De institutione musica III.9: De toni partibus per consonantias sumendis.
But enough of these things. Now it seems we must say how, by means of the musical consonances, we can at will extend or lower prescribed spaces. Let this be done by lines, and let the lines we draw be taken in the place of the voice. But let the reasoning itself now show the matter. Let it be proposed to take the space of a tone by means of consonances, upward and downward. Let there be a sound B; from this I raise another sound, which is distant from it by the space of a fifth, to C. From this I lower the consonance of a fourth to D; and since the tone makes the difference between the fifth and the fourth, the space BD is found to be a tone.
In the downward direction we shall produce the tone thus. From B I raise a fourth to F, and from F I lower a fifth to E. EB will therefore be a tone. The careful reader will observe, then, that DB is a tone made upward, and EB downward.
Let it be proposed to take the lesser part of a tone by means of consonances, upward and downward. The lesser part of a tone is the space by which the consonance of the fourth exceeds two tones. Let there be a sound A. I raise a fourth from A to B. Again I raise a fourth from B to C. And from C I lower a fifth to D. BD is therefore a tone. Again from D I raise a fourth to E. I lower again from E a fifth to F. DF is therefore a tone. There are therefore two tones, BD and DF. And BA was a whole fourth; FA will therefore be the lesser part of a tone, which is called a semitone.
In the downward direction thus. Let there be a sound A. I raise two tones by means of consonances to G, and I lower a fourth from G to K. KA will therefore be the lesser part of the semitone, which was to be done.
If we take a fourth away from three tones, the apotome is what remains. Let there be three tones AB, BC, CD. From these let the fourth AE be taken away. EC will therefore be the lesser semitone, and the apotome is ED. This apotome, if convenient, we shall take thus. And first upward. I raise the three tones from A, those which are AB, and from B to C I lower the consonance of a fourth, and CA is the apotome that remains of the three tones. If we wish to make the same space toward the low sound, it is done thus: Let there be a sound A. I raise the lesser semitone, that is AD; I lower from D the tone DE. AE will therefore be the apotome we seek.
Let it be proposed to take the comma upward. Let there be a sound A. I raise the apotome AB; I lower the lesser semitone BC. And since the lesser semitone is less than the apotome by a comma, the comma will be CA. Again, downward, in this way. I raise from the sound A the lesser semitone, that is AD, and from D I lower the apotome, that is DE. EA will therefore be the comma.