Miracle - Revision history

FloraC: Cleanup on infobox

2026-02-09T09:27:22Z

Cleanup on infobox

← Older revision		Revision as of 09:27, 9 February 2026
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	\| de = Miracle		\| de = Miracle
	}}		}}
	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Miracle		\| Title = Miracle
	\| Subgroups = 2.3.5.7, 2.3.5.7.11		\| Subgroups = 2.3.5.7, 2.3.5.7.11
	\| Comma basis = [[225/224]], [[1029/1024]] (7-limit); <br> [[225/224]], [[243/242]], [[385/384]] (11-limit)		\| Comma basis = [[225/224]], [[1029/1024]] (7-limit); <br>[[225/224]], [[243/242]], [[385/384]] (11-limit)
	\| Edo join 1 = 31 \| Edo join 2 = 41		\| Edo join 1 = 31 \| Edo join 2 = 41
	\| ~~Generator~~ = 15/14 \| ~~Generator~~ tuning = 116.7 \| Optimization method = CTE		\| Generators = 15/14 \| Generators tuning = 116.7 \| Optimization method = CTE
	\| MOS scales = …, [[1L 9s]], [[10L 1s]], [[10L 11s]], [[10L 21s]]		\| MOS scales = …, [[1L 9s]], [[10L 1s]], [[10L 11s]], [[10L 21s]]
	\| Mapping = 1; 6 -7 -2 15		\| Mapping = 1; 6 -7 -2 15

Overthink: 9-odd-limit is better than 7

2026-02-08T17:28:09Z

9-odd-limit is better than 7

← Older revision		Revision as of 17:28, 8 February 2026
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	\| Mapping = 1; 6 -7 -2 15		\| Mapping = 1; 6 -7 -2 15
	\| Pergen = (P8, P5/6)		\| Pergen = (P8, P5/6)
	\| Odd limit 1 = 7 \| Mistuning 1 = 2.43 \| Complexity 1 = 21		\| Odd limit 1 = 9 \| Mistuning 1 = 3.32 \| Complexity 1 = 21
	\| Odd limit 2 = (11-limit) 21 \| Mistuning 2 = 4.86 \| Complexity 2 = 31		\| Odd limit 2 = 11-limit 21 \| Mistuning 2 = 4.86 \| Complexity 2 = 31
	}}		}}
	{{Wikipedia\|Miracle temperament}}		{{Wikipedia\|Miracle temperament}}

FloraC: Interwiki

2026-01-19T09:20:45Z

Interwiki

← Older revision		Revision as of 09:20, 19 January 2026
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	[[de:Miracle]]		{{Interwiki
			\| en = Miracle
			\| de = Miracle
			}}
	{{Infobox Regtemp		{{Infobox Regtemp
	\| Title = Miracle		\| Title = Miracle

Squib: mention revelation here instead of in extensions page

2026-01-18T17:07:10Z

mention revelation here instead of in extensions page

← Older revision		Revision as of 17:07, 18 January 2026
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	}}		}}
	{{Wikipedia\|Miracle temperament}}		{{Wikipedia\|Miracle temperament}}
	'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It ~~extends~~ naturally ~~to the~~ [[11-limit]] by treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]]. It is supported by the highly notable [[EDO\|edos]] [[31edo\|31]], [[41edo\|41]], and [[72edo\|72]], with 72edo being an especially good tuning.		'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It is naturally a full [[11-limit]] temperament, treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]]. It is supported by the highly notable [[EDO\|edos]] [[31edo\|31]], [[41edo\|41]], and [[72edo\|72]], with 72edo being an especially good tuning. (There is an alternative mapping for 11 known as [[revelation]], but there is little reason to use it unless you are using [[31edo]], in which case it is identical to miracle anyway.)

	Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.		Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.
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	Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]]~[[49/48]]~[[50/49]]~[[55/54]]~[[56/55]]~[[64/63]] all [[tempered]] together.		Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]]~[[49/48]]~[[50/49]]~[[55/54]]~[[56/55]]~[[64/63]] all [[tempered]] together.

	See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.		See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.

	== Interval chain ==		== Interval chain ==

Overthink: Note about supporting edos

2025-11-16T18:54:21Z

Note about supporting edos

← Older revision		Revision as of 18:54, 16 November 2025
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	}}		}}
	{{Wikipedia\|Miracle temperament}}		{{Wikipedia\|Miracle temperament}}
	'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]].		'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]]. It is supported by the highly notable [[EDO\|edos]] [[31edo\|31]], [[41edo\|41]], and [[72edo\|72]], with 72edo being an especially good tuning.

	Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.		Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.

FloraC: /* Tunings */ + destretched tunings

2025-11-16T12:44:59Z

Tunings: + destretched tunings

← Older revision		Revision as of 12:44, 16 November 2025
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	\|-		\|-
	! rowspan="2" \|		! rowspan="2" \|
	! colspan="2" \| Euclidean		! colspan="3" \| Euclidean
	\|-		\|-
	! Constrained		! Constrained
	! Constrained & skewed		! Constrained & skewed
			! Destretched
	\|-		\|-
	! Equilateral		! Equilateral
	\| CEE: ~15/14 = 116.~~516~~{{c}}		\| CEE: ~15/14 = 116.5155{{c}}
	\| CSEE: ~15/14 = 116.~~561~~{{c}}		\| CSEE: ~15/14 = 116.5612{{c}}
			\| POEE: ~15/14 = 116.6465{{c}}
	\|-		\|-
	! Tenney		! Tenney
	\| CTE: ~15/14 = 116.~~677~~{{c}}		\| CTE: ~15/14 = 116.6772{{c}}
	\| CWE: ~15/14 = 116.~~676~~{{c}}		\| CWE: ~15/14 = 116.6756{{c}}
			\| POTE: ~15/14 = 116.6752{{c}}
	\|-		\|-
	! Benedetti, <br>Wilson		! Benedetti, <br>Wilson
	\| CBE: ~15/14 = 116.~~730~~{{c}}		\| CBE: ~15/14 = 116.7297{{c}}
	\| CSBE: ~15/14 = 116.~~714~~{{c}}		\| CSBE: ~15/14 = 116.7136{{c}}
			\| POBE: ~15/14 = 116.6903{{c}}
	\|}		\|}

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	\|-		\|-
	! rowspan="2" \|		! rowspan="2" \|
	! colspan="2" \| Euclidean		! colspan="3" \| Euclidean
	\|-		\|-
	! Constrained		! Constrained
	! Constrained & skewed		! Constrained & skewed
			! Destretched
	\|-		\|-
	! Equilateral		! Equilateral
	\| CEE: ~15/14 = 116.~~687~~{{c}}		\| CEE: ~15/14 = 116.6868{{c}}
	\| CSEE: ~15/14 = 116.~~630~~{{c}}		\| CSEE: ~15/14 = 116.6304{{c}}
			\| POEE: ~15/14 = 116.5817{{c}}
	\|-		\|-
	! Tenney		! Tenney
	\| CTE: ~15/14 = 116.~~711~~{{c}}		\| CTE: ~15/14 = 116.7112{{c}}
	\| CWE: ~15/14 = 116.~~647~~{{c}}		\| CWE: ~15/14 = 116.6469{{c}}
			\| POTE: ~15/14 = 116.6327{{c}}
	\|-		\|-
	! Benedetti, <br>Wilson		! Benedetti, <br>Wilson
	\| CBE: ~15/14 = 116.~~736~~{{c}}		\| CBE: ~15/14 = 116.7355{{c}}
	\| CSBE: ~15/14 = 116.~~677~~{{c}}		\| CSBE: ~15/14 = 116.6768{{c}}
			\| POBE: ~15/14 = 116.6643{{c}}
	\|}		\|}

Overthink at 04:40, 19 October 2025

2025-10-19T04:40:55Z

← Older revision		Revision as of 04:40, 19 October 2025
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	\| Mapping = 1; 6 -7 -2 15		\| Mapping = 1; 6 -7 -2 15
	\| Pergen = (P8, P5/6)		\| Pergen = (P8, P5/6)
	\| Odd limit 1 = 7 \| Mistuning 1 = 2.43 \| Complexity 1 = 31		\| Odd limit 1 = 7 \| Mistuning 1 = 2.43 \| Complexity 1 = 21
	\| Odd limit 2 = (11-limit) 21 \| Mistuning 2 = 4.86 \| Complexity 2 = 72		\| Odd limit 2 = (11-limit) 21 \| Mistuning 2 = 4.86 \| Complexity 2 = 31
	}}		}}
	{{Wikipedia\|Miracle temperament}}		{{Wikipedia\|Miracle temperament}}

Overthink: added links to rastmic and gamelismic clan

2025-09-27T16:23:30Z

added links to rastmic and gamelismic clan

← Older revision		Revision as of 16:23, 27 September 2025
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	Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.		Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.

	Some temperaments have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and other temperaments have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts.		[[Rastmic clan\|Some temperaments]] have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and [[Gamelismic clan\|other temperaments]] have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts.

	Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]]~[[49/48]]~[[50/49]]~[[55/54]]~[[56/55]]~[[64/63]] all [[tempered]] together.		Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]]~[[49/48]]~[[50/49]]~[[55/54]]~[[56/55]]~[[64/63]] all [[tempered]] together.

FloraC: /* Target tunings */ consolidate tables for readability

2025-09-25T10:54:55Z

Target tunings: consolidate tables for readability

← Older revision		Revision as of 10:54, 25 September 2025
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	=== Target tunings ===		=== Target tunings ===
	{\| class="wikitable center-all mw-collapsible mw-collapsed"		{\| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
	\|+ style="white-space: nowrap;" \| ~~Minimax~~ tunings		\|+ style="white-space: nowrap;" \| Target tunings
	\|-		\|-
	! Target		! rowspan="2" \| Target
			! colspan="2" \| Minimax
			! colspan="2" \| Least squares
			\|-
			! Generator
			! Eigenmonzo*
	! Generator		! Generator
	! Eigenmonzo*		! Eigenmonzo*
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	\| ~16/15 = 116.588{{c}}		\| ~16/15 = 116.588{{c}}
	\| 5/3		\| 5/3
			\| ~16/15 = 116.578{{c}}
			\| {{Monzo\| 0 -19 20 }}
	\|-		\|-
	\| 7-odd-limit		\| 7-odd-limit
	\| ~15/14 = 116.588{{c}}		\| ~15/14 = 116.588{{c}}
	\| 5/3		\| 5/3
			\| ~15/14 = 116.573{{c}}
			\| {{Monzo\| 0 -27 25 5 }}
	\|-		\|-
	\| 9-odd-limit		\| 9-odd-limit
	\| ~15/14 = 116.716{{c}}		\| ~15/14 = 116.716{{c}}
	\| 9/5		\| 9/5
			\| ~15/14 = 116.721{{c}}
			\| {{Monzo\| 0 117 -44 -19 }}
	\|-		\|-
	\| 11-odd-limit		\| 11-odd-limit
	\| ~15/14 = 116.716{{c}}		\| ~15/14 = 116.716{{c}}
	\| 9/5		\| 9/5
	\|}

	~~{\| class="wikitable center-all left-3 mw-collapsible mw-collapsed"~~
	~~\|+ style="white-space: nowrap;" \| Least squares tunings~~
	\|-
	~~! Target~~
	~~! Generator~~
	! Eigenmonzo*
	\|-
	~~\| 5-odd-limit~~
	~~\| ~16/15 = 116.578{{c}}~~
	~~\| {{Monzo\| 0 -19 20 }}~~
	\|-
	~~\| 7-odd-limit~~
	~~\| ~15/14 = 116.573{{c}}~~
	~~\| {{Monzo\| 0 -27 25 5 }}~~
	\|-
	~~\| 9-odd-limit~~
	~~\| ~15/14 = 116.721{{c}}~~
	~~\| {{Monzo\| 0 117 -44 -19 }}~~
	\|-
	~~\| 11-odd-limit~~
	\| ~15/14 = 116.672{{c}}		\| ~15/14 = 116.672{{c}}
	\| {{Monzo\| 0 17 -11 -6 11 }}		\| {{Monzo\| 0 17 -11 -6 11 }}

Overthink: Added minimax errors, 11-limit 15- and 21-odd-limit minimax

2025-09-20T19:46:36Z

Added minimax errors, 11-limit 15- and 21-odd-limit minimax

← Older revision		Revision as of 19:46, 20 September 2025
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	\| Mapping = 1; 6 -7 -2 15		\| Mapping = 1; 6 -7 -2 15
	\| Pergen = (P8, P5/6)		\| Pergen = (P8, P5/6)
	\| Odd limit 1 = 7 \| Mistuning 1 = ? \| Complexity 1 = 31		\| Odd limit 1 = 7 \| Mistuning 1 = 2.43 \| Complexity 1 = 31
	\| Odd limit 2 = (11-limit) 21 \| Mistuning 2 = ? \| Complexity 2 = 72		\| Odd limit 2 = (11-limit) 21 \| Mistuning 2 = 4.86 \| Complexity 2 = 72
	}}		}}
	{{Wikipedia\|Miracle temperament}}		{{Wikipedia\|Miracle temperament}}
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	\| 5/3		\| 5/3
	\| 116.588		\| 116.588
	\| 5- and 7-odd-limit minimax		\| 5- and 7-odd-limit, 11-limit 15- and 21-odd-limit minimax
	\|-		\|-
	\|		\|