On the Multiple-superparticular Genus

The fourth genus, called Multiple-superparticular, arises by adding the lesser term of any proportion of the Superparticular genus to the greater, always adding the same lesser to the number that comes from such addition. Whence, if we add the Binary — the lesser term of the Sesquialtera — to the greater, which is the Ternary, there will come the Quinary; to which, the same Binary being likewise added, will arise the Septenary; and so the others to infinity. So that, observing the same rule in the others, one may have infinite species, as may be understood in the figure set below.

[Editorial note: Here Zarlino’s original contains a woodcut wheel diagram of the multiple-superparticular genus, in the manner of the earlier genus wheels. It is divided into three families — labelled “prima specie,” “Seconda specie,” and “Terza specie” — built respectively upon the Sesquialtera, Sesquiterza, and Sesquiquarta, whose fixed lesser terms (2, 3, and 4) stand at the centre. Around the rim each family’s species are named: Duplasesquialtera (5:2), Triplasesquialtera (7:2), and Quadruplasesquialtera (9:2); Duplasesquiterza (7:3), Triplasesquiterza (10:3), and Quadruplasesquiterza (13:3); Duplasesquiquarta (9:4), Triplasesquiquarta (13:4), and Quadruplasesquiquarta (17:4) — the greater terms (5, 7, 9, 10, 13, 17) being set at the rim.]