[Footnote 1: _The Introduction to Harmonics_ ([Greek: eisagôgê harmonikê]) which bears the name of Euclid in modern editions (beginning with J. Pena, Paris, 1557) cannot be his work. In some MSS. it is ascribed to Cleonides, in others to Pappus, who was probably of the fourth century A.D. The author is one of the [Greek: harmonikoi] or Aristoxeneans, who adopt the method of equal temperament. He may perhaps be assigned to a comparatively early period on the ground that he recognises only the thirteen keys ascribed to Aristoxenus--not the fifteen keys given by most later writers (Aristides Quint., p. 22 Meib.). For some curious evidence connecting it with the name of the otherwise unknown writer Cleonides, see K. von Jan, _Die Harmonik des Aristoxenianers Kleonides_ (Landsberg, 1870). The _Section of the Canon_ ([Greek: kanonos katatomê]) belongs to the mathematical or Pythagorean school, dividing the tetrachord into two major tones and a [Greek: leimma] which is somewhat less than a semitone. In point of form it is decidedly Euclidean: but we do not find it referred to by any writer before the third century A.D.--the earliest testimony being that of Porphyry (pp. 272-276 in Wallis' edition).] § 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 [Greek: harmonia] and [Greek: tonos]. With regard to the [Greek: 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 ([Greek: 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 [Greek: harmoniai]?