De institutione musica I.6: Cur multiplicitas et superparticularitas consonantiis deputentur.
For those things are judged suited to joining together that are simple by nature. And since lowness and highness consist in quantity, those will appear to preserve most the nature of concord that can keep the property of discrete quantity. For while one quantity is discrete and another continuous, the discrete is bounded at its least but proceeds without limit toward the greater. For in it the smallest is the unit, and this is bounded, but the measure of plurality increases without limit, as with number, which, though it begins from a bounded unit, has no end of growing. Conversely, the continuous is wholly bounded, yet is diminished without limit. For a line, which is continuous, is divided by an unending partition, although its whole may be a foot or some other fixed measure. Hence number always increases without limit, while continuous quantity is diminished without limit. The multiple, then, since it has no end of growing, preserves above all the nature of number. The superparticular, since it diminishes the lesser without limit, preserves the property of continuous quantity. And it diminishes the lesser when it always contains it and with it either a half part or a third or a fourth or a fifth. For the part named from the greater number is itself always the smaller. For since the third is named from three and the fourth from four, and four exceeds three, the fourth is found to be smaller than the third. The superpartient, however, already departs in a certain way from simplicity; for it has two or three or four parts over and above, and departing from simplicity it overflows into a certain plurality of parts. Again, every multiple keeps itself in wholeness. For the double has the lesser whole twice, the triple likewise contains the lesser whole three times, and so on in the same manner. The superparticular, however, keeps nothing whole, but exceeds either by a half, or a third, or a fourth, or a fifth; yet it effects the division by single and simple parts. The superpartient inequality, though, neither keeps the whole nor removes single parts, and therefore, according to the Pythagoreans, it is not at all applied to musical consonances. Ptolemy nevertheless places this proportion also among the consonances, as I shall show later.