The Pure and Secret Harmonies

The pure and secret harmonies are insensible to us.

I explained the archetypes or paradigms of sensible harmonies. If the archetypes existed outside the soul, then we would be deprived of a great argument for asserting the necessity of the soul, in order to establish the essence of harmony.

But to establish them outside the soul is self-contradictory.

This is because the basic elements of true and archetypal harmony, which exists its terms, without any tinge of sensible emanation, are nevertheless divided and several in number.

For since it is also a proportion, it therefore itself requires its pair of terms. However, these terms, as we have assumed in the previous Books, are the complete circle and an aliquot part or aliquot parts of it, which are constructible by division of the arc.

This is the specific distinguishing feature of harmonic proportion, by which not only is harmonic proportion differentiated from other proportions which are classified in the same category, but also the pure and archetypal harmony from the sensible ones, except insofar as in the familiar common usage only the congruity of sounds is called harmony.

For pure harmony is most clearly differentiated from sensible or concrete harmony by the very fact that in pure harmony the terms come from mathematical categories, the circle and the arc, formed in a certain way, as the circle takes its form and shape from itself, and the arc takes its terms from its chord, its shape from the circle; whereas in the case of the sensible harmonies there is no need of this special formation.

For they can be either straight lines or sensible quantities shaped in some other way, provided they are faithful copies of this archetypal harmony of theirs, each in its own quantity, or as much of a faithful imitation as is possible in sensible things.

For in their case what is close to the truth, more or less, is accepted as the truth itself. These, then, are the terms of the archetypal harmony.

Second, as well as the terms, again as has been stated previously in the case of sensible harmonies, a mind is also required, which compares the terms, and assesses whether they, being of course arcs of a circle, are such as some side of a constructible figure divides off from the whole circle.

Thus in a sense there are three basic elements of the archetypal harmony, two with respect to the terms, the material, to express it by an analogy, the circle and its part, and the formal, the division of the part by a constructible figure; and one with respect to the actual relation of the terms, that is to say the mind which in a sense brings it into being.

Since every proportion, and so also this proportion of a part of a circle to the whole, is predicated of a relation, in this case this definite and prescribed form of a proportion is to be found to be a quality of the fourth kind.*® For harmony is a relation, in a sense, of quality or shape, being formed from the regular figures.

But if it belonged to the essence of sensible harmony previously that sensible things should flow into the soul by means of an emanation, how much more is it also necessary in this case than what we have stated as the terms of the secret harmony, the circle and its arc, should be within the mind itself, whether that is said to have come about by the reception into it of emanations, or whether they have been with the mind always, and present before anything was received into it. From now on this must be examined with the full power of my talents.

However, since we have now arrived at this point, we cannot in fact without considerable injustice both to the reader who is eager for philosophy and to the ancients, who handled this part of philosophy before us, conceal their opinions on the same topics, insofar as they are known to us.

I judge that only one thing must be said by way of preface: that a distinction must be drawn between the actual mathematical species, the circle and its arc, and a comparison between them. And the reason for that is that if the actual species, as terms, must be located in the mind without being received, it will be even more important to locate harmony, which occurs between those parts, in the mind, so that it does not have its essence outside it, inasmuch as its essence consists of some action of the mind on those species. Also the circle along with its arcs is in the soul in such a way that beyond controversy they are in sensible things as well; but har­ mony, which is between the circle and its part, as far as its formal aspect is concerned, is in no way outside the soul, as was made clear above by the example of number. In this case the arguments of the ancients are especially about the actual species, which is a simpler matter; whereas harmony is more a compound matter.

Also there is this difference between Aristotle, Plato, and Proclus on the one hand, and Ptolemy on the other. For the former deal with essence of the emanations, Ptolemy with the essence of harmony.*’ But we shall defer the text of Ptolemy until the Appendix of the whole work, in case, as we feared at the start of this chapter, he incites the crowds against us; whereas we shall now listen to those who say what is most apposite to the present investigation. Now Plato’s view on mathematical things was that the human mind is in itself thoroughly informed on species or figures, and axioms and conclusions about things. However, when it seems to learn, it is merely being reminded by sensible diagrams of those things which it knows on its own account. He conveys that with singular ingenuity in the Dialogues by introducing a slave who when questioned by his master makes all the replies as desired.*® Aristotle on the contrary in his Metaphysics^^ calls this a fabrica­ tion, “a fictitious argument, twisted to fit the hypothesis,” as neither do these Mathematicals ever exist independently of sensible things, nor is their character, even in the mind, on any different basis from that on which other universals are in the mind, as the species of the actual essence of individual sensible things is formed in the mind by definition.

Thus they are indeed prior to sensible things, and abstracted from sensible things, yet not in reality, but in mental conception.

When Aristotle names any mathematical category by way of example, he always names either a point, line, surface, body, or a number.

• These are the chief categories for quantity

But he very rarely mentions quantities as:

• shapes, and
• the fourth species of quality (where their material aspect, quantity, is one thing, but their formal aspect, shape, another)

He does not mention them as relations at all.

• He even proposes investigating harmonic knowledge through voices only
• He regards voices solely as lines, exactly as in optics.

However, of the interval between the lines, which are a proportion (a relation, of course, and that indeed of quality and shape) Aristotle never even dreams.

Thus, he would have made greater progress in this investigation if he had been imbued with more profound mathematics (concerning the intelligible difference between possible and impossible figures, 2 ’ with which we have dealt in Book I).

Therefore, as far as he is discussing the chief categories of quantity, he is an easy victor, with no opposition.

However, where he draws a universal* conclusion, and convicts Plato of the stupidity which is his own fantasy, and finally where to the Platonic picture of the “selftaught” slave he opposes a contrary picture of his own, asserting that the mind in itself is empty, not only of other knowledge and of mathematical categories but also of species,22 and is just a blank sheet, so that nothing is written on it, not even any mathematicals, but everything can be written on it; from this aspect, I say, he is not to be tolerated in the Christian religion, and several centuries later he found many “correctors,” as Proclus says, including Proclus himself as opponent, though here he does not cite Aristotle himself by name, but he names Plato, whom he defends, and openly admits him as his leader.

Therefore, the philosophy of this Proclus on the species of mathematical things, which I confess to be the terms of harmonic proportion which is pure and hidden from sensible things, is worth transcribing here word for word from his Book I on Euclid.

He writes as follows:

It remains for us to see what character or essence should be assigned to mathematical categories and species. Should it be conceded that they receive their character from sensible things, whether by abstraction, as it is customarily called, or by bringing together things which have been dispersed in various parts into one common concept (or definition), or even prior to that must a character be granted to them, as Plato claims, and the development of the totality of things shows'?

First, then, if we assert that mathematical species are established from sensible things, when the soul forms within itself from material triangles or circles the circular and triangular species, by a kind of secondary generation, I ask where the concepts (or definitions) get such great certainty, and such great accuracy.

For it will be either from sensible things or from the soul itself. But is it not impossible for it to be from sensible things, as there is much greater subtlety and exactness in these concepts? Therefore, it is from the soul, which procures perfection for imperfect things, and that accurate subtlety for things which are greatly inaccurate.

For tell me where among sensible things is to be found the nature which is indivisible (a point) or which has no breadth (such as a line) or depth (such as a surface) or where the equality of lines from the center, or where the perpetually constant proportions of sides (the material of my Book I) or the rightness of angles is to be found. For my part, as all divisible things are mingled and confused with each other, I see nothing pure, nothing unmixed with its contrary: all things, both those which are in scattered positions and those which are united are divisible. How therefore shall we procure their enduring essencefor immovable things from things which are movable, and are different in different places?

For they allow that whatever takes its character from mobile essences takes an alterable character. And how shall we bring about their accuracy for accurate and certain species from things which are not accurate? For everything which is the cause of a perpetually changeable notion is much more so itself.

Then it will have to be supposed that the soul itself is the generator of mathematical species and concepts. But if, containing them in itself as first patterns or paradigms, it makes them take their essential character, in such a way that their generation (the Christian understands, the creation of sensible things) is nothing but the propagation of species which were previously in it (that the mathematical reasons for the creation of bodies were coeternal with God, and that God is pre-eminently soul and mind, whereas human souls are images of God the Creator, even in essentials in their own way, is known to Christians) then we shall agree with Plato, in saying this, and the true essence of mathematics will have been discovered by us.

But if on the other hand the soul, since it did not have the mathematical concepts and had not previously received them from anywhere (if it was because they had not been created along with it) nevertheless weaves this wonderful immaterial equipage, it promotes this very splendid speculation:

how, then, does it in that case distinguish whether the things which it has generated are real and stable (I read poviga, “stable,” not yovipa, “generative”) or gone with the wind, and specters rather than true? What norms will it use to measure their truth?

How, since it did not measure their essence, does it generate such a great variety of concepts? For on this basis we shall make their character accidental, and not aiming at any goal or end.

Then mathematical species are the offspring of the soul itself, and it does not take the concepts which it establishes from sensible things. On the contrary, the latter are propagated from the former; these are its progeny, and clearly this is how permanent and perennial species are born.

Second, if we assemble the concepts of mathematics (or definitions) from below and from sensible things, surely the proofs which sensible things establish will he better than the proofs of universal and simple species.

For we say that, for investigating the subject of an enquiry, wherever there are basic assumptions and propositions, they are related to the proofs or conclusions.

If, therefore, individual things are the causes for universals, sensible things for objects of thought, how can it come about that the end of the proof is referred to the more universal, not to the particular, and that we show that the essence of intelligible things is more closely related to the proofs than that of sensible things.

For what they say is this: if someone proves that an isosceles triangle has angles equal to 2 right angles, and the same of an equilateral one, and the same of a scalene one, that is not legitimately knowing; but someone will have knowledge properly so called if he has proved that of every triangle without qualification. Again: universals are better for proof than particulars.

Further: proofs refer more to universals, and indeed the things of which proofs are constructed are prior, and by nature precede singulars, and are the causes of that which is proved.

Therefore, the demonstrative branches of knowledge are very farfrom assembling their propositions like beggars around sensible things which are posterior in origin 0and more obscure.

I shall say it for a third time: those who say this make the soul baser than the actual species.

For if matter takes from nature things which are essential, and closer to reality, and more obvious, whereas the soul receives them from it by a secondary act, and constructs the patterns and images in itself with a secondary origin, looking to baser essences and abstracting from matter things which are inseparable according to its own nature, are they not making soul more needy and more obscure than matter'?

For matter is also the place for material concepts, just as soul is for species (immaterial).

For the former would be the place for that which is prior, the latter for that which is secondary, and the former for that which holds the lead in existence, the latter for that which takes its character from it, and finally the former for that which is made in accordance with its essence, the latter for that which is named according to its implications. How therefore can soul, which shares in mind and the intelligible first essence, and which thence has complete knowl­ edge, how, I say, can it also be the receptacle for the most obscure species of the whole of life, that is to say of the lowest degree among things, and of all that is most imperfect in its existence?

It is entirely superfluous to mount more attacks on this opinion, which has long since deservedly been flogged by many.

But if mathematical species do not have their existence by abstraction from material things, nor by assembling common features which are found in individual things, and are not at all secondary in origin, or drawn from sensible things, it will indeed necessarily be the case that the soul adopts them either from itself or from Mind, or indeed from both itself and Mind at the same time.

Yet if it is from itself alone, how will they he images of intelligible species, how will they be intermediate between divisible and indivisible nature, though they obtain no integration of the first-named for their existence? Lastly, how are the first patterns, the paradigms or ideas, which are in the mind, the originators of universals?

But if on the other hand it is from Mind alone that they are adopted by the soul, how can the soul remain itself, functioning of itself and moving itself, if the concepts which are in it have flowed into itselffrom elsewhere, according to the standard of the natures of those things which are activated by something else? And in what way will the soul differ from Matter, which is everything only in potential, but generates nothing among material species?

The remaining possibility, therefore, is that the .soul adopts them both from itself andfrom Mind, and so that it is itself an absolute integration of the species which take their nature and obtain entry to existence from the intelligible first patterns or paradigms which are generated by themselves.

Then the soul is in no way a blank sheet, empty of all thoughts; but it is always a written sheet, and it both writes of itself on itself, and is filled with writings by Mind.

For the soul is a Mind, or a kind of Intellect, which reflects on itself in accordance with an Intellect which is prior to itself, having become an image of it and a representation or external copy of it. If, therefore, it is everything intellectually, soul will also be all things spiritually; and if Intellect is everything as a pattern, soul will be in the manner of an image; and if Intellect is everything in combination and unity, soul will be in a divided way.

Since Plato also had understood that, he constituted soul from all things, and divided it according to numbers, and linked it to proportions and harmonic ratios; and he related to it the first basic principles, which are responsible for the figures (I mean the straight and the curved), and he sets in motion the circles, which are in it intellectually.

Then everything mathematical is first of all in the soul, and before numbers there are the numbers which set themselves in motion, and before the seen figures there are the lifegiving figures; and before the consonant and the melodic there are the actual ratios of the consonances, or harmonic ratios; and before the bodies which are moved in a circle the actual invisible orbits were established.

Soul is the integration of all things; and it is a sort of embellishment of a different kind (from the sensible) which both applies itself of itself (to things) and is applied from a basic principle which is related and akin; and it both fills itself with life of itself, and is filled by the Creator by a means which is incor­ poreal and unlocalized.

(He is not far from every one of us; in Him we live, move and have our being.)

When He puts forward and expounds its ratios.

He then also reveals all bodies of knowledge and all virtues. Therefore, Soul has its essence in these species; and it must not be supposed that Number, which is in it, is a multiplicity of units, nor that the form and idea of those things which may be scattered in space must be understood as corporeal; but all things must be taken as lifegiving and intellectual, and as the first originals of visible numbers and figures and ratios and motions.

Here we must follow Timaeus, who integrates and completes the whole source and structure from the mathematical types, and locates in it the causes of all things.

Those 7 terms of all numbers pre-existed in it, as far as cause is concerned.

The basic principles of the figures have been located in it in the architectonic or structural sense; and the first and chiefmost of all the motions, which surrounds and rouses all other motions, existed along with it.

For of all that moves, the basic principle is the circle, and circular motion. Therefore, the concepts of mathematical things, which integrate souls, are essential and self-moving; and Soul putting them forward and propagating them and unfolding them, causes the whole variety of mathematical knowledge to persist.

For it will never happen that it ceases to engender and bring to light one thing after another continuously, while it uncovers its concepts which are indivisible in their simplicity.

For it has previously received everything, by way of chief and primeval species; and in accordance with its limitless ability, from the basic principles which it has previously adopted, it constructs the development of contemplations of every kind. So much for Proclus.

I wanted to transcribe the whole passage, not only because he set the genuine terms of the harmonies, the circles and the arcs cut off by the figures, among other mathematical things, in the soul and in the mind essentially, in such a way that these mathematical things be­ come for the soul and correspondingly the soul for them (insofar as they are separated from individual things) like a sort of essence, but also because he removes from me, who am putting forward similar views, the blame for rejecting Aristotle in both directions, and gives an outstanding recommendation to this philosophy.