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number of keys is precisely seven and no more. These are sufficient to account for modulation, so long as each key has a counterpart at the distance of a fourth. So let the series be constructed from a cycle of descending fourths, f''-c"-g'-d'-a-e-B. Arrange these in order in one octave, keeping F at the top, and it emerges as the old series g a b c' d' e' f'. g-c' make a fourth, and so do a-d', b-e', c'-f'. Ptolemy keeps to the old names, Hypodorian, Hypophrygian, Hypolydian, Dorian, Phrygian, Lydian, Mixolydian. To complete the octave with Hypermixolydian, as Aristoxenus did, is condemned as mere duplication; going beyond the octave, as his successors did, compounded the error. Filling in the semitone steps is redundant, because these sharp or flat keys (Ionian and Aeolian with their Hypos and Hypers) cannot bring new species into view but merely reproduce, a semitone higher or lower, the species associated with an adjacent key.34 |
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Ptolemy's penetrating intellect went to the heart of the matter. But he came too late. The fifteen-key system had become firmly established in the theoretical tradition and, as we shall see in the next chapter, enshrined in notational practice. |
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Calculation of harmonic ratios |
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A separate branch of theory concerned itself with the mathematical values of the intervals making up an octave. In the Classical period this approach was particularly associated with the Pythagorean tradition. Aristoxenus and his later followers disregarded it, sticking to the simple empirical equation, a fourth = 2 1/2 tones, and to the belief that all musical intervals are to be measured in tones and fractions of a tone. |
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The Pythagoreans were fascinated by number and inclined to see it as the key to the universe. One of their old sayings went: 'What is cleverest?-Number; and in second place whoever gave things their names.'35 Philolaus wrote that everything has a number which gives it definition; without this we would have no perception or conception of things.36 The realization that numbers underlie the basic concords of fourth, fifth, and octave must have been, if not the source, at any rate a powerful reinforcement of this exaltation of number. These musical phenomena provided concepts and formulae that could be |
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34 Ptol. Harm. 2.7-11. His criticism finds an echo at Ath. 625d. |
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35 Iambl. De vita Pythagorica 82, Ael. VH 4. 17. |
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36 Diels, Vorsokr. 44 B 4; Burkert, LS 261-6. |
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