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 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

# every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 is approximated using two fifths of the same size), and

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

In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.

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''N''-edo is '''strongly consistent''' with respect to a chord ''C'' if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:

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

# every instance of an interval in C is mapped to the same size in C', and

# 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.)

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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 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
+# every instance of an interval in C is mapped to the same size in C' (for example, 4:6:9 is approximated using two fifths of the same size), and
 # no interval within the chord is off by more than 50% of an edo step.
 In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.
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 ''N''-edo is '''strongly consistent''' with respect to a chord ''C'' if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:
-# the same interval in C is always mapped to the same size in C', and
+# every instance of an interval in C is mapped to the same size in C', and
 # 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.)