Consistency: Difference between revisions

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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). Going even further, an equal-step tuning is '''purely consistent''' 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%).

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). Going even further, 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%).

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.

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-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&mdash;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). Going even further, an equal-step tuning is '''purely consistent''' 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%).
+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&mdash;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). Going even further, 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%).
 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.