Miracle extensions: Difference between revisions
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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]]. | ||