What Proportion Is, and Its Division

Sounds and voices, then, proportioned to one another — which without any doubt have their being from natural things — generate and bring forth in act the Consonance, governess of all melodic movement, by whose means one arrives at the use of Melodies, wherein consists the whole perfection of Music. It is indeed true that to its generation there concur (as we shall see elsewhere) two dissimilar sounds, which, according to the form and the ratio of the harmonic numbers, are proportionately distant one from another in the low and in the high.

But one must know that all those things from which sound can be born — such as strings, sinews, breathed air, and other like things — the Musician calls Distance; while the Form, or ratio, of the Numbers, which is drawn from the measure of the sounding strings, he calls Proportion.

Now Proportion divides immediately into two parts, that is, into Common and Proper. The first is the comparison of two things together, made under one and the same attribute, or univocal predicate — as when one compares Gioseffo with respect to whiteness, or to some other quality in which they [the two things] agree. The second (as Euclid would have it) is that certain relation, or correspondence, which two finite quantities of one and the same proximate genus have, whether they be equal or unequal to one another. And it has been said “of one and the same proximate genus,” because one cannot with reason say that a Line is greater, or lesser, or equal to a Surface, nor to a Body; nor that Time is greater, or lesser, or equal to a Place; but one may well say that a Line is greater, or lesser, or equal to another Line, and likewise a Body to another body, and other such cases. For (as the Philosopher [Aristotle] teaches us) comparison must be made only among things that have a single signification and that are of one and the same proximate genus, and not among those that have several significations and are of diverse genera, or, absolutely, of a single remote genus.

Nor is Proportion found only in the aforesaid quantities, but also in Weights, in Measures, and (as Plato would have it) in Powers and in Sounds, as we shall see — which proportion is never found in anything except insofar as the one is equal to, or greater than, or lesser than the other; for it is proper to Quantity to be called Equal or Unequal. And such proportion is found first in Quantity, and thereafter in the other things named.

I shall now leave off speaking of the Common, since it makes nothing to our purpose, and shall again divide the Proper into the Rational and the Irrational. And I shall say that the Rational is that which takes its denomination from numbers that contain or are contained — as from the 2, which, being compared to Unity in the relation of containing, is named the Duple proportion. Whence such quantities are called commensurable and communicating, because the one and the other can always be measured by a common measure. The Irrational, then, is that which can be denominated by no rational number — such as that of the Diameter and the Side of the Square; for there can be given no common measure, certain and entire, that measures both the one and the other; and these are therefore called incommensurable Quantities.

[Editorial note: Here Zarlino’s original contains a geometric figure: a square with a single diagonal drawn from corner to corner. The vertical side is labelled “Lato” (Side) and the diagonal “Diametro” (Diameter). The figure illustrates the example just given — the diagonal and the side of a square share no common measure, and so stand in an irrational proportion that no rational number can name.]

We must note, however, that every proportion found in numbers, which are discrete quantity, is found also in the continuous; for all numbers are commensurable and communicating, since they are at the least numbered by Unity. This does not happen in the continuous, in which are found infinite ratios that are not found in the discrete; and this because every proportion found in one genus of continuous quantity is found also in another. Whence, just as two straight lines agree the one with the other, so likewise do two Surfaces agree, two Bodies, two Times, two Places, two Sounds, and other like things; but the same does not occur in Numbers, or discrete Quantity.

Whence it is manifest that the proportions in the continuous are of greater abstraction than those found in the discrete; for every Arithmetical proportion is rational, whereas the Geometrical ones are both rational and irrational. But because the Irrational ones make nothing to our purpose, I shall set them aside and take up the Rational, which likewise divide into the proportion of equality and that of inequality.

I say, then, that the proportion of Equality is that which is found between two quantities that are equal to one another — as 1 to 1, 2 to 2, 3 to 3, and so on with the rest; or two sounds, or two lines, or two surfaces, or two bodies equal to one another. This truly makes nothing to our purpose, being by nature indivisible; for in its extremes no difference whatever is found, and one cannot say that the one quantity is greater than the other. And this comes about because Equality, or likeness, with the Musician gives birth to no consonance.

The proportion of Inequality, then — which is that of which I intend to treat — is when two quantities, the one greater than the other, are set in comparison, in such a way that the one contains, or is contained by, the other: as the Binary compared to Unity, or the contrary. And this likewise divides into two parts, that is, into that of Greater inequality and into that of Lesser. For when the greater number is compared to the lesser, if the greater contains the lesser simply, without any other consideration, then is born that of greater inequality; but comparing the lesser to the greater, if the lesser, without any other regard, is contained by the greater, then is born that of lesser inequality.