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The earliest Pythagoreans appear to have been entranced by the simple ratios of the octave, fifth, and fourth, and to have treated them as an exclusive set with a mystical significance. One of their catechistical propositions ran: 'What is the oracle at Delphi?Tetraktys, which is the octave (harmonia), which has the Sirens in it.'40 Tetraktys (tes dekados), 'tetradizing (of ten)', was the name they gave to an arrangement of the numbers 1, 2, 3, and 4 in a triangular figure: |
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Among other properties, it embodied the concord ratios 4:3, 3:2, 2:1; that is why it is identified with the octave.41 |
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Philolaus, a couple of generations later than Hippasus, is the first we know of who attempted to establish mathematical values for other intervals. He states that the difference between a fifth and a fourth is epogdoon, i.e. 9:8. The interval remaining when two of these epogdoa are subtracted from a fourth he calls diesis.42 According to Boethius, he calculated the diesis (correctly) to be 256:243, and constructed a series of further small units by subtractions and subdivisions: |
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tone minus diesis = apotome 'offcut'.
apotome minus diesis = komma 'chip'.
half of diesis = diaschisma 'splinter'.
half of komma = schisma 'crack'.43 |
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The purpose of bisecting the diesis and the minute komma must have been to produce values for the two equal small intervals of the |
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40 Iambl. De vita Pythagorica 82. For the Sirens cf. p. 224. |
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41 See Burkert, LS 72, 186f.; 'The tetractys has within it the secret of the world; and in this manner we can also understand the connection with Delphi, the seat of the highest and most secret wisdom. Perhaps Pythagorean speculation touched upon that focal point, or embodiment, of Delphic wisdom, the bronze tripod of Apollo' (187). We recall that Pythagoras of Zacynthus' triple kithara, upon which the Dorian, Phrygian, and Lydian modes could all be played, was built in the likeness of the Delphic tripod. |
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42 Philolaus fr. 6. Diesis means literally 'a letting-through', probably from the way in which an aulete raised a note slightly by half-stopping. Aristoxenus and his followers use the word to mean a quarter-tone or other microtone as used in the enharmonic and chromatic genera. |
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43 Boeth. Inst. Mus. 3.8 p. 278.11 Friedl. |
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