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'''Miracle temperament''' is a regular temperament discovered by [[George Secor]] in 1974 which has as a generator an interval, called '''secor''', that serves as both [[15/14]] and [[16/15]] semitones.
[[de:Miracle]]
{{Wikipedia|Miracle temperament}}
'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]].  


Miracle is an exceptionally efficient linear temperament which is a member of both the [[Marvel_temperaments|marvel temperaments]] and the [[Gamelismic_clan|gamelismic clan]]. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-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 following table:
Miracle is an exceptionally efficient linear temperament. 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.


{| class="wikitable"
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 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.
 
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.
 
See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.
 
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
|-
! | number of <br>secors
! #
! | cents value* <br>(octave-reduced)
! Cents*
! | JI intervals <br>represented
! Approximate ratios
|-
|-
| | 0
| 0
| | 0.00
| 0.0
| | '''1/1'''
| '''1/1'''
|-
|-
| | 1
| 1
| | 116.63
| 116.6
| | 16/15, 15/14
| 15/14, '''16/15'''
|-
|-
| | 2
| 2
| | 233.27
| 233.3
| | '''8/7'''
| '''8/7'''
|-
|-
| | 3
| 3
| | 349.90
| 349.9
| | '''11/9'''
| 11/9
|-
|-
| | 4
| 4
| | 466.53
| 466.6
| | 21/16
| '''21/16'''
|-
|-
| | 5
| 5
| | 583.16
| 583.2
| | '''7/5'''
| 7/5
|-
|-
| | 6
| 6
| | 699.80
| 699.9
| | '''3/2'''
| '''3/2'''
|-
|-
| | 7
| 7
| | 816.43
| 816.5
| | '''8/5'''
| '''8/5'''
|-
|-
| | 8
| 8
| | 933.06
| 933.2
| | '''12/7'''
| 12/7
|-
|-
| | 9
| 9
| | 1049.69
| 1049.8
| | '''11/6'''
| 11/6
|-
|-
| | 10
| 10
| | 1166.33
| 1166.5
| | 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
| | 82.96
| 83.1
| | 22/21, 21/20
| 21/20, 22/21
|-
|-
| | 12
| 12
| | 199.59
| 199.8
| | '''9/8'''
| '''9/8'''
|-
|-
| | 13
| 13
| | 316.23
| 316.4
| | '''6/5'''
| 6/5
|-
|-
| | 14
| 14
| | 432.86
| 433.1
| | '''9/7'''
| 9/7
|-
|-
| | 15
| 15
| | 549.49
| 549.7
| | '''11/8'''
| '''11/8'''
|-
|-
| | 16
| 16
| | 666.12
| 666.3
| | 22/15
| 22/15
|-
|-
| | 17
| 17
| | 782.76
| 783.0
| | '''11/7'''
| 11/7
|-
|-
| | 18
| 18
| | 899.39
| 899.6
| | 42/25, 27/16
| 27/16, 42/25
|-
|-
| | 19
| 19
| | 1016.02
| 1016.3
| | '''9/5'''
| 9/5
|-
|-
| | 20
| 20
| | 1132.65
| 1132.9
| | 48/25, 27/14
| 27/14, 48/25
|-
|-
| | 21
| 21
| | 49.29
| 49.6
| | 36/35, 33/32
| 33/32, 36/35
|-
|-
| | 22
| 22
| | 165.92
| 166.2
| | '''11/10'''
| 11/10
|-
|-
| | 23
| 23
| | 282.55
| 282.9
| | 33/28
| 33/28
|-
|-
| | 24
| 24
| | 399.19
| 399.5
| | 44/35
| 44/35
|-
|-
| | 25
| 25
| | 515.82
| 516.2
| | 27/20
| 27/20
|-
|-
| | 26
| 26
| | 632.45
| 632.8
| | 36/25
| 36/25
|-
|-
| | 27
| 27
| | 749.08
| 749.5
| | 54/35, 77/50
| 54/35, 77/50
|-
|-
| | 28
| 28
| | 865.72
| 866.1
| | 33/20
| 33/20
|-
|-
| | 29
| 29
| | 982.35
| 982.8
| | 44/25
| 44/25
|-
|-
| | 30
| 30
| | 1098.98
| 1099.4
| | 66/35
| 66/35
|-
|-
| | 31
| 31
| | 15.62
| 16.1
| | 81/80
| 81/80, 99/98, 121/120
|}
|}
<span style="font-family: Arial,Helvetica,sans-serif;">*in 11-limit POTE tuning</span>
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


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 (in the [[Gamelismic_clan|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.
== Chords ==
{{Main| Chords of miracle }}


Miracle can also be thought of as a [[Cluster_temperament|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.
== Scales ==
{{See also| Miracle 10 MODMOS }}
 
; Mos scales
* [[Miracle 10|Miracle{{lbrack}}10{{rbrack}}]] – 72edo tuning
* [[Blackjack|Blackjack (miracle{{lbrack}}21{{rbrack}})]] – 72edo tuning
* [[Blackwoo]]
; Transversal scales
* [[Miracle21trans]]
* [[Miracle21trans511]]
* [[Miracle31trans]]
* [[Miracle31trans511]]
; 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
 
== Tunings ==
[[File:Derivation of the secor.png|thumb|600px|right|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.]]
 
Displayed on the right is 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.
 
=== Prime-optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-optimized tunings
|-
! rowspan="2" |
! colspan="2" | Euclidean
|-
! Constrained
! Constrained & skewed
|-
! Equilateral
| CEE: ~15/14 = 116.516{{c}}
| CSEE: ~15/14 = 116.561{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.677{{c}}
| CWE: ~15/14 = 116.676{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~15/14 = 116.730{{c}}
| CSBE: ~15/14 = 116.714{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="2" | Euclidean
|-
! Constrained
! Constrained & skewed
|-
! Equilateral
| CEE: ~15/14 = 116.687{{c}}
| CSEE: ~15/14 = 116.630{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.711{{c}}
| CWE: ~15/14 = 116.647{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~15/14 = 116.736{{c}}
| CSBE: ~15/14 = 116.677{{c}}
|}
 
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings
|-
! Target
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.588{{c}}
| 5/3
|-
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
|-
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
|-
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
|}


=Spectrum of Miracle Tunings by Eigenmonzos=
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings
|-
! Target
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
| 7-odd-limit
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
| 9-odd-limit
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
| 11-odd-limit
| ~15/14 = 116.672{{c}}
| {{Monzo| 0 17 -11 -6 11 }}
|}


{| class="wikitable"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|-
! | Eigenmonzo
| [[21edo|2\21]]
! | Secor
|
| 114.286
| Lower bound of 7-odd-limit diamond monotone
|-
|-
| | 8/7
|  
| | 115.587
| 7/4
| 115.587
|
|-
|-
| | 11/9
|  
| | 115.803
| 11/9
| 115.803
|
|-
|-
| | 5/4
| [[31edo|3\31]]
| | 116.241
|  
| 116.129
| Lower bound of 9- and 11-odd-limit, <br>11-limit 15- and 21-odd-limit diamond monotone
|-
|-
| | 7/5
|  
| | 116.502
| 5/4
| 116.241
|
|-
|-
| | 6/5
|  
| | 116.588 (5 and 7 limit minimax)
| 21/11
| 116.412
|
|-
|-
| | 11/10
|  
| | 116.591
| 15/11
| 116.441
|
|-
|-
| | 12/11
|  
| | 116.596
| 7/5
| 116.502
|
|-
|-
| | 14/11
| [[103edo|10\103]]
| | 116.617
|  
| 116.505
|
|-
|-
| | 7/6
|  
| | 116.641
| 5/3
| 116.588
| 5- and 7-odd-limit minimax
|-
|-
| | 10/9
|  
| | 116.716 (9 and 11 limit minimax, <br>Secor's definition of secor)
| 11/10
| 116.591
|
|-
|-
| | 11/8
|  
| | 116.755
| 11/6
| 116.596
|
|-
|-
| | 9/7
|  
| | 116.792
| 11/7
| 116.617
|
|-
|-
| | 4/3
|  
| | 116.993
| 7/6
| 116.641
|  
|-
| [[72edo|7\72]]
|
| 116.667
|
|-
|
| 9/5
| 116.716
| 9- and 11-odd-limit minimax, <br>Secor's definition of secor
|-
|
| 11/8
| 116.755
|
|-
|
| 21/20
| 116.770
|
|-
|
| 9/7
| 116.792
|
|-
| [[113edo|11\113]]
|
| 116.814
|
|-
|
| 3/2
| 116.993
|
|-
| [[41edo|4\41]]
|
| 117.073
| Upper bound of 11-odd-limit, <br>11-limit 15- and 21-odd-limit diamond monotone
|-
|
| 21/16
| 117.695
|
|-
|
| 15/14
| 119.443
|
|-
| [[10edo|1\10]]
|
| 120.000
| Upper bound of 7- and 9-odd-limit diamond monotone
|}
|}
<nowiki/>* Besides the octave
== Music ==
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – in Miracle[31] with a 116.72-cent generator and 1200.53-cent octave
; [[Joseph Pehrson]]
* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – in [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]
* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – in Blackjack (Miracle[21])
* ''Black and Jill'' (2003) – in Blackjack (Miracle[21])
** Soprano version – [https://web.archive.org/web/20201127012730/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 play] | [https://soundclick.com/share.cfm?id=2373400 SoundClick]
** [https://soundclick.com/share.cfm?id=9583778 Udderbot version]
* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – in Blackjack (Miracle[21])
; [[Gene Ward Smith]]
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning


[[Category:miracle]]
[[Category:Miracle| ]] <!-- main article -->
[[Category:temperament]]
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Gamelismic clan]]
Retrieved from "https://en.xen.wiki/w/Miracle"