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 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 span 1" is equivalent to "consistent".

Non-technically, a chord is '''consistent to span''' ''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 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.

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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 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 span 1" is equivalent to "consistent".
+Non-technically, a chord is '''consistent to span''' ''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 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.