Consistency: Difference between revisions

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== Generalization ==

=== Pure consistency ===

Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all odd harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of less than 25%).

Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all odd harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of less than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.

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

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.)

A chord is '''consistent to distance''' {{nowrap|''d'' ≥ 1}} or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) {{w|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 {{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.

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.

Therefore, consistency to large distances 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 long-range consistency of a small number of intervals.

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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''.

{{~~proof~~

{{Proof

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

@@ Line 59: / Line 59: @@
 == Generalization ==
 === Pure consistency ===
-Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all odd harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of less than 25%).
+Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all odd harmonics from 1 up to and including ''q'' within one quarter of a step (in other words, maintaining [[relative interval error|relative errors]] of less than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.
 === Consistency to distance ''d'' ===
-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.)
+A chord is '''consistent to distance''' {{nowrap|''d'' &ge; 1}} or '''consistent to''' ''d'' '''copies''' in an edo (or other equal division) {{w|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 {{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.
+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.
 Therefore, consistency to large distances 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 long-range consistency of a small number of intervals.
@@ Line 78: / Line 78: @@
 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''.
-{{proof
+{{Proof
 | title=Consistency to distance ''d'' can be interpreted as allowing stacking ''d'' copies of a chord C, including the original chord, via intervals 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.