Consistency: Difference between revisions
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Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to distance 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to distance 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 distance 1", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to distance 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 | For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:6:7:11 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 2.3.5.7.11. | ||
Formally, for some real ''d'' > 0, a chord C is consistent to distance ''d'' in ''n'' ED''k'' if the consistent approximation C' of C in ''n'' ED''k'' satisfies the property that all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n'' ED''k''. | Formally, for some real ''d'' > 0, a chord C is consistent to distance ''d'' in ''n'' ED''k'' if the consistent approximation C' of C in ''n'' ED''k'' satisfies the property that all intervals in C' are off from their corresponding intervals in C by less than 1/(2''d'') steps of ''n'' ED''k''. | ||