(See the Appendix, Table I.)
In the Harmonics of Aristoxenus an account of the seven species of the Octave followed the elaborate theory of Systems already referred to (p. 48), and doubtless exhibited the application of that general theory to the particular cases of the Fourth, Fifth, and Octave. Unfortunately the existing manuscripts have only preserved the first few lines of this chapter of the Aristoxenean work (p. 74, ll. 10-24 Meib.).
The next source from which we learn anything of this part of the subject is the pseudo-Euclidean Introductio Harmonica. The writer enumerates the species of the Fourth, the Fifth, and the Octave, first in the Enharmonic and then in the Diatonic genus. He shows that if we take Fourths on a Diatonic scale, beginning with HypatÊ HypatÔn (our b), we get successively b c d e (a scale with the intervals ½ 1 1), c d e f (1 1 ½) and d e f g (1 ½ 1). Similarly on the Enharmonic scale we get—
| HypatÊ HypatÔn to HypatÊ MesÔn | | b b* c e | | (¼ ¼ 2 ) |
| ParhypatÊ " " ParhypatÊ " | | b* c e e* | | (¼ 2 ¼) |
| Lichanos " " Lichanos " | | c e e* f | | (2 ¼ ¼) |
In the case of the Octave the species is distinguished on the Enharmonic scale by the place of the tone which separates the tetrachords, the so-called Disjunctive Tone (tonos diazeuktikos). Thus in the octave from HypatÊ HypatÔn to ParamesÊ (b-b) this tone (a-b) is the highest interval; in the next octave, from ParhypatÊ HypatÔn to TritÊ DiezeugmenÔn (c-c), it is the second highest; and so on. These octaves, or species of the Octave, the writer goes on to tell us, were anciently called by the same names as the seven oldest Keys, as follows:
| Mixo-lydian | | b - b | | ¼ ¼ 2 ¼ ¼ 2 1 |
| Lydian | | b*- b* | | ¼ 2 ¼ ¼ 2 1 ¼ |
| Phrygian | | c - c | | 2 ¼ ¼ 2 1 ¼ ¼ |
| Dorian | | e - e | | ¼ ¼ 2 1 ¼ ¼ 2 |
| Hypo-lydian | | e*- e* | | ¼ 2 1 ¼ ¼ 2 ¼ |
| Hypo-phrygian | | f - f | | 2 1 ¼ ¼ 2 ¼ ¼ |
| Hypo-dorian | | a - a | | 1 ¼ ¼ 2 ¼ ¼ 2 |
On the Diatonic scale, according to the same writer, the species of an Octave is distinguished by the places of the two semitones. Thus in the first species, b-b, the semitones are the first and fourth intervals (b-c and e-f): in the second, c-c, they are the third and the seventh, and so on. He does not however say, as he does in the case of the Enharmonic scale, that these species were known by the names of the Keys. This statement is first made by Gaudentius (p. 20 Meib.), a writer of unknown date. If we adopt it provisionally, the species of the Diatonic octave will be as follows:
| [Mixo-lydian] | | b - b | | ½ 1 1 ½ 1 1 1 |
| [Lydian] | | c - c | | 1 1 ½ 1 1 1 ½ |
| [Phrygian] | | d - d | | 1 ½ 1 1 1 ½ 1 |
| [Dorian] | | e - e | | ½ 1 1 1 ½ 1 1 |
| [Hypo-lydian] | | f - f | | 1 1 1 ½ 1 1 ½ |
| [Hypo-phrygian] | | g - g | | 1 1 ½ 1 1 ½ 1 |
| [Hypo-dorian] | | a - a | | 1 ½ 1 1 ½ 1 1 |
