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A116474
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Equal divisions of the octave with progressively increasing consistency levels.
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3
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1, 3, 4, 5, 22, 26, 29, 58, 80, 94, 282, 311, 17461
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listen;
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OFFSET
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3,2
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COMMENTS
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An equal temperament is consistent at level N (odd integer) if all the intervals in the N-limit tonality diamond (set of ratios with odd factors of numerator and denominator not exceeding N) are approximated consistently, i.e. the composition of the approximations is the closest approximation of the composition.
These EDOs are not necessarily any good for musical purposes. Even though 4-EDO is consistent through the 7 limit, no one would seriously consider using it for 7-limit music because the approximations are so bad.
While for the smallest values these EDOs are not directly usable, their consistency is even so a valuable feature. For example, 4-EDO is consistent through the 7 limit, but is not usable directly for 7-limit music. However, indirectly, by means of subsequently adjusting the harmony, it can be and has been useful as a compositional tool for composing music in the 7-limit. The same comment applies to 3 in the 5-limit and 5 in the 9-limit. Any of the values above 5 are usable directly as equal temperament approximations to the corresponding limit. - Gene Ward Smith, Mar 29 2006
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LINKS
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EXAMPLE
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3-EDO is consistent through the 5 limit because 6/5, 5/4 and 4/3 map to 1 step and 3/2, 8/5 and 5/3 map to 2 steps and all the compositions work out, for example 6/5 * 5/4 = 3/2 and 1 step + 1 step = 2 steps. It is not consistent through the 7 limit because 8/7 and 7/6 both map to 1 step, but 8/7 * 7/6 = 4/3 also maps to 1 step.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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