559edo - Revision history

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"

2026-03-13T13:31:57Z

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
Line 99:		Line 99:
	\| [[Meridic]]		\| [[Meridic]]
	\|}		\|}
	<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

ArrowHead294 at 22:50, 20 February 2025

2025-02-20T22:50:20Z

← Older revision		Revision as of 22:50, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|559~~}}		{{ED intro}}

	== Theory ==		== Theory ==
	559edo is a very strong 5-limit system. ~~The equal temperament~~ [[tempering out\|tempers out]] the luna comma, {{monzo\| 38 -2 -15 }} and the minortone comma, {{monzo\| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo\| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].		559edo is a very strong 5-limit system. It [[tempering out\|tempers out]] the luna comma, {{monzo\| 38 -2 -15 }} and the minortone comma, {{monzo\| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo\| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].

	=== Prime harmonics ===		=== Prime harmonics ===

ArrowHead294 at 19:22, 15 January 2025

2025-01-15T19:22:44Z

← Older revision		Revision as of 19:22, 15 January 2025
Line 99:		Line 99:
	\| [[Meridic]]		\| [[Meridic]]
	\|}		\|}
	<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

ArrowHead294 at 19:05, 15 January 2025

2025-01-15T19:05:50Z

← Older revision		Revision as of 19:05, 15 January 2025
Line 26:		Line 26:
	\| {{monzo\| 886 -559 }}		\| {{monzo\| 886 -559 }}
	\| {{mapping\| 559 886 }}		\| {{mapping\| 559 886 }}
	\| ~~−0~~.0040		\| −0.0040
	\| 0.0040		\| 0.0040
	\| 0.19		\| 0.19
Line 33:		Line 33:
	\| {{monzo\| 38 -2 -15 }}, {{monzo\| -16 35 -17 }}		\| {{monzo\| 38 -2 -15 }}, {{monzo\| -16 35 -17 }}
	\| {{mapping\| 559 886 1298 }}		\| {{mapping\| 559 886 1298 }}
	\| ~~−0~~.0157		\| −0.0157
	\| 0.0168		\| 0.0168
	\| 0.78		\| 0.78

ArrowHead294: Partial undo

2024-11-16T12:56:41Z

Partial undo

← Older revision		Revision as of 12:56, 16 November 2024
Line 12:		Line 12:

	== 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
Line 41:		Line 50:
	\| 0.1175		\| 0.1175
	\| 5.48		\| 5.48
	~~{{comma basis end}~~}		\|}
	* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo\|441]] and before [[612edo\|612]].		* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo\|441]] and before [[612edo\|612]].

	=== 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
Line 82:		Line 98:
	\| 4/3<br />(385/384)		\| 4/3<br />(385/384)
	\| [[Meridic]]		\| [[Meridic]]
	~~{{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

ArrowHead294 at 03:09, 16 November 2024

2024-11-16T03:09:34Z

← Older revision		Revision as of 03:09, 16 November 2024
Line 12:		Line 12:

	== 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
	\| {{monzo\| 886 -559 }}		\| {{monzo\| 886 -559 }}
	\| {{mapping\| 559 886 }}		\| {{mapping\| 559 886 }}
	\| -0.0040		\| −0.0040
	\| 0.0040		\| 0.0040
	\| 0.19		\| 0.19
Line 32:		Line 24:
	\| {{monzo\| 38 -2 -15 }}, {{monzo\| -16 35 -17 }}		\| {{monzo\| 38 -2 -15 }}, {{monzo\| -16 35 -17 }}
	\| {{mapping\| 559 886 1298 }}		\| {{mapping\| 559 886 1298 }}
	\| -0.0157		\| −0.0157
	\| 0.0168		\| 0.0168
	\| 0.78		\| 0.78
Line 49:		Line 41:
	\| 0.1175		\| 0.1175
	\| 5.48		\| 5.48
	\|}		{{comma basis end}}
	* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo\|441]] and before [[612edo\|612]].		* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo\|441]] and before [[612edo\|612]].

	=== 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
Line 85:		Line 71:
	\| [[Acrokleismic]]		\| [[Acrokleismic]]
	\|-		\|-
	\|13		\| 13
	\|232\559<br>(17\559)		\| 232\559<br />(17\559)
	\|498.03<br>(36.494)		\| 498.03<br />(36.494)
	\|4/3<br>(?)		\| 4/3<br />(?)
	\|[[Aluminium]]		\| [[Aluminium]]
	\|-		\|-
	\| 43		\| 43
	\| 232\559<br>(2\559)		\| 232\559<br />(2\559)
	\| 498.03<br>(4.29)		\| 498.03<br />(4.29)
	\| 4/3<br>(385/384)		\| 4/3<br />(385/384)
	\| [[Meridic]]		\| [[Meridic]]
	\|}		{{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}}

FloraC: 5-limit notability

2023-11-06T16:58:09Z

5-limit notability

← Older revision		Revision as of 16:58, 6 November 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	559edo is a strong 5-limit system. The equal temperament [[tempering out\|tempers out]] the luna comma, {{monzo\| 38 -2 -15 }} and the minortone comma, {{monzo\| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo\| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].		559edo is a very strong 5-limit system. The equal temperament [[tempering out\|tempers out]] the luna comma, {{monzo\| 38 -2 -15 }} and the minortone comma, {{monzo\| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo\| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 50:		Line 50:
	\| 5.48		\| 5.48
	\|}		\|}
			* 559et has a lower relative error than any previous equal temperaments in the 5-limit, past [[441edo\|441]] and before [[612edo\|612]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===

FloraC: +subsets and supersets

2023-11-02T13:43:14Z

+subsets and supersets

← Older revision		Revision as of 13:43, 2 November 2023
Line 7:		Line 7:
	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|559\|columns=11}}		{{Harmonics in equal\|559\|columns=11}}

			=== Subsets and supersets ===
			Since 559 factors into {{factorization\|559}}, 559edo contains [[13edo]] and [[43edo]] as subsets.

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

Eliora: /* Rank-2 temperaments */ + aluminium

2023-11-01T15:34:27Z

Rank-2 temperaments: + aluminium

← Older revision		Revision as of 15:34, 1 November 2023
Line 80:		Line 80:
	\| 6/5		\| 6/5
	\| [[Acrokleismic]]		\| [[Acrokleismic]]
			\|-
			\|13
			\|232\559<br>(17\559)
			\|498.03<br>(36.494)
			\|4/3<br>(?)
			\|[[Aluminium]]
	\|-		\|-
	\| 43		\| 43

FloraC: Cleanup; clarify the title row of the rank-2 temp table; -redundant categories

2023-10-29T09:01:07Z

Cleanup; clarify the title row of the rank-2 temp table; -redundant categories

← Older revision		Revision as of 09:01, 29 October 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	559edo [[tempering out\|tempers out]] the luna comma, {{monzo\| 38 -2 -15 }} and the minortone comma, {{monzo\| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo\| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].		559edo is a strong 5-limit system. The equal temperament [[tempering out\|tempers out]] the luna comma, {{monzo\| 38 -2 -15 }} and the minortone comma, {{monzo\| -16 35 -17 }} in the [[5-limit]], as well as the [[monzisma]], {{monzo\| 54 -37 2 }}; [[4375/4374]], [[2100875/2097152]], and 282475249/281250000 in the [[7-limit]]; 12005/11979, [[41503/41472]], 160083/160000, and 172032/171875 in the [[11-limit]]. Rank-2 temperaments it [[support]]s include [[mitonic]], [[lunatic]], [[acrokleismic]], [[monzism]], and [[meridic]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 21:		Line 21:
	\| 2.3		\| 2.3
	\| {{monzo\| 886 -559 }}		\| {{monzo\| 886 -559 }}
	\| [{{~~val~~\| 559 886 }}]		\| {{mapping\| 559 886 }}
	\| -0.0040		\| -0.0040
	\| 0.0040		\| 0.0040
Line 28:		Line 28:
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 38 -2 -15 }}, {{monzo\| -16 35 -17 }}		\| {{monzo\| 38 -2 -15 }}, {{monzo\| -16 35 -17 }}
	\| [{{~~val~~\| 559 886 1298 }}]		\| {{mapping\| 559 886 1298 }}
	\| -0.0157		\| -0.0157
	\| 0.0168		\| 0.0168
Line 35:		Line 35:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 2100875/2097152, {{monzo\| -4 -2 -9 10 }}		\| 4375/4374, 2100875/2097152, {{monzo\| -4 -2 -9 10 }}
	\| [{{~~val~~\| 559 886 1298 1569 }}]		\| {{mapping\| 559 886 1298 1569 }}
	\| +0.0478		\| +0.0478
	\| 0.1109		\| 0.1109
Line 42:		Line 42:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 4375/4374, 12005/11979, 41503/41472, 172032/171875		\| 4375/4374, 12005/11979, 41503/41472, 172032/171875
	\| [{{~~val~~\| 559 886 1298 1569 1934 }}]		\| {{mapping\| 559 886 1298 1569 1934 }}
	\| 0.0161		\| 0.0161
	\| 0.1175		\| 0.1175
Line 51:		Line 51:
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per ~~Octave~~		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 87:		Line 87:
	\| [[Meridic]]		\| [[Meridic]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct
	[[~~Category:Equal divisions of~~ the octave\|~~###~~]] ~~<!-- 3-digit number -->~~