Miracle extensions

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

Tuning spectra

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

Gencom mapping: [⟨1 1 3 3 2 4], ⟨0 6 -7 -2 15 -3]]

ET generator	eigenmonzo (unchanged-interval)	secor (¢)	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

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

Gencom mapping: [⟨1 1 3 3 2 7], ⟨0 6 -7 -2 15 -34]]

ET generator	eigenmonzo (unchanged-interval)	secor (¢)	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

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

Gencom mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 38]]

ET generator	eigenmonzo (unchanged-interval)	secor (¢)	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

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

Gencom mapping: [⟨1 1 3 3 5 4], ⟨0 6 -7 -2 -16 -3]]

ET generator	eigenmonzo (unchanged-interval)	secor (¢)	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