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'' ==

== Consistency to distance ''d'' ==

Non-technically, a chord is '''consistent to distance''' ''m'' in an edo, if the chord is consistent and you can "walk away" up to distance ''m'' from the chord consistently. So an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.

Non-technically, a chord is '''consistent to distance''' ''d'' in an edo, if the chord is consistent and you can "walk away" up to distance ''m'' from the chord consistently. So an approximation consistent to distance ''d'' would play more nicely in a regular temperament-style [[subgroup]] context.

Formally, 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:

Formally, if ''d'' ≥ 0, a chord ''C'' is ''consistent to distance'' ''d'' 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.

# no interval within ''C' '' has [[relative error]] 1/(2(''d''+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.)

(The 1/(2(''m''+1)) threshold is meant to allow stacking ''d'' dyads that occur in the chord without having the sum of the dyads have over 50% relative error.)

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.

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-''d'' consistency.

For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:11 is only consistent to distance 0 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 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'' ==
+== Consistency to distance ''d'' ==
-Non-technically, a chord is '''consistent to distance''' ''m'' in an edo, if the chord is consistent and you can "walk away" up to distance ''m'' from the chord consistently. So an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.
+Non-technically, a chord is '''consistent to distance''' ''d'' in an edo, if the chord is consistent and you can "walk away" up to distance ''m'' from the chord consistently. So an approximation consistent to distance ''d'' would play more nicely in a regular temperament-style [[subgroup]] context.
-Formally, 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:
+Formally, if ''d'' ≥ 0, a chord ''C'' is ''consistent to distance'' ''d'' 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.
+# no interval within ''C' '' has [[relative error]] 1/(2(''d''+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.)
+(The 1/(2(''m''+1)) threshold is meant to allow stacking ''d'' dyads that occur in the chord without having the sum of the dyads have over 50% relative error.)
-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.
+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-''d'' consistency.
 For example, 4:5:6:7 is consistent to distance 2 in [[31edo]]. However, 4:5:6:7:11 is only consistent to distance 0 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.