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.

Miraculous, benediction, and manna can all be extended to the 17-limit by recognizing 21/16~17/13, tempering out 273/272. For miraculous it implies the generator also represents 17/16, which is supported by 72fg.

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	17-limit extensions
		11-limit	Miraculous	Benediction	Manna
0	0.0	1/1
1	116.6	15/14, 16/15	14/13, 17/16
2	233.3	8/7	15/13, 17/15
3	349.9	11/9	16/13, 17/14, 21/17
4	466.6	21/16	13/10, 17/13	17/13	17/13
5	583.2	7/5	24/17
6	699.9	3/2
7	816.5	8/5	21/13
8	933.2	12/7	17/10
9	1049.8	11/6	24/13
10	1166.5	49/25, 55/28, 63/32, 88/45, 96/49, 108/55	39/20, 51/26, 77/39, 128/65, 135/68, 168/85	51/26, 100/51	51/26, 65/33
11	83.1	21/20, 22/21	18/17, 26/25
12	199.8	9/8
13	316.4	6/5
14	433.1	9/7	22/17
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	35/34, 40/39	34/33
22	166.2	11/10
23	282.9	33/28		20/17	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	17/11
28	866.1	33/20		28/17
29	982.8	44/25		30/17
30	1099.4	66/35		32/17	17/9
31	16.1	81/80, 99/98, 121/120	66/65	105/104, 120/119, 136/135, 144/143, 154/153, 170/169	65/64, 78/77, 85/84, 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, 21/17	26/21
35	482.6	33/25
36	599.3	99/70		24/17	17/12
37	715.9	121/80
38	832.6	121/75		21/13	13/8, 34/21
39	949.2	121/70		45/26	26/15
40	1065.9	231/125		24/13	13/7
41	1182.5	99/50		77/39, 128/65, 168/85, 180/91	119/60, 143/72, 135/68, 153/77, 169/85, 195/98

* In 11-limit CWE tuning, octave reduced

Tunings

Prime-optimized tunings

5-limit POTE: ~16/15 = 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 ¢

Target tunings

Minimax tunings (miracle)
Target	Generator	Eigenmonzo*
5-odd-limit	~16/15 = 116.588 ¢	5/3
7-odd-limit	~15/14 = 116.588 ¢	5/3
9-odd-limit	~15/14 = 116.716 ¢	9/5
11-odd-limit	~15/14 = 116.716 ¢	9/5

Least squares tunings (miracle)
Target	Generator	Eigenmonzo*
5-odd-limit	~16/15 = 116.578 ¢	[0 -19 20⟩
7-odd-limit	~15/14 = 116.573 ¢	[0 -27 25 5⟩
9-odd-limit	~15/14 = 116.721 ¢	[0 117 -44 -19⟩
11-odd-limit	~15/14 = 116.672 ¢	[0 17 -11 -6 11⟩

Minimax tunings (miraculous)
Target	Generator	Eigenmonzo*
13-odd-limit	~15/14 = 116.716 ¢	9/5
15-odd-limit	~15/14 = 116.993 ¢	3/2

Least squares tunings (miraculous)
Target	Generator	Eigenmonzo
13-odd-limit	~15/14 = 116.846 ¢	[0 141 -70 -35 84 -42⟩
15-odd-limit	~15/14 = 116.820 ¢	[0 127 -84 -36 100 -44⟩

Minimax tunings (benediction)
Target	Generator	Eigenmonzo*
13-odd-limit	~15/14 = 116.595 ¢	13/9
15-odd-limit	~15/14 = 116.588 ¢	5/3

Least squares tunings (benediction)
Target	Generator	Eigenmonzo
13-odd-limit	~15/14 = 116.56309 ¢	[0 -234 39 4 -115 228⟩
15-odd-limit	~15/14 = 116.56348 ¢	[0 -251 22 5 -131 261⟩

Minimax tunings (manna)
Target	Generator	Eigenmonzo*
13-odd-limit	~15/14 = 116.760 ¢	13/10
15-odd-limit	~15/14 = 116.725 ¢	15/13

Least squares tunings (manna)
Target	Generator	Eigenmonzo
13-odd-limit	~15/14 = 116.780 ¢	[0 18 -111 -76 43 204⟩
15-odd-limit	~15/14 = 116.764 ¢	[0 -37 -166 -77 59 243⟩

Minimax tunings (revelation)
Target	Generator	Eigenmonzo*
11-odd-limit	~15/14 = 116.164 ¢	11/9
13-odd-limit	~15/14 = 116.164 ¢	11/9
15-odd-limit	~15/14 = 116.164 ¢	11/9

Least squares tunings (revelation)
Target	Generator	Eigenmonzo
11-odd-limit	~15/14 = 116.198 ¢	[0 -195 35 5 89⟩
13-odd-limit	~15/14 = 116.249 ¢	[0 -234 39 4 102 11⟩
15-odd-limit	~15/14 = 116.229 ¢	[0 -251 22 5 117 13⟩

Tuning spectra

Miraculous

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	17/16	104.955
	17/15	108.343
	15/8	111.731
	17/14	112.043
	13/10	113.553
	17/10	114.830
	7/4	115.587
	11/9	115.803
	17/13	116.107
3\31		116.129	Lower bound of 11- to 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	5/4	116.241
	15/11	116.441
	7/5	116.502
10\103		116.505	103fg val
17\175		116.571	175ffggg val
	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	72fg val
	9/5	116.716	9-, 11- and 13-odd-limit minimax
	11/8	116.755
18\185		116.757	185cffggg val
	9/7	116.792
11\113		116.814	113fgg val
	3/2	116.993	15-odd-limit minimax
4\41		117.073	Upper bound of 11- to 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	13/11	117.266
	13/9	117.559
	17/11	117.597
	13/12	117.936
	17/9	118.087
	17/12	119.400
	15/14	119.443
	13/8	119.824
	21/17	121.942
	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
	17/13	116.107
3\31		116.129	Lower bound of 11- to 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	5/4	116.241
	15/11	116.441
	13/8	116.455
	17/16	116.501
	7/5	116.502
10\103		116.505
	13/7	116.509
	13/10	116.511
	13/12	116.536
	13/11	116.547
	17/14	116.567
17\175		116.571	175f val
	17/10	116.581
	17/12	116.583
	17/11	116.586
	5/3	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
	17/9	116.642
	21/17	116.642
	17/15	116.666
7\72		116.667	Upper bound of 13- to 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	9/5	116.716	9- and 11-odd-limit minimax
	11/8	116.755
18\185		116.757	185cfffgg val
	9/7	116.792
11\113		116.814	113ffg val
	3/2	116.993
4\41		117.073	41fg val, upper bound of 11-odd-limit diamond monotone
	15/14	119.443

Manna

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	15/8	111.731
	7/4	115.587
	11/9	115.803
	17/13	116.107
3\31		116.129	31fg val, lower bound of 11-odd-limit diamond monotone
	5/4	116.241
	15/11	116.441
	7/5	116.502
10\103		116.505	103ffgg val
17\175		116.571	175fffgg val
	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	Lower bound of 13- to 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	17/15	116.667
	21/17	116.689
	17/9	116.702
	17/10	116.707
	9/5	116.716	9- and 11-odd-limit minimax
	15/13	116.725	15-odd-limit minimax
	17/14	116.730
	17/12	116.750
	11/8	116.755
18\185		116.757	185cf val
	13/10	116.760	13-odd-limit minimax
	17/16	116.785
	9/7	116.792
	13/7	116.79254
	13/9	116.79299
	17/11	116.801
11\113		116.814
	13/12	116.830
	13/8	116.856
	13/11	116.922
	3/2	116.993
4\41		117.073	Upper bound of 11- to 17-odd-limit, and 17-limit 21-odd-limit diamond monotone
	15/14	119.443

Revelation

Edo generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	15/8	111.731
	13/10	113.553
2\21		114.286
	13/11	114.555
	11/10	115.000
5\52		115.385	52f val
	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- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone (singleton)
	11/9	116.164	11-, 13- and 15-odd-limit minimax
	5/4	116.241
	7/5	116.502
	5/3	116.588	5- and 7-odd-limit minimax
	7/6	116.641
7\72		116.667	72ee val
	9/5	116.716	9-odd-limit minimax
	9/7	116.792
	3/2	116.993
4\41		117.073	41ef val
	13/9	117.559
	13/12	117.936
	15/14	119.443
	13/8	119.824
	15/13	123.871
	13/7	128.298