Consistency: Difference between revisions
mNo edit summary |
Correction (tempering -> tempering out). Move "pure consistency" together with consistency to distance d as instances of generalization. Request clarification on "the above equation" |
||
| Line 5: | Line 5: | ||
| ja = 一貫性 | | ja = 一貫性 | ||
}} | }} | ||
An [[edo]] (or other [[equal-step tuning]]) represents the ''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, the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An [[equal-step tuning]] is '''distinctly consistent''' (uniquely consistent) in the ''q''-odd-limit if every interval in that odd limit is mapped to a distinct/unique step (so for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step—This would correspond to tempering [[50/49]], and in the case of edos, would mean the edo must be a multiple, or superset, of 2edo | An [[edo]] (or other [[equal-step tuning]]) represents the ''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, the difference between the closest 7/4 and the closest 5/4 is also the closest 7/5. An [[equal-step tuning]] is '''distinctly consistent''' (uniquely consistent) in the ''q''-odd-limit if every interval in that odd limit is mapped to a distinct/unique step (so for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps [[7/5]] and [[10/7]] to the same step—This would correspond to [[tempering out]] [[50/49]], and in the case of edos, would mean the edo must be a multiple, or superset, of 2edo). | ||
Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent. | Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent. | ||
| Line 56: | Line 56: | ||
An example of the difference between consistency vs distinct 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 distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct 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 distinctly consistent only up to the [[11-odd-limit]]. | An example of the difference between consistency vs distinct 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 distinctly consistent only up to the [[5-odd-limit]]. Another example of non-distinct 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 distinctly consistent only up to the [[11-odd-limit]]. | ||
== Consistency to distance ''d'' == | == Generalization == | ||
A chord is '''consistent to distance''' ''d'' ≥ 1 or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to d intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.) | === Pure consistency === | ||
Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all odd 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%). | |||
=== Consistency to distance ''d'' === | |||
A chord is '''consistent to distance''' ''d'' ≥ 1 or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to ''d'' intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.) | |||
For ''d'' ≥ 1, this implies consistency in the ordinary sense. | For ''d'' ≥ 1, this implies consistency in the ordinary sense. | ||
| Line 88: | Line 92: | ||
Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the dyads in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via dyads that occur in C) such that adding another interval adjacent to ''S'' via a dyad in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''. | Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the dyads in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via dyads that occur in C) such that adding another interval adjacent to ''S'' via a dyad in ''C'' results in a chord that is inconsistent in ''N''-edo. The ''maximal connected neighborhood'' (MCN) of ''C'' is a maximal consistent set containing ''C''. | ||
== | == For non-octave tunings == | ||
It is possible to generalize the concept of consistency to non-edo equal-step tunings. Because octaves are no longer equivalent, instead of an odd limit we might use an integer limit, and the term 2<sup>''n''</sup> in the above equation is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ ''q'' and ''v'' ≤ ''q'' (''q'' is the largest integer harmonic in S). | It is possible to generalize the concept of consistency to non-edo equal-step tunings. Because octaves are no longer equivalent, instead of an odd limit we might use an integer limit, and the term 2<sup>''n''</sup> in the above equation{{clarify}} <!-- which? --> is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ ''q'' and ''v'' ≤ ''q'' (''q'' is the largest integer harmonic in S). | ||
This also means that the concept of octave inversion no longer applies: in this example, [[13/9]] is in S, but [[18/13]] is not. | This also means that the concept of octave inversion no longer applies: in this example, [[13/9]] is in S, but [[18/13]] is not. | ||