De institutione musica III.2: Ex sesquitertia proportione sublatis duobus tonis toni dimidium non relinqui.
...that no half is left. II. Now if we set out those numbers which are left from the sesquitertian proportion when two tones are withdrawn, we can consider in them whether the proportion which is left after two tones is to be reckoned as a whole semitone. If this is found to be so, it is also proved that the consonance of the fourth is joined from two tones and a whole semitone. Above, then, the first term was 192; to this 256 held the sesquitertian proportion. But 216 to the first term make a tone, and again 243 to 216 holds the place of a tone. What is left of the whole fourth proportion is therefore that relation which stands in the units 243 and 256. If this is proved to be a half of a whole tone, it cannot be doubted that the fourth consists of two tones and a semitone. Since, then, it has been demonstrated that the half of a tone is placed between the sesquisextadecimal and the sesquiseptendecimal proportion, by this comparison also this proportion must be measured. For, not to go further, I take from 243 its eighteenth part. This makes 13 and a half. If I add this to the same, they make 256 and a half. It is plain, therefore, that the proportion of 256 to 243 is less than the sesquioctadecimal relation. But if the half tone is less than the sesquisextadecimal and greater than the sesquiseptendecimal proportion, and the sesquioctadecimal is less than the sesquiseptendecimal relation, and the comparison of 256 to 243, which is what is left of the fourth when two tones are withdrawn, is less than the sesquioctadecimal, there is no doubt that this proportion of two numbers is by far less than a semitone.