De institutione musica IV.8: Monochordi netarum diezeugmenon per tria genera partitio.
If I take half of nete diezeugmenon, which is 3,072, it will be 1,536, which added to the same make 4,608, which is mese, which we have designated by the letter O. And if from the same nete diezeugmenon, that is DD, namely 3,072, I take away the third part, it will be 1,024, which joined to the same will make 4,096, which will be called paramese, marked by the letter X. Nete therefore, diezeugmenon, that is 3,072, to mese, that is 4,608, since it stands in the sesquialter comparison, will sound the symphony of the fifth. The same nete diezeugmenon, that is 3,072, to paramese, that is 4,096, which is composed to it in the sesquitertian proportion, holds the consonance of the fourth. If, then, from nete diezeugmenon, 3,072, I take away the eighth part, that is 384, and add it to them, there will be 3,456. This will be paranete diezeugmenon diatonos, marked with the letters CC, holding a tone to nete diezeugmenon. If from this I take away the eighth part, that is of 3,456, which is 432, and join it to the same, there will be 3,888. This will be trite diezeugmenon diatonos. But since nete diezeugmenon held to paramese the sesquitertian proportion, and trite diezeugmenon diatonos is two tones from nete diezeugmenon, a lesser semitone will be contained between trite diezeugmenon and paramese. The diatonic kind has therefore been completed in this tetrachord also and in the pentachord, so that of the tetrachord of nete diezeugmenon to paramese there is the consonance of the fourth, and of the pentachord of nete diezeugmenon to mese there is the consonance of the fifth. The enharmonic and chromatic kind we shall weave by this method. I take the distance of nete and paranete diezeugmenon of the diatonic, that is 3,072 and 3,456; their difference is 384. I divide this; it will be 192. If I take this and join it to paranete diezeugmenon diatonos, that is 3,456, there will be 3,648. This will be paranete diezeugmenon chromatica, marked with the doubled letters BB, distant from nete diezeugmenon by a tone and a semitone, that is three semitones, containing to trite diezeugmenon, hitherto diatonic but now chromatic, that is 3,888, the semitone remaining of that tone which was divided between paranete diatonos diezeugmenon and trite diatonos diezeugmenon. And another semitone is left over from the tetrachord between trite diezeugmenon chromatica and paramese, which is left, namely, from the consonance of the fourth between nete diezeugmenon and paramese when the two tones are subtracted which nete diezeugmenon and trite diezeugmenon chromatica contained. That which in the diatonic kind is trite diezeugmenon diatonica, and in the chromatic trite diezeugmenon chromatica, is called in the enharmonic kind paranete diezeugmenon enarmonios—for it is distant two whole tones from nete diezeugmenon—and is marked AA. And between nete diezeugmenon and paranete enarmonios diezeugmenon no string intervenes, and therefore it is named paranete. The semitone, however, which lies between paranete enarmonios diezeugmenon and paramese, that is between AA and X, we divide in this way so that there are two dieses. I take the difference of paranete enarmonios diezeugmenon and paramese, that is 3,888 and 4,096. It is 208. I divide this; it will be 104.
These I add to 3,888, and they will be 3,992. This will be trite diezeugmenon enarmonios, marked by the letter Z. The description of this tetrachord through the three kinds I have added below, and I have gathered together with it the hyperboleon tetrachord arranged above, so that there might be one description of both, and the form of the whole arrangement might rise gradually, joined together.