header: Chapter III. concerning Numbers,

...that any movement of the air is not sound, but rather air affected by some external force, enclosed and compressed, is the efficient cause of sound. However, let this be the investigation of Physicists; for the Musician does not consider sound as such, or (as they say) in the abstract, but relatively, and taken under a certain comparison, by which sound differs from sound by reason of pitch and depth original: "acuminis & gravitatis"; literally sharpness and heaviness, referring to high and low musical pitches.. Since this comparison is made primarily with the help of numbers, it follows that:

CHAPTER III.

Concerning Numbers, their Proportions, and Differences.

By Numbers, I mean the Elements of Arithmetic:

The Musician uses these as handmaidens to find their intervals, comparing number with number, and inquiring into their distances, differences, excesses, and deficits.

There are many kinds of numbers. However, having left aside others which do not serve our purpose, we shall mention only the following, namely: rational, irrational; radical, and irradical. Rational numbers are those which can be divided by a certain number so that nothing remains, such as 6 with 2 and 3; or 12 with 2, 3, 4, and 6. Irrational numbers are those which cannot be measured or divided in such a way, such as 3, 5, 7, and 9 with respect to 2 In this context, "irrational" refers to numbers that are not evenly divisible by a specific divisor, rather than the modern mathematical definition of non-repeating decimals.. Radical numbers are those which cannot be reduced into smaller numbers while preserving the proportion they constitute These are what we would call "lowest terms" or "prime to each other" in a ratio, such as 2:3.; Irradical numbers, on the other hand, are the opposite.

It is established that from proportions, or the comparison of one quantity to another, various distances and differences arise of their own accord. The question is now: what is this Proportion? It is defined by Euclid Euclid (c. 300 BCE), the Greek mathematician whose "Elements" provided the foundational definitions for ratios and proportions used in music theory. and others as the relation of two quantities, one to the other, in the same proximate kind. Quantity, which is mentioned here, is of two kinds: continuous, which exists in a physical body, such as a line, a board, a surface, etc.; and discrete, which is found in some multitude, such as a people, numbers, etc. By "relation" original: "habitudinem" is understood that connection which exists from one quantity to another; furthermore, it is said: in the same proximate kind; by which we are cautioned not to establish a proportion between a continuous quantity and a discrete one—such as from a physical line to a number—but rather from line to line, and from number to number.