Miracle extensions: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Expand
Squib (talk | contribs)
392 edits
put 495/494 back
Tags: Mobile edit Mobile web edit Advanced mobile edit
 
(27 intermediate revisions by 5 users not shown)
Line 1: Line 1:
The basic 7-limit [[miracle]] temperament has various [[extension]]s to the 11- and 13-limit. The following temperaments are discussed in this article:
{{Breadcrumb|Miracle}}
* '''Miraculous''' (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;
* '''Manna''' (31f & 41f) – tempering out 225/224, 243/242, 325/324, and 385/384;


In addition, we also consider the only alternative 11-limit extension:  
The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:  
* '''Revelation''' (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.
* '''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]].  
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.  


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]].  
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]].


For technical information see [[Gamelismic clan #Miracle]].  
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.
 
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 ==
== Interval chain ==
Line 23: Line 31:
|-
|-
! rowspan="2" | 11-limit
! rowspan="2" | 11-limit
! colspan="3" | 13-limit extensions
! colspan="3" | 17-limit extensions
|-
|-
! Miraculous
! Miraculous
Line 39: Line 47:
| 116.6
| 116.6
| 15/14, '''16/15'''
| 15/14, '''16/15'''
| 14/13
| 14/13, '''17/16'''
|  
|  
|  
|  
Line 46: Line 54:
| 233.3
| 233.3
| '''8/7'''
| '''8/7'''
| 15/13
| 15/13, 17/15
|  
|  
|  
|  
Line 53: Line 61:
| 349.9
| 349.9
| 11/9
| 11/9
| '''16/13'''
| '''16/13''', 17/14, 21/17
|  
|  
|  
|  
Line 60: Line 68:
| 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 96:
| 933.2
| 933.2
| 12/7
| 12/7
|  
| 17/10
|  
|  
|  
|  
Line 101: Line 109:
| 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 138:
| 433.1
| 433.1
| 9/7
| 9/7
|  
| 22/17
|  
|  
|  
|  
Line 180: Line 188:
| 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 202:
| 33/28
| 33/28
|  
|  
|  
| 20/17
| 13/11
| 13/11
|-
|-
Line 223: Line 231:
|  
|  
| 20/13
| 20/13
|  
| 17/11
|-
|-
| 28
| 28
Line 229: Line 237:
| 33/20
| 33/20
|  
|  
|  
| 28/17
|  
|  
|-
|-
Line 236: Line 244:
| 44/25
| 44/25
|  
|  
|  
| 30/17
|  
|  
|-
|-
Line 243: Line 251:
| 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 279:
| 99/80
| 99/80
|  
|  
| '''16/13'''
| '''16/13''', 21/17
| 26/21
| 26/21
|-
|-
Line 285: Line 293:
| 99/70
| 99/70
|  
|  
|  
| 24/17
|  
| 17/12
|-
|-
| 37
| 37
Line 300: Line 308:
|  
|  
| 21/13
| 21/13
| '''13/8'''
| '''13/8''', 34/21
|-
|-
| 39
| 39
Line 320: Line 328:
| 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


== Tuning spectra ==
== Tunings ==
=== Miraculous ===
=== 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"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 17/16
| 104.955
|
|-
|
| 17/15
| 108.343
|
|-
|
| 15/8
| 111.731
| 111.731
|
|-
|
| 17/14
| 112.043
|  
|  
|-
|-
Line 345: Line 525:
|-
|-
|  
|  
| 8/7
| 17/10
| 114.830
|
|-
|
| 7/4
| 115.587
| 115.587
|  
|  
Line 352: Line 537:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 357: Line 547:
|  
|  
| 116.129
| 116.129
|  
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 377: Line 567:
|  
|  
| 116.505
| 116.505
|  
| 103fg val
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
|  
| 175ffggg val
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7 limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5 limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5 and 7 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
|  
|  
Line 405: Line 585:
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
|-
|
|
| 116.633
| POTE11
|-
|-
|  
|  
Line 427: Line 602:
|  
|  
| 116.667
| 116.667
|  
| 72fg val
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 9/5
| 116.672
| 11 limit least squares
|-
|
|
| 116.673
| POTE5
|-
|
|
| 116.675
| POTE7
|-
|
| 10/9
| 116.716
| 116.716
| 9, 11 and 13 limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9 limit least squares
|-
|-
|  
|  
| 11/8
| 116.755
|  
|  
| 116.747
| POTE13
|-
|-
| 18\185
|  
|  
| 11/8
| 116.757
| 116.755
| 185cffggg val
|  
|-
|-
|  
|  
Line 472: Line 627:
|  
|  
| 116.814
| 116.814
|  
| 113fgg val
|-
|
| {{monzo| 0 127 -84 -36 100 -44 }}
| 116.820
| 15 limit least squares
|-
|
| {{monzo| 0 141 -70 -35 84 -42 }}
| 116.846
| 13 limit least squares
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
| 15 limit minimax
| 15-odd-limit minimax
|-
|-
| 4\41
| 4\41
|  
|  
| 117.073
| 117.073
|  
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 500: Line 645:
|-
|-
|  
|  
| 18/13
| 13/9
| 117.559
| 117.559
|
|-
|
| 17/11
| 117.597
|  
|  
|-
|-
Line 507: Line 657:
| 13/12
| 13/12
| 117.936
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|  
|  
|-
|-
Line 515: Line 675:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|
|-
|
| 21/17
| 121.942
|  
|  
|-
|-
Line 525: Line 690:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
|}
|}


=== Benediction ===
==== Benediction ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 551: Line 716:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 556: Line 726:
|  
|  
| 116.129
| 116.129
|  
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 569: Line 739:
|-
|-
|  
|  
| 16/13
| 13/8
| 116.455
| 116.455
|
|-
|
| 17/16
| 116.501
|  
|  
|-
|-
Line 584: Line 759:
|-
|-
|  
|  
| 14/13
| 13/7
| 116.509
| 116.509
|  
|  
Line 604: Line 779:
|-
|-
|  
|  
| {{monzo| 0 -234 39 4 -115 228 }}
| 17/14
| 116.56309
| 116.567
| 13 limit least squares
|-
|  
|  
| {{monzo| 0 -251 22 5 -131 261 }}
| 116.56348
| 15 limit least squares
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
| 175f val
|-
|  
|  
|-
| 17/10
| 116.581
|  
|  
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7 limit least squares
|-
|-
|  
|  
| 17/12
| 116.583
|  
|  
| 116.574
| POTE13
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| 17/11
| 116.578
| 116.586
| 5 limit least squares
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5, 7 and 15 limit minimax
| 5-, 7- and 15-odd-limit minimax
|-
|-
|  
|  
Line 644: Line 814:
|-
|-
|  
|  
| 18/13
| 13/9
| 116.595
| 116.595
| 13 limit minimax
| 13-odd-limit minimax
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
Line 659: Line 829:
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
|-
|
|
| 116.633
| POTE11
|-
|-
|  
|  
Line 673: Line 838:
|  
|  
|-
|-
| 7\72
|  
|  
| 116.667
| 17/9
| 116.642
|  
|  
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 21/17
| 116.672
| 116.642
| 11 limit least squares
|  
|-
|-
|  
|  
| 17/15
| 116.666
|  
|  
| 116.673
| POTE5
|-
|-
| 7\72
|  
|  
|
| 116.667
| 116.675
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| POTE7
|-
|-
|  
|  
| 10/9
| 9/5
| 116.716
| 116.716
| 9 and 11 limit minimax
| 9- and 11-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9 limit least squares
|-
|-
|  
|  
Line 707: Line 867:
| 116.755
| 116.755
|  
|  
|-
| 18\185
|
| 116.757
| 185cfffgg val
|-
|-
|  
|  
Line 716: Line 881:
|  
|  
| 116.814
| 116.814
|  
| 113ffg val
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
|  
|  
Line 726: Line 891:
|  
|  
| 117.073
| 117.073
|  
| 41fg val, upper bound of 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 734: Line 899:
|}
|}


=== Manna ===
==== Manna ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 755: Line 920:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 760: Line 930:
|  
|  
| 116.129
| 116.129
|  
| 31fg val, lower bound of 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 780: Line 950:
|  
|  
| 116.505
| 116.505
|  
| 103ffgg val
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
|  
| 175fffgg val
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7 limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5 limit least squares
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5 and 7 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
|  
|  
Line 808: Line 968:
|-
|-
|  
|  
| 12/11
| 11/6
| 116.596
| 116.596
|  
|  
|-
|-
|  
|  
| 14/11
| 11/7
| 116.617
| 116.617
|  
|  
|-
|
|
| 116.633
| POTE11
|-
|-
|  
|  
Line 829: Line 984:
| 7\72
| 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
| 116.667
|  
|  
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 21/17
| 116.672
| 116.689
| 11 limit least squares
|  
|-
|-
|  
|  
| 17/9
| 116.702
|  
|  
| 116.673
| POTE5
|-
|-
|  
|  
| 17/10
| 116.707
|  
|  
| 116.675
| POTE7
|-
|-
|  
|  
| 10/9
| 9/5
| 116.716
| 116.716
| 9 and 11 limit minimax
| 9- and 11-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 117 -44 -19 }}
| 15/13
| 116.721
| 116.725
| 9 limit least squares
| 15-odd-limit minimax
|-
|-
|  
|  
| 15/13
| 17/14
| 116.725
| 116.730
| 15 limit minimax
|  
|-
|-
|  
|  
| 17/12
| 116.750
|  
|  
| 116.739
| POTE13
|-
|-
|  
|  
Line 871: Line 1,031:
| 116.755
| 116.755
|  
|  
|-
| 18\185
|
| 116.757
| 185cf val
|-
|-
|  
|  
| 13/10
| 13/10
| 116.760
| 116.760
| 13 limit minimax
| 13-odd-limit minimax
|-
|-
|  
|  
| {{monzo| 0 -37 -166 -77 59 243 }}
| 17/16
| 116.764
| 116.785
| 15 limit least squares
|-
|  
|  
| {{monzo| 0 18 -111 -76 43 204 }}
| 116.780
| 13 limit least squares
|-
|-
|  
|  
Line 893: Line 1,053:
|-
|-
|  
|  
| 14/13
| 13/7
| 116.79254
| 116.79254
|  
|  
|-
|-
|  
|  
| 18/13
| 13/9
| 116.79299
| 116.79299
|
|-
|
| 17/11
| 116.801
|  
|  
|-
|-
Line 913: Line 1,078:
|-
|-
|  
|  
| 16/13
| 13/8
| 116.856
| 116.856
|  
|  
Line 923: Line 1,088:
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
|  
|  
Line 930: Line 1,095:
|  
|  
| 117.073
| 117.073
|  
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 938: Line 1,103:
|}
|}


=== Revelation ===
==== Revelation ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 16/15
| 15/8
| 111.731
| 111.731
|  
|  
Line 954: Line 1,119:
| 13/10
| 13/10
| 113.553
| 113.553
|
|-
| 2\21
|
| 114.286
|  
|  
|-
|-
Line 965: Line 1,135:
| 115.000
| 115.000
|  
|  
|-
| 5\52
|
| 115.385
| 52f val
|-
|-
|  
|  
| 14/11
| 11/7
| 115.536
| 115.536
|  
|  
Line 977: Line 1,152:
|-
|-
|  
|  
| 8/7
| 7/4
| 115.587
| 115.587
|  
|  
Line 987: Line 1,162:
|-
|-
|  
|  
| 12/11
| 11/6
| 115.938
| 115.938
|  
|  
Line 994: Line 1,169:
|  
|  
| 116.129
| 116.129
|  
| 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
| 11/9
| 11/9
| 116.164
| 116.164
| 11, 13 and 15 limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 -195 35 5 89 }}
| 116.198
| 11 limit least squares
|-
|
| {{monzo| 0 -251 22 5 117 13 }}
| 116.229
| 15 limit least squares
|-
|-
|  
|  
Line 1,015: Line 1,180:
| 116.241
| 116.241
|  
|  
|-
|
| {{monzo| 0 -234 39 4 102 11 }}
| 116.249
| 13 limit least squares
|-
|
|
| 116.268
| POTE13
|-
|
|
| 116.277
| POTE11
|-
|-
|  
|  
Line 1,036: Line 1,186:
|  
|  
|-
|-
| 10\103
|
| 116.505
|  
|  
|-
| 5/3
| 17\175
|
| 116.571
|
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7 limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5 limit least squares
|-
|
| 6/5
| 116.588
| 116.588
| 5 and 7 limit minimax
| 5- and 7-odd-limit minimax
|-
|-
|  
|  
Line 1,069: Line 1,199:
|  
|  
| 116.667
| 116.667
|  
| 72ee val
|-
|-
|  
|  
|  
| 9/5
| 116.673
| POTE5
|-
|
|
| 116.675
| POTE7
|-
|
| 10/9
| 116.716
| 116.716
| 9 limit minimax
| 9-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9 limit least squares
|-
|-
|  
|  
| 9/7
| 9/7
| 116.792
| 116.792
|
|-
| 11\113
|
| 116.814
|  
|  
|-
|-
|  
|  
| 4/3
| 3/2
| 116.993
| 116.993
|  
|  
Line 1,109: Line 1,219:
|  
|  
| 117.073
| 117.073
|  
| 41ef val
|-
|-
|  
|  
| 18/13
| 13/9
| 117.559
| 117.559
|  
|  
Line 1,127: Line 1,237:
|-
|-
|  
|  
| 16/13
| 13/8
| 119.824
| 119.824
|  
|  
Line 1,137: Line 1,247:
|-
|-
|  
|  
| 14/13
| 13/7
| 128.298
| 128.298
|  
|  
Line 1,144: Line 1,254:
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Latest revision as of 17:58, 8 March 2026

< Miracle

The 11-limit miracle temperament has various extensions 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;
  • Benediction (31 & 41f) – tempering out 225/224, 243/242, 351/350, and 385/384;
  • Manna (31f & 41) – tempering out 225/224, 243/242, 325/324, and 385/384;

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.

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 supported by 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 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.

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 62, 72, and 82. Finally, there is mirage, the rank-3 expansion of miracle with the addition of an independent generator for prime 13.

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.

# Cents* Approximate ratios
11-limit 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,
88/45, 96/49, 108/55 		39/20, 51/26, 77/39,
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,
144/143, 154/153, 170/169 		65/64, 78/77,
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,
168/85, 180/91 		119/60, 143/72, 135/68,
153/77, 169/85, 195/98

* In 11-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

  • 5-limit POTE: ~16/15 = 116.673 ¢
  • 7-limit POTE: ~15/14 = 116.675 ¢
  • 11-limit POTE
    • Miracle: ~15/14 = 116.633 ¢
    • Revelation: ~15/14 = 116.277 ¢
  • 13-limit POTE
    • Miraculous: ~15/14 = 116.747 ¢
    • Benediction: ~15/14 = 116.574 ¢
    • Manna: ~15/14 = 116.739 ¢
    • Revelation: ~15/14 = 116.268 ¢

Target tunings

Target tunings (miracle)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
5-odd-limit ~16/15 = 116.588 ¢ 5/3 ~16/15 = 116.578 ¢ [0 -19 20⟩
7-odd-limit ~15/14 = 116.588 ¢ 5/3 ~15/14 = 116.573 ¢ [0 -27 25 5⟩
9-odd-limit ~15/14 = 116.716 ¢ 9/5 ~15/14 = 116.721 ¢ [0 117 -44 -19⟩
11-odd-limit ~15/14 = 116.716 ¢ 9/5 ~15/14 = 116.672 ¢ [0 17 -11 -6 11⟩
Target tunings (miraculous)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.716 ¢ 9/5 ~15/14 = 116.846 ¢ [0 141 -70 -35 84 -42⟩
15-odd-limit ~15/14 = 116.993 ¢ 3/2 ~15/14 = 116.820 ¢ [0 127 -84 -36 100 -44⟩
Target tunings (benediction)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.595 ¢ 13/9 ~15/14 = 116.56309 ¢ [0 -234 39 4 -115 228⟩
15-odd-limit ~15/14 = 116.588 ¢ 5/3 ~15/14 = 116.56348 ¢ [0 -251 22 5 -131 261⟩
Target tunings (manna)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.760 ¢ 13/10 ~15/14 = 116.780 ¢ [0 18 -111 -76 43 204⟩
15-odd-limit ~15/14 = 116.725 ¢ 15/13 ~15/14 = 116.764 ¢ [0 -37 -166 -77 59 243⟩
Target tunings (revelation)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
11-odd-limit ~15/14 = 116.164 ¢ 11/9 ~15/14 = 116.198 ¢ [0 -195 35 5 89⟩
13-odd-limit ~15/14 = 116.164 ¢ 11/9 ~15/14 = 116.249 ¢ [0 -234 39 4 102 11⟩
15-odd-limit ~15/14 = 116.164 ¢ 11/9 ~15/14 = 116.229 ¢ [0 -251 22 5 117 13⟩

Tuning spectra

Miraculous

Edo
generator 		Unchanged interval
(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,
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,
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

Edo
generator 		Unchanged interval
(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,
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,
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

Edo
generator 		Unchanged interval
(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,
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,
and 17-limit 21-odd-limit diamond monotone
15/14 119.443

Revelation

Edo
generator 		Unchanged interval
(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,
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
Retrieved from "https://en.xen.wiki/index.php?title=Miracle_extensions&oldid=225441"