Consistency: Difference between revisions

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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.

~~== Consistency to distance ''m'' ==~~

~~If ''m'' ≥ 0, a chord ''C'' is '''consistent to distance''' ''m'' in ''N''-edo if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:~~

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

~~# no interval within ''C' '' has [[relative error]] 1/(2(''m''+1)) or more.~~

~~"Consistent to distance 0" is equivalent to "consistent".~~

(The 1/(2(''m''+1)) threshold is meant to allow stacking ''m'' dyads that occur in the chord without having the sum of the dyads have over 50% relative error. Since "consistent to distance ''m''" conveys the idea that a local neighborhood of the consonant chord in the JI lattice is mapped nicely, an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.)

~~Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of that one chord to check distance-''m'' consistency.~~

For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:11 is only consistent and not to distance 1 because 11/5 is mapped too inaccurately (rel error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.

~~An example of a more advanced concept that builds on this is [[telicity]].~~

==Examples==

@@ Line 19: / Line 19: @@
 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.
-== Consistency to distance ''m'' ==
-If ''m'' ≥ 0, a chord ''C'' is '''consistent to distance''' ''m'' in ''N''-edo if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:
-# every instance of an interval in C is mapped to the same size in C', and
-# no interval within ''C' '' has [[relative error]] 1/(2(''m''+1)) or more.
-"Consistent to distance 0" is equivalent to "consistent".
-(The 1/(2(''m''+1)) threshold is meant to allow stacking ''m'' dyads that occur in the chord without having the sum of the dyads have over 50% relative error. Since "consistent to distance ''m''" conveys the idea that a local neighborhood of the consonant chord in the JI lattice is mapped nicely, an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.)
-Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of that one chord to check distance-''m'' consistency.
-For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:11 is only consistent and not to distance 1 because 11/5 is mapped too inaccurately (rel error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.
-An example of a more advanced concept that builds on this is [[telicity]].
 ==Examples==