On the Origin of Consonances and Their Relationship

Sound is to Sound as string is to string.* Now each string contains within itself all other strings that are smaller than it, but not those that are larger; consequently in each Sound all the acute sounds are contained within the grave, but not reciprocally all the grave within the acute; from which it is evident that one must seek the most acute term by the division of the most grave: this division must be Arithmetic, that is to say in equal parts, etc. Let therefore A B be the most grave, from which I wish to find the most acute term in order to form the first of all the Consonances; I divide it into two (this number being the first of all) as you see has been done at point C, and then A C, A B are distant one from the other by the interval of the Octave or Diapason. If I wish to have the other Consonances that follow immediately upon the first, I divide A B into three equal parts, and there will result not one acute term alone, but two: namely A D and A E, from which will arise two Consonances of the same kind — to wit, a Twelfth and a Fifth — and I can further divide the line A B into 4, into 5, or into 6 parts, but no further, because the capacity of the ears does not extend beyond this, etc.

To render this proposition more evident, we shall take seven strings, whose divisions will be marked by numbers — supposing them to be all tuned to the Unison, without troubling ourselves otherwise with any other equality; the numbers will then be placed in their natural order beside each string, as has been observed in the following demonstration, each number marking the division into equal parts of the string that corresponds to it; where one will remark only that the number 7, being unable to give any agreeable interval (as this is evident to Connoisseurs), we suppose the number 8 to be the first after 7 that is double one of those contained in the Senaire, making the triple Octave with 1, which does not increase the quantity of intervals, since 6 and 8 give the same interval as 3 and 4, every number always representing the double of the one it is derived from.

[Site Editorial Note: Here Rameau’s original text contains a plate captioned Démonstration. It consists of seven horizontal lines of varying length, each marked by moveable bridge notches and labelled at left with a solmisation name and a number (Ut…♩8, Sol…♩6, Mi…♩5, Ut…♩4, Sol…♩3, Ut…♩2, Ut…♩1). Brackets at right identify the intervals generated between adjacent strings: Octave, Douzième, Double Octave, Triple Octave on one side; and reading the inner brackets, Quinte, Tierce majeure, Tierce mineure, Quarte, Sixte majeure, Sixte mineure on the other. The plate demonstrates that all consonances are contained within the first six divisions of a string (the senaire 1–6), with 8 added as the triple octave.]

It must first be remembered that the numbers mark throughout the division of unity, just as the whole string corresponds to 1.

The order of the origin and perfection of these Consonances is found to be determined by the numbers themselves; so that the Octave, found between 1 and 2, which is engendered first, is more perfect than the Fifth, which is found between 2 and 3; thence to the Fourth which is between 3 and 4; and so on following the natural progression of numbers — and admitting the Sixths only as the last.

The name of the Notes must make one perceive that string 1, its Octave 2, its double and triple Octave 4 and 8, render, so to speak, only one and the same Sound; moreover, the disposition of these Notes conforms to the order of numbers and to the divisions of the string, and gives the most perfect Harmony one can imagine, as everyone is at liberty to verify for themselves. For what concerns the particular properties of each Sound or each Consonance, we shall distribute them by Articles, to give a more distinct idea.

Article I — On the Principle of Harmony, or the Fundamental Sound

We must suppose first that the whole string corresponding to 1 renders a certain Sound, whose properties must be examined by relating them to those of this unique string, or equally to those of Unity, which is the principle of all numbers.

1°. The different divisions marked on all the strings that are equal to the first, and determined by the quantity contained in each number that corresponds to them, prove to us evidently that each part of these strings proceeds from the first, unique string — since these parts are contained within this first and unique string; thus the Sounds that these divided strings must render are engendered by the first Sound, which is consequently the principle and the foundation.

2°. From the different distances that are found between this fundamental Sound and those it has engendered by its division, there are formed different intervals, whose principle is consequently this fundamental Sound.

3°. And finally, from the union of these different intervals, there are formed different Consonances, of which Harmony cannot be perfect if this first Sound does not reign beneath them, being the Bass and the Fondement, according to what appears in the Demonstration; thus this first Sound is still the principle of these Consonances and of the Harmony they form.

We shall see in the following Articles the Sounds that have the most correspondence with this principle, and the use that is made of it.

Article II — On the Unison

The Unison is strictly speaking only a single Sound, which can be rendered by several voices or several instruments, as appears in the seven strings of the preceding demonstration before they are divided; from which one says that the Unison is not a Consonance, because it does not meet the necessary condition for making one — namely the difference of Sounds in regard to the grave and the acute — but that it bears the same relationship to the Consonances that Unity bears to numbers.

Article III — On the Octave

The proportion of the whole to its half, or of the half to the whole, is so natural that it conceives itself at once; which must predispose us in favour of the Octave, whose ratio is as 1 to 2. Unity is the principle of numbers, and 2 is the first, there being found a great relationship between these two numbers — the Epithets Principle and First, whose application is very just. Also in practice, the Octave is distinguished only under the name of replique; every replique being thenceforth confounded with its principle, as appears by the name of the Notes of the preceding demonstration; and this replique being regarded less as a chord than as a supplement to the chords, which is why some compare it to zero. Male and female voices naturally intone the Octave, believing they intone the Unison or the same Sound. In flutes this Octave depends only on the force of the wind; and if one takes a Viole whose strings are long enough to make the beats distinguishable, one will remark that in causing a string to resonate with some violence, those that will be lower or higher by an Octave will themselves tremble, whereas it is not so with the acute Sound of the Fifth that trembles, and not the grave; which proves that the principle of the Octave is confounded within the two Sounds that form it, and that by consequence of all the other intervals, it resides uniquely in the grave and fundamental Sound.* Descartes has here been deceived by the false proof he draws from a Lute, with regard to the Octave.

Furthermore, the Octave serves as a boundary to all intervals, and everything that is engendered by the division of the principle, having been compared to that principle, can equally be compared to its Octave; this double comparison producing in Harmony only the single diversity that can come from the different position of two terms, as 2. 3. or 3. 2., which one calls in terms of Geometry the ratio or inverted comparison. Now since this inverted comparison is nothing other in Harmony than the transposition of a grave Sound to the acute, since if 2 marks the grave Sound, being the first, it will mark by consequence the acute Sound, being the last; one must distinguish this transposition from the number that represents the Octave, by putting 4 in the place of 3. 2., which must make us perceive that every number multiplied Geometrically always represents the same Sound, or rather that it gives the replique of the one of which it is the root — as this is proved by the preceding demonstration, in beginning this multiplication at number 2, which is engendered the first by the division of Unity, this latter yielding to this number the privilege of engendering in its place all the rest, without losing any of its force; for that which accords with 2 accords equally with 1, the Octave, the Double, the Triple Octave, and so on, are in the end only one and the same interval, which one distinguishes only under the name of doubled or replique — thus the Fifth with the Twelfth and the Fourth with the Eleventh, etc. — and it is only in order to find the mean numbers which can accord with each term of this ratio 1. 2, that one multiplies it as many times as is necessary, finding for example 3 between 2 and 4; 5. 6. 7. between 4 and 8; and so on ever further to infinity — 2. 4. or 4. 8. being in the same ratio as 1. to 2.

From this conformity found between the intervals that are born of numbers compared indifferently with 1 and with 2, although it is always 1 above and 2 below, we can judge that these same numbers compared above 1 and below 2 will form intervals whose ratio will be nearly equal; but much more, from this inverted comparison which comes only from the transposition of a Sound to its Octave, or of a number to its double, we must judge that the ratio of these Sounds thus transposed cannot be altered except by a difference of proportion that causes almost none to the ear, since the proportion of 2 to 4 has nearly the same effect as that of 2 to 2, as all that we have just said, joined to experience, proves sufficiently; which has given occasion to attribute to the Octave the same force as to the principal and fundamental Sound, and to regard it as the mother, the source, and the origin of all the intervals. The Octave, says Zarlin, is the mother, the source, and the origin of all the Harmonic chords; it is by the division of these two terms that all the Harmonic chords are engendered.** However, although this may be true in some fashion, it is always from the division of the unique and fundamental Sound that all the other Sounds are engendered, and by consequence all the intervals and all the chords; so that in order to make the sentiment of Zarlin prevail, one cannot dispense with adding to it that the fundamental Sound makes use of its Octave as a second term within which all the intervals engendered by its division must answer, to mark better that it is the commencement and the end; that this Octave has no other properties than those communicated to it by the fundamental Sound from which it is engendered — or, to express it yet better, that it is always the same Sound that is transposed into its Octave or into its replique, or that multiplies itself, so as to determine from all sides the particular intervals of each Sound it has engendered, without altering any of the properties fallen to the lot of these engendered Sounds in the first comparison that had to be made first with this fundamental Sound. Thus: such a Sound forms a perfect Consonance with this fundamental Sound, which forms it equally with its Octave; such another has had to form an imperfect Consonance, or a Dissonance, which forms it equally with the other; such another has had to ascend or descend from one side as it mounts and descends from the other; finally all that accords from one side accords also from the other, and nothing is altered in any way; except that the perfection attached to the chords formed by the principal Consonances — or rather when the fundamental Sound occupies its natural place, which is the most grave — is found duly altered when this Sound is transposed into its Octave to introduce the diversity by the different order that these same Consonances hold among themselves, as one can verify in the preceding demonstration, where one will receive a very great satisfaction from the present disposition of all the Consonances, and where this satisfaction will diminish without however hurting the ear at all, if one removes the Sounds 1. 2. and then the Sounds 1. 2. 3. 4., although this be even more sensible in the course of a piece of Music.

From all these remarks we can conclude that any Sound whatsoever is always understood in its Octave;*** Descartes agrees in part (when he says, one never hears any Sound but its Octave seems to strike the ears in some fashion) and he would perhaps have added l’Octave au dessous, had he not been deceived in the proof he draws from a Lute (as we have said) or had he taken account of the sentiment of Aristotle, who says in his 24th Problem, and in his 43rd Problem (on the relationship of Desermes****) if one touches the string nete which makes the acute of the Octave, one will also hear the string hypate which makes the grave, because the commencement of the grave Sound resembles the echo or the image of the acute Sound; there is perhaps not a Musician who does not make use of this expression: such a Sound, such a Note, or such an interval is under-heard in the Bass; adding sometimes dans la Basse; so that the expression often prevails in this case over the one who knows least its force. The Harmonic reasons that the accord parfait offers us, not being able then to admit the accords de Sixte and the Sixte Quarte that derive from it, without supposing that the fundamental Sound of this perfect chord is under-heard in its Octave, otherwise one must destroy all principle; and beyond all this, experience makes us feel that a chord composed of the Third and the Fifth is always perfect and complete without the Octave, which leaves us to think that this Octave is understood, since it being engendered first, it is always under-heard; then this Octave placed above this Third and this Fifth, with which it forms for us a Sixth and a Fourth, we have nonetheless to hear an accord that is always good, although the fundamental Sound has no further place there — thus this fundamental Sound is transposed or under-heard in its Octave; from which comes the fact that this last accord is less perfect than the first, although composed of the same Sounds; thus these different ways of expressing oneself, the principle is reversed, transposed, or under-heard in its Octave, always come to the same; so that the acute Sound of the Octave must by no means be regarded as a principle different from the one from which it is immediately engendered, but as representing it and making a whole with it in which all the Sounds, all the intervals, and all the chords must begin and end — without forgetting however that all the properties of this Octave, of the Sounds in general, of the intervals and of the chords depend absolutely on this unique and fundamental principle, which is represented to us by the whole string or by unity.

Article IV — On the Fifth and the Fourth

The Sounds that form the Fifth and the Fourth are all contained within the divisions of the whole string, and by consequence are engendered by the fundamental Sound; however with regard to intervals, it is only the Octave and the Fifth that are engendered immediately by the fundamental Sound; for the Fourth is only a consequence of the Octave — this Fourth not proceeding except from the difference found between this Octave and the Fifth; also it is not even mentioned in the original articles, where all the force is attributed to the Fifth alone, the Octave not even being recalled, although it has preceded the Fifth in its origin, and that by consequence the Fifth can have no place without it: so that if one does not recall this Octave in the chords, it is apparent that it is understood, otherwise the Fourth could never be admitted, since it can subsist only with the Octave.

It is here that one must give full attention to this renversement de comparaison of which we have spoken in the preceding article. This renversement is the knot of all the diversity in which Harmony can participate; it suffices to know it in order to overcome the greatest difficulties; and this knowledge consists only in knowing how to distinguish the intervals that can be born of the reciprocal comparison of a mean number with each term of the Octave: so that if we take 3, which is the Arithmetic middle of the Octave 2. 4, comparing it on one side we shall find the Fifth with 2, and on the other the Fourth with 4; no difference being found in these intervals, except in what proceeds from the comparison made with the grave and fundamental Sound of the Octave, which must without doubt be more perfect than that which proceeds from the comparison made with the acute Sound of the same Octave; for the difference of proportion that one encounters in these intervals does not need to stop us, since it comes only from the difference between the Octave and the Unison, as if one compared 3. to 2, and again to 2; which would cause no difference. And therefore this great relationship between the two Sounds of the Octave, which scarcely distinguish themselves from the Unison and seem to be no more than one, must make us regard at the same time as nearly equal two intervals, whose difference consists only in one of these terms 2. 4 — giving preference to the one where the fundamental Sound occupies its natural place, since it proceeds immediately from this Sound; which has given occasion to make use in this case of the proportion Arithmetique, which is very simple, since it consists only in finding the middle of two proposed numbers, as we have found 3 between 2 and 4, and which has given occasion further to those who have followed the order of multiplications of inventing a new proportion that they have called Harmonique, and which is nothing other than a renversement of the preceding one, as we shall see in the following chapter; if both of these proportions applied to the same object give us the Fifth on the side of the grave Sound of the Octave, and the Fourth on the side of the acute; and if applying afterwards one of these proportions to the object of the other, it gives us the Fourth on the side of the grave, and the Fifth on the side of the acute; this renversement discovering itself more and more as one penetrates into the secrets of Harmony: for example, if one begins with the numbers, whose natural progression is to go increasing, one will see that in Harmony this progression must go diminishing; if on one side the Arithmetic proportion can be favourable, on the other the one called Harmonique does the same effect; if to conform to the first proportion it is necessary to suppose that the numbers mark the division of unity; for conforming to the second it is necessary to reverse the order of the progression of the numbers; if to conform to the natural progression of numbers (in always supposing they mark the division of unity) it is necessary to divide a proposed string; for conforming to the renversement of the progression of these numbers it is necessary to multiply this proposed string; if all the Sounds that are born of divisions are found to the acute against the natural order of gravity, all those that are born of multiplications find themselves to the contrary at the grave against the natural order — which is however made good by means of the Harmonic proportion. Finally if the Octave has all the relationship we have remarked, and which we cannot dispute, without destroying what reason and experience offer us on this subject, we see here that its first interval within its species gives us the Fifth first for the interval in the grave Sound and fundamental of this Octave, and that it gives us the Fourth as the shadow* (it is the expression of Descartes) of this Fifth — which comes only from the renversement of the two Sounds that have composed this Fifth in the first place, by the transposition of the grave and fundamental Sound 2 into its Octave in the acute; this last renversement being the principal object of this Work.

Article V — On the Thirds and the Sixths

The Sounds that form the Thirds and the Sixths are all contained within the divisions of the whole string, and by consequence are engendered by the fundamental Sound; however with regard to intervals, it is only the Octave, the Fifth, and the major Third that are engendered immediately by the fundamental Sound, the minor Third and the Sixths being only a consequence of the Fifth and the Octave, in that this minor Third and these Sixths proceed only from the difference found between the major Third and the Fifth, and between the two Thirds and the Octave; which merits some reflections, above all with regard to the minor Third.

Since all these intervals are engendered by the Octave, and it is there that all commence and end; thus the minor Third must be comprised there, and not indirectly as we find the major Third and the Fifth here, but reporting itself directly to the fundamental Sound or to its Octave; otherwise this Third could no longer change place, the middle would be its share in the chords, and it could never occupy the extremities — which would be altogether contrary to what experience proves to us, and to the properties that one attributes in this case to the Arithmetic and Harmonic proportions; the first dividing the Fifth by the major Third at the grave, and the minor at the acute; and the second dividing it on the contrary by the minor Third at the grave, and the major at the acute; a new species of renversement in the order of these Thirds, which proves that all the diversity of Harmony is principally founded on this renversement.

To persuade oneself better of this, one need only remark the agreeable effect that all the Consonances of the preceding demonstration produce in the order they hold there, and the properties attached to each of them: first the Octave presents itself as so inseparably united to the principle from which it draws its origin, that it becomes inseparable from it; next the fundamental Sound appropriates the Fifth to itself to form all the chords, in determining immediately thereafter, by its union with the Third, the construction of these chords; so that the Fifth being composed of a major Third and a minor Third, it is impossible that either of these Thirds can

report itself at the same time to its principle; but it suffices also that one of them appear to be engendered immediately, for one to be unable to dispense with attributing to the other the same privilege, because the difference of the major to the minor that is encountered in them causes no difference in the genre of the interval, which is always a Third from one side and the other; which is moreover, the Fifth can serve as a boundary to the intervals only by virtue of the Octave, this quality belonging to it only since the Octave is inseparable from the principle; so that all that can be found between the principle and its Fifth is always dependent on the Octave, since this Fifth is inseparable from this principle, as we have proved up to the present; and moreover, since one cannot judge an interval by another, if it is not by the help of the Octave; it is therefore necessary to abandon the Fifth and the minor Third, thus the Octave of the grave and fundamental Sound of this minor Third will be at that point under-heard, and will enjoy the same privileges that are attached to it in the origin of all intervals — that is to say, that just as the Fifth between 2 and 3 engendered immediately by the fundamental Sound of the Octave 2. 4, has produced the Fourth between 3 and 4 by renversement, or by the transposition of the fundamental Sound 2 into its Octave 4, which is equal; in the same way the major Third between 4 and 5, engendered immediately by the fundamental Sound of the Octave 4. 8, will produce by its renversement a minor Sixth between 5 and 8; and in the same way again the minor Third between 5 and 6, engendered immediately by the fundamental Sound of the Octave 5. 10, will produce by its renversement a major Sixth between 6 and 10, or between 3 and 5; so that there is no point here of difference between the immediate origin of the Fifth and that of the two Thirds, nor between the mediate origin of the Fourth and that of the two Sixths; and as one could yet oppose that the principle of the minor Third seems to be different from that of the major Third, of the Fifth or of the Octave, in that 5 is not a multiple of 2 (taking here 2 for unity) it is good to warn that it is only in order to avoid fractions, in conforming to the natural order of numbers, which prescribes in such cases the division of the string, that one makes the ratio of this minor Third to be found between 5 and 6; since this ratio could be rendered in the same proportion between 1 and 1 5/6 of which unity would be the principle; which is perceived in the following Article.

We must conclude from all that we have just said that there are only three primary Consonances, which are the Fifth and the two Thirds, from which is composed a chord that is called natural or perfect, and from which come three secondary Consonances, which are the Fourth

and the two Sixths, from which are composed two new chords that are nonetheless reversed from the first — leaving aside the Octave which must be under-heard in each of these chords — and for which the term of Consonance is not as proper as that of Equisonance, with which the majority of the best Authors have adorned it.

If we have given an equal force to each Third in relation to the fundamental Sound, it is not to say that the place determined for them by the natural division of the Fifth is not the most convenient; and we shall see throughout that the acute is less suited to the major Third than to the minor.

Article VI — Abridgement of the Contents of This Chapter, in Which the Properties of the Preceding Demonstration Are Found Enclosed Within a Single String

Since a part of each string of the preceding Demonstration suffices for the proof of all that we have just said, we shall mark this part on a single string by the number that determines the division into equal parts, and we shall take this part from the number onwards to the end of the string by drawing to the right.

[Site Editorial Note: Here Rameau’s original text contains a plate captioned Démonstration — Du rapport des Consonances dans les longueurs prises à gauche. It shows a single horizontal line marked with the numbers 1, 2., 3., 4., 5., 6., 8., 10. at points along it. Above the line, brackets identify Octave, Quinte, Tierce majeure, and Tierce min. reading inward from the right. Below the line, brackets identify Quarte, Sixte min., and Sixte maj. reading outward. The plate condenses the full seven-string Démonstration from earlier in the chapter into a single string showing all consonances simultaneously.]

One need here pay attention only to the Octaves 2. 4., 4. 8., and 5. 10., to compare reciprocally with each number of each of these Octaves those that are found at the middle; where one will find that the first interval will always be enclosed within the comparison of the mean number to the one that represents the grave and fundamental Sound of the Octave, and that the interval reversed of the first will be enclosed within the comparison of the same mean number to the one that represents the acute Sound of the same Octave. For example, if one takes the Octave 2. 4., one will find there that the Fifth 2. 3. is the principle of the Fourth 3. 4.; if one takes next the Octave 4. 8., one will find there that the major Third 4. 5. is the principle of the minor Sixth 5. 8.; and if one takes finally the Octave 5. 10., one will find there that the minor Third 5. 6. is the principle of the major Sixth 6. 10., or between 3 and 5; the whole proceeding only from the transposition of the fundamental Sounds 2. 4. and 5. into their Octaves 4. 8. and 10.

To render the whole still more evident, one need only take the lengths that result from the same division, by drawing to the left from the number as far as the end of the string, where for then each length will be comparable to the whole string which is the principle, and to its Octave 2. which serves it as a term; so that 3. will give the Fifth with 1., and the Fourth with 2.; 5. will give the major Third with 1., and the minor Sixth with 2.; and 6. will give the minor Third with 1., and the major Sixth with 2.; unity being here (as one sees) the immediate principle of the Fifth and of the two Thirds, and the Octave 2 their mediate principle for deriving the Fourth and the two Sixths by the transposition of the fundamental Sounds 2. 4. and 5. into their Octaves.

This renversement that we have just remarked among the Consonances has been for the most part considered by Theoricians as merely the simple difference there is from one interval to another; however the difference of a Consonance to the Octave must be distinguished from that of two Consonances, in that the Octave representing the principle, nothing can accord with one of its terms (as Descartes* says) without also according with the other; but in Harmony we pay no more attention to the principal Sound of this Octave, the acute Sound counting for nothing there; also we shall remark that the primary Consonances, and those that come from their renversement, can always be taken on our string by drawing to the right, which is the most natural side, because their difference proceeds only from that of the two terms of the Octave; but the difference of two consecutive Consonances can only be taken by drawing to the left (as we shall see at Chapter V) because they have been engendered only from the principal Sound, to which it is necessary to return in order to know this difference, since it is the origin.

If one makes a reflection on the manner of finding the ratios of the intervals engendered by the transposition of the two Sounds of the Octave, or of those that come from the distance there is from one interval to another, we shall see that for having a reversed interval, one need only double the smaller term of a given ratio, or divide the larger by half. (This is the same thing;) as for example, the minor Third 5. 6. gives us the major Sixth, in doubling 5, or in dividing 6: the major Sixth being thus rendered between the same proportion between 1 and 1 5/6, of which unity would be the principle, as we also see that the Octave can be formed from the Unison by the division or the multiplication of one of the terms of its ratio as 2. to 2., as well as dividing, one thus has the Octave, which subsists always as such, whether in the grave or in the acute.