On the Production of the Superpartient Genus

The species of the third genus, called Superpartient, are infinite; for some are called Superbipartient, some Supertripartient, and some Superquadripartient, proceeding thus to infinity according to the natural order of the numbers.

Whence the Superbipartient is found between two numbers differing from one another by the Binary, which are greater than it, and of which it cannot be a common measure; and such numbers must be prime to one another (Contra se primi), whose nature and property is such that they are the radical terms of whatever proportion they contain. Leaving aside, then, the Binary, as something that little serves the purpose, we shall take the Ternary and the Quinary, which are, in the natural order of the numbers, the first that observe such a law; for if we compare the greater to the lesser, we shall have the proportion called Superbipartienteterza, since 5 contains 3 once, and besides a non-aliquot part of it, that is, two third-parts.

By this same difference, between 7 and 5 is generated the proportion Superbipartientequinta, and between 9 and 7 the Superbipartientesettima; and so the other species one after another. But between 7 and 4 arises the Supertripartientequarta, which is the first species among the Supertripartients; whence it is necessary that, just as in the first ones the difference of the Binary was observed, so in these second ones that of the Ternary be observed, and in those called Superquadripartients, that of the Quaternary. By which means, observing such a rule in the others in order, one might go on to infinity, as is seen here below.

[Editorial note: Here Zarlino’s original contains a complex woodcut figure of three large interlocking rings, representing the three families of the superpartient genus named in the text — the Superbipartient, Supertripartient, and Superquadripartient — here labelled “prima specie,” “Seconda specie,” and “Terza specie.” Around the rings their individual species are named, among them Superbipartienteterza, Superbipartientequinta, Supertripartientequarta, Supertripartientesettima, Superquadripartientequinta, and Superquadripartientenona, with the governing numbers (3, 4, 5, 6, 7, 10) set in the inner field. The nested, interlocking form depicts how each family extends without end through the natural numbers.]