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down a fourth. Converted into cents, Philolaus' scheme of intervals would appear thus: |
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or |
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Archytas in the first half of the fourth century achieved greater mathematical rigour.29 He expressed all his intervals as ratios, but I shall again convert them into cents so that their magnitudes can be appreciated; we are not concerned in this context with the beauty of Archytas' arithmetic. |
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There is one interesting peculiarity which sets Archytas' system apart from everyone else's. According to him the lower 'movable' note is not movable at all: the lowest interval in the tetrachord remains constant in all three genera at approximately a third of a tone.30 The Greek system of musical notation, which will be described in another chapter, agrees with Archytas inasmuch as it uses the same symbol for the lower movable note in every genus.31 In the chromatic, Archytas agrees with Philolaus in making the two smaller intervals together exactly equal to a tone.32 His scheme for the enharmonic is also strikingly close to one of the two postulated for Philolaus. |
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Aristoxenus' approach is very different. According to him the two inner notes of the tetrachord can be pitched anywhere within a continuous band, and it is necessary to lay down boundaries to demarcate one genus from another. The lower movable note may be anything from a quarter to a third of a tone above the bottom note of the tetrachord in the enharmonic genus, and anything from a third to half a tone in the chromatic and diatonic. The upper one may be any- |
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29 His results are reported by Ptolemy, Harm. 1. 13 p. 30. 9ff.; see Barker, GMW ii. 43-52. |
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30 In the enharmonic it is a shade wider than the next interval above it, something that Aristoxenus (Harm. 1. 27, 2. 52) says never happens. |
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31 For the upper one, to be sure, it uses the same symbol in chromatic as in enharmonic, but a different one in diatonic. |
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32 This feature of chromatic tuning also appears in Thrasyllus ap. Theon. Smyrn. p. 91. 19f., Gaud. Harm. p. 343. 15, and Anecdota Varia, pp. 5-7 Studemund. |
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