Miracle

Miracle temperament is a regular temperament discovered by George Secor in 1974 which has as a generator an interval, called secor, that serves as both 15/14 and 16/15 semitones. In terms of 13-limit extensions, it is discussed in miraculous, benediction, and manna.

Miracle is an exceptionally efficient linear temperament which is a member of both the marvel temperaments (→ Marvel temperaments #Miracle) and the gamelismic clan. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-limit tonality diamond are represented with only one or two cents of error. Yet it is also very "low complexity" (efficient), as evidenced by the high density of 11-odd-limit ratios (in bold) in the following table:

number of secors	cents value (octave-reduced)	JI intervals represented
0	0.00	1/1
1	116.63	16/15, 15/14
2	233.27	8/7
3	349.90	11/9
4	466.53	21/16
5	583.16	7/5
6	699.80	3/2
7	816.43	8/5
8	933.06	12/7
9	1049.69	11/6
10	1166.33	88/45, 96/49, 49/25, 108/55, 55/28, 63/32
11	82.96	22/21, 21/20
12	199.59	9/8
13	316.23	6/5
14	432.86	9/7
15	549.49	11/8
16	666.12	22/15
17	782.76	11/7
18	899.39	42/25, 27/16
19	1016.02	9/5
20	1132.65	48/25, 27/14
21	49.29	36/35, 33/32
22	165.92	11/10
23	282.55	33/28
24	399.19	44/35
25	515.82	27/20
26	632.45	36/25
27	749.08	54/35, 77/50
28	865.72	33/20
29	982.35	44/25
30	1098.98	66/35
31	15.62	81/80

Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the gamelismic clan) have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a "secor", and it represents both 16/15 and 15/14.

Miracle can also be thought of as a cluster temperament with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together.

Spectrum of miracle tunings by eigenmonzos

Eigenmonzo	Secor	Comments
8/7	115.587
11/9	115.803
5/4	116.241
7/5	116.502
[0 -27 25 5⟩	116.573	7-odd-limit least squares
[0 -19 20⟩	116.578	5-odd-limit least squares
6/5	116.588	5- and 7-odd-limit minimax
11/10	116.591
12/11	116.596
14/11	116.617
7/6	116.641
[0 17 -11 -6 11⟩	116.672	11-odd-limit least squares
10/9	116.716	9- and 11-odd-limit minimax, Secor's definition of secor
[0 117 -44 -19⟩	116.721	9-odd-limit least squares
11/8	116.755
9/7	116.792
4/3	116.993