The enquiry must start from the remarkable fact that the two octaves represented by the fifteen original letters are in the _Hypo-lydian_ key--the key which down to the time of Aristoxenus was called the Hypo-dorian. Are we to suppose that the scheme was devised in the first instance for that key only? This assumption forms the basis of the ingenious and elaborate theory by which M. Gevaert explains the development of the notation (_Musique de l'Antiquité_, t. I. pp. 244 ff.). It is open to the obvious objection that the Hypo-lydian (or Hypo-dorian) cannot have been the oldest key. M. Gevaert meets this difficulty by supposing that the original scale was in the Dorian key, and that subsequently, from some cause the nature of which we cannot guess, a change of pitch took place by which the Dorian scale became a semitone higher. It is perhaps simpler to conjecture that the original Dorian became split up, so to speak, into two keys by difference of local usage, and that the lower of the two came to be called Hypo-dorian, but kept the original notation. A more serious difficulty is raised by the high antiquity which M. Gevaert assigns to the Perfect System. He supposes that the inventor of the notation made use of an instrument (the _magadis_) which 'magadised' or repeated the notes an octave higher. But this would give us a repetition of the primitive octave _e - e_, rather than an enlargement by the addition of tetrachords at both ends. M. Gevaert regards the adaptation of the scheme to the other keys as the result of a gradual process of extension. Here we may distinguish between the recourse to the modified characters--which served essentially the same purpose as the 'sharps' and 'flats' in the signature of a modern key--and the additional notes obtained either by means of new characters ([Symbols: a] and [Symbols: e]), or by the use of accents ([Symbols:?'], &c.). The Hypo-dorian and Hypo-phrygian, which employ the new characters [Symbols: a] and [Symbols: e], are known to be comparatively recent. The Phrygian and Lydian, it is true, employ the accented notes; but they do so only in the highest tetrachord (Hyperbolaiôn), which may not have been originally used in these high keys. The modified characters doubtless belong to an earlier period. They are needed for the three oldest keys--Dorian, Phrygian, Lydian--and also for the Enharmonic and Chromatic genera. If they are not part of the original scheme, the musician who devised them may fairly be counted as the second inventor of the instrumental notation.