Consistency: Difference between revisions

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(If such an approximation exists, it must be the only such approximation, since changing one interval would make that interval go over the 50% threshold.)

In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]. This is a feature, not a bug, as the distinction can be useful in some circumstances.

In this formulation, 12edo represents the chord 1:3:5:7:9:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]. This is a feature, not a bug, as the distinction can be useful in some circumstances.

The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

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Therefore, consistency to large spans 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 high-span consistency of a small number of intervals.

Note that if the chord comprised of all the odd harmonics up to the ''k''-th is "consistent to span 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it 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); namely, the ones present in the chord.

Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to span 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it 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); namely, the ones present in the chord.

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 (relative 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.

@@ Line 13: / Line 13: @@
 (If such an approximation exists, it must be the only such approximation, since changing one interval would make that interval go over the 50% threshold.)
-In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]. This is a feature, not a bug, as the distinction can be useful in some circumstances.
+In this formulation, 12edo represents the chord 1:3:5:7:9:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]. This is a feature, not a bug, as the distinction can be useful in some circumstances.
 The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).
@@ Line 42: / Line 42: @@
 Therefore, consistency to large spans 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 high-span consistency of a small number of intervals.
-Note that if the chord comprised of all the odd harmonics up to the ''k''-th is "consistent to span 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it 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); namely, the ones present in the chord.
+Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to span 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it 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); namely, the ones present in the chord.
 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 (relative 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.