On Dividing, or Partitioning, the Proportions; and What Proportionality Is

It is to be observed that, by the fourth operation, I intend nothing other than the Division, or Partitioning, of any proportion, which is done by the placing of some found number between its extremes — and it is named the Divisor — since it divides that proportion proportionately into two parts; which division the Mathematicians call Proportionality, or Progression. Whence it has seemed to me fitting to declare first what this name Proportionality imports, and then to come to the operations.

Proportionality, then, according to the mind of Euclid, is a similitude of proportions, which is found at the least in the mean of three terms that contain two proportions. And although among the Mathematicians (as Boethius shows) the proportionalities are ten — or (according to the mind of Giordano [Nemorarius]) eleven — nevertheless the first three, which are the most famous, and approved by the ancient Philosophers Pythagoras, Plato, and Aristotle, are considered and embraced by the Musician, as those which serve his purpose more than the others. Of these, the first is called Arithmetic, the second Geometric, and the third Harmonic.

And, wishing to reason somewhat of each of them, we shall first see what each one is separately. Beginning, then, from the first, I say that the Arithmetic division, or proportionality, is that which, between two terms of any proportion, will have a mean term so accommodated that, the differences of its terms being equal, its proportions will be unequal. Conversely, I say that the Geometric division, or proportionality, is that whose proportions, by virtue of the said mean term, being equal, its differences will be unequal.

The Harmonic, then, I call that which, with such a mean term, will make unequal not only its differences but its proportions also — in such a way that the same proportion that is found between its differences is found also in its extreme terms, as is seen here below.

[Editorial note: Here Zarlino’s original sets out a three-column table comparing the three proportionalities. Under “Arithmetica”: equal differences (1, 1), the terms 4 · 3 · 2 (a Sesquiterza, 4:3, then a Sesquialtera, 3:2), and unequal proportions. Under “Geometrica”: unequal differences (2, 1), the terms 4 · 2 · 1 (a Dupla and a Dupla, 4:2 and 2:1), and equal proportions. Under “Harmonica”: unequal differences (2, 1), the terms 6 · 4 · 3 (a Sesquialtera, 6:4, then a Sesquiterza, 4:3), and unequal proportions — the proportion between the differences (2:1) being the same as that between the extreme terms (6:3).]

The proportions, then, being divided regularly by one of the modes shown, it is needful to show separately in what manner we may easily find the mean term of each — which is its Divisor; therefore, beginning from the first, we shall see how the Arithmetic Divisor may be found, and in what manner every proportion may be divided by it.