De institutione musica II.23: Demonstratio diapente, diatessaron et tonum in superparticulari esse.
...is superparticular. XXIII. It remains, then, for us to show that the fifth and the fourth and the tone are to be placed in the superparticular. For although this also became quite clear, in a certain manner of reasoning, in the first proof by which we showed that the octave is not to be placed in the superparticular kind, let us nevertheless treat of it singly and more carefully. For if anyone says that these relations are not to be placed in the superparticular, he will admit that they must be placed in the multiple kind. For why they cannot be placed in the superpartient or the other mixed kinds has, I think, been explained above. Let them be placed, then, if it can be done, in the multiple kind. And since the consonance of the fourth is the lesser, the fifth the greater, let the fourth be fitted to the double and the fifth to the triple proportion of multiplicity. For it is likely that, as the consonance of the fourth is continuous with the consonance of the fifth, so, if the fourth is established in the double, the fifth is placed in the continuation of the double, that is, in the triple. The tone, however, since it is placed after the fourth in the musical relations, should doubtless be placed in that proportion which is less than the double. But this cannot be found in the kind of multiplicity. It remains, then, that it falls into the relation of the superparticular. Let the first, then, that is, the sesquialter, be the proportion of the tone. For if we take the double from the triple, what remains is sesquialter. But if the fourth is double and the fifth triple, and when the fourth is taken from the fifth the tone is what remains, there can be no doubt that the tone ought to be established in the sesquialter proportion. But two sesquialter proportions exceed the double, as anyone instructed in arithmetic can gather for himself. Two tones, therefore, will exceed the fourth, which is unfitting. For the fourth exceeds two tones by the space of a semitone. It cannot therefore be that the fifth and the fourth are not placed in the superparticular kind of inequality. But if anyone should write that the tone also is in the multiple kind, since the tone is less than the fourth, and the fourth less than the fifth, let the fifth be placed in the quadruple, the fourth in the triple, the tone in the double. But the fifth consists of the fourth and a tone; according to this reasoning, therefore, the quadruple will consist of the triple and the double, which cannot be.
Again, let the fourth be established in the triple and the fifth in the quadruple. If then we take the triple from the quadruple, the sesquitertian is left. Again, if you subtract the fourth from the consonance of the fifth, the tone remains. According to this reasoning, therefore, the tone will consist in the sesquitertian proportion. But three sesquitertians are less than one triple; three tones, therefore, will by no reasoning make up one fourth, which is utterly false. For two tones and a semitone make up the consonance of the fourth, less a little. From these things, therefore, it is demonstrated that the consonance of the fourth is not multiple. I say, moreover, that the consonance of the fifth also cannot be placed in the multiple kind. For if it is established in it, since the continuous lesser consonance, that is, the fourth, is less than it, the fifth will not be placed in the least multiple, that is, in the double, so that there may be a place in which the consonance of the fourth could be fitted. But the consonance of the fourth is not of the multiple kind; therefore the fifth also cannot be fitted to a greater relation of the multiple than the double, which is the least. Let the fifth, then, be in the least, namely the double. The fourth, which is lesser, cannot be fitted in the multiple—for there is nothing less than the double—let it be, therefore, sesquialter, and the tone sesquitertian; for it will be placed in the continuous proportion. But two sesquitertians are greater than one sesquialter. Two tones, therefore, will exceed one consonance of the fourth, which will happen by no reasoning. From these things, therefore, it is proved that the fifth and the fourth cannot be placed in the multiple kind. Therefore they are rightly placed in the superparticular kind of inequality.