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Miracle extensions: Difference between revisions

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+17-limit ratios
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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]].  


All of them 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.  
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]].  
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! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 17/16
| 104.955
|
|-
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| 17/15
| 108.343
|
|-
|-
|  
|  
| 15/8
| 15/8
| 111.731
| 111.731
|
|-
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| 17/14
| 112.043
|  
|  
|-
|-
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| 13/10
| 13/10
| 113.553
| 113.553
|
|-
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| 17/10
| 114.830
|  
|  
|-
|-
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| 11/9
| 11/9
| 115.803
| 115.803
|
|-
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| 17/13
| 116.107
|  
|  
|-
|-
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| 13/9
| 13/9
| 117.559
| 117.559
|
|-
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| 17/11
| 117.597
|  
|  
|-
|-
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| 13/12
| 13/12
| 117.936
| 117.936
|
|-
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| 17/9
| 118.087
|
|-
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| 17/12
| 119.400
|  
|  
|-
|-
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| 13/8
| 13/8
| 119.824
| 119.824
|
|-
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| 21/17
| 121.942
|  
|  
|-
|-
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| 11/9
| 11/9
| 115.803
| 115.803
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|-
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| 17/13
| 116.107
|  
|  
|-
|-
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| 13/8
| 13/8
| 116.455
| 116.455
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|-
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| 17/16
| 116.501
|  
|  
|-
|-
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| 13/11
| 13/11
| 116.547
| 116.547
|
|-
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| 17/14
| 116.567
|  
|  
|-
|-
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| 116.571
| 116.571
| 175f val
| 175f val
|-
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| 17/10
| 116.581
|
|-
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| 17/12
| 116.583
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|-
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| 17/11
| 116.586
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|-
|-
|  
|  
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| 7/6
| 7/6
| 116.641
| 116.641
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|-
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| 17/9
| 116.642
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|-
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| 21/17
| 116.642
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|-
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| 17/15
| 116.666
|  
|  
|-
|-
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| 11/9
| 11/9
| 115.803
| 115.803
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|-
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| 17/13
| 116.107
|  
|  
|-
|-
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| 116.667
| 116.667
| Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
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| 17/15
| 116.667
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|-
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| 21/17
| 116.689
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|-
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| 17/9
| 116.702
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|-
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| 17/10
| 116.707
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|-
|-
|  
|  
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| 116.725
| 116.725
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
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| 17/14
| 116.730
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|-
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| 17/12
| 116.750
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|-
|-
|  
|  
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| 116.760
| 116.760
| 13-odd-limit minimax
| 13-odd-limit minimax
|-
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| 17/16
| 116.785
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|-
|-
|  
|  
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| 13/9
| 13/9
| 116.79299
| 116.79299
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|-
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| 17/11
| 116.801
|  
|  
|-
|-
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