Consistency: Difference between revisions

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An [[edo]] represents the q-[[odd limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19. A different formulation: an edo ~~represents~~ a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:

An [[edo]] represents the q-[[odd limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19. A different formulation: an edo approximates a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:

# the same interval in C is always mapped to the same size in C', and

# no interval within the chord is off by more than 50% of an edo step.

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# no interval within ''C' '' has [[relative error]] 25% or more.

(The 25% threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have over 50% relative error; thus a strongly consistent approximation would play more nicely in a regular temperament-style [[subgroup]] context.)

An equivalent definition is that ''N''-edo ~~represents~~ ''C'' strongly consistently if 2''N''-edo ~~represents~~ ''C'' consistently.

An equivalent definition is that ''N''-edo approximates ''C'' strongly consistently if 2''N''-edo approximates ''C'' consistently.

The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.

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-An [[edo]] represents the q-[[odd limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19. A different formulation: an edo represents a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:
+An [[edo]] represents the q-[[odd limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19. A different formulation: an edo approximates a chord C '''consistently''' if there exists an approximation of the chord in the edo such that:
 # the same interval in C is always mapped to the same size in C', and
 # no interval within the chord is off by more than 50% of an edo step.
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 # no interval within ''C' '' has [[relative error]] 25% or more.
 (The 25% threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have over 50% relative error; thus a strongly consistent approximation would play more nicely in a regular temperament-style [[subgroup]] context.)
-An equivalent definition is that ''N''-edo represents ''C'' strongly consistently if 2''N''-edo represents ''C'' consistently.
+An equivalent definition is that ''N''-edo approximates ''C'' strongly consistently if 2''N''-edo approximates ''C'' consistently.
 The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.