De institutione musica III.3: Adversum Aristoxenum demonstrationes diatessaron consonantiam ex duobus tonis et semitonio non constare integro nec diapason tonis sex.
...nor the octave six tones.
III. But if, as Aristoxenus says, the consonance of the fourth is joined from two tones and a semitone, two consonances of the fourth will necessarily make five tones, and the fifth and fourth joined, just as they join one octave, so are equal to six tones in continuous proportion. And since a little before we set out six tones, of which the least number was 262,144, and the last number in the sixth tone was 531,441, and the fifth tone was held by 472,392: let them be set out in this way:
Now, then, let us speak of the lesser numbers, that is, of five tones. If then the fourth consisted of two tones and a semitone, and twice the fourth of five tones, when I raised a fourth from 262,144 and lowered another fourth from 472,392, the same number would be found between the two raisings or lowerings. This comes about in the following way. From the number 262,144 I raise a fourth, that is, a sesquitertian, which comes out at 349,525⅓. Again, from the numbers 472,392 I lower a sesquitertian proportion, which comes out at 354,294. Let us set out these proportions in this way, and let the first number be A, the second B, the third C, the fourth D: 349,525⅓, 354,294, 472,392.
Since, then, the term A is removed from the term D by five tones, and since the fourth is joined in two tones and a semitone, as Aristoxenus thinks, and one fourth is placed between A and B, another between C and D, the terms B and C ought not to be different, but one and the same, so that the whole of the tones would seem to consist of two consonances of the fourth. But now, since there is a difference of 4,768⅔, it is argued that the fourth is by no means joined of two tones and a semitone.