Consistency: Difference between revisions

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An example of the difference between 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'' ==

== Consistency to ''d'' copies ==

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:

A chord is '''consistent to''' ''d'' '''copies''' 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% (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 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.

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 d copies'' 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.

Therefore, consistency to many copies 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 ''q''-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.

Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to 1 copy", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to 1 copy" 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.

For example, 4:5:6:7 is consistent to 3 copies in [[31edo]]. However, 4:5:6:7:11 is only consistent to 1 copy 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, 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:

Formally, for some real ''d'' > 0, a chord C is consistent to ''d'' copies 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'') steps of ''n'' ED''k''.

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

This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to 1/2 copies'' 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 1/2 copies" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.

'''Theorem:''' A (1/(2''d'') steps of ''n'' ED''k'') threshold 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 will always satisfy condition #2 of chord consistency.

@@ Line 33: / Line 33: @@
 An example of the difference between 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'' ==
+== Consistency to ''d'' copies ==
-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:
+A chord is '''consistent to''' ''d'' '''copies''' 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% (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 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.
+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 d copies'' 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.
+Therefore, consistency to many copies 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 ''q''-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.
+Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to 1 copy", this is equivalent to the EDO (or ED''k'') being consistent in the ''q''-[[odd-limit]], and more generally, as "consistent to 1 copy" 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.
+For example, 4:5:6:7 is consistent to 3 copies in [[31edo]]. However, 4:5:6:7:11 is only consistent to 1 copy 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, 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:
+Formally, for some real ''d'' > 0, a chord C is consistent to ''d'' copies 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'') steps of ''n'' ED''k''.
-This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to distance 1/2'' can be interpreted as meaning that all intervals in C are ''at worst'' represented using their second-best mapping, which can be tolerable for some purposes assuming sufficiently small steps. "Consistency to distance 1/2" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.
+This more formal definition also provides an interesting generalisation of ''d'' from the naturals to the positive reals, as ''consistency to 1/2 copies'' 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 1/2 copies" can be nicknamed "semiconsistency", in which case ''C' '' is said to be a "semiconsistent" representation/approximation of C.
 '''Theorem:''' A (1/(2''d'') steps of ''n'' ED''k'') threshold 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 will always satisfy condition #2 of chord consistency.