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[1] (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 |