Consistency: Difference between revisions

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An [[edo]] represents the q-[[odd limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, ~~i.e.~~ the odd harmonics 3, 5, 7, 9, 15, 17, and 19.

An [[edo]] represents the q-[[odd limit|odd-limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd-limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's [[19-odd-limit]], meaning for the set of the odd harmonics 3, 5, 7, 9, 15, 17, and 19.

A different formulation is that an edo approximates a chord C '''consistently''' if the following hold for the best approximation C' of the chord in the edo:

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(If such an approximation exists, it must be the only such approximation, since changing one interval would make that interval go over the 50% threshold.)

In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]

In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]. This is a feature, not a bug, as the distinction can be useful in some circumstances.

The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).

Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[~~Odd~~ limit|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. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.

Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[odd limit|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. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.

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

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

The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best 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 best 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 best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best 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 best 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]].

One 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. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.

One 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. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.

Examples on consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]].

== Consistency to span ''d'' ==

~~Non-technically, a~~ chord is '''consistent to span''' ''d'' in an edo, if the chord is ~~consistent and error~~ accrues slowly enough that you can ~~move up~~ to distance ''d'' from the chord consistently. ~~So an approximation~~ consistent to some reasonable distance would play more nicely in a regular temperament-style [[subgroup]] context. "~~Consistent~~ to span 1" is equivalent to "consistent".

A chord is '''consistent to span''' ''d'' in an edo (or other equal division) [https://en.wikipedia.org/wiki/If_and_only_if iff] all of the following are true:

* The chord is "consistent", meaning every instance of an interval in the chord is represented using the same number of steps.

* Error accrues slowly enough that '''any''' 0 to d intervals can be combined (multiplied or divided) in '''any''' order without accruing 50% (AKA 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.)

If helpful/for the mathematically/geometrically inclined, you can think of the set of all ''n'' [https://en.wikipedia.org/wiki/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 span d'' means that all points that are a [https://en.wikipedia.org/wiki/Taxicab_geometry taxicab distance] of at most ''d'' from the origin (which represents unison) have the [[direct mapping]] 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 spans 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 high-span consistency of a small number of intervals.

Note that if the chord comprised of all the odd harmonics up to the ''k''th is "consistent to span 1", this is equivalent to the EDO (or ED''k'') being consistent in the q-[[odd limit|odd-limit]], and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it is intuitively equivalent to the idea of "consistency" with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times); namely, the ones present in the chord.

For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 1 because 11/5 is mapped too inaccurately (~~rel~~ error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.

For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.

Formally, if ''d'' ≥ 0, a chord ''C'' is ''consistent to span'' ''d'' in ''N''~~-edo~~ if there exists an approximation ''C' '' of ''C'' in ''N''~~-edo~~ such that:

Formally, for some real ''d'' > 0, a chord ''C'' is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation ''C' '' of ''C'' in ''n'' ED''k'' such that:

# every instance of an interval in C is mapped to the same size in C', and

* every instance of an interval in C is mapped to the same size in C', and

~~# no interval within~~ ''C' '' ~~has [[relative error]]~~ 1/(2(''d'')~~) or more~~.

* all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than (1/2''d'') EDk.

~~(The 1/(2(~~''d''~~)) threshold is meant~~ to ~~allow stacking~~ ''d+1'' ~~chords~~, ~~including the original chord~~, ~~via dyads that occur~~ in ~~the chord without having the sum of the dyads have over 50% relative error. That~~ is~~, you can make~~ a ~~copy~~ of ~~a chord up to distance d away from the original chord without inconsistency~~.)

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''.

~~Since~~ a ~~consistent approximation must~~ be ~~unique~~, ~~it suffices to find~~ the ~~consistent approximation and check~~ the ~~relative error of that one~~ chord to ~~check span-''d'' consistency.~~

Question: Is it true that a 1/(2''d'') ED''k'' threshold can be interpreted as allowing stacking ''d'' copies of a chord, including the original chord, via dyads that occur in the chord without causing the resulting chord to be inconsistent?

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

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-An [[edo]] represents the q-[[odd limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's 19-odd limit, i.e. the odd harmonics 3, 5, 7, 9, 15, 17, and 19.
+An [[edo]] represents the q-[[odd limit|odd-limit]] '''consistently''' if the best approximations of the odd harmonics of the q-odd-limit in that edo also give the best approximations of all the differences between these odd harmonics; for example, the difference between the best 7/4 and the best 5/4 is also the best 7/5. This word can actually be used with any set of odd harmonics: e.g. [[12edo]] is consistent in the no-11's, no 13's [[19-odd-limit]], meaning for the set of the odd harmonics 3, 5, 7, 9, 15, 17, and 19.
 A different formulation is that an edo approximates a chord C '''consistently''' if the following hold for the best approximation C' of the chord in the edo:
@@ Line 12: / Line 12: @@
 (If such an approximation exists, it must be the only such approximation, since changing one interval would make that interval go over the 50% threshold.)
-In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]
+In this formulation, 12edo represents the chord 1:3:5:7:17:19 consistently. Note: The chord definition disagrees with the subgroup definition for some chords such as 1:3:81:243 in [[80edo]]. This is a feature, not a bug, as the distinction can be useful in some circumstances.
 The concept only makes sense for edos and not for non-edo rank-2 (or higher) temperaments, since in these tunings you can get any ratio you want to arbitary accuracy by piling up a lot of generators (assuming the generator is an irrational fraction of the octave).
-Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[Odd limit|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. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.
+Stated more mathematically, if N-edo is an [[equal division of the octave]], and if for any interval r, N(r) is the best N-edo approximation to r, then N is '''consistent''' with respect to a set of intervals S if for any two intervals a and b in S where ab is also in S, N(ab) = N(a) + N(b). Normally this is considered when S is the set of [[odd limit|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. N is then said to be ''q limit consistent''. If each interval in the q-limit is mapped to a unique value by N, then it said to be ''uniquely q limit consistent''.
 The page ''[[Minimal consistent EDOs]]'' shows the smallest edo that is consistent or uniquely consistent in a given odd limit while the page ''[[Consistency levels of small EDOs]]'' shows the largest odd limit that a given edo is consistent or uniquely consistent in.
@@ Line 24: / Line 24: @@
 An example for a system that is ''not'' consistent in a particular odd limit is [[25edo]]:
-The best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best 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 best 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 best approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the best 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 best 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]].
-One 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. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.
+One 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. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.
 Examples on consistency vs. unique 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 uniquely consistent only up to the [[5-odd-limit]]. Another example or non-unique 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 uniquely consistent only up to the [[11-odd-limit]].
 == Consistency to span ''d'' ==
-Non-technically, a chord is '''consistent to span''' ''d'' in an edo, if the chord is consistent and error accrues slowly enough that you can move up to distance ''d'' from the chord consistently. So an approximation consistent to some reasonable distance would play more nicely in a regular temperament-style [[subgroup]] context. "Consistent to span 1" is equivalent to "consistent".
+A chord is '''consistent to span''' ''d'' in an edo (or other equal division) [https://en.wikipedia.org/wiki/If_and_only_if iff] all of the following are true:
+* The chord is "consistent", meaning every instance of an interval in the chord is represented using the same number of steps.
+* Error accrues slowly enough that '''any''' 0 to d intervals can be combined (multiplied or divided) in '''any''' order without accruing 50% (AKA 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.)
+If helpful/for the mathematically/geometrically inclined, you can think of the set of all ''n'' [https://en.wikipedia.org/wiki/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 span d'' means that all points that are a [https://en.wikipedia.org/wiki/Taxicab_geometry taxicab distance] of at most ''d'' from the origin (which represents unison) have the [[direct mapping]] 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 spans 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 high-span consistency of a small number of intervals.
+Note that if the chord comprised of all the odd harmonics up to the ''k''th is "consistent to span 1", this is equivalent to the EDO (or ED''k'') being consistent in the q-[[odd limit|odd-limit]], and more generally, as "consistent to span 1" means that the direct mappings agree with how the intervals are reached arithmetically, it is intuitively equivalent to the idea of "consistency" with respect to a set of "basis intervals" (intervals you can combine how you want up to ''d'' times); namely, the ones present in the chord.
-For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 1 because 11/5 is mapped too inaccurately (rel error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.
+For example, 4:5:6:7 is consistent to span 3 in [[31edo]]. However, 4:5:6:7:11 is only consistent to span 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is especially strong in the 2.3.5.7 subgroup and weaker in 2.3.5.7.11.
-Formally, if ''d'' ≥ 0, a chord ''C'' is ''consistent to span'' ''d'' in ''N''-edo if there exists an approximation ''C' '' of ''C'' in ''N''-edo such that:
+Formally, for some real ''d'' > 0, a chord ''C'' is consistent to span ''d'' in ''n'' ED''k'' if there exists an approximation ''C' '' of ''C'' in ''n'' ED''k'' such that:
-# every instance of an interval in C is mapped to the same size in C', and
+* every instance of an interval in C is mapped to the same size in C', and
-# no interval within ''C' '' has [[relative error]] 1/(2(''d'')) or more.
+* all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than (1/2''d'') EDk.
-(The 1/(2(''d'')) threshold is meant to allow stacking ''d+1'' chords, including the original chord, via dyads that occur in the chord without having the sum of the dyads have over 50% relative error. That is, you can make a copy of a chord up to distance d away from the original chord without inconsistency.)
+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''.
-Since a consistent approximation must be unique, it suffices to find the consistent approximation and check the relative error of that one chord to check span-''d'' consistency.
+Question: Is it true that a 1/(2''d'') ED''k'' threshold can be interpreted as allowing stacking ''d'' copies of a chord, including the original chord, via dyads that occur in the chord without causing the resulting chord to be inconsistent?
 Examples of more advanced concepts that build on this are [[telicity]] and [[Consistent#Maximal consistent set|maximal consistent set]]s.