De institutione musica II.24: Demonstratio diapente et diatessaron in maximis superparticularibus esse.
This also must necessarily be added, that if the fifth and the fourth hold superparticular proportions, they are placed in the greatest superparticular proportions. The greatest are the sesquialter and the sesquitertian. This is proved in the following way. For if the consonances of the fifth and the fourth were placed in proportions smaller than the sesquialter or the sesquitertian, there is no doubt that, just as any other superparticular proportions besides the sesquialter and sesquitertian, when joined, do not make one double, so the fifth and the fourth would by no reasoning complete one octave. For since it has been shown that the octave is in the double proportion, and the double proportion is composed of the sesquialter and the sesquitertian, and the octave is joined from the fourth and the fifth, there is no doubt that, if we establish the whole octave in the double, the fifth and the fourth must be placed in the sesquialter and sesquitertian proportions. For otherwise, when joined, they will not be able to complete the octave, a consonance which consists in the double proportion, unless they stand in these two proportions, the sesquialter and the sesquitertian. For other superparticular proportions will by no reasoning make this union.