Consistency: Difference between revisions
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{{ | {{Interwiki | ||
| | | en = Consistency | ||
| | | 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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}} | }} | ||
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 == | ||