Consistency: Difference between revisions

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(The 1\(2N(''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.)

For example, 4:5:6:7~~:9:15:21~~ is consistent to distance 1 in [[31edo]]. ~~(The worst interval is 9/7, which is just below the 25% threshold at 24.0% (9.278c) relative error.)~~ However, 4:5:6:7:9:11:15 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.

For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:9:11:15 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.

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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 (The 1\(2N(''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.)
-For example, 4:5:6:7:9:15:21 is consistent to distance 1 in [[31edo]]. (The worst interval is 9/7, which is just below the 25% threshold at 24.0% (9.278c) relative error.) However, 4:5:6:7:9:11:15 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.
+For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:9:11:15 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.
 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.