De institutione musica III.14: Semitonium minus maius quidem esse tribus commatibus, minus vero quattuor.
...but less than four. XIV. We therefore propose to demonstrate that the lesser semitone is greater than three commas, but less than four, which you can most easily recognize from this. Let three numbers be so arranged that they hold among themselves the proportion of the octave and that which is called the proportion of six tones. Let A be 262,144. Let five continuous tones be raised to B, and let B be 472,392; and let the consonance of the octave be referred to C, and let C be 524,288; and to D let six tones be raised, and let D be 531,441. With these so set and established, it is manifest that the comma is constituted between C and D, and that their difference is 7,153. Let this be K. Let two tones be lowered from B to E, and let E be 373,248. Again, from E I raise a fourth, which is F, 497,664. Since, then, between E and B there are two tones, and between E and F a fourth, between B and F the lesser semitone is found. For when two tones are taken from the consonance of the fourth, the lesser semitone remains, which I said above stands in the first numbers 256 and 243. If you multiply these same numbers 1,944 times (a thousand, nine hundred, forty, and four times), you will unfold the numbers B and F. These necessarily hold the same proportion as the numbers named above, which, multiplied equally by one and the same number, that is 1,944, grew. Likewise from F I raise a fourth, namely to G, and let G be 663,552. Again from the same G I lower two tones to P, and let P be 524,288. This P must show the same sound as the number C; for it has proceeded to equality with it by such reasoning. For the consonance of the octave AC, which consists of tones and two lesser semitones, is exceeded by the six tones by a comma. From the same term A, then, the number P has withdrawn by five tones and two semitones in this way. From A up to B there are gathered, of course, five tones. From B up to F the lesser semitone is noted. F and P again include the lesser semitone. A up to P has therefore produced five tones and two lesser semitones. P and C are rightly written with the same numbers. But since between F and C there is a lesser semitone, let us see what their difference is, so that we may compare it with the comma. Their difference is 26,624, and let this be M. K, then, is the difference of the comma, and M that of the lesser semitone. If, then, we increase the number K three times, the number will be 21,459, and let this be L. If you wish to multiply the same number K four times, it will be 28,612, and let this be N. M is therefore greater than L; the same M is less than N. But N has grown by the comma taken four times, L by three, and M holds the difference of the lesser semitone. It was rightly said, therefore, that the lesser semitone is less than four commas, but greater than three.