Consistency: Difference between revisions

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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 a approximation ''C' '' of ''C'' in ''N''-edo such that no interval within ''C' '' has [[relative error]] more than 25% (The threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have more than >50% relative error.)

''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]] more than 25% (The threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have more than >50% relative error.)

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.

@@ Line 11: / Line 11: @@
 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 a approximation ''C' '' of ''C'' in ''N''-edo such that no interval within ''C' '' has [[relative error]] more than 25% (The threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have more than >50% relative error.)
+''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]] more than 25% (The threshold is meant to allow stacking two dyads ''a'', ''b'' once without having the sum ''a + b'' of the dyads have more than >50% relative error.)
 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.