Dimostrationi Harmoniche

...what we wish to call them. From this arises the diversity of the Sciences, stemming from the variety of things considered in many ways. These can be considered in three modes. Firstly, there are some whose existence depends on Matter; they cannot be defined without it. Secondly, there are some that cannot exist apart from this matter, yet such matter is not included in their definitions. Beyond these, there are others that not only do not depend on such matter according to their existence, but also not according to reason or definition; and these are the things that fall under that Divine Science which we call Metaphysics. This is because the things it considers are either never found in matter—such as God (the Blessed), and the other separate substances—or because they are not universally found within it—such as Substance, Potency, and Act, and that which philosophers call Being original: "Ente"; therefore, since these are not corporeal things, matter is not included in their definition. But the first group, which are natural things, are considered in the Natural Science called Physics original: "Fisica"; these have their existence in sensible matter and are subject to movement; they are defined by the named matter. Thus, when we define what a Man is, we say: he is a rational and mortal animal original: "Animal rationale & mortale", and the Animal is without any doubt a natural thing and has its existence among the things of nature. The other group then consists of all things belonging to the Mathematical sciences, such as points, lines, surfaces, and bodies—all those things together that belong to multitude and magnitude. Therefore, in their definition, sensible matter is not included, even though they cannot exist without it; for one does not say that a Triangle is a figure made of wood, or stone, or iron, or any other metal or material one might wish; rather, one says that it is a figure which has within it three angles equal to two right angles, even though the Triangle has its existence in matter, as you may recognize through reasoning.

ADRI. I liked this discourse very much. DESI. And I as well; all the more because I see something almost new: that in discussions of Music, one also speaks of things belonging to Philosophy. GIOS. Do you not know, Sir, that Music—being a Mathematical Science in part, and a Natural Science in part, as you have seen in my Institutions—is subject to Philosophy?
Part 1, chapter 20.

DESI. I know it for certain; but these things delight me greatly, especially when I see them applied to some beautiful purpose. GIOS. Know that this entire discourse was made so that—since Music is, as I already said in the Institutions, a Science of Relation original: "scienza di Relatione" and has as its subject the Sonorous Number The "Sonorous Number" is Zarlino's concept that music is governed by mathematical ratios applied to physical sound.—it is not without purpose that it comes to be part mathematical and part natural. Considered in the first state, since its existence depends on numbers, it is counted among the things already mentioned in the third place [Metaphysics/Mathematics]. But considered in the second way, since Sounds have their existence among natural things, it is placed among those things that possess the second [Physics]. However, I believe you remember what I call the Matter and what I call the Form of consonances.
Part 1, chapters 18 & 19.

ADRI. Continue your speaking, for we understand those things perfectly. GIOS. You must know then that although through the means of quantities we have true knowledge of musical things, since Music is not purely mathematical, musicians are sometimes accustomed to include Matter in the definition of a Consonance and of any other interval. We have seen this in many definitions where Sounds or Voices are included; these are the Matter of consonances and every other interval. This is because musicians contemplate such intervals in actuality original: "in atto"; meaning as they are heard in real time, not just as abstract concepts., which do not exist without matter. This is not what Arithmeticians or Geometers do; for the former contemplate Number and the latter measurable Quantities insofar as they are distant from matter. DESI. Is it for this reason, then, that some who have defined Consonance have said that it is the distance between a low and a high sound? GIOS. It is so; but there have also been others who have said Consonance is "formed air." However, take note that in explaining what things are (I speak to those who do not know), we use two kinds of Declaration: the first is called Definition, which explains the thing by its essential components; and the second is called Description, which does not describe the thing by its essentials, but by its accidents In Aristotelian logic, an "accident" is a property that a thing has but which is not part of its essential definition (e.g., being tall is an accident of a man).. CLAV. Give us, if you please, an example of one and the other. GIOS. The first is when we define Man and say that he is an Animal...