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''' (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, 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.

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~~ [[19-odd-limit]], meaning ~~for~~ the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, and 19, 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-11s, no-13s, no-23s [[27-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, 21, 25, and 27, where we deliberately skip 11, 13, and 23. (Alternatively, with the octave tuned 2–6{{c}} flat, 12-[[ET]] is consistent in the 2.3.5.7.17.19-[[subgroup]] 48-[[integer-limit]].)

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''.

The concept only ~~makes sense~~ for [[equal-step tuning]]s and not for unequal multirank tunings, since for ~~some~~ 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).

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).

The page ''[[Minimal consistent edos]]'' shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page ''[[Consistency limits of small edos]]'' shows the largest odd limit that a given edo is consistent or distinctly consistent in.

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(''Under construction'')

Non-technically, a '''maximal consistent set'''(MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.

Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.

Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the intervals in ''C'', a ''maximal consistent set'' is a connected set ''S''(connected via intervals that occur in C) such that adding another interval adjacent to ''S'' via an interval 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{{clarify}} <!~~-- which~~? -->~~ is ~~no longer present. Instead,~~ the set ''S'' ~~consists~~ of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (''q'' is the largest integer harmonic in ''S'').

In non-edo [[equal-step tuning]]s, octaves are not perfectly tuned, and thus an infinite [[odd limit]] cannot fully be consistently represented. Instead, we measure consistency in the ''q''-[[integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an EDO describes the highest odd limit it represents consistently, while the consistency limit of any other equal-step tuning (or [[ET]] without an exact octave) instead describes the highest integer limit it represents consistently.

~~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.

The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not.

~~Alternatively~~, ~~we can use~~ "~~modulo-~~''n'' limit" if the [[~~equave~~]] is ''n''/1~~. Thus the tritave analogue of odd~~ limit ~~would only allow integers not divisible by 3 below~~ a ~~given number, assuming tritave equivalence and tritave invertibility~~.

It is possible to extend the concept of [[odd limits]] to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[EDT]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of [[EDO]]s with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for EDOs anyways (an EDO that is consistent to the ''q''-odd-limit will be consistent to the (''q'' + 1)-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.

== External links ==

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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''' (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.
+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&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.
-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 [[19-odd-limit]], meaning for the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, and 19, 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-11s, no-13s, no-23s [[27-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, 21, 25, and 27, where we deliberately skip 11, 13, and 23. (Alternatively, with the octave tuned 2&ndash;6{{c}} flat, 12-[[ET]] is consistent in the 2.3.5.7.17.19-[[subgroup]] 48-[[integer-limit]].)
 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'' &minus; 2)}}:''q''.
-The concept only makes sense for [[equal-step tuning]]s and not for unequal multirank tunings, since for some 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).
+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).
 The page ''[[Minimal consistent edos]]'' shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page ''[[Consistency limits of small edos]]'' shows the largest odd limit that a given edo is consistent or distinctly consistent in.
@@ Line 90: / Line 90: @@
 (''Under construction'')
-Non-technically, a '''maximal consistent set'''(MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.
+Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.
 Formally, given ''N''-edo, a consistent chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the intervals in ''C'', a ''maximal consistent set'' is a connected set ''S''(connected via intervals that occur in C) such that adding another interval adjacent to ''S'' via an interval 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{{clarify}} <!-- which? --> is no longer present. Instead, the set ''S'' consists of all intervals ''u''/''v'' where {{nowrap|''u'' &le; ''q''}} and {{nowrap|''v'' &le; ''q''}} (''q'' is the largest integer harmonic in ''S'').
+In non-edo [[equal-step tuning]]s, octaves are not perfectly tuned, and thus an infinite [[odd limit]] cannot fully be consistently represented. Instead, we measure consistency in the ''q''-[[integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' &le; ''q''}} and {{nowrap|''v'' &le; ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an EDO describes the highest odd limit it represents consistently, while the consistency limit of any other equal-step tuning (or [[ET]] without an exact octave) instead describes the highest integer limit it represents consistently.
-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.
+The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not.
-Alternatively, we can use "modulo-''n'' limit" if the [[equave]] is ''n''/1. Thus the tritave analogue of odd limit would only allow integers not divisible by 3 below a given number, assuming tritave equivalence and tritave invertibility.
+It is possible to extend the concept of [[odd limits]] to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[EDT]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of [[EDO]]s with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for EDOs anyways (an EDO that is consistent to the ''q''-odd-limit will be consistent to the (''q''&#x202F;+&#x202F;1)-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.
 == External links ==