Miracle extensions: Difference between revisions

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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]].

[[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]].

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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]].
-[[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]].