De institutione musica II.18: De consonantiarum merito vel modo secundum Nicomachum.
XVIII. But enough of these matters. Now it seems that this should be added: how the Pythagoreans prove that the musical consonances are found in the aforesaid proportions. In this matter Ptolemy evidently does not agree with them, and of him we shall speak a little later. For that must be held to be above all the first and sweetest consonance whose property the sense grasps more plainly. For whatever each thing is in itself, such it is also perceived by the sense. If therefore that consonance which consists in the double is better known to all, there is no doubt that the diapason, the octave, is the first of all consonances and rightly excels, since it precedes in being known. The remaining ones, according to the Pythagoreans, necessarily hold the order which the increases of the multiple and the decreases of the superparticular relation give them. For it has been shown that multiple inequality surpasses the superparticular proportions in the antiquity of its worth. Therefore let the natural numbers be arranged from unity up to four: 1, 2, 3, 4. Two, then, compared with one, makes the double proportion and yields the consonance of the octave, which is the greatest and, by its simplicity, the best known. But if three is compared with unity, it will sound the concord of the octave and fifth. Four, compared with unity, holds the quadruple, producing twice the octave symphony. But if three is compared with two, it supplies the fifth; if four with three, the concord of the fourth. And this is the order of these when all are compared with one another. For one comparison remains. If we compare four with two, it falls into the double proportion, which two held when compared with unity. Thus sounds are farthest apart in the double octave, when they depart from each other by an interval measured as quadruple. But sounds seem to be least far apart and still consonant when the higher exceeds the lower by a third part of the lower. And so the measure of concords stands, which can neither be extended beyond the quadruple nor contracted within the third part. And according to Nicomachus the order of the consonances is this: the first is the octave, the second the octave and fifth, the third the double octave, the fourth the fifth, the fifth the fourth.