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,<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.

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 tuning]]s 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).

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).

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

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== Generalizations ==

=== Pure consistency ===

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.

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 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.)

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.

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== For non-octave tunings ==

In non-edo [[equal-step tuning]]s, octaves are not perfectly tuned, and thus an infinite odd limit cannot fully be consistently represented. Instead, we measure consistency in the [[integer limit|''q''-integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an edo describes the highest odd limit it represents consistently, while the consistency limit of any other equal-step tuning (or [[equal temperament]] without an exact octave) instead describes the highest integer limit it represents consistently.

In non-edo [[equal step tuning]]s, octaves are not perfectly tuned, and thus an infinite odd limit cannot fully be consistently represented. Instead, we measure consistency in the [[integer limit|''q''-integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an edo describes the highest odd limit it represents consistently, while the consistency limit of any other equal step tuning (or [[equal temperament]] without an exact octave) instead describes the highest integer limit it represents consistently.

The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not.

@@ 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 [[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.
+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 tuning]]s 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).
+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).
 The page ''[[Minimal consistent edos]]'' shows the smallest edo that is consistent or distinctly consistent in a given odd limit while the page ''[[Consistency limits of small edos]]'' shows the largest odd limit that a given edo is consistent or distinctly consistent in.
@@ Line 59: / Line 59: @@
 == Generalizations ==
 === Pure consistency ===
-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.
+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 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.)
+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.
@@ Line 95: / Line 95: @@
 == For non-octave tunings ==
-In non-edo [[equal-step tuning]]s, octaves are not perfectly tuned, and thus an infinite odd limit cannot fully be consistently represented. Instead, we measure consistency in the [[integer limit|''q''-integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an edo describes the highest odd limit it represents consistently, while the consistency limit of any other equal-step tuning (or [[equal temperament]] without an exact octave) instead describes the highest integer limit it represents consistently.
+In non-edo [[equal step tuning]]s, octaves are not perfectly tuned, and thus an infinite odd limit cannot fully be consistently represented. Instead, we measure consistency in the [[integer limit|''q''-integer-limit]], which is simply the set ''S'' consisting of all intervals ''u''/''v'' where {{nowrap|''u'' ≤ ''q''}} and {{nowrap|''v'' ≤ ''q''}} (and ''q'' is the largest integer harmonic in ''S''). Accordingly, the '''consistency limit''' of an edo describes the highest odd limit it represents consistently, while the consistency limit of any other equal step tuning (or [[equal temperament]] without an exact octave) instead describes the highest integer limit it represents consistently.
 The concept of integer limits means that octave inversion and octave equivalence no longer apply: for example, [[13/10]] and [[11/7]] are in the 16-integer-limit, but [[20/13]] and [[22/7]] are not.