Miracle

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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 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

*in 11-limit POTE tuning

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
8/7 115.587
11/9 115.803
5/4 116.241
7/5 116.502
|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
7/6 116.641
|0 17 -11 -6 11> 116.672 (11 limit least squares)
10/9 116.716 (9 and 11 limit minimax,
Secor's definition of secor)
|0 117 -44 -19> 116.721 (9 limit least squares)
11/8 116.755
9/7 116.792
4/3 116.993