On Harmonic Division, or Proportionality

Harmonic Division, or Proportionality, is made when, between the terms of some proportion, a Divisor is so placed that — besides the conditions touched upon in chapter 35 — among the greater terms the greater proportions are found, and among the lesser the lesser: a property that is found only in this proportionality, which is properly called Mediocrity [a mean]. For, in sounds, the middle string of three strings drawn out according to its terms brings forth, together with its extreme strings, that sweet concord called Harmony. Whence Pietro d’Abano, commentator on the Problems of Aristotle, said very well that the mean is that which generates the harmony.

Such a Divisor, then, we shall be able to find easily when, the radical terms of the proportion we wish to divide being taken, we first divide them by the Arithmetic Proportionality; then, its extreme terms being multiplied by the mean term, the products will come to be the extremes of the Harmonic; and likewise, the greater being multiplied by the least, there will be produced the mean of such Proportionality — that is, the Divisor; for such terms will come to be placed under the conditions narrated above.

Then, if we wish to divide harmonically a Sesquialtera, contained between these radical terms, 3 and 2, we shall first divide it Arithmetically, according to the mode shown in chapter 36, and we shall have such proportionality between these terms: 6, 5, 4. We shall then reduce this to the harmonic, multiplying the 6 and the 4 by the 5, then the 6 by the 4; and from the products we shall have the sought division, contained between these terms: 30, 24, 20, as is seen in the following figure.

For as great is the proportion found between 6 and 4 — which are the differences of the harmonic terms — as is that found between 30 and 20 — which are the extremes of the Sesquialtera that was to be divided; which remains divided into a Sesquiquarta, contained between 30 and 24, and into a Sesquiquinta, contained between 24 and 20. And thus among the greater terms the greater proportions are found, and among the lesser the lesser, as is proper to such proportionality.

[Editorial note: Here Zarlino’s original contains a figure headed “Proportione da dividere secondo la Proportionalità harmonica.” It shows the Sesquialtera in its radical terms (3 . 2), then sets the two divisions side by side for comparison: the “Divisione arithmetica,” 6 . Sesquiquinta . 5 . Sesquiquarta . 4; and the “Divisione harmonica,” 30 . Sesquiquarta . 24 . Sesquiquinta . 20, each with an arc marking the “Divisore.” Beneath, labelled “Differenze inequali di i termini harmonici” (unequal differences of the harmonic terms), stand the differences 6 and 4 — themselves a Sesquialtera, the same proportion as the extremes, 30 : 20.]