Miracle extensions

From Xenharmonic Wiki
Revision as of 14:41, 8 January 2025 by FloraC (talk | contribs) (Expand)
Jump to navigation Jump to search

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

Tuning spectra

Miraculous

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
13/10 113.553
8/7 115.587
11/9 115.803
3\31 116.129
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505
17\175 116.571
[0 -27 25 5⟩ 116.573 7 limit least squares
[0 -19 20⟩ 116.578 5 limit least squares
6/5 116.588 5 and 7 limit minimax
11/10 116.591
12/11 116.596
14/11 116.617
116.633 POTE11
7/6 116.641
7\72 116.667
[0 17 -11 -6 11⟩ 116.672 11 limit least squares
116.673 POTE5
116.675 POTE7
10/9 116.716 9, 11 and 13 limit minimax
[0 117 -44 -19⟩ 116.721 9 limit least squares
116.747 POTE13
11/8 116.755
9/7 116.792
11\113 116.814
[0 127 -84 -36 100 -44⟩ 116.820 15 limit least squares
[0 141 -70 -35 84 -42⟩ 116.846 13 limit least squares
4/3 116.993 15 limit minimax
4\41 117.073
13/11 117.266
18/13 117.559
13/12 117.936
15/14 119.443
16/13 119.824
15/13 123.871
14/13 128.298

Benediction

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
8/7 115.587
11/9 115.803
3\31 116.129
5/4 116.241
15/11 116.441
16/13 116.455
7/5 116.502
10\103 116.505
14/13 116.509
13/10 116.511
13/12 116.536
13/11 116.547
[0 -234 39 4 -115 228⟩ 116.56309 13 limit least squares
[0 -251 22 5 -131 261⟩ 116.56348 15 limit least squares
17\175 116.571
[0 -27 25 5⟩ 116.573 7 limit least squares
116.574 POTE13
[0 -19 20⟩ 116.578 5 limit least squares
6/5 116.588 5, 7 and 15 limit minimax
11/10 116.591
18/13 116.595 13 limit minimax
12/11 116.596
15/13 116.598
14/11 116.617
116.633 POTE11
7/6 116.641
7\72 116.667
[0 17 -11 -6 11⟩ 116.672 11 limit least squares
116.673 POTE5
116.675 POTE7
10/9 116.716 9 and 11 limit minimax
[0 117 -44 -19⟩ 116.721 9 limit least squares
11/8 116.755
9/7 116.792
11\113 116.814
4/3 116.993
4\41 117.073
15/14 119.443

Manna

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
8/7 115.587
11/9 115.803
3\31 116.129
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505
17\175 116.571
[0 -27 25 5⟩ 116.573 7 limit least squares
[0 -19 20⟩ 116.578 5 limit least squares
6/5 116.588 5 and 7 limit minimax
11/10 116.591
12/11 116.596
14/11 116.617
116.633 POTE11
7/6 116.641
7\72 116.667
[0 17 -11 -6 11⟩ 116.672 11 limit least squares
116.673 POTE5
116.675 POTE7
10/9 116.716 9 and 11 limit minimax
[0 117 -44 -19⟩ 116.721 9 limit least squares
15/13 116.725 15 limit minimax
116.739 POTE13
11/8 116.755
13/10 116.760 13 limit minimax
[0 -37 -166 -77 59 243⟩ 116.764 15 limit least squares
[0 18 -111 -76 43 204⟩ 116.780 13 limit least squares
9/7 116.792
14/13 116.79254
18/13 116.79299
11\113 116.814
13/12 116.830
16/13 116.856
13/11 116.922
4/3 116.993
4\41 117.073
15/14 119.443

Revelation

Edo
generator 		Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
16/15 111.731
13/10 113.553
13/11 114.555
11/10 115.000
14/11 115.536
11/8 115.543
8/7 115.587
15/11 115.797
12/11 115.938
3\31 116.129
11/9 116.164 11, 13 and 15 limit minimax
[0 -195 35 5 89⟩ 116.198 11 limit least squares
[0 -251 22 5 117 13⟩ 116.229 15 limit least squares
5/4 116.241
[0 -234 39 4 102 11⟩ 116.249 13 limit least squares
116.268 POTE13
116.277 POTE11
7/5 116.502
10\103 116.505
17\175 116.571
[0 -27 25 5⟩ 116.573 7 limit least squares
[0 -19 20⟩ 116.578 5 limit least squares
6/5 116.588 5 and 7 limit minimax
7/6 116.641
7\72 116.667
116.673 POTE5
116.675 POTE7
10/9 116.716 9 limit minimax
[0 117 -44 -19⟩ 116.721 9 limit least squares
9/7 116.792
11\113 116.814
4/3 116.993
4\41 117.073
18/13 117.559
13/12 117.936
15/14 119.443
16/13 119.824
15/13 123.871
14/13 128.298
Retrieved from "https://en.xen.wiki/index.php?title=Miracle_extensions&oldid=175247"