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:

Non-technically, a chord ''C'' is '''consistent to distance''' ''m'' if a small piece of the JI lattice surrounding the chord (namely, all intervals up to distance ''m'' from notes that occur in the chord) is mapped consistently. So an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.

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.)

(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 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.

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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:
+Non-technically, a chord ''C'' is '''consistent to distance''' ''m'' if a small piece of the JI lattice surrounding the chord (namely, all intervals up to distance ''m'' from notes that occur in the chord) is mapped consistently. So an approximation consistent to distance ''m'' would play more nicely in a regular temperament-style [[subgroup]] context.
+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.)
+(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 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.