Consistency: Difference between revisions

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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:…:(''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'', ''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 ''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

| title=Proof for equivalence

| 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 '''m'''''i'' + '''m'''''j'', respectively.

| 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:

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=== Consistency to distance ''d'' ===

A chord is '''consistent to distance''' ''d'' ≥ 1 or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to ''d'' intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.)

A chord is '''consistent to distance''' {{nowrap|''d'' ≥ 1}} or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to ''d'' intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.)

For ''d'' ≥ 1, this implies consistency in the ordinary sense.

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.

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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 ''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''.

~~'''Theorem:'''~~ Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads 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).

{{proof

| title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads 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.

Proof: Consider the union ''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 dyad {{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 dyad ''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 dyad ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency. {{qed}}

}}

Consider any dyad ''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 C''i'' + ''m'', where 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 dyad ''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 1 - ε > 1/2 and 1 + ɛ respectively as approximations to the JI dyad ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency. ~~QED.~~

Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s.

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== 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''n'' in the above equation{{clarify}}  is no longer present. Instead, the set ''S'' consists of all intervals ''u''/''v'' where ''u'' ≤ ''q'' and ''v'' ≤ ''q'' (''q'' is the largest integer harmonic in ''S'').

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''n'' in the above equation{{clarify}}  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'').

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.

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[[Category:EDO theory pages]]

[[Category:Terms]]

[[Category:Consistency| ]]

[[Category:Consistency| ]]

[[Category:Odd limit]]

[[Category:Pages with proofs]]

@@ 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'' - 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:…:(''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'', ''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 ''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
 | title=Proof for equivalence
-| 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 '''m'''<sub>''i''</sub> + '''m'''<sub>''j''</sub>, respectively.
+| 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:
@@ Line 62: / Line 62: @@
 === Consistency to distance ''d'' ===
-A chord is '''consistent to distance''' ''d'' &ge; 1 or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to ''d'' intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.)
+A chord is '''consistent to distance''' {{nowrap|''d'' &ge; 1}} or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) [[Wikipedia: If and only if|iff]] the following holds: error accrues slowly enough that ''any'' 0 to ''d'' intervals can be combined (multiplied or divided) in ''any'' order without accruing 50% (i.e. half a step) or more of [[relative error]], ''as long as all the intervals chosen are ones present in the chord''. (Note that you may use the same interval ''d'' times even if only one instance of that interval is present in the chord.)
-For ''d'' &ge; 1, this implies consistency in the ordinary sense.
+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.
@@ Line 74: / Line 74: @@
 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 ''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'' &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''.
-'''Theorem:''' Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads 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).
+{{proof
+| title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via dyads 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.
-Proof: Consider the union ''C' '' = ''C''<sub>1</sub> ∪ ''C''<sub>2</sub> ∪ … ∪ ''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 dyad {{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 dyad ''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 dyad ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency. {{qed}}
+}}
-Consider any dyad ''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 C<sub>''i'' + ''m''</sub>, where 1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''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 dyad ''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 1 - ε > 1/2 and 1 + ɛ respectively as approximations to the JI dyad ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency. QED.
 Examples of more advanced concepts that build on this are [[telicity]] and [[#Maximal consistent set|maximal consistent set]]s.
@@ Line 94: / Line 95: @@
 == 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 ''u'' ≤ ''q'' and ''v'' ≤ ''q'' (''q'' is the largest integer harmonic in ''S'').
+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'').
 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.
@@ Line 106: / Line 107: @@
 [[Category:EDO theory pages]]
 [[Category:Terms]]
-[[Category:Consistency| ]] <!-- main article -->
+[[Category:Consistency| ]] <!-- Main article -->
 [[Category:Odd limit]]
 [[Category:Pages with proofs]]