundamentals ut and re are simple dominants.

188. In every chord, whether perfect, or of the seventh, or of the sixth, if you have a mind that the third above the fundamental note should be major, though it is naturally minor, you must place a sharp above the fundamental note. For example, if I would mark the perfect major chord re fa la re, as the third fa above re is naturally minor, I place above re a sharp, as you may see in example IV. In the same manner the chord of the seventh re fa la ut, and the chord of the great sixth re fa la fa, is marked with a ♯ above re, and above the ♯ a 7 or a 6, (see V. and VI.)

On the contrary, when the third is naturally major, and if you should incline to render it minor, you must place above the fundamental note a b. Thus the examples VII. VIII. IX. shew the chords sol fa re sol, sol fa re fa, sol fa re mi, (tt).

CHAP. V. Of the Fundamental Bafs.

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

Thus (Exam. XVIII.) you will find that this modulation, ut re mi fa sol la si ut, has or may admit for its fundamental bafs, ut sol fa ut re fa ut.

In reality, the first note ut in the upper-part is found in the chord of the first note ut in the bafs, which chord is ut mi sol ut; the second note re in the treble is found in the chord sol fa re fa, which is the chord of the second note in the bafs, &c. and the bafs is composed only of notes which carry a perfect chord, or that of the seventh, or that of the great fifth. Moreover, it is formed according to the rules which we are now about to give.

CHAP. VI. Rules for the Fundamental Bafs.

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

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

Rule I.

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

Rule II.

192. In every chord of the simple dominant, it is necessary that the note which constitutes the seventh, or

(tt) 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 the clef of fa (see Exam. X.), one may satisfy himself with simply writing re, without a sharp to mark the perfect chord major of re, re fa la re. In the same manner, in the Example XII. where the flat is at the cliff upon fa, one may satisfy himself with simply writing sol, to mark the perfect chord minor of sol fa re sol.

But if a case occurs where there is a sharp or a flat at the cliff, if any one should wish to render the chord minor which is major, or vice versa, he must place above the fundamental note a b, or natural. Thus the Example XII. marks the minor chord re fa la re, and Example XIII. the major chord sol fa re sol.—Frequently, in lieu of a natural, a flat is used to signify the minor chord, and a sharp to signify the major. Thus Example XIV. marks the minor chord re fa la re, and Example XV. the major chord sol fa re sol.

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

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, they write by the side of the seventh a b 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 b; but M. Rameau suppresses these characters. The reason shall be given below, when we speak of chords by supposition.

If there be a sharp on the cliff of fa, and if I should incline to mark the chord sol fa re fa, or the chord la ut mi fa, I would place before the seventh or the sixth a b or a ♯.

In the same manner, if there be a flat on the cliff at fa, and if I should incline to mark the chord ut mi sol fa, I would place before the seventh a ♯ or a b, and so of the rest. 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 fa la ut re, it is necessary that fa, la, or ut, which are the consonances of the chord, should be found in the chord preceding. The dissonance re may either be found in it or not.

RULE IV.

194. Every simple or tonic dominant ought to descend by a fifth. In the first case, that is to say, when the dominant is simple, the note which follows can only be a dominant; in the second it may be any one you choose; 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 you will presently see. For let us assume the succession of the two chords la ut mi fol re fa la ut, (see Exam. XIX.) this succession is by no means legitimate, though in it the first dominant descends by a fifth; because the ut 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, one should take away the sharp carried by the ut in the first; or if, without meddling with the first chord, one should render ut or fa sharp in the second (uv); or in short, if one should simply render the re of the second chord a tonic dominant, in causing it to carry fa instead of fa natural (119. & 122.).

It is likewise by the same rule that we ought to reject the succession of the two following chords, re fa la ut, fol si re fa; (see Exam. XX.).

RULE V.

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

REMARK.

Of the five fundamental rules which have now been given, instead of the three first, one may substitute the three following, which are nothing but consequences from them, and which you may pass unnoticed if you think it proper.

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 XXI. and XXII. (xx) a tonic dominant, (124.) see XXIII. and XXIV.; or a sub-dominant, (124.) see XXV. and XXVI.; or, to express the rule more simply, that second note may be any one you please, 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. XXVII. and XXVIII.; or a tonic dominant, or a simple dominant, yet in such a manner that the rule of art. 192. may be observed, (124.) see XXIX. XXX. XXXI. XXXII.; or a sub-dominant (124.), see XXXIII. and XXXIV.

The procedure of the bass ut mi fol ut, fa la ut mi, from the tonic ut to the dominant fa (Ex. XXXV.), 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 XXXVI. and XXXVII. (yy).

We must here advertise our readers, that the examples XXXVIII. XXXIX. XL. XLI. belong to the fourth rule above, art. 194.; and the examples XLII. XLIII. XLIV. 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); how ever the cadences are formed at the distance of four bars employed, one from another, whilst the tonic then falls within the first time of the bar. See XLV. and XLVI.

REMARK II.

197. We must avoid as much as we can, syncopations in the fundamental bass; that, within the first time only of which a bar is constituted, the ear may be entertained with a harmony different from that which it had before perceived in the last time of the preceding bar, cence.

(uv) In this chord it is necessary that the ut and fa should be sharp at the same time; for the chord re fa la ut, in which ut would be sharp without the fa, is excluded by art. 179.

(xx) 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 I ascend from the tonic ut to the tonic mi, the major mode of ut, ut mi fol ut, will be changed into the minor mode of mi, mi fol si mi. For what remains, we must never ascend from one tonic to another, when there is no sound common to both their modes; for example, you cannot rise to the mode of ut, ut mi fol ut, from the minor mode of mi, mi fol si mi (91.).

(yy) By this we may see, that all the intervals, viz. the third, the fifth, and second, may be admitted in the fundamental bass, except that of a second in descending. For what remains, it is very proper to remark, that the rules immediately given for the fundamental bass are not without exception, as approved compositions in music will certainly discover; but these exceptions being in reality licences, and for the most part in opposition to the great principle of connection, which prescribes that there should be at least one note in common between a preceding and a subsequent chord, it does not seem necessary to entertain initiates with 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. Nevertheless, syncopation may be sometimes admitted in the fundamental bass, but it is by a licence (zz).

CHAP. VII. Of the Rules which ought to be observed in the Treble with relation to the Fundamental Bass.

198. The treble is nothing else but a modulation above the fundamental bass, and whose notes are found in the chords of that bass which corresponds with it, (189.) Thus in Ex. XVIII. the scale ut re mi fa sol la si ut, is a treble with respect to the fundamental bass ut sol fa ut re sol ut.

199. We are just 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 ut mi sol mi ut, may have for its fundamental bass the note ut alone, because the chord of that note comprehends the sounds ut, mi, sol, 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, sol alone may answer to these three notes in the bass, ut sol ut (AAA).

(zz) There are notes which may be found several times in the fundamental bass in succession with a different harmony. For instance, the tonic ut, after having carried the chord ut mi sol ut, may be followed by another ut which carries the chord of the seventh, provided that this chord be the chord of the tonic dominant ut mi sol si. See LXXII. In the same manner, the tonic ut may be followed by the same tonic ut, which may be rendered a sub-dominant, by causing it to carry the chord ut mi sol la. See LXXIII.

A dominant, whether tonic or simple, sometimes descends or rises upon one another by the interval of a tritone or false fifth. For example, the dominant fa, carrying the chord fa la ut mi, may be followed by another dominant fa, carrying the chord si re fa la. 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 ut to which fa and si belong. If one should descend from fa to si, by the interval of a just fifth, he would then depart from that scale, because si is no part of it.

(AAA) 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 Exam. XLVII. in which this modulation ut re mi fa sol, may be regarded as equivalent to this other, ut mi sol, as re and fa are no more than notes of passage. So that the bass of this modulation may be simply ut sol.

When the notes are of equal duration, and arranged in a diatonic order, the notes which occupy the perfect part of each time, or those which are accented, ought each of them to carry chords. Those which occupy the imperfect part, or which are unaccented, are no more than mere notes of passage. Sometimes, however, the note which occupies the imperfect part may be made to carry harmony; but the value of this note is then commonly increased by a point which is placed after it, which proportionally diminishes the continuance of the note that occupies the perfect time, 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.

(BBB) 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, as long as this dominant may be retained. For instance, let us imagine this modulation,

\[ \begin{array}{cccc} \text{ut} & \text{re} & \text{ut} & \text{si} \\ \text{ut} & \text{re} & \text{sol} & \text{ut} \\ \end{array} \]

and this fundamental bass,

\[ \begin{array}{cccc} \text{ut} & \text{re} & \text{ut} & \text{re} \\ \text{ut} & \text{re} & \text{sol} & \text{ut} \\ \end{array} \]

(see Example L.I.); the re of the fundamental bass answers to the two notes re ut of the treble. The dissonance ut has no need of preparation, because the note re of the fundamental bass having already been employed for the re which precedes ut, the dissonance ut is afterwards presented, below which the chord re may be preserved, or re fa la ut. 203. One may likewise observe here, that, according to the rules for the fundamental bass which we have given, the note upon which the dissonance ought to descend or rise will always be found in the subsequent chord (ccc).

**CHAP. VIII. Of the Continued Bass, and its Rules.**

204. A continued* or thorough bass, is nothing else but 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 *sol fa re fa*, I should say *fa re fa sol*, or *re fa sol fa*, &c., the chord is inverted. Let us see then, in the first place, all the possible ways in which a chord may be inverted.

The ways in which a Perfect Chord may be Inverted.

205. The perfect chord *ut mi sol ut* may be inverted in two different ways.

1. *Mi sol ut mi*, which we call a chord of the sixth, composed of a third, a sixth, and an octave and in this case the note *mi* is marked with a 6. (See LVI.)

2. *Sol ut mi sol*, 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 4. (See LVII.)

The perfect minor chord is inverted in the same manner.

The ways in which the Chord of the Seventh may be Inverted.

206. In the chord of the tonic dominant, as *sol fa re fa*, the third major *si* above the fundamental note *sol* is called a sensible note (77.) and the inverted chord *fa re fa sol*, composed of a third, a false fifth, and sixth, is called the chord of the false fifth, and is marked with an 8 or a 5 (see LVIII. and LIX.)

The chord *re fa sol fa*, composed of a third, a fourth, and a sixth, is called the chord of the sensible fifth, and marked with a 6 or a 6. In this chord thus figured, of composition the third is minor, and the sixth major, as it is easy to perceive. (See LX.)

The chord *fa sol fa re*, composed of a second, a tritone, and a sixth, is called the chord of the tritone, and is marked thus 4, thus 4, or thus 4. (See LXI.)

207. In the chord of the simple dominant *re fa la ut*, we find,

1. *Fa la ut re*, a chord of the great fifth, which is composed of a third, a fifth, and a sixth, and which is figured with a 5. See LXIII. (ddd).

2. *La ut re fa*, a chord of the lesser fifth, which is figured with a 6. See LXIV. (eee).

3. *Ut re fa la*, a chord of the second, composed of a second, a fourth, and a sixth, and which is marked with a 2. See LXII. (fff).

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

208. The chord of the sub-dominant, as *fa la ut re*, may be inverted in three different manners; but the method of inverting it which is most in practice is the chord of the lesser fifth *la ut re fa*, which is marked with a 6, and the chord of the seventh *re fa la ut*. See LXIV.

Rules for the Continued Bass.

209. The continued bass is a fundamental bass, whose chords are only inverted in order to render it more in the taste of singing, and suitable to the voice. See LXV. in which the fundamental bass which in itself is monotonic and little suited for singing, *ut sol ut sol ut sol ut*, produces, by inverting its chords, this continued bass highly proper to be sung, *ut si ut re mi fa mi*, &c. (ggg.)

The continued bass then is properly nothing else but

(ccc) When the treble syncopates in descending diatonically, it is common enough to make the second part of the syncopate carry a discord, and the first a concord. See Example LV. where the first part of the syncopated note *sol* is in concord with the notes *ut mi sol ut*, which answer to it in the fundamental bass, and where the second part is a dissonance in the subsequent chord *la ut mi sol*. In the same manner, the first part of the syncopated note *fa* is in concord with the notes *re fa la ut*, which answer to it; and the second part is a dissonance in the subsequent chord *sol fa re fa*, which answer to it, &c.

(ddd) We are obliged to mark likewise, in the continued bass, the chord of the sub-dominant with a 5, which in the fundamental bass is figured with a 6 alone; and this to distinguish it from the chords of the sixth and of the lesser sixth. (See Examples LVI. and LXIV.) For what remains, the chord of the great fifth in the fundamental bass carries always the sixth major, whereas in the continued bass it may carry the sixth minor. For instance, the chord of the seventh *ut mi sol fa*, gives the chord of the great fifth *mi sol fa ut*, thus improperly called, since the sixth from *mi* to *ut* is minor.

(eee) M. Rameau has justly observed, that we ought rather to figure this lesser sixth with a 5, to distinguish it from the sensible sixth which arises from the chord of the tonic dominant, and from the sixth which arises from the perfect chord. In the mean time he figures in his works with a 6 alone, the lesser sixths which do not arise from the tonic dominant; that is to say, he figures them as those which arise from the perfect chord; and we have followed him in that, though we thought with him, that it would be better to mark this chord by a particular figure.

(fff) The chord of the seventh *si re fa la* gives, when inverted, the chord *fa la si re*, composed of a third, a tritone, and a sixth. This chord is commonly marked with a 6, as if the tritone were a just fourth. It is his business who performs the accompaniment, to know whether the fourth above *fa* be a tritone or a fourth redundant. One may, as to what remains, figure this chord thus 4.

(ggg) The continued bass is proportionably better adapted to singing, as the sounds which form it more scrupulously 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 LXV. is much more in the taste of singing, and more agreeable, than the fundamental bass which answers to it. Part II.

Principles but a treble with respect to the fundamental bass. Its rules immediately follow; which are properly no other than those already given for the treble.

RULE I.

210. Every note which carries the chord of the false fifth, and which of consequence must be what we have called a sensible note, ought (77) to rise diatonically upon the note which follows it. Thus in example LXV. the note fa, carrying the chord of the false fifth marked with an 8, rises diatonically upon ut (HHH).

RULE II.

211. Every note carrying the chord of the tritone should descend diatonically upon the subsequent note. Thus in the same example LXV. fa, which carries the chord of the tritone figured with a 4+, descends diatonically upon mi. (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, 202.) See the example LXVI. where the second ut, which is syncopated, and which descends afterwards upon fa, carries the chord of the second (III).

CHAP. IX. Of some Licences assumed in the Fundamental Basses.

§ 1. Of Broken and Interrupted Cadences.

213. The broken cadence is executed by means of a dominant which rises diatonically upon another, or hence how upon a tonic by a licence. See, in the example LXXIV. sol la, (132, and 134).

214. The interrupted cadence is formed by a dominant which descends by a third upon another (136). Sec, in the example LXXV, sol mi (LLL).

These cadences ought not to be permitted but rarely.

(HHH) 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 re, carrying a chord of the seventh re fa la ut, to which the chord of the sub-dominant fa la ut re corresponds in the fundamental bass, ought to rise diatonically upon mi, (art. 129, n° 2. and art. 202.)

(III) 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 LXVII.) In the cloves which are found in the treble at fa and ut (bars third and fourth), the notes in the fundamental and continued bass are the same, viz. sol for the first cadence, and ut for the second. This rule ought above all to be observed in final 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 second note of the second bar in example LXVII. mi is found at the same time in the continued bass and in the treble: but the treble descends from fa to mi, whilst the bass rises from re to mi.

Two octaves, or two fifths, in succession, must likewise be shunned. For instance, in the treble sounds sol mi, the bass must be prevented from sounding sol mi, ut la, or re fa; because in the first case there are two octaves in succession, sol against sol, and mi against mi; and because in the second case there are two fifths in succession, ut against sol, and la against mi, or re against sol, and fa against mi. 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 LXVIII. where the fundamental bass ut gives the continued bass ut mi sol mi: the two mi's ought in this bass to carry the chord 6, and sol the chord 5; but as these chords are comprehended in the perfect chord ut mi sol ut, which is the first of the continued bass, we place nothing above ut, only we draw a line over ut mi sol mi.

In like manner, in the second bar of the same example, the notes fa and re of the continued bass, rising from the note sol alone of the fundamental bass which carries the chord sol fa re fa; we think it sufficient to figure fa with the number of the tritone 4x, and to draw a line above fa and re.

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

In general, whilst the same chord subsists in passing through different notes, the chord is reckoned the same as if the first note of the chord had subsisted; in such a manner, that, if the first note of the chord is, for instance, the sensible note, we ought to find the tonic when the chord changes. See example LXIX. or this continued bass, ut fa sol fa re ut, is reckoned the same with this ut fa ut. (Example LXXX.)

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 sol in a continued bass may answer to this fundamental bass ut sol ut, (see example LXXI.); in this case, we may regard the note sol as divided into three parts, of which the first carries the chord 6, the second the chord 7, and the third the chord 4.

We shall repeat here, with respect to the rules of the continued bass, what we have formerly said concerning the rules of the fundamental bass in the note upon the third rule, art. 193. The rules of the continued bass have exceptions, which practice and the perusal of good authors will teach. There are likewise several other rules which might require a considerable detail, and which will be found in the Treatise of Harmony by M. Rameau, and elsewhere. These rules, which are proper for a complete dissertation, did not appear to me indispensably necessary in an elementary essay upon 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.

(LLL) One may sometimes, but very rarely, cause several tonics in succession to follow one another in ascending... § 2. Of Supposition.

215. When a dominant is preceded by a tonic in the fundamental bass, we add sometimes, in the continued bass to the chord of that dominant, a new note which is a third or a fifth below; and the chord which results from it in this continued bass is called a chord by supposition.

For example, let us suppose that in the fundamental bass we have a dominant sol carrying the chord of the seventh sol si re fa; let us add to this chord the note ut, which is a fifth below this dominant, and we shall have the total chord ut sol si re fa, or ut re fa sol si, which is called a chord by supposition (mm).

Of the different kinds of chords by supposition.

216. It is easy to perceive, that chords by supposition are of different kinds. For instance, the chord of the tonic sol si re fa gives,

1. By adding the fifth ut, the chord ut sol si re fa, called a chord of the seventh redundant, and composed of a fifth, seventh, ninth, and eleventh. It is figured with a \( \frac{7}{3} \); see LXXVI. (nnn). This chord is not practised but upon the tonic. They sometimes leave ing or descending diatonically, as ut mi sol ut, re fa la re, si re fa si; 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 fa, a fifth below the first tonic ut, is found in the chord re fa la re, and in the chord si re fa si (37 and note G).

\( \text{(mmm)} \) Though supposition be a kind of licence, yet it is in some measure founded on the experiment related in the note (r), 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, ut added beneath sol, causes sol to resound. Thus sol is found in some measure to be implied in ut.

If the third added beneath the fundamental sound be minor, for example, if to the chord sol si re fa, we add the third mi, 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.

\( \text{(nnn)} \) Many musicians figure this chord with a \( \frac{7}{3} \); M. Rameau suppresses this \( \frac{7}{3} \), and merely marks it to be the seventh redundant by a \( \frac{7}{3} \) or \( \frac{7}{3} \). But it may be said, how shall we distinguish this chord from the seventh major, which, as it would seem, ought to be marked with a \( \frac{7}{3} \)? M. Rameau answers, that there is no danger 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 re sol, according to M. Rameau, would indicate re fa \( \frac{7}{3} \) la ut, sol si re fa \( \frac{7}{3} \). If we would change fa \( \frac{7}{3} \) of the second chord into fa, it would then be necessary to write re sol. In notes such as ut, whose natural seventh is major, the figure \( \frac{7}{3} \) preceded or followed by a sharp will sufficiently serve to distinguish the chord of the seventh redundant ut sol si re fa, from the simple chord of the seventh ut mi sol si, which is marked with a \( \frac{7}{3} \) alone. All this appears just and well-founded.

\( \text{(ooo)} \) Supposition introduces into a chord dissonances which were not in it before. For instance, if to the chord mi sol si re, we should add the note of supposition ut descending by a third, it is plain that, besides the dissonance between mi and re which was in the original chord, we have two new dissonances, ut si and ut re; 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.)

\( \text{(ppp)} \) Several musicians call this last chord the chord of the ninth; and that which, with M. Rameau, we have 218. What is more, in the chord of the simple dominant, as re fa la ut, when the fifth sol is added they frequently obliterate the sounds fa and la, that too great a number of dissonances may be avoided, which reduces the chord to sol ut re. 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. (See LXXX.)

219. Sometimes they only remove the note la, and then the chord ought to be figured with 2 or 4 (QQQ). 220. Finally, in the minor mode, for example, in that of la, where the chord of the tonic dominant (109), is mi sol fa re; if we add to this chord the third ut below, we shall have ut mi sol fa re, called the chord of the fifth redundant, and composed of a third, a fifth redundant, a seventh, and a ninth. It is figured with a 5, or a +5. See LXXXI. (RRR).

§ 3. Of the Chord of the Diminished Seventh.

221. In the minor mode, for instance, in that of la, mi a fifth from la is the tonic dominant (109), and carries the chord mi sol fa re, in which sol is the sensible note. For this chord they sometimes substitute that other sol fa re fa (116), all composed of minor thirds; and which has for its fundamental found the sensible note sol. This chord is called a chord of the flats, or diminished seventh, and is figured with a 7 in the fundamental bas, (see LXXXII.) but it is always considered as representing the chord of the tonic dominant.

222. This chord in the fundamental bas produces in the continued bas the following chords:

1. The chord fa re fa sol, composed of a third, false

have simply called a chord of the ninth, they term a chord of the ninth and seventh. This last chord they mark with a 7; 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.

(QQQ) They often remove some dissonances from chords of supposition, either to soften the harshness of the chord, or to remove discords which can neither be prepared nor resolved. For instance, let us suppose, that in the continued bas the note ut is preceded by the sensible note si, carrying the chord of the false fifth, and that we should choose to form upon this note ut the chord ut mi sol fa re, we must obliterate the seventh si, because in retaining it we should destroy the effect of the sensible note si, which ought to rise to ut.

In the same manner, if to the harmony of a tonic dominant sol fa re fa, one should add the note by supposition ut, it is usual to retrench from this chord the sensible note si; because, as the re ought to defend diatonically to ut, and the si to rise to it, the effect of the one would destroy that of the other. This above all takes place in the supposes, concerning which we shall presently treat.

(RRR) Supposition produces what we call suspenso; 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, if this fundamental bas be given ut sol ut, and this continued bas above it ut ut ut,

it is a supposition; but if we have this fundamental bas ut sol sol ut, and this continued bas above it ut sol ut ut,

it is a supposo; because the perfect chord of ut, which we naturally expect after sol in the continued bas, is

suspended and retarded by the chord ut, which is formed by retaining the sounds sol fa re fa of the preceding

chord to join them to the note ut in this manner, ut sol fa re fa; but this chord ut does nothing in this case but suspend for a moment the perfect chord ut mi sol ut, which ought to follow it.

(SSS) The chord of the diminished seventh, such as sol fa re fa, and the three derived from it, are termed chords of substitution. They are in general barbs, and proper for imitating melancholy objects.

(PPP) As the chord of the diminished seventh sol fa re fa, and the chord of the tonic dominant mi sol fa re, only differ one from the other by the notes mi and fa; one may form a diatonic modulation of these two notes, and then the fundamental bas does nothing but pass from the tonic dominant to the sensible note, and from that note to the tonic dominant, till it arrives at the tonic. (See XCII.)

For the same reason, as the chord of the diminished seventh sol fa re fa, and the chord fa re fa la, which carries In the Treatise of Harmony by M. Rameau, and elsewhere, may be seen a much longer detail of the chords by supposition: But here we delineate the elements alone.

Chap. X. Of some Licences used in the Treble and Continued Basso.

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 licence ought to be rarely practised.

In like manner, in a continued bass, the dissonance in a chord of the sub-dominant inverted, as la in the chord la ut mi sol, inverted from ut mi sol la, may sometimes descend diatonically instead of rising as it ought to do, art. 129, n° 2; but in that case the note ought to be repeated in another part, that the dissonance may be there resolved in ascending.

225. Sometimes likewise, to render a continued bass more agreeable by causing it to proceed diatonically, they place between two sounds of that bass a note which belongs to the chord of neither. See example XCIV, in which the fundamental bass sol ut produces the continued bass sol la fa sol ut, where la is added on account of the diatonic modulation. This la has a line drawn above it to show its resolution by passing under the chord sol fa re fa.

In the same manner, (see XCV), this fundamental bass ut fa may produce the continued bass ut re mi ut fa, where the note re which is added passes under

carries the fifth fa of the tonic dominant mi, only differs by the sensible note fa, and the tonic la; one may sometimes, while the treble modulates fa la fa la fa la, ascend in the fundamental bass, from the sensible note to the third above, provided one descend at last from thence to the tonic dominant, and from thence to the tonic; (see XCIII.) As to what remains, this and the preceding examples are licences.

(uuu) I say a tonic, or reckoned such, because it may perhaps be a dominant from which the dissonance has been removed. But in that case one may know that it is a real dominant by the note which precedes it. For instance, if the note fa, carrying a perfect chord, is preceded by re a simple dominant, carrying the chord re fa la ut, that note fa is not a real tonic; because, in order to this, it would have been necessary that re should have been a tonic dominant, and should have carried the chord re fa la ut; and that a simple dominant, as re, carrying the chord re fa la ut, should only naturally descend to a dominant, (art. 194.)

(xxx) Sometimes a note which carries a 7 in the continued bass, gives in the fundamental bass its third above, figured with a 6. For example, this continued bass la fa ut gives this fundamental bass ut sol ut; but in this case it is necessary that the note figured with a 6 should rise by a fifth, as we see here ut rite to sol.

(yyy) 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 xxx).

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

M. P'Abbe Rouffier, to redress this deficiency, had invented a new manner of figuring the continued bass. His method is most simple for those who know the fundamental bass. It consists in expressing each chord by only signifying the fundamental sound with that letter of the scale by which it is denominated, to which is joined a 7 or 6, or a 6, in order to mark all the discords. Thus the fundamental chord of the seventh re fa la ut is expressed by a D; and the same chord, when it is inverted from that of the sub-dominant fa la ut re, is characterized by F; the chord of the second ut re fa la, inverted from the dominant re fa la ut, is likewise represented by D; and the same chord ut re fa la inverted from that of the sub-dominant fa la ut re is signified by F; the case is 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. 7. Every note figured with an 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 LXXX.)

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 2, 4, or 5: which shews 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 8 gives the third minor below, figured with a 7. (See LXXXIII.)

15. Every note marked with a 8 gives the tritone above, figured with a 7. (See LXXXIV.)

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

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

18. Every note marked with a 7 gives the seventh redundant above, figured with a 7. See LXXXVII. (zzz).

Remark.

228. We have omitted two cases 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 re in the fundamental bass, the note which answers to it in the continued bass may be la carrying the figure 6, (see zzz) We may only add, that here and in the preceding articles, 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, as may be seen in the article Chiffre, in Vol. III. of the Encyclopedie; an admirable article, of which M. Rousseau of Geneva is the author: but here we do not treat of accompaniments. For every reason, then, we should advise initiates to prefer the continued basses of M. Rameau to all the others, as by them they will most successfully study the fundamental bass.

It is even necessary to advertize the reader, and I have already done it (note eee), that M. Rameau only marks the lesser sixth by a 6 without a line, when this lesser sixth does not result from the chord of the tonic dominant; in such a manner that the 6 renders it uncertain whether in the fundamental bass we ought to choose the third or the fifth below; but it will be easy to see whether the third or the fifth is signified by that figure. This may be distinguished, 1. In observing which of the two notes is excluded by the rules of the fundamental bass. 2. If the two notes may with equal propriety be placed in the fundamental bass, the preference must be determined by the tone or mode of the treble in that particular passage. In the following chapter we shall give rules for determining the mode.

There is a chord of which we have not spoken in this enumeration, and which is called the chord of the sixth redundant. This chord is composed of a note, of its third major, of its redundant fourth or tritone, and its redundant sixth, as fa la si re. It is marked with a 6. It appears difficult to find a fundamental bass for this chord; nor is it indeed much in use among us. (See the note upon the art. 115.) Principles to do but to substitute for ut sharp ut natural; so that of Compo the major third la ut may become minor la ut; I sition shall have then

la si ut re mi fa sol la,

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

la sol fa mi ut re si la,

in which the sol and fa 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 la, we place three sharps at the clef upon fa, ut, and sol; and on the contrary, in the minor mode of la, we place none, because the minor mode of la, 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 Mode.

| Maj. Mode | ut re mi fa sol la si ut | |-----------|-------------------------| | of ut | ut re mi fa sol la si ut | | of sol | sol la si ut re mi fa sol | | of re | re mi fa sol la si ut re | | of la | la si ut re mi fa sol la | | of mi | mi fa sol la si ut re mi | | of si | si ut re mi fa sol la si | | of fa | fa sol la si ut re mi fa | | of ut | ut re mi fa sol la si ut | | of sol | sol la si ut re mi fa sol | | of re | re mi fa sol la si ut re | | of la | la si ut re mi fa sol la | | of mi | mi fa sol la si ut re mi | | of si | si ut re mi fa sol la si | | of fa | fa sol la si ut re mi fa |

(AAAA) The major mode of fa, of ut or re, and of sol or lab, are not much practised. In the opera of Pyramus and Thisbe, p. 267, there is a passage in the scene, of which one part is in the major mode of fa, and the other in the major mode of ut, and there are six sharps at the clef.

When a piece begins upon ut, 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 reb, which is almost the same thing with ut. It is for this reason that we substitute here the mode of reb for that of ut.

It is still much more necessary to substitute the mode of lab for that of sol; for the scale of the major mode of sol is

sol; la, fa, ut, re, mi, sol, sol,

in which you may see that there are at the same time both a sol natural and a sol: it would then be necessary, even at the same time, that upon sol there should and should not be a sharp at the clef; which is shocking and inconsistent. It is true that this inconvenience may be avoided by placing a sharp upon sol at the clef, and by marking the note sol 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. One might likewise in this series, instead of sol natural, which is the note immediately before the last, substitute fa, that is to say, fa twice sharp; which, however, is not absolutely the same found with sol natural, especially upon instruments whose scales are fixed, or whose intervals are invariable. But in that case two sharps may be placed at the clef upon fa, which would produce another inconvenience. But by substituting lab for sol, the trouble is eluded.

(BBBB) We have already seen, that in each mode the principal note is called a tonic; that the fifth above that note 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.

1. That when there are neither sharps nor flats at the cleft, it is a token that the piece begins in the major mode of ut, or in the minor mode of la.

2. That when there is one single sharp, it will always be placed upon fa, and that the piece begins in the major mode of sol, or the minor of mi, in such a manner that it may be sung as if there were no sharp, by fingering fi instead of fa, and in fingering the tune as if it had been in another cleft. For instance, let there be a sharp upon fa in the cleft of sol upon the note is called a tonic dominant, or the dominant of the mode, or simply a dominant; that the fifth beneath 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 beneath 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 re in the mode of ut, and fi in that of la, is termed a sub-tonic; the following note, which is a third major or minor from the tonic, according as the chord is major or minor, such as mi in the major mode of ut, and ut in the minor mode of la, is called a mediant; in short, the note which is a tone above the dominant, such as la in the mode of ut, and fa in that of la, is called a sub-dominant.

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

It will easily occur to every reader, that if each of the intervals through the whole diatonic series were equal in a mathematical sense, it would be absolutely indifferent upon what note any air were begun, if within the compass of the gammut; because the same equal intervals must always have the same effects. But since, besides the natural semitones, there is another distinction of diatonic intervals into greater and lesser tones; and since these vary their positions in the series of an octave, according as the note from whence you begin is placed, that note is consequently the best key for any tune whose natural series is most exactly correspondent with the intervals which that melody or harmony requires. But in instruments whose scales are fixed, notwithstanding the temperament and other expedients of the same kind, such a series is far from being easily found, and is indeed in common practice almost totally neglected. All that can frequently be done is, to take care that the ear may not be sensibly shocked. This, however, would be the case, if, in transposing any tune, the situation of the semitones, whether natural or artificial, were not exactly correspondent in the series to which your air must be transposed, with their positions in the scale from which you transpose it. Suppose, for instance, your air should begin upon ut or C, requiring the natural diatonic series through the whole gammut, in which the distance between mi and fa, or E and F, as also that between fi and ut, or 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 sol or 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 places in the octave to which you transpose. Now, from sol or G, the note with which you propose to begin, the three tones immediately succeeding are full; but the fourth, ut or C, is only a semitone; it may therefore be kept in its place. But from fa or F, the seventh note above, to sol or G the eighth, the interval is a full tone, which must consequently be redressed by raising your fa 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. But the order in which you must proceed, and the intervals required in every mode, are minutely and ingeniously delineated by our author.

(cccc) Two sharps, fa and ut, indicate the major mode of re, or the minor of fi; and then, by transposition, the ut is changed into fi, and of consequence, re into ut, and fi into la.

Three sharps, fa ut sol, indicate the major mode of la, or the minor of fa; and it is then sol, which must be changed into fi, and of consequence la into ut, and fa into la.

Four sharps, fa ut sol re, indicate the major mode of mi, or the minor of ut; then the re is changed into fi, and of consequence mi into ut, and ut into la.

Five sharps, fa ut sol re la, indicate the major mode of fi, or the minor of sol; la then is changed into fi, and of consequence fi into ut, and sol into la.

Six sharps, fa ut sol re la mi, indicate the major mode of fa; mi then is changed into fi, and of consequence fa into ut.

Six flats, fi mi la reb sol ut, indicate the minor mode of mi; ut is changed into fa, and of consequence mi into la. CHAP. XIII. To find the Fundamental Bass of a given Modulation.

238. As we have reduced to a very small number the rules of the fundamental bass, and those which in the treble ought to be observed with relation to this bass, it should no longer be difficult to find the fundamental bass of a given modulation, nay, frequently to find several; for every fundamental bass will be legitimate, when it is formed according to the rules which we have given (chap. VI.); and that, besides this, the dissonances which the modulation may form with this bass will both be prepared, if it is necessary that they should be so, and always resolved (DDDD).

239. It

Five flats, \( \text{fa} \) \( \text{mi} \) \( \text{la} \) \( \text{re} \) \( \text{sol} \), indicate the major mode of \( \text{re} \), or the minor mode of \( \text{fa} \); then the \( \text{fa} \) is changed into \( \text{fa} \), and of consequence the \( \text{re} \) into \( \text{ut} \), and the \( \text{mi} \) into \( \text{la} \).

Four flats, \( \text{fa} \) \( \text{mi} \) \( \text{la} \) \( \text{re} \), indicate the major mode of \( \text{la} \), or the minor mode of \( \text{fa} \); \( \text{re} \) then is changed into \( \text{fa} \), and of consequence \( \text{la} \) into \( \text{ut} \), and \( \text{fa} \) into \( \text{la} \).

Three flats, \( \text{fa} \) \( \text{mi} \) \( \text{la} \), indicate the major mode of \( \text{mi} \), or the minor of \( \text{ut} \); the \( \text{la} \) then is changed into \( \text{fa} \), and of consequence \( \text{mi} \) into \( \text{ut} \), and the \( \text{fa} \) into \( \text{la} \).

Two flats, \( \text{fa} \) \( \text{mi} \), indicate the major mode of \( \text{fa} \), or the minor of \( \text{sol} \); \( \text{mi} \) then is changed into \( \text{fa} \), and of consequence \( \text{fa} \) into \( \text{ut} \), and the \( \text{fa} \) into \( \text{la} \).

One flat, \( \text{fa} \), indicates the major mode of \( \text{fa} \), or the minor mode of \( \text{re} \), and \( \text{fa} \) is changed into \( \text{fa} \); of consequence the \( \text{fa} \) is changed into \( \text{ut} \), and the \( \text{re} \) into \( \text{la} \).

All the major modes then may be reduced to that of \( \text{ut} \), and the modes minor to that of \( \text{la} \) minor.

It only remains to remark, that many musicians, and amongst others the ancient musicians of France, as Lulli, Campara, &c. place one flat less in the minor mode: so that in the minor mode of \( \text{re} \), they place neither sharp nor flat at the clef; in the minor mode of \( \text{fa} \), one flat only; in the minor mode of \( \text{ut} \), 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.

However this be, I here present you with a rule for transposition, which appears to me more simple than the rule in common use.

For the Sharps.

Suppose \( \text{sol} \), \( \text{re} \), \( \text{la} \), \( \text{mi} \), \( \text{fa} \), and change \( \text{sol} \) into \( \text{ut} \) if there is one sharp at the clef, \( \text{re} \) into \( \text{ut} \) if there are two sharps, \( \text{la} \) into \( \text{ut} \) if there are three, &c.

For the Flats.

Suppose \( \text{fa} \), \( \text{fa} \), \( \text{mi} \), \( \text{la} \), \( \text{re} \), \( \text{sol} \), and change \( \text{fa} \) into \( \text{ut} \) if there is only one flat at the clef, \( \text{fa} \) into \( \text{ut} \) if there are two flats, \( \text{mi} \) into \( \text{ut} \) if there are three, &c.

(DDDD) We often say, that we are upon a particular key, instead of saying that we are in a particular mode. The following expressions therefore are synonymous; such a piece is in \( \text{ut} \) major, or in the mode of \( \text{ut} \) major, or in the key of \( \text{ut} \) major.

We have seen that the diatonic scale or gammut of the Greeks was \( \text{la} \) \( \text{fa} \) \( \text{ut} \) \( \text{re} \) \( \text{mi} \) \( \text{fa} \) \( \text{sol} \) \( \text{la} \), (art. 49.) A method has likewise been invented of representing each of the sounds in this scale by a letter of the alphabet; \( \text{la} \) by A, \( \text{fa} \) by B, \( \text{ut} \) by C, &c. It is from hence that these forms of speaking proceed: Such a piece is upon A, with \( \text{mi} \), \( \text{la} \), and its third minor; or, simply, it is upon A, with \( \text{mi} \), \( \text{la} \), and its minor; such another piece upon C, with \( \text{sol} \), \( \text{ut} \), and its third major; or, simply, upon C, with \( \text{sol} \), \( \text{ut} \), and its major; to signify that the one is in the mode of \( \text{la} \) minor, or that the other is in that of \( \text{ut} \) major; this last manner of speaking is more concise, and on this account it begins to become general.

They likewise call the clef of \( \text{ut} \) \( \text{fa} \), the clef of \( \text{re} \) \( \text{sol} \) G, &c. to denominate the clef of \( \text{fa} \), the clef of \( \text{sol} \), &c.

They say likewise to take the A \( \text{mi} \) \( \text{la} \), to give the A \( \text{mi} \) \( \text{la} \); that is to say, to take the union of a certain note called \( \text{la} \) in the harpichord, which \( \text{la} \) is the same that occupies the fifth line, or the highest line in the first clef of \( \text{fa} \). This \( \text{la} \) divides in the middle the two octaves which subside (note r8) between the \( \text{sol} \) which occupies the first line in the clef of \( \text{sol} \) upon that same line, and that \( \text{sol} \) which occupies the first line in the clef of \( \text{fa} \) upon the fourth; and as it possetles (if we may speak so) the middle station between the sharpest and lowest sounds, it has been chosen to be the sound with relation to which all the voices and instruments ought to be tuned in a concert (§).

(§) Thus far our author; and though the note is no more than an illustration of the technical phraseology in his native language, we did not think it consistent with the fidelity of a translation to omit it. We have little reason to envy, and till let us follow, the French in their abbreviations of speech; the native energy of our tongue supercedes this necessity in a manner so effectual, that, in proportion as we endeavour to become succinct, our style, without the smallest sacrifice of perspicuity, becomes more agreeable to the genius of our language: whereas, in French, laconic diction is equally ambiguous and disagreeable. Of this we cannot give a more flagrant instance than the note upon which these observations are made, in its original. We must, however, follow the author's example, in reciting a few technical phrases upon the same subject, which occur in our language, and which, if we are not mistaken, will be found equally concise, at the same time that they are more natural and intelligible. When we mean to express the fundamental note of that series within the diatonic octave which any piece of music demands, we call that note the key. When we intend to signify its mode, whether major or minor, we denominate the harmony \( \text{sharp} \) or \( \text{flat} \). When in a concert we mean to try how instruments are in tune by that note upon which, according to the genius of each particular instrument, they may best agree in unison, we desire the musicians who join us to sound A. 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 fol ut may belong to all the modes, as well major as minor, in which fol and ut are found together; and each of these two sounds may even be considered as belonging to a different mode.

For what remains, one 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 (Ch. VI.) for the procedure of the fundamental bas.

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 indispensible 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 ut is ut, and la in that of la, &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.

To know upon what mode or in what key a piece commences, our inquiry may be entirely reduced to distinguish the major mode of ut from the minor of la. 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 a piece. We shall now therefore give a detail of the different means by which these two modes may be distinguished.

1. From the principal and characteristic sounds of the mode, which are ut mi fol in the one, and la ut mi in the other; so that if a piece should, for instance,

(EEEE) It is certain that the minor mode of mi has an extremely natural connection with the mode of ut, as has been proved (art. 92.) both by arguments and by examples. It has likewise appeared in the note upon the art. 93. that the minor mode of re may be joined to the major mode of ut; and thus in a particular sense, this mode may be considered as relative to the mode of ut: but it is still less so than the major modes of fol and fa, or than those of la and mi minor; because we cannot immediately, and without licence, pass in a fundamental bas from the perfect minor chord of ut to the minor chord of re; and if you pass immediately from the major mode of ut, to the minor mode of re in a fundamental bas, it is by passing, for instance, from the tonic ut, or from ut mi fol ut, to the tonic dominant of re, carrying the chord la ut mi fol, in which there are two sounds, mi fol, which are found in the preceding chord; or otherwise from ut mi fol ut to fol mi re mi, a chord of the sub-dominant in the minor mode of re; which chord has likewise two sounds, fol and mi, in common with that which went immediately before it.

(FFFF) 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, mi ut fol, we must not with too much precipitation conclude from thence that we are in the major mode of ut, although these three sounds, mi ut fol, be the principal and characteristic sounds in the major mode of ut; we may be in the minor mode of mi, especially if the note mi should be long. You may see an example in the fourth act of Zoroaster, where the first air sung by the priests of Arimanes begins thus with two times, fol mi b, each of these notes being a crotchet. This air is in the minor mode of fol, and not in the major mode of mi, as one would at first be tempted to believe it. Now we may be sensible that it is in fol minor, by the relative modes which follow, and by the notes where the cadences fall. When a person is once able to ascertain the mode, and can render himself sure of it by the different means which we have pointed out, the fundamental bals will cost little pains. For in each mode there are three fundamental sounds.

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

Major mode of UT. ut mi sol ut. Minor mode of LA. la ut mi la.

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 UT. sol fa re fa. Minor mode of LA. mi sol fa re.

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 UT. fa la ut re. Minor mode of LA. re fa la fa.

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

To remedy this, the chord of the sub-dominant fa la ut re must be inverted into a fundamental chord of the seventh in this manner, re fa la ut; 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 la fa, this fundamental bals re sol; which procedure is agreeable to rules.

Here then are four chords, ut mi sol ut, sol fa re fa, fa la ut re, re fa la ut, which may be employed in the major mode of ut. We shall find in like manner, for the minor mode of la, four chords,

(4444) 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 ut, viz. ut carrying the perfect chord, fa carrying that of the sub-dominant, and sol that of the tonic dominant, to which we may join the chord of the seventh, re fa la ut (art. 105.) but we here regard as extended the series of dominants in question, as belonging to the mode of ut, because it prefers 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 la belongs naturally to the mode of sol, the simple dominant fa to that of la, &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 ut.

And in this mode we sometimes change the last of these chords into fa re fa la, substituting the fa for fa. For instance, if we have this melody in the minor mode of la, mi fa sol la, we would cause the first note mi to carry the perfect chord la ut mi la, the second note fa to carry the chord of the seventh, fa re fa la, the third note sol the chord of the tonic dominant mi sol fa re, and in short, the last the perfect chord la ut mi la.

On the contrary, if this melody is given always in the minor mode la la sol la, the second la being syncopated, it might have the same bals as the modulation mi fa sol la; with this difference alone, that fa might be substituted for fa in the chord fa re fa la, the better to mark out the minor mode.

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,

ut la re sol ut fa fa mi la re sol ut,

which are terminated equally in the tonic ut, either entirely belong, or at least may be reckoned as belonging (4444) to the mode of ut; because none of these dominants are tonic dominants except sol, which is the tonic dominant of the mode of ut; and besides, because the chord of each of these dominants forms no other sounds than such as belong to the scale of ut.

But if I were to form this fundamental bals,

ut la re sol ut,

considering the last ut as a tonic dominant in this manner, ut mi sol fa; the mode would then be changed at the second ut, and we should enter into the mode of fa; because the chord ut mi sol fa indicates the tonic dominant of the mode of fa; besides, it is evident that the mode is changed, because fa does not belong to the scale of ut.

In the same manner, were I to form this fundamental bals

ut la re sol ut,

considering the last ut as a tonic dominant, in this manner, ut mi sol la; this last ut would indicate the mode of sol, of which ut is the sub-dominant.

In like manner, still, if in the first series of dominants, I caused the first re to carry the third major, in this manner, re fa la ut; this re having become a tonic dominant, would signify to me the major mode of sol; and the sol which should follow it, carrying the chord fa re fa, would relapse into the mode of ut, from whence we had departed.

Finally, in the same manner, if in this series of dominants, one should cause fa to carry fa in this manner, fa re fa la; this fa would show that we had departed from the mode of ut, to enter into that of sol.

From From 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, one cannot be secure by the assistance of this dominant alone, whether the mode be major or minor; but it is only necessary for the composer to cast his eye upon the following note, which must be the tonic of the mode in which he is; by the third of this tonic he will discover 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 (HHHH).

I here subjoin the folioquy of Armida, with the continued and fundamental bases. The changes of the mode will be easily distinguished in the fundamental bas, by the rules which we have just given at the end of the article 242. This folioquy will serve for a lesson to beginners. M. Rameau quotes it in his New System of Music, as an example of modulation highly just and extremely simple. (See Plate VI. and the following.) (HHHH)

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 ut and the major of sol, or the major mode of ut and the minor of la; 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 ut and the minor of si, &c.

When you find yourself 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, you must continue in it but for a short time, because the ear is always impatient to return to the former mode.

(1111) It is extremely proper to remark, that we have given the fundamental, the continued bas, and in general the modulation of this folioquy, merely as a lesson in composition extremely suitable to beginners; not that we recommend the folioquy in itself as a model of expression. Upon this last object what we have said may be seen in what we have written concerning the liberties to be taken in music, Vol. IV. p. 435. of our Literary Miscellany. It is precisely because this folioquy is a proper lesson for initiates, that it would be a bad one for the mature and ingenious artist. The novice should learn tenaciously to observe his rules; the man of art and genius ought to know on what occasions and in what manner they may be violated when this expedient becomes necessary.

(LLLL) 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 XC. 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 re to sol, and a fifth redundant from sol to ut.

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

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

Now this last fundamental bas is formed according to the common rules, unless that there is a broken cadence from re to mi, and an interrupted cadence from mi to ut, which are licenses (art. 213 and 214.). Principles enharmonic in the trio of the Fatal Sisters, in Hippolitus and Aricia, at the words, "Ou cours-tu malheureux," and that there are no examples of the chromatic enharmonic, at least in our French operas. M. Rameau had imitated an earthquake by this species of music, in the second act of the Gallant Indians; but he informs us, that in 1735 he could not cause it to be executed by the band. The trio of the Fatal Sisters in Hippolitus has never been sung in the opera as it is composed. But M. Rameau affirms, (and we have heard it elsewhere by people of taste, before whom the piece was performed), that the trial had succeeded when made by able hands that were not mercenary, and that its effect was astonishing.

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 of several bars 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. 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.

251. Imitation and fugue are sometimes conducted by rules merely deducible from taste, which may be seen in the 332d and following pages of M. Rameau's Treatise on Harmony; where will likewise be found a detail of the rules for composition in several parts. The chief rules for composition in several parts are, that the discords should be found, as much as possible, prepared and resolved in the same part; that a discord should not be heard at the same time in several parts, because its harshness would disgust the ear; and that in no particular part there should be found two octaves or two fifths in succession (Mmmm) with respect to the bass. Musicians, however, do not hesitate sometimes to violate this precept, when taste or occasion require. In music, as in all the other fine arts, it is the business of the artist to assign and to observe rules; the province of men who are adorned with taste and genius is to find the exceptions.

Chap. XVI. Definitions of the Different Airs.

252. We shall finish this treatise by giving in a few words the characteristic distinctions of the different airs to which names have been given, as chacoon, minuet, rigadoon, &c.

The chacoon is a long piece of music, containing three times in each bar, of which the movement is regular, and the bars sensibly distinguished. It consists of several couplets, which are varied as much as possible.

Formerly the bass of the chacoon was a constrained bass, or regulated by a rhythmus terminating in 4 bars, and proceeding again by the same number; at present composers of this species no longer confine themselves to that practice. The chacoon begins, for the most part, not with the perfect time, which is struck by the hand or foot, but with the imperfect, which passes while the hand or foot is elevated.

The villanelle is a chacoon a little more lively, with its movement somewhat more brisk than the ordinary chacoon.

The paffacaille only differs from a chacoon as it is more slow, more tender, and beginning for ordinary with a perfect time.

The minuet is an air in triple time, whose movement is regular, and neither extremely brisk nor slow, consisting of two parts or strains, which are each of them repeated; and for which reason they are called by the French reprises: each strain of the minuet begins with a time which is struck, and ought to consist of 4, of 8, or of 12 bars; so that the cadences may be easily distinguished, and recur at the end of each 4 bars.

The farabando is properly a slow minuet; and the courant a very slow farabando: this last is no longer in use. The passepied is properly a very brisk minuet, which does not begin like the common minuet, with a stroke of the foot or hand; but in which each strain begins in the last of the three times of which the bar consists.

The loure is an air whose movement is slow, whose time is marked with 4, and where two of the times in which the bar consists are beaten; it generally begins with that in which the foot is raised. For ordinary the note in the middle of each time is shortened, and the first note of the same time pointed.

The jig is properly nothing else but a loure very brisk, and whose movement is extremely quick.

The furlana is a moderate movement, and in a mediocrity between the loure and the jig.

The rigadoon has two times in a bar, is composed of two strains, each to be repeated, and each consisting of 4, of 8, or of 12 bars: its movement is lively; each strain begins, not with a stroke of the foot, but at the last note of the second time.

The bourée is almost the same thing with the rigadoon.

The gavotte has two times in each bar, is composed of two strains, each to be repeated, and each consisting of 4, of 8, or of 12 bars: the movement is sometimes slow, sometimes brisk; but never extremely quick, nor very slow.

The tambourin has two strains, each to be repeated, and each consisting of 4, of 8, or of 12 bars. Two of the times that make up each bar are beaten, and are very lively; and each strain generally begins in the second time.

The mufette consists of two or three times in each bar; its movement is neither very quick nor very slow; and for its bass it has often no more than a single note, which may be continued through the whole piece.

(Mmmm) 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 fa re, and the other ut la in rising, ut is the fifth of fa, and la the twelfth of re.