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The basic 7-limit [[miracle]] temperament has various [[extension]]s to the 11- and 13-limit. The following temperaments are discussed in this article:
{{Breadcrumb|Miracle}}
* '''Miraculous''' (31 & 41) – tempering out 105/104, 144/143, 196/195, and 243/242;
* '''Benediction''' (31 & 41f) – tempering out 225/224, 243/242, 351/350, and 385/384;
* '''Manna''' (31f & 41f) – tempering out 225/224, 243/242, 325/324, and 385/384;


In addition, we also consider the only alternative 11-limit extension:  
The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:  
* '''Revelation''' (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.
* '''Miraculous''' ({{nowrap| 31 & 41 }}) – tempering out 105/104, 144/143, 196/195, and 243/242;
* '''Benediction''' ({{nowrap| 31 & 41f }}) – tempering out 225/224, 243/242, 351/350, and 385/384;
* '''Manna''' ({{nowrap| 31f & 41 }}) – tempering out 225/224, 243/242, 325/324, and 385/384;


As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].  
In addition, we also consider the only alternative 11-limit mapping:
* '''Revelation''' ({{nowrap| 21 & 31 }}) – tempering out 66/65, 99/98, 105/104, and 512/507.  


All of them can be extended to the 17-limit by recognizing 21/16~17/13, tempering out [[273/272]]. For miraculous it implies the generator also represents 17/16, which is supported by 72fg.  
As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing [[15/14]], and [[16/15]], goes one step further to stand in for [[~]][[14/13]], and you can find [[11/9]]~[[16/13]] just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].


Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]].  
Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.  


For technical information see [[Gamelismic clan #Miracle]].  
The 13-limit extensions can all be extended to the [[17-limit]] by recognizing [[21/16]]~[[17/13]], tempering out [[273/272]] (and many other commas such as [[715/714]] and [[833/832]]). For miraculous it implies the generator also represents [[17/16]], which is supported by 72fg. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].
 
[[225/224]] factors into ([[400/399]])·([[513/512]]) in the [[19-limit]], suggesting that miracle can be extended to include prime 19 by tempering out both commas. However, this means 31edo is no longer in the valid [[diamond monotone]] range. Alternatively, 19 can be reached by tempering out [[324/323]] and [[495/494]], which is called prism. The two merge in manna.
 
There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]]. For prism, this also tempers out [[760/759]], which conflates [[23/19]] with [[40/33]].
 
For technical information see [[Gamelismic clan #Miracle]].


== Interval chain ==
== Interval chain ==
Line 328: Line 334:


== Tunings ==
== Tunings ==
=== Prime-optimized tunings ===
=== Norm-based tunings ===
* 5-limit POTE: ~16/15 = 116.673{{c}}
* 5-limit POTE: ~16/15 = 116.673{{c}}
* 7-limit POTE: ~15/14 = 116.675{{c}}
* 7-limit POTE: ~15/14 = 116.675{{c}}
Line 341: Line 347:


=== Target tunings ===
=== Target tunings ===
* 5-odd-limit least squares: ~16/15 = 116.578{{c}} (eigenmonzo: {{monzo| 0 -19 20 }})
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
* 7-odd-limit least squares: ~15/14 = 116.573{{c}} (eigenmonzo: {{monzo| 0 -27 25 5 }})
|+ style="white-space: nowrap;" | Target tunings (miracle)
* 9-odd-limit least squares: ~15/14 = 116.721{{c}} (eigenmonzo: {{monzo| 0 117 -44 -19 }})
|-
* 11-odd-limit least squares
! rowspan="2" | Target
** Miracle: ~15/14 = 116.672{{c}} (eigenmonzo: {{monzo| 0 17 -11 -6 11 }})
! colspan="2" | Minimax
** Revelation: ~15/14 = 116.198{{c}} (eigenmonzo: {{monzo| 0 -195 35 5 89 }})
! colspan="2" | Least squares
* 13-odd-limit least squares
|-
** Miraculous: ~15/14 = 116.846{{c}} (eigenmonzo: {{monzo| 0 141 -70 -35 84 -42 }})
! Generator
** Benediction: ~15/14 = 116.56309{{c}} (eigenmonzo: {{monzo| 0 -234 39 4 -115 228 }})
! Eigenmonzo*
** Manna: ~15/14 = 116.780{{c}} (eigenmonzo: {{monzo| 0 18 -111 -76 43 204 }})
! Generator
** Revelation: ~15/14 = 116.249{{c}} (eigenmonzo: {{monzo| 0 -234 39 4 102 11 }})
! Eigenmonzo*
* 15-odd-limit least squares
|-
** Miraculous: ~15/14 = 116.820{{c}} (eigenmonzo: {{monzo| 0 127 -84 -36 100 -44 }})
| 5-odd-limit
** Benediction: ~15/14 = 116.56348{{c}} (eigenmonzo: {{monzo| 0 -251 22 5 -131 261 }})
| ~16/15 = 116.588{{c}}
** Manna: ~15/14 = 116.764{{c}} (eigenmonzo: {{monzo| 0 -37 -166 -77 59 243 }})
| 5/3
** Revelation: ~15/14 = 116.229{{c}} (eigenmonzo: {{monzo| 0 -251 22 5 117 13 }})
| ~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 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (miraculous)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.846{{c}}
| {{Monzo| 0 141 -70 -35 84 -42 }}
|-
| 15-odd-limit
| ~15/14 = 116.993{{c}}
| 3/2
| ~15/14 = 116.820{{c}}
| {{Monzo| 0 127 -84 -36 100 -44 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (benediction)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.595{{c}}
| 13/9
| ~15/14 = 116.56309{{c}}
| {{Monzo| 0 -234 39 4 -115 228 }}
|-
| 15-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
| ~15/14 = 116.56348{{c}}
| {{Monzo| 0 -251 22 5 -131 261 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (manna)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.760{{c}}
| 13/10
| ~15/14 = 116.780{{c}}
| {{Monzo| 0 18 -111 -76 43 204 }}
|-
| 15-odd-limit
| ~15/14 = 116.725{{c}}
| 15/13
| ~15/14 = 116.764{{c}}
| {{Monzo| 0 -37 -166 -77 59 243 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (revelation)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
| ~15/14 = 116.198{{c}}
| {{Monzo| 0 -195 35 5 89 }}
|-
| 13-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
| ~15/14 = 116.249{{c}}
| {{Monzo| 0 -234 39 4 102 11 }}
|-
| 15-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
| ~15/14 = 116.229{{c}}
| {{Monzo| 0 -251 22 5 117 13 }}
|}


=== Tuning spectra ===
=== Tuning spectra ===
Line 363: Line 495:
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 17/16
| 104.955
|
|-
|
| 17/15
| 108.343
|
|-
|-
|  
|  
| 15/8
| 15/8
| 111.731
| 111.731
|
|-
|
| 17/14
| 112.043
|  
|  
|-
|-
Line 375: Line 522:
| 13/10
| 13/10
| 113.553
| 113.553
|
|-
|
| 17/10
| 114.830
|  
|  
|-
|-
Line 385: Line 537:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 490: Line 647:
| 13/9
| 13/9
| 117.559
| 117.559
|
|-
|
| 17/11
| 117.597
|  
|  
|-
|-
Line 495: Line 657:
| 13/12
| 13/12
| 117.936
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|  
|  
|-
|-
Line 505: Line 677:
| 13/8
| 13/8
| 119.824
| 119.824
|
|-
|
| 21/17
| 121.942
|  
|  
|-
|-
Line 522: Line 699:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 539: Line 716:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 559: Line 741:
| 13/8
| 13/8
| 116.455
| 116.455
|
|-
|
| 17/16
| 116.501
|  
|  
|-
|-
Line 589: Line 776:
| 13/11
| 13/11
| 116.547
| 116.547
|
|-
|
| 17/14
| 116.567
|  
|  
|-
|-
Line 597: Line 789:
|-
|-
|  
|  
| 6/5
| 17/10
| 116.581
|
|-
|
| 17/12
| 116.583
|
|-
|
| 17/11
| 116.586
|
|-
|
| 5/3
| 116.588
| 116.588
| 5-, 7- and 15-odd-limit minimax
| 5-, 7- and 15-odd-limit minimax
Line 629: Line 836:
| 7/6
| 7/6
| 116.641
| 116.641
|
|-
|
| 17/9
| 116.642
|
|-
|
| 21/17
| 116.642
|
|-
|
| 17/15
| 116.666
|  
|  
|-
|-
Line 681: Line 903:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 698: Line 920:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 759: Line 986:
| 116.667
| 116.667
| Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 17/15
| 116.667
|
|-
|
| 21/17
| 116.689
|
|-
|
| 17/9
| 116.702
|
|-
|
| 17/10
| 116.707
|
|-
|-
|  
|  
Line 769: Line 1,016:
| 116.725
| 116.725
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|
| 17/14
| 116.730
|
|-
|
| 17/12
| 116.750
|
|-
|-
|  
|  
Line 784: Line 1,041:
| 116.760
| 116.760
| 13-odd-limit minimax
| 13-odd-limit minimax
|-
|
| 17/16
| 116.785
|
|-
|-
|  
|  
Line 798: Line 1,060:
| 13/9
| 13/9
| 116.79299
| 116.79299
|
|-
|
| 17/11
| 116.801
|  
|  
|-
|-
Line 840: Line 1,107:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 987: Line 1,254:
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Miracle_extensions"