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[[de:Miracle]]
[[de:Miracle]]
{{Wikipedia|Miracle temperament}}
{{Wikipedia|Miracle temperament}}
'''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 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]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. 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.  
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.  


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 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.
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.  
See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.  


== Interval chain ==
== Interval chain ==
Line 25: Line 25:
|-
|-
| 1
| 1
| 116.7
| 116.6
| 15/14, '''16/15'''
| 15/14, '''16/15'''
|-
|-
| 2
| 2
| 233.4
| 233.3
| '''8/7'''
| '''8/7'''
|-
|-
| 3
| 3
| 350.1
| 349.9
| 11/9
| 11/9
|-
|-
| 4
| 4
| 466.8
| 466.6
| '''21/16'''
| '''21/16'''
|-
|-
| 5
| 5
| 583.6
| 583.2
| 7/5
| 7/5
|-
|-
| 6
| 6
| 700.3
| 699.9
| '''3/2'''
| '''3/2'''
|-
|-
| 7
| 7
| 817.0
| 816.5
| '''8/5'''
| '''8/5'''
|-
|-
| 8
| 8
| 933.7
| 933.2
| 12/7
| 12/7
|-
|-
| 9
| 9
| 1050.4
| 1049.8
| 11/6
| 11/6
|-
|-
| 10
| 10
| 1167.1
| 1166.5
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
|-
|-
| 11
| 11
| 83.8
| 83.1
| 21/20, 22/21
| 21/20, 22/21
|-
|-
| 12
| 12
| 200.5
| 199.8
| '''9/8'''
| '''9/8'''
|-
|-
| 13
| 13
| 317.2
| 316.4
| 6/5
| 6/5
|-
|-
| 14
| 14
| 434.0
| 433.1
| 9/7
| 9/7
|-
|-
| 15
| 15
| 550.7
| 549.7
| '''11/8'''
| '''11/8'''
|-
|-
| 16
| 16
| 667.4
| 666.3
| 22/15
| 22/15
|-
|-
| 17
| 17
| 784.1
| 783.0
| 11/7
| 11/7
|-
|-
| 18
| 18
| 900.8
| 899.6
| 27/16, 42/25
| 27/16, 42/25
|-
|-
| 19
| 19
| 1017.5
| 1016.3
| 9/5
| 9/5
|-
|-
| 20
| 20
| 1134.2
| 1132.9
| 27/14, 48/25
| 27/14, 48/25
|-
|-
| 21
| 21
| 50.9
| 49.6
| 33/32, 36/35
| 33/32, 36/35
|-
|-
| 22
| 22
| 167.6
| 166.2
| 11/10
| 11/10
|-
|-
| 23
| 23
| 284.4
| 282.9
| 33/28
| 33/28
|-
|-
| 24
| 24
| 401.1
| 399.5
| 44/35
| 44/35
|-
|-
| 25
| 25
| 517.8
| 516.2
| 27/20
| 27/20
|-
|-
| 26
| 26
| 634.5
| 632.8
| 36/25
| 36/25
|-
|-
| 27
| 27
| 751.2
| 749.5
| 54/35, 77/50
| 54/35, 77/50
|-
|-
| 28
| 28
| 867.9
| 866.1
| 33/20
| 33/20
|-
|-
| 29
| 29
| 984.6
| 982.8
| 44/25
| 44/25
|-
|-
| 30
| 30
| 1101.3
| 1099.4
| 66/35
| 66/35
|-
|-
| 31
| 31
| 18.0
| 16.1
| 81/80, 121/120
| 81/80, 99/98, 121/120
|}
|}
<nowiki/>* In 11-limit [[CTE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


== Chords ==
== Chords ==
Line 174: Line 174:
* [[Miracle 24lo]] – 24-tone scale, 72edo tuning
* [[Miracle 24lo]] – 24-tone scale, 72edo tuning


== Tuning spectrum ==
== 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
|}
{| 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 }}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Secor (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
Line 228: Line 326:
|  
|  
|-
|-
| [[103edo|10\103]]
|  
|  
| {{monzo| 0 -27 25 5 }}
| 116.505
| 116.573
| 7-odd-limit least squares
|-
|  
|  
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 267: Line 360:
| 116.667
| 116.667
|  
|  
|-
|
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 11-odd-limit least squares
|-
|-
|  
|  
Line 277: Line 365:
| 116.716
| 116.716
| 9- and 11-odd-limit minimax, <br>Secor's definition of secor
| 9- and 11-odd-limit minimax, <br>Secor's definition of secor
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 296: Line 379:
| 9/7
| 9/7
| 116.792
| 116.792
|
|-
| [[113edo|11\113]]
|
| 116.814
|  
|  
|-
|-
Line 323: Line 411:
| Upper bound of 7- and 9-odd-limit diamond monotone
| Upper bound of 7- and 9-odd-limit diamond monotone
|}
|}
 
<nowiki/>* Besides the octave
== Images ==
[[File:Derivation of the secor.png|thumb|600px|center|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 ==
== Music ==
; [[Herman Miller]]
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/realm-of-possibility Realm of Possibility]'' (2021, [[Miracle31trans]] scale)
* [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]]
; [[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] – Blackjack (miracle{{lbrack}}21{{rbrack}})
* ''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] – 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) – Blackjack (miracle{{lbrack}}21{{rbrack}})
* ''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]
** 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=9583778 Udderbot version]
* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – Blackjack (miracle{{lbrack}}21{{rbrack}})
* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – Blackjack (miracle{{lbrack}}21{{rbrack}})
* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – Blackjack (miracle{{lbrack}}21{{rbrack}})
* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – in Blackjack (Miracle[21])


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – [[Blackjack|Blackjack (miracle{{lbrack}}21{{rbrack}})]] in 175edo tuning
* ''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:Temperaments]]
[[Category:Miracle| ]] <!-- main article -->
[[Category:Miracle| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
Retrieved from "https://en.xen.wiki/w/Miracle"