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Cleanup. Note the caveat in the "19-limit extension"
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{{Breadcrumb|Miracle}}
{{Breadcrumb|Miracle}}


The 11-limit [[miracle]] temperament has various [[extension]]s to the 13-limit. The following temperaments are discussed in this article:  
The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:  
* '''Miraculous''' ({{nowrap|31 & 41}}) – tempering out 105/104, 144/143, 196/195, and 243/242;  
* '''Miraculous''' ({{nowrap| 31 & 41 }}) – tempering out 105/104, 144/143, 196/195, and 243/242;  
* '''Benediction''' ({{nowrap|31 & 41f}}) – tempering out 225/224, 243/242, 351/350, and 385/384;  
* '''Benediction''' ({{nowrap| 31 & 41f }}) – tempering out 225/224, 243/242, 351/350, and 385/384;  
* '''Manna''' ({{nowrap|31f & 41}}) – tempering out 225/224, 243/242, 325/324, and 385/384;  
* '''Manna''' ({{nowrap| 31f & 41 }}) – tempering out 225/224, 243/242, 325/324, and 385/384;  


In addition, we also consider the only alternative 11-limit mapping:  
In addition, we also consider the only alternative 11-limit mapping:  
* '''Revelation''' ({{nowrap|21 & 31}}) – tempering out 66/65, 99/98, 105/104, and 512/507.  
* '''Revelation''' ({{nowrap| 21 & 31 }}) – tempering out 66/65, 99/98, 105/104, and 512/507.  


As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].
As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing [[15/14]], and [[16/15]], goes one step further to stand in for [[~]][[14/13]], and you can find [[11/9]]~[[16/13]] just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].


Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], implies 72, and [[82edo|82]].
Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.  


Miraculous, benediction, manna, and semimiracle all support {{nowrap|31 & 41 & 72}}, the [[rank-3]] [[expansion]] of miracle. Since they can all be extended to higher limits in the same way, it is natural to first extend {{nowrap|31 & 41 & 72}}, then temper it back into a [[rank-2 temperament]] using your extension of choice.
Miraculous, benediction, and manna can all be extended to the [[17-limit]] by recognizing [[21/16]]~[[17/13]], tempering out [[273/272]]. For miraculous it implies the generator also represents [[17/16]], which is supported by 72fg. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].  


{{nowrap|31 & 41 & 72}} can be extended to the 17-limit by recognizing 21/16~17/13, tempering out [[273/272]]. For miraculous it implies the generator also represents 17/16, which is supported by 72fg. For semimiracle it implies the half-octave also represents 17/12~24/17.
[[225/224]] factors into ([[400/399]])·([[513/512]]) in the [[19-limit]], suggesting that miracle can be extended to include prime 19 by tempering out both commas. However, this means 31edo is no longer in the valid [[diamond monotone]] range. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].
 
225/224 factors into ([[400/399]])·([[513/512]]) in the 19-limit, suggesting that miracle (including 11-limit miracle) can be extended to prime 19 by tempering both commas.
 
There is also a natural extension to the 23-limit that tempers out [[300/299]] and [[392/391]].


For technical information see [[Gamelismic clan #Miracle]].
For technical information see [[Gamelismic clan #Miracle]].

Revision as of 08:45, 22 December 2025

< Miracle

The 11-limit miracle temperament has various extensions to the 13-limit. The following temperaments are discussed in this article:

  • Miraculous (31 & 41) – tempering out 105/104, 144/143, 196/195, and 243/242;
  • Benediction (31 & 41f) – tempering out 225/224, 243/242, 351/350, and 385/384;
  • Manna (31f & 41) – tempering out 225/224, 243/242, 325/324, and 385/384;

In addition, we also consider the only alternative 11-limit mapping:

  • Revelation (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.

As we will see in #Interval chain, miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is supported by 72f. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned patent val mapping of prime 13 in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as 103edo whereas manna benefits from a sharper tuning such as 113edo.

Another possible path which relates a sense of compromise is to temper out 169/168, leading to semimiracle. This has the effect of slicing the period in two, and is supported by 62, 72, and 82. Finally, there is mirage, the rank-3 expansion of miracle with the addition of an independent generator for prime 13.

Miraculous, benediction, and manna can all be extended to the 17-limit by recognizing 21/16~17/13, tempering out 273/272. For miraculous it implies the generator also represents 17/16, which is supported by 72fg. For semimiracle it implies the half-octave period represents 17/12~24/17.

225/224 factors into (400/399)·(513/512) in the 19-limit, suggesting that miracle can be extended to include prime 19 by tempering out both commas. However, this means 31edo is no longer in the valid diamond monotone range. There is also a natural extension to the 23-limit that tempers out 300/299 and 392/391.

For technical information see Gamelismic clan #Miracle.

Interval chain

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

# Cents* Approximate ratios
11-limit 17-limit extensions
Miraculous Benediction Manna
0 0.0 1/1
1 116.6 15/14, 16/15 14/13, 17/16
2 233.3 8/7 15/13, 17/15
3 349.9 11/9 16/13, 17/14, 21/17
4 466.6 21/16 13/10, 17/13 17/13 17/13
5 583.2 7/5 24/17
6 699.9 3/2
7 816.5 8/5 21/13
8 933.2 12/7 17/10
9 1049.8 11/6 24/13
10 1166.5 49/25, 55/28, 63/32,
88/45, 96/49, 108/55 		39/20, 51/26, 77/39,
128/65, 135/68, 168/85 		51/26, 100/51 51/26, 65/33
11 83.1 21/20, 22/21 18/17, 26/25
12 199.8 9/8
13 316.4 6/5
14 433.1 9/7 22/17
15 549.7 11/8 18/13
16 666.3 22/15
17 783.0 11/7
18 899.6 27/16, 42/25 22/13
19 1016.3 9/5
20 1132.9 27/14, 48/25 52/27
21 49.6 33/32, 36/35 27/26 35/34, 40/39 34/33
22 166.2 11/10
23 282.9 33/28 20/17 13/11
24 399.5 44/35
25 516.2 27/20
26 632.8 36/25 13/9
27 749.5 54/35, 77/50 20/13 17/11
28 866.1 33/20 28/17
29 982.8 44/25 30/17
30 1099.4 66/35 32/17 17/9
31 16.1 81/80, 99/98, 121/120 66/65 105/104, 120/119, 136/135,
144/143, 154/153, 170/169 		65/64, 78/77,
85/84, 91/90
32 132.7 27/25 14/13 13/12
33 249.3 81/70 15/13 52/45
34 366.0 99/80 16/13, 21/17 26/21
35 482.6 33/25
36 599.3 99/70 24/17 17/12
37 715.9 121/80
38 832.6 121/75 21/13 13/8, 34/21
39 949.2 121/70 45/26 26/15
40 1065.9 231/125 24/13 13/7
41 1182.5 99/50 77/39, 128/65,
168/85, 180/91 		119/60, 143/72, 135/68,
153/77, 169/85, 195/98

* In 11-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

  • 5-limit POTE: ~16/15 = 116.673 ¢
  • 7-limit POTE: ~15/14 = 116.675 ¢
  • 11-limit POTE
    • Miracle: ~15/14 = 116.633 ¢
    • Revelation: ~15/14 = 116.277 ¢
  • 13-limit POTE
    • Miraculous: ~15/14 = 116.747 ¢
    • Benediction: ~15/14 = 116.574 ¢
    • Manna: ~15/14 = 116.739 ¢
    • Revelation: ~15/14 = 116.268 ¢

Target tunings

Target tunings (miracle)
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⟩
Target tunings (miraculous)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.716 ¢ 9/5 ~15/14 = 116.846 ¢ [0 141 -70 -35 84 -42⟩
15-odd-limit ~15/14 = 116.993 ¢ 3/2 ~15/14 = 116.820 ¢ [0 127 -84 -36 100 -44⟩
Target tunings (benediction)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.595 ¢ 13/9 ~15/14 = 116.56309 ¢ [0 -234 39 4 -115 228⟩
15-odd-limit ~15/14 = 116.588 ¢ 5/3 ~15/14 = 116.56348 ¢ [0 -251 22 5 -131 261⟩
Target tunings (manna)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.760 ¢ 13/10 ~15/14 = 116.780 ¢ [0 18 -111 -76 43 204⟩
15-odd-limit ~15/14 = 116.725 ¢ 15/13 ~15/14 = 116.764 ¢ [0 -37 -166 -77 59 243⟩
Target tunings (revelation)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
11-odd-limit ~15/14 = 116.164 ¢ 11/9 ~15/14 = 116.198 ¢ [0 -195 35 5 89⟩
13-odd-limit ~15/14 = 116.164 ¢ 11/9 ~15/14 = 116.249 ¢ [0 -234 39 4 102 11⟩
15-odd-limit ~15/14 = 116.164 ¢ 11/9 ~15/14 = 116.229 ¢ [0 -251 22 5 117 13⟩

Tuning spectra

Miraculous

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
17/16 104.955
17/15 108.343
15/8 111.731
17/14 112.043
13/10 113.553
17/10 114.830
7/4 115.587
11/9 115.803
17/13 116.107
3\31 116.129 Lower bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505 103fg val
17\175 116.571 175ffggg val
5/3 116.588 5- and 7-odd-limit minimax
11/10 116.591
11/6 116.596
11/7 116.617
7/6 116.641
7\72 116.667 72fg val
9/5 116.716 9-, 11- and 13-odd-limit minimax
11/8 116.755
18\185 116.757 185cffggg val
9/7 116.792
11\113 116.814 113fgg val
3/2 116.993 15-odd-limit minimax
4\41 117.073 Upper bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
13/11 117.266
13/9 117.559
17/11 117.597
13/12 117.936
17/9 118.087
17/12 119.400
15/14 119.443
13/8 119.824
21/17 121.942
15/13 123.871
13/7 128.298

Benediction

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
17/13 116.107
3\31 116.129 Lower bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
5/4 116.241
15/11 116.441
13/8 116.455
17/16 116.501
7/5 116.502
10\103 116.505
13/7 116.509
13/10 116.511
13/12 116.536
13/11 116.547
17/14 116.567
17\175 116.571 175f val
17/10 116.581
17/12 116.583
17/11 116.586
5/3 116.588 5-, 7- and 15-odd-limit minimax
11/10 116.591
13/9 116.595 13-odd-limit minimax
11/6 116.596
15/13 116.598
11/7 116.617
7/6 116.641
17/9 116.642
21/17 116.642
17/15 116.666
7\72 116.667 Upper bound of 13- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
9/5 116.716 9- and 11-odd-limit minimax
11/8 116.755
18\185 116.757 185cfffgg val
9/7 116.792
11\113 116.814 113ffg val
3/2 116.993
4\41 117.073 41fg val, upper bound of 11-odd-limit diamond monotone
15/14 119.443

Manna

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
17/13 116.107
3\31 116.129 31fg val, lower bound of 11-odd-limit diamond monotone
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505 103ffgg val
17\175 116.571 175fffgg val
5/3 116.588 5- and 7-odd-limit minimax
11/10 116.591
11/6 116.596
11/7 116.617
7/6 116.641
7\72 116.667 Lower bound of 13- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
17/15 116.667
21/17 116.689
17/9 116.702
17/10 116.707
9/5 116.716 9- and 11-odd-limit minimax
15/13 116.725 15-odd-limit minimax
17/14 116.730
17/12 116.750
11/8 116.755
18\185 116.757 185cf val
13/10 116.760 13-odd-limit minimax
17/16 116.785
9/7 116.792
13/7 116.79254
13/9 116.79299
17/11 116.801
11\113 116.814
13/12 116.830
13/8 116.856
13/11 116.922
3/2 116.993
4\41 117.073 Upper bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
15/14 119.443

Revelation

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
13/10 113.553
2\21 114.286
13/11 114.555
11/10 115.000
5\52 115.385 52f val
11/7 115.536
11/8 115.543
7/4 115.587
15/11 115.797
11/6 115.938
3\31 116.129 11- to 15-odd-limit,
and 13-limit 21-odd-limit diamond monotone (singleton)
11/9 116.164 11-, 13- and 15-odd-limit minimax
5/4 116.241
7/5 116.502
5/3 116.588 5- and 7-odd-limit minimax
7/6 116.641
7\72 116.667 72ee val
9/5 116.716 9-odd-limit minimax
9/7 116.792
3/2 116.993
4\41 117.073 41ef val
13/9 117.559
13/12 117.936
15/14 119.443
13/8 119.824
15/13 123.871
13/7 128.298
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