Miracle extensions

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.

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