On the Institution of Music (De institutione musica)

Of these original: "Quorum", referring to the types of quantity, there is a distinct and almost perfect property. For multitude the concept of discrete numbers, like 1, 2, 3, starting from a finite amount, increases and proceeds toward infinity, so that no end to its growth is ever found. It is bounded at its minimum, but is limitless at its maximum. Its beginning is unity the number 1, than which nothing is smaller. It grows through numbers and proceeds toward infinity, and no force can create a limit to prevent it from growing. But magnitude the concept of continuous size or space, like a line, on the other hand, admits a finite quantity in its measure, yet it may become infinite. For if there is a line one foot long, or of any other kind, it can be divided into two equal parts; and its half can be cut into another half; and that half again into another half, so that there is never any limit to the cutting of the magnitude. Thus, magnitude is limited at its largest scale, but becomes infinite when it begins to be divided. Conversely, number is finite at its smallest scale, but begins to be infinite as it increases. Since these things are thus infinite, philosophy treats them as if they were finite things. Within them, it finds some limit about which the mind can rightly apply its own specialized observation. For of magnitude, some things are immobile, like the earth; others are mobile, like the sphere of the heavens and whatever revolves within it with swift rotation. Of discrete quantities, some are absolute original: "per se", meaning they exist on their own, like the number 4, while others are relative original: "ad aliquid", meaning they exist in relation to another number, like "double", such as those born from mathematical operations. Geometry holds the study of immobile magnitude, while astronomy pursues the science of mobile magnitude. Arithmetic is the authority on absolute discrete quantity, while Music is proven to possess the skill related to relative quantity.

On the division of relative quantity. Chapter 4.

We have already spoken sufficiently in the Arithmetic regarding that discrete quantity which is absolute. Regarding relative quantity, there are three simple types.

One is the multiple; another is the superparticular; and the third is the superpartient. When the multiple is mixed with the superparticular and the superpartient, two others are formed from these: namely, the multiple superparticular and the multiple superpartient. The rule for all of these is as follows: If you wish to compare unity the number 1 to all numbers in their natural order, the multiple order is woven by the ratio. For two to one is double. Three to the same is triple. Four is quadruple, and in others in the same way, as the description placed below teaches.

If, however, you wish to find the superparticular proportion, compare the natural numbers to one another with a difference of one literally "with unity removed": for example, 3 to 2 is the one-and-a-half original: "sesquialter" ratio; 4 to 3 is the one-and-a-third original: "sesquitertius" ratio; 5 to 4 is the one-and-a-fourth original: "sesquiquartus" ratio; and in others in the same way, as the following description demonstrates:

You will find the superpartients in this way. Arrange the natural numbers starting from three. If you skip one step, you will note a superpartient specifically the "two-parts-over" ratio. If you skip two, a "three-parts-over" ratio; if three, a "four-parts-over" ratio; and so on for the others.

A semicircular diagram positioned below a row of Roman numerals (3 through 9). Three curved arcs connect specific numbers: the first arc connects 3 to 5 and is labeled "Two-parts-over" original: "Superbiptiens"; a ratio where the larger number contains the smaller once plus two-thirds of the smaller; the second arc connects 4 to 7 and is labeled "Three-parts-over" original: "Supertripartiens"; a ratio where the larger number contains the smaller once plus three-quarters of the smaller; the third arc connects 5 to 9 and is labeled "Four-parts-over" original: "Superquadrupartiens"; a ratio where the larger number contains the smaller once plus four-fifths of the smaller. The labels are written along the curves of the arcs.

The diligent reader will further observe the order and the proportions composed of multiples and superparticulars, or multiples and superpartients. But all these things have been explained more quickly in the Arithmetic.