836edo - Revision history

Tristanbay: /* Theory */Added sentence about midpoint between 19edo and 22edo perfect fifth

2026-04-11T04:27:48Z

Theory: Added sentence about midpoint between 19edo and 22edo perfect fifth

← Older revision		Revision as of 04:27, 11 April 2026
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	== Theory ==		== Theory ==
	836edo is a strong 11-limit system, having the record of lowest absolute error and beating [[612edo]].		836edo is a strong 11-limit system, having the record of lowest absolute error and beating [[612edo]]. Its perfect fifth is notable as the midpoint between that of [[19edo]] and [[22edo]], which are nearly the same distance away from a pure 3/2 but in opposite directions, thus being quite an accurate approximation.

	As an equal temperament, it [[tempering out\|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.		As an equal temperament, it [[tempering out\|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:31:54Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:31, 13 March 2026
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	\| [[Ruthenium]]		\| [[Ruthenium]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	[[Category:Quasithird]]		[[Category:Quasithird]]

Francium: changed EDO intro to ED intro

2025-02-21T06:32:15Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:32, 21 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|836~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T16:39:17Z

Theory: cleanup

← Older revision		Revision as of 16:39, 17 January 2025
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	== Theory ==		== Theory ==
	836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].		836edo is a strong 11-limit system, having the record of lowest absolute error and beating [[612edo]].

	It [[tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.		As an equal temperament, it [[tempering out\|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.

	Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.		Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 836 factors into ~~{{factorization\|836}}~~, 836edo has subset edos {{EDOs\| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. [[1672edo]], which doubles it, provides a good correction for [[harmonic]] [[13/1\|13]].		Since 836 factors into 2<sup>2</sup> × 11 × 19, 836edo has subset edos {{EDOs\| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. [[1672edo]], which doubles it, provides a good correction for [[harmonic]] [[13/1\|13]].

	== Regular temperament properties ==		== Regular temperament properties ==

ArrowHead294 at 14:02, 16 January 2025

2025-01-16T14:02:46Z

← Older revision		Revision as of 14:02, 16 January 2025
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	836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].		836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].

	~~The equal temperament~~ [[~~tempering out\|~~tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.		It [[tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.

	Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.		Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.
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	\| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213		\| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
	\| {{mapping\| 836 1325 1941 2347 2892 3094 }} (836)		\| {{mapping\| 836 1325 1941 2347 2892 3094 }} (836)
	\| ~~−0~~.0085		\| −0.0085
	\| 0.0785		\| 0.0785
	\| 5.47		\| 5.47
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	\| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374		\| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
	\| {{mapping\| 836 1325 1941 2347 2892 3094 3417 }} (836)		\| {{mapping\| 836 1325 1941 2347 2892 3094 3417 }} (836)
	\| ~~−0~~.0014		\| −0.0014
	\| 0.0747		\| 0.0747
	\| 5.21		\| 5.21
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	\| [[Ruthenium]]		\| [[Ruthenium]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	[[Category:Quasithird]]		[[Category:Quasithird]]

ArrowHead294 at 12:07, 16 November 2024

2024-11-16T12:07:58Z

← Older revision		Revision as of 12:07, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.0747		\| 0.0747
	\| 5.20		\| 5.20
	~~{{comma basis end}~~}		\|}
	* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo\|612]] and before [[1084edo\|1084]].		* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo\|612]] and before [[1084edo\|1084]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
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	\| 4/3<br />(18375/18304)		\| 4/3<br />(18375/18304)
	\| [[Ruthenium]]		\| [[Ruthenium]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	[[Category:Quasithird]]		[[Category:Quasithird]]

ArrowHead294 at 05:30, 16 November 2024

2024-11-16T05:30:23Z

← Older revision		Revision as of 05:30, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.0337		\| 0.0337
	\| 2.35		\| 2.35
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	~~\| style="border-top: double;"~~ \| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213		\| 2200/2197, 3025/3024, 4096/4095, 4375/4374, 31250/31213
	~~\| style="border-top: double;"~~ \| {{mapping\| 836 1325 1941 2347 2892 3094 }} (836)		\| {{mapping\| 836 1325 1941 2347 2892 3094 }} (836)
	\| ~~style="border-top: double~~;~~" \| -~~0.0085		\| −0.0085
	~~\| style="border-top: double;"~~ \| 0.0785		\| 0.0785
	~~\| style="border-top: double;"~~ \| 5.47		\| 5.47
	\|-		\|-
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374		\| 1275/1274, 2200/2197, 2500/2499, 3025/3024, 4096/4095, 4375/4374
	\| {{mapping\| 836 1325 1941 2347 2892 3094 3417 }} (836)		\| {{mapping\| 836 1325 1941 2347 2892 3094 3417 }} (836)
	\| -0.0014		\| −0.0014
	\| 0.0747		\| 0.0747
	\| 5.21		\| 5.21
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	~~\| style="border-top: double;"~~ \| 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256		\| 1716/1715, 2080/2079, 3025/3024, 15379/15360, 234375/234256
	~~\| style="border-top: double;"~~ \| {{mapping\| 836 1325 1941 2347 2892 3093 }} (836f)		\| {{mapping\| 836 1325 1941 2347 2892 3093 }} (836f)
	~~\| style="border-top: double;"~~ \| +0.0561		\| +0.0561
	~~\| style="border-top: double;"~~ \| 0.0805		\| 0.0805
	~~\| style="border-top: double;"~~ \| 5.60		\| 5.60
	\|-		\|-
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
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	\| 0.0747		\| 0.0747
	\| 5.20		\| 5.20
	\|}		{{comma basis end}}
	* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo\|612]] and before [[1084edo\|1084]].		* 836et is notable in the 11-limit with a lower absolute error than any previous equal temperaments, past [[612edo\|612]] and before [[1084edo\|1084]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 2		\| 2
	\| 265\836<br>(56\836)		\| 265\836<br />(56\836)
	\| 380.38<br>(80.38)		\| 380.38<br />(80.38)
	\| 81/65<br>(22/21)		\| 81/65<br />(22/21)
	\| [[Quasithird]] (836)		\| [[Quasithird]] (836)
	\|-		\|-
	\| 19		\| 19
	\| 347\836<br>(5\836)		\| 347\836<br />(5\836)
	\| 498.09<br>(7.18)		\| 498.09<br />(7.18)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\| 22		\| 22
	\| 347\836<br>(5\836)		\| 347\836<br />(5\836)
	\| 498.09<br>(7.18)		\| 498.09<br />(7.18)
	\| 4/3<br>({{monzo\| 16 -13 2 }})		\| 4/3<br />({{monzo\| 16 -13 2 }})
	\| [[Major arcana]]		\| [[Major arcana]]
	\|-		\|-
	\| 38		\| 38
	\| 347\836<br>(5\836)		\| 347\836<br />(5\836)
	\| 498.09<br>(7.18)		\| 498.09<br />(7.18)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|-		\|-
	\| 44		\| 44
	\| 347\836<br>(5\836)		\| 347\836<br />(5\836)
	\| 498.09<br>(7.18)		\| 498.09<br />(7.18)
	\| 4/3<br>(18375/18304)		\| 4/3<br />(18375/18304)
	\| [[Ruthenium]]		\| [[Ruthenium]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	[[Category:Quasithird]]		[[Category:Quasithird]]

Eliora: /* Theory */

2023-12-10T16:19:48Z

Theory

← Older revision		Revision as of 16:19, 10 December 2023
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 836 factors into {{factorization\|836}}, 836edo has subset edos {{EDOs\| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for [[harmonic]] [[13/1\|13]].		Since 836 factors into {{factorization\|836}}, 836edo has subset edos {{EDOs\| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. [[1672edo]], which doubles it, provides a good correction for [[harmonic]] [[13/1\|13]].

	== Regular temperament properties ==		== Regular temperament properties ==

FloraC: Adopt template: Factorization; misc. cleanup

2023-11-02T11:42:51Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 11:42, 2 November 2023
Line 5:		Line 5:
	836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].		836edo is a strong 11-limit system, having the lowest absolute error and beating [[612edo]].

	The equal temperament [[~~Tempering~~ out\|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.		The equal temperament [[tempering out\|tempers out]] the [[counterschisma]] and the [[enneadeca]] in the 5-limit; [[4375/4374]], [[703125/702464]] in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit. It supports [[enneadecal]] in the 7-limit as well as [[hemienneadecal]] in the 11-limit. It also tunes [[orga]] and [[quasithird]]. In addition, it is divisible by 44 and in light of that it tunes [[ruthenium]] in the 7-limit and also 11-limit.

	Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.		Extending it to the 13-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], it tempers out [[2200/2197]], [[4096/4095]], 31250/31213 in the 13-limit; and [[1275/1274]], [[2500/2499]], [[2601/2600]] in the 17-limit. It provides the [[optimal patent val]] for 13-limit quasithird. Using the 836f [[val]], it tempers out [[1716/1715]], [[2080/2079]], 15379/15360 in the 13-limit; and [[2431/2430]], 2500/2499, [[4914/4913]], [[5832/5831]], 11271/11264 in the 17-limit. It gives a good tuning for 13-limit orga.
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 836 factors into ~~2<sup>2</sup> × 11 × 19~~, 836edo has subset edos {{EDOs\| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for harmonic 13.		Since 836 factors into {{factorization\|836}}, 836edo has subset edos {{EDOs\| 2, 4, 11, 19, 22, 38, 44, 76, 209, 418 }}. 1672edo, which doubles it, provides a good correction for [[harmonic]] [[13/1\|13]].

	== Regular temperament properties ==		== Regular temperament properties ==
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	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\|22		\| 22
	\|347\836<br>(5\836)		\| 347\836<br>(5\836)
	\|498.09<br>(7.18)		\| 498.09<br>(7.18)
	\|4/3<br>({{monzo\|16 -13 2}})		\| 4/3<br>({{monzo\| 16 -13 2 }})
	\|[[Major arcana]]		\| [[Major arcana]]
	\|-		\|-
	\| 38		\| 38

Eliora: /* Rank-2 temperaments */ monzo

2023-10-24T10:53:55Z

Rank-2 temperaments: monzo

← Older revision		Revision as of 10:53, 24 October 2023
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	\|347\836<br>(5\836)		\|347\836<br>(5\836)
	\|498.09<br>(7.18)		\|498.09<br>(7.18)
	\|4/3<br>([16 -13 2⟩)		\|4/3<br>({{monzo\|16 -13 2}})
	\|[[Major arcana]]		\|[[Major arcana]]
	\|-		\|-