2.3.5.11 subgroup: Difference between revisions

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[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction.

[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. [[65edo]] is the intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80|S9 = 81/80]], [[100/99|S10 = 100/99]], and [[121/120|S11 = 121/120]], thus tempering out [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

=== Rank-3 temperaments ===

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 [[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction.
-[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. [[65edo]] is the intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.
+[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80|S9 = 81/80]], [[100/99|S10 = 100/99]], and [[121/120|S11 = 121/120]], thus tempering out [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.
 === Rank-3 temperaments ===