Miracle: Difference between revisions

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[[de:Miracle]]

{{Interwiki

{{Infobox ~~Regtemp~~

| en = Miracle

| de = Miracle

}}

{{Infobox regtemp

| Title = Miracle

| 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

| ~~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]]

| Mapping = 1; 6 -7 -2 15

| Pergen = (P8, P5/6)

| Odd limit 1 = 7 | Mistuning 1 = 2.43 | Complexity 1 = 31

| Odd limit 1 = 9 | Mistuning 1 = 3.32 | 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

}}

'''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 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.

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.

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

== Interval chain ==

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Line 199:

|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! 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

| 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

| 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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Line 225:

|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! 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

| 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

| 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}}

|}

@@ Line 1: / Line 1: @@
-[[de:Miracle]]
+{{Interwiki
-{{Infobox Regtemp
+| en = Miracle
+| de = Miracle
+}}
+{{Infobox regtemp
 | Title = Miracle
 | 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
-| 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]]
 | Mapping = 1; 6 -7 -2 15
 | Pergen = (P8, P5/6)
-| Odd limit 1 = 7 | Mistuning 1 = 2.43 | Complexity 1 = 31
+| Odd limit 1 = 9 | Mistuning 1 = 3.32 | 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}}
-'''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 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.
-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.
 See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.
 == Interval chain ==
@@ Line 196: / Line 199: @@
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
 ! Constrained
 ! Constrained & skewed
+! Destretched
 |-
 ! 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
-| 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
-| 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}}
 |}
@@ Line 218: / Line 225: @@
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
 ! Constrained
 ! Constrained & skewed
+! Destretched
 |-
 ! 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
-| 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
-| 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}}
 |}