Miracle extensions - Revision history

Squib: put 495/494 back

2026-03-08T17:58:05Z

put 495/494 back

← Older revision		Revision as of 17:58, 8 March 2026
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	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]].		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]], which is called prism. The two merge in manna.		[[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]].		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]].

Squib at 02:11, 8 March 2026

2026-03-08T02:11:33Z

← Older revision		Revision as of 02:11, 8 March 2026
Line 15:		Line 15:
	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]].		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 [[~~495~~/~~494~~]]. The two merge in manna.		[[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]], 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]]. ~~Using the alternative 19-limit extension~~, this also tempers out [[760/759]], which conflates [[23/19]] with [[40/33]].		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]].		For technical information see [[Gamelismic clan #Miracle]].

FloraC at 17:57, 26 February 2026

2026-02-26T17:57:18Z

← Older revision		Revision as of 17:57, 26 February 2026
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	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]]. 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 [[715/714]]. 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]].		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 [[495/494]]. The two merge in manna.		[[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 [[495/494]]. The two merge in manna.

Overthink: better language

2026-02-26T17:42:34Z

better language

← Older revision		Revision as of 17:42, 26 February 2026
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	[[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 [[495/494]]. The two merge in manna.		[[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 [[495/494]]. The two merge in manna.

	There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]]. Using the alternative 19-limit extension, this also tempers out [[760/759]], ~~mapping~~ [[23/19]] to [[40/33]].		There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]]. Using the alternative 19-limit extension, this also tempers out [[760/759]], which conflates [[23/19]] with [[40/33]].

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

Squib: another 19-limit extension

2026-02-26T15:42:29Z

another 19-limit extension

← Older revision		Revision as of 15:42, 26 February 2026
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	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]]. 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]]. 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]].		The 13-limit extensions can all be extended to the [[17-limit]] by recognizing [[21/16]]~[[17/13]], tempering out [[273/272]] and [[715/714]]. 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].		[[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 [[495/494]]. The two merge in manna.

			There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]]. Using the alternative 19-limit extension, this also tempers out [[760/759]], mapping [[23/19]] to [[40/33]].

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

FloraC: Mirage isn't (formally) in the scope of this article

2026-02-20T20:40:11Z

Mirage isn't (formally) in the scope of this article

← Older revision		Revision as of 20:40, 20 February 2026
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	[[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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].		[[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. 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~~]] and [[Rastmic rank-3 clan #Mirage~~]].		For technical information see [[Gamelismic clan #Miracle]].

	== Interval chain ==		== Interval chain ==

Squib at 20:48, 19 February 2026

2026-02-19T20:48:53Z

← Older revision		Revision as of 20:48, 19 February 2026
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	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]]. Finally, there is [[mirage]], the [[rank-3 temperament\|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.

	~~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. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].		The 13-limit extensions 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. 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].		[[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. 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]].		For technical information see [[Gamelismic clan #Miracle]] and [[Rastmic rank-3 clan #Mirage]].

	== Interval chain ==		== Interval chain ==

FloraC: Undo revision 221913 by Squib (talk). Put revelation back. Strong extension of miracle.

2026-01-18T20:24:02Z

Undo revision 221913 by Squib (talk). Put revelation back. Strong extension of miracle.

← Older revision		Revision as of 20:24, 18 January 2026
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	* '''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 mapping:
			* '''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]].
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	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]]. Finally, there is [[mirage]], the [[rank-3 temperament\|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.

	~~All of these~~ 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. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].		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. 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].		[[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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].

Squib: moved revelation to its own page. it does not belong here

2026-01-18T17:40:03Z

moved revelation to its own page. it does not belong here

← Older revision		Revision as of 17:40, 18 January 2026
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	* '''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 mapping:~~
	* '''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]].
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	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]]. Finally, there is [[mirage]], the [[rank-3 temperament\|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.

	~~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. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].		All of these 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. 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].		[[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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].

FloraC: Cleanup. Note the caveat in the "19-limit extension"

2025-12-22T08:45:37Z

Cleanup. Note the caveat in the "19-limit extension"

← Older revision		Revision as of 08:45, 22 December 2025
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	{{Breadcrumb\|Miracle}}		{{Breadcrumb\|Miracle}}

	The 11-limit [[miracle]] temperament has various [[extension]]s to the 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 mapping:		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]].

	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]], ~~implies~~ 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]]. Finally, there is [[mirage]], the [[rank-3 temperament\|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.

	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~~.		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. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].

	~~{{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. For semimiracle it implies the half-octave also 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].

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

← Older revision		Revision as of 08:45, 22 December 2025
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	{{Breadcrumb\|Miracle}}		{{Breadcrumb\|Miracle}}

	The 11-limit [[miracle]] temperament has various [[extension]]s to the 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 mapping:		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]].

	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]], ~~implies~~ 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]]. Finally, there is [[mirage]], the [[rank-3 temperament\|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.

	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~~.		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. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].

	~~{{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. For semimiracle it implies the half-octave also 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. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].

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