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[[de:Miracle]]
[[de:Miracle]]
{{Wikipedia|Miracle temperament}}
{{Wikipedia|Miracle temperament}}
'''Miracle temperament''' 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. In terms of [[13-limit]] extensions, it is discussed in [[Miracle extensions|miraculous, benediction, and manna]].
'''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 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.  


Some temperaments have [[11/9]] as a "neutral third", meaning it's 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]].


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.
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 ==
== Interval chain ==
Line 150: Line 152:


== Scales ==
== Scales ==
; MOS scales
{{See also| Miracle 10 MODMOS }}
* [[Miracle 10]]
 
* [[Blackjack]]
; Mos scales
* [[Miracle 10|Miracle{{lbrack}}10{{rbrack}}]] – 72edo tuning
* [[Blackjack|Blackjack (miracle{{lbrack}}21{{rbrack}})]] – 72edo tuning
* [[Blackwoo]]
* [[Blackwoo]]
; Transversal scales
; Transversal scales
Line 160: Line 164:
* [[Miracle31trans511]]
* [[Miracle31trans511]]
; Others
; Others
* [[Mir1]]
* [[Mir1]] – 6-tone scale, 72edo tuning
* [[Mir2]]
* [[Mir2]] – 6-tone scale, 72edo tuning
* [[Miracle 8]]
* [[Miracle 8]] – 8-tone scale, 72edo tuning
* [[Miracle 12]]
* [[Miracle 12]] – 12-tone scale, 72edo tuning
* [[Miracle 12a]]
* [[Miracle 12a]] – 12-tone scale, 72edo tuning
* [[Miracle 24hi]]
* [[Miracle 24hi]] – 24-tone scale, 72edo tuning
* [[Miracle 24lo]]
* [[Miracle 24lo]] – 24-tone scale, 72edo tuning
* [[Miracle 10 MODMOS]]


== Tuning spectrum ==
== Tuning spectrum ==


{| class="wikitable center-all"
{| class="wikitable center-all left-4"
|-
|-
! ET<br>generator
! Edo<br>Generator
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! secor<br>(¢)
! Secor (¢)
! comments
! Comments
|-
|-
|  
|  

Revision as of 11:03, 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 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.

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

# of
secors 		Cents value[1]
(octave-reduced) 		JI intervals
represented
0 0.00 1/1
1 116.63 16/15, 15/14
2 233.27 8/7
3 349.90 11/9
4 466.53 21/16
5 583.16 7/5
6 699.80 3/2
7 816.43 8/5
8 933.06 12/7
9 1049.69 11/6
10 1166.33 88/45, 96/49, 49/25,
108/55, 55/28, 63/32
11 82.96 22/21, 21/20
12 199.59 9/8
13 316.23 6/5
14 432.86 9/7
15 549.49 11/8
16 666.12 22/15
17 782.76 11/7
18 899.39 42/25, 27/16
19 1016.02 9/5
20 1132.65 48/25, 27/14
21 49.29 36/35, 33/32
22 165.92 11/10
23 282.55 33/28
24 399.19 44/35
25 515.82 27/20
26 632.45 36/25
27 749.08 54/35, 77/50
28 865.72 33/20
29 982.35 44/25
30 1098.98 66/35
31 15.62 81/80
  1. ↑ in 11-limit POTE tuning

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
8/7 115.587
11/9 115.803
3\31 116.129
5/4 116.241
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
6/5 116.588 5- and 7-odd-limit minimax
11/10 116.591
12/11 116.596
14/11 116.617
7/6 116.641
7\72 116.667
[0 17 -11 -6 11⟩ 116.672 11-odd-limit least squares
10/9 116.716 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
9/7 116.792
4/3 116.993
4\41 117.073

Music

Gene Ward Smith
Joseph Pehrson
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