De institutione musica IV.10: Monochordi meson per tria genera partitio.
From what has been said, then, I think we need not labor longer in the rest; for after the pattern of these the remaining tetrachords, the meson and hypaton, are to be woven. And first we shall describe the meson tetrachord of the diatonic kind in this order. Of mese, which is O, 4,608, I take the third part. It is 1,536.
I join this to the same; they will be 6,144. Let this be H, hypate meson, containing the consonance of the fourth to mese. This is divided by two tones and a semitone thus. I take the eighth part of mese, that is 4,608, which is 576. This I join to the same; they will be 5,184. This is lichanos meson diatonos, that is M. Of this again let the eighth part be taken. It is 648. This I join to the same; it will be 5,832. Let this be L, parhypate meson diatonos, holding a tone to lichanos meson diatonos, and being distant from mese by two tones. There remains, then, a semitone constituted between hypate meson diatonos and parhypate meson diatonos, that is between 6,144 and 5,832. The same tetrachord of mese and hypate meson we divide in the chromatic kind by this method.
I take the difference of mese to lichanos meson diatonos, that is 4,608 to 5,184. It is 576. This I divide in half; it makes 288. The same I add to the greater number, that is 5,184; it makes 5,472, which let be N, lichanos meson chromatice. There remain, then, two semitones, one between lichanos meson chromatice and parhypate meson chromatice, that is between 5,472 and 5,832, and another between parhypate meson chromatice and hypate meson, that is between 5,832 and 6,144. The enharmonic kind we divide in this way. Since that which was parhypate meson diatonos, or that which was parhypate meson chromatice, was distant two tones from mese, holding the number 5,832, this will be in the enharmonic kind lichanos meson enarmonios, marked with the letter L, holding nonetheless two tones to mese. The remaining semitone, which is between lichanos meson enarmonios and hypate meson, that is between 5,832 and 6,144, we divide into two dieses in this way. I take the difference of 5,832 to 6,144. It is 312. This I divide in half; it will be 156. This to 5,832 I join; it will be 5,988. And let this be R, parhypate meson enarmonios. There are two dieses between lichanos meson enarmonios and parhypate meson enarmonios, that is between 5,832 and 5,988, and between parhypate meson enarmonios and hypate meson, that is between 5,988 and 6,144. The meson tetrachord has thus been divided, and let it be so placed in the description that it is gathered together with the tetrachords described above,