On the Nature and Properties of the Named Genera

From what has been shown above, then, one may understand that the genera and the species of the proportions of lesser inequality arise among the Numbers in the very same manner as do those of greater; nor is any other difference found between the one and the other, save that in the one the comparison is made of the lesser term to the greater, inasmuch as the one is contained by the other, and in the other the comparison is made of the greater term to the lesser, inasmuch as the one contains the other. And thus both that of greater and that of lesser inequality come to be produced at one time, and to be in the same subject.

But in my judgment the proportions of lesser inequality may be considered otherwise, and also called Rational (so to speak) and Privative, and those of greater, Real and Positive. And for a greater understanding of this, and also to know the nature of these genera, one must understand that, Equality being as it were the element of the proportions, it comes to be the principle of Inequality (as Boethius and Jordanus [Nemorarius] would have it), and to hold the middle place between the genus of greater inequality and that of lesser. And this being so, it is by its nature simple; for (as may be seen), whether multiplied or divided, the proportion that is found in the whole is found also in each of its parts; and it is ever permanent, and retains its being in whatsoever genus of inequality.

This is plainly seen to be true; for by removing a Dupla from another Dupla in the genus of greater inequality (in the manner we shall see further on), and likewise in that of lesser a Subdupla from another, one comes immediately to Equality; for (according to the opinion of Boethius) every Inequality resolves into Equality, as into the element of its own genus. Which does not happen with the proportions of inequality, which are mutable; for, multiplied or divided, the proportions of the whole are different from those of their parts; and the greater proportions have no place among the terms of the lesser — as may be seen from the Dupla, which, being greater than the Sesquialtera, has no place among its terms.

As is manifest; for, wishing to take the Dupla — contained between these terms, 2 and 1 — from the Sesquialtera — contained between these, 3 and 2 — in the manner I intend to show, there arises the Subsesquiterza between these two, 3 and 4, contained in the second genus of lesser inequality, called Subsuperparticular; which, being of a genus different from the two first proposed, gives manifest sign that the Sesquialtera is deprived of just so much quantity as that by which the Sesquialtera is exceeded by the Dupla — that is, it is deprived of a Sesquiterza. And this is most true; for, adding the Sesquialtera to the Sesquiterza, immediately the Dupla arises. Whence the Subsesquiterza comes to be only the ratio of that proportion which is lacking between the extremes of the Sesquialtera, for it to ascend to the sum and quantity of the Dupla; which defect is made manifest by the particle “Sub” that is added to it, which in composition sometimes denotes diminution — whence, from its effect, we may call it Privative.

I say Privative, not because it has the power to deprive any proportion of its quantity, but because it declares the proportion to which it is added to be deprived in its terms, and diminished by just so much quantity as is its denomination. And this is not said beside the purpose; for just as it is impossible that from a lesser number one could in fact take away any greater, so likewise it is impossible that from a proportion which is lesser one could in fact remove a greater — it being needful that the quantity from which another is taken be either greater than, or equal to, that which we intend to remove. Therefore, operating in the manner I am about to show, from a Dupla we shall always be able to take some Sesquialtera, and there will remain over a Sesquiterza; and from a Sesquialtera we shall be able to remove another, and there will come Equality. But we shall never be able to take a Dupla from a Sesquialtera without some quantity being lacking — which will always appear in the product of subtracting the one from the other, as we shall see, and will demonstrate such a deficiency — the Dupla being greater than it by a Sesquiterza, and the Sesquialtera diminished by that quantity, as has been seen.

Whence no one ought to marvel if I shall liken the proportions of greater inequality to Habit, having called them Positive — since they give the reason of the proportions, that is, of the form and of the being of a real, determinate subject — and those of lesser to Privation, naming them Rational and Privative; for they deny the proportion that they represent in the named subject, and are deprived of one of their real terms, because they do not pass beyond Equality, but are lesser than it. Whence, the genus of greater inequality being different from and opposed to the genus of lesser — taken in this way — it is necessary that the one and the other be considered under different aspects: that is, the first under the aspect of Habit, or of Position, and the second under the aspect of Privation, as I have said.

And therefore they ought also to be considered as two opposites corresponding to one another in the third mode of Opposition; for the genera and the species subordinate to the one correspond (considered under the aspect of Habit) to the genera and the species subordinate to the other, considered under the aspect of Privation — almost in the same way that Ignorance corresponds to Knowledge, Darkness to Light, and the like. They ought also to be considered as two opposites corresponding to their mean — that is, to Equality — which is, as it were, the subject of the habit and of the privation, since about it such things come to pass. Nor would I have said this without some foundation; for just as the subject of a non-natural habit and of an imperfect privation is apt to receive now the one, now the other, by succession — and retains that which is presented to it until it is deprived of it — as we see of the Air, which is apt to receive now the light and now the darkness, and is lucid for just so long as the light stays near it and does not separate from it: so Equality is apt to receive now the proportion of greater, now that of lesser inequality.

And just as the subject maintains in its quality the thing that it receives, and for this reason is not altered in its substance, so Equality does not change that proportion, of whatsoever genus, that accompanies it; nor yet is it itself altered when any proportion of whatsoever genus is added to it or removed from it — its terms being (as I have shown) immutable and invariable. And because, just as in the subject there is always privation when the habit is removed, and the habit (or the aptitude) when the privation is removed, in like manner, a proportion — any whatsoever of greater inequality — being removed from Equality, there comes immediately one almost similar but contrary, of those of lesser; and that of greater inequality is introduced when that of lesser is removed from it — as, removing a Dupla from it there comes a Subdupla, and removing the Subdupla, the Dupla arises. But because every extreme has its mean, and the mean is that which is equally distant from its extremes — the two genera of inequality being two extremes equidistant from Equality — therefore I have said that Equality holds the middle place between the one and the other of the two named genera of inequality, in the manner that may be clearly seen in the figure set below.

[Editorial note: Here Zarlino’s original contains a ladder-form table headed “Principio della Inequalità” (The Principle of Inequality). Down the centre runs a column of equalities — the rungs 1—1, 2—2, 3—3 … 10—10 — labelled vertically for the proportion of Equality. In the spaces between successive rungs the proportions are named: on the left, headed “Proportioni Positive & Reali,” the major-inequality series (Dupla, Sesquialtera, Sesquiterza … Sesquinona); on the right, headed “Proportioni Privative & Rationali,” their minor-inequality counterparts (Subdupla, Subsesquialtera … Subsesquinona). The table closes “Et più oltra in infinito” (and so onward to infinity), figuring Equality as the middle term and origin from which both inequalities descend.]

And although such examples are set down only in the terms of some species of the two first genera of greater and of lesser inequality, nevertheless those of the other species are also to be understood therein, which I have omitted for brevity, thinking that these alone suffice to show what we have proposed. Yet anyone who should be desirous of seeing the other species of such genera will be able to investigate them for himself, having regard to what has been shown above.

Now, from what has been said, we may understand for what reason we may call the proportions of greater inequality Real and Positive, and those of lesser, Rational and Privative; or say also that they are two extremes between which Equality is found placed in the middle; and likewise know the nature and property of each of such genera, and what is their true office. When, therefore, we wish to name some proportion of the genus of lesser inequality, we may accompany it with that particle “Sub”; those then that are of the other genus we shall set down without such an addition. And so that the proportions of one of the two opposite genera may be known from those of the other, we shall observe this order, when it shall be needful: that we shall set the greater terms of those proportions which are of the genus of greater inequality on the left side, and the lesser on the right, in this manner, 3 . 2; and the terms of those which are of the genus of lesser inequality we shall set contrariwise, in this manner, 2 . 3; for those of Equality may be set down without any difference of place, being by their nature invariable.