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Eratosthenes' diatonic is the same as Philolaus', Plato's, and Euclid's, and his enharmonic is also based on the practice of constructing two tones down from the top of the tetrachord. Didymus', on the other hand, like Archytas', is a mathematical refinement of the other Philolaic form of enharmonic in which the ditone was slightly reduced. Didymus' chromatic is somewhat anomalous. Like Archytas' enharmonic, it breaks Aristoxenus' rule that the lowest interval is never bigger than the middle one. Ptolemy criticizes this feature of Didymus' scheme as unmelodic in principle and out of accord with the evidence of our ears.35 |
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Ptolemy's own analyses are more clearly related to the actual musical practice of his time. Only two varieties of chromatic are to be distinguished, he maintains, as against Aristoxenus' three; on the other hand, two species of diatonic are not enough, as 'it is obvious that those that are sung are more'. He also disagrees with Aristoxenus' treatment of the two small intervals in the enharmonic and chromatic pyknon as equal, stating that the upper one is always perceived as being greater than the lower.36 In his own tables it is about twice as large. This evidently represents a historical change since Aristoxenus' time. Aristoxenus' two forms of diatonic both have counterparts in Ptolemy's system, though his 'soft diatonic' corresponds to what Ptolemy calls 'tense chromatic'. The unnamed forms of chromatic and diatonic that Aristoxenus mentions, with respectively  and  tones, correspond to what Ptolemy calls 'soft chromatic' and 'tonic diatonic'. |
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Ptolemy's categories are as follows.37 Besides giving the intervals in cents I show what each tetrachord looks like when expressed in modern notes with appropriate modifiers. |
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35Harm. 2. 13 p. 68. 24ff. Eratosthenes' and Didymus' ratios are set out in 2. 14 pp. 71-3; see Barker, GMW ii. 346-9. |
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36Harm. 1. 14 p. 32. 15ff., cf. 1. 15 p. 33. 22; Barker, GMW ii. 305-7. |
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37Harm 1. 15-16. 2. 14. |
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