Miracle extensions: Difference between revisions

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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]], 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]], implies 72, and [[82edo|82]].

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.

{{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.

{{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 (including 11-limit miracle) can be extended to prime 19 by tempering both commas.

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 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]], 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]], implies 72, and [[82edo|82]].
 Miraculous, benediction, manna, and semimiracle all support {{nowrap|31 &amp; 41 &amp; 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 &amp; 41 &amp; 72}}, then temper it back into a [[rank-2 temperament]] using your extension of choice.
-{{nowrap|31 &amp; 41 &amp; 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.
+{{nowrap|31 &amp; 41 &amp; 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.
 /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.