Miracle
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Miracle is a regular temperament discovered by George Secor in 1974 which splits a tempered 3/2 into six generators, called secors (after George), that serve as both 15/14 and 16/15 semitones. A stack of two generators represents 8/7, and a stack of seven generators represents 8/5. It is a member of both the marvel temperaments, by tempering out 225/224, and the gamelismic clan, by tempering out 1029/1024. It is naturally a full 11-limit temperament, treating the neutral third from three generators as 11/9, tempering out 243/242, 385/384, 441/440, and 540/539. It is supported by the highly notable edos 31, 41, and 72, with 72edo being an especially good tuning. (There is an alternative mapping for 11 known as revelation, but there is little reason to use it unless you are using 31edo, in which case it is identical to miracle anyway.)
Miracle is an exceptionally efficient linear temperament. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-odd-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 the #Interval chain. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.
Some temperaments have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and other temperaments 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.
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.
See Miracle extensions for 13-limit and 17-limit extensions. See Gamelismic clan #Miracle for technical data.
Interval chain
In the following table, odd harmonics and subharmonics 1–21 are labeled in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 116.6 | 15/14, 16/15 |
| 2 | 233.3 | 8/7 |
| 3 | 349.9 | 11/9 |
| 4 | 466.6 | 21/16 |
| 5 | 583.2 | 7/5 |
| 6 | 699.9 | 3/2 |
| 7 | 816.5 | 8/5 |
| 8 | 933.2 | 12/7 |
| 9 | 1049.8 | 11/6 |
| 10 | 1166.5 | 49/25, 55/28, 63/32, 88/45, 96/49, 108/55 |
| 11 | 83.1 | 21/20, 22/21 |
| 12 | 199.8 | 9/8 |
| 13 | 316.4 | 6/5 |
| 14 | 433.1 | 9/7 |
| 15 | 549.7 | 11/8 |
| 16 | 666.3 | 22/15 |
| 17 | 783.0 | 11/7 |
| 18 | 899.6 | 27/16, 42/25 |
| 19 | 1016.3 | 9/5 |
| 20 | 1132.9 | 27/14, 48/25 |
| 21 | 49.6 | 33/32, 36/35 |
| 22 | 166.2 | 11/10 |
| 23 | 282.9 | 33/28 |
| 24 | 399.5 | 44/35 |
| 25 | 516.2 | 27/20 |
| 26 | 632.8 | 36/25 |
| 27 | 749.5 | 54/35, 77/50 |
| 28 | 866.1 | 33/20 |
| 29 | 982.8 | 44/25 |
| 30 | 1099.4 | 66/35 |
| 31 | 16.1 | 81/80, 99/98, 121/120 |
* In 11-limit CWE tuning, octave reduced
Chords
Scales
- Mos scales
- Miracle[10] – 72edo tuning
- Blackjack (miracle[21]) – 72edo tuning
- Blackwoo
- Transversal scales
- Others
- Mir1 – 6-tone scale, 72edo tuning
- Mir2 – 6-tone scale, 72edo tuning
- Miracle 8 – 8-tone scale, 72edo tuning
- Miracle 12 – 12-tone scale, 72edo tuning
- Miracle 12a – 12-tone scale, 72edo tuning
- Miracle 24hi – 24-tone scale, 72edo tuning
- Miracle 24lo – 24-tone scale, 72edo tuning
Tunings
Displayed on the right is a chart of the tuning spectrum of miracle by how the odd harmonics up to 11 are tuned, showing the minimax generator, i.e. the secor.
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~15/14 = 116.5155 ¢ | CSEE: ~15/14 = 116.5612 ¢ | POEE: ~15/14 = 116.6465 ¢ |
| Tenney | CTE: ~15/14 = 116.6772 ¢ | CWE: ~15/14 = 116.6756 ¢ | POTE: ~15/14 = 116.6752 ¢ |
| Benedetti, Wilson |
CBE: ~15/14 = 116.7297 ¢ | CSBE: ~15/14 = 116.7136 ¢ | POBE: ~15/14 = 116.6903 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~15/14 = 116.6868 ¢ | CSEE: ~15/14 = 116.6304 ¢ | POEE: ~15/14 = 116.5817 ¢ |
| Tenney | CTE: ~15/14 = 116.7112 ¢ | CWE: ~15/14 = 116.6469 ¢ | POTE: ~15/14 = 116.6327 ¢ |
| Benedetti, Wilson |
CBE: ~15/14 = 116.7355 ¢ | CSBE: ~15/14 = 116.6768 ¢ | POBE: ~15/14 = 116.6643 ¢ |
Target tunings
| Target | Minimax | Least squares | ||
|---|---|---|---|---|
| Generator | Eigenmonzo* | Generator | Eigenmonzo* | |
| 5-odd-limit | ~16/15 = 116.588 ¢ | 5/3 | ~16/15 = 116.578 ¢ | [0 -19 20⟩ |
| 7-odd-limit | ~15/14 = 116.588 ¢ | 5/3 | ~15/14 = 116.573 ¢ | [0 -27 25 5⟩ |
| 9-odd-limit | ~15/14 = 116.716 ¢ | 9/5 | ~15/14 = 116.721 ¢ | [0 117 -44 -19⟩ |
| 11-odd-limit | ~15/14 = 116.716 ¢ | 9/5 | ~15/14 = 116.672 ¢ | [0 17 -11 -6 11⟩ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 15/8 | 111.731 | ||
| 2\21 | 114.286 | Lower bound of 7-odd-limit diamond monotone | |
| 7/4 | 115.587 | ||
| 11/9 | 115.803 | ||
| 3\31 | 116.129 | Lower bound of 9- and 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone | |
| 5/4 | 116.241 | ||
| 21/11 | 116.412 | ||
| 15/11 | 116.441 | ||
| 7/5 | 116.502 | ||
| 10\103 | 116.505 | ||
| 5/3 | 116.588 | 5- and 7-odd-limit, 11-limit 15- and 21-odd-limit minimax | |
| 11/10 | 116.591 | ||
| 11/6 | 116.596 | ||
| 11/7 | 116.617 | ||
| 7/6 | 116.641 | ||
| 7\72 | 116.667 | ||
| 9/5 | 116.716 | 9- and 11-odd-limit minimax, Secor's definition of secor | |
| 11/8 | 116.755 | ||
| 21/20 | 116.770 | ||
| 9/7 | 116.792 | ||
| 11\113 | 116.814 | ||
| 3/2 | 116.993 | ||
| 4\41 | 117.073 | Upper bound of 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone | |
| 21/16 | 117.695 | ||
| 15/14 | 119.443 | ||
| 1\10 | 120.000 | Upper bound of 7- and 9-odd-limit diamond monotone |
* Besides the octave
Music
- Realm of Possibility (2021) – in Miracle[31] with a 116.72-cent generator and 1200.53-cent octave
- Blackjack (2001) – play | SoundClick – in Blackjack (Miracle[21])
- Blacklight (2002) – play | SoundClick – in Blackjack (Miracle[21])
- Black and Jill (2003) – in Blackjack (Miracle[21])
- Soprano version – play | SoundClick
- Udderbot version
- Inner Voices (2005) – in Blackjack (Miracle[21])
- Transpian (2006) – in Blackjack (Miracle[21])
- microproj (2007) – in Blackjack (Miracle[21])