Miracle: Difference between revisions

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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 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]] and the [[gamelismic clan]]. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[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 bold) 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.  
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 [[cent]]s/[[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]].
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]].
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== Interval chain ==
== Interval chain ==
11-odd-limit ratios are labeled in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! # of<br>secors
! #
! Cents value<ref>in 11-limit [[POTE tuning]]</ref><br>(octave-reduced)
! Cents*
! JI intervals <br>represented
! Approximate Ratios
|-
|-
| 0
| 0
| 0.00
| 0.0
| '''1/1'''
| '''1/1'''
|-
|-
| 1
| 1
| 116.63
| 116.7
| 16/15, 15/14
| 15/14, 16/15
|-
|-
| 2
| 2
| 233.27
| 233.4
| '''8/7'''
| '''8/7'''
|-
|-
| 3
| 3
| 349.90
| 350.1
| '''11/9'''
| '''11/9'''
|-
|-
| 4
| 4
| 466.53
| 466.8
| 21/16
| 21/16
|-
|-
| 5
| 5
| 583.16
| 583.6
| '''7/5'''
| '''7/5'''
|-
|-
| 6
| 6
| 699.80
| 700.3
| '''3/2'''
| '''3/2'''
|-
|-
| 7
| 7
| 816.43
| 817.0
| '''8/5'''
| '''8/5'''
|-
|-
| 8
| 8
| 933.06
| 933.7
| '''12/7'''
| '''12/7'''
|-
|-
| 9
| 9
| 1049.69
| 1050.4
| '''11/6'''
| '''11/6'''
|-
|-
| 10
| 10
| 1166.33
| 1167.1
| 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
| 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
|-
|-
| 11
| 11
| 82.96
| 83.8
| 22/21, 21/20
| 22/21, 21/20
|-
|-
| 12
| 12
| 199.59
| 200.5
| '''9/8'''
| '''9/8'''
|-
|-
| 13
| 13
| 316.23
| 317.2
| '''6/5'''
| '''6/5'''
|-
|-
| 14
| 14
| 432.86
| 434.0
| '''9/7'''
| '''9/7'''
|-
|-
| 15
| 15
| 549.49
| 550.7
| '''11/8'''
| '''11/8'''
|-
|-
| 16
| 16
| 666.12
| 667.4
| 22/15
| 22/15
|-
|-
| 17
| 17
| 782.76
| 784.1
| '''11/7'''
| '''11/7'''
|-
|-
| 18
| 18
| 899.39
| 900.8
| 42/25, 27/16
| 27/16, 42/25
|-
|-
| 19
| 19
| 1016.02
| 1017.5
| '''9/5'''
| '''9/5'''
|-
|-
| 20
| 20
| 1132.65
| 1134.2
| 48/25, 27/14
| 27/14, 48/25
|-
|-
| 21
| 21
| 49.29
| 50.9
| 36/35, 33/32
| 33/32, 36/35
|-
|-
| 22
| 22
| 165.92
| 167.6
| '''11/10'''
| '''11/10'''
|-
|-
| 23
| 23
| 282.55
| 284.4
| 33/28
| 33/28
|-
|-
| 24
| 24
| 399.19
| 401.1
| 44/35
| 44/35
|-
|-
| 25
| 25
| 515.82
| 517.8
| 27/20
| 27/20
|-
|-
| 26
| 26
| 632.45
| 634.5
| 36/25
| 36/25
|-
|-
| 27
| 27
| 749.08
| 751.2
| 54/35, 77/50
| 54/35, 77/50
|-
|-
| 28
| 28
| 865.72
| 867.9
| 33/20
| 33/20
|-
|-
| 29
| 29
| 982.35
| 984.6
| 44/25
| 44/25
|-
|-
| 30
| 30
| 1098.98
| 1101.3
| 66/35
| 66/35
|-
|-
| 31
| 31
| 15.62
| 18.0
| 81/80
| 81/80, 121/120
|}
|}
<references/>
<nowiki>*</nowiki> in 11-limit [[CTE tuning]], octave reduced


== Chords ==
== Chords ==

Revision as of 11:50, 18 February 2024

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 and the gamelismic clan. 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

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

Music

Gene Ward Smith
Joseph Pehrson
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