Miracle extensions: Difference between revisions
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{{Breadcrumb|Miracle}} | {{Breadcrumb|Miracle}} | ||
The | 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; | * '''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; | * '''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; | * '''Manna''' ({{nowrap|31f & 41}}) – tempering out 225/224, 243/242, 325/324, and 385/384; | ||
In addition, we also consider the only alternative 11-limit | 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. | * '''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]]. | ||
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]]. | |||
Miraculous, benediction, manna, and semimiracle all support {{nowrap|31 & 41 & 72}}, the [[rank-3]] [[expansion]] of miracle. Since they can all be extended to higher limits in the same way, it is natural to first extend {{nowrap|31 & 41 & 72}}, then temper it back into a [[rank-2 temperament]] using your extension of choice. | |||
For technical information see [[Gamelismic clan #Miracle]]. | {{nowrap|31 & 41 & 72}} 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. | ||
225/224 factors into ([[400/399]])·([[513/512]]) in the 19-limit, suggesting that miracle (including 11-limit miracle) can be extended to prime 19 by tempering both commas. | |||
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 == | == Interval chain == | ||