1. Aristoxenus, _Harm._ p. 2, 15 Meib.: [Greek: ta gar diagrammata autois tôn enarmoniôn] ([Greek: harmoniôn] MSS.) [Greek: 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] ([Greek: harmoniôn] MSS.) [Greek: monon elegon, peri de tôn allôn genôn te kai schêmatôn en autô te tô genei tontô kai tois loipois oud' epecheirei oudeis katamanthanein.] [Footnote 1: The investigation occupies a considerable space in his _Harmonics_, viz. pp. 27-29 Meib. (from the words [Greek: peri de synecheias kai tou hexês]), and again pp. 58-72 Meib.] '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.: [Greek: 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. [Greek: proapod.]) [Greek: 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.