Consistency - Revision history

Currywurst44: Move 'maximal consistent set' to user space as it hasn't been used in any example on the Wiki or been mentioned on either the Xengrove Discord or Xenharmonic Alliance Discord.

2026-05-13T20:29:45Z

Move 'maximal consistent set' to user space as it hasn't been used in any example on the Wiki or been mentioned on either the Xengrove Discord or Xenharmonic Alliance Discord.

← Older revision		Revision as of 20:29, 13 May 2026
Line 85:		Line 85:
	}}		}}

	Examples of more advanced concepts that build on this are [[telicity]] and [[#~~Maximal consistent set~~\|maximal consistent set]]s.		Examples of more advanced concepts that build on this are [[telicity]] and [[#User:Inthar/Maximal_consistent_set\|maximal consistent set]]s.

	~~== Maximal consistent set ==~~
	Non-technically, a '''maximal consistent set''' (MCS) is a piece of a [[JI subgroup]] such that when you add another interval which is adjacent to the piece (viewed as a chord), then the piece becomes inconsistent in the edo.

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

	== For non-octave tunings ==		== For non-octave tunings ==

FloraC: Re-organize. Put the alt definition in the math section

2026-04-09T20:36:16Z

Re-organize. Put the alt definition in the math section

← Older revision		Revision as of 20:36, 9 April 2026
Line 1:		Line 1:
	{{~~interwiki~~		{{Interwiki
			\| en = Consistent
	\| de = konsistent		\| de = konsistent
	~~\| en = Consistent~~
	\| es =		\| es =
	\| ja = 一貫性		\| ja = 一貫性
	}}		}}
	An [[edo]] represents the [[odd limit\|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]).		An [[edo]] represents the [[odd limit\|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]).

	This is equivalent to looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] that does the same approximation, so that the intervals are lined up by the val. If there is such a val, then the edo is consistent within that odd limit, otherwise it is inconsistent.

	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.		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.
Line 19:		Line 17:
	== Mathematical definition ==		== Mathematical definition ==
	Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', {{nowrap\|''T''(''r''<sub>''i''</sub>''r''<sub>''j''</sub>) {{=}} ''T''(''r''<sub>''i''</sub>) + ''T''(''r''<sub>''j''</sub>).}}		Formally, if ''T'' is an equal tuning, and if for an interval ''r'', ''T''(''r'') is the closest approximation to ''r'' in ''T'', then ''T'' is '''consistent''' with respect to a set of intervals ''S'' if for any two intervals ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in ''S'' where ''r''<sub>''i''</sub>''r''<sub>''j''</sub> is also in ''S'', {{nowrap\|''T''(''r''<sub>''i''</sub>''r''<sub>''j''</sub>) {{=}} ''T''(''r''<sub>''i''</sub>) + ''T''(''r''<sub>''j''</sub>).}}

			This is equivalent to looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] that does the same approximation, so that the intervals are lined up by the val. If there is such a val, then the edo is consistent within that odd limit, otherwise it is inconsistent.

	; Alternative formulation using val		; Alternative formulation using val

ArrowHead294 at 16:08, 28 March 2025

2025-03-28T16:08:42Z

← Older revision		Revision as of 16:08, 28 March 2025
Line 59:		Line 59:
	== Generalizations ==		== Generalizations ==
	=== Pure consistency ===		=== 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 no greater than 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]]s of no greater than than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2, introduced next.

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

ArrowHead294: /* Generalizations */

2025-03-19T13:00:08Z

Generalizations

← Older revision		Revision as of 13:00, 19 March 2025
Line 59:		Line 59:
	== Generalizations ==		== Generalizations ==
	=== Pure consistency ===		=== 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 no greater than than 25%). Pure consistency is stronger than consistency but weaker than consistency to distance 2 , introduced next.

	=== Consistency to distance ''d'' ===		=== Consistency to distance ''d'' ===
Line 70:		Line 70:
	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.		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 the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit\|''q''-integer-limit]] (as well as the {{nowrap\|(~~2⋅ceil(~~''q''/2~~) -~~ 1)}}-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]]).		Note that if the chord comprised of the harmonic series up to ''q'' is "consistent to distance 1", this is equivalent to the tuning being consistent in the [[integer limit\|''q''-integer-limit]] (as well as the {{nowrap\|(2{{ceil\|''q''/2}} − 1)}}-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.		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.

FloraC: Clarify, cleanup, and consistency in style

2025-02-21T11:00:58Z

Clarify, cleanup, and consistency in style

Show changes

Recentlymaterialized: oops

2025-02-21T07:14:00Z

oops

← Older revision		Revision as of 07:14, 21 February 2025
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.		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.		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 ==		== Generalizations ==
	=== Pure consistency ===		=== 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'' ===		=== 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.		For {{nowrap\|''d'' ≥ 1}}, this implies consistency in the ordinary sense.
Line 95:		Line 95:

	== For non-octave tunings ==		== 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.		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.

Recentlymaterialized at 07:11, 21 February 2025

2025-02-21T07:11:52Z

Recentlymaterialized at 01:21, 21 February 2025

2025-02-21T01:21:07Z

← Older revision		Revision as of 01:21, 21 February 2025
Line 5:		Line 5:
	\| ja = 一貫性		\| ja = 一貫性
	}}		}}
	An [[edo]] represents the [[odd limit\|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An ~~[[equal-step tuning]]~~ is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is mapped to a distinct~~/unique step~~. ~~So for~~ example, an ~~equal-step tuning~~ cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same ~~step—this would correspond to~~ [[tempering out]] [[50/49]]~~, and in the case of edos, would mean the edo must be a multiple, or superset, of 2edo~~.		An [[edo]] represents the [[odd limit\|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, if the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An edo is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is consistent and mapped to a distinct edostep. For example, an edo cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step (in this case, the semi-octave of [[2edo]], [[tempering out]] [[50/49]]).

	Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.		Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up.{{clarify}} If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.

	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.		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 ~~edo~~ is '''consistent relative to a chord ''C''''', or that '''chord ''C'' is consistent in some ~~edo~~''', if its best approximation to all the notes in the chord, relative to the root, 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''.		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]]).

	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).

FloraC: Move the 46edo compression example to the right section

2025-01-05T10:55:48Z

Move the 46edo compression example to the right section

← Older revision		Revision as of 10:55, 5 January 2025
Line 50:		Line 50:

	The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives {{nowrap\|(3/2)(7/6) {{=}} [[7/4]]}}, the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.		The closest approximation for the interval of [[7/6]] (the septimal subminor third) in 25edo is 6 steps, and the closest approximation for the just perfect fifth ([[3/2]]) is 15 steps. Adding the two just intervals gives {{nowrap\|(3/2)(7/6) {{=}} [[7/4]]}}, the harmonic seventh, for which the closest approximation in 25edo is 20 steps. Adding the two approximated intervals, however, gives 21 steps. This means that 25edo is not consistent in 7-odd-limit. The 4:6:7 triad cannot be mapped to 25edo without one of its three component intervals being inaccurately mapped.

			As another notable example, [[46edo]] is not consistent in the [[15-odd-limit]]. The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees.

	An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].		An example for a system that ''is'' consistent in the [[7-odd-limit]] is [[12edo]]: 3/2 maps to 7\12, 7/6 maps to 3\12, and 7/4 maps to 10\12, which equals 7\12 plus 3\12. 12edo is also consistent in the [[9-odd-limit]], but not in the [[11-odd-limit]].

	One notable example: [[46edo]] is not consistent in the [[15-odd-limit]]. The 15/13 interval is slightly closer to 9 degrees of 46edo than to 10 degrees, but the ''functional'' [[15/13]] (the difference between 46edo's versions of [[15/8]] and [[13/8]]) is 10 degrees. However, if we compress the octave slightly (by about a cent), this discrepancy no longer occurs, and we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating mode 8 of the harmonic series.

	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]].		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]].
Line 98:		Line 98:

	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.		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.

			As a result, [[stretched and compressed tuning\|octave stretch and compression]] can be employed to improve an equal tuning's consistency limits: if we compress the octave of 46edo slightly (by about a cent), we end up with an 18-''integer''-limit consistent system, which makes it ideal for approximating Mode 8 of the harmonic series.

	It is possible to extend the concept of odd limits to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[edt]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the ''q''-odd-limit will be consistent to the {{nowrap\|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.		It is possible to extend the concept of odd limits to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[edt]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the ''q''-odd-limit will be consistent to the {{nowrap\|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.

FloraC: Skipping three primes is too much for most in the 12edo example and its integer-limit consistency that follows immediately isn't useful before introducing the very concept of integer-limit consistency. Misc. cleanup

2025-01-05T10:47:14Z

Skipping three primes is too much for most in the 12edo example and its integer-limit consistency that follows immediately isn't useful before introducing the very concept of integer-limit consistency. Misc. cleanup

← Older revision		Revision as of 07:11, 21 February 2025
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.		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).		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]].		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 ===		=== 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'' ===		=== 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 {{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.		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.		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		{{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).		\| 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.		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.

← Older revision		Revision as of 10:47, 5 January 2025
Line 5:		Line 5:
	\| ja = 一貫性		\| ja = 一貫性
	}}		}}
	An [[edo]] represents the [[odd limit\|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An [[equal-step tuning]] is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is mapped to a distinct/unique step. So for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same ~~step—this~~ would correspond to [[tempering out]] [[50/49]], and in the case of edos, would mean the edo must be a multiple, or superset, of 2edo.		An [[edo]] represents the [[odd limit\|''q''-odd-limit]] '''consistently''' if the closest approximations of the odd harmonics of the ''q''-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics; for example, the difference between the closest [[7/4]] and the closest [[5/4]] is also the closest [[7/5]]. An [[equal-step tuning]] is '''distinctly consistent''' (or '''uniquely consistent''') in the ''q''-odd-limit if every interval in that odd limit is mapped to a distinct/unique step. So for example, an equal-step tuning cannot be distinctly consistent in the [[7-odd-limit]] if it maps 7/5 and [[10/7]] to the same step—this would correspond to [[tempering out]] [[50/49]], and in the case of edos, would mean the edo must be a multiple, or superset, of 2edo.

	Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.		Note that we are looking at the [[direct approximation]] (i.e. the closest approximation) for each interval, and trying to find a [[val]] to line them up. If there is such a val, then the edo is consistent within that odd-limit, otherwise it is inconsistent.

	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-~~11s~~, no-~~13s, no-23s~~ [[27-odd-limit]], meaning the set of the odd harmonics 1, 3, 5, 7, 9, 15, 17, 19, 21~~, 25, and 27~~, where we deliberately skip 11, 13~~, and 23. (Alternatively, with the octave tuned 2–6{{c}} flat, 12-[[ET]] is consistent in the 2.3.5.7.17.19-[[subgroup]] 48-[[integer-limit]]~~.)		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 edo is '''consistent relative to a chord C''', or that '''chord C is consistent in some edo''', if its best approximation to all the notes in the chord, relative to the root, 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''.		In general, we can say that some edo is '''consistent relative to a chord ''C''''', or that '''chord ''C'' is consistent in some edo''', if its best approximation to all the notes in the chord, relative to the root, 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''.

	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).
Line 62:		Line 62:

	=== Consistency to distance ''d'' ===		=== 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 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 {{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.		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 74:		Line 74:
	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.		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 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''.

	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		{{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).		\| 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 temperament, 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.		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.
	}}		}}

Line 95:		Line 95:

	== For non-octave tunings ==		== 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 ''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 [[ET]] 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.		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.

	It is possible to extend the concept of [[odd limits]] to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[~~EDT~~]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of ~~[[EDO]]s~~ with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for ~~EDOs~~ anyways (an ~~EDO~~ that is consistent to the ''q''-odd-limit will be consistent to the (''q''~~ ~~+~~ ~~1)-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.		It is possible to extend the concept of odd limits to other equaves, such as the "''q''-throdd-limit" with 3/1 (tritave) equivalence, but because an [[edt]] that is consistent to a certain throdd limit will also be consistent to the corresponding integer-limit, there is little reason to complicate the analysis with additional types of infinite interval sets. This wiki measures consistency in the special case of edos with odd limits instead of integer limits for ease of explanation, but the two types of consistency are effectively equivalent for edos anyways (an edo that is consistent to the ''q''-odd-limit will be consistent to the {{nowrap\|(''q'' + 1)}}-integer-limit and vice versa) unless intervals or primes are skipped or if a [[JI subgroup]] is used.

	== External links ==		== External links ==