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enharmonic pyknon, which together made up a diesis or an apotome (= diesis plus komma). The purpose of the apotome was to make the chromatic pyknon up to a tone, its lower interval being a diesis. The pattern of the tetrachords for Philolaus is shown in Fig. 8.2. (For cent values see p. 168.) Philolaus' mathematics went completely off the rails when he tried to assign numbers to these new units. He did not understand that only ratios between numbers were musically significant, and that absolute quantities were not to be sought in them. He could only visualize a set of ratios in terms of a set of whole numbers in which they were embodied. For the intervals of the diatonic tetra-chord the required numbers were 192:216: 243:256. From this Philolaus mistakenly concluded that the last interval, the diesis, could meaningfully be equated with the number 13 (256 minus 243). |
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FIG. 8.2.
Philolaus' pattern of tetrachords |
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On the same principle he subtracted 216 from 243 to get 27 for a tone (243: 216 = 9:8), and then he subtracted his 13 from his 27 to get a figure for the apotome. He thought it significant that 13 = 1 + 3 + 32 (three being the first odd number by Greek reckoning), while 27 = 33, and also 27:24 = a tone. This is as much number-mysticism as mathematics.44 |
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Archytas of Tarentum moved on an altogether more sophisticated plane. He worked out a coherent system of ratios for all the intervals |
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44 Boeth. Inst. Mus. 3.5 p. 276.15ff. Friedl. See Burkert, LS 394-9. |
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