400edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"

2026-03-13T13:34:37Z

Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"

← Older revision		Revision as of 13:34, 13 March 2026
Line 206:		Line 206:
	\| [[Octogintic]]		\| [[Octogintic]]
	\|}		\|}
	<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms\|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 form]] in parentheses if distinct

	== Scales ==		== Scales ==

FloraC: /* Regular temperament properties */ emmthird

2025-09-04T15:13:01Z

Regular temperament properties: emmthird

← Older revision		Revision as of 15:13, 4 September 2025
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	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[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
	\|-		\|-
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	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| -7 22 -12 }}, {{monzo\| 47 -15 -10 }}		\| {{Monzo\| -7 22 -12 }}, {{monzo\| 47 -15 -10 }}
	\| {{~~mapping~~\| 400 634 929 }}		\| {{Mapping\| 400 634 929 }}
	\| ~~−0~~.1080		\| −0.1080
	\| 0.1331		\| 0.1331
	\| 4.44		\| 4.44
Line 101:		Line 101:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 1959552/1953125, 14348907/14336000		\| 2401/2400, 1959552/1953125, 14348907/14336000
	\| {{~~mapping~~\| 400 634 929 1123 }}		\| {{Mapping\| 400 634 929 1123 }}
	\| ~~−0~~.0965		\| −0.0965
	\| 0.1170		\| 0.1170
	\| 3.90		\| 3.90
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 5632/5625, 9801/9800, 46656/46585		\| 2401/2400, 5632/5625, 9801/9800, 46656/46585
	\| {{~~mapping~~\| 400 634 929 1123 1384 }}		\| {{Mapping\| 400 634 929 1123 1384 }}
	\| ~~−0~~.1166		\| −0.1166
	\| 0.1121		\| 0.1121
	\| 3.74		\| 3.74
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325		\| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
	\| {{~~mapping~~\| 400 634 929 1123 1384 1480 }}		\| {{Mapping\| 400 634 929 1123 1384 1480 }}
	\| ~~−0~~.0734		\| −0.0734
	\| 0.1407		\| 0.1407
	\| 4.69		\| 4.69
Line 122:		Line 122:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095		\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
	\| {{~~mapping~~\| 400 634 929 1123 1384 1480 1635 }}		\| {{Mapping\| 400 634 929 1123 1384 1480 1635 }}
	\| ~~−0~~.0645		\| −0.0645
	\| 0.1321		\| 0.1321
	\| 4.40		\| 4.40
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	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715		\| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
	\| {{~~mapping~~\| 400 634 929 1123 1384 1480 1635 1699 }}		\| {{Mapping\| 400 634 929 1123 1384 1480 1635 1699 }}
	\| ~~−0~~.0413		\| −0.0413
	\| 0.1380		\| 0.1380
	\| 4.60		\| 4.60
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	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperament		! Temperament
	\|-		\|-
Line 149:		Line 149:
	\| 83\400		\| 83\400
	\| 249.00		\| 249.00
	\| {{~~monzo~~\| -26 18 -1 }}		\| {{Monzo\| -26 18 -1 }}
	\| [[Monzismic]]		\| [[Monzismic]]
	\|-		\|-
Line 163:		Line 163:
	\| 25/21		\| 25/21
	\| [[Quinmite]]		\| [[Quinmite]]
			\|-
			\| 1
			\| 131\400
			\| 393.00
			\| 2744/2187
			\| [[Emmthird]] (7-limit)
	\|-		\|-
	\| 1		\| 1
Line 183:		Line 189:
	\|-		\|-
	\| 5		\| 5
	\| 123\400<br />(37\400)		\| 123\400<br>(37\400)
	\| 369.00<br />(111.00)		\| 369.00<br>(111.00)
	\| 1024/891<br />(16/15)		\| 1024/891<br>(16/15)
	\| [[Quintosec]]		\| [[Quintosec]]
	\|-		\|-
	\| 10		\| 10
	\| 83\400<br />(3\400)		\| 83\400<br>(3\400)
	\| 249.00<br />(9.00)		\| 249.00<br>(9.00)
	\| 15/13<br />(176/175)		\| 15/13<br>(176/175)
	\| [[Decoid]]		\| [[Decoid]]
	\|-		\|-
	\| 80		\| 80
	\| 166\400<br />(1\400)		\| 166\400<br>(1\400)
	\| 498.00<br />(3.00)		\| 498.00<br>(3.00)
	\| 4/3<br />(245/243)		\| 4/3<br>(245/243)
	\| [[Octogintic]]		\| [[Octogintic]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[Normal ~~lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==

Tristanbay: /* Subsets and supersets */Added 1600edo and 2000edo as notable 400edo supersets

2025-08-14T12:33:21Z

Subsets and supersets: Added 1600edo and 2000edo as notable 400edo supersets

← Older revision		Revision as of 12:33, 14 August 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 400 factors into 2<sup>4</sup> × 5<sup>2</sup>, 400edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.		Since 400 factors into 2<sup>4</sup> × 5<sup>2</sup>, 400edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.

			Of edos that are a multiple of 400, {{EDOs\| 1600 and 2000}} are notable for their high consistency limits, as [[Interval size measure\|interval size measures]], and perhaps as ways of tuning various temperaments.

	== Interval table ==		== Interval table ==

ArrowHead294 at 14:52, 20 February 2025

2025-02-20T14:52:50Z

← Older revision		Revision as of 14:52, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|400~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Interval table */ cleanup

2025-01-17T16:09:50Z

Interval table: cleanup

← Older revision		Revision as of 16:09, 17 January 2025
Line 16:		Line 16:
	== Interval table ==		== Interval table ==
	=== All intervals ===		=== All intervals ===
	~~''see~~ [[Table of 400edo intervals]]''		See [[Table of 400edo intervals]].

	=== Selected intervals ===		=== Selected intervals ===
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	\|-		\|-
	! Step		! Step
	! Eliora's ~~Naming System~~		! Eliora's naming system
	! Associated ~~Ratio~~		! Associated ratio
	\|-		\|-
	\| 0		\| 0

FloraC: /* Theory */ cleanup; shorten the prime harmonics table (this was insane)

2025-01-17T16:09:04Z

Theory: cleanup; shorten the prime harmonics table (this was insane)

← Older revision		Revision as of 16:09, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	400edo is a strong 17- and 19-limit system, [[distinctly consistent]] to the [[21-odd-limit]] ~~and [[purely consistent]] to the 21-odd-limit~~. It shares its excellent [[harmonic]] [[3/1\|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.		400edo is a strong 17- and 19-limit system, [[consistency\|distinctly and purely consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1\|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.

	It [[tempers out]] the unidecma, {{monzo\| -7 22 -12 }}, and the quintosec comma, {{monzo\| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.		As an equal temperament, it [[tempering out\|tempers out]] the unidecma, {{monzo\| -7 22 -12 }}, and the quintosec comma, {{monzo\| 47 -15 -10 }}, in the [[5-limit]]; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the [[7-limit]]; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the [[13-limit]], [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|400\|columns=13}}		{{Harmonics in equal\|400\|columns=13}}
	{{Harmonics in equal\|400~~\|start=14~~\|columns=13~~\|title=Approximation of prime harmonics in 400edo (continued)}}~~		{{Harmonics in equal\|400\|columns=13\|start=14\|collapsed=true\|title=Approximation of prime harmonics in 400edo (continued)}}
	~~{{Harmonics in equal\|400\|start=27\|columns=11\|collapsed=1\|title=Approximation of prime harmonics in 400edo (103 to 157)}}~~
	~~{{Harmonics in equal\|400~~\|start~~=38\|columns~~=14\|collapsed=1\|title=Approximation of prime harmonics in 400edo (~~163 to 229~~)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 400 factors into ~~{{factorization\|400}}~~, 400edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.		Since 400 factors into 2<sup>4</sup> × 5<sup>2</sup>, 400edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}.

	== Interval table ==		== Interval table ==

ArrowHead294 at 15:34, 16 January 2025

2025-01-16T15:34:49Z

← Older revision		Revision as of 15:34, 16 January 2025
Line 200:		Line 200:
	\| [[Octogintic]]		\| [[Octogintic]]
	\|}		\|}
	<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

	== Scales ==		== Scales ==

ArrowHead294 at 15:29, 16 January 2025

2025-01-16T15:29:30Z

← Older revision		Revision as of 15:29, 16 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	400edo is a strong 17- and 19-limit system, [[~~consistency\|~~distinctly consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1\|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.		400edo is a strong 17- and 19-limit system, [[distinctly consistent]] to the [[21-odd-limit]] and [[purely consistent]] to the 21-odd-limit. It shares its excellent [[harmonic]] [[3/1\|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities.

	~~The equal temperament~~ [[~~tempering out\|~~tempers out]] the unidecma, {{monzo\| -7 22 -12 }}, and the quintosec comma, {{monzo\| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.		It [[tempers out]] the unidecma, {{monzo\| -7 22 -12 }}, and the quintosec comma, {{monzo\| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; [[5632/5625]], [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, [[support]]ing the [[decoid]] temperament and the [[quinmite]] temperament. It tempers out [[936/935]], [[1156/1155]], [[2058/2057]], [[2601/2600]], [[4914/4913]] and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit.

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

ArrowHead294 at 11:49, 16 November 2024

2024-11-16T11:49:13Z

← Older revision		Revision as of 11:49, 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]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.1380		\| 0.1380
	\| 4.60		\| 4.60
	~~{{comma basis end}~~}		\|}
	* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo\|354]] in the 17-limit, and [[311edo\|311]] in the 19-limit; it is bettered by [[422edo\|422]] in either subgroup.		* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo\|354]] in the 17-limit, and [[311edo\|311]] in the 19-limit; it is bettered by [[422edo\|422]] in either subgroup.

	=== 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*
			! Temperament
	\|-		\|-
	\| 1		\| 1
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	\| 4/3<br />(245/243)		\| 4/3<br />(245/243)
	\| [[Octogintic]]		\| [[Octogintic]]
	~~{{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

	== Scales ==		== Scales ==

ArrowHead294 at 05:50, 16 November 2024

2024-11-16T05:50:24Z

← Older revision		Revision as of 05:50, 16 November 2024
Line 22:		Line 22:
	=== Selected intervals ===		=== Selected intervals ===
	{\| class="wikitable center-1"		{\| class="wikitable center-1"
	\|+		\|-
	! Step		! Step
	! Eliora's Naming System		! Eliora's Naming System
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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]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| -7 22 -12 }}, {{monzo\| 47 -15 -10 }}		\| {{monzo\| -7 22 -12 }}, {{monzo\| 47 -15 -10 }}
	\| {{mapping\| 400 634 929 }}		\| {{mapping\| 400 634 929 }}
	\| -0.1080		\| −0.1080
	\| 0.1331		\| 0.1331
	\| 4.44		\| 4.44
Line 101:		Line 93:
	\| 2401/2400, 1959552/1953125, 14348907/14336000		\| 2401/2400, 1959552/1953125, 14348907/14336000
	\| {{mapping\| 400 634 929 1123 }}		\| {{mapping\| 400 634 929 1123 }}
	\| -0.0965		\| −0.0965
	\| 0.1170		\| 0.1170
	\| 3.90		\| 3.90
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	\| 2401/2400, 5632/5625, 9801/9800, 46656/46585		\| 2401/2400, 5632/5625, 9801/9800, 46656/46585
	\| {{mapping\| 400 634 929 1123 1384 }}		\| {{mapping\| 400 634 929 1123 1384 }}
	\| -0.1166		\| −0.1166
	\| 0.1121		\| 0.1121
	\| 3.74		\| 3.74
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	\| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325		\| 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325
	\| {{mapping\| 400 634 929 1123 1384 1480 }}		\| {{mapping\| 400 634 929 1123 1384 1480 }}
	\| -0.0734		\| −0.0734
	\| 0.1407		\| 0.1407
	\| 4.69		\| 4.69
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	\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095		\| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095
	\| {{mapping\| 400 634 929 1123 1384 1480 1635 }}		\| {{mapping\| 400 634 929 1123 1384 1480 1635 }}
	\| -0.0645		\| −0.0645
	\| 0.1321		\| 0.1321
	\| 4.40		\| 4.40
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	\| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715		\| 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715
	\| {{mapping\| 400 634 929 1123 1384 1480 1635 1699 }}		\| {{mapping\| 400 634 929 1123 1384 1480 1635 1699 }}
	\| -0.0413		\| −0.0413
	\| 0.1380		\| 0.1380
	\| 4.60		\| 4.60
	\|}		{{comma basis end}}
	* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo\|354]] in the 17-limit, and [[311edo\|311]] in the 19-limit; it is bettered by [[422edo\|422]] in either subgroup.		* 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat [[354edo\|354]] in the 17-limit, and [[311edo\|311]] in the 19-limit; it is bettered by [[422edo\|422]] in either subgroup.

	=== 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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	\|-		\|-
	\| 5		\| 5
	\| 123\400<br>(37\400)		\| 123\400<br />(37\400)
	\| 369.00<br>(111.00)		\| 369.00<br />(111.00)
	\| 1024/891<br>(16/15)		\| 1024/891<br />(16/15)
	\| [[Quintosec]]		\| [[Quintosec]]
	\|-		\|-
	\| 10		\| 10
	\| 83\400<br>(3\400)		\| 83\400<br />(3\400)
	\| 249.00<br>(9.00)		\| 249.00<br />(9.00)
	\| 15/13<br>(176/175)		\| 15/13<br />(176/175)
	\| [[Decoid]]		\| [[Decoid]]
	\|-		\|-
	\| 80		\| 80
	\| 166\400<br>(1\400)		\| 166\400<br />(1\400)
	\| 498.00<br>(3.00)		\| 498.00<br />(3.00)
	\| 4/3<br>(245/243)		\| 4/3<br />(245/243)
	\| [[Octogintic]]		\| [[Octogintic]]
	\|}		{{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}}

	== Scales ==		== Scales ==