552edo - 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"

2026-03-13T13:31:52Z

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 109:		Line 109:
	\| [[Palladium]] (5-limit)		\| [[Palladium]] (5-limit)
	\|}		\|}
	<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

Francium: changed EDO intro to ED intro

2025-02-21T06:31:33Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:31, 21 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:27, 15 January 2025

2025-01-15T19:27:45Z

← Older revision		Revision as of 19:27, 15 January 2025
Line 28:		Line 28:
	\| {{monzo\| 875 -552 }}		\| {{monzo\| 875 -552 }}
	\| {{mapping\| 552 875 }}		\| {{mapping\| 552 875 }}
	\| ~~−0~~.0691		\| −0.0691
	\| 0.0691		\| 0.0691
	\| 3.18		\| 3.18
Line 35:		Line 35:
	\| {{monzo\| 8 14 -13 }}, {{monzo\| 71 -36 -6 }}		\| {{monzo\| 8 14 -13 }}, {{monzo\| 71 -36 -6 }}
	\| {{mapping\| 552 875 1282 }}		\| {{mapping\| 552 875 1282 }}
	\| ~~−0~~.1383		\| −0.1383
	\| 0.1130		\| 0.1130
	\| 5.20		\| 5.20
Line 42:		Line 42:
	\| 250047/250000, 589824/588245, 33554432/33480783		\| 250047/250000, 589824/588245, 33554432/33480783
	\| {{mapping\| 552 875 1282 1550 }}		\| {{mapping\| 552 875 1282 1550 }}
	\| ~~−0~~.1696		\| −0.1696
	\| 0.1118		\| 0.1118
	\| 5.15		\| 5.15
Line 49:		Line 49:
	\| 5632/5625, 9801/9800, 151263/151250, 161280/161051		\| 5632/5625, 9801/9800, 151263/151250, 161280/161051
	\| {{mapping\| 552 875 1282 1550 1910 }}		\| {{mapping\| 552 875 1282 1550 1910 }}
	\| ~~−0~~.1851		\| −0.1851
	\| 0.1048		\| 0.1048
	\| 4.82		\| 4.82
Line 56:		Line 56:
	\| 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449		\| 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449
	\| {{mapping\| 552 875 1282 1550 1910 2043 }}		\| {{mapping\| 552 875 1282 1550 1910 2043 }}
	\| ~~−0~~.1892		\| −0.1892
	\| 0.0961		\| 0.0961
	\| 4.42		\| 4.42
Line 63:		Line 63:
	\| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379		\| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379
	\| {{mapping\| 552 875 1282 1550 1910 2043 2345 }}		\| {{mapping\| 552 875 1282 1550 1910 2043 2345 }}
	\| ~~−0~~.1727		\| −0.1727
	\| 0.0977		\| 0.0977
	\| 4.50		\| 4.50
Line 109:		Line 109:
	\| [[Palladium]] (5-limit)		\| [[Palladium]] (5-limit)
	\|}		\|}
	<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

2024-11-16T12:22:30Z

Partial undo

← Older revision		Revision as of 12:22, 16 November 2024
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		\| 2.3
Line 57:		Line 66:
	\| 0.0977		\| 0.0977
	\| 4.50		\| 4.50
	~~{{comma basis end}~~}		\|}
	* 552et is notable for being the first equal temperament to beat [[270edo\|270]] in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is [[581edo\|581]].		* 552et is notable for being the first equal temperament to beat [[270edo\|270]] in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is [[581edo\|581]].

	=== 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 />(?)		\| 4/3<br />(?)
	\| [[Palladium]] (5-limit)		\| [[Palladium]] (5-limit)
	~~{{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 04:55, 16 November 2024

2024-11-16T04:55:40Z

← Older revision		Revision as of 04:55, 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		\| 2.3
	\| {{monzo\| 875 -552 }}		\| {{monzo\| 875 -552 }}
	\| {{mapping\| 552 875 }}		\| {{mapping\| 552 875 }}
	\| -0.0691		\| −0.0691
	\| 0.0691		\| 0.0691
	\| 3.18		\| 3.18
Line 34:		Line 26:
	\| {{monzo\| 8 14 -13 }}, {{monzo\| 71 -36 -6 }}		\| {{monzo\| 8 14 -13 }}, {{monzo\| 71 -36 -6 }}
	\| {{mapping\| 552 875 1282 }}		\| {{mapping\| 552 875 1282 }}
	\| -0.1383		\| −0.1383
	\| 0.1130		\| 0.1130
	\| 5.20		\| 5.20
Line 41:		Line 33:
	\| 250047/250000, 589824/588245, 33554432/33480783		\| 250047/250000, 589824/588245, 33554432/33480783
	\| {{mapping\| 552 875 1282 1550 }}		\| {{mapping\| 552 875 1282 1550 }}
	\| -0.1696		\| −0.1696
	\| 0.1118		\| 0.1118
	\| 5.15		\| 5.15
Line 48:		Line 40:
	\| 5632/5625, 9801/9800, 151263/151250, 161280/161051		\| 5632/5625, 9801/9800, 151263/151250, 161280/161051
	\| {{mapping\| 552 875 1282 1550 1910 }}		\| {{mapping\| 552 875 1282 1550 1910 }}
	\| -0.1851		\| −0.1851
	\| 0.1048		\| 0.1048
	\| 4.82		\| 4.82
Line 55:		Line 47:
	\| 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449		\| 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449
	\| {{mapping\| 552 875 1282 1550 1910 2043 }}		\| {{mapping\| 552 875 1282 1550 1910 2043 }}
	\| -0.1892		\| −0.1892
	\| 0.0961		\| 0.0961
	\| 4.42		\| 4.42
Line 62:		Line 54:
	\| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379		\| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379
	\| {{mapping\| 552 875 1282 1550 1910 2043 2345 }}		\| {{mapping\| 552 875 1282 1550 1910 2043 2345 }}
	\| -0.1727		\| −0.1727
	\| 0.0977		\| 0.0977
	\| 4.50		\| 4.50
	\|}		{{comma basis end}}
	* 552et is notable for being the first equal temperament to beat [[270edo\|270]] in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is [[581edo\|581]].		* 552et is notable for being the first equal temperament to beat [[270edo\|270]] in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is [[581edo\|581]].

	=== 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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	\|-		\|-
	\| 6		\| 6
	\| 229\552<br>(45\552)		\| 229\552<br />(45\552)
	\| 497.83<br>(97.83)		\| 497.83<br />(97.83)
	\| 4/3<br>(128/121)		\| 4/3<br />(128/121)
	\| [[Sextile]]		\| [[Sextile]]
	\|-		\|-
	\| 24		\| 24
	\| 232\552<br>(2\552)		\| 232\552<br />(2\552)
	\| 504.348<br>(4/348)		\| 504.348<br />(4/348)
	\| 7/5<br>(?)		\| 7/5<br />(?)
	\| [[Chromium]]		\| [[Chromium]]
	\|-		\|-
	\| 46		\| 46
	\| 229\552<br>(1\552)		\| 229\552<br />(1\552)
	\| 497.83<br>(97.83)		\| 497.83<br />(97.83)
	\| 4/3<br>(?)		\| 4/3<br />(?)
	\| [[Palladium]] (5-limit)		\| [[Palladium]] (5-limit)
	\|}		{{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: Notable as a 2.3.5.7.11.13.19 subgroup temp

2024-03-17T09:29:45Z

Notable as a 2.3.5.7.11.13.19 subgroup temp

← Older revision		Revision as of 09:29, 17 March 2024
Line 4:		Line 4:
	== Theory ==		== Theory ==
	552edo is [[consistency\|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. The equal temperament [[tempering out\|tempers out]] {{monzo\| 8 14 -3 }} ([[parakleisma]]) in the 5-limit; 250047/250000 ([[landscape comma]]), 589824/588245 ([[hewuermera comma]]), 26873856/26796875, and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[5632/5625]], [[9801/9800]], 46656/46585, 151263/151250, and 161280/161051 in the 11-limit; and [[1716/1715]], [[2080/2079]], [[10648/10647]], and 20480/20449 in the 13-limit. It [[support]]s [[sextile]] and gives a good tuning for it.		552edo is [[consistency\|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. The equal temperament [[tempering out\|tempers out]] {{monzo\| 8 14 -3 }} ([[parakleisma]]) in the 5-limit; 250047/250000 ([[landscape comma]]), 589824/588245 ([[hewuermera comma]]), 26873856/26796875, and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[5632/5625]], [[9801/9800]], 46656/46585, 151263/151250, and 161280/161051 in the 11-limit; and [[1716/1715]], [[2080/2079]], [[10648/10647]], and 20480/20449 in the 13-limit. It [[support]]s [[sextile]] and gives a good tuning for it.

			It is also consistent in the no-17 [[23-odd-limit]] and the no-17 no-25 [[33-odd-limit]]. In the 2.3.5.7.11.13.19 subgroup, it tempers out [[1216/1215]], [[2376/2375]], [[2926/2925]], [[3136/3135]], 3328/3325, [[3971/3969]] among other commas.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 56:		Line 58:
	\| 0.0961		\| 0.0961
	\| 4.42		\| 4.42
			\|-
			\| 2.3.5.7.11.13.19
			\| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379
			\| {{mapping\| 552 875 1282 1550 1910 2043 2345 }}
			\| -0.1727
			\| 0.0977
			\| 4.50
	\|}		\|}
			* 552et is notable for being the first equal temperament to beat [[270edo\|270]] in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is [[581edo\|581]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
Line 95:		Line 105:
	\| 497.83<br>(97.83)		\| 497.83<br>(97.83)
	\| 4/3<br>(?)		\| 4/3<br>(?)
	\| [[Palladium]]		\| [[Palladium]] (5-limit)
	\|}		\|}
	<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

Eliora: /* Rank-2 temperaments */

2024-03-16T13:01:23Z

Rank-2 temperaments

@@ Line 84: / Line 84: @@
 | 4/3<br>(128/121)
 | [[Sextile]]
 |}
 <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: Created page with "{{Infobox ET}} {{EDO intro}} == Theory == 552edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through..."

2024-03-15T10:42:03Z

Created page with "{{Infobox ET}} {{EDO intro}} == Theory == 552edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through..."

New page

{{Infobox ET}}
{{EDO intro}}

== Theory ==
552edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. The equal temperament [[tempering out|tempers out]] {{monzo| 8 14 -3 }} ([[parakleisma]]) in the 5-limit; 250047/250000 ([[landscape comma]]), 589824/588245 ([[hewuermera comma]]), 26873856/26796875, and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[5632/5625]], [[9801/9800]], 46656/46585, 151263/151250, and 161280/161051 in the 11-limit; and [[1716/1715]], [[2080/2079]], [[10648/10647]], and 20480/20449 in the 13-limit. It [[support]]s [[sextile]] and gives a good tuning for it.

=== Prime harmonics ===
{{Harmonics in equal|552}}

=== Subsets and supersets ===
Since 552 factors into {{factorization|552}}, 552edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, and 276 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 875 -552 }}
| {{mapping| 552 875 }}
| -0.0691
| 0.0691
| 3.18
|-
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| 71 -36 -6 }}
| {{mapping| 552 875 1282 }}
| -0.1383
| 0.1130
| 5.20
|-
| 2.3.5.7
| 250047/250000, 589824/588245, 33554432/33480783
| {{mapping| 552 875 1282 1550 }}
| -0.1696
| 0.1118
| 5.15
|-
| 2.3.5.7.11
| 5632/5625, 9801/9800, 151263/151250, 161280/161051
| {{mapping| 552 875 1282 1550 1910 }}
| -0.1851
| 0.1048
| 4.82
|-
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449
| {{mapping| 552 875 1282 1550 1910 2043 }}
| -0.1892
| 0.0961
| 4.42
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods per 8ve
! Generator*
! Cents*
! Associated Ratio*
! Temperaments
|-
| 1
| 145\552
| 315.22
| 6/5
| [[Parakleismic]] (5-limit)
|-
| 1
| 229\552
| 497.83
| 4/3
| [[Gary]] (2.3.7 subgroup)
|-
| 6
| 229\552 (45\552)
| 497.83 (97.83)
| 4/3 (128/121)
| [[Sextile]]
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct