161edo - Revision history

Overthink: inconsistencies in 25-odd-limit

2026-04-10T20:25:46Z

inconsistencies in 25-odd-limit

← Older revision		Revision as of 20:25, 10 April 2026
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	== Theory ==		== Theory ==
	161edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1\|5]] being sharp, while [[3/1\|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s. Prime [[29/1\|29]] is also accurate, though harmonic [[27/1\|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the [[25-odd-limit]].		161edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1\|5]] being sharp, while [[3/1\|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s, and additional inconsistencies in the [[25-odd-limit]] include [[25/18]], [[25/23]], and their octave complements. Prime [[29/1\|29]] is also accurate, though harmonic [[27/1\|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the 25-odd-limit.

	As an equal temperament, 161et [[tempering out\|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.		As an equal temperament, 161et [[tempering out\|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.

Overthink: /* Theory */ expand description

2026-04-10T20:24:06Z

Theory: expand description

← Older revision		Revision as of 20:24, 10 April 2026
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	== Theory ==		== Theory ==
	161et [[tempering out\|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.		161edo has a [[perfect fifth]] slightly sharp of that of [[12edo]], such that it maps the [[Pythagorean comma]] to one step. It approximates many of the low primes fairly well; however, it is only consistent to the [[7-odd-limit]], due to [[10/9]] being mapped too sharply from prime [[5/1\|5]] being sharp, while [[3/1\|3]] is flat. Nonetheless it does well for its size in higher limits, with the inconsistent intervals in the [[23-odd-limit]] being 9/5, [[13/9]], [[23/13]], and their [[octave complement]]s. Prime [[29/1\|29]] is also accurate, though harmonic [[27/1\|27]] is mapped inconsistently flat, causing many of its intervals to be inconsistent. Additionally, the flatness of 27 causes [[28/27]] to be mapped wider than [[27/26]], meaning 161edo is at most [[diamond monotone]] in the [[25-odd-limit]].

			As an equal temperament, 161et [[tempering out\|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	~~In the range of edos from 100 to 200, 161edo is notable as being low in [[29-limit]] relative error.~~

	{{Harmonics in equal\|161}}		{{Harmonics in equal\|161}}

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"

2026-03-13T13:32:33Z

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 13:32, 13 March 2026
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	\| [[Absurdity]]		\| [[Absurdity]]
	\|}		\|}
	<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:Mintone]]		[[Category:Mintone]]

Overthink: + interval table

2026-03-02T00:37:53Z

+ interval table

← Older revision		Revision as of 00:37, 2 March 2026
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.		Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.

			== Intervals ==
			{{Interval table}}

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

Francium: changed EDO intro to ED intro

2025-02-19T18:02:33Z

changed EDO intro to ED intro

← Older revision		Revision as of 18:02, 19 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T15:26:24Z

Theory: cleanup

← Older revision		Revision as of 15:26, 17 January 2025
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	== Theory ==		== Theory ==
	161et [[tempers out]] the [[würschmidt comma]], 393216/390625, in the 5-limit; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the 11-limit; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the 13-limit. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.		161et [[tempering out\|tempers out]] the [[würschmidt comma]], 393216/390625, in the [[5-limit]]; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the [[7-limit]]; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the [[11-limit]]; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the [[13-limit]]. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	In the range of edos from 100 to 200, 161edo is notable as being low in [[29-limit]] relative error.		In the range of edos from 100 to 200, 161edo is notable as being low in [[29-limit]] relative error.

	{{Harmonics in equal\|161}}		{{Harmonics in equal\|161}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 161 factors into ~~{{factorization\|161}}~~, 161edo contains [[7edo]] and [[23edo]] as its subsets.		Since 161 factors into 7 × 23, 161edo contains [[7edo]] and [[23edo]] as its subsets.

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

ArrowHead294 at 19:51, 15 January 2025

2025-01-15T19:51:20Z

← Older revision		Revision as of 19:51, 15 January 2025
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	== Theory ==		== Theory ==
	161et [[~~tempering out\|~~tempers out]] the [[würschmidt comma]], 393216/390625, in the 5-limit; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the 11-limit; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the 13-limit. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.		161et [[tempers out]] the [[würschmidt comma]], 393216/390625, in the 5-limit; [[3136/3125]], [[6144/6125]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], [[540/539]] and [[5632/5625]] in the 11-limit; and [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1575/1573]] and [[1716/1715]] in the 13-limit. It serves as the [[optimal patent val]] for the [[mintone]] temperament in the 5-, 7-, 11- and 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
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	\| 243/242, 351/350, 441/440, 847/845, 3136/3125		\| 243/242, 351/350, 441/440, 847/845, 3136/3125
	\| {{mapping\| 161 255 374 452 557 596 }}		\| {{mapping\| 161 255 374 452 557 596 }}
	\| ~~−0~~.046		\| −0.046
	\| 0.449		\| 0.449
	\| 6.03		\| 6.03
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	\| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088		\| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
	\| {{mapping\| 161 255 374 452 557 596 658 }}		\| {{mapping\| 161 255 374 452 557 596 658 }}
	\| ~~−0~~.018		\| −0.018
	\| 0.422		\| 0.422
	\| 5.66		\| 5.66
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	\| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512		\| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
	\| {{mapping\| 161 255 374 452 557 596 658 684 }}		\| {{mapping\| 161 255 374 452 557 596 658 684 }}
	\| ~~−0~~.034		\| −0.034
	\| 0.397		\| 0.397
	\| 5.32		\| 5.32
	\|}		\|}
	* 161et has a lower [[TE error\|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo~~\|183et~~]].		* 161et has a lower [[TE error\|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
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	\| [[Absurdity]]		\| [[Absurdity]]
	\|}		\|}
	<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:Mintone]]		[[Category:Mintone]]

ArrowHead294: Partial undo

2024-11-16T12:24:39Z

Partial undo

← Older revision		Revision as of 12:24, 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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	\| 0.397		\| 0.397
	\| 5.32		\| 5.32
	~~{{comma basis end}~~}		\|}
	* 161et has a lower [[TE error\|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo\|183et]].		* 161et has a lower [[TE error\|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo\|183et]].

	=== 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 />(81/80)		\| 4/3<br />(81/80)
	\| [[Absurdity]]		\| [[Absurdity]]
	~~{{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:Mintone]]		[[Category:Mintone]]

ArrowHead294 at 04:46, 16 November 2024

2024-11-16T04:46:56Z

← Older revision		Revision as of 04:46, 16 November 2024
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	\| [[Absurdity]]		\| [[Absurdity]]
	{{rank-2 end}}		{{rank-2 end}}
	{{orf}]		{{orf}}

	[[Category:Mintone]]		[[Category:Mintone]]

ArrowHead294 at 04:46, 16 November 2024

2024-11-16T04:46:46Z

← Older revision		Revision as of 04:46, 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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	\| 243/242, 351/350, 441/440, 847/845, 3136/3125		\| 243/242, 351/350, 441/440, 847/845, 3136/3125
	\| {{mapping\| 161 255 374 452 557 596 }}		\| {{mapping\| 161 255 374 452 557 596 }}
	\| -0.046		\| −0.046
	\| 0.449		\| 0.449
	\| 6.03		\| 6.03
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	\| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088		\| 243/242, 351/350, 441/440, 561/560, 847/845, 1089/1088
	\| {{mapping\| 161 255 374 452 557 596 658 }}		\| {{mapping\| 161 255 374 452 557 596 658 }}
	\| -0.018		\| −0.018
	\| 0.422		\| 0.422
	\| 5.66		\| 5.66
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	\| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512		\| 243/242, 324/323, 351/350, 441/440, 456/455, 495/494, 513/512
	\| {{mapping\| 161 255 374 452 557 596 658 684 }}		\| {{mapping\| 161 255 374 452 557 596 658 684 }}
	\| -0.034		\| −0.034
	\| 0.397		\| 0.397
	\| 5.32		\| 5.32
	\|}		{{comma basis end}}
	* 161et has a lower [[TE error\|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo\|183et]].		* 161et has a lower [[TE error\|absolute error]] than any previous equal temperaments in the 19-limit, even though it is inconsistent in the corresponding odd limit. The same subgroup is only better tuned by [[183edo\|183et]].

	=== 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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	\|-		\|-
	\| 7		\| 7
	\| 67\161<br>(2\161)		\| 67\161<br />(2\161)
	\| 499.38<br>(14.91)		\| 499.38<br />(14.91)
	\| 4/3<br>(81/80)		\| 4/3<br />(81/80)
	\| [[Absurdity]]		\| [[Absurdity]]
	\|}		{{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:Mintone]]		[[Category:Mintone]]