Miracle extensions: Difference between revisions

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<nowiki/>* In 11-limit [[CWE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


== Tuning spectra ==
== Tunings ==
=== Miraculous ===
* 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 ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
Line 335: Line 347:
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
Line 345: Line 357:
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 387: Line 399:
| {{monzo| 0 -27 25 5 }}
| {{monzo| 0 -27 25 5 }}
| 116.573
| 116.573
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| {{monzo| 0 -19 20 }}
| 116.578
| 116.578
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5 and 7 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
|  
|  
Line 405: Line 417:
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
|-
|
|
| 116.633
| POTE11
|-
|-
|  
|  
Line 432: Line 439:
| {{monzo| 0 17 -11 -6 11 }}
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 116.672
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
|  
| 9/5
| 116.673
| POTE5
|-
|
|
| 116.675
| POTE7
|-
|
| 10/9
| 116.716
| 116.716
| 9, 11 and 13 limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 117 -44 -19 }}
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 116.721
| 9 limit least squares
| 9-odd-limit least squares
|-
|
|
| 116.747
| POTE13
|-
|-
|  
|  
Line 477: Line 469:
| {{monzo| 0 127 -84 -36 100 -44 }}
| {{monzo| 0 127 -84 -36 100 -44 }}
| 116.820
| 116.820
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 141 -70 -35 84 -42 }}
| {{monzo| 0 141 -70 -35 84 -42 }}
| 116.846
| 116.846
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
| 15 limit minimax
| 15-odd-limit minimax
|-
|-
| 4\41
| 4\41
Line 500: Line 492:
|-
|-
|  
|  
| 18/13
| 13/9
| 117.559
| 117.559
|  
|  
Line 515: Line 507:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 525: Line 517:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
|}
|}


=== Benediction ===
==== Benediction ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
Line 539: Line 531:
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 569: Line 561:
|-
|-
|  
|  
| 16/13
| 13/8
| 116.455
| 116.455
|  
|  
Line 584: Line 576:
|-
|-
|  
|  
| 14/13
| 13/7
| 116.509
| 116.509
|  
|  
Line 606: Line 598:
| {{monzo| 0 -234 39 4 -115 228 }}
| {{monzo| 0 -234 39 4 -115 228 }}
| 116.56309
| 116.56309
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 -251 22 5 -131 261 }}
| {{monzo| 0 -251 22 5 -131 261 }}
| 116.56348
| 116.56348
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
| 17\175
| 17\175
Line 621: Line 613:
| {{monzo| 0 -27 25 5 }}
| {{monzo| 0 -27 25 5 }}
| 116.573
| 116.573
| 7 limit least squares
| 7-odd-limit least squares
|-
|
|
| 116.574
| POTE13
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| {{monzo| 0 -19 20 }}
| 116.578
| 116.578
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 6/5
| 116.588
| 116.588
| 5, 7 and 15 limit minimax
| 5-, 7- and 15-odd-limit minimax
|-
|-
|  
|  
Line 644: Line 631:
|-
|-
|  
|  
| 18/13
| 13/9
| 116.595
| 116.595
| 13 limit minimax
| 13-odd-limit minimax
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
Line 659: Line 646:
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
|-
|
|
| 116.633
| POTE11
|-
|-
|  
|  
Line 681: Line 663:
| {{monzo| 0 17 -11 -6 11 }}
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 116.672
| 11 limit least squares
| 11-odd-limit least squares
|-
|
|
| 116.673
| POTE5
|-
|
|
| 116.675
| POTE7
|-
|-
|  
|  
| 10/9
| 9/5
| 116.716
| 116.716
| 9 and 11 limit minimax
| 9- and 11-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 117 -44 -19 }}
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 116.721
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
Line 719: Line 691:
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
|  
|  
Line 734: Line 706:
|}
|}


=== Manna ===
==== Manna ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
Line 743: Line 715:
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 790: Line 762:
| {{monzo| 0 -27 25 5 }}
| {{monzo| 0 -27 25 5 }}
| 116.573
| 116.573
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| {{monzo| 0 -19 20 }}
| 116.578
| 116.578
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5 and 7 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
|  
|  
Line 808: Line 780:
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
|-
|
|
| 116.633
| POTE11
|-
|-
|  
|  
Line 835: Line 802:
| {{monzo| 0 17 -11 -6 11 }}
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 116.672
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
|  
| 9/5
| 116.673
| POTE5
|-
|
|
| 116.675
| POTE7
|-
|
| 10/9
| 116.716
| 116.716
| 9 and 11 limit minimax
| 9- and 11-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 117 -44 -19 }}
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 116.721
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
| 15/13
| 15/13
| 116.725
| 116.725
| 15 limit minimax
| 15-odd-limit minimax
|-
|
|
| 116.739
| POTE13
|-
|-
|  
|  
Line 875: Line 827:
| 13/10
| 13/10
| 116.760
| 116.760
| 13 limit minimax
| 13-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 -37 -166 -77 59 243 }}
| {{monzo| 0 -37 -166 -77 59 243 }}
| 116.764
| 116.764
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 18 -111 -76 43 204 }}
| {{monzo| 0 18 -111 -76 43 204 }}
| 116.780
| 116.780
| 13 limit least squares
| 13-odd-limit least squares
|-
|-
|  
|  
Line 893: Line 845:
|-
|-
|  
|  
| 14/13
| 13/7
| 116.79254
| 116.79254
|  
|  
|-
|-
|  
|  
| 18/13
| 13/9
| 116.79299
| 116.79299
|  
|  
Line 913: Line 865:
|-
|-
|  
|  
| 16/13
| 13/8
| 116.856
| 116.856
|  
|  
Line 923: Line 875:
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
|  
|  
Line 938: Line 890:
|}
|}


=== Revelation ===
==== Revelation ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
Line 947: Line 899:
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
Line 967: Line 919:
|-
|-
|  
|  
| 14/11
| 11/7
| 115.536
| 115.536
|  
|  
Line 977: Line 929:
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 987: Line 939:
|-
|-
|  
|  
| 12/11
| 11/6
| 115.938
| 115.938
|  
|  
Line 999: Line 951:
| 11/9
| 11/9
| 116.164
| 116.164
| 11, 13 and 15 limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 -195 35 5 89 }}
| {{monzo| 0 -195 35 5 89 }}
| 116.198
| 116.198
| 11 limit least squares
| 11-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 -251 22 5 117 13 }}
| {{monzo| 0 -251 22 5 117 13 }}
| 116.229
| 116.229
| 15 limit least squares
| 15-odd-limit least squares
|-
|-
|  
|  
Line 1,019: Line 971:
| {{monzo| 0 -234 39 4 102 11 }}
| {{monzo| 0 -234 39 4 102 11 }}
| 116.249
| 116.249
| 13 limit least squares
| 13-odd-limit least squares
|-
|
|
| 116.268
| POTE13
|-
|
|
| 116.277
| POTE11
|-
|-
|  
|  
Line 1,049: Line 991:
| {{monzo| 0 -27 25 5 }}
| {{monzo| 0 -27 25 5 }}
| 116.573
| 116.573
| 7 limit least squares
| 7-odd-limit least squares
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| {{monzo| 0 -19 20 }}
| 116.578
| 116.578
| 5 limit least squares
| 5-odd-limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5 and 7 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
|  
|  
Line 1,072: Line 1,014:
|-
|-
|  
|  
|  
| 9/5
| 116.673
| POTE5
|-
|
|
| 116.675
| POTE7
|-
|
| 10/9
| 116.716
| 116.716
| 9 limit minimax
| 9-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 117 -44 -19 }}
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 116.721
| 9 limit least squares
| 9-odd-limit least squares
|-
|-
|  
|  
Line 1,102: Line 1,034:
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
|  
|  
Line 1,112: Line 1,044:
|-
|-
|  
|  
| 18/13
| 13/9
| 117.559
| 117.559
|  
|  
Line 1,127: Line 1,059:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 1,137: Line 1,069:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  

Revision as of 14:54, 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
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-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
[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
9/7 116.792
11\113 116.814
[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
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
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
[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
[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
9/7 116.792
11\113 116.814
3/2 116.993
4\41 117.073
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
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-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
[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
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
13/11 114.555
11/10 115.000
11/7 115.536
11/8 115.543
7/4 115.587
15/11 115.797
11/6 115.938
3\31 116.129
11/9 116.164 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
10\103 116.505
17\175 116.571
[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
9/5 116.716 9-odd-limit minimax
[0 117 -44 -19⟩ 116.721 9-odd-limit least squares
9/7 116.792
11\113 116.814
3/2 116.993
4\41 117.073
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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