Consistency: Difference between revisions
Clarify, cleanup, and consistency in style |
Currywurst44 (talk | contribs) Move 'maximal consistent set' to user space as it hasn't been used in any example on the Wiki or been mentioned on either the Xengrove Discord or Xenharmonic Alliance Discord. |
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{{ | {{Interwiki | ||
| en = Consistent | |||
| de = konsistent | | de = konsistent | ||
| es = | | es = | ||
| ja = 一貫性 | | ja = 一貫性 | ||
}} | }} | ||
An [[edo]] represents the [[odd limit|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]). | An [[edo]] represents the [[odd limit|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]). | ||
While the term '''consistency''' is most frequently used to refer to some odd limit, sometimes one may only care about 'some' of the intervals in some odd limit; this situation often arises when working in [[JI subgroup]]s. We can also skip certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[21-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, and 21, where we deliberately skip 11 and 13. | While the term '''consistency''' is most frequently used to refer to some odd limit, sometimes one may only care about 'some' of the intervals in some odd limit; this situation often arises when working in [[JI subgroup]]s. We can also skip certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[21-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, and 21, where we deliberately skip 11 and 13. | ||
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== Mathematical definition == | == Mathematical definition == | ||
Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', {{nowrap|''T''(''r''<sub>''i''</sub>''r''<sub>''j''</sub>) {{=}} ''T''(''r''<sub>''i''</sub>) + ''T''(''r''<sub>''j''</sub>).}} | Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', {{nowrap|''T''(''r''<sub>''i''</sub>''r''<sub>''j''</sub>) {{=}} ''T''(''r''<sub>''i''</sub>) + ''T''(''r''<sub>''j''</sub>).}} | ||
This is equivalent to looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] that does the same approximation, so that the intervals are lined up by the val. If there is such a val, then the edo is consistent within that odd limit, otherwise it is inconsistent. | |||
; Alternative formulation using val | ; Alternative formulation using val | ||
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== Generalizations == | == Generalizations == | ||
=== Pure consistency === | === 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 | 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]]s 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'' === | === 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. | 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|( | 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. | 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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}} | }} | ||
Examples of more advanced concepts that build on this are [[telicity]] and [[# | Examples of more advanced concepts that build on this are [[telicity]] and [[#User:Inthar/Maximal_consistent_set|maximal consistent set]]s. | ||
== For non-octave tunings == | == For non-octave tunings == | ||