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The basic 7-limit [[Miracle|miracle]] temperament has various extensions to the 11 and 13 limits.
{{Breadcrumb|Miracle}}


== Tuning spectra ==
The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
* '''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;


=== Miraculous ===
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.


Gencom: [2 15/14; 105/104 144/143 196/195 275/273]
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]].


Gencom mapping: [{{map|1 1 3 3 2 4}}, {{map|0 6 -7 -2 15 -3}}]
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.


{| class="wikitable center-all"
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 ==
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'''
|
|
|
|-
|-
! | ET<br>generator
| 7
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 816.5
! | secor<br>(¢)
| '''8/5'''
! | comments
| 21/13
|  
|  
|-
|-
| |
| 8
| | 16/15
| 933.2
| | 111.731
| 12/7
| |
| 17/10
|  
|  
|-
|-
| |
| 9
| | 13/10
| 1049.8
| | 113.553
| 11/6
| |
| 24/13
|  
|  
|-
|-
| |
| 10
| | 8/7
| 1166.5
| | 115.587
| 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
| | 11/9
| 83.1
| | 115.803
| 21/20, 22/21
| |
| 18/17, 26/25
|  
|  
|-
|-
| | 3\31
| 12
| |
| 199.8
| | 116.129
| '''9/8'''
| |
|  
|  
|  
|-
|-
| |
| 13
| | 5/4
| 316.4
| | 116.241
| 6/5
| |
|  
|  
|  
|-
|-
| |
| 14
| | 15/11
| 433.1
| | 116.441
| 9/7
| |
| 22/17
|  
|  
|-
|-
| |
| 15
| | 7/5
| 549.7
| | 116.502
| '''11/8'''
| |
| 18/13
|  
|  
|-
|-
| | 10\103
| 16
| |
| 666.3
| | 116.505
| 22/15
| |
|  
|  
|  
|-
|-
| | 17\175
| 17
| |
| 783.0
| | 116.571
| 11/7
| |
|  
|  
|  
|-
|-
| |
| 18
| | {{vector|0 -27 25 5}}
| 899.6
| | 116.573
| 27/16, 42/25
| | 7 limit least squares
| 22/13
|  
|  
|-
|-
| |
| 19
| | {{vector|0 -19 20}}
| 1016.3
| | 116.578
| 9/5
| | 5 limit least squares
|  
|  
|  
|-
|-
| |
| 20
| | 6/5
| 1132.9
| | 116.588
| 27/14, 48/25
| | 5 and 7 limit minimax
|  
|  
| 52/27
|-
|-
| |
| 21
| | 11/10
| 49.6
| | 116.591
| 33/32, 36/35
| |
| 27/26
| 35/34, 40/39
| 34/33
|-
|-
| |
| 22
| | 12/11
| 166.2
| | 116.596
| 11/10
| |
|  
|  
|  
|-
|-
| |
| 23
| | 14/11
| 282.9
| | 116.617
| 33/28
| |
|  
| 20/17
| 13/11
|-
|-
| |
| 24
| |
| 399.5
| | 116.633
| 44/35
| | POTE11
|  
|  
|  
|-
|-
| |
| 25
| | 7/6
| 516.2
| | 116.641
| 27/20
| |
|  
|  
|  
|-
|-
| | 7\72
| 26
| |
| 632.8
| | 116.667
| 36/25
| |
|  
|  
| 13/9
|-
|-
| |
| 27
| | {{vector|0 17 -11 -6 11}}
| 749.5
| | 116.672
| 54/35, 77/50
| | 11 limit least squares
|  
| 20/13
| 17/11
|-
|-
| |
| 28
| |
| 866.1
| | 116.673
| 33/20
| | POTE5
|  
| 28/17
|  
|-
|-
| |
| 29
| |
| 982.8
| | 116.675
| 44/25
| | POTE7
|  
| 30/17
|  
|-
|-
| |
| 30
| | 10/9
| 1099.4
| | 116.716
| 66/35
| | 9, 11 and 13 limit minimax
|  
| '''32/17'''
| 17/9
|-
|-
| |
| 31
| | {{vector|0 117 -44 -19}}
| 16.1
| | 116.721
| 81/80, 99/98, 121/120
| | 9 limit least squares
| 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
| | 116.747
| 27/25
| | POTE13
|  
| 14/13
| 13/12
|-
|-
| |
| 33
| | 11/8
| 249.3
| | 116.755
| 81/70
| |
|  
| 15/13
| 52/45
|-
|-
| |
| 34
| | 9/7
| 366.0
| | 116.792
| 99/80
| |
|  
| '''16/13''', 21/17
| 26/21
|-
|-
| | 11\113
| 35
| |
| 482.6
| | 116.814
| 33/25
| |
|  
|  
|  
|-
|-
| |
| 36
| | {{vector|0 127 -84 -36 100 -44}}
| 599.3
| | 116.820
| 99/70
| | 15 limit least squares
|  
| 24/17
| 17/12
|-
|-
| |
| 37
| | {{vector|0 141 -70 -35 84 -42}}
| 715.9
| | 116.846
| 121/80
| | 13 limit least squares
|  
|  
|  
|-
|-
| |
| 38
| | 4/3
| 832.6
| | 116.993
| 121/75
| | 15 limit minimax
|  
| 21/13
| '''13/8''', 34/21
|-
|-
| | 4\41
| 39
| |
| 949.2
| | 117.073
| 121/70
| |
|  
| 45/26
| 26/15
|-
|-
| |
| 40
| | 13/11
| 1065.9
| | 117.266
| 231/125
| |
|  
| 24/13
| 13/7
|-
|-
| |
| 41
| | 18/13
| 1182.5
| | 117.559
| 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
| | 13/12
! colspan="2" | Minimax
| | 117.936
! colspan="2" | Least squares
| |
|-
|-
| |
! Generator
| | 15/14
! Eigenmonzo*
| | 119.443
! Generator
| |
! Eigenmonzo*
|-
|-
| |
| 5-odd-limit
| | 16/13
| ~16/15 = 116.588{{c}}
| | 119.824
| 5/3
| |
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
|-
| |
| 7-odd-limit
| | 15/13
| ~15/14 = 116.588{{c}}
| | 123.871
| 5/3
| |
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
|-
| |
| 9-odd-limit
| | 14/13
| ~15/14 = 116.716{{c}}
| | 128.298
| 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 }}
|}
|}


=== Benediction ===
{| 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 }}
|}


Gencom: [2 15/14; 225/224 243/242 351/350 441/440]
{| 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 }}
|}


Gencom mapping: [{{map|1 1 3 3 2 7}}, {{map|0 6 -7 -2 15 -34}}]
{| 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"
{| 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
|-
|-
! | ET<br>generator
! Generator
! | eigenmonzo<br>(unchanged interval)
! Eigenmonzo*
! | secor<br>(¢)
! Generator
! | comments
! Eigenmonzo*
|-
|-
| |
| 11-odd-limit
| | 16/15
| ~15/14 = 116.164{{c}}
| | 111.731
| 11/9
| |
| ~15/14 = 116.198{{c}}
| {{Monzo| 0 -195 35 5 89 }}
|-
|-
| |
| 13-odd-limit
| | 8/7
| ~15/14 = 116.164{{c}}
| | 115.587
| 11/9
| |
| ~15/14 = 116.249{{c}}
| {{Monzo| 0 -234 39 4 102 11 }}
|-
|-
| |
| 15-odd-limit
| | 11/9
| ~15/14 = 116.164{{c}}
| | 115.803
| 11/9
| |
| ~15/14 = 116.229{{c}}
| {{Monzo| 0 -251 22 5 117 13 }}
|}
 
=== Tuning spectra ===
==== Miraculous ====
{| class="wikitable center-all left-4"
|-
|-
| | 3\31
! Edo<br>generator
| |
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
| | 116.129
! Generator (¢)
| |
! Comments
|-
|-
| |
|  
| | 5/4
| 17/16
| | 116.241
| 104.955
| |
|  
|-
|-
| |
|  
| | 15/11
| 17/15
| | 116.441
| 108.343
| |
|  
|-
|-
| |
|  
| | 16/13
| 15/8
| | 116.455
| 111.731
| |
|  
|-
|-
| |
|  
| | 7/5
| 17/14
| | 116.502
| 112.043
| |
|  
|-
|-
| | 10\103
|  
| |
| 13/10
| | 116.505
| 113.553
| |
|  
|-
|-
| |
|  
| | 14/13
| 17/10
| | 116.509
| 114.830
| |
|  
|-
|-
| |
|  
| | 13/10
| 7/4
| | 116.511
| 115.587
| |
|  
|-
|-
| |
|  
| | 13/12
| 11/9
| | 116.536
| 115.803
| |
|  
|-
|-
| |
|  
| | 13/11
| 17/13
| | 116.547
| 116.107
| |
|  
|-
|-
| |
| 3\31
| | {{vector|0 -234 39 4 -115 228}}
|  
| | 116.56309
| 116.129
| | 13 limit least squares
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
| |
|  
| | {{vector|0 -251 22 5 -131 261}}
| 5/4
| | 116.56348
| 116.241
| | 15 limit least squares
|  
|-
|-
| | 17\175
|  
| |
| 15/11
| | 116.571
| 116.441
| |
|  
|-
|-
| |
|  
| | {{vector|0 -27 25 5}}
| 7/5
| | 116.573
| 116.502
| | 7 limit least squares
|  
|-
|-
| |
| 10\103
| |
|  
| | 116.574
| 116.505
| | POTE13
| 103fg val
|-
|-
| |
| 17\175
| | {{vector|0 -19 20}}
|  
| | 116.578
| 116.571
| | 5 limit least squares
| 175ffggg val
|-
|-
| |
|  
| | 6/5
| 5/3
| | 116.588
| 116.588
| | 5, 7 and 15 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
| |
|  
| | 11/10
| 11/10
| | 116.591
| 116.591
| |
|  
|-
|-
| |
|  
| | 18/13
| 11/6
| | 116.595
| 116.596
| | 13 limit minimax
|  
|-
|-
| |
|  
| | 12/11
| 11/7
| | 116.596
| 116.617
| |
|  
|-
|-
| |
|  
| | 15/13
| 7/6
| | 116.598
| 116.641
| |
|  
|-
|-
| |
| 7\72
| | 14/11
|  
| | 116.617
| 116.667
| |
| 72fg val
|-
|-
| |
|  
| |
| 9/5
| | 116.633
| 116.716
| | POTE11
| 9-, 11- and 13-odd-limit minimax
|-
|-
| |
|  
| | 7/6
| 11/8
| | 116.641
| 116.755
| |
|  
|-
|-
| | 7\72
| 18\185
| |
|  
| | 116.667
| 116.757
| |
| 185cffggg val
|-
|-
| |
|  
| | {{vector|0 17 -11 -6 11}}
| 9/7
| | 116.672
| 116.792
| | 11 limit least squares
|  
|-
|-
| |
| 11\113
| |
|  
| | 116.673
| 116.814
| | POTE5
| 113fgg val
|-
|-
| |
|  
| |
| 3/2
| | 116.675
| 116.993
| | POTE7
| 15-odd-limit minimax
|-
|-
| |
| 4\41
| | 10/9
|  
| | 116.716
| 117.073
| | 9 and 11 limit minimax
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
| |
|  
| | {{vector|0 117 -44 -19}}
| 13/11
| | 116.721
| 117.266
| | 9 limit least squares
|  
|-
|-
| |
|  
| | 11/8
| 13/9
| | 116.755
| 117.559
| |
|  
|-
|-
| |
|  
| | 9/7
| 17/11
| | 116.792
| 117.597
| |
|  
|-
|-
| | 11\113
|  
| |
| 13/12
| | 116.814
| 117.936
| |
|  
|-
|-
| |
|  
| | 4/3
| 17/9
| | 116.993
| 118.087
| |
|  
|-
|-
| | 4\41
|  
| |
| 17/12
| | 117.073
| 119.400
| |
|  
|-
|-
| |
|  
| | 15/14
| 15/14
| | 119.443
| 119.443
| |
|
|-
|
| 13/8
| 119.824
|
|-
|
| 21/17
| 121.942
|
|-
|
| 15/13
| 123.871
|
|-
|
| 13/7
| 128.298
|  
|}
|}


=== Manna ===
==== Benediction ====
 
{| class="wikitable center-all left-4"
Gencom: [2 15/14; 225/224 243/242 325/324 385/384]
 
Gencom mapping: [{{map|1 1 3 3 2 0}}, {{map|0 6 -7 -2 15 38}}]
 
{| class="wikitable center-all"
|-
|-
! | ET<br>generator
! Edo<br>generator
! | eigenmonzo<br>(unchanged interval)
! Unchanged interval<br>(eigenmonzo)
! | secor<br>(¢)
! Generator (¢)
! | comments
! Comments
|-
|-
| |
|  
| | 16/15
| 15/8
| | 111.731
| 111.731
| |
|  
|-
|-
| |
|  
| | 8/7
| 7/4
| | 115.587
| 115.587
| |
|  
|-
|-
| |
|  
| | 11/9
| 11/9
| | 115.803
| 115.803
| |
|  
|-
|-
| | 3\31
|  
| |
| 17/13
| | 116.129
| 116.107
| |
|  
|-
|-
| |
| 3\31
| | 5/4
|  
| | 116.241
| 116.129
| |
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
| |
|  
| | 15/11
| 5/4
| | 116.441
| 116.241
| |
|  
|-
|-
| |
|  
| | 7/5
| 15/11
| | 116.502
| 116.441
| |
|  
|-
|-
| | 10\103
|  
| |
| 13/8
| | 116.505
| 116.455
| |
|  
|-
|-
| | 17\175
|  
| |
| 17/16
| | 116.571
| 116.501
| |
|  
|-
|-
| |
|  
| | {{vector|0 -27 25 5}}
| 7/5
| | 116.573
| 116.502
| | 7 limit least squares
|  
|-
|-
| |
| 10\103
| | {{vector|0 -19 20}}
|  
| | 116.578
| 116.505
| | 5 limit least squares
|  
|-
|-
| |
|  
| | 6/5
| 13/7
| | 116.588
| 116.509
| | 5 and 7 limit minimax
|  
|-
|-
| |
|  
| | 11/10
| 13/10
| | 116.591
| 116.511
| |
|  
|-
|-
| |
|  
| | 12/11
| 13/12
| | 116.596
| 116.536
| |
|  
|-
|-
| |
|  
| | 14/11
| 13/11
| | 116.617
| 116.547
| |
|  
|-
|-
| |
|  
| |
| 17/14
| | 116.633
| 116.567
| | POTE11
|  
|-
|-
| |
| 17\175
| | 7/6
|  
| | 116.641
| 116.571
| |
| 175f val
|-
|-
| | 7\72
|  
| |
| 17/10
| | 116.667
| 116.581
| |
|  
|-
|-
| |
|  
| | {{vector|0 17 -11 -6 11}}
| 17/12
| | 116.672
| 116.583
| | 11 limit least squares
|  
|-
|-
| |
|  
| |
| 17/11
| | 116.673
| 116.586
| | POTE5
|  
|-
|-
| |
|  
| |
| 5/3
| | 116.675
| 116.588
| | POTE7
| 5-, 7- and 15-odd-limit minimax
|-
|-
| |
|  
| | 10/9
| 11/10
| | 116.716
| 116.591
| | 9 and 11 limit minimax
|  
|-
|-
| |
|  
| | {{vector|0 117 -44 -19}}
| 13/9
| | 116.721
| 116.595
| | 9 limit least squares
| 13-odd-limit minimax
|-
|-
| |
|  
| | 15/13
| 11/6
| | 116.725
| 116.596
| | 15 limit minimax
|  
|-
|-
| |
|  
| |
| 15/13
| | 116.739
| 116.598
| | POTE13
|  
|-
|-
| |
|  
| | 11/8
| 11/7
| | 116.755
| 116.617
| |
|  
|-
|-
| |
|  
| | 13/10
| 7/6
| | 116.760
| 116.641
| | 13 limit minimax
|  
|-
|-
| |
|  
| | {{vector|0 -37 -166 -77 59 243}}
| 17/9
| | 116.764
| 116.642
| | 15 limit least squares
|  
|-
|-
| |
|  
| | {{vector|0 18 -111 -76 43 204}}
| 21/17
| | 116.780
| 116.642
| | 13 limit least squares
|  
|-
|-
| |
|  
| | 9/7
| 17/15
| | 116.792
| 116.666
| |
|  
|-
|-
| |
| 7\72
| | 14/13
|  
| | 116.79254
| 116.667
| |
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
| |
|  
| | 18/13
| 9/5
| | 116.79299
| 116.716
| |
| 9- and 11-odd-limit minimax
|-
|-
| | 11\113
|  
| |
| 11/8
| | 116.814
| 116.755
| |
|  
|-
|-
| |
| 18\185
| | 13/12
|  
| | 116.830
| 116.757
| |
| 185cfffgg val
|-
|-
| |
|  
| | 16/13
| 9/7
| | 116.856
| 116.792
| |
|  
|-
|-
| |
| 11\113
| | 13/11
|  
| | 116.922
| 116.814
| |
| 113ffg val
|-
|-
| |
|  
| | 4/3
| 3/2
| | 116.993
| 116.993
| |
|  
|-
|-
| | 4\41
| 4\41
| |
|  
| | 117.073
| 117.073
| |
| 41fg val, upper bound of 11-odd-limit diamond monotone
|-
|-
| |
|  
| | 15/14
| 15/14
| | 119.443
| 119.443
| |
|  
|}
|}


=== Revelation ===
==== Manna ====
 
{| class="wikitable center-all left-4"
Gencom: [2 15/14; 66/65 99/98 105/104 1001/1000]
|-
 
! Edo<br>generator
Gencom mapping: [{{map|1 1 3 3 5 4}}, {{map|0 6 -7 -2 -16 -3}}]
! Unchanged interval<br>(eigenmonzo)
 
! Generator (¢)
{| class="wikitable center-all"
! 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
|
|-
|-
! | ET<br>generator
|  
! | eigenmonzo<br>(unchanged interval)
| 9/7
! | secor<br>(¢)
| 116.792
! | comments
|  
|-
|-
| |
|  
| | 16/15
| 13/7
| | 111.731
| 116.79254
| |
|  
|-
|-
| |
|  
| | 13/10
| 13/9
| | 113.553
| 116.79299
| |
|  
|-
|-
| |
|  
| | 13/11
| 17/11
| | 114.555
| 116.801
| |
|  
|-
|-
| |
| 11\113
| | 11/10
|  
| | 115.000
| 116.814
| |
|  
|-
|-
| |
|  
| | 14/11
| 13/12
| | 115.536
| 116.830
| |
|  
|-
|-
| |
|  
| | 11/8
| 13/8
| | 115.543
| 116.856
| |
|  
|-
|-
| |
|  
| | 8/7
| 13/11
| | 115.587
| 116.922
| |
|  
|-
|-
| |
|  
| | 15/11
| 3/2
| | 115.797
| 116.993
| |
|  
|-
|-
| |
| 4\41
| | 12/11
|  
| | 115.938
| 117.073
| |
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
| | 3\31
|  
| |
| 15/14
| | 116.129
| 119.443
| |
|  
|}
 
==== Revelation ====
{| class="wikitable center-all left-4"
|-
|-
| |
! Edo<br>generator
| | 11/9
! Unchanged interval<br>(eigenmonzo)
| | 116.164
! Generator (¢)
| | 11, 13 and 15 limit minimax
! Comments
|-
|-
| |
|  
| | {{vector|0 -195 35 5 89}}
| 15/8
| | 116.198
| 111.731
| | 11 limit least squares
|  
|-
|-
| |
|  
| | {{vector|0 -251 22 5 117 13}}
| 13/10
| | 116.229
| 113.553
| | 15 limit least squares
|  
|-
|-
| |
| 2\21
| | 5/4
|  
| | 116.241
| 114.286
| |
|  
|-
|-
| |
|  
| | {{vector|0 -234 39 4 102 11}}
| 13/11
| | 116.249
| 114.555
| | 13 limit least squares
|  
|-
|-
| |
|  
| |
| 11/10
| | 116.268
| 115.000
| | POTE13
|  
|-
|-
| |
| 5\52
| |
|  
| | 116.277
| 115.385
| | POTE11
| 52f val
|-
|-
| |
|  
| | 7/5
| 11/7
| | 116.502
| 115.536
| |
|  
|-
|-
| | 10\103
|  
| |
| 11/8
| | 116.505
| 115.543
| |
|  
|-
|-
| | 17\175
|  
| |
| 7/4
| | 116.571
| 115.587
| |
|  
|-
|-
| |
|  
| | {{vector|0 -27 25 5}}
| 15/11
| | 116.573
| 115.797
| | 7 limit least squares
|  
|-
|-
| |
|  
| | {{vector|0 -19 20}}
| 11/6
| | 116.578
| 115.938
| | 5 limit least squares
|  
|-
|-
| |
| 3\31
| | 6/5
|  
| | 116.588
| 116.129
| | 5 and 7 limit minimax
| 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
|-
|-
| |
|  
| | 7/6
| 11/9
| | 116.641
| 116.164
| |
| 11-, 13- and 15-odd-limit minimax
|-
|-
| | 7\72
|  
| |
| 5/4
| | 116.667
| 116.241
| |
|  
|-
|-
| |
|  
| |
| 7/5
| | 116.673
| 116.502
| | POTE5
|  
|-
|-
| |
|  
| |
| 5/3
| | 116.675
| 116.588
| | POTE7
| 5- and 7-odd-limit minimax
|-
|-
| |
|  
| | 10/9
| 7/6
| | 116.716
| 116.641
| | 9 limit minimax
|  
|-
|-
| |
| 7\72
| | {{vector|0 117 -44 -19}}
|  
| | 116.721
| 116.667
| | 9 limit least squares
| 72ee val
|-
|-
| |
|  
| | 9/7
| 9/5
| | 116.792
| 116.716
| |
| 9-odd-limit minimax
|-
|-
| | 11\113
|  
| |
| 9/7
| | 116.814
| 116.792
| |
|  
|-
|-
| |
|  
| | 4/3
| 3/2
| | 116.993
| 116.993
| |
|  
|-
|-
| | 4\41
| 4\41
| |
|  
| | 117.073
| 117.073
| |
| 41ef val
|-
|-
| |
|  
| | 18/13
| 13/9
| | 117.559
| 117.559
| |
|  
|-
|-
| |
|  
| | 13/12
| 13/12
| | 117.936
| 117.936
| |
|  
|-
|-
| |
|  
| | 15/14
| 15/14
| | 119.443
| 119.443
| |
|  
|-
|-
| |
|  
| | 16/13
| 13/8
| | 119.824
| 119.824
| |
|  
|-
|-
| |
|  
| | 15/13
| 15/13
| | 123.871
| 123.871
| |
|  
|-
|-
| |
|  
| | 14/13
| 13/7
| | 128.298
| 128.298
| |
|  
|}
|}
[[Category:benediction]]
 
[[Category:eigenmonzo]]
[[Category:Miracle]]
[[Category:manna]]
[[Category:Temperament extensions]]
[[Category:miracle]]
[[Category:Rank-2 temperaments]]
[[Category:miraculous]]
[[Category:revelation]]
[[Category:spectrum]]
Retrieved from "https://en.xen.wiki/w/Miracle_extensions"