Xenharmonic Wiki

Miracle: Difference between revisions

← Older edit
VisualWikitext
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
Wikipedia box
m →Music: cleanup
 
(26 intermediate revisions by 5 users not shown)
Line 1: Line 1:
[[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 '''secor''', 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 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]] 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. 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'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 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.
See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.


== Interval chain ==
== Interval chain ==
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"
|-
|-
! # 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
| 116.6
| 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
|}
|}
<references/>
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


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


== 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 166:
* [[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]]
 
== 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}}
|}


== Tuning spectrum ==
{| 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"
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings
|-
|-
! ET<br>generator
! Target
! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
! Generator
! secor<br>(¢)
! Eigenmonzo*
! comments
|-
| 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"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|-
|  
|  
| 8/7
| 15/8
| 111.731
|
|-
| [[21edo|2\21]]
|
| 114.286
| Lower bound of 7-odd-limit diamond monotone
|-
|
| 7/4
| 115.587
| 115.587
|  
|  
Line 191: Line 304:
|  
|  
| 116.129
| 116.129
|  
| Lower bound of 9- and 11-odd-limit, <br>11-limit 15- and 21-odd-limit diamond monotone
|-
|-
|  
|  
| 5/4
| 5/4
| 116.241
| 116.241
|
|-
|
| 21/11
| 116.412
|
|-
|
| 15/11
| 116.441
|  
|  
|-
|-
Line 203: 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
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5- and 7-odd-limit minimax
| 5- and 7-odd-limit minimax
Line 224: Line 342:
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
Line 244: Line 362:
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 9/5
| 116.672
| 11-odd-limit least squares
|-
|
| 10/9
| 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 }}
| 11/8
| 116.721
| 116.755
| 9-odd-limit least squares
|  
|-
|-
|  
|  
| 11/8
| 21/20
| 116.755
| 116.770
|  
|  
|-
|-
Line 268: Line 381:
|  
|  
|-
|-
| [[113edo|11\113]]
|  
|  
| 4/3
| 116.814
|
|-
|
| 3/2
| 116.993
| 116.993
|  
|  
Line 276: Line 394:
|  
|  
| 117.073
| 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 ==
== Music ==
By [[Gene Ward Smith]]
; [[Herman Miller]]
* [http://www.archive.org/details/RachmaninoffPlaysBlackjack Rachmaninoff Plays Blackjack] [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play]
* [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])


By [[Joseph Pehrson]]
; [[Gene Ward Smith]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 Black and Jill]
* ''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
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 Blacklight]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 Blackjack]


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