De institutione musica IV.9: Monochordi netarum synemmenon per tria genera partitio.
The foregoing description of the three kinds has shown how two tetrachords are placed, which are joined to each other but disjoined from mese. Now we must come to that tetrachord which is called synemmenon, which is joined to mese. For since we said before that between nete diezeugmenon and mese there is the consonance of the fifth, and the consonance of the fifth is of three tones and a semitone, and there are three tones in this pentachord, of which one is from nete diezeugmenon to paranete diezeugmenon diatonos, another from paranete diezeugmenon diatonos to trite diezeugmenon diatonos, the third from paramese to mese, and the remaining semitone from trite diezeugmenon diatonos to paramese; and since the tetrachord of nete diezeugmenon and paramese is disjoined from mese by the tone which lies between paramese and mese: if from that pentachord, which is from nete diezeugmenon to mese, we take away one tone, namely that which is contained between nete diezeugmenon and paranete diezeugmenon diatonos, we shall be able to join another tetrachord to mese, so that synemmenon results, which is the conjoined, in this way. Since the number of paranete diezeugmenon diatonos, which is CC, is 3,456, a third of this added to the same will make mese. This number, then, marked in the diezeugmenon tetrachord with CC, was distant by a tone from nete diezeugmenon in the diatonic kind and was called paranete diezeugmenon diatonos. In the synemmenon tetrachord, however, that is of the conjoined, let nete synemmenon be constituted in the three kinds, marked by the letter V, and let the eighth part be taken from it, which is 432, and added to them; there will be 3,888, which is paranete synemmenon diatonos, marked with the letter T. Let the eighth part of this be taken, which is 486. If this sum is added to the same of which it is the eighth, there will be 4,374, which is trite synemmenon diatonos, that is Q. But since nete synemmenon to mese, that is 3,456 to 4,608 holds the sesquitertian proportion, which is the fourth, and trite synemmenon to nete synemmenon, that is 4,374 to 3,456, holds the proportion of two tones, there remains the proportion of a semitone of trite synemmenon diatonos to mese; and this tetrachord is joined with mese, and therefore is called synemmenon, as continuous and conjoined. The proportion of the diatonic kind is made in this way. The division of the chromatic is as follows. I take the difference of nete synemmenon and paranete synemmenon diatonos, that is 3,456 and 3,888. It is 432. I divide this so that a semitone results; it makes 216. I add this to 3,888, so that three semitones result; they will be 4,104, which is paranete synemmenon chromatica, over which the letter S is placed. From this, then, that is from paranete synemmenon chromatica to trite synemmenon, hitherto diatonic but now chromatic, there is a semitone, and from trite synemmenon chromatica to mese another semitone is found. But since from nete synemmenon to trite synemmenon diatonos or chromatica there are two tones, that which in the diatonic or chromatic kinds is trite synemmenon diatonos or chromatica is in the enharmonic kind paranete synemmenon enarmonios, having the sum 4,374, and let it be R. From this to mese there is a semitone. This I divide into two dieses in this way. I take the difference of paranete synemmenon enarmonios and mese, that is 4,374 and 4,608. It is 234. I divide this; it will be 117. I add this to paranete synemmenon enarmonios, that is 4,374; there will be 4,491, which let be marked by the letter P and be trite synemmenon enarmonios. And the semitone will be that which is contained between paranete synemmenon enarmonios and mese, that is between 4,374 and 4,608, divided by trite synemmenon enarmonios, that is, that which is 4,491. The reasoning of this tetrachord also is therefore explained. Now the description is to be made, joined however with the rest, that is the hyperboleon and the diezeugmenon, so that gradually the fixed progress of the arrangement may be made.