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'' ==

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.

Non-technically, a chord 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.

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:

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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'' ==
-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.
+Non-technically, a chord 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.
 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: