De institutione musica II.16: De armonica medietate deque ea uberior speculatio.
But having entered upon the harmonic discussion, I do not think that what can be said of it more carefully should be passed over in silence. Let the harmonic proportionality, then, be placed, and in that description let the differences of the terms be set out among themselves in the upper order.
Do you see, then, how 4 to 3 give the diatessaron consonance, how 6 to 4 agree in the diapente, how 6 to 3 mingle the diapason symphony, and how their differences again establish the same consonance? For two is double to the unit, constituted in the diapason consonance. But if the extremes are multiplied by themselves, and likewise the middle increases by its own multiplication, the numbers compared will keep the relation and concord of the tone. For three times 6 makes 18, four times 4 makes 16. But the number 18 exceeds 16 by an eighth part of the lesser. Again, the least term, if it multiplies itself, will make 9. If the greater term grows by its own multiplication, it will make 36, and these compared with each other keep the quadruple, that is, the double diapason concord. If we look at these more carefully, this will be the whole multiplication of the differences or terms among themselves. Let the least term be multiplied by the middle: it will make 12. Likewise let the least term be multiplied by the greatest: it will make 18. Let the middle term be increased by the number of the greatest: there will be 24. Again let the least term grow by itself: there will be 9; and in the same manner the middle: it will make 16. And six, which is the greatest, if it multiplies itself, will give 36. Let these be set out in order:
The diatessaron consonance, then, is sounded by 24 to 18 and 12 to 9; the diapente by 18 to 12 and 24 to 16 and 36 to 24; the triple, which is diapason and diapente, by 36 to 12; the quadruple, which is the double diapason, by 36 to 9; and the epogdous (sesquioctave), which is the tone, is kept by the comparison of 18 to 16.