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Miracle extensions: Difference between revisions

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Line 1: Line 1:
{{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]].  
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 23: Line 27:
|-
|-
! rowspan="2" | 11-limit
! rowspan="2" | 11-limit
! colspan="3" | 13-limit extensions
! colspan="3" | 17-limit extensions
|-
|-
! Miraculous
! Miraculous
Line 39: Line 43:
| 116.6
| 116.6
| 15/14, '''16/15'''
| 15/14, '''16/15'''
| 14/13
| 14/13, '''17/16'''
|  
|  
|  
|  
Line 46: Line 50:
| 233.3
| 233.3
| '''8/7'''
| '''8/7'''
| 15/13
| 15/13, 17/15
|  
|  
|  
|  
Line 53: Line 57:
| 349.9
| 349.9
| 11/9
| 11/9
| '''16/13'''
| '''16/13''', 17/14, 21/17
|  
|  
|  
|  
Line 60: Line 64:
| 466.6
| 466.6
| '''21/16'''
| '''21/16'''
| 13/10
| 13/10, 17/13
|  
| 17/13
|  
| 17/13
|-
|-
| 5
| 5
| 583.2
| 583.2
| 7/5
| 7/5
|  
| 24/17
|  
|  
|  
|  
Line 88: Line 92:
| 933.2
| 933.2
| 12/7
| 12/7
|  
| 17/10
|  
|  
|  
|  
Line 101: Line 105:
| 10
| 10
| 1166.5
| 1166.5
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
| 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
| 83.1
| 83.1
| 21/20, 22/21
| 21/20, 22/21
| 26/25
| 18/17, 26/25
|  
|  
|  
|  
Line 130: Line 134:
| 433.1
| 433.1
| 9/7
| 9/7
|  
| 22/17
|  
|  
|  
|  
Line 180: Line 184:
| 33/32, 36/35
| 33/32, 36/35
| 27/26
| 27/26
| 40/39
| 35/34, 40/39
|  
| 34/33
|-
|-
| 22
| 22
Line 194: Line 198:
| 33/28
| 33/28
|  
|  
|  
| 20/17
| 13/11
| 13/11
|-
|-
Line 223: Line 227:
|  
|  
| 20/13
| 20/13
|  
| 17/11
|-
|-
| 28
| 28
Line 229: Line 233:
| 33/20
| 33/20
|  
|  
|  
| 28/17
|  
|  
|-
|-
Line 236: Line 240:
| 44/25
| 44/25
|  
|  
|  
| 30/17
|  
|  
|-
|-
Line 243: Line 247:
| 66/35
| 66/35
|  
|  
|  
| '''32/17'''
|  
| 17/9
|-
|-
| 31
| 31
| 16.1
| 16.1
| 81/80, 99/98, 121/120
| 81/80, 99/98, 121/120
|  
| 66/65
| 105/104, 144/143, 196/195
| 105/104, 120/119, 136/135, <br>144/143, 154/153, 170/169
| 91/90
| 65/64, 78/77, <br>85/84, 91/90
|-
|-
| 32
| 32
Line 271: Line 275:
| 99/80
| 99/80
|  
|  
| '''16/13'''
| '''16/13''', 21/17
| 26/21
| 26/21
|-
|-
Line 285: Line 289:
| 99/70
| 99/70
|  
|  
|  
| 24/17
|  
| 17/12
|-
|-
| 37
| 37
Line 300: Line 304:
|  
|  
| 21/13
| 21/13
| '''13/8'''
| '''13/8''', 34/21
|-
|-
| 39
| 39
Line 320: Line 324:
| 99/50
| 99/50
|  
|  
| 180/91
| 77/39, 128/65, <br>168/85, 180/91
| 143/72, 195/98, 208/105
| 119/60, 143/72, 135/68, <br>153/77, 169/85, 195/98
|}
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


== 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 342: 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 354: Line 559:
| 13/10
| 13/10
| 113.553
| 113.553
|
|-
|
| 17/10
| 114.830
|  
|  
|-
|-
Line 364: Line 574:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 369: Line 584:
|  
|  
| 116.129
| 116.129
| Lower bound of 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 389: Line 604:
|  
|  
| 116.505
| 116.505
| 103f val
| 103fg val
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
| 175ff 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 434: Line 639:
|  
|  
| 116.667
| 116.667
| 72f val
| 72fg val
|-
|
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 11-odd-limit least squares
|-
|-
|  
|  
Line 445: 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 459: Line 654:
|  
|  
| 116.757
| 116.757
| 185cff val
| 185cffggg val
|-
|-
|  
|  
Line 469: Line 664:
|  
|  
| 116.814
| 116.814
| 113f 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 489: Line 674:
|  
|  
| 117.073
| 117.073
| Upper bound of 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 499: Line 684:
| 13/9
| 13/9
| 117.559
| 117.559
|
|-
|
| 17/11
| 117.597
|  
|  
|-
|-
Line 504: Line 694:
| 13/12
| 13/12
| 117.936
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|  
|  
|-
|-
Line 514: Line 714:
| 13/8
| 13/8
| 119.824
| 119.824
|
|-
|
| 21/17
| 121.942
|  
|  
|-
|-
Line 531: Line 736:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 548: Line 753:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 553: Line 763:
|  
|  
| 116.129
| 116.129
| Lower bound of 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 568: Line 778:
| 13/8
| 13/8
| 116.455
| 116.455
|
|-
|
| 17/16
| 116.501
|  
|  
|-
|-
Line 601: 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 616: Line 826:
|-
|-
|  
|  
| {{monzo| 0 -27 25 5 }}
| 17/10
| 116.573
| 116.581
| 7-odd-limit least squares
|  
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| 17/12
| 116.578
| 116.583
| 5-odd-limit least squares
|
|-
|
| 17/11
| 116.586
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5-, 7- and 15-odd-limit minimax
| 5-, 7- and 15-odd-limit minimax
Line 658: Line 873:
| 7/6
| 7/6
| 116.641
| 116.641
|
|-
|
| 17/9
| 116.642
|
|-
|
| 21/17
| 116.642
|
|-
|
| 17/15
| 116.666
|  
|  
|-
|-
Line 663: Line 893:
|  
|  
| 116.667
| 116.667
| Upper bound of 13- and 15-odd-limit, and 13-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 674: 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 688: Line 908:
|  
|  
| 116.757
| 116.757
| 185cfff val
| 185cfffgg val
|-
|-
|  
|  
Line 698: Line 918:
|  
|  
| 116.814
| 116.814
| 113ff val
| 113ffg val
|-
|-
|  
|  
Line 708: Line 928:
|  
|  
| 117.073
| 117.073
| 41f val, upper bound of 11-odd-limit diamond monotone
| 41fg val, upper bound of 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 720: Line 940:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 737: Line 957:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 742: Line 967:
|  
|  
| 116.129
| 116.129
| 31f val, lower bound of 11-odd-limit diamond monotone
| 31fg val, lower bound of 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 762: Line 987:
|  
|  
| 116.505
| 116.505
| 103ff val
| 103ffgg val
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
| 175fff 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 807: Line 1,022:
|  
|  
| 116.667
| 116.667
| Lower bound of 13- and 15-odd-limit, and 13-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
|
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 17/10
| 116.672
| 116.707
| 11-odd-limit least squares
|  
|-
|-
|  
|  
Line 818: 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 828: Line 1,053:
| 116.725
| 116.725
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|
| 17/14
| 116.730
|
|-
|
| 17/12
| 116.750
|
|-
|-
|  
|  
Line 845: 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 867: Line 1,097:
| 13/9
| 13/9
| 116.79299
| 116.79299
|
|-
|
| 17/11
| 116.801
|  
|  
|-
|-
Line 897: Line 1,132:
|  
|  
| 117.073
| 117.073
|  
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 909: Line 1,144:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 971: Line 1,206:
|  
|  
| 116.129
| 116.129
| 11- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone (singleton)
| 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 977: 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 992: Line 1,217:
| 116.241
| 116.241
|  
|  
|-
|
| {{monzo| 0 -234 39 4 102 11 }}
| 116.249
| 13-odd-limit least squares
|-
|-
|  
|  
Line 1,002: 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 1,032: 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 1,086: 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"