The Origin Of The Harmonic Proportions And The Nature of Melody.

Mental endeavor is the preparation for theology.

For those features which to the uninitiated in the truth of divine matters seem difficult to grasp and lofty are by mathematical reasoning shown to be trustworthy, manifest and uncontroversial, by means of certain images.

For they show proof of the supernatural properties in numbers; and they make clear the powers of the intelligible forms in reasoning.

Thus Plato teaches us many remarkable things about the nature of the gods through the appearance of mathematical things; and the Pythagorean philosophy disguises its teaching on divine matters with these, so to speak, veils. For of this kind is the whole of that sacred writing,^ both Philolaus on the Bacchae,^ and the whole Pythagorean system of teaching about God.

Again, it perfects us in moral philosophy, implanting in our behavior order, propriety and harmony in social relations. It also informs us what figures, what songs, what motions are appropriate to virtue; and also the teaching by which the Athenian"* would have those who will pay attention to moral virtues from their youth cultivated and perfected. Furthermore he makes plain the proportions of numbers which are associated with the virtues, some in arithmetic, some in geometry and others in harmony; and he shows the excesses and deficiencies of the vices.

By all of these we are guided to the middle way in behavior and in morals.

So far, following the natural order, we have spoken first of Regular Plane Figures, and next we passed on to their congruences.

In what follows it will not be right to depart at all from the natural method, so that the learning of the human mind, which quite often uses a different route, may be given all the more assistance.

For what the nature of the subject requires is that we should now thirdly expound in the abstract those proportions which occur between a circle and a part cut off on any side,’’ and the other kinds of cases which arise from the combination and separation of such proportions; then fourthly that we should pass to the operations of the world, which either God himself the Creator has adjusted to proportions of that kind,*’ or Sublunary Nature applies daily according to the rule of those proportions in the angles between the stellar rays; and finally we should add human music, showing how the human mind, shaping our Judgement of what we hear, by its natural instinct imitates the Creator by showing delight and approval for the same proportions in notes which have pleased God in the adjustment of the celestial motions.

For it is indeed difficult to abstract mentally the distinctions, types, and modes of the harmonic proportions from musical notes and sounds, since the only vocabulary which comes to our aid, as is necessary to expound matters, is the musical one.

Therefore in this Book we shall have to combine the third section with the fifth and last one,** and we shall not only have to speak of harmonic proportions in the abstract, but also to deal in anticipation with this human imitation of the creation in melody, though it has been specified in the title of the Book that it should deal at that point with the work of the creation of the heavens, which must be postponed to the last section on account of its sublime and incredible nature.

So much for the order of treatment.

In order to throw more light on the contraries which are opposed to each other, it is desirable to inaugurate this dissertation on human melody by recalling what the ancients have said about the origin of consonances.

Certainly, just as it is ordained in all human affairs that in those things which are bestowed on us by nature, use precedes understanding of causes, similarly as far as melody is concerned it happened to the human race that from its very beginning it used without speculating or knowing about their causes the same rhythms and intervals between notes as we commonly use today, in the chanting of melodies, not only in churches and in choirs of musicians, but everywhere with­ out applying any art, even at crossroads and in the fields.

This antiquity of melody is apparent from the first book, of Genesis.

For great must the delight in the melody of the human voice have been (when I say delight, I mean the harmonious and melodic intervals) which moved Jubal,^^’ eighth in line from Adam, to learn and teach how to imitate the melodies of men with inanimate instruments.

Unless I am mistaken, this Jubal is the Apollo, by a slight change of letters, who defeated his brother Jabel, the originator of cattle breeding, whose joy was in the shepherd’s pipe (and who was believed by the Greeks to be the god Pan), by the clear ringing of the lyre which he had invented, having borrowed the material for the strings from his brother Tubalcain (and let him be Vulcan for us, by a play on the name).

Yet however ancient be the pattern of human melody, made up of consonant or melodic intervals, yet the causes of the intervals have remained unknown to men — so much so that before Pythagoras they were not even sought; and after they have been sought for 2,000 years, I shall be the first, unless I am mistaken, to reveal them with such accuracy."

Pythagoras was the first, when he was passing through a smithy, and had noticed that the sounds of the hammers were in harmony, to realize that the difference in the sounds depended on the size of the hammers, in such a way that the big ones gave out low sounds, and the little ones high sounds.

A proportion is properly speaking observed between sizes, he measured the hammers, and readily perceived the proportions at which consonant or dissonant intervals occurred, and melodic or unmelodic intervals occurred between notes.

He passed at once from the hammers to the length of strings, where the ear indicates more exactly what fractions of the string are consonant with the whole, and which are dissonant with it.

Having discovered definite proportions, or “the fact that,” it remained to track down the causes as well, or “the reason why”’"’ some proportions marked out melodic, pleasant, and consonant intervals between notes, and other proportions those which are dissonant, ab­ horrent to the ear, and strange.

After 2,000 years the opinion has been reached that the causes are to be looked for in the properties of the proportions themselves, as they are contained within the boundaries of a discrete quantity, that is to say of Numbers.*’’ For the Pythagoreans saw that perfect harmonies were es- i i • i i • r i ^ tablished II cords under equal tensions have their lengths in double proportion, or triple, or quadruple, as between the numbers 1 and 2 or 1 and 3 or 1 and 4. Such proportions are called in arithmetic multiple. Further, slightly more imperfect consonances occurred be­ tween the strings which make up the sesquialterate proportion, the Hemiholia, and the sesquitertiate, the Epitriton, that is between the numbers 2 and 3 and 3 and 4. These two proportions combined make the double proportion, as between the numbers 2 and 4 or 1 and 2; but the smaller proportion, between the numbers 3 and 4, divided 1 . into the greater, that between 2 and 3, left the proportion of one to 2 . 3 . one and an eighth, that is between 8 and 9. And this, they discovered, 4 . 6 . 9 . 8. 12 . 18 . 27 . was the size of the interval of a tone, the commonest of all in melody. But the number 8 is the cube of 2, and the number 9 is the square of 3. Then the following numbers were already before them: 1, 2, 3, 4, 8, 9. However since Unity is the same as its square and its cube, whereas the binary had as its square 4 and as its cube 8, to the ternary they also added its cube 27 as well as its square 9, because they sup­ posed that it was right always go as far as the cubes on account of the fact that the whole world and everything that gives notes consisted not of empty surfaces but of solid bodies. Eventually from that be­ ginning such a strong opinion grew up about these numbers, on ac­ count of the fact that they were Primes, and their squares and their cubes, that the Pythagoreans resolved that the whole of Philosophy should be composed of them. For Unity represented for them Idea and Mind and Form, because just as Unity is indivisible and remains the same when it is squared or cubed, so the Ideas also were irreducible and universal and always the same. Therefore they made Unity the symbol of the nature of Identity, but the other numbers the symbols of the nature of otherness. Then the Binary signified otherness and matter, because the former admits of division, and so does the latter; and as the Binary squared becomes 4, and cubed becomes 8, which are numbers distinct from 2, so matter can be unstable and multiform. On the other hand, the Binary also signified Soul, because although Mind is immobile, or takes Joy in uniform, that is circular motion. Soul on the contrary receives multiple motions from Body, and is more amenable to rectilinear motions, which are differentiated in six ways. Lastly the Ternary denoted for them Substance, which is made up of Form and Matter, Just as 3 is made up of 2 and 1, and because bodies in the real world have the same number of dimensions as the Ternary has Unities.

Nor were the numbers symbols only of the three basic principles, but moreover Soul itself was made up of these very numbers, and of all their proportions, and the subdivision of their proportions into sesquialterates,*’ sesquitertiates, and sesquioctaves; so that Soul, the bond between Mind and Body, was in its essence nothing but H ar­ mony, and made up of harmonies. Undoubtedly they were led to this doctrine by contemplating the fact that the human soul is so greatly delighted by notes which form and contain some harmonious pro­ portions between their magnitudes.