User:Currywurst44/Consistency Rewrite: Difference between revisions

Line 7:

'''''3c''''' 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]]).

'''''9c'''''The concept is only defined for equal-step tunings 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).

'''''10c''''' 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 28:

Line 30:

'''''12a''''' ''S'' shall be a set of intervals and ''M'' a tuning's pitch mapping of these intervals. ''r''''1'' and ''r''''2'' shall be in ''S'' with ''r''''1'' * ''r''''2'' = ''r''''3'' also in S. A tuning is '''consistent''' to distance ''d'' when the error of all ''M''(''r''''1'' * ''r''''2'') < 1/(2''d'') and the interval mapping is linear with ''M''(''r''''1'' * ''r''''2'') = ''M''(''r''''1'') + ''M''(''r''''2'').

A tuning is '''distinctly consistent''' ~~when~~ all ''M''(''r''''3'') are different.

A tuning is '''distinctly consistent''' if all ''M''(''r''''3'') are different.

==Examples==

'''''20c''''' An example for a system that is ''not'' consistent in the 7-odd-limit is [[25edo]]:

'''''21c''''' 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.

'''''22c''''' 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.

'''''23c''''' Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit.

'''''24c''''' The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

'''''27c''''' 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.

'''''28c'''''12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

'''''29c'''''An example of the difference between consistency vs distinct consistency:

'''''30c''''' In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]].

26edo integer vs odd distinct consistency

==Application==

Line 36:

Line 58:

===Interval Sets===

'''''65a!''''' ~~Odd~~-limit, integer-limit, >1-integer-limit or ~~odd-harmonic-chords~~ are all equivalent when used with an edo ~~for~~ consistency but differ for distinct consistency.

Depending on the application, a variety of interval sets may be checked for consistency. Below are common ones but other sets are possible.

'''''65a!''''' An odd-limit, integer-limit, >1-integer-limit or chord of the same limit are all equivalent when used with an edo in regards to consistency but differ for distinct consistency.

====Odd Limit====

'''''58a''''' [[Odd limit]]s are infinite sets so a tuning without pure octaves will always accrue infinite error as well and be inconsistent. It is only applicable to [[EDO]]s.

'''''58a''''' [[Odd limit]]s are infinite sets so a tuning without pure octaves will always accrue infinite error as well and be inconsistent. Because of this it is only applicable to [[EDO]]s.

====Other Equave Limits====

'''''64c''''' It is possible to extend the concept of odd limits to other [[equave limit]]s, 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.

====Integer Limit====

'''''25c''''' [[46edo]] is not consistent in the 15-integer-limit.

'''''26c''''' The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.

'''''63a'''''If we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system.

====>1-Integer Limit====

It might be desirable to investigate a tunings consistency above some root instead of considering arbitrarily removed modulations. This is accomplished by only including intervals larger than 1/1 in the interval set ~~(excluding 1/1)~~.

It might be desirable to investigate a tunings consistency above some root instead of considering arbitrarily removed modulations. This is accomplished by only including intervals larger than 1/1 in the interval set.

~~In practice~~ >1-integer-limit consistency can be checked by ~~checking~~ the relative error of intervals with a denominator two or more less less than the numerator. In the 6-limit for example

It can be proven that >1-integer-limit consistency can be checked by looking at the relative error of intervals with a denominator two or more less less than the numerator. In the 6-limit for example

{3/1, 4/2=2/1, 4/1, 5/3, 5/2, 5/1, 6/1}

====Chords====

'''''6,7,8a!''''' An odd harmonics chord (1:3:5:7:9:…) provides an alternative construction for odd-limit consistency. Using all possible differences between notes of the chord produces an interval set on which consistency is equivalent to the full odd-limit consistency for edos. For example in the 9-limit {9/7, 9/5, 9/1, 7/5, 7/3, 7/1, 5/3, 5/1, 3/1, 1/1}. This provides an interval set that has a few intervals as possible while still being equivalent to consistency of other interval sets.

'''''6,7,8a!''''' An odd harmonics chord (1:3:5:7:9:…) provides an alternative construction for odd-limit consistency. Using all possible differences between notes of the chord produces an interval set on which consistency is equivalent to the full odd-limit consistency for edos. For example in the 9-limit {9/7, 9/5, 9/1, 7/5, 7/3, 7/1, 5/3, 5/1, 3/1, 1/1}. This construction provides an interval set that has a few intervals as possible while still being equivalent to consistency of other common interval sets.

Distinct consistency for an odd-harmonic-chord-set can be interpreted as the tuning being distinct in a [[Non-over-1 temperament|non-over-1]] context.

Using a full harmonic series chord (1:2:3:4:5:6:..) simply produces all intervals of the corresponding integer limit.

~~====Integer Harmonics====~~

Consistency to distance d over the integer harmonics {1/1, 2/1, 3/1, 4/1, 5/1, 6/1,…} implies consistency to distance d/2 in the corresponding integer-limit. A distance of 2 or 25% relative error is sometimes called pure consistency{{idiosyncratic}}.

====JI-Subgroups====

'''''4,5a!''''' Interval sets are often modified to be compatible with a JI-subgroup by removing all intervals that contain a certain prime in its factorization.

Removing all powers of 2 results in the same interval set as an odd-harmonic-chord-set.

====Integer Harmonics====

Consistency to distance d over the integer harmonics {1/1, 2/1, 3/1, 4/1, 5/1, 6/1,…} implies consistency to distance d/2 in the corresponding integer-limit. A distance of 2 or 25% relative error is sometimes called pure consistency{{idiosyncratic}} and implies consistency to distance 1 in the same odd/integer-limit.

===Distance===

=Unrewritten Original Article=

Line 72:

Line 105:

#'''''A''''' In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the [[chord of nature|chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q'']].

#'''''A''''' By convention, when assessing a tuning's '''consistency limit''', this type of odd-integer harmonic series chord (limited to an [[odd limit]]) is used in edos, while in other equal-step tunings the unmodified harmonic series (limited to an [[integer limit]]) is used instead.

#The concept is only defined for equal-step tunings 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).

#'''''C''''' The concept is only defined for equal-step tunings 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).

#'''''C''''' 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.

#'''''C''''' '''==Mathematical definition=='''

Line 85:

Line 118:

#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''.

#'''==Examples=='''

#'''''C''''' '''==Examples=='''

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

#'''''C'''''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.

#'''''C''''' 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.

#'''''C''''' 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.

#'''''C''''' 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.

#'''''C''''' The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

#As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]].

#'''''C''''' As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]].

#The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.

#'''''C''''' The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.

#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.

#'''''C''''' 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]].

#'''''C'''''12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

#An example of the difference between consistency vs distinct consistency:

#'''''C'''''An example of the difference between consistency vs distinct consistency:

#In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]].

#'''''C''''' In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]].

#Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps.

#'''''D''''' Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps.

#Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]].

#'''''D''''' Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]].

#'''==Generalizations=='''

#'''=Pure consistency='''

Line 132:

Line 165:

#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.

#As a result, [[stretched and compressed tuning|octave stretch and compression]] can be employed to improve an equal tuning's consistency limits:

#if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series.

#'''''A''''' if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series.

#'''''C''''' 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 edos 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 {{nowrap|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.

@@ Line 7: / Line 7: @@
 '''''3c''''' 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]]).
+'''''9c'''''The concept is only defined for equal-step tunings 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).
 '''''10c''''' 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 28: / Line 30: @@
 '''''12a''''' ''S'' shall be a set of intervals and ''M'' a tuning's pitch mapping of these intervals. ''r''<sub>''1''</sub> and ''r''<sub>''2''</sub> shall be in ''S'' with ''r''<sub>''1''</sub> * ''r''<sub>''2''</sub> = ''r''<sub>''3''</sub> also in S. A tuning is '''consistent''' to distance ''d'' when the error of all ''M''(''r''<sub>''1''</sub> * ''r''<sub>''2''</sub>) < 1/(2''d'') and the interval mapping is linear with ''M''(''r''<sub>''1''</sub> * ''r''<sub>''2''</sub>) = ''M''(''r''<sub>''1''</sub>) + ''M''(''r''<sub>''2''</sub>).
-A tuning is '''distinctly consistent''' when all ''M''(''r''<sub>''3''</sub>) are different.
+A tuning is '''distinctly consistent''' if all ''M''(''r''<sub>''3''</sub>) are different.
 ==Examples==
+'''''20c''''' An example for a system that is ''not'' consistent in the 7-odd-limit is [[25edo]]:
+'''''21c''''' 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.
+'''''22c''''' 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.
+'''''23c''''' Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit.
+'''''24c''''' The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
+'''''27c''''' 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.
+'''''28c'''''12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].
+'''''29c'''''An example of the difference between consistency vs distinct consistency:
+'''''30c''''' In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]].
+edo integer vs odd distinct consistency
 ==Application==
@@ Line 36: / Line 58: @@
 ===Interval Sets===
-'''''65a!''''' Odd-limit, integer-limit, >1-integer-limit or odd-harmonic-chords are all equivalent when used with an edo for consistency but differ for distinct consistency.
+Depending on the application, a variety of interval sets may be checked for consistency. Below are common ones but other sets are possible.
+'''''65a!''''' An odd-limit, integer-limit, >1-integer-limit or chord of the same limit are all equivalent when used with an edo in regards to consistency but differ for distinct consistency.
 ====Odd Limit====
-'''''58a''''' [[Odd limit]]s are infinite sets so a tuning without pure octaves will always accrue infinite error as well and be inconsistent. It is only applicable to [[EDO]]s.
+'''''58a''''' [[Odd limit]]s are infinite sets so a tuning without pure octaves will always accrue infinite error as well and be inconsistent. Because of this it is only applicable to [[EDO]]s.
 ====Other Equave Limits====
 '''''64c''''' It is possible to extend the concept of odd limits to other [[equave limit]]s, 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.
 ====Integer Limit====
+'''''25c''''' [[46edo]] is not consistent in the 15-integer-limit.
+'''''26c''''' The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.
+'''''63a'''''If we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system.
 ====>1-Integer Limit====
-It might be desirable to investigate a tunings consistency above some root instead of considering arbitrarily removed modulations. This is accomplished by only including intervals larger than 1/1 in the interval set (excluding 1/1).
+It might be desirable to investigate a tunings consistency above some root instead of considering arbitrarily removed modulations. This is accomplished by only including intervals larger than 1/1 in the interval set.
-In practice >1-integer-limit consistency can be checked by checking the relative error of intervals with a denominator two or more less less than the numerator. In the 6-limit for example
+It can be proven that >1-integer-limit consistency can be checked by looking at the relative error of intervals with a denominator two or more less less than the numerator. In the 6-limit for example
 {3/1, 4/2=2/1, 4/1, 5/3, 5/2, 5/1, 6/1}
 ====Chords====
-'''''6,7,8a!''''' An odd harmonics chord (1:3:5:7:9:…) provides an alternative construction for odd-limit consistency. Using all possible differences between notes of the chord produces an interval set on which consistency is equivalent to the full odd-limit consistency for edos. For example in the 9-limit {9/7, 9/5, 9/1, 7/5, 7/3, 7/1, 5/3, 5/1, 3/1, 1/1}.  This provides an interval set that has a few intervals as possible while still being equivalent to consistency of other interval sets.
+'''''6,7,8a!''''' An odd harmonics chord (1:3:5:7:9:…) provides an alternative construction for odd-limit consistency. Using all possible differences between notes of the chord produces an interval set on which consistency is equivalent to the full odd-limit consistency for edos. For example in the 9-limit {9/7, 9/5, 9/1, 7/5, 7/3, 7/1, 5/3, 5/1, 3/1, 1/1}.  This construction provides an interval set that has a few intervals as possible while still being equivalent to consistency of other common interval sets.
+Distinct consistency for an odd-harmonic-chord-set can be interpreted as the tuning being distinct in a [[Non-over-1 temperament|non-over-1]] context.
 Using a full harmonic series chord (1:2:3:4:5:6:..) simply produces all intervals of the corresponding integer limit.
-====Integer Harmonics====
-Consistency to distance d over the integer harmonics {1/1, 2/1, 3/1, 4/1, 5/1, 6/1,…} implies consistency to distance d/2 in the corresponding integer-limit. A distance of 2 or 25% relative error is sometimes called pure consistency{{idiosyncratic}}.
 ====JI-Subgroups====
 '''''4,5a!''''' Interval sets are often modified to be compatible with a JI-subgroup by removing all intervals that contain a certain prime in its factorization.
+Removing all powers of 2 results in the same interval set as an odd-harmonic-chord-set.
+====Integer Harmonics====
+Consistency to distance d over the integer harmonics {1/1, 2/1, 3/1, 4/1, 5/1, 6/1,…} implies consistency to distance d/2 in the corresponding integer-limit. A distance of 2 or 25% relative error is sometimes called pure consistency{{idiosyncratic}} and implies consistency to distance 1 in the same odd/integer-limit.
 ===Distance===
 =Unrewritten Original Article=
@@ Line 72: / Line 105: @@
 #'''''A''''' In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the [[chord of nature|chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q'']].
 #'''''A''''' By convention, when assessing a tuning's '''consistency limit''', this type of odd-integer harmonic series chord (limited to an [[odd limit]]) is used in edos, while in other equal-step tunings the unmodified harmonic series (limited to an [[integer limit]]) is used instead.
-#The concept is only defined for equal-step tunings 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).
+#'''''C''''' The concept is only defined for equal-step tunings 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).
 #'''''C''''' 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.
 #'''''C''''' '''==Mathematical definition=='''
@@ Line 85: / Line 118: @@
 #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''.
-#'''==Examples=='''
+#'''''C''''' '''==Examples=='''
-#An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:
+#'''''C'''''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.
+#'''''C''''' 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.
+#'''''C''''' 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.
+#'''''C''''' 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.
+#'''''C''''' The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.
-#As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]].
+#'''''C''''' As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]].
-#The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.
+#'''''C''''' The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.
-#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.
+#'''''C''''' 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]].
+#'''''C'''''12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].
-#An example of the difference between consistency vs distinct consistency:
+#'''''C'''''An example of the difference between consistency vs distinct consistency:
-#In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]].
+#'''''C''''' In 12edo the [[7-odd-limit]] intervals 6/5 and 7/6 are both consistently mapped to 3 steps, and although 12edo is consistent up to the [[9-odd-limit]], it is distinctly consistent only up to the [[5-odd-limit]].
-#Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps.
+#'''''D''''' Another example of non-distinct consistency is given by the intervals [[14/13]] and [[13/12]] in [[72edo]] where they are both mapped to 8 steps.
-#Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]].
+#'''''D''''' Although 72edo is consistent up to the [[17-odd-limit]], it is distinctly consistent only up to the [[11-odd-limit]].
 #'''==Generalizations=='''
 #'''=Pure consistency='''
@@ Line 132: / Line 165: @@
 #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.
 #As a result, [[stretched and compressed tuning|octave stretch and compression]] can be employed to improve an equal tuning's consistency limits:
-#if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series.
+#'''''A''''' if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series.
 #'''''C''''' 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 edos 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 {{nowrap|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.