With regard to the order in which the new notes obtained a place in the schemes of theoretical musicians we have no trustworthy information. The name proslambanomenos, applied to the lowest note of the Perfect System, points to a time when it was the last new addition to the scale. Plutarch in his work on the Timaeus of Plato (peri tês en Timaiô psychogonias) speaks of the Proslambanomenos as having been added in comparatively recent times (p. 1029 c hoi de neôteroi ton proslambanomenon tonô diapheronta tês hypatês epi to bary taxantes to men holon diastêma dis dia pasôn epoiêsan). The rest of the Perfect System he ascribes to 'the ancients' (tous palaious ismen hypatas men dyo, treis de nêtas, mian de mesên kai mian paramesên tithemenous). An earlier addition—perhaps the first made to the primitive octave—was a note called Hyperhypatê, which was a tone below the old Hypatê, in the place afterwards occupied on the Diatonic scale by Lichanos Hypatôn. It naturally disappeared when the tetrachord Hypatôn came into use. It is only mentioned by one author, Thrasyllus (quoted by Theon Smyrnaeus, cc. 35-36 [13]).
The notes of the Perfect System, with the intervals of the scale which they formed, are fully set out in the two treatises that pass under the name of the geometer Euclid, viz. the Introductio Harmonica and the Sectio Canonis. Unfortunately the authorship of both these works is doubtful [14]. All that we can say is that if the Perfect System was elaborated in the brief interval between the time of Aristotle and that of Euclid, the materials for it must have already existed in musical practice.
§ 19. Relation of System and Key.
Let us now consider the relation between this fixed or standard scale and the varieties denoted by the terms harmonia and tonos.
With regard to the tonoi or Keys of Aristoxenus we are not left in doubt. A system, as we have seen, is a series of notes whose relative pitch is fixed. The key in which the System is taken fixes the absolute pitch of the series. As Aristoxenus expresses it, the Systems are melodies set at the pitch of the different keys (tous tonous, eph' hôn tithemena ta systêmata melôdeitai). If then we speak of Hypatê or Mesê (just as when we speak of a moveable Do), we mean as many different notes as there are keys: but the Dorian Hypatê or the Lydian Mesê has an ascertained pitch. The Keys of Aristoxenus, in short, are so many transpositions of the scale called the Perfect System.
Such being the relation of the standard System to the key, can we suppose any different relation to have subsisted between the standard System and the ancient 'modes' known to Plato and Aristotle under the name of harmoniai?
It appears from the language used by Plato in the Republic that Greek musical instruments differed very much in the variety of modes or harmoniai of which they were susceptible. After Socrates has determined, in the passage quoted above (p. 7), that he will admit only two modes, the Dorian and Phrygian, he goes on to observe that the music of his state will not need a multitude of strings, or an instrument of all the modes (panarmonion) [15]. 'There will be no custom therefore for craftsmen who make triangles and harps and other instruments of many notes and many modes. How then about makers of the flute (aulos) and players on the flute? Has not the flute the greatest number of notes, and are not the scales which admit all the modes simply imitations of the flute? There remain then the lyre and the cithara for use in our city; and for shepherds in the country a syrinx (pan's pipes).' The lyre, it is plain, did not admit of changes of mode. The seven or eight strings were tuned to furnish the scale of one mode, not of more. What then is the relation between the mode or harmonia of a lyre and the standard scale or systêma which (as we have seen) was based upon the lyre and its primitive gamut?