Consistency: Difference between revisions

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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, 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~~.

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]]).

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.{{clarify}} If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.

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.

In general, we can say that some ~~edo~~ is '''consistent relative to a chord ''C''''', or that '''chord ''C'' is consistent in some ~~edo~~''', if its best approximation to all the notes in the chord, relative to the root, also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''.

In general, we can say that some [[equal-step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal-step tuning''', if its best approximation to all the notes in the chord, relative to the root,<sup style="white-space: nowrap;">{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}}</sup> also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''. By convention, when assessing a tuning's '''consistency limit''', the chord used is an odd limit in edos, or an [[integer limit]] in other equal step tunings ([[#For non-octave tunings]]).

The concept is only defined for [[equal-step tuning]]s and not for unequal multirank tunings, since for most choices of generator sizes in these temperaments, you can get any ratio you want to arbitrary precision by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

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-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, 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.
+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]]).
-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.{{clarify}} If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.
 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.
-In general, we can say that some edo is '''consistent relative to a chord ''C''''', or that '''chord ''C'' is consistent in some edo''', if its best approximation to all the notes in the chord, relative to the root, also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''.
+In general, we can say that some [[equal-step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal-step tuning''', if its best approximation to all the notes in the chord, relative to the root,<sup style="white-space: nowrap;">{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}}</sup> also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''. By convention, when assessing a tuning's '''consistency limit''', the chord used is an odd limit in edos, or an [[integer limit]] in other equal step tunings ([[#For non-octave tunings]]).
 The concept is only defined for [[equal-step tuning]]s and not for unequal multirank tunings, since for most choices of generator sizes in these temperaments, you can get any ratio you want to arbitrary precision by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).