Theoretical and Practical (c).

PRELIMINARY DISCOURSE.

MUSIC may be considered, either as an art, which has for its object one of the greatest pleasures of which our senses (D) 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 man: 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 affluency. But there remains to us much uncertainty concerning the degree of perfection to which they brought it. Almost every question which has been propounded with respect to the music of the ancients has divided the learned; and probably may 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. In the preceding history we have stated a few facts respecting the nature of ancient music, and the inventors of the several musical instruments; but it were to be wished, that, in order to elucidate, as much 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 The difficulties in the same work the most probable opinions story of established or propounded by the learned, upon a subject, difficult and curious. This philosophical history of ancient music is a work which might highly embellish the literature of our times.

In the mean time, 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 accessions 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 (E); 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

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(c) 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 to have written anything so fit for our purpose as M. d'Alembert, whose treatise on music is the most methodical, perspicuous, concise, and elegant dissertation on that subject with which we are acquainted. As it was unknown to most English readers before a former edition of this work, it ought to have all the merit of an original. We have given a translation of it; and in the notes, we have added, from the works of succeeding authors, and from our own observation, such explanations as appeared necessary, to adapt the work to the present day.

(d) In this passage, and in the definitions of melody and harmony, our author seems to have adopted the vulgar error, that the pleasures of music terminate 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 Avilion. In the mean time 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.

(e) 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 supporting 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, Preliminary 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 incontrovertible traces of harmony which appear in the succeeding ages, and to pursue those 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 evolution 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: they 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.

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 traditionary manner through the Roman provinces from a more remote period of antiquity.

(r) See M. Rameau's letter upon this subject, Merc. de Mai, 1752. One end which we have proposed in this treatise, was not only to elucidate, but to simplify the discoveries of M. Rameau.—For instance, besides the fundamental experiment mentioned above, that celebrated musician, to facilitate the explication of certain phenomena, had recourse to another experiment; 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 account for some other rules established in harmony; but 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 questions foreign to it.

In some other points also, (as, the origin of the chord of the sub-dominant*, and the explication of the seventh in certain cases) it is imagined that we have simplified, and perhaps in some measure extended, the principles of the celebrated artist.

We have likewise banished every consideration of geometrical, arithmetical, and harmonical proportions and progressions, which have been sought in the mixture and proportion 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 theory of music, and to reduce it to a form more complete and more luminous than has hitherto been done, we ought to caution our readers against misapprehension either of the nature of our subject or of the purpose of our endeavours.

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 vain to expect what is called demonstration: it is much too reduced the principal facts to a consistent and connected system; 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 if the intimate and unalterable conviction which can only be produced by the strongest evidence is not here to be required, we must also doubt whether a clearer elucidation of our subject be possible.

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

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

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

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

We would the more willingly persuade those who are skilled in theory and eminent in practice to extend and improve the views of him who before them pursued and pointed out the career, because many amongst them have already made laudable attempts, and have even been in some measure successful in diffusing new light through the theory of this enchanting art. It was with this view that the celebrated Tartini has presented us in 1754 with a treatise of harmony, founded on a principle different from that of M. Rameau. This principle is the result of a most beautiful experiment (c). If at once two different sounds are produced from two instruments of the same kind, these two sounds generate a third different from both the others. We have inferred in the Encyclopédie, under the article Fundamental, a detail of this experiment according to M. Martini; and we owe to the public an information, of which in composing this article we were its discoverer: M. Romien, a member of the Royal Society very originally at Montpelier, had presented to that society in the year 1753, before the work of M. Tartini had appeared, a memorial printed the same year, and where may be found the same experiment displayed at full length. In relating this fact, which it was necessary for us to do, it is by no means our intention to detract in any degree from the reputation of M. Tartini; we are persuaded that he owes this discovery to his own researches alone: but we think ourselves obliged in honour to give public testimony in favour of him who was the first in exhibiting this discovery.

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

(c) Had the utility of the preliminary discourse in which we are now engaged been less important and obvious than it really is, we should not have given ourselves the trouble of translating, or our readers that of perusing it. But it must be evident to every one, that the cautions here given, and the advices offered, are 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 justified in a contrary opinion. It may certainly be doubted with great justice, whether the solid contents of honourable bodies, and their degrees of cohesion or elasticity, can be ascertained with sufficient accuracy to render them the subjects of musical speculation, and to determine their effects with such precision as may render the conclusions deduced from them geometrically true. It is admitted, that sound is a secondary quality of matter, and that secondary qualities have no obvious 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 fill be so credulous as to pay them much attention), the English reader may consult Smith's Harmonics, Malcolm's Dissertation on Music, and Pleydel's Treatise on the same subject inserted in a former edition of this work. Our author next treats of the famous discovery made by Signor Tartini, of which the reader may accept the following compendious account.

If two sounds be produced at the same time properly tuned and with due force, from their conjunction a third sound is generated, so much more distinctly to be perceived by delicate ears as the relation between the generating sounds is more simple; yet from this rule we must except the unison and octave. From the fifth is produced a sound unison with its lowest generator; from the fourth, one which is an octave lower than the highest of its generators; from the third major, one which is an octave lower than its lowest; and from the sixth minor (whose highest note forms an octave with the lowest in the third formerly mentioned) will be produced a sound lower by a double octave than the highest of the lesser sixth; from the third minor, one which is double the distance of a greater third from its lowest; but from the sixth major (whole 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 that of a minor or diatonic or greater semitone, a sound lower by 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. Elements.

Preliminary which he has not communicated with sufficient perspicuity, and from whence the art might perhaps derive considerable advantage if they were placed in a proper light. Of this we are 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 caution them against mistaking 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 showing 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 a theory of music, at once great, complete and luminous. The enlightened philosopher will not attempt the explanation of facts, because he knows how little mechanical such explanations are to be relied on. To estimate them according to their proper value, it is only necessary to consider 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. Some 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, 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? Others pretend that there are particles in the air, which, by their different degrees of tension, being naturally susceptible of different oscillations, produce the multiplicity of sound in question. But what do we know of all this? 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?

If philosophical musicians ought not to lose their preliminary time in searching for mechanical explications of the Difficult phenomena in music, explications which will always be found vague and unsatisfactory; much less is it their conclusion to exhaust their powers in vain attempts to rise above their sphere into a region still more remote adequate from the prospect of their faculties, and to lose themselves in a labyrinth of metaphysical speculations upon the causes of that pleasure which we feel from harmony. In vain would they accumulate hypotheses on 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 it should be granted, would it not follow, that the same chord should successively and rapidly affect us with contrary sensations, since the vibrations are alternately coincident and 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 judgements? Besides, why should certain chords which are extremely pleasing in themselves, such as the fifth, lose almost nothing of the pleasure which they give us, when they are altered, and of consequence when the simplicity of their relations are destroyed; whilst other chords, which are likewise extremely agreeable, such as the third, become harsh almost by the smallest alteration; nay, whilst the most perfect and the most agreeable of all chords, the octave, cannot suffer the most inconsiderable change? Let us in sincerity confess our ignorance concerning the genuine causes of these effects (H). The metaphysical

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

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

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

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

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

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

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

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

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

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

We have therefore, in our present edition, preferred to the French syllables the British nomenclature by the letters C, D, E, F, G, A, B, as being more simple, more familiar to British musicians, and equally applicable to instrumental as to vocal music. Preliminary believing all this arithmetic necessary to form an artist.

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 of so little importance, that nothing can require a less ostentatious display.

Let us not imitate those musicians, who, believing themselves geometers, or those geometers who, believing themselves musicians, fill their writings with figures upon figures; imagining, perhaps, that this apparatus is necessary to the art. The propensity of adorning their works with a false air of science, can only impose upon ignorance, and render their treatises more obscure and less instructive.

This abuse of geometry in music may be condemned with 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 any thing 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 be useless, and contrary to found philosophy, to invent other relations in order to form the basis of any system of music less easy and simple than that which we have delineated in this treatise.

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

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

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

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

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

DEFINITIONS.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Since then, in the scale thus divided, C, D, E, F, G, A, B, 'c', the first division is perfectly like the last; and since the intervals between C and D, between D and E, and between F and G, are equal; it follows, that the intervals between G and A, and between A and B, are likewise equal to every one of the three intervals between C and D, between D and E, and between F and G; and that the intervals between E and F and between B and 'c' are also equal, but that they only constitute one half of the others. Elements.

Definitions.

7. It is for this reason that they have called the interval from E to F, and from B to C, a semitone; whereas those between C and D, D and E, F and G, G and A, A and B, are tones.

* Plate CC.XXIII.

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 found to another by the interval of a tone or of a semitone, or in general by seconds, whether major or minor; as from D to C, or from C to D, from F to E, or from E to F.

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

Third minor, what.

Third major, what.

Fourth, what.

Triton, what.

Fifth, what.

Sixth minor, what.

Sixth major, what.

Seventh minor, what.

Seventh major, what.

Octave, what.

Unison, what.

ever unequal in their force, are said to be in unison one with the other.

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

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

III. Of Intervals greater than the Octave.

13. If, after having sung the scale C, D, E, F, G, Fig. 21 A, B, c, one would carry this scale still farther in ascent, it would be discovered without difficulty that a new scale would be formed, c, d, e, f, &c. entirely similar to the former, and of which the founds will be an octave ascending, each to its correspondent note in the former scale; thus 'd', the second note of the second scale, will be an octave in ascent to the D of the first scale; in the same manner 'e' shall be the octave to E, &c. and so of the rest.

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

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

The octave above the octave of any found 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. (r).

IV.

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

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

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

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

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

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

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

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

V. What is meant by Consonances and Dissonances.

16. A chord composed of sounds whose union or coalescence pleases the ear is called a consonance; and the sounds which form this chord are said to be consonant one with relation to the other. The reason of this definition's 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 dissonance, whose union is displeasing to the ear is called a dissonance, and the sounds which form it are said to be dissonant one with relation to the other. The second, the tritone, and the seventh of a sound, are dissonants with relation to it. Thus the sounds C D, C B, or F B, &c. simultaneously heard, form a dissonance. The reason which renders dissonance disagreeable, is, that the sounds which compose it, seem by no means coalescent to the ear, and are heard each of them by itself as distinct sounds, though produced at the same time.

PART I. THEORY OF HARMONY.

CHAP. I. Preliminary and Fundamental Experiments.

EXPERIMENT I.

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

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

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

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

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

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

| Quantity | Description | |----------|-------------| | 1 | 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 | | ½ | Its double octave below | | ¼ | Its triple octave below | | ⅛ | And so of the rest. | | 3 | Its twelfth above | | ⅓ | Its twelfth below | | 5 | Its 17th major above | | ⅕ | Its 17th major below |

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

Experiment II.

22. There is no person insensible of the resemblance 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

VOL. XIV. Part II.

served, \( \frac{3}{4} \); which signifies that this string performs \( \frac{3}{4} \) vibrations; that is to say, one vibration and a half during a single vibration of the string which gives the sound 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 C, for instance, is F, of which the double octave f is the fourth above c. Since then the twelfth below 1 is \( \frac{3}{4} \), it follows that the double octave above this twelfth, that is to say, the fourth from the sound 1 in ascending, will be \( \frac{3}{4} \) multiplied by 4, or \( \frac{3}{4} \).

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 \( \frac{5}{4} \) divided by 4, or in other words \( \frac{5}{4} \).

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

To determine this relation, it is necessary to remark, that \( \frac{3}{4} \) are the same thing with \( \frac{3}{8} \), and that \( \frac{5}{4} \) are the same thing with \( \frac{5}{8} \): so that \( \frac{3}{4} \) shall be to \( \frac{5}{4} \) in the same relation as \( \frac{3}{8} \) to \( \frac{5}{8} \); 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 E and G, 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.

Let us suppose, for example, that the third major of the fifth \( \frac{3}{4} \) is sought. That third major ought to be, by what has been shown above, the \( \frac{3}{4} \) of the fifth; for the third major of any sound whatever is the \( \frac{3}{4} \) of that sound. We must then look for a fraction which expresses the \( \frac{3}{4} \) of \( \frac{3}{4} \); which is done by multiplying the numerators and denominators of both fractions one by the other, from whence results the new fraction \( \frac{9}{16} \). It will likewise be found that the fifth of the fifth is \( \frac{5}{4} \), because the fifth of the fifth is the \( \frac{5}{4} \) of \( \frac{5}{4} \).

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

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

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

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

26. It is on this account that, after having sung the tone C, we naturally modulate the third E, and the fifth G, instead of the double octave of E, and the octave of G; from whence we form, by joining the octave of the sound G, this modulation, C, E, G, 'c', 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 C, E, G, 'c', in which the chord C, E, is a third major, constitutes that kind of harmony or melody which we call the mode major; from whence it follows, that this mode results from the immediate operation of nature.

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

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

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

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

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

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

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