Consistency limits of small EDOs - Revision history

FloraC: Cleanup & style

2025-08-07T11:49:45Z

Cleanup & style

← Older revision		Revision as of 11:49, 7 August 2025
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	An [[~~EDO~~]] N is [[consistent]] with respect to the q-odd-limit 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. It is [[distinctly consistent]] if every one of those closest approximations is a distinct value. Below is a table of every ~~EDO~~ up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the ~~EDO~~ is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the ~~EDO~~ is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance d\|~~"Consistency~~ distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.		An [[edo]] ''N'' is [[consistent]] with respect to the [[odd limit\|''q''-odd-limit]] 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. It is [[distinctly consistent]] if every one of those closest approximations is a distinct value. Below is a table of every edo up to 99. "Consistent" gives its consistency limit, i.e. the highest odd limit to which the edo is consistent, and "Distinct" gives the distinct consistency limit, i.e. the highest odd limit to which the edo is distinctly consistent. The remaining columns give the [[Consistent #Consistency to distance d\|consistency distance]] (also called ''consistency level''<ref group="note">This term was coined by [[Paul Hahn]] in 1996. See [https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html Yahoo! Tuning Group \| ''Consistency generalized'']. </ref>) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible~~" style="text~~-~~align:right~~"		{\| class="wikitable sortable mw-collapsible right-all"
	\|-		\|-
	! rowspan="2~~" style="text-align:center~~" \| ~~EDO~~		! rowspan="2" \| Edo
	! colspan="2~~" style="text-align:center~~" \| Consistency limit		! colspan="2" \| Consistency limit
	! colspan="11~~" style="text-align:center~~" \| Consistency distance		! colspan="11" \| Consistency distance
	\|-		\|-
	! ~~style="text-align:center" \|~~ Consistent		! Consistent
	! ~~style="text-align:center" \|~~ Distinct		! Distinct
	! ~~style="text-align:center" \|~~ 3-limit		! 3-limit
	! ~~style="text-align:center" \|~~ 5-limit		! 5-limit
	! ~~style="text-align:center" \|~~ 7-limit		! 7-limit
	! ~~style="text-align:center" \|~~ 9-limit		! 9-limit
	! ~~style="text-align:center" \|~~ 11-limit		! 11-limit
	! ~~style="text-align:center" \|~~ 13-limit		! 13-limit
	! ~~style="text-align:center" \|~~ 15-limit		! 15-limit
	! ~~style="text-align:center" \|~~ 17-limit		! 17-limit
	! ~~style="text-align:center" \|~~ 19-limit		! 19-limit
	! ~~style="text-align:center" \|~~ 21-limit		! 21-limit
	! ~~style="text-align:center" \|~~ 23-limit		! 23-limit
	\|-		\|-
	\| 1 \|\| style="background-color: #fefefe;"\| 3 \|\| 1 \|\| style="background-color: #fefefe;"\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0		\| 1 \|\| style="background-color: #fefefe;"\| 3 \|\| 1 \|\| style="background-color: #fefefe;"\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0
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	\| 99 \|\| style="background-color: #fefefe;"\| 9 \|\| 9 \|\| 5 \|\| 3 \|\| 3 \|\| style="background-color: #fefefe;"\| 2 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0		\| 99 \|\| style="background-color: #fefefe;"\| 9 \|\| 9 \|\| 5 \|\| 3 \|\| 3 \|\| style="background-color: #fefefe;"\| 2 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0
	\|}		\|}

			== Notes ==
			<references group="note"/>

	[[Category:Mapping]]		[[Category:Mapping]]
	[[Category:Consistency]]		[[Category:Consistency]]
	[[Category:Odd limit]]		[[Category:Odd limit]]

Xenllium at 07:50, 12 May 2023

2023-05-12T07:50:47Z

← Older revision		Revision as of 07:50, 12 May 2023
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	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"
	\|-		\|-
	! rowspan="2" \| EDO		! rowspan="2" style="text-align:center" \| EDO
	! colspan="2" \| Consistency limit		! colspan="2" style="text-align:center" \| Consistency limit
	! colspan="11" \| Consistency distance		! colspan="11" style="text-align:center" \| Consistency distance
	\|-		\|-
	! Consistent		! style="text-align:center" \| Consistent
	! Distinct		! style="text-align:center" \| Distinct
	! 3-limit		! style="text-align:center" \| 3-limit
	! 5-limit		! style="text-align:center" \| 5-limit
	! 7-limit		! style="text-align:center" \| 7-limit
	! 9-limit		! style="text-align:center" \| 9-limit
	! 11-limit		! style="text-align:center" \| 11-limit
	! 13-limit		! style="text-align:center" \| 13-limit
	! 15-limit		! style="text-align:center" \| 15-limit
	! 17-limit		! style="text-align:center" \| 17-limit
	! 19-limit		! style="text-align:center" \| 19-limit
	! 21-limit		! style="text-align:center" \| 21-limit
	! 23-limit		! style="text-align:center" \| 23-limit
	\|-		\|-
	\| 1 \|\| style="background-color: #fefefe;"\| 3 \|\| 1 \|\| style="background-color: #fefefe;"\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0		\| 1 \|\| style="background-color: #fefefe;"\| 3 \|\| 1 \|\| style="background-color: #fefefe;"\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-02-26T07:34:50Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 07:34, 26 February 2023
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	\|}		\|}

	~~[[Category:Theory]]~~
	[[Category:Mapping]]		[[Category:Mapping]]
	[[Category:Consistency]]		[[Category:Consistency]]
	[[Category:Odd limit]]		[[Category:Odd limit]]

Xenwolf: typo

2022-08-01T11:27:25Z

typo

← Older revision		Revision as of 11:27, 1 August 2022
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	~~AAn~~ [[EDO]] N is [[consistent]] with respect to the q-odd-limit 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. It is [[distinctly consistent]] if every one of those closest approximations is a distinct value. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance d\|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.		An [[EDO]] N is [[consistent]] with respect to the q-odd-limit 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. It is [[distinctly consistent]] if every one of those closest approximations is a distinct value. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance d\|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"

Martins: mirrored main consistency definition (my previous attempt wrong)

2022-07-31T10:33:51Z

mirrored main consistency definition (my previous attempt wrong)

← Older revision		Revision as of 10:33, 31 July 2022
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	An [[EDO]] N is [[consistent]] with respect to ~~a set of rational numbers s~~ if the ~~[[direct approximation]]~~ of ~~every element~~ of ~~s is~~ the closest ~~N-EDO approximation~~. It is [[distinctly consistent]] if every ~~element~~ of s is ~~mapped to~~ a distinct value~~. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively~~. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance d\|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.		AAn [[EDO]] N is [[consistent]] with respect to the q-odd-limit 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. It is [[distinctly consistent]] if every one of those closest approximations is a distinct value. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance d\|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"

Cmloegcmluin: fix link

2022-07-29T20:12:38Z

fix link

← Older revision		Revision as of 20:12, 29 July 2022
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	An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance ''d''\|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.		An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the [[Consistent#Consistency to distance d\|"Consistency distance"]] (also called "consistency level"<ref>This term was coined by [[Paul Hahn]] in 1996: https://yahootuninggroupsultimatebackup.github.io/mills-tuning-list/topicId_884.html</ref>) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"

Cmloegcmluin: add historical information about "consistency level" terminology; add subtle highlighting to table to help illuminate relationship between limit and level

2022-07-29T16:14:25Z

add historical information about "consistency level" terminology; add subtle highlighting to table to help illuminate relationship between limit and level

Show changes

Martins at 21:36, 28 July 2022

2022-07-28T21:36:30Z

← Older revision		Revision as of 21:36, 28 July 2022
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	An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the "Consistency distance", see [[~~consistent~~#Consistency to distance ''d'']] (also called Consistency level) for every odd limit from 3 to 23.		An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the "Consistency distance", see [[Consistent#Consistency to distance ''d'']] (also called Consistency level) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"

Martins at 21:28, 28 July 2022

2022-07-28T21:28:39Z

← Older revision		Revision as of 21:28, 28 July 2022
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	An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the "Consistency distance", see [[Consistency to distance ''d'']] (also called Consistency level) for every odd limit from 3 to 23.		An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the "Consistency distance", see [[consistent#Consistency to distance ''d'']] (also called Consistency level) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"

Martins: re-worded definition to be closer to "consistency" page

2022-07-28T21:12:59Z

re-worded definition to be closer to "consistency" page

← Older revision		Revision as of 21:12, 28 July 2022
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	An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[~~patent val~~]] ~~mapping~~ of every element of s is the ~~nearest~~ N-EDO approximation. It is [[distinctly~~/uniquely~~ consistent]] if every element of s is mapped to a distinct~~/unique~~ value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly~~/uniquely~~ consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives ~~the~~ consistency limit, and "Distinct" the distinct~~/unique~~ consistency limit. "Consistency ~~level~~" ~~gives the~~ [[~~consistency level~~]] (~~consistency diameter? consistency distance?~~) of every odd limit from 3 to 23.		An [[EDO]] N is [[consistent]] with respect to a set of rational numbers s if the [[direct approximation]] of every element of s is the closest N-EDO approximation. It is [[distinctly consistent]] if every element of s is mapped to a distinct value. If the set s is the q [[odd limit]], we say N is q-limit consistent and q-limit distinctly consistent, respectively. Below is a table of every EDO up to 99. "Consistent" gives its "consistency limit", i.e. the highest odd limit to which the EDO is consistent, and "Distinct" gives the "distinct consistency limit" i.e. the highest odd limit to which the EDO is distinctly consistent. The remaining columns give the "Consistency distance", see [[Consistency to distance ''d'']] (also called Consistency level) for every odd limit from 3 to 23.

	{\| class="wikitable sortable mw-collapsible" style="text-align:right"		{\| class="wikitable sortable mw-collapsible" style="text-align:right"