Consistency: Difference between revisions

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While the term '''consistency''' is most frequently used to refer to some odd limit, sometimes one may only care about 'some' of the intervals in some odd limit; this situation often arises when working in [[JI subgroup]]s. We can also skip certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[21-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, and 21, where we deliberately skip 11 and 13.

In general, we can say that some [[equal-step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal-step tuning''', if its best approximation to all the notes in the chord, relative to the root,{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}} also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''. By convention, when assessing a tuning's '''consistency limit''', ~~the~~ chord ~~used is~~ an odd limit in edos, ~~or an~~ [[~~integer limit]] in other~~ equal step tunings ([[~~#For non-octave tunings~~]]).

In general, we can say that some [[equal-step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal-step tuning''', if its best approximation to all the notes in the chord, relative to the root,{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}} also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the [[Overtone scale|chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q'']]. 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 tuning]]s 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).

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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]]. 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]].

== ~~Generalization~~ ==

== Generalizations ==

=== 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%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.

Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer 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~~) {{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.)

A chord is '''consistent to distance''' {{nowrap|''d'' ≥ 1}} or '''consistent to''' ''d'' '''copies''' in an equal-step tuning ([[EST]]) {{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 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.

Note that if the chord comprised of ~~all~~ the ~~odd harmonics~~ up to ~~the~~ ''q''~~-th~~ is "consistent to distance 1", this is equivalent to the ~~edo (or ed-''k'')~~ being consistent in the [[~~odd~~ limit|''q''-odd-limit]]~~, and~~ more generally, as "consistent to distance 1" means that the direct approximations 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 the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the [[EST]] being consistent in the [[integer limit|''q''-integer-limit]] (as well as the [[odd limit|<math>2 \lceil \frac{q}{2} \rceil-1</math>-odd-limit]] if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept 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) — in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).

For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.

Formally, for some real {{nowrap| ''d'' > 0 }}, a chord ''C'' is consistent to distance ''d'' in ''~~n''-ed-''k~~'' if the consistent approximation ''C'' of ''C'' in ''~~n''-ed-''k~~'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''C'' by less than 1/(2''d'') steps of ''~~n''-ed-''k~~''.

Formally, for some real {{nowrap| ''d'' > 0 }}, a JI chord ''c'' is consistent to distance ''d'' in an [[equal tuning]] ''T'' if the consistent approximation ''C'' of ''c'' in ''T'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''c'' by less than 1/(2''d'') steps of ''T''.

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 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

| 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.

| contents=Consider the union {{nowrap|''C'' {{=}} ''C''1 ∪ ''C''2 ∪ … ∪ ''C''''d''}} in the equal tuning, where the ''C''''i'' are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.

Consider any interval {{nowrap|''D'' {{=}} {{(}}''x'', ''y''{{)}}}} consisting of two notes ''x'' and ''y'' that occur in ''C''. We may assume that the notes ''x'' and ''y'' belong in two different copies of ''C'', ''C''''i'' and {{nowrap|C''i'' + ''m''}}, where {{nowrap|1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''}}. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each interval ''D''''j'' in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ''ε'' on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap| 1 − ''ε'' > 1/2 }} and {{nowrap| 1 + ε }} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.

@@ Line 11: / Line 11: @@
 While the term '''consistency''' is most frequently used to refer to some odd limit, sometimes one may only care about 'some' of the intervals in some odd limit; this situation often arises when working in [[JI subgroup]]s. We can also skip certain intervals when evaluating consistency. For instance, [[12edo]] is consistent in the no-11's, no-13's [[21-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, and 21, where we deliberately skip 11 and 13.
-In general, we can say that some [[equal-step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal-step tuning''', if its best approximation to all the notes in the chord, relative to the root,<sup style="white-space: nowrap;">{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}}</sup> also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q''. By convention, when assessing a tuning's '''consistency limit''', the chord used is an odd limit in edos, or an [[integer limit]] in other equal step tunings ([[#For non-octave tunings]]).
+In general, we can say that some [[equal-step tuning]] is '''consistent relative to a [[chord]] ''C''''', or that a '''chord ''C'' is consistent in some equal-step tuning''', if its best approximation to all the notes in the chord, relative to the root,<sup style="white-space: nowrap;">{{lbrack}}[[:Category:Todo:clarify|''which?'']]{{rbrack}}</sup> also gives the best approximation to all of the intervals between the pairs of notes in the chord. In particular, an edo is consistent in the ''q''-odd-limit if and only if it is consistent relative to the [[Overtone scale|chord 1:3:…:{{nowrap|(''q'' − 2)}}:''q'']]. 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 tuning]]s 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).
@@ Line 57: / Line 57: @@
 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]]. 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]].
-== Generalization ==
+== Generalizations ==
 === 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%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.
+Going even further than consistency, an equal-step tuning is '''purely consistent'''{{idiosyncratic}} if it approximates all integer 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) {{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.)
+A chord is '''consistent to distance''' {{nowrap|''d'' ≥ 1}} or '''consistent to''' ''d'' '''copies''' in an equal-step tuning ([[EST]]) {{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 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.
-Note that if the chord comprised of all the odd harmonics up to the ''q''-th is "consistent to distance 1", this is equivalent to the edo (or ed-''k'') being consistent in the [[odd limit|''q''-odd-limit]], and more generally, as "consistent to distance 1" means that the direct approximations 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 the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the [[EST]] being consistent in the [[integer limit|''q''-integer-limit]] (as well as the [[odd limit|<math>2 \lceil \frac{q}{2} \rceil-1</math>-odd-limit]] if it is an edo); more generally, because "consistent to distance 1" means that the direct approximations agree with how the intervals are reached arithmetically, the concept 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) — in this case, intervals between the "basis" harmonics of a truncated harmonic series (an [[integer limit]]).
 For example, 4:5:7 is consistent to distance 10 in [[31edo]]. However, 4:5:7:11 is only consistent to distance 1 because 11/5 is mapped too inaccurately (relative error 26.2%). This shows that 31edo is extremely strong in the 2.5.7 subgroup and much weaker in 2.5.7.11.
-Formally, for some real {{nowrap| ''d'' > 0 }}, a chord ''C'' is consistent to distance ''d'' in ''n''-ed-''k'' if the consistent approximation ''C'' of ''C'' in ''n''-ed-''k'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''C'' by less than 1/(2''d'') steps of ''n''-ed-''k''.
+Formally, for some real {{nowrap| ''d'' > 0 }}, a JI chord ''c'' is consistent to distance ''d'' in an [[equal tuning]] ''T'' if the consistent approximation ''C'' of ''c'' in ''T'' satisfies the property that all intervals in ''C'' are off from their corresponding intervals in ''c'' by less than 1/(2''d'') steps of ''T''.
-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 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
 | 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> ∪ ''C''<sub>2</sub> ∪ … ∪ ''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.
+| contents=Consider the union {{nowrap|''C'' {{=}} ''C''<sub>1</sub> ∪ ''C''<sub>2</sub> ∪ … ∪ ''C''<sub>''d''</sub>}} in the equal tuning, where the ''C''<sub>''i''</sub> are copies of the (approximations of) chord ''C''. We need to show that this chord is consistent.
 Consider any interval {{nowrap|''D'' {{=}} {{(}}''x'', ''y''{{)}}}} consisting of two notes ''x'' and ''y'' that occur in ''C''. We may assume that the notes ''x'' and ''y'' belong in two different copies of ''C'', ''C''<sub>''i''</sub> and {{nowrap|C<sub>''i'' + ''m''</sub>}}, where {{nowrap|1 ≤ ''i'' ≤ ''i'' + ''m'' ≤ ''d''}}. Thus ''x'' and ''y'' are separated by a path of at most ''d'' steps (at most ''d'' − 1 for the different copies of ''C'', and 1 for the additional step within C). By consistency to distance ''d'', each interval ''D''<sub>''j''</sub> in the path has relative error 1/(2''d''). Hence by the triangle inequality, the total relative error ''ε'' on ''D'' is strictly less than 1/2 (50%). Since the adjacent intervals to the approximation of ''D'' must have relative error {{nowrap| 1 − ''ε'' > 1/2 }} and {{nowrap| 1 + ε }} respectively as approximations to the JI interval ''D'', the approximation we got must be the best one. Since ''D'' is arbitrary, we have proved chord consistency.