De institutione musica II.28: De semitonio, in quibus minimis numeris constet.
For semitones seem to have been so named, not because they are truly halves of tones, but because they are not whole tones; and the measure of this space, which we now call a semitone, but which among the ancients was called limma or diesis, is this. When two sesquioctave relations, which are those of tones, are taken from the sesquitertian proportion, which is the fourth, there remains the space which is called a semitone. Let us therefore seek two tones set down in continuous arrangement. But since these, as has been said, consist in the sesquioctave proportion, and we cannot employ two continuous sesquioctave proportions unless that multiple is found from which they can be derived, let unity be first and its eightfold, eight, the first octuple. From this I shall be able to derive one sesquioctave. But since we seek two, let eight times eight be made, and from it 64 be unfolded. This, then, will be the second octuple, from which we can draw out two sesquioctave proportions. For eight, which is the eighth part of 64 units, added to them completes the whole sum, 72. If its own eighth part, which is nine, is likewise added to this, they make 81. And these two continuous tones will be written in the principal arrangement: 64, 72, 81. Now let us seek the sesquitertian of 64 units. But since 64 is shown to have no third part, if all these numbers are multiplied by three, they at once receive a third part, and all will remain in the same proportion in which they were before the multiplier three was brought to them. Let three times 64 be made, that is, 192. A third of this, 64, added to the same, will make 256. This will therefore be the sesquitertian proportion, holding the consonance of the fourth. Now let us place in their fixed order the two sesquioctave proportions, held within two numbers, in relation to 192. Let three times 72 be made, that is, 216; again three times 81, which is 243. Let these be placed between the two terms written above in this way: 192, 216, 243, 256. In this arrangement of proportions, therefore, the first number to the last constitutes the consonance of the fourth, and the same first to the second and the second to the third continue twin tones. The space that remains, then, is from 243 to 256, in which least numbers the form of the semitone consists.