De institutione musica II.20: Sententia Nicomachi, quae quibus consonantiis opponantur.
But Nicomachus does not think that the opposition among them is the same, but rather that, as unity in arithmetic was the principle of increase and decrease, so the symphony of the octave is the principle of the remaining ones, and that these can be constituted in a contrary division one to another. This will be easier to know if it is first seen in numbers. Let unity be set down, and let two parts flow from it, one of multiplication, the other of division, and let this be the figure:
And the progression goes on in this manner without end. For two is the double of unity; but its contrary part shows the half of unity; three is the triple and the contrary part a third; four the quadruple and the contrary part a fourth; and so in simple unity lies the principle of growing and of decreasing. Let us now turn the same to the consonances. The octave, which is double, will hold the place of the supreme principle; the remaining ones will be in the contrary division in this way: the sesquialter to the triple, the sesquitertian to the quadruple. This will be proved by the following argument. For the first sesquialter is the same as the first triple, namely of the principal unity. For three is the first triple if compared with unity, and the first sesquialter if compared with two. Again, three is the triple of that difference which it makes with two, of which it is proved to be by nature the sesquialter. Since therefore the sesquialter is rightly opposed to the triple, the consonance of the fifth is reasonably thought to be opposed to the consonance of the fifth and octave. Again, the quadruple holds the contrary division to the sesquitertian. For the first quadruple is found again to be the first sesquitertian in this way. Four is the first quadruple if compared with unity, and the first sesquitertian if compared with three. Again, it is itself the quadruple of the difference which it holds between itself and three. Hence it comes about that the sesquitertian proportion, which is the fourth, is divided in contrary fashion against the quadruple proportion, which is the double octave. But the double, since it has no opposed proportion, and is itself not the sesquialter of anything, nor is there any number to which two, which is the first double, could be joined in a superparticular proportion, goes beyond such a form of contrary proportion. And for this reason, according to Nicomachus, the octave holds the principle of the consonances in this way:
But although this is so, he says, it is nevertheless better that all the multiple proportions precede the consonances and the superparticular follow, as we described a little before. Since then a consonance is a fixed blending of two voices, a sound is the fall of a modulated voice produced by a single tension, and the same is the least particle of modulation, and every sound consists in a stroke, and every stroke arises from motion, and of motions some are equal and others unequal, and of the unequal some are very unequal, others less so, and others moderately unequal: from equality arises equality of sounds, and from inequality, namely that in which the inequalities are of moderate distance, there arise the manifest, first and simpler proportions, which are those of the multiple and superparticular, the consonances of the double, triple, quadruple, sesquialter and sesquitertian. But from the inequalities that arise in the remaining proportions, whether multiple in many ways, or not so clear, or wholly far distant from one another, dissonances come to be, and no concord of sounds is produced.