De institutione musica IV.17: Ratio superius dispositae modorum descriptionis.
We said before that there are seven modes, but it should seem nothing incongruous that an eighth has been attached above them. For I shall give the reason for this addition a little later. For now this must be considered: that in these little columns, which a straight order of lines distinguishes from one another, some have musical notes and others do not. For example, in the mode inscribed Hypermixolydian, the first little column is marked with a sign, the third with the letter O, and the second lacks notes. In this interval between the notes, then, a tone is shown to lie between them. But where a note of the third and of the fourth column is not divided by a little column but by a line drawn in a straight row, this declares that they differ by a semitone. This is proved as follows. For if the one sign is the proslambanomenos, and O the hypate hypaton, and the next the parhypate hypaton, it is necessary that between the proslambanomenos and the hypate hypaton, which is O, there be the distance of a tone, and that between the hypate hypaton, which is O, and the parhypate hypaton, there be contained the difference of a semitone. So this must be observed as a rule in all cases: if a whole column separates the notes of the voices, we know that the distance between them is a tone; but if a line and not a column distinguishes the notes, we are not to doubt that the distance is a semitone. With these things premised, if two orders constituted in the double octave are compared with each other, so that it may be known which order is the lower, then if the proslambanomenos is lower than the other proslambanomenos, or if any other voice is marked as lower than the voice of the same place, being set in the same genus, it is necessary that the whole order also be the lower. Yet this is better taken from the middle string, which is the mese. For of two orders constituting a double octave, the one whose mese is lower will have its whole order lower also. For the other strings, compared one with one, are found no less lower. So if one middle string seems higher or lower than another middle string by a tone, all the strings too, if they are in the same genus, compared one with one, will be seen to be higher or lower by a tone. Now of four middle strings, if the first keeps the distance of a diatessaron to the fourth, while the first differs from the second by a tone, and the second differs from the third by the same tone, the third to the fourth will make the difference of a semitone, thus: let there be four middle strings A, B, C, D, and let A, compared with D, keep the sesquitertian proportion, which is the diatessaron; likewise let A be distant from B by a tone, and B from C by a tone; it remains that C keep to D the distance of a semitone.
And if there are five middle strings, it is the same. For if the first is distant from the fifth by the sesquialter proportion, and the first from the second, the second from the third, and the third from the fourth are each distant by a tone, the fourth to the fifth will make the difference of a semitone. Likewise, whatever middle strings of other modes approach the proslambanomenoi make the modes lower, and whatever approach the netai make them higher. Since, then, in the modes described on the page above, the proslambanomenoi hold the left part for the reader, while the right is closed by the extreme netai, the highest of all will be the mode inscribed Hypermixolydian, and the lowest of all that which is the Hypodorian. But we, beginning from the lowest, that is, the Hypodorian, shall set out what difference the others have among themselves. For in the Hypodorian mode the mese is distant by a tone from the mese which is in the Hypophrygian mode. This will easily be seen in the following way, if one compares with the mese of the Hypophrygian, which is O, the sign of that same Hypophrygian, which is the mese of the Hypodorian but in the Hypophrygian the lichanos meson. For the two differ by a tone, as the interposed column shows. Likewise the mese of the Hypolydian makes a difference of a tone from the mese of the Hypophrygian. For C, which is the mese of the Hypolydian, is distant by a tone from O, which in the Hypolydian is the lichanos meson, but in the Hypophrygian the mese. Again, the mese of the Hypolydian, which is C, is distant by a semitone from the mese of the Dorian. This can be recognized from the fact that a single line and not a column separates the order of its mese proceeding upward, that is, of the Hypolydian, and the order of the mese of the Dorian proceeding upward. Hence it follows that the mese of the Hypodorian is distant from the mese of the Dorian by a whole diatessaron consonance. And this is proved thus. For the mese which is in the Hypodorian is the same as the hypate meson in the Dorian, which differs from the mese, in any mode or genus whatever, by a diatessaron consonance. Likewise the mese of the Dorian is distant by a tone from the mese of the Phrygian, that is, M. For the mese in the Dorian is the lichanos meson in the Phrygian. Again, the mese of the Phrygian, which is M, is distant by a tone from the mese of the Lydian, that is L. For the mese M in the Phrygian is the lichanos meson in the Lydian. Again, the mese of the Lydian mode is distant by a semitone from the mese of the Mixolydian, that is H. For the order which contains the mese of the Lydian, compared with the order which has the mese of the Mixolydian, is disjoined not by a column but by a line. The mese of the Mixolydian, that is H, makes a difference of a tone to the mese of the Hypermixolydian, because H, which is the mese in the Mixolydian, is the lichanos meson in the Hypermixolydian. Hence it follows that the mese of the Dorian is distant from the mese of the Mixolydian by a diatessaron consonance. This is proved thus. For the mese of the Dorian is the same as the hypate meson of the Mixolydian, which keeps a diatessaron consonance to the mese of any mode whatever. Likewise the mese of the Dorian keeps a diapente consonance to the mese of the Hypermixolydian. For that mese of the Dorian is the lichanos hypaton in the order of the Hypermixolydian. And the lichanos hypaton, if compared to the mese in the diatonic genus in any mode, is distant by a diapente consonance. Why the eighth mode, the Hypermixolydian, was added is clear from this. Let this be the double octave:
A therefore keeps the diapason consonance to H, for it is contained by eight voices. We have said, then, that the first species of the diapason is that which is AH, the second BI, the third CK, the fourth DL, the fifth EM, the sixth FN, the seventh GO. There remains outside the series HP, which was added so that the whole order might be filled up. And this is the eighth mode, which Ptolemy attached above.