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 is '''consistent to distance''' ''m'' if 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''' ''m'' 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.

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 is '''consistent to distance''' ''m'' if 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''' ''m'' 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.
 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: