Jean-Philippe Rameau · section 7 of 109

Chapter four

Observations on the Properties of Harmonic and Arithmetic Proportions

Descartes proposes dividing a string into equal parts as evidence of the origin of consonances, which we have not set out here, because this proof can be drawn only by reversing the natural progression of numbers. The numbers then express the multiplication of the lengths resulting from that division. This entirely disturbs the order of harmony: the octave, which ought naturally to be divided with the fifth below, is instead divided with the fourth below. This also led those who followed that reversed progression to invent a new proportion, which they called harmonic, to restore chords to their natural form. Indeed, anyone who understands this proportion’s nature cannot deny that it reproduces arithmetic proportion point for point. It is already probable that, if the progression of numbers is reversed, its proportion must also be reversed, so that the reversal may imitate in every respect the perfections belonging to the natural progression. The uniformity of these two proportions is so evident when their objects differ only by inversion that further discussion is unnecessary. Hence most arithmeticians and geometers who have not studied music have been content to mention harmonic proportion without defining its properties, apparently because they knew none belonging to it. The Reverend Father Pardie’s words prove this:* “Everything said until now about this progression or proportion is of little use, and I do not wish to undertake to say extraordinary things here.” * Desermes, who discussed the matter extensively, says explicitly that the movements of air producing consonances and dividing the octave so that the fifth lies below and the fourth above follow not harmonic proportion, but arithmetic proportion, as in the numbers 2, 3, 4, and so forth. Further on, he says that what is called harmonic proportion must therefore be called arithmetic proportion or progression. This may have been why the Greeks did not occupy themselves with the former. Without examining whether the Greeks concerned themselves with harmonic proportion, let us now see whether Zarlino had good reason to dwell upon it. We must chiefly attend to this author: he served as a model for those who followed him; we are always referred to him concerning practice; he is still the oracle of some musicians; and Monsieur de Brossard himself calls him the prince of modern musicians.

* Zarlino observes that music is subordinate to arithmetic, that unity, the principle of numbers, represents the sounding body from which proof of the relations of sounds is drawn, and that unison is the principle of consonances. Yet he forgets all this in his demonstrations and rules. Far from following the principle he has just declared, the deeper he goes, the further he departs from it. He cannot help revealing it in a whole string that he proposes to divide, the sounding body just mentioned. But he effaces this object from our thought by making a new, separate comparison of every length resulting from the division. In this comparison the whole string is no longer distinguished: far from serving as the principle, it becomes dependent on what formerly depended on it. As though the main business were constructing instruments, he requires us to measure lengths already determined by the same numbers that determined the string’s division into equal parts. He does not foresee that the relations among those numbers suffice to give us the most perfect understanding of harmony we could desire. To obtain the proof, we need only attach a new idea to the numbers. Since music is subordinate to arithmetic, and harmonic progression must decrease whereas arithmetic progression must increase, imagine that numbers expressing the multiplication of unity in arithmetic express, in harmony, the division of that unity into as many equal parts as they contain units. Anyone concerned only with the properties of numbers then finds music simple and natural, and proves it just as easily this way as the other. But rather than risk this supposition, Zarlino prefers to weary our minds with a second operation. He reverses not only the natural progression of numbers but also the beautiful order of harmony first offered by the string’s division. This will be evident to anyone who tests it. The test will also show that he has, in a sense, fallen into the very fault he sought to avoid. Consider the common measure applied to each length, which the numbers then determine through their quantities of units. That measure must be applied—and the string consequently lengthened—as many times as the number contains units. Thus the numbers here express multiplication, not division, of the given string, the sounding body represented by unity. It is true that, if the largest number represents the whole string, smaller numbers will represent its divisions. But that largest number cannot serve as the principle everywhere. It changes in quantity as the string is divided into more or fewer parts. As its divisions increase, the principle it should represent recedes further and further until it is lost from view. Consider the numbers 6, 5, 4, 3, 2, 1. If 6 is regarded as the principle, one need only hear the effect of all the sounds produced by the lengths these numbers determine to be disabused at once. The same holds if 6 is removed and 5 taken as the principle, or if 5 is then removed and 4 taken, and so on. In short, this numerical order contains as many imperfections as the opposite order contains perfections, considering the properties we suppose to belong to each. To remedy the defect of his second operation, Zarlino was compelled to undertake a third. To recover what he had lost, he resorted to a certain multiplication of the numbers, explained in our Chapter XI. From it he formed a new progression, which he or others called harmonic proportion. It gives us only what arithmetic proportion offered at the very first divisions, but with this difference: all the simplicity of arithmetic proportion becomes obscurity in harmonic proportion. The original root numbers and the lengths they determined no longer concern us. We must begin again with new operations, as though everything found so far had become useless, although those operations merely return us to the path we had lost. Having strayed too far, however, we have lost sight of the principle and can scarcely recognize it here. * The Reverend Father Mersenne makes all these truths clear when he sets out to prove that the harmonic number is simply the number of movements of the air stirred by the string’s vibrations, and that this number makes arithmetic division sweeter, more agreeable, easier, and more familiar than harmonic division.

* The Reverend Father Pardie, Book VIII, p. 100. ]

[ * Desermes, Book I on Music, Theorem 28, p. 237.

* Zarlino, Prima parte, chapter 20, fol. 37; chapter 40, fol. 61. Terza parte, chapter 11, fol. 183. ]

* All the difficulties Zarlino creates in his harmonic operations would still be of little account if he recalled the principle he first proposed. Far from keeping it before us everywhere, he immediately abandons it. If he recalls it in the octave, he does so only in passing. If he says the octave is the origin of all intervals, he forgets that it is also the origin of their inversion, discussed in his Harmonic Demonstrations. If he acknowledges that inversion, he forgets the inversion of chords, which is merely its consequence. If he gives the perfect chord as the principle, since it is the only chord presented by harmonic ratios, he no longer mentions the principle of that chord—or at least his applications have no relation to it. If he discusses the properties of the bass, the position where this principle must always reside, as his comparison of the bass with the earth makes sufficiently clear, he treats it quite differently in his rules and examples. If he discusses the perfect cadence and the bass’s progression in it, he makes no valid connection with his modes, although a piece can end only with a perfect cadence in some mode. Finally, he discusses dissonances without any foundation, and the principle is obscured everywhere in his demonstrations, rules, and examples. We shall treat this more particularly in the second book.

* The Reverend Father Mersenne, Harm., book 1, De numero, pondere et mensura, article I, proposition VI.

* Zarlino, Dem. Harm., p. 2, definition x, fols. 83 and 84.

These are the great fruits Zarlino has gathered from this harmonic proportion. By attaching to numbers the meaning we have described, however, everything is simple, familiar, precise, just, and correct. Nothing is simpler or more familiar than the natural progression of numbers and the arithmetic operations that alone suffice here for the proof; nothing is more precise than all the properties of harmony contained within the number six; and nothing is juster or more correct than finding the principle everywhere in unity, as we shall explain.

1. If we find chords in which unity does not appear, we must seek it in one of its geometric multiples, or rather in one of those of the number 2, which represents it. Observe that, if this multiple does not stand at the head of the chord, it will at least form part of it. We need then only halve it to give it its proper place and, at the same time, to recognize the true chord in question. Chords reduced in this way will certainly always be the fundamental forms of those in which the multiples of unity do not come first. For example, if we find 5, 6, 8, or 6, 8, 10, we need only halve 8, and in either case we shall have 4, 5, 6, which forms the perfect chord arising from the division of the fifth; we have also halved 10, which does not change the substance of the chord.

2. Since the number 5 or its geometric multiples may sometimes represent unity—provided, of course, that neither unity nor its multiples then appears—we must treat the multiples of this number 5 as we have treated those of unity. Thus the origin of the chord 12, 15, 20 will be found by reducing 20 to 10, and so forth.

3. When the fifth and major third occupy the lower position, unity, taken in one of its multiples, always comes first; and when the minor third occupies the lower position with this fifth, unity, taken in the multiples of 5, always comes first. As we have said,* this is done only to avoid fractions. But if the fifth is not in the lower position, the numbers that should represent unity no longer come first. Hence neither 3 nor its multiples, in which the fourth is generated, can represent unity or consequently stand at the head of chords without reversing their natural order. The number 3 is a harmonic mean and must everywhere remain such. When unity is represented by one of its multiples, 3 is likewise represented by one of its multiples; and when unity is represented by a multiple of 5, 3 is multiplied by 5 or by a multiple of 5. Thus neither 3 nor its multiples can occupy the lower position without in some way destroying the foundation; for if the foundation could not be understood as implied there, it would certainly be destroyed altogether. It is only from this consequence that we can prove the perfection of inverted chords: they derive this perfection from a truly perfect chord from which they originate. The rules we establish on this subject will complete our conviction.

* Chapter III, Article V, p. 13.

See the demonstrations in chapter 6.

All this admits only a very slight exception, found in two chords in which the diminished fifth occupies the lower position, according to the demonstrations in Chapter VII, Articles VI and VII. Unity is then represented by the square or the cube of the number 5.

Once we have thus attained a perfect knowledge of all the properties of harmony through the operations most closely related to it, we may convey the same idea through other operations, according to the subject to which we wish to apply them. But since harmony alone concerns us here, we shall retain our first system. Nevertheless, the last chapter shows the close relationship between the numbers that mark the divisions or the multiplications of the string: the whole difference consists in a simple inversion.