Miracle

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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
Transversal scales
Others

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

Gene Ward Smith
Joseph Pehrson