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.

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