De institutione musica II.21: Quid oporteat praemitti, ut diapason in multiplici genere demonstretur.
With this thus distinguished, it will be demonstrated that the octave, which is the best of all consonances, is found in the multiple kind of inequality and in the relation of the double. And first, it must be demonstrated how the octave can be recognized in the multiple kind. We must therefore run briefly ahead to a small point, which, once known, will make the demonstration easier. If from any superparticular proportion one takes away a superparticular proportion continuous with it, which is of course the lesser, what remains is less than the half of the proportion which was taken away. As in the sesquialter and the sesquitertian: since the sesquialter is the greater, let us take the sesquitertian from the sesquialter; there remains the sesquioctave proportion, which when doubled does not make the whole sesquitertian proportion, but is less by that distance which is found in the semitone. But if the doubled sesquioctave comparison is not the whole sesquitertian, the single sesquioctave is not the full half of the sesquitertian proportion. And if you take the sesquiquartan from the sesquitertian, what remains does not make the half of the sesquiquartan. And the same holds in the others.