1. Aristoxenus, Harm. p. 2, 15 Meib.: ta gar diagrammata autois tôn enarmoniôn (harmoniôn MSS.) ekkeitai monon systêmatôn, diatonôn d' ê chrômatikôn oudeis pôpoth' heôraken; kaitoi ta diagrammata g' autôn edêlou tên pasan tês melôdias taxin, en hois peri systêmatôn oktachordôn enarmoniôn (harmoniôn MSS.) monon elegon, peri de tôn allôn genôn te kai schêmatôn en autô te tô genei tontô kai tois loipois oud' epecheirei oudeis katamanthanein.
'The diagrams of the earlier writers set forth Systems in the Enharmonic genus only, never in the Diatonic or Chromatic: and yet these diagrams professed to give the whole scheme of their music, and in them they treated of Enharmonic octave Systems only; of other genera and other forms of this or any genus no one attempted to discover anything.'
2. Ibid. p. 6, 20 Meib.: tôn d' allôn katholou men kathaper emprosthen eipomen oudeis hêptai, henos de systêmatos Eratoklês epecheirêse kath' hen genos exarithmêsai ta schêmata tou dia pasôn apodeiktikôs tê periphora tôn diastêmatôn deiknys; ou katamathôn hoti, mê prosapodeichthentôn (qu. proapod.) tôn de tou dia pente schêmatôn kai tôn tou dia tessarôn pros de toutois kai tês syntheseôs autôn tis pot' esti kath' hên emmelôs syntithentai, pollaplasia tôn hepta symbainein gignesthai deiknytai.
'The other Systems no one has dealt with by a general method: but Eratocles has attempted in the case of one System, in one genus, to enumerate the forms or species of the Octave, and to determine them mathematically by the periodic recurrence of the intervals: not perceiving that unless we have first demonstrated the forms of the Fifth and the Fourth, and the manner of their melodious combination, the forms of the Octave will come to be many more than seven.'
The 'periodic recurrence of intervals' here spoken of may be illustrated on the key-board of a piano. If we take successive octaves of white notes, a-a, b-b, and so on, we obtain each time a different order of intervals (i.e. the semitones occur in different places), until we reach a-a again, when the series begins afresh. In this way it is shown that only seven species of the Octave can be found on any particular scale. Aristoxenus shows how to prove this from first principles, viz. by analysing the Octave as the combination of a Fifth with a Fourth.
3. Ibid. p. 36, 29 Meib.: tôn de systêmatôn tas diaphoras hoi men holôs ouk epecheiroun exarithmein, alla peri autôn monon tôn heptachordôn ha ekaloun harmonias tên episkepsin epoiounto, hoi de epicheirêsantes oudena tropon exêrithmounto.
For heptachordôn Meibomius and other editors read hepta oktachordôn—a correction strongly suggested by the parallel words systêmatôn oktachordôn in the first passage quoted.
'Some did not attempt to enumerate the differences of the Systems, but confined their view to the seven octachord Systems which they called harmoniai; others who did make the attempt did not succeed.'
It appears from these passages that before the time of Aristoxenus musicians had framed diagrams or tables showing the division of the octave scale according to the Enharmonic genus: and that a certain Eratocles—of whom nothing else is known—had recognised seven forms or species of the octachord scale, and had shown how the order of the intervals in the several species passes through a sort of cycle. Finally, if the correction proposed in the third passage is right, the seven species of the Octave were somehow shown in the diagrams of which the first passage speaks. In what respect Eratocles failed in his treatment of the seven species can hardly be conjectured.
Elsewhere the diagrams are described by Aristoxenus somewhat differently, as though they exhibited a division into Enharmonic dieses or quarter-tones, without reference to the melodious character of the scale. Thus we find him saying—