Miracle extensions: Difference between revisions

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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.

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.

@@ Line 13: / Line 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.