Consistency: Difference between revisions

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~~ ''d'' ==

== Consistency to span ''d'' ==

Non-technically, a chord is '''consistent to distance''' ''d'' in an edo, if the chord is consistent and error accrues slowly enough that you can move up to distance ''d'' from the chord consistently. So an approximation consistent to some reasonable distance would play more nicely in a regular temperament-style [[subgroup]] context. "Consistent to ~~distance 0~~" is equivalent to "consistent".

Non-technically, a chord is '''consistent to distance''' ''d'' in an edo, if the chord is consistent and error accrues slowly enough that you can move up to distance ''d'' from the chord consistently. So an approximation consistent to some reasonable distance would play more nicely in a regular temperament-style [[subgroup]] context. "Consistent to span 1" is equivalent to "consistent".

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.

For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 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.

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:

Formally, if ''d'' ≥ 0, a chord ''C'' is ''consistent to span'' ''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(''d''+1)) or more.

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

(The 1/(2(''d''+1)) threshold is meant to allow stacking ''d+1'' chords, including the original chord, via dyads that occur in the chord without having the sum of the dyads have over 50% relative error. That is, you can make a copy of a chord up to distance d away from the original chord without inconsistency.)

(The 1/(2(''d'')) threshold is meant to allow stacking ''d+1'' chords, including the original chord, via dyads that occur in the chord without having the sum of the dyads have over 50% relative error. That is, you can make a copy of a chord up to distance d away from the original chord without inconsistency.)

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.

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 span-''d'' consistency.

Examples of more advanced concepts that build on this are [[telicity]] and [[Consistent#Maximal consistent set|maximal consistent set]]s.

@@ 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 ''d'' ==
+== Consistency to span ''d'' ==
-Non-technically, a chord is '''consistent to distance''' ''d'' in an edo, if the chord is consistent and error accrues slowly enough that you can move up to distance ''d'' from the chord consistently. So an approximation consistent to some reasonable distance would play more nicely in a regular temperament-style [[subgroup]] context. "Consistent to distance 0" is equivalent to "consistent".
+Non-technically, a chord is '''consistent to distance''' ''d'' in an edo, if the chord is consistent and error accrues slowly enough that you can move up to distance ''d'' from the chord consistently. So an approximation consistent to some reasonable distance would play more nicely in a regular temperament-style [[subgroup]] context. "Consistent to span 1" is equivalent to "consistent".
-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.
+For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 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.
-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:
+Formally, if ''d'' ≥ 0, a chord ''C'' is ''consistent to span'' ''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(''d''+1)) or more.
+# no interval within ''C' '' has [[relative error]] 1/(2(''d'')) or more.
-(The 1/(2(''d''+1)) threshold is meant to allow stacking ''d+1'' chords, including the original chord, via dyads that occur in the chord without having the sum of the dyads have over 50% relative error. That is, you can make a copy of a chord up to distance d away from the original chord without inconsistency.)
+(The 1/(2(''d'')) threshold is meant to allow stacking ''d+1'' chords, including the original chord, via dyads that occur in the chord without having the sum of the dyads have over 50% relative error. That is, you can make a copy of a chord up to distance d away from the original chord without inconsistency.)
-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.
+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 span-''d'' consistency.
 Examples of more advanced concepts that build on this are [[telicity]] and [[Consistent#Maximal consistent set|maximal consistent set]]s.