Of the Properties of the Senary Number and of Its Parts; and How within Them Every Musical Consonance Is Found

The First Property: All Consonances within the Senario

Although many are the properties of the Senary number, nonetheless so as not to go on too long I shall recount only those which are relevant to our purpose. And the first will be that it is among the perfect numbers the first, and contains within itself parts which are proportionate to each other in such a way that, taking any two of them whatever, they have such a relation that they give us the ratio or form of one of the proportions of the musical consonances — whether simple or compound — as can be seen in the following figure.

[Site Editorial Note: Here Zarlino’s original text contains a diagram of overlapping concentric circles mapping all the consonances derivable from the six numbers 1–6. The numbers 1, 2, 3, 4, 5, and 6 are placed within the innermost circles. Labelled “Numeri” (Numbers) on the left and “Sonori” (Sonorous) and “Harmonici” (Harmonic) on the right, the diagram shows how each pair of numbers from the Senario yields a musical interval: 2:1 = Diapason, 3:2 = Diapente, 4:3 = Diatessaron, 5:4 = Ditone, 6:5 = Semiditone. The outer rings then display all compound consonances derivable from these same six numbers: Diapason diapente (3:1), Disdiapason (4:1), Diapason with the Ditone, Diapason with the Semiditone, and the Disdiapason with the Ditone and Semiditone respectively. The diagram demonstrates visually that the Senary contains within itself the complete system of musical consonance.]

The Second Property: Harmonic Division of the Perfect Consonances

The parts of the Senario are furthermore arranged and ordered in such a way that the forms of each of the two greater simple consonances — which are called Perfect by musicians — being contained within the parts of the Ternary, are divided into two parts in harmonic proportionality by a mean term.

For finding the Diapason first in the form and proportion that is between 2 and 1, without any mean, it is then between 4 and 1 divided into two parts — that is, into two consonances — by the Ternary: in the Diatessaron first, which is found between 4 and 3, and in the Diapente placed between 3 and 2.

This Diapente is then found between 6 and 4, divided by 5 into two consonant parts — that is, into a Ditone contained between 5 and 4, and a Semiditone contained between 6 and 5.

Beyond this is seen the major Hexachord, contained in such order between the terms 5 and 3 — which I say to be a consonance composed of the Diatessaron and the Ditone, because it is contained between terms mediated by 4, as can be seen in the figure shown.

The Third Property: Products and Dissonances

And these parts are arranged in such a way that if one were to take six strings on any instrument whatever, tuned according to the ratio of the numbers shown, and strike them together, in the sounds that would arise from the said strings not only would no discrepancy be heard, but from them there would result such a harmony that the hearing would take the greatest pleasure from it — and the contrary would occur if such order were altered in any part.

These parts have beyond this yet another property: that multiplied one by the other in all the ways possible, and the products placed in order, there will be found without any doubt among them harmonic relation, comparing the greater to the nearest lesser. And if to this order we add the square of each part — that is, the products of its multiplication by itself, placing them in the aforesaid order in their proper place according as they are arranged in natural disposition — we shall have not only the ratio of every consonance apt for harmonies and melodies, but the ratios of the Dissonances as well — by which I mean the forms of the Dissonant intervals, which are the Tones, and the major and minor Semitones: differences of the aforesaid consonances, since they show how much the one surpasses or is surpassed by the other. And these differences are not only useful but necessary in modulations, as we shall see. All of which can be seen in order in the following figure.

[Site Editorial Note: Here Zarlino’s original text contains a wheel diagram with “NVMERI” (Numbers) on the left and “SONORI” (Sonorous) on the right. The inner ring carries the six numbers of the Senario (1–6) along with their squares (36 at the top). Radiating outward, each segment is labelled with a musical interval — consonances on the upper-left half (Diapason, Diapente, Diatessaron, Ditone, Semiditone) and dissonances on the lower-right half (Tuono maggiore, Tuono minore, Semituono maggiore, Semituono minore). The outer ring of numbers (10, 11, 12, 15, 18, 20, 25, 30, 36) represents the products of multiplying the parts of the Senario together. The diagram demonstrates that all musical intervals — both consonant and dissonant — are generated by the products of the six numbers 1–6, confirming that the Senario contains the complete material of musical proportion.]

These are therefore the properties of the Senary number and of its parts, which it is impossible to find in any other number whether lesser or greater than it.