De institutione musica I.16: De consonantiis proportionum et tono et semitonio.
For if one voice is double another in highness or lowness, the diapason consonance results; if a voice is higher or lower than another by the sesquialter proportion or the sesquitertian or the sesquioctave, it yields the diapente or the diatessaron or the tone. Likewise, if a diapason, as 2 to 4, and a diapente, as 6 to 4, are joined, they produce the triple symphony, which is diapason and diapente; and if the diapason is made twice, as 2 to 4 and 4 to 8, the quadruple consonance results, which is the double diapason; and if the sesquialter and the sesquitertian, that is, diapente and diatessaron, as 2 to 3 and 3 to 4, are joined, the double concord is born, the diapason of course. For 4 to 3 holds the sesquitertian proportion, and 3 to the number two are joined by the sesquialter comparison; and the same four, set beside two, is joined to it by the double comparison; but the sesquitertian creates the diatessaron, the sesquialter proportion the diapente consonance, and the double makes the diapason symphony. The diatessaron and the diapente, therefore, join to make one diapason concord. Again, the tone cannot be divided into equal parts; why, will be clear later; for now it suffices to know that the tone is never divided into two equal parts.
And to prove this most easily, let the sesquioctave proportion be 8 and 9. No number naturally falls between them. Let us therefore multiply them by two, and twice 8 is 16, twice 9 is 18. Between 16 and 18 one number naturally falls, namely 17. Let them be set in order: 16, 17, 18. Now 16 and 18 compared hold the sesquioctave proportion and therefore the tone. But the middle number 17 does not divide this proportion into equal parts. For compared with 16 it contains 16 whole and a sixteenth part of it, namely the unit. If, however, the third number 18 is compared with it, that is, with 17, it contains it whole and a seventeenth part of it; so it does not exceed the lesser and is not exceeded by the greater by the same parts. And the lesser part is the seventeenth, the greater the sixteenth. Yet both are called semitones, not because they are altogether equal halves, but because semum is wont to be said of what does not reach wholeness. But of these, one is called the greater semitone, the other the lesser.