Consistency: Difference between revisions

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(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 one chord.

Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of one chord to show 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.

@@ Line 27: / Line 27: @@
 (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 one chord.
+Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of one chord to show 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.