Note 14 — § 18. The Perfect System.

The Introduction to Harmonics (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 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 (kanonos katatomê) belongs to the mathematical or Pythagorean school, dividing the tetrachord into two major tones and a 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).