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{{Breadcrumb|Miracle}}
This page is an alternative to [[Miracle extensions]], describing the extensions as ways to collapse [[mirage]] back into a rank-2.


The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
{{Editable user page}}
* '''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]].
* '''Miraculous''' (31 & 41)
* '''Benediction''' (31 & 72)
* '''Manna''' (41 & 72)
* '''Semimiracle''' (62 & 82)
* '''Hemisecordite''' (41 & 62)
* '''31 & 82'''
* '''Phicordial''' (103 & 113)


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.
{{stub}}
 
All of these 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. 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].
 
For technical information see [[Gamelismic clan #Miracle]].
 
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
|-
! rowspan="2" | 11-limit
! colspan="3" | 17-limit extensions
|-
! Miraculous
! Benediction
! Manna
|-
| 0
| 0.0
| '''1/1'''
|
|
|
|-
| 1
| 116.6
| 15/14, '''16/15'''
| 14/13, '''17/16'''
|
|
|-
| 2
| 233.3
| '''8/7'''
| 15/13, 17/15
|
|
|-
| 3
| 349.9
| 11/9
| '''16/13''', 17/14, 21/17
|
|
|-
| 4
| 466.6
| '''21/16'''
| 13/10, 17/13
| 17/13
| 17/13
|-
| 5
| 583.2
| 7/5
| 24/17
|
|
|-
| 6
| 699.9
| '''3/2'''
|
|
|
|-
| 7
| 816.5
| '''8/5'''
| 21/13
|
|
|-
| 8
| 933.2
| 12/7
| 17/10
|
|
|-
| 9
| 1049.8
| 11/6
| 24/13
|
|
|-
| 10
| 1166.5
| 49/25, 55/28, 63/32, <br>88/45, 96/49, 108/55
| 39/20, 51/26, 77/39, <br>128/65, 135/68, 168/85
| 51/26, 100/51
| 51/26, 65/33
|-
| 11
| 83.1
| 21/20, 22/21
| 18/17, 26/25
|
|
|-
| 12
| 199.8
| '''9/8'''
|
|
|
|-
| 13
| 316.4
| 6/5
|
|
|
|-
| 14
| 433.1
| 9/7
| 22/17
|
|
|-
| 15
| 549.7
| '''11/8'''
| 18/13
|
|
|-
| 16
| 666.3
| 22/15
|
|
|
|-
| 17
| 783.0
| 11/7
|
|
|
|-
| 18
| 899.6
| 27/16, 42/25
| 22/13
|
|
|-
| 19
| 1016.3
| 9/5
|
|
|
|-
| 20
| 1132.9
| 27/14, 48/25
|
|
| 52/27
|-
| 21
| 49.6
| 33/32, 36/35
| 27/26
| 35/34, 40/39
| 34/33
|-
| 22
| 166.2
| 11/10
|
|
|
|-
| 23
| 282.9
| 33/28
|
| 20/17
| 13/11
|-
| 24
| 399.5
| 44/35
|
|
|
|-
| 25
| 516.2
| 27/20
|
|
|
|-
| 26
| 632.8
| 36/25
|
|
| 13/9
|-
| 27
| 749.5
| 54/35, 77/50
|
| 20/13
| 17/11
|-
| 28
| 866.1
| 33/20
|
| 28/17
|
|-
| 29
| 982.8
| 44/25
|
| 30/17
|
|-
| 30
| 1099.4
| 66/35
|
| '''32/17'''
| 17/9
|-
| 31
| 16.1
| 81/80, 99/98, 121/120
| 66/65
| 105/104, 120/119, 136/135, <br>144/143, 154/153, 170/169
| 65/64, 78/77, <br>85/84, 91/90
|-
| 32
| 132.7
| 27/25
|
| 14/13
| 13/12
|-
| 33
| 249.3
| 81/70
|
| 15/13
| 52/45
|-
| 34
| 366.0
| 99/80
|
| '''16/13''', 21/17
| 26/21
|-
| 35
| 482.6
| 33/25
|
|
|
|-
| 36
| 599.3
| 99/70
|
| 24/17
| 17/12
|-
| 37
| 715.9
| 121/80
|
|
|
|-
| 38
| 832.6
| 121/75
|
| 21/13
| '''13/8''', 34/21
|-
| 39
| 949.2
| 121/70
|
| 45/26
| 26/15
|-
| 40
| 1065.9
| 231/125
|
| 24/13
| 13/7
|-
| 41
| 1182.5
| 99/50
|
| 77/39, 128/65, <br>168/85, 180/91
| 119/60, 143/72, 135/68, <br>153/77, 169/85, 195/98
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced
 
== Tunings ==
=== Norm-based tunings ===
* 5-limit POTE: ~16/15 = 116.673{{c}}
* 7-limit POTE: ~15/14 = 116.675{{c}}
* 11-limit POTE
** Miracle: ~15/14 = 116.633{{c}}
** Revelation: ~15/14 = 116.277{{c}}
* 13-limit POTE
** Miraculous: ~15/14 = 116.747{{c}}
** Benediction: ~15/14 = 116.574{{c}}
** Manna: ~15/14 = 116.739{{c}}
** Revelation: ~15/14 = 116.268{{c}}
 
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (miracle)
|-
! 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 }}
|}
 
{| 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 ===
==== Miraculous ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|
| 17/16
| 104.955
|
|-
|
| 17/15
| 108.343
|
|-
|
| 15/8
| 111.731
|
|-
|
| 17/14
| 112.043
|
|-
|
| 13/10
| 113.553
|
|-
|
| 17/10
| 114.830
|
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
|
| 17/13
| 116.107
|
|-
| 3\31
|
| 116.129
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 15/11
| 116.441
|
|-
|
| 7/5
| 116.502
|
|-
| 10\103
|
| 116.505
| 103fg val
|-
| 17\175
|
| 116.571
| 175ffggg val
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 11/6
| 116.596
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
| 7\72
|
| 116.667
| 72fg val
|-
|
| 9/5
| 116.716
| 9-, 11- and 13-odd-limit minimax
|-
|
| 11/8
| 116.755
|
|-
| 18\185
|
| 116.757
| 185cffggg val
|-
|
| 9/7
| 116.792
|
|-
| 11\113
|
| 116.814
| 113fgg val
|-
|
| 3/2
| 116.993
| 15-odd-limit minimax
|-
| 4\41
|
| 117.073
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 13/11
| 117.266
|
|-
|
| 13/9
| 117.559
|
|-
|
| 17/11
| 117.597
|
|-
|
| 13/12
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
|
| 21/17
| 121.942
|
|-
|
| 15/13
| 123.871
|
|-
|
| 13/7
| 128.298
|
|}
 
==== Benediction ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
|
| 17/13
| 116.107
|
|-
| 3\31
|
| 116.129
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 15/11
| 116.441
|
|-
|
| 13/8
| 116.455
|
|-
|
| 17/16
| 116.501
|
|-
|
| 7/5
| 116.502
|
|-
| 10\103
|
| 116.505
|
|-
|
| 13/7
| 116.509
|
|-
|
| 13/10
| 116.511
|
|-
|
| 13/12
| 116.536
|
|-
|
| 13/11
| 116.547
|
|-
|
| 17/14
| 116.567
|
|-
| 17\175
|
| 116.571
| 175f val
|-
|
| 17/10
| 116.581
|
|-
|
| 17/12
| 116.583
|
|-
|
| 17/11
| 116.586
|
|-
|
| 5/3
| 116.588
| 5-, 7- and 15-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 13/9
| 116.595
| 13-odd-limit minimax
|-
|
| 11/6
| 116.596
|
|-
|
| 15/13
| 116.598
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
|
| 17/9
| 116.642
|
|-
|
| 21/17
| 116.642
|
|-
|
| 17/15
| 116.666
|
|-
| 7\72
|
| 116.667
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 9/5
| 116.716
| 9- and 11-odd-limit minimax
|-
|
| 11/8
| 116.755
|
|-
| 18\185
|
| 116.757
| 185cfffgg val
|-
|
| 9/7
| 116.792
|
|-
| 11\113
|
| 116.814
| 113ffg val
|-
|
| 3/2
| 116.993
|
|-
| 4\41
|
| 117.073
| 41fg val, upper bound of 11-odd-limit diamond monotone
|-
|
| 15/14
| 119.443
|
|}
 
==== Manna ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
|
| 17/13
| 116.107
|
|-
| 3\31
|
| 116.129
| 31fg val, lower bound of 11-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 15/11
| 116.441
|
|-
|
| 7/5
| 116.502
|
|-
| 10\103
|
| 116.505
| 103ffgg val
|-
| 17\175
|
| 116.571
| 175fffgg val
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 11/6
| 116.596
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
| 7\72
|
| 116.667
| 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
|
|-
|
| 9/5
| 116.716
| 9- and 11-odd-limit minimax
|-
|
| 15/13
| 116.725
| 15-odd-limit minimax
|-
|
| 17/14
| 116.730
|
|-
|
| 17/12
| 116.750
|
|-
|
| 11/8
| 116.755
|
|-
| 18\185
|
| 116.757
| 185cf val
|-
|
| 13/10
| 116.760
| 13-odd-limit minimax
|-
|
| 17/16
| 116.785
|
|-
|
| 9/7
| 116.792
|
|-
|
| 13/7
| 116.79254
|
|-
|
| 13/9
| 116.79299
|
|-
|
| 17/11
| 116.801
|
|-
| 11\113
|
| 116.814
|
|-
|
| 13/12
| 116.830
|
|-
|
| 13/8
| 116.856
|
|-
|
| 13/11
| 116.922
|
|-
|
| 3/2
| 116.993
|
|-
| 4\41
|
| 117.073
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 15/14
| 119.443
|
|}
 
==== Revelation ====
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 13/10
| 113.553
|
|-
| 2\21
|
| 114.286
|
|-
|
| 13/11
| 114.555
|
|-
|
| 11/10
| 115.000
|
|-
| 5\52
|
| 115.385
| 52f val
|-
|
| 11/7
| 115.536
|
|-
|
| 11/8
| 115.543
|
|-
|
| 7/4
| 115.587
|
|-
|
| 15/11
| 115.797
|
|-
|
| 11/6
| 115.938
|
|-
| 3\31
|
| 116.129
| 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
|-
|
| 11/9
| 116.164
| 11-, 13- and 15-odd-limit minimax
|-
|
| 5/4
| 116.241
|
|-
|
| 7/5
| 116.502
|
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit minimax
|-
|
| 7/6
| 116.641
|
|-
| 7\72
|
| 116.667
| 72ee val
|-
|
| 9/5
| 116.716
| 9-odd-limit minimax
|-
|
| 9/7
| 116.792
|
|-
|
| 3/2
| 116.993
|
|-
| 4\41
|
| 117.073
| 41ef val
|-
|
| 13/9
| 117.559
|
|-
|
| 13/12
| 117.936
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
|
| 15/13
| 123.871
|
|-
|
| 13/7
| 128.298
|
|}


[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/User:Squib/Drafts/Miracle_extensions_and_mirage"