De institutione musica III.16: Superius dictorum per numeros demonstratio.
But although it has been demonstrated by this reasoning how the tone is compared with commas, we must not weary as if sluggish, lest the tone itself be shown not to hold this comparison against commas by itself. Let A, then, be 262,144.
Let B, distant from it by five tones, be 472,392, and C, containing the symphony of the octave to A, be in numbers 524,288. And let D, differing from A by six whole tones, be 531,441. D, then, is distant from C by the comma of the sixth tone over the consonance of the octave. Let this be E, 7,153. D is distant from B by a whole tone, namely the sixth tone after five tones.
Let this be F, 59,049. If, then, I multiply E by nine, I get 64,377; if by eight, 57,224. Let that be G. But H, compared with the number F, exceeds it, and G is exceeded by it, and F is the difference of the tone, H the comma multiplied nine times, G the comma multiplied eight times. It has therefore been demonstrated that the tone is less than nine commas, and greater than eight commas.
With these premised, although it has been shown that the greater semitone differs from the lesser semitone by a comma, the same will nevertheless be proved by itself and by the numbers set below, by this reasoning. Let A be the number 497,664; let B be the number, distant from it by a lesser semitone, which has also been described above, 524,288. Let the apotome be distant from A by the number gathered in 531,441 units. And let this be C. Since AB is the lesser semitone and AC the greater, the difference of that which is B from that which is C must be sought. It is 7,153. Let this be D. But this number a while ago showed the comma. Between the greater semitone and the lesser, therefore, the comma makes the difference.
Again I propose to demonstrate that the tone is greater than two lesser semitones by a comma alone. Let A be the number 472,392. From this let a tone be raised, 531,441, and let this be D. From A let a lesser semitone be raised, which is B, and let B be 497,664. Likewise from B let another lesser semitone be raised, which is C, and let C be 524,288. Since, then, AD is a tone, and AC contains two lesser semitones, let us see what difference is constituted between the numbers C and D. It is, namely E, 7,153 units. It has therefore been demonstrated that the tone is greater than two lesser semitones by a comma.
But since all that we promised to prove has now been shown by its proper reasoning, what remains for the instruction in music is the division of the regular monochord. Since a longer treatise extends this matter, we have decided to transfer it to the discussion of a later book.