Miracle: Difference between revisions
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{{Interwiki | |||
{{Infobox | | en = Miracle | ||
| de = Miracle | |||
}} | |||
{{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 | ||
| | | 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 | ||
| Pergen = (P8, P5/6) | | Pergen = (P8, P5/6) | ||
| Odd limit 1 = | | Odd limit 1 = 9 | Mistuning 1 = 3.32 | Complexity 1 = 21 | ||
| Odd limit 2 = | | Odd limit 2 = 11-limit 21 | Mistuning 2 = 4.86 | Complexity 2 = 31 | ||
}} | }} | ||
{{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 | '''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 == | ||