On the Production of the Multiple Genus

Although the said five last-named genera of the proportions of greater inequality (as we have seen above) are finite, one must not therefore think that their species are finite; for, in the manner of numbers (following their natural order to infinity), they can be increased infinitely. And although such species may be infinite, nonetheless Music contents itself with a small portion that is finite and nearer to simplicity, and does not admit the infinite; for whatsoever thing is farther from its origin is less pure and less simple, and is less grasped by sense and less understood by the intellect — just as the contrary comes about when it is nearer, for then not only does sense comprehend it, but the intellect also apprehends it.

Whence one sees in numbers that, the farther they are from Unity, which is simple, the less simple and less pure they are, and the less grasped by sense and understood by the intellect; but on the contrary, the nearer they are, the simpler they are found to be, and the better known to the senses and to the intellect, since they partake of that simplicity. The same befalls the extreme sounds, or voices, of any consonance: the nearer they are to one another and united, the more intelligible they are; but if it happens that they extend too far into the high or into the low, sense abhors it and cannot have so ready a cognition of it — for so great a distance, whether by natural or by artificial instruments, is grasped only with difficulty. And although toward the high and toward the low they might extend a great deal, nonetheless they could proceed no farther than nature and art permit.

But because all harmonic sounds, which are rational — that is, which have between them a determinate and rational interval, or proportion — are necessarily subject to the reason of number (since their extremes, compared one to another, necessarily fall under the reason of one of the species of the named genera), therefore, having reasoned thus far concerning them, I shall now come to reason of the manner in which their species are generated. Whence, beginning from the first, which is simpler than every other and called Multiple, we shall be able to gain cognition of all its species by first laying out the natural order of the Numbers, beginning from Unity and proceeding to infinity, were it needful; and then making the comparison of the Binary, the Ternary, the Quaternary, and the other numbers, in order, to that same Unity. And doing thus, we shall find in each relation various species of proportions: for, comparing the Binary to Unity, such proportion will be called Duple, after its Denominator, which is 2; then, comparing the Ternary, there will arise a proportion that will be named Triple, likewise after its Denominator, which is 3; and so following in order — so that, always making the comparison of each number to Unity, we shall have in this way the species of the first genus, called Multiple, as are those set out below.

[Editorial note: Here Zarlino’s original contains a woodcut wheel diagram of the species of the multiple genus. Unity (1) stands at the centre; nine spokes radiate outward, each naming a species and bearing its number at the rim — Dupla (2), Tripla (3), Quadrupla (4), Quintupla (5), Sestupla (6), Settupla (7), Ottupla (8), Nonupla (9), and Decupla (10) — each being the proportion of that number to Unity (2:1, 3:1, and so on up to 10:1).]