Consistency: Difference between revisions

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== Generalizations ==

=== Pure consistency ===

Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of ~~less~~ than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.

Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer harmonics from 1 up to and including ''q'' within one-quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of no greater than than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2 , introduced next.

=== Consistency to distance ''d'' ===

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Therefore, consistency to large distances represent very accurate (relative to the step size) [[subgroup]] interpretations because a large "space" of the arithmetic is captured "correctly" (without causing contradictions; consistently). Approximations consistent to some reasonable distance (ideally at least 2) would play more nicely in a regular-temperament-style subgroup context where you might prefer a larger variety of low complexity intervals to be consistent to a lesser degree rather than focusing on long-range consistency of a small number of intervals.

Note that if the chord comprised of the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit|''q''-integer-limit]] (as well as the {{nowrap|(~~2⋅ceil(~~''q''/2~~) -~~ 1)}}-odd-limit if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept is intuitively equivalent to the idea of consistency with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times) – in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).

Note that if the chord comprised of the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit|''q''-integer-limit]] (as well as the {{nowrap|(2{{ceil|''q''/2}} − 1)}}-odd-limit if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept is intuitively equivalent to the idea of consistency with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times) – in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).

For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.

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 == Generalizations ==
 === Pure consistency ===
-Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of less than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.
+Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer harmonics from 1 up to and including ''q'' within one-quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of no greater than than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2 , introduced next.
 === Consistency to distance ''d'' ===
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 Therefore, consistency to large distances represent very accurate (relative to the step size) [[subgroup]] interpretations because a large "space" of the arithmetic is captured "correctly" (without causing contradictions; consistently). Approximations consistent to some reasonable distance (ideally at least 2) would play more nicely in a regular-temperament-style subgroup context where you might prefer a larger variety of low complexity intervals to be consistent to a lesser degree rather than focusing on long-range consistency of a small number of intervals.
-Note that if the chord comprised of the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit|''q''-integer-limit]] (as well as the {{nowrap|(2⋅ceil(''q''/2) - 1)}}-odd-limit if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept is intuitively equivalent to the idea of consistency with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times) – in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).
+Note that if the chord comprised of the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit|''q''-integer-limit]] (as well as the {{nowrap|(2{{ceil|''q''/2}} − 1)}}-odd-limit if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept is intuitively equivalent to the idea of consistency with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times) – in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).
 For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.