210edo - 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"

2026-03-13T13:33:06Z

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 13:33, 13 March 2026
Line 62:		Line 62:
	\| [[Countdown]] (210e)		\| [[Countdown]] (210e)
	\|}		\|}
	<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 19:27, 20 February 2025

2025-02-20T19:27:28Z

← Older revision		Revision as of 19:27, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|210~~}}		{{ED intro}}

	== Theory ==		== Theory ==
	Since {{nowrap\|210 {{=}} 3 ~~×~~ 70}}, 210edo shares its [[3/2\|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit.		Since {{nowrap\|210 {{=}} 3 × 70}}, 210edo shares its [[3/2\|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit.

	Using the 210e val, which does the best, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit. Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit.		Using the 210e val, which does the best, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit. Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit.

ArrowHead294 at 17:46, 15 January 2025

2025-01-15T17:46:35Z

← Older revision		Revision as of 17:46, 15 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 210 factors into ~~2 × 3 × 5 × 7~~, 210edo has subset edos {{EDOs\| 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105 }}.		Since 210 factors into {{factorisation\|210}}, 210edo has subset edos {{EDOs\| 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| {{monzo\| 26 -12 -3 }}, {{monzo\| 19 10 -15 }}		\| {{monzo\| 26 -12 -3 }}, {{monzo\| 19 10 -15 }}
	\| {{mapping\| 210 333 488 }}		\| {{mapping\| 210 333 488 }}
	\| ~~−0~~.5138		\| −0.5138
	\| 0.3987		\| 0.3987
	\| 6.98		\| 6.98
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	\| 3136/3125, 5120/5103, 118098/117649		\| 3136/3125, 5120/5103, 118098/117649
	\| {{mapping\| 210 333 488 590 }}		\| {{mapping\| 210 333 488 590 }}
	\| ~~−0~~.6170		\| −0.6170
	\| 0.3888		\| 0.3888
	\| 6.80		\| 6.80
Line 62:		Line 62:
	\| [[Countdown]] (210e)		\| [[Countdown]] (210e)
	\|}		\|}
	<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: Partial undo

2024-11-16T13:01:46Z

Partial undo

← Older revision		Revision as of 13:01, 16 November 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	210 = 3 × 70, ~~and~~ 210edo shares its [[3/2\|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit.		Since {{nowrap\|210 {{=}} 3 × 70}}, 210edo shares its [[3/2\|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit.

	Using the 210e val, which does the best, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit. Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit.		Using the 210e val, which does the best, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit. Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit.
Line 14:		Line 14:

	== 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]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.3888		\| 0.3888
	\| 6.80		\| 6.80
	~~{{comma basis end}~~}		\|}

	=== 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
	\|-		\|-
	\| 3		\| 3
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	\| 25/24		\| 25/24
	\| [[Countdown]] (210e)		\| [[Countdown]] (210e)
	~~{{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 02:50, 16 November 2024

2024-11-16T02:50:35Z

← Older revision		Revision as of 02:50, 16 November 2024
Line 14:		Line 14:

	== 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]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 26 -12 -3 }}, {{monzo\| 19 10 -15 }}		\| {{monzo\| 26 -12 -3 }}, {{monzo\| 19 10 -15 }}
	\| {{mapping\| 210 333 488 }}		\| {{mapping\| 210 333 488 }}
	\| -0.5138		\| −0.5138
	\| 0.3987		\| 0.3987
	\| 6.98		\| 6.98
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	\| 3136/3125, 5120/5103, 118098/117649		\| 3136/3125, 5120/5103, 118098/117649
	\| {{mapping\| 210 333 488 590 }}		\| {{mapping\| 210 333 488 590 }}
	\| -0.6170		\| −0.6170
	\| 0.3888		\| 0.3888
	\| 6.80		\| 6.80
	\|}		{{comma basis end}}

	=== 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~~
	\|-		\|-
	\| 3		\| 3
	\| 123\210<br>(17\210)		\| 123\210<br />(17\210)
	\| 702.86<br>(97.14)		\| 702.86<br />(97.14)
	\| 3/2<br>(18/17)		\| 3/2<br />(18/17)
	\| [[Misty]] (210gh)		\| [[Misty]] (210gh)
	\|-		\|-
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	\| 25/24		\| 25/24
	\| [[Countdown]] (210e)		\| [[Countdown]] (210e)
	\|}		{{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: Rework

2024-04-03T09:18:28Z

Rework

← Older revision		Revision as of 09:18, 3 April 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~210et~~ [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 ([[trisedodge comma]]) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit~~. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency for [[harmonic]]s [[3/1\|3]], [[5/1\|5]], and [[7/1\|7]]~~.		210 = 3 × 70, and 210edo shares its [[3/2\|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit.

	Using the ~~patent~~ val, it tempers out [[~~176~~/~~175~~]]~~, 1375/1372~~, [[~~8019~~/~~8000~~]], and ~~41503~~/~~41472~~ in the 11-limit; [[351/350]], [[~~352~~/~~351~~]], [[~~847~~/~~845~~]], 2197/2187, and ~~16900~~/~~16807~~ in the 13-limit. Using the ~~210e~~ val, it tempers out [[~~540~~/~~539~~]], [[~~4000~~/~~3993~~]]~~, 6912/6875~~, and ~~15488~~/~~15435~~ in the 11-limit; [[351/350]], [[~~364~~/~~363~~]], [[~~1001~~/~~1000~~]], [[2197/2187]], and ~~3584~~/~~3575~~ in the 13-limit.		Using the 210e val, which does the best, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit. Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit.

	=== Odd harmonics ===		=== Odd harmonics ===
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	\| 123\210<br>(17\210)		\| 123\210<br>(17\210)
	\| 702.86<br>(97.14)		\| 702.86<br>(97.14)
	\| 3/2<br>(~~135\128~~)		\| 3/2<br>(18/17)
	\| [[Misty]]		\| [[Misty]] (210gh)
	\|-		\|-
	\| 5		\| 5
	\| ~~98\210<br>(~~13\210)		\| 13\210
	\| ~~560.00<br>(~~74.29)		\| 74.29
	\| ~~864/625<br>(~~25/24)		\| 25/24
	\| [[~~Trisedodge~~]]		\| [[Countdown]] (210e)
	\|}		\|}
	<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct		<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

FloraC: Cleanup

2024-04-03T09:13:50Z

Cleanup

← Older revision		Revision as of 09:13, 3 April 2024
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|210}}		{{EDO intro\|210}}
	==Theory==
	210et tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for ~~harmonics~~ 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.		== Theory ==
	===Odd harmonics===		210et [[tempering out\|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 ([[trisedodge comma]]) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency for [[harmonic]]s [[3/1\|3]], [[5/1\|5]], and [[7/1\|7]].

			Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit.

			=== Odd harmonics ===
	{{Harmonics in equal\|210}}		{{Harmonics in equal\|210}}
	===Subsets and supersets===
	~~210edo~~ factors into 2 × 3 × 5 × 7, ~~with~~ subset edos {{EDOs\|2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105}}.		=== Subsets and supersets ===
	==Regular temperament properties==		Since 210 factors into 2 × 3 × 5 × 7, 210edo has subset edos {{EDOs\| 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105 }}.

			== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \|[[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \|[[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|Comma List]]
	! rowspan="2" \|[[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \|Optimal<br>8ve Stretch (¢)		! rowspan="2" \| Optimal<br>8ve Stretch (¢)
	! colspan="2" \|Tuning Error		! colspan="2" \| Tuning Error
	\|-		\|-
	![[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-
	~~\|2.3~~
	~~\|{{monzo\|111 -70}}~~
	~~\|{{val\|210 333}}~~
	~~\| -0.2846~~
	~~\| 0.2845~~
	~~\| 4.98~~
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|26 -12 -3}}, {{monzo\|19 10 -15}}		\| {{monzo\| 26 -12 -3 }}, {{monzo\| 19 10 -15 }}
	\|{{~~val~~\|210 333 488}}		\| {{mapping\| 210 333 488 }}
	\| -0.5138		\| -0.5138
	\| 0.3987		\| 0.3987
	\| 6.98		\| 6.98
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|3136/3125, 5120/5103, 118098/117649		\| 3136/3125, 5120/5103, 118098/117649
	\|{{~~val~~\|210 333 488 590}}		\| {{mapping\| 210 333 488 590 }}
	\| -0.6170		\| -0.6170
	\| 0.3888		\| 0.3888
	\| 6.80		\| 6.80
	\|}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| 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 8ve		! Periods<br>per 8ve
	! Generator~~<br>(reduced)~~		! Generator*
	! Cents~~<br>(reduced)~~		! Cents*
	! Associated<br>~~ratio~~		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
	\|3		\| 3
	\|123\210<br>(17\210)		\| 123\210<br>(17\210)
	\|702.86<br>(97.14)		\| 702.86<br>(97.14)
	\|3/2<br>(135\128)		\| 3/2<br>(135\128)
	\|[[Misty]]		\| [[Misty]]
	\|-		\|-
	\|5		\| 5
	\|98\210<br>(13\210)		\| 98\210<br>(13\210)
	\|560.00<br>(74.29)		\| 560.00<br>(74.29)
	\|864/625<br>(25/24)		\| 864/625<br>(25/24)
	\|[[Trisedodge]]		\| [[Trisedodge]]
	\|}		\|}
			<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 -->~~

Francium: /* Rank-2 temperaments */ typo

2023-10-19T05:12:13Z

Rank-2 temperaments: typo

← Older revision		Revision as of 05:12, 19 October 2023
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	\|560.00<br>(74.29)		\|560.00<br>(74.29)
	\|864/625<br>(25/24)		\|864/625<br>(25/24)
	\|[[~~Trisendodge~~]]		\|[[Trisedodge]]
	\|}		\|}

	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->

Francium at 05:11, 19 October 2023

2023-10-19T05:11:45Z

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''210edo''' is the [[EDO|equal division of the octave]] into 210 parts of 5.7143 cents each. It tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.
+{{EDO intro|210}}
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Plumtree: Infobox ET added

2022-10-04T21:22:11Z

Infobox ET added

← Older revision		Revision as of 21:22, 4 October 2022
Line 1:		Line 1:
			{{Infobox ET}}
	'''210edo''' is the [[EDO\|equal division of the octave]] into 210 parts of 5.7143 cents each. It tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.		'''210edo''' is the [[EDO\|equal division of the octave]] into 210 parts of 5.7143 cents each. It tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.

	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->