Miracle: Difference between revisions

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== Interval chain ==
== Interval chain ==
11-odd-limit ratios are labeled in '''bold'''.  
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
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! #
! #
! Cents*
! Cents*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
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| 1
| 1
| 116.7
| 116.7
| 15/14, 16/15
| 15/14, '''16/15'''
|-
|-
| 2
| 2
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| 3
| 3
| 350.1
| 350.1
| '''11/9'''
| 11/9
|-
|-
| 4
| 4
| 466.8
| 466.8
| 21/16
| '''21/16'''
|-
|-
| 5
| 5
| 583.6
| 583.6
| '''7/5'''
| 7/5
|-
|-
| 6
| 6
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| 8
| 8
| 933.7
| 933.7
| '''12/7'''
| 12/7
|-
|-
| 9
| 9
| 1050.4
| 1050.4
| '''11/6'''
| 11/6
|-
|-
| 10
| 10
| 1167.1
| 1167.1
| 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
|-
|-
| 11
| 11
| 83.8
| 83.8
| 22/21, 21/20
| 21/20, 22/21
|-
|-
| 12
| 12
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| 13
| 13
| 317.2
| 317.2
| '''6/5'''
| 6/5
|-
|-
| 14
| 14
| 434.0
| 434.0
| '''9/7'''
| 9/7
|-
|-
| 15
| 15
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| 17
| 17
| 784.1
| 784.1
| '''11/7'''
| 11/7
|-
|-
| 18
| 18
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| 19
| 19
| 1017.5
| 1017.5
| '''9/5'''
| 9/5
|-
|-
| 20
| 20
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| 22
| 22
| 167.6
| 167.6
| '''11/10'''
| 11/10
|-
|-
| 23
| 23
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| 81/80, 121/120
| 81/80, 121/120
|}
|}
<nowiki>*</nowiki> in 11-limit [[CTE tuning]], octave reduced
<nowiki/>* In 11-limit [[CTE tuning]], octave reduced


== Chords ==
== Chords ==
{{main| Chords of miracle }}
{{Main| Chords of miracle }}


== Scales ==
== Scales ==

Revision as of 11:32, 7 January 2025

English Wikipedia has an article on:

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

In the following table, odd harmonics and subharmonics 1–21 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 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
11 83.8 21/20, 22/21
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

Main article: Chords of miracle

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

Images

A diagram taken from George Secor's article "The Miracle Temperament and Decimal Keyboard" which was published in Xenharmonikôn 18 (2006). Highlighting the error band and adding arrows was done for clarity by Douglas Blumeyer on Dave Keenan's request.

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.

Music

Herman Miller
Joseph Pehrson
Gene Ward Smith
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