Miracle extensions

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The basic 7-limit miracle temperament has various extensions to the 11 and 13 limits.

Tuning Spectra

Spectrum of Miraculous Tunings by Eigenmonzos

Gencom: [2 15/14; 105/104 144/143 196/195 275/273]

Gencom map: [<1 1 3 3 2 4|, <0 6 -7 -2 15 -3|]

Eigenmonzo Secor
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
POTE11 116.633
7/6 116.641
7\72 116.667
|0 17 -11 -6 11> 116.672 (11 limit least squares)
POTE5 116.673
POTE7 116.675
10/9 116.716 (9, 11 and 13 limit minimax)
|0 117 -44 -19> 116.721 (9 limit least squares)
POTE13 116.747
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

Spectrum of Benediction Tunings by Eigenmonzos

Gencom: [2 15/14; 225/224 243/242 351/350 441/440]

Gencom map: [<1 1 3 3 2 7|, <0 6 -7 -2 15 -34|]

Eigenmonzo Secor
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)
POTE13 116.574
|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
POTE11 116.633
7/6 116.641
7\72 116.667
|0 17 -11 -6 11> 116.672 (11 limit least squares)
POTE5 116.673
POTE7 116.675
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

Spectrum of Manna Tunings by Eigenmonzos

Gencom: [2 15/14; 225/224 243/242 325/324 385/384]

Gencom map: [<1 1 3 3 2 0|, <0 6 -7 -2 15 38|]

Eigenmonzo Secor
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
POTE11 116.633
7/6 116.641
7\72 116.667
|0 17 -11 -6 11> 116.672 (11 limit least squares)
POTE5 116.673
POTE7 116.675
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)
POTE13 116.739
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

Spectrum of Revelation Tunings by Eigenmonzos

Gencom: [2 15/14; 66/65 99/98 105/104 1001/1000]

Gencom map: [<1 1 3 3 5 4|, <0 6 -7 -2 -16 -3|]

Eigenmonzo Secor
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)
POTE13 116.268
POTE11 116.277
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
POTE5 116.673
POTE7 116.675
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