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{{Breadcrumb|Miracle}}
{{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 [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[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 & 41 }}) – 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 mapping:  
* '''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]].


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]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.  


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]].  
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]].  


For technical information see [[Gamelismic clan #Miracle]].  
[[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 330: 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 343: Line 347:


=== Target tunings ===
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (miracle)
|+ style="white-space: nowrap;" | Target tunings (miracle)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
! Target
! Generator
! Eigenmonzo*
! Generator
! Generator
! Eigenmonzo*
! Eigenmonzo*
Line 353: Line 362:
| ~16/15 = 116.588{{c}}
| ~16/15 = 116.588{{c}}
| 5/3
| 5/3
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
|-
| 7-odd-limit
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| ~15/14 = 116.588{{c}}
| 5/3
| 5/3
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
|-
| 9-odd-limit
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| ~15/14 = 116.716{{c}}
| 9/5
| 9/5
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
|-
| 11-odd-limit
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| ~15/14 = 116.716{{c}}
| 9/5
| 9/5
| ~15/14 = 116.672{{c}}
| {{Monzo| 0 17 -11 -6 11 }}
|}
|}


{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (miracle)
|+ style="white-space: nowrap;" | Target tunings (miraculous)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
! Target
! Generator
! Generator
! Eigenmonzo*
! 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
! Generator
! Eigenmonzo*
! Eigenmonzo*
Line 401: Line 399:
| ~15/14 = 116.716{{c}}
| ~15/14 = 116.716{{c}}
| 9/5
| 9/5
| ~15/14 = 116.846{{c}}
| {{Monzo| 0 141 -70 -35 84 -42 }}
|-
|-
| 15-odd-limit
| 15-odd-limit
| ~15/14 = 116.993{{c}}
| ~15/14 = 116.993{{c}}
| 3/2
| 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}}
| ~15/14 = 116.820{{c}}
| {{Monzo| 0 127 -84 -36 100 -44 }}
| {{Monzo| 0 127 -84 -36 100 -44 }}
|}
|}


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (benediction)
|+ style="white-space: nowrap;" | Target tunings (benediction)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
! Target
! Generator
! Eigenmonzo*
! Generator
! Generator
! Eigenmonzo*
! Eigenmonzo*
Line 433: Line 424:
| ~15/14 = 116.595{{c}}
| ~15/14 = 116.595{{c}}
| 13/9
| 13/9
| ~15/14 = 116.56309{{c}}
| {{Monzo| 0 -234 39 4 -115 228 }}
|-
|-
| 15-odd-limit
| 15-odd-limit
| ~15/14 = 116.588{{c}}
| ~15/14 = 116.588{{c}}
| 5/3
| 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}}
| ~15/14 = 116.56348{{c}}
| {{Monzo| 0 -251 22 5 -131 261 }}
| {{Monzo| 0 -251 22 5 -131 261 }}
|}
|}


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (manna)
|+ style="white-space: nowrap;" | Target tunings (manna)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
! Target
! Generator
! Eigenmonzo*
! Generator
! Generator
! Eigenmonzo*
! Eigenmonzo*
Line 465: Line 449:
| ~15/14 = 116.760{{c}}
| ~15/14 = 116.760{{c}}
| 13/10
| 13/10
| ~15/14 = 116.780{{c}}
| {{Monzo| 0 18 -111 -76 43 204 }}
|-
|-
| 15-odd-limit
| 15-odd-limit
| ~15/14 = 116.725{{c}}
| ~15/14 = 116.725{{c}}
| 15/13
| 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}}
| ~15/14 = 116.764{{c}}
| {{Monzo| 0 -37 -166 -77 59 243 }}
| {{Monzo| 0 -37 -166 -77 59 243 }}
|}
|}


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (revelation)
|+ style="white-space: nowrap;" | Target tunings (revelation)
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
|-
! Target
! Generator
! Eigenmonzo*
! Generator
! Generator
! Eigenmonzo*
! Eigenmonzo*
Line 497: Line 474:
| ~15/14 = 116.164{{c}}
| ~15/14 = 116.164{{c}}
| 11/9
| 11/9
| ~15/14 = 116.198{{c}}
| {{Monzo| 0 -195 35 5 89 }}
|-
|-
| 13-odd-limit
| 13-odd-limit
| ~15/14 = 116.164{{c}}
| ~15/14 = 116.164{{c}}
| 11/9
| 11/9
| ~15/14 = 116.249{{c}}
| {{Monzo| 0 -234 39 4 102 11 }}
|-
|-
| 15-odd-limit
| 15-odd-limit
| ~15/14 = 116.164{{c}}
| ~15/14 = 116.164{{c}}
| 11/9
| 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}}
| ~15/14 = 116.229{{c}}
| {{Monzo| 0 -251 22 5 117 13 }}
| {{Monzo| 0 -251 22 5 117 13 }}
Line 532: Line 495:
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 736: Line 699:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 940: Line 903:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 1,144: Line 1,107:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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