Miracle extensions: Difference between revisions

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→Tunings: +warts and ranges. Adjust edo tunings (+185edo tuning for miraculous, manna, and benediction. +21- and 52edo, -103-, 113- and 175edo for revelation)
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| 41f val, upper bound of 11-odd-limit diamond monotone
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| 52f val
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| 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone (singleton)
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Revision as of 15:28, 8 January 2025

The basic 7-limit miracle temperament has various extensions to the 11- and 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 & 41f) – tempering out 225/224, 243/242, 325/324, and 385/384;

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

  • 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.

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 13-limit extensions
Miraculous Benediction Manna
0 0.0 1/1
1 116.6 15/14, 16/15 14/13
2 233.3 8/7 15/13
3 349.9 11/9 16/13
4 466.6 21/16 13/10
5 583.2 7/5
6 699.9 3/2
7 816.5 8/5 21/13
8 933.2 12/7
9 1049.8 11/6 24/13
10 1166.5 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
11 83.1 21/20, 22/21 26/25
12 199.8 9/8
13 316.4 6/5
14 433.1 9/7
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 40/39
22 166.2 11/10
23 282.9 33/28 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
28 866.1 33/20
29 982.8 44/25
30 1099.4 66/35
31 16.1 81/80, 99/98, 121/120 105/104, 144/143, 196/195 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 26/21
35 482.6 33/25
36 599.3 99/70
37 715.9 121/80
38 832.6 121/75 21/13 13/8
39 949.2 121/70 45/26 26/15
40 1065.9 231/125 24/13 13/7
41 1182.5 99/50 180/91 143/72, 195/98, 208/105

* In 11-limit CWE tuning, octave reduced

Tunings

  • 5-limit POTE: ~15/14 = 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

Tuning spectra

Miraculous

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
15/8 111.731
13/10 113.553
7/4 115.587
11/9 115.803
3\31 116.129 Lower bound of 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505 103f val
17\175 116.571 175ff val
[0 -27 25 5⟩ 116.573 7-odd-limit least squares
[0 -19 20⟩ 116.578 5-odd-limit least squares
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 72f val
[0 17 -11 -6 11⟩ 116.672 11-odd-limit least squares
9/5 116.716 9-, 11- and 13-odd-limit minimax
[0 117 -44 -19⟩ 116.721 9-odd-limit least squares
11/8 116.755
18\185 116.757 185cff val
9/7 116.792
11\113 116.814 113f val
[0 127 -84 -36 100 -44⟩ 116.820 15-odd-limit least squares
[0 141 -70 -35 84 -42⟩ 116.846 13-odd-limit least squares
3/2 116.993 15-odd-limit minimax
4\41 117.073 Upper bound of 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
13/11 117.266
13/9 117.559
13/12 117.936
15/14 119.443
13/8 119.824
15/13 123.871
13/7 128.298

Benediction

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
3\31 116.129 Lower bound of 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
5/4 116.241
15/11 116.441
13/8 116.455
7/5 116.502
10\103 116.505
13/7 116.509
13/10 116.511
13/12 116.536
13/11 116.547
[0 -234 39 4 -115 228⟩ 116.56309 13-odd-limit least squares
[0 -251 22 5 -131 261⟩ 116.56348 15-odd-limit least squares
17\175 116.571 175f val
[0 -27 25 5⟩ 116.573 7-odd-limit least squares
[0 -19 20⟩ 116.578 5-odd-limit least squares
6/5 116.588 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
7\72 116.667 Upper bound of 13- and 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
[0 17 -11 -6 11⟩ 116.672 11-odd-limit least squares
9/5 116.716 9- and 11-odd-limit minimax
[0 117 -44 -19⟩ 116.721 9-odd-limit least squares
11/8 116.755
18\185 116.757 185cfff val
9/7 116.792
11\113 116.814 113ff val
3/2 116.993
4\41 117.073 41f val, upper bound of 11-odd-limit diamond monotone
15/14 119.443

Manna

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
3\31 116.129 31f 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 103ff val
17\175 116.571 175fff val
[0 -27 25 5⟩ 116.573 7-odd-limit least squares
[0 -19 20⟩ 116.578 5-odd-limit least squares
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- and 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
[0 17 -11 -6 11⟩ 116.672 11-odd-limit least squares
9/5 116.716 9- and 11-odd-limit minimax
[0 117 -44 -19⟩ 116.721 9-odd-limit least squares
15/13 116.725 15-odd-limit minimax
11/8 116.755
18\185 116.757 185cf val
13/10 116.760 13-odd-limit minimax
[0 -37 -166 -77 59 243⟩ 116.764 15-odd-limit least squares
[0 18 -111 -76 43 204⟩ 116.780 13-odd-limit least squares
9/7 116.792
13/7 116.79254
13/9 116.79299
11\113 116.814
13/12 116.830
13/8 116.856
13/11 116.922
3/2 116.993
4\41 117.073
15/14 119.443

Revelation

Edo
generator 		Eigenmonzo
(unchanged-interval) 		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
[0 -195 35 5 89⟩ 116.198 11-odd-limit least squares
[0 -251 22 5 117 13⟩ 116.229 15-odd-limit least squares
5/4 116.241
[0 -234 39 4 102 11⟩ 116.249 13-odd-limit least squares
7/5 116.502
[0 -27 25 5⟩ 116.573 7-odd-limit least squares
[0 -19 20⟩ 116.578 5-odd-limit least squares
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
[0 117 -44 -19⟩ 116.721 9-odd-limit least squares
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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