How the Superparticular Genus Arises

The second genus of the proportions of greater inequality arises in this manner: leaving aside only Unity in the aforesaid natural order of the numbers, and beginning from the Binary, following that order step by step — if we make the comparison of the greater number to the nearest lesser, from such comparison will be produced the Superparticular genus; of which the first species is the Sesquialtera, comparing the Ternary to the Binary. For then, the Quaternary being compared to the Ternary, there arises the second species, called Sesquiterza; and so the others in order, each of which (as I have said) is denominated by its own proper denominator, or aliquot part.

Whence one sees that if, in any proportion, the part by which the greater number exceeds the lesser is the half of that lesser, it is called Sesquialtera; and if it is the third part, it is called Sesquiterza; and, briefly, all the other species, however infinite they might be, are denominated from their parts, as may be seen in the example set out below.

[Editorial note: Here Zarlino’s original contains a woodcut wheel diagram of the species of the superparticular genus, in the manner of the multiple-genus wheel above. The natural numbers from 2 to 10 are arranged around the rim, and each scalloped segment lying between two successive numbers names a species — Sesquialtera (3:2), Sesquiterza (4:3), Sesquiquarta (5:4), Sesquiquinta (6:5), Sesquisesta (7:6), Sesquisettima (8:7), Sesquiottava (9:8), and Sesquinona (10:9) — each being the proportion of two consecutive numbers.]