Consistency: Difference between revisions

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

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:

A chord is '''consistent to span''' ''d'' in an edo (or other equal division) [[Wikipedia: 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.)

* 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.)

~~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.

For the mathematically/geometrically inclined, you can think of the set of all ''n'' [[Wikipedia: 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 [[Wikipedia: 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.

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]], 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 (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.

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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

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

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

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Formally, given ''N''-edo, a chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the dyads in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via dyads that occur in C) such that adding another interval adjacent to ''S'' via a dyad 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''.

==Generalization to non-octave scales==

== Generalization to non-octave scales ==

It is possible to generalize the concept of consistency to non-edo equal temperaments. 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 is no longer present. Instead, the set S consists of all intervals u/v where u ~~<=~~ q ~~>=~~ v.

It is possible to generalize the concept of consistency to non-edo equal temperaments. 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 is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ q ≥ v.

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.

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

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

==Links==

== Links ==

* [http://www.tonalsoft.com/enc/c/consistent.aspx ~~consistent~~ (TonalSoft encyclopedia)]

* [http://www.tonalsoft.com/enc/c/consistent.aspx Consistent (TonalSoft encyclopedia)]

* [https://docs.google.com/spreadsheets/d/1yt239Aeh26RwktiI9Nkli87A0nmBcRri1MXrFb4hE-g/edit?usp=sharing Consistency and relative error of EDO]

@@ Line 34: / Line 34: @@
 == Consistency to span ''d'' ==
-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:
+A chord is '''consistent to span''' ''d'' in an edo (or other equal division) [[Wikipedia: 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.)
+* 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.)
-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.
+For the mathematically/geometrically inclined, you can think of the set of all ''n'' [[Wikipedia: 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 [[Wikipedia: 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.
+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]], 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 (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.
@@ Line 48: / Line 48: @@
 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
-* all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than (1/2''d'') EDk.
+* all intervals in ''C' '' are off from their corresponding intervals in ''C'' by less than (1/2''d'') 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''.
@@ Line 66: / Line 66: @@
 Formally, given ''N''-edo, a chord ''C'' and a [[JI subgroup]] ''G'' [[generator|generated]] by the octave and the dyads in ''C'', a ''maximal consistent set'' is a connected set ''S'' (connected via dyads that occur in C) such that adding another interval adjacent to ''S'' via a dyad 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''.
-==Generalization to non-octave scales==
+== Generalization to non-octave scales ==
-It is possible to generalize the concept of consistency to non-edo equal temperaments. 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 is no longer present. Instead, the set S consists of all intervals u/v where u &lt;= q &gt;= v.
+It is possible to generalize the concept of consistency to non-edo equal temperaments. 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 is no longer present. Instead, the set S consists of all intervals ''u''/''v'' where ''u'' ≤ q ≥ v.
 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.
-Alternatively, we can use "modulo-n limit" if the [[equave]] is n/1. Thus the tritave analogue of odd limit would only allow integers not divisible by 3 under a given limit, assuming that tritave equivalence and tritave inversion applies.
+Alternatively, we can use "modulo-''n'' limit" if the [[equave]] is ''n''/1. Thus the tritave analogue of odd limit would only allow integers not divisible by 3 under a given limit, assuming that tritave equivalence and tritave inversion applies.
-==Links==
+== Links ==
-* [http://www.tonalsoft.com/enc/c/consistent.aspx consistent (TonalSoft encyclopedia)]
+* [http://www.tonalsoft.com/enc/c/consistent.aspx Consistent (TonalSoft encyclopedia)]
 * [https://docs.google.com/spreadsheets/d/1yt239Aeh26RwktiI9Nkli87A0nmBcRri1MXrFb4hE-g/edit?usp=sharing Consistency and relative error of EDO]