De institutione musica IV.14: De consonantiarum speciebus.
Now we must treat of the species of the first consonances. The first consonances are the octave, the fifth and the fourth. A species is a certain position having its own form, according to each kind, set between the terms of the one making the consonance of each proportion; as in the diatonic kind. For if we place the diezeugmenon tetrachord between the hyperboleon tetrachord and mese, the synemmenon tetrachord being taken away, there will be fifteen strings. And if proslambanomenos is taken from these, there will be fourteen. Let these be arranged in this way. Let A be hypate hypaton, B parhypate hypaton, C lichanos hypaton, D hypate meson, E parhypate meson, F lichanos meson, G mese, H paramese, I trite diezeugmenon, K paranete diezeugmenon, L nete diezeugmenon, M trite hyperboleon, N paranete hyperboleon, O nete hyperboleon. From hypate to paramese there is the consonance of the octave, from the same paramese to hypate meson the fifth, and from mese to hypate meson the fourth. The octave will therefore be of eight strings, the fourth of four, the fifth of five. And for this reason the fourth will have three species, the fifth four species, and the octave seven species; and there will always be one species fewer than there are voices. For, to begin the rest from mese, there are three species of the consonance of the fourth in this way. One species will be from G to D, the second from F to C, the third from E to B; and thus far the species of the fourth proceed, because thus far the species contain two strings of the same fourth, as GD those which are E and F, and FC those which are E and D, and EB those which are C and D. But if I add to these the fourth DA, it will be different from that which is GD; for GD will contain only one string of the consonance, that is D alone. It has therefore passed beyond the consonance GD. And therefore the fourth is said to have three species. And in the other consonances it is the same.
The fifth will have four species in this way. One from H to D, another from G to C, another from F to B, another from E to A. The octave will have seven species in this way. The first from O to G, the second from N to F, the third from M to E, the fourth from L to D, the fifth from K to C, the sixth from I to B, the seventh from H to A. It is plain, therefore, from what has been said, that the consonance of the fourth is contained only once by fixed and established voices. For if I begin from hypate hypaton, AD, that is from hypate hypaton to hypate meson, will be the first in this order. For the others are not terminated by established voices, as BE and CF. For parhypate hypaton and parhypate meson and lichanos hypaton and lichanos meson have been shown to be movable. But if again we begin the consonance of the fourth from hypate meson, the species of the fourth terminated by established voices will be DG, which is the first, that is from hypate meson to mese; the rest not at all, as EH and FI. For parhypate meson and lichanos meson and trite diezeugmenon are not proved to be fixed. Again, if paramese undertakes to begin the same fourth, the species of the fourth confined by established sounds will be HL, that is from paramese to nete diezeugmenon, which is the first. For the others, which are IM and KN, are terminated by movable sounds. For we have said before that trite diezeugmenon, paranete diezeugmenon and trite hyperboleon are movable voices. Likewise the consonance of the fifth holds only two species which are enclosed by established voices. Thus if we begin from hypate meson, one is DH, that is from hypate meson to paramese, which is the first, the other GL, that is from mese to nete diezeugmenon. This is the fourth. The rest, that is EL and FK, are not at all closed by established voices. For parhypate and lichanos and trite and paranete are proved unstable. A like reasoning will hold if from nete diezeugmenon toward the lower part, that is to mese, the species of this consonance are considered. For they will be contained by the same fixed voices which were named above. Whether we lead the consonances toward the lower part from hypate meson or from mese or from paramese or also from nete hyperboleon, there cannot be a drawing out of more than two that are confined by established voices. But as for the consonance of the octave, whether the order is taken from hypate hypaton to paramese, or from nete hyperboleon to mese, it will have only three species which are confined by fixed voices. For beginning from hypate hypaton, one is AH, which is the first, from hypate hypaton to paramese; another DL, which is the fourth, from hypate meson to nete diezeugmenon; and then GO—this is the seventh—that is from mese to nete hyperboleon. The extreme voices of the remaining species are in no way established. For parhypate and lichanos and trite and paranete, as was also said above, are not fixed. Likewise, and through the same voices, if we begin from nete hyperboleon, the order of the species is woven. The description set below will make the understanding of all these the better known.