De institutione musica III.10: Regula semitonii sumendi.
It is fitting that all these consonances be duly known to the mind and to the ears. For in vain are these things gathered by reason and science unless they have become well known by use and practice. But so that what we began in the instruction of music may be judged not at once by the ears, which belongs to those already advanced in music, but for the present by reason, we shall give one example of finding a space which seems somewhat more difficult, namely that of the lesser semitone, so that it may be found in both directions, upward and downward, in a fixed order. Let there be a fourth AB. Around the consonance AB, then, we must lead a lesser semitone both to the lower and to the higher part. I raise, therefore, the fourth BC. I lower again the fifth CD. BD will therefore be a tone. For the consonance of the fourth is exceeded by the consonance of the fifth by a tone, and the space CB is exceeded by the space DC by the space BD. Again I raise a fourth DE, and lower a fifth EF. DF is therefore a tone. But BD was also a tone. AF is therefore the lesser semitone, which is left when the two tones FD and BD are subtracted from the space of the fourth AB. Again I lower a fourth AG, and raise a fifth GH. AH will therefore be a tone. But AF was a semitone. FH will therefore be the apotome. Again I lower a fourth HK, and raise a fifth KL. HL is therefore a tone. But the tone was HA; the lesser semitone is therefore LB. But DB was a tone; LD will therefore be the apotome. Again I raise a fourth FM; BM is therefore a semitone. I lower a fourth LN; NA is therefore a semitone. By means of consonances, therefore, two semitones have been taken around the fourth AB, BM upward and NA downward, and the whole MN is less than a fifth; for it consists of five semitones and a doubled apotome, therefore of two tones and three lesser semitones. And since two semitones cannot fill one tone, but a comma is left, the whole space MN is less than the space of the consonance of the fifth by one comma, which the careful reader will most easily understand.
But since we have said a little beforehand about the nature of the comma, we must not shrink from also showing in what proportion that same comma is contained—for the comma is the last thing that the hearing can comprehend—and we must say how many commas the lesser semitone and the greater semitone are each seen to consist of, and how many commas the tone itself is again joined of. And first let a fitting beginning be taken from here.