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[[de:Miracle]]
{{Interwiki
{{Infobox Regtemp
| 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
| 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
| Pergen = (P8, P5/6)
| Pergen = (P8, P5/6)
| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = 31
| Odd limit 1 = 9 | Mistuning 1 = 3.32 | Complexity 1 = 21
| Odd limit 2 = (11-limit) 21 | Mistuning 2 = ? | Complexity 2 = 72
| 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 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.  
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.


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 ==
Line 196: Line 199:
|-
|-
! 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}}
|}
|}


=== 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
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
! Target
! Generator
! Eigenmonzo*
! Generator
! Generator
! Eigenmonzo*
! Eigenmonzo*
Line 247: Line 263:
| ~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 }}
Line 346: Line 346:
| 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
|-
|-
|  
|  
Line 441: Line 441:
; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning
== External links ==
* [https://x31eq.com/decimal_lattice.htm ''Lattices with Decimal Notation''] by [[Graham Breed]]


[[Category:Miracle| ]] <!-- main article -->
[[Category:Miracle| ]] <!-- main article -->

Latest revision as of 09:27, 9 February 2026

Miracle
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 225/224, 1029/1024 (7-limit);
225/224, 243/242, 385/384 (11-limit)
Reduced mapping ⟨1; 6 -7 -2 15]
ET join 31 & 41
Generators (CTE) ~15/14 = 116.7 ¢
MOS scales …, 1L 9s, 10L 1s, 10L 11s, 10L 21s
Ploidacot hexacot
Pergen (P8, P5/6)
Minimax error 9-odd-limit: 3.32 ¢;
11-limit 21-odd-limit: 4.86 ¢
Target scale size 9-odd-limit: 21 notes;
11-limit 21-odd-limit: 31 notes
English Wikipedia has an article on:

Miracle is a regular temperament discovered by George Secor in 1974 which splits a tempered 3/2 into six generators, called secors (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 edos 31, 41, and 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 cents/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.

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

In the following table, odd harmonics and subharmonics 1–21 are labeled in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 116.6 15/14, 16/15
2 233.3 8/7
3 349.9 11/9
4 466.6 21/16
5 583.2 7/5
6 699.9 3/2
7 816.5 8/5
8 933.2 12/7
9 1049.8 11/6
10 1166.5 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
11 83.1 21/20, 22/21
12 199.8 9/8
13 316.4 6/5
14 433.1 9/7
15 549.7 11/8
16 666.3 22/15
17 783.0 11/7
18 899.6 27/16, 42/25
19 1016.3 9/5
20 1132.9 27/14, 48/25
21 49.6 33/32, 36/35
22 166.2 11/10
23 282.9 33/28
24 399.5 44/35
25 516.2 27/20
26 632.8 36/25
27 749.5 54/35, 77/50
28 866.1 33/20
29 982.8 44/25
30 1099.4 66/35
31 16.1 81/80, 99/98, 121/120

* In 11-limit CWE tuning, octave reduced

Chords

Main article: Chords of miracle

Scales

See also: Miracle 10 MODMOS
Mos scales
Transversal scales
Others

Tunings

A diagram taken from George Secor's article "The Miracle Temperament and Decimal Keyboard" which was published in Xenharmonikôn 18 (2006). Highlighting the error band and adding arrows was done for clarity by Douglas Blumeyer on Dave Keenan's request.

Displayed on the right is a chart of the tuning spectrum of miracle by how the odd harmonics up to 11 are tuned, showing the minimax generator, i.e. the secor.

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~15/14 = 116.5155 ¢ CSEE: ~15/14 = 116.5612 ¢ POEE: ~15/14 = 116.6465 ¢
Tenney CTE: ~15/14 = 116.6772 ¢ CWE: ~15/14 = 116.6756 ¢ POTE: ~15/14 = 116.6752 ¢
Benedetti,
Wilson 		CBE: ~15/14 = 116.7297 ¢ CSBE: ~15/14 = 116.7136 ¢ POBE: ~15/14 = 116.6903 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~15/14 = 116.6868 ¢ CSEE: ~15/14 = 116.6304 ¢ POEE: ~15/14 = 116.5817 ¢
Tenney CTE: ~15/14 = 116.7112 ¢ CWE: ~15/14 = 116.6469 ¢ POTE: ~15/14 = 116.6327 ¢
Benedetti,
Wilson 		CBE: ~15/14 = 116.7355 ¢ CSBE: ~15/14 = 116.6768 ¢ POBE: ~15/14 = 116.6643 ¢

Target tunings

Target tunings
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
5-odd-limit ~16/15 = 116.588 ¢ 5/3 ~16/15 = 116.578 ¢ [0 -19 20⟩
7-odd-limit ~15/14 = 116.588 ¢ 5/3 ~15/14 = 116.573 ¢ [0 -27 25 5⟩
9-odd-limit ~15/14 = 116.716 ¢ 9/5 ~15/14 = 116.721 ¢ [0 117 -44 -19⟩
11-odd-limit ~15/14 = 116.716 ¢ 9/5 ~15/14 = 116.672 ¢ [0 17 -11 -6 11⟩

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
15/8 111.731
2\21 114.286 Lower bound of 7-odd-limit diamond monotone
7/4 115.587
11/9 115.803
3\31 116.129 Lower bound of 9- and 11-odd-limit,
11-limit 15- and 21-odd-limit diamond monotone
5/4 116.241
21/11 116.412
15/11 116.441
7/5 116.502
10\103 116.505
5/3 116.588 5- and 7-odd-limit, 11-limit 15- and 21-odd-limit minimax
11/10 116.591
11/6 116.596
11/7 116.617
7/6 116.641
7\72 116.667
9/5 116.716 9- and 11-odd-limit minimax,
Secor's definition of secor
11/8 116.755
21/20 116.770
9/7 116.792
11\113 116.814
3/2 116.993
4\41 117.073 Upper bound of 11-odd-limit,
11-limit 15- and 21-odd-limit diamond monotone
21/16 117.695
15/14 119.443
1\10 120.000 Upper bound of 7- and 9-odd-limit diamond monotone

* Besides the octave

Music

Herman Miller
Joseph Pehrson
Gene Ward Smith
  • Rachmaninoff Plays Blackjack (archived 2010) – detail | play – in Blackjack (Miracle[21]), 175edo tuning

External links

Retrieved from "https://en.xen.wiki/index.php?title=Miracle&oldid=223647"