De institutione musica III.13: Quod semitonium minus maius quidem sit quam XX ad XVIIII, minus vero quam XVIIII S ad XVIII S.
If a like consideration is turned to the lesser semitone, we shall most easily find its proportion also, which stands between 256 and 243. Let 256 be A, and 243 be B. Their difference is 13, C. I say that A to B holds a less proportion than 19½ to 18½. For let C measure A nineteen and a half times, that is, let C be multiplied nineteen and a half times; it makes 253½, which let be D; and compared with A it is exceeded by A by two and a half; let this difference be F, namely 2½. Again let the same difference C measure the number B eighteen and a half times, that is, let it be multiplied eighteen and a half times; it will make 240½, which let be E. E compared with B is exceeded by B by the same F, that is, two and a half. D is therefore less than A, and E less than B, by the same difference F. When F is taken from A and B, D and E result; D and E therefore hold a greater proportion between themselves than A and B. But D and E hold between themselves the same proportion as 19½ to 18½. A to B therefore holds a less proportion than 19½ to 18½, which was to be shown.
Yet the same proportion of 256 to 243 seems to be greater than that which 20 and 19 contain. For let A, B, C be the same as were described above. Let the difference C measure the term A twenty times; it will make 260, which let be D. Compared with A, it exceeds the same by four. Let this be F. Again let the same C measure B nineteen times; it will make 247. Let this be E. Compared with B, it exceeds it by the same F. The number D therefore exceeds the number A, and the number E exceeds the number B, by the same F. With F added to the numbers A and B, D and E result. The proportion of A and B is therefore greater than that of D and E. But D and E are the number C multiplied by twenty and by nineteen. The proportion of A and B, therefore, which contain the semitone, is greater than that of 20 to 19.
It has therefore been demonstrated that the lesser semitone has a greater proportion than 20 to 19 and a less than 19½ to 18½. Now let us compare the same lesser semitone with the comma, which is the last thing subject to the hearing and the last proportion.