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 no interval within the chord has to be off by more than 50% of an edo step. In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.

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:

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

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

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

The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators.

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Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.

''N''-edo is '''strongly consistent''' with respect to a chord ''C'' if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that no interval within ''C' '' has [[relative error]] 25% or more. (The 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.)

''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

# 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.)

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 no interval within the chord has to be off by more than 50% of an edo step. In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.
+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:
+# the same interval in C is always mapped to the same size in C', and
+# no interval within the chord has to be off by more than 50% of an edo step.
+In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently.
 The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators.
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 Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|q odd limit intervals]], consisting of everything of the form 2^n u/v, where u and v are odd integers less than or equal to q. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.
-''N''-edo is '''strongly consistent''' with respect to a chord ''C'' if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that no interval within ''C' '' has [[relative error]] 25% or more. (The 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.)
+''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
+# 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.)
 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.