Consistency: Difference between revisions

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| ja = 一貫性

}}

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'''(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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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.

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:…:(''q'' − 2):''q''.

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).

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== 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''''i'' and ''r''''j'' in ''S'' where ''r''''i''''r''''j'' is also in ''S'', {{nowrap|''T'' (''r''''i''''r''''j'') {{=}} ''T'' (''r''''i'') + ''T'' (''r''''j'').}}

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''''i'' and ''r''''j'' in ''S'' where ''r''''i''''r''''j'' is also in ''S'', {{nowrap|''T''(''r''''i''''r''''j'') {{=}} ''T''(''r''''i'') + ''T''(''r''''j'').}}

; Alternative formulation using val

If for any interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that {{nowrap|''T'' (''r'') {{=}} ''V'' (''r'')}} for any ''r'' in ''S''.

If for any interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', and if ''V''(''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that {{nowrap|''T''(''r'') {{=}} ''V''(''r'')}} for any ''r'' in ''S''.

{{Proof

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| contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''''i'', ''r''''j'', and ''r''''i''''r''''j'' in ''S'', their monzos are '''m'''''i'', '''m'''''j'', and {{nowrap|'''m'''''i'' + '''m'''''j''}}, respectively.

The ratio ''r'' mapped by the val ''V'' is the tempered step number ''V'' (''r'') ~~=~~ ''V''·'''m''', with the following identity:

The ratio ''r'' mapped by the val ''V'' is the tempered step number {{nowrap|''V''(''r'') {{=}} ''V''·'''m'''}}, with the following identity:

Hence,

If ''T'' satisfies

then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that ''T'' (''r'') ~~=~~ ''V'' (''r'') for any ''r'' in ''S''.

then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that {{nowrap|''T''(''r'') {{=}} ''V''(''r'')}} for any ''r'' in ''S''.

}}

Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form 2''n'' ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''T'' is then said to be ''q-odd-limit consistent''.

Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form {{nowrap|2''n'' ''u''/''v''}}, where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''T'' is then said to be ''q-odd-limit consistent''.

If each interval in the ''q''-odd-limit is mapped to a unique value by ''T'', then it is said to be ''uniquely q-odd-limit consistent''.

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An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:

The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives (3/2)(7/6) = [[7/4]], the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives {{nowrap|(3/2)(7/6) {{=}} [[7/4]]}}, the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

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For {{nowrap|''d'' ≥ 1}}, this implies consistency in the ordinary sense.

For the mathematically/geometrically inclined, you can think of the set of all ''n'' {{w|equality (mathematics)|distinct}} intervals in the chord as forming ''n'' (mutually perpendicular) axes of length 1 that form a (hyper)cubic grid of points (existing in ''n''-dimensional space) representing intervals. Then moving in the direction of one of these axes by 1 unit of distance represents multiplying by the corresponding interval once, and going in the opposite direction represents division by that interval. Then, to be ''consistent to distance d'' means that all points that are a [[taxicab distance]] of at most ''d'' from the origin (which represents unison) have the [[direct approximation]] of their associated intervals agree with the sum of the steps accumulated through how they were reached in terms of moving along axes, with each axis representing the whole number of steps that closest fits the associated interval present in the chord.

For the mathematically/geometrically inclined, you can think of the set of all ''n'' {{w|equality (mathematics)|distinct}} intervals in the chord as forming ''n''(mutually perpendicular) axes of length 1 that form a (hyper)cubic grid of points (existing in ''n''-dimensional space) representing intervals. Then moving in the direction of one of these axes by 1 unit of distance represents multiplying by the corresponding interval once, and going in the opposite direction represents division by that interval. Then, to be ''consistent to distance d'' means that all points that are a [[taxicab distance]] of at most ''d'' from the origin (which represents unison) have the [[direct approximation]] of their associated intervals agree with the sum of the steps accumulated through how they were reached in terms of moving along axes, with each axis representing the whole number of steps that closest fits the associated interval present in the chord.

Therefore, consistency to large distances represent very accurate (relative to the step size) [[subgroup]] interpretations because a large "space" of the arithmetic is captured "correctly" (without causing contradictions; consistently). Approximations consistent to some reasonable distance (ideally at least 2) would play more nicely in a regular-temperament-style subgroup context where you might prefer a larger variety of low complexity intervals to be consistent to a lesser degree rather than focusing on long-range consistency of a small number of intervals.

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For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.

Formally, for some real {{nowrap|''d'' > 0}}, a chord ''C'' is consistent to distance ''d'' in ''n''-ed-''k'' if the consistent approximation ''C' '' of ''C'' in ''n''-ed-''k'' satisfies the property that all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than 1/(2''d'') steps of ''n''-ed-''k''.

Formally, for some real {{nowrap|''d'' > 0}}, a chord ''C'' is consistent to distance ''d'' in ''n''-ed-''k'' if the consistent approximation ''C'' of ''C'' in ''n''-ed-''k'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''C'' by less than 1/(2''d'') steps of ''n''-ed-''k''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in ''C'' are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of ''C''.

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in ''C'' are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C'' is said to be a "semiconsistent" representation/approximation of ''C''.

{{Proof

| title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via intervals that occur in the chord, so that the resulting chord (the union of the ''d'' copies) will always be consistent in the temperament (no matter which intervals are used to stack the ''d'' copies).

| contents=Consider the union {{nowrap|''C' '' {{=}} ''C''1 ∪ ''C''2 ∪ … ∪ ''C''''d''}} in the equal temperament, where the ''C''''i'' are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.

| contents=Consider the union {{nowrap|''C'' {{=}} ''C''1 ∪ ''C''2 ∪ … ∪ ''C''''d''}} in the equal temperament, where the ''C''''i'' are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.

Consider any interval {{nowrap|''D'' {{=}} {{(}}''x'', ''y''{{)}}}} consisting of two notes ''x'' and ''y'' that occur in ''C' ''. We may assume that the notes ''x'' and ''y'' belong in two different copies of ''C'', ''C''''i'' and {{nowrap|C''i'' + ''m''}}, where {{nowrap|1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''}}. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each interval ''D''''j'' in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 − ε > 1/2}} and {{nowrap|1 + ɛ}} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.

Consider any interval {{nowrap|''D'' {{=}} {{(}}''x'', ''y''{{)}}}} consisting of two notes ''x'' and ''y'' that occur in ''C''. We may assume that the notes ''x'' and ''y'' belong in two different copies of ''C'', ''C''''i'' and {{nowrap|C''i'' + ''m''}}, where {{nowrap|1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''}}. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each interval ''D''''j'' in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 − ε > 1/2}} and {{nowrap|1 + ɛ}} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.

}}

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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''.

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 ==

@@ Line 5: / Line 5: @@
 | ja = 一貫性
 }}
-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'''(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.
@@ Line 11: / Line 11: @@
 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.
-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:…:(''q'' &minus; 2):''q''.
+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).
@@ Line 18: / Line 18: @@
 == 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>).}}
 ; Alternative formulation using val
-If for any interval ''r'', ''T'' (''r'') is the closest approximation to ''r'' in ''T'', and if ''V'' (''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that {{nowrap|''T'' (''r'') {{=}} ''V'' (''r'')}} for any ''r'' in ''S''.
+If for any interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', and if ''V''(''r'') is ''r'' mapped by a val ''V'', then ''T'' is consistent with respect to a set of intervals ''S'' if there exists a val ''V'' such that {{nowrap|''T''(''r'') {{=}} ''V''(''r'')}} for any ''r'' in ''S''.
 {{Proof
@@ Line 27: / Line 27: @@
 | contents=Let us denote the monzo of any ratio ''r'' by '''m'''. Due to the linearity of the interval space, for any intervals ''r''<sub>''i''</sub>, ''r''<sub>''j''</sub>, and ''r''<sub>''i''</sub>''r''<sub>''j''</sub> in ''S'', their monzos are '''m'''<sub>''i''</sub>, '''m'''<sub>''j''</sub>, and {{nowrap|'''m'''<sub>''i''</sub> + '''m'''<sub>''j''</sub>}}, respectively.
-The ratio ''r'' mapped by the val ''V'' is the tempered step number ''V'' (''r'') &#61; ''V''·'''m''', with the following identity:
+The ratio ''r'' mapped by the val ''V'' is the tempered step number {{nowrap|''V''(''r'') {{=}} ''V''&#xB7;'''m'''}}, with the following identity:
-<math>V\cdot(\vec {m_i} + \vec {m_j}) &#61; V\cdot\vec {m_i} + V\cdot\vec {m_j}</math>
+<math>V\cdot(\vec {m_i} + \vec {m_j}) = V\cdot\vec {m_i} + V\cdot\vec {m_j}</math>
 Hence,
-<math>V (r_i r_j) &#61; V (r_i) + V (r_j)</math>
+<math>V (r_i r_j) = V (r_i) + V (r_j)</math>
 If ''T'' satisfies
-<math>T (r_i r_j) &#61; T (r_i) + T (r_j)</math>
+<math>T (r_i r_j) = T (r_i) + T (r_j)</math>
-then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that ''T'' (''r'') &#61; ''V'' (''r'') for any ''r'' in ''S''.
+then ''T'' is an element of the function space formed by all vals {''V''}. Therefore, there exists a val ''V'' such that {{nowrap|''T''(''r'') {{=}} ''V''(''r'')}} for any ''r'' in ''S''.
 }}
-Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form 2<sup>''n''</sup> ''u''/''v'', where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''T'' is then said to be ''q-odd-limit consistent''.
+Normally, ''S'' is considered to be some set of ''q''-odd-limit intervals, consisting of everything of the form {{nowrap|2<sup>''n''</sup> ''u''/''v''}}, where ''u'' and ''v'' are odd integers less than or equal to ''q''. ''T'' is then said to be ''q-odd-limit consistent''.
 If each interval in the ''q''-odd-limit is mapped to a unique value by ''T'', then it is said to be ''uniquely q-odd-limit consistent''.
@@ Line 49: / Line 49: @@
 An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:
-The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives (3/2)(7/6) = [[7/4]], the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
+The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives {{nowrap|(3/2)(7/6) {{=}} [[7/4]]}}, the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
 An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].
@@ Line 66: / Line 66: @@
 For {{nowrap|''d'' &ge; 1}}, this implies consistency in the ordinary sense.
-For the mathematically/geometrically inclined, you can think of the set of all ''n'' {{w|equality (mathematics)|distinct}} intervals in the chord as forming ''n'' (mutually perpendicular) axes of length 1 that form a (hyper)cubic grid of points (existing in ''n''-dimensional space) representing intervals. Then moving in the direction of one of these axes by 1 unit of distance represents multiplying by the corresponding interval once, and going in the opposite direction represents division by that interval. Then, to be ''consistent to distance d'' means that all points that are a [[taxicab distance]] of at most ''d'' from the origin (which represents unison) have the [[direct approximation]] of their associated intervals agree with the sum of the steps accumulated through how they were reached in terms of moving along axes, with each axis representing the whole number of steps that closest fits the associated interval present in the chord.
+For the mathematically/geometrically inclined, you can think of the set of all ''n'' {{w|equality (mathematics)|distinct}} intervals in the chord as forming ''n''(mutually perpendicular) axes of length 1 that form a (hyper)cubic grid of points (existing in ''n''-dimensional space) representing intervals. Then moving in the direction of one of these axes by 1 unit of distance represents multiplying by the corresponding interval once, and going in the opposite direction represents division by that interval. Then, to be ''consistent to distance d'' means that all points that are a [[taxicab distance]] of at most ''d'' from the origin (which represents unison) have the [[direct approximation]] of their associated intervals agree with the sum of the steps accumulated through how they were reached in terms of moving along axes, with each axis representing the whole number of steps that closest fits the associated interval present in the chord.
 Therefore, consistency to large distances represent very accurate (relative to the step size) [[subgroup]] interpretations because a large "space" of the arithmetic is captured "correctly" (without causing contradictions; consistently). Approximations consistent to some reasonable distance (ideally at least 2) would play more nicely in a regular-temperament-style subgroup context where you might prefer a larger variety of low complexity intervals to be consistent to a lesser degree rather than focusing on long-range consistency of a small number of intervals.
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 For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.
-Formally, for some real {{nowrap|''d'' &gt; 0}}, a chord ''C'' is consistent to distance ''d'' in ''n''-ed-''k'' if the consistent approximation ''C' '' of ''C'' in ''n''-ed-''k'' satisfies the property that all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than 1/(2''d'') steps of ''n''-ed-''k''.
+Formally, for some real {{nowrap|''d'' &gt; 0}}, a chord ''C'' is consistent to distance ''d'' in ''n''-ed-''k'' if the consistent approximation ''C'' of ''C'' in ''n''-ed-''k'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''C'' by less than 1/(2''d'') steps of ''n''-ed-''k''.
-This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in ''C'' are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of ''C''.
+This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in ''C'' are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C'' is said to be a "semiconsistent" representation/approximation of ''C''.
 {{Proof
 | title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via intervals that occur in the chord, so that the resulting chord (the union of the ''d'' copies) will always be consistent in the temperament (no matter which intervals are used to stack the ''d'' copies).
-| contents=Consider the union {{nowrap|''C' '' {{=}} ''C''<sub>1</sub> &cup; ''C''<sub>2</sub> &cup; … &cup; ''C''<sub>''d''</sub>}} in the equal temperament, where the ''C''<sub>''i''</sub> are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.
+| contents=Consider the union {{nowrap|''C'' {{=}} ''C''<sub>1</sub> &cup; ''C''<sub>2</sub> &cup; … &cup; ''C''<sub>''d''</sub>}} in the equal temperament, where the ''C''<sub>''i''</sub> are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.
-Consider any interval {{nowrap|''D'' {{=}} {{(}}''x'', ''y''{{)}}}} consisting of two notes ''x'' and ''y'' that occur in ''C' ''. We may assume that the notes ''x'' and ''y'' belong in two different copies of ''C'', ''C''<sub>''i''</sub> and {{nowrap|C<sub>''i'' + ''m''</sub>}}, where {{nowrap|1 &le; ''i'' &le; ''i'' + ''m'' &le; ''d''}}. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' &minus; 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each interval ''D''<sub>''j''</sub> in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 &minus; ε &gt; 1/2}} and {{nowrap|1 + ɛ}} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.
+Consider any interval {{nowrap|''D'' {{=}} {{(}}''x'', ''y''{{)}}}} consisting of two notes ''x'' and ''y'' that occur in ''C''. We may assume that the notes ''x'' and ''y'' belong in two different copies of ''C'', ''C''<sub>''i''</sub> and {{nowrap|C<sub>''i'' + ''m''</sub>}}, where {{nowrap|1 &le; ''i'' &le; ''i'' + ''m'' &le; ''d''}}. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' &minus; 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each interval ''D''<sub>''j''</sub> in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ε on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap|1 &minus; ε &gt; 1/2}} and {{nowrap|1 + &#603;}} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.
 }}
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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''.
+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 ==