For What Reason Music Is Called Subalternate to Arithmetic, and Mean Between Mathematical and Natural Science

Music as a Subalternate Science

Since the science of Music takes (as we have been able to see) from Arithmetic the Numbers, and from Geometry the measurable Quantities — that is, the Sonorous Bodies — for this reason it makes itself subject to the two named sciences and is called a subalternate science.

Now it is to be known that there are two sorts of sciences: for some are called Principal or Subalternating, and some Non-principal or Subalternate. The first are those which depend on principles known by natural light and sensitive cognition — as Arithmetic and Geometry — which have certain principles known through the cognition of certain terms acquired by way of the senses: such as that a Line is length without breadth, which is a proper principle of Geometry; and that Number is a multitude composed of several unities, which is the proper principle of Arithmetic; beyond the common principles, which are those that say the whole is greater than the part, the part is lesser than its whole, and many others from which the Arithmetician and the Geometer draw their conclusions.

The second sort are those which, beyond their own proper principles acquired through the senses, have some others that proceed from the principles known in one of the superior and principal sciences — and these are called Subalternate to the first: as Perspective is to Geometry, since beyond its own proper principles it has some others that are known and approved in the science superior to it, which is Geometry.

The Subalternate Takes the Same Subject with an Added Accident

And the non-principal and subalternate is of such a nature that it takes from the principal the same subject — but by its own difference adds an accident to it. For if it were otherwise, there would be between the one and the other some difference of subject, as is seen in Perspective, which takes for its subject the Line in itself — which Geometry also uses — and adds to it by its accident Visibility; and so the Visual Line comes to be its subject.

Music Subalternate to Arithmetic

The same occurs in Music: for having in common with Arithmetic the Number as its subject, and adding to it by its own difference Sonority, it makes itself subalternate to Arithmetic, holding the Sonorous Number as its subject. Nor does Music have only its own proper principles — it also takes others from Arithmetic, through the means of its demonstrations, for by them we then have the true knowledge of the science.

It is indeed true that such principles and means are not all the conclusions that are found in Arithmetic, but only a part of them — those which the Musician requires — and these are of Relation, that is of Proportions: and this in order to show the properties of the Sonorous Numbers, which also serves our purpose. Wherefore we too shall cite only those conclusions that are necessary to us, and shall apply them to Sound or to Voice, which are considered by natural science (as the Philosopher demonstrates).

Music Also Subalternate to Natural Science

And I shall have the boldness to say that Music is subalternate not only to Mathematics but to Natural Science as well — not as regards the part of Numbers, but rather as regards the part of Sound, which is natural, from which arises every modulation, every consonance, every harmony, and every melody. Which thing is confirmed also by Avicenna, saying that Music has its principles from natural science and from the science of numbers.

And just as in natural things no thing is perfect while it remains in potency — but only when it is reduced to act — so Music cannot be perfect unless it is made to sound through the medium of natural or artificial instruments. Which thing cannot be done with Number alone, nor with Voices alone, but only by accompanying both these and that together — especially since Number is inseparable from consonance.

Music as Mean Between the Two Sciences

For this reason therefore it will be manifest that Music can be called neither simply mathematical nor simply natural, but rather part natural and part mathematical, and consequently mean between the one and the other.

But because from natural science the Musician has the account of the matter of Consonance — which are the Sounds and the Voices — and from Mathematics he has the account of its form, that is of its proportion: therefore, since all things are to be denominated from the nobler thing, we more reasonably say Music to be a mathematical science than a natural one — since form is nobler than matter.