Miracle
Miracle is a regular temperament discovered by George Secor in 1974 which has as a generator an interval, called a secor (after George), that serves as both 15/14 and 16/15 semitones.
Miracle is an exceptionally efficient linear temperament which is a member of both the marvel temperaments, by tempering out 225/224, and the gamelismic clan, by tempering out 1029/1024. 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 (→ 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.
In terms of 13-limit extensions, it is discussed in Miracle extensions. See Gamelismic clan #Miracle for technical data.
Interval chain
11-odd-limit ratios are labeled in bold.
# | Cents* | Approximate Ratios |
---|---|---|
0 | 0.0 | 1/1 |
1 | 116.7 | 15/14, 16/15 |
2 | 233.4 | 8/7 |
3 | 350.1 | 11/9 |
4 | 466.8 | 21/16 |
5 | 583.6 | 7/5 |
6 | 700.3 | 3/2 |
7 | 817.0 | 8/5 |
8 | 933.7 | 12/7 |
9 | 1050.4 | 11/6 |
10 | 1167.1 | 88/45, 96/49, 49/25, 108/55, 55/28, 63/32 |
11 | 83.8 | 22/21, 21/20 |
12 | 200.5 | 9/8 |
13 | 317.2 | 6/5 |
14 | 434.0 | 9/7 |
15 | 550.7 | 11/8 |
16 | 667.4 | 22/15 |
17 | 784.1 | 11/7 |
18 | 900.8 | 27/16, 42/25 |
19 | 1017.5 | 9/5 |
20 | 1134.2 | 27/14, 48/25 |
21 | 50.9 | 33/32, 36/35 |
22 | 167.6 | 11/10 |
23 | 284.4 | 33/28 |
24 | 401.1 | 44/35 |
25 | 517.8 | 27/20 |
26 | 634.5 | 36/25 |
27 | 751.2 | 54/35, 77/50 |
28 | 867.9 | 33/20 |
29 | 984.6 | 44/25 |
30 | 1101.3 | 66/35 |
31 | 18.0 | 81/80, 121/120 |
* in 11-limit CTE tuning, octave reduced
Chords
Scales
- See also: Miracle 10 MODMOS
- 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
Tuning spectrum
Edo Generator |
Eigenmonzo (Unchanged-interval) |
Secor (¢) | 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 | ||
[0 -27 25 5⟩ | 116.573 | 7-odd-limit least squares | |
[0 -19 20⟩ | 116.578 | 5-odd-limit least squares | |
5/3 | 116.588 | 5- and 7-odd-limit minimax | |
11/10 | 116.591 | ||
11/6 | 116.596 | ||
11/7 | 116.617 | ||
7/6 | 116.641 | ||
7\72 | 116.667 | ||
[0 17 -11 -6 11⟩ | 116.672 | 11-odd-limit least squares | |
9/5 | 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 | ||
21/20 | 116.770 | ||
9/7 | 116.792 | ||
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 |
Music
- Rachmaninoff Plays Blackjack (archived 2010) – detail | play – Blackjack (miracle[21]) in 175edo tuning
- Blackjack (2001) – play | SoundClick – Blackjack (miracle[21])
- Blacklight (2002) – play | SoundClick – Blackjack (miracle[21])
- Black and Jill (2003) – Blackjack (miracle[21])
- Soprano version – play | SoundClick
- Udderbot version
- Inner Voices (2005) – Blackjack (miracle[21])
- Transpian (2006) – Blackjack (miracle[21])
- microproj (2007) – Blackjack (miracle[21])