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 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 |