On the Production of the Fifth and Last Genus, called Multiple-superpartient

But if we observe the manner that we observed in the production of the Multiple-superparticular — that is, of adding the lesser term of the proportions of the Superpartient genus to the greater term, and to the product always adding that same lesser term, continuing to infinity (could it be done) — there will, by such addition, be created the Fifth and last genus, called Multiple-superpartient; of which (since it is not a very difficult matter) I shall not extend myself to reason further, it sufficing only to set down the examples, that they may be a guide and a light to the understanding of such a rule; and they will be those set below.

And just as, in the manners shown, the Superbipartienteterza, the Supertripartientequarta, and the Superquadripartientequinta are composed, so likewise are composed the other species, which (as I have said) are infinite. And what has been said of the genera and species of Greater inequality is said also of those of Lesser, whose species will be found set among their radical terms, as are the species shown above.

Whence it is to be noted that those numbers are called Radical terms, or Roots, of any proportion, of which it is impossible to find lesser numbers in that same proportion; and such numbers are prime to one another (Contraseprimi), as has been shown above, and as Euclid in Book 7 of his Elements (or Principles, as we may call them), and also Boethius in chapter 8 of the second book of the Music, make manifest.

And the Musicians, in the prolation of the singable figures, mark the Numbers of the proportions of Greater inequality in such a way that they set the greater term of the proportion they wish to show above the lesser — as, wishing to show the prolation of the Dupla, they mark it in this manner, 2/1, and that of the Sesquialtera thus, 3/2. But in those of Lesser inequality they mark such numbers contrariwise, that is, the lesser term of the proportion above the greater — as is seen in the prolation of the Subdupla and of the Subsesquialtera, which they mark in this manner, 1/2 and 2/3; and so likewise in the others in each genus.

And although I have set the examples only in the genera shown, in the radical terms of the proportions, one must not therefore believe that such proportions are not found also in other numbers — as in the Composite ones, which are not radical terms of the proportions; for the Dupla is found as much between 8 and 4, and between 12 and 6, as between 2 and 1. Which is to be understood likewise of the others, in the other genera; as in those of the Sesquialtera, which is found as much between 6 and 4 as between 3 and 2, as we shall see further on.

[Editorial note: Here Zarlino’s original contains a woodcut wheel diagram of the multiple-superpartient genus, matching the form of the multiple-superparticular wheel above. It is divided into three families — “prima specie,” “Seconda specie,” and “Terza specie” — built respectively upon the Superbipartienteterza, Supertripartientequarta, and Superquadripartientequinta, whose fixed lesser terms (3, 4, and 5) stand at the centre. Around the rim each family’s species are named: Duplasuperbipartienteterza (8:3), Triplasuperbipartienteterza (11:3), and Quadruplasuperbipartienteterza (14:3); Duplasupertripartientequarta (11:4), Triplasupertripartientequarta (15:4), and Quadruplasupertripartientequarta (19:4); Duplasuperquadripartientequinta (14:5), Triplasuperquadripartientequinta (19:5), and Quadruplasuperquadripartientequinta (24:5).]