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


In addition, we also consider the only alternative 11-limit extension:  
In addition, we also consider the only alternative 11-limit extension:  
* '''Revelation''' (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.  
* '''Revelation''' ({{nowrap|21 & 31}}) – tempering out 66/65, 99/98, 105/104, and 512/507.  


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]].  
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]].  


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.  
Miraculous, benediction, and manna can all 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.  


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]].  
Line 328: Line 330:


== Tunings ==
== Tunings ==
* 5-limit POTE: ~15/14 = 116.673
=== Prime-optimized tunings ===
* 7-limit POTE: ~15/14 = 116.675
* 5-limit POTE: ~16/15 = 116.673{{c}}
* 7-limit POTE: ~15/14 = 116.675{{c}}
* 11-limit POTE
* 11-limit POTE
** Miracle: ~15/14 = 116.633
** Miracle: ~15/14 = 116.633{{c}}
** Revelation: ~15/14 = 116.277
** Revelation: ~15/14 = 116.277{{c}}
* 13-limit POTE
* 13-limit POTE
** Miraculous: ~15/14 = 116.747
** Miraculous: ~15/14 = 116.747{{c}}
** Benediction: ~15/14 = 116.574
** Benediction: ~15/14 = 116.574{{c}}
** Manna: ~15/14 = 116.739
** Manna: ~15/14 = 116.739{{c}}
** Revelation: ~15/14 = 116.268
** Revelation: ~15/14 = 116.268{{c}}
 
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (miracle)
|-
! 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 (miracle)
|-
! 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 center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (miraculous)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
|-
| 15-odd-limit
| ~15/14 = 116.993{{c}}
| 3/2
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (miraculous)
|-
! Target
! Generator
! Eigenmonzo
|-
| 13-odd-limit
| ~15/14 = 116.846{{c}}
| {{Monzo| 0 141 -70 -35 84 -42 }}
|-
| 15-odd-limit
| ~15/14 = 116.820{{c}}
| {{Monzo| 0 127 -84 -36 100 -44 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (benediction)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.595{{c}}
| 13/9
|-
| 15-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (benediction)
|-
! Target
! Generator
! Eigenmonzo
|-
| 13-odd-limit
| ~15/14 = 116.56309{{c}}
| {{Monzo| 0 -234 39 4 -115 228 }}
|-
| 15-odd-limit
| ~15/14 = 116.56348{{c}}
| {{Monzo| 0 -251 22 5 -131 261 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (manna)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.760{{c}}
| 13/10
|-
| 15-odd-limit
| ~15/14 = 116.725{{c}}
| 15/13
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (manna)
|-
! Target
! Generator
! Eigenmonzo
|-
| 13-odd-limit
| ~15/14 = 116.780{{c}}
| {{Monzo| 0 18 -111 -76 43 204 }}
|-
| 15-odd-limit
| ~15/14 = 116.764{{c}}
| {{Monzo| 0 -37 -166 -77 59 243 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (revelation)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
|-
| 13-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
|-
| 15-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (revelation)
|-
! Target
! Generator
! Eigenmonzo
|-
| 11-odd-limit
| ~15/14 = 116.198{{c}}
| {{Monzo| 0 -195 35 5 89 }}
|-
| 13-odd-limit
| ~15/14 = 116.249{{c}}
| {{Monzo| 0 -234 39 4 102 11 }}
|-
| 15-odd-limit
| ~15/14 = 116.229{{c}}
| {{Monzo| 0 -251 22 5 117 13 }}
|}


=== Tuning spectra ===
=== Tuning spectra ===
Line 344: Line 532:
|-
|-
! 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 356: Line 559:
| 13/10
| 13/10
| 113.553
| 113.553
|
|-
|
| 17/10
| 114.830
|  
|  
|-
|-
Line 366: Line 574:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 397: Line 610:
| 116.571
| 116.571
| 175ffggg val
| 175ffggg val
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 437: Line 640:
| 116.667
| 116.667
| 72fg val
| 72fg val
|-
|
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 11-odd-limit least squares
|-
|-
|  
|  
Line 447: Line 645:
| 116.716
| 116.716
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 472: Line 665:
| 116.814
| 116.814
| 113fgg val
| 113fgg val
|-
|
| {{monzo| 0 127 -84 -36 100 -44 }}
| 116.820
| 15-odd-limit least squares
|-
|
| {{monzo| 0 141 -70 -35 84 -42 }}
| 116.846
| 13-odd-limit least squares
|-
|-
|  
|  
Line 501: Line 684:
| 13/9
| 13/9
| 117.559
| 117.559
|
|-
|
| 17/11
| 117.597
|  
|  
|-
|-
Line 506: Line 694:
| 13/12
| 13/12
| 117.936
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|  
|  
|-
|-
Line 516: Line 714:
| 13/8
| 13/8
| 119.824
| 119.824
|
|-
|
| 21/17
| 121.942
|  
|  
|-
|-
Line 533: Line 736:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 550: Line 753:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 570: Line 778:
| 13/8
| 13/8
| 116.455
| 116.455
|
|-
|
| 17/16
| 116.501
|  
|  
|-
|-
Line 603: Line 816:
|-
|-
|  
|  
| {{monzo| 0 -234 39 4 -115 228 }}
| 17/14
| 116.56309
| 116.567
| 13-odd-limit least squares
|-
|  
|  
| {{monzo| 0 -251 22 5 -131 261 }}
| 116.56348
| 15-odd-limit least squares
|-
|-
| 17\175
| 17\175
Line 618: Line 826:
|-
|-
|  
|  
| {{monzo| 0 -27 25 5 }}
| 17/10
| 116.573
| 116.581
| 7-odd-limit least squares
|
|-
|
| 17/12
| 116.583
|  
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| 17/11
| 116.578
| 116.586
| 5-odd-limit least squares
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5-, 7- and 15-odd-limit minimax
| 5-, 7- and 15-odd-limit minimax
Line 660: Line 873:
| 7/6
| 7/6
| 116.641
| 116.641
|
|-
|
| 17/9
| 116.642
|
|-
|
| 21/17
| 116.642
|
|-
|
| 17/15
| 116.666
|  
|  
|-
|-
Line 666: Line 894:
| 116.667
| 116.667
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 11-odd-limit least squares
|-
|-
|  
|  
Line 676: Line 899:
| 116.716
| 116.716
| 9- and 11-odd-limit minimax
| 9- and 11-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 722: Line 940:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 739: Line 957:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 770: Line 993:
| 116.571
| 116.571
| 175fffgg val
| 175fffgg val
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 812: Line 1,025:
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 17/15
| 116.672
| 116.667
| 11-odd-limit least squares
|  
|-
|
| 21/17
| 116.689
|
|-
|
| 17/9
| 116.702
|
|-
|
| 17/10
| 116.707
|  
|-
|-
|  
|  
Line 820: Line 1,048:
| 116.716
| 116.716
| 9- and 11-odd-limit minimax
| 9- and 11-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 830: Line 1,053:
| 116.725
| 116.725
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|
| 17/14
| 116.730
|
|-
|
| 17/12
| 116.750
|
|-
|-
|  
|  
Line 847: Line 1,080:
|-
|-
|  
|  
| {{monzo| 0 -37 -166 -77 59 243 }}
| 17/16
| 116.764
| 116.785
| 15-odd-limit least squares
|-
|  
|  
| {{monzo| 0 18 -111 -76 43 204 }}
| 116.780
| 13-odd-limit least squares
|-
|-
|  
|  
Line 869: Line 1,097:
| 13/9
| 13/9
| 116.79299
| 116.79299
|
|-
|
| 17/11
| 116.801
|  
|  
|-
|-
Line 911: Line 1,144:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 979: Line 1,212:
| 116.164
| 116.164
| 11-, 13- and 15-odd-limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 -195 35 5 89 }}
| 116.198
| 11-odd-limit least squares
|-
|
| {{monzo| 0 -251 22 5 117 13 }}
| 116.229
| 15-odd-limit least squares
|-
|-
|  
|  
Line 994: Line 1,217:
| 116.241
| 116.241
|  
|  
|-
|
| {{monzo| 0 -234 39 4 102 11 }}
| 116.249
| 13-odd-limit least squares
|-
|-
|  
|  
Line 1,004: Line 1,222:
| 116.502
| 116.502
|  
|  
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 1,034: Line 1,242:
| 116.716
| 116.716
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 1,088: Line 1,291:
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Miracle_extensions"