The Essence of the Harmonic Proportions, Both Sensible and Intelligible

Being on the point of examining the essence of the harmonies, I find myself in doubt whether it will be clearer

Is it better to seek the opinions of the ancients first on and then to compare them with mine, or to start by expounding my own opinion.

The former method in philosophy is accepted by everyone, and is frequently recommended by Aristotle:

the latter seems more suitable for the matter in hand.

For few have attempted to examine the nature of the harmonies as a class; and if anything has been said on the substance of the mathematical types and classes, whatever can be applied to the harmonies as a class must necessarily in the philosophy of the Gentiles be full of obscurity, and on that account the mind of the reader must be tossed with doubt between love for such pretty theories and suspicion of falsehood.

In the company of Christians, and for anyone who embraces with a firm faith the sacrosanct mystery of the Trinity and the origin of all things in the Mosaic story, the heads of the examination can both be expounded more clearly, and find the minds of readers more inclined to believe them.

Therefore, to organize the procedure for stating them, we shall have to start with the classification. For sensible harmony, or things which are analogous to it,is one thing, harmony which is apart from and purified of sensible things is another.

The former are many, both in respect of their subjects, which are different in kind, and individually; but genuine harmony which is apart from sensible subjects is one and the same in whatever kind. For instance, the harmony which arises from double proportion is one and the same kind.

If it is in sounds, it is called a diapason; if in r a d ia tio n ,it is named the aspect of opposition; and in the musical system indeed it is either further up or down, it is either higher or lower in pitch, it is either in human voices or in sounds from instruments.

Nor is it less various in the study of the heavens, for it is the same whether it belongs to Saturn andjupiter or to another couple, whether it is among the signs around the equinoxes or those near the solstices.

On each kind of harmony, therefore, the question is, what is their basis, each in its own right, whether that basis is in themselves or in other things.

As far as sensible harmonies are concerned, then, the following four features are involved in their essence: I. Two sensible things of the same kind, and of a certain size, so that they can be com pared with each other in respect of size. 2. The soul which compares them.

3. The reception of the sensible things within [the soul].

4. An appropriate proportion, which is defined as a harmony. If one of these is taken away, the sensible harmony is taken away.

For it is easy to understand that the nature of harmony is not to be defined by means of sensible things alone, such as a sound or a ray from a star.

For a sound is one thing: a definite order among different sounds is another. Now I say order at this point, not with reference to some physical point, not with reference to time, but with reference to height and depth of pitch.

Therefore, there can be different sounds, but unless there is a definite order among them, which is defined by definite proportions, as a matter of mathematics, there will be no harmony among the parts.

On the other hand, if the sound is taken away, what audible harmony, or if the rays of the planets are taken away, what harmony between the configurations could be devised? Furthermore, as musical harmony is not a sound, but order among several sounds, it follows from that that it is in the category of relations. For the order of which we are speaking here is a relation, and the things which are ordered are related to each other.

Therefore, sounds are a harmony in virtue of something which is an accident, inasmuch as it is found in the things subject to it, and does not exist separately, and its absence does not involve the dissolution of the things subject to it.

Second, just as quantity is universally inseparable from the bodies to which it belongs, but can quite well be increased and diminished along with the bodies themselves, and yet is an accident; in the same way also the actual order among sounds, to which we have made harmony subject as to its category, cannot be taken away from sounds which are more than one in number and differentiated from each other by the quantity of their height and depth in pitch. For either they will cease to be more than one, or if they are more than one, there will be an order among them in respect of excess and deficiency— a changeable order indeed, if one of the things which are related by being subject to it changes.

Therefore, order goes hand in hand with quantities, and in the same species as number.

Third, just as number is, it is defined as being a multiplicity assembled from units, in which Aristotle’* recognizes something analogous to matter, that is to say its units, and something formal, the conception in the mind, which recognizes one multiplicity of those individual units, in whatever respect it may be separated from the rest.

Thus the same philosopher says elsewhere that if the soul which counts is taken away, all number is taken away, but not the individual units.

Hence number is found in many things themselves materially, but is nothing apart from them, unless a mind is present to count. For only then is number apart from things themselves, and abstracted from things, something different from them, that is to say it is the concep- tion in the mind of the plurality of indivisible things.

In the same way also the actual order of sounds and of other sensible things with which we are here dealing is nothing other than several sounds, unless mind is present, comparing with each other sounds which are different in pitch; and in general every relation is nothing without mind apart from the things which it relates, because they do not have the relation which they are said to have unless the presence of some mind is assumed, to relate one to another.

Therefore, what is true in general of order and of relation is to be presumed by far the most strongly of harmony, which is based on proportion, and on the counting of parts which are equal in quantity.

That is to say, for some sensible harmony to exist, and for its essence to be possible, there must be in addition to two sensible terms a soul as well which compares them.

For if that is taken away, there will indeed be two terms which are sensible things, but they will not be a single harmony, which is a thing of reason.