Consistency: Difference between revisions

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Examples on consistency vs. unique consistency: In [[12edo]] the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is uniquely consistent only up to the [[11-odd-limit]].

== 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]].

==Generalization to non-octave scales==

@@ Line 31: / Line 31: @@
 Examples on consistency vs. unique consistency: In [[12edo]] the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps. Although 72edo is consistent up to the [[17-odd-limit]], it is uniquely consistent only up to the [[11-odd-limit]].
+== 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]].
 ==Generalization to non-octave scales==

Consistency: Difference between revisions

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