Consistency: Difference between revisions

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An [[edo]] (or other [[equal-step tuning]]) 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, 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.

An [[edo]] (or other [[equal-step tuning]]) 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, 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''' (or '''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.

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-An [[edo]] (or other [[equal-step tuning]]) 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, 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 out]] [[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 [[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, 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''' (or '''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 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.