De institutione musica II.27: Diatessaron ac diapason non esse secundum Pythagoricos consonantias.
But in these matters the careful reader should observe that consonances placed upon consonances have produced certain other consonances. For the fifth and fourth joined create the octave, as has been said. And if the symphony of the fifth is again joined to this, that is, to the octave, there results the consonance named from both terms, namely the octave and fifth. If the fourth is added to it, the double octave results, which holds the quadruple proportion. What, then, if we join the consonances of the fourth and the octave? Will they, according to the Pythagoreans, produce any consonance? By no means. For at once it falls into the superpartient kind of inequality, and observes neither the order of multiplicity nor the simplicity of the superparticular. Come, let numbers be set down by which we may prove this more easily. Let there be three, of which six is the double, standing, that is, in the proportion of the octave. To this let the sesquitertian be fitted, which we said above is the fourth, as eight. For this stands to six in the proportion of the fourth. This eight, compared with three, has it twice, but, so that it may not be multiple, it has also some parts of it, and not simple ones. For it exceeds it by two units, which are two thirds of three, which we set as the first and least term. Let these terms be, then, 3, 6, 8. This also holds of what falls between two consonances continuous with each other. For it is neither a whole double, so as to give the consonance of the octave, nor a triple, so as to make the symphony of the octave and fifth. If a tone is added to it, it will at once make the triple measure of proportion. For since the octave and fifth joined to each other make the triple, and the fourth and a tone join to make the consonance of the fifth, if the fourth is added to the consonance of the octave it becomes dissonant, since between the double and the triple no multiple proportion can naturally be understood. But if I add a tone to it, the octave, fourth and tone will result, which will differ in nothing from the octave and fifth. For the fourth and the tone constitute the fifth. Let the octave be 3 and 6, the fourth 6 and 8, the tone 8 and 9, the fifth 6 and 9, the octave and fifth 3 to 9. The triple proportion will therefore be thus: 3, 6, 8, 9. But although Nicomachus says much on these matters, we have, with such brevity as we could, partly set forth the very things the Pythagoreans affirm, and partly proved, arguing certain consequences from the same, that if the octave is added to the consonance of the fourth, a consonance cannot be joined from them. What Ptolemy thinks of these matters I shall set down later. But enough of these things. Now we must consider the semitones.