De institutione musica IV.6: Monochordi netarum hyperboleon per tria genera partitio.
The description of the diatonic kind has now been made, namely in that mode which is simpler and first, which we call Lydian. Of these modes we need not now speak. But that a mixed description may run through the three kinds, and in all of them the proper multitude of numbers be set down, in order to preserve the proportions of the tones and of the dieses, a number has been devised which can fulfil all these, so that the greatest is written for proslambanomenos, which is 9,216, and the least 2,304, and the proportions of the remaining sounds are woven in the space between these. We proceed from the lower, and show the names of all the strings not only by names but also by letters set beside them; but since the division is to be made for three kinds, and the number of strings exceeds the number of the letters, where the letters fail we double them again in this way: when we have come to Z, we write the remaining strings with A twice, that is AA, and B twice, that is BB, and C twice, that is CC. Let the first and greatest number, which holds the place of proslambanomenos, be 9,216, and let the extent of the whole chord be from A to LL. This A, that is proslambanomenos, 9,216, I divide in half at O, so that the whole A is double of O. Likewise let O be double of LL. A will therefore be proslambanomenos, O mese, and LL nete hyperboleon. A will therefore have 9,216, and O the half of this, that is 4,608, so that mese agrees with proslambanomenos in the consonance of the octave; and LL the half of mese, so that proslambanomenos is quadruple of nete hyperboleon and sounds the double octave with it; and let LL be 2,304.
If, then, I take from 2,304 the eighth part, that is 288, and add it to the same, I shall have 2,592, and KK will be 2,592, which is paranete hyperboleon, holding a distance of a tone from nete hyperboleon. Again, I take from KK, that is 2,592, the eighth, which is 324, and add it to those of which it is the eighth, and they will be 2,916, and I shall have FF, trite hyperboleon diatonos in the diatonic kind, namely 2,916, a tone distant from paranete hyperboleon diatonos, and a ditone from nete hyperboleon. The same FF will be in the chromatic kind trite hyperboleon chromatica, and in the enharmonic paranete hyperboleon enarmonios; why this happens will be more easily recognized when we have described the three first tetrachords of the three kinds beginning from nete hyperboleon. But since, if I take two sesquioctaves from the sesquitertian proportion, the lesser semitone is left to me, I take the third of that which is LL, that is nete hyperboleon; it is 768. These I add to the same, and I shall have 3,072, of which DD is nete diezeugmenon holding the lesser semitone to trite hyperboleon. For since nete diezeugmenon contains the consonance of the fourth with nete hyperboleon, and trite hyperboleon diatonos is a ditone from nete hyperboleon, there remains the space between nete diezeugmenon and trite hyperboleon of the lesser semitone. Since, then, we have completed the hyperboleon tetrachord of the diatonic kind, now the tetrachords of the chromatic and enharmonic must be supplied in this way. Since paranete hyperboleon is distant from nete hyperboleon in the diatonic kind by a tone, in the chromatic by three semitones, in the enharmonic by two tones, if we take the distance of paranete hyperboleon and nete hyperboleon of the diatonic kind and its half, and add it to paranete hyperboleon of the diatonic kind, we shall have a number distant from nete hyperboleon by three semitones; and this will be in the chromatic kind paranete hyperboleon. I take away, therefore, from 2,592, that is paranete hyperboleon of the diatonic kind, 2,304, that is nete hyperboleon; there remain 288. I divide these; they will be 144. I join the same to 2,592, and I shall have 2,736. This will be paranete hyperboleon chromatica. Again, since trite hyperboleon, whether diatonic or chromatic, is distant two tones from nete hyperboleon, and in the enharmonic kind paranete hyperboleon is distant two tones from nete hyperboleon, the same will be in the enharmonic kind paranete hyperboleon which in the diatonic or chromatic is trite hyperboleon. But since trite hyperboleon of the diatonic and chromatic kind preserves the lesser semitone to nete diezeugmenon, and the tetrachord of the enharmonic kind consists of two whole tones and a diesis and diesis, which are half spaces of the lesser semitone, I take the distance between nete diezeugmenon and paranete hyperboleon enarmonios. But since nete diezeugmenon is 3,072, and paranete hyperboleon enarmonios 2,916, their distance will be 156. Of these I take the half part, which is 78. I add these to 2,916, and the trite hyperboleon enarmonios is 2,994. The tetrachord in the three kinds has been described, the form of which we have added below.