Alphatricot family: Difference between revisions

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== Alphatrillium ==

Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] is the 123-~~tone~~ one. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].

Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] that contains it is the 123-note one, though otonal and utonal tetrads don't occur until the 176-note mos due to 7/5 being mapped to -124 generators. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].

It can be extended to the 11-limit by tempering out [[131072/130977]], and to the 13-limit by tempering out [[2080/2079]], [[4096/4095]] and [[4225/4224]]. The optimal tunings in the 11- and 13-limit lean towards [[494edo]]; [[935edo]] and especially [[1429edo]] are recommendable tunings.

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 == Alphatrillium ==
-Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] is the 123-tone one. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].
+Alphatrillium, named by [[Xenllium]] in 2021 as ''trillium'' but renamed following the specifications of ploidacot, can be described as the {{nowrap| 53 & 441 }} temperament, tempering out the [[ragisma]] aside from the alphatricot comma. [[441edo]] is a good tuning for this temperament, with generator 233\441. The harmonic 7 is found at -95 generator steps, so that the smallest [[mos scale]] that contains it is the 123-note one, though otonal and utonal tetrads don't occur until the 176-note mos due to 7/5 being mapped to -124 generators. For much simpler mappings of 7 at the cost of higher errors, you could try [[#Alphatrident|alphatrident]] and [[#Alphatrimot|alphatrimot]].
 It can be extended to the 11-limit by tempering out [[131072/130977]], and to the 13-limit by tempering out [[2080/2079]], [[4096/4095]] and [[4225/4224]]. The optimal tunings in the 11- and 13-limit lean towards [[494edo]]; [[935edo]] and especially [[1429edo]] are recommendable tunings.