De institutione musica II.22: Demonstratio per inpossibile diapason in multiplici genere esse.
Come now, let us return to the octave. If it is not in the multiple kind of inequality, it will fall into the superparticular kind of inequality. Let the octave, then, be a superparticular proportion. Let the continuous consonance, that is, the fifth, be taken from it; the fourth remains. Twice the fourth is therefore less than one fifth, and the fourth itself does not fill the half of the consonance of the fifth, which is impossible. For it will be shown that twice the fourth exceeds the consonance of the fifth by a tone and a semitone. Therefore the octave also cannot be placed in the superparticular kind of inequality.