Xenharmonic Wiki

Miracle: Difference between revisions

← Older edit
VisualWikitext
→Music: +SoundClick links
m Cleanup on infobox
 
(35 intermediate revisions by 6 users not shown)
Line 1: Line 1:
[[de:Miracle]]
{{Interwiki
| en = Miracle
| de = Miracle
}}
{{Infobox regtemp
| Title = Miracle
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[225/224]], [[1029/1024]] (7-limit); <br>[[225/224]], [[243/242]], [[385/384]] (11-limit)
| Edo join 1 = 31 | Edo join 2 = 41
| Generators = 15/14 | Generators tuning = 116.7 | Optimization method = CTE
| MOS scales = …, [[1L 9s]], [[10L 1s]], [[10L 11s]], [[10L 21s]]
| Mapping = 1; 6 -7 -2 15
| Pergen = (P8, P5/6)
| Odd limit 1 = 9 | Mistuning 1 = 3.32 | Complexity 1 = 21
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 4.86 | Complexity 2 = 31
}}
{{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 is naturally a full [[11-limit]] temperament, treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]]. It is supported by the highly notable [[EDO|edos]] [[31edo|31]], [[41edo|41]], and [[72edo|72]], with 72edo being an especially good tuning. (There is an alternative mapping for 11 known as [[revelation]], but there is little reason to use it unless you are using [[31edo]], in which case it is identical to miracle anyway.)


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.  
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 (&rarr; [[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]].
[[Rastmic clan|Some temperaments]] have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and [[Gamelismic clan|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 ==
11-odd-limit ratios are labeled in '''bold'''.  
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"
Line 18: Line 33:
! #
! #
! Cents*
! Cents*
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
Line 25: Line 40:
|-
|-
| 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
| 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
| 83.8
| 83.1
| 22/21, 21/20
| 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>*</nowiki> in 11-limit [[CTE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


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


== Scales ==
== Scales ==
Line 174: Line 189:
* [[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.
 
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~15/14 = 116.5155{{c}}
| CSEE: ~15/14 = 116.5612{{c}}
| POEE: ~15/14 = 116.6465{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.6772{{c}}
| CWE: ~15/14 = 116.6756{{c}}
| POTE: ~15/14 = 116.6752{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~15/14 = 116.7297{{c}}
| CSBE: ~15/14 = 116.7136{{c}}
| POBE: ~15/14 = 116.6903{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~15/14 = 116.6868{{c}}
| CSEE: ~15/14 = 116.6304{{c}}
| POEE: ~15/14 = 116.5817{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.7112{{c}}
| CWE: ~15/14 = 116.6469{{c}}
| POTE: ~15/14 = 116.6327{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~15/14 = 116.7355{{c}}
| CSBE: ~15/14 = 116.6768{{c}}
| POBE: ~15/14 = 116.6643{{c}}
|}


=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.588{{c}}
| 5/3
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~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 338:
|  
|  
|-
|-
| [[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
|-
|-
|  
|  
| 5/3
| 5/3
| 116.588
| 116.588
| 5- and 7-odd-limit minimax
| 5- and 7-odd-limit, 11-limit 15- and 21-odd-limit minimax
|-
|-
|  
|  
Line 267: Line 372:
| 116.667
| 116.667
|  
|  
|-
|
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 11-odd-limit least squares
|-
|-
|  
|  
Line 277: Line 377:
| 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 391:
| 9/7
| 9/7
| 116.792
| 116.792
|
|-
| [[113edo|11\113]]
|
| 116.814
|  
|  
|-
|-
Line 323: Line 423:
| Upper bound of 7- and 9-odd-limit diamond monotone
| Upper bound of 7- and 9-odd-limit diamond monotone
|}
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==
; [[Gene Ward Smith]]
; [[Herman Miller]]
* ''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
* [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) – 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
== External links ==
* [https://x31eq.com/decimal_lattice.htm ''Lattices with Decimal Notation''] by [[Graham Breed]]


[[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"