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 [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:

* '''Miraculous''' ({{nowrap|31 &~~amp;~~ 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 &~~amp;~~ 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 &~~amp;~~ 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 ~~extension~~:

In addition, we also consider the only alternative 11-limit mapping:

* '''Revelation''' ({{nowrap|21 &~~amp;~~ 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]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.

~~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~~]].

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]].

For technical information see [[Gamelismic clan #Miracle]].

[[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 ==

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 {{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 [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
-* '''Miraculous''' ({{nowrap|31 &amp; 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 &amp; 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 &amp; 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 extension:
+In addition, we also consider the only alternative 11-limit mapping:
-* '''Revelation''' ({{nowrap|21 &amp; 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]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.
-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]].
+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]].
-For technical information see [[Gamelismic clan #Miracle]].
+[[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 ==