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of melodies as the only aim of harmonic analysis. So far from being its goal, he protests, it is not even a part of the science. He speaks as if there was just one notational system. He complains that it merely indicates intervals without defining functions. For instance, it tells you that two notes are a fourth apart, but the notation is the same whether they are the outer notes of a tetrachord (of whatever genus) or not, and, if they are the outer notes, it is the same whether the tetrachord is Hyperbolaion or Meson. There is no agreement among scholars as to whether Aristoxenus is referring to the system of notation that we know or to some other. In view of our conclusions about the age of the common system and its general currency from the third century BC onwards, it would be surprising if Aristoxenus was unacquainted with it and regarded some other system as the only one available. Although his criticisms seem somewhat tendentious, I agree with Macran and Pöhlmann that they can be understood in relation to the standard system.24 |
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We do, nevertheless, know of another system. Aristides Quintilianus, after stating that the ancients (i.e. pre-Aristoxenian theorists) divided the octave into twenty-four quarter-tone steps, appends a table of note-symbols associated with this division. It covers two octaves, and appears to be based on the letters of the Ionic alphabet in their normal order, though some of them have become displaced in the course of transmission. Each letter provided two symbols, one of which was the other reversed or inverted. The primary forms (some of them amputated or otherwise altered in the interests of reversibility) were assigned to the twenty-four semitone steps contained in the double octave, and the reversed forms to the intervening quarter-tones.25 The twenty-four quarter-tone steps into which each octave was divided, therefore, were not treated as all of equal status, but the twelve semitone steps were. The system differs significantly in this respect from the usual one based on diatonic steps. It |
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(Footnote continued from previous page) |
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there is no sure ground for supposing them to be more than libretti; see Winnington-Ingram, Gnomon 33 (1961), 693. |
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24 H. S. Macran, The Harmonics of Aristoxenus (Oxford, 1902), 270-2; Pöhlmann, Beiträge, 74-6. For example, the notes  (46, 37) represent the outer notes of the tetrachord Hyperbolaion in Hypodorian, and of the tetrachord Meson in Hyperphrygian (or Hypermixolydian, as it was called in Aristoxenus' key-system); or Paranete synemmenon and Lichanos in diatonic Lydian. See also Barker, GMW ii. 156 n. 46, who takes a different view. |
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25 For detailed interpretation see Winnington-Ingram, Philol. 117 (1973), 243-9; M. L. West, ZPE 92 (1992), 42-6. |
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