De institutione musica III.15: Apotomen maiorem esse quam quattuor commata, minorem quam quinque, tonum maiorem quam octo, minorem quam novem.
By the same reasoning we can find also how many commas the greater semitone, which we said above is called the apotome, comprises, in this way: Let A be 262,144; let B, distant from it by five tones, be 472,392; let D, distant from A by six tones, be 531,441. Between B and D, then, is a tone; and let the lesser semitone be distant from B to C, and let C be 497,664. There remains, therefore, between C and D the proportion of the apotome. For since BD is a tone, if you take from it the lesser semitone BC, the greater, CD, remains, which we said above is the apotome. Between D and C, then, the difference is 33,777. Let this be E. But the difference of the comma was 7,153. Let this be F. If I multiply F, that is the comma, five times, I get 35,765, and let this be G. If I multiply the same F four times, K results, which is 28,612. G is therefore greater than E, K less. But G is the comma increased five times, K four times. The difference of the apotome is E. It was rightly said, therefore, that the apotome is less than five commas, but greater than four. From this it is proved that the tone is greater than eight commas, but less than nine. For if the lesser semitone is greater than three commas but less than four, and the apotome is greater than four commas but less than five, the lesser semitone joined to the greater semitone, which is the apotome, will be altogether greater than eight commas but less than nine. But the apotome and the lesser semitone make one tone. The tone is therefore greater than eight commas, but less than nine.