219edo: Difference between revisions

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== Theory ==

219edo is in[[consistent]] in the [[5-odd-limit]] ~~as well as higher odd limits. Its approximations to lower [[harmonic]]s are ''exceptionally bad'':~~ [[5/1|5]], [[11/1|11]], and [[13/1|13]] ~~are~~ about halfway between its steps, and [[19/1|19]] and [[23/1|23]] ~~are~~ off by about a ~~third~~ step. ~~If anything~~, it can be considered as a 2.3.7.17.29.31 [[subgroup]] tuning. ~~One can see~~ that ~~there are much~~ better ~~alternatives to 219edo if the goal is to mimick just intonation, for example~~ [[212edo]] (~~being~~ a superset of [[53edo]]) or [[217edo]] (~~being~~ a superset of [[31edo]]).

219edo is in[[consistent]] in the [[5-odd-limit]], with primes [[5/1|5]], [[11/1|11]], and [[13/1|13]] being about halfway between its steps, and [[19/1|19]] and [[23/1|23]] being off by about 1/3 of a step. In terms of [[just intonation]], it can be considered as a 2.3.7.17.29.31 [[subgroup]] tuning. Edos of a similar size that do better at approximating JI include [[212edo]] (which is a superset of [[53edo]]) and [[217edo]] (which is a superset of [[31edo]]).

The [[patent val]] for 219edo is {{val| 214 347 509 615 758 810 }}, which [[tempering out|tempers out]] the following [[comma]]s up to the 13-limit: [[32805/32768]] in the 5-limit; [[243/242]], [[441/440]] and [[65536/65219]] in the 11-limit; [[364/363]] in the 13-limit.

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 == Theory ==
-edo is in[[consistent]] in the [[5-odd-limit]] as well as higher odd limits. Its approximations to lower [[harmonic]]s are ''exceptionally bad'': [[5/1|5]], [[11/1|11]], and [[13/1|13]] are about halfway between its steps, and [[19/1|19]] and [[23/1|23]] are off by about a third step. If anything, it can be considered as a 2.3.7.17.29.31 [[subgroup]] tuning. One can see that there are much better alternatives to 219edo if the goal is to mimick just intonation, for example [[212edo]] (being a superset of [[53edo]]) or [[217edo]] (being a superset of [[31edo]]).
+edo is in[[consistent]] in the [[5-odd-limit]], with primes [[5/1|5]], [[11/1|11]], and [[13/1|13]] being about halfway between its steps, and [[19/1|19]] and [[23/1|23]] being off by about 1/3 of a step. In terms of [[just intonation]], it can be considered as a 2.3.7.17.29.31 [[subgroup]] tuning. Edos of a similar size that do better at approximating JI include [[212edo]] (which is a superset of [[53edo]]) and [[217edo]] (which is a superset of [[31edo]]).
 The [[patent val]] for 219edo is {{val| 214 347 509 615 758 810 }}, which [[tempering out|tempers out]] the following [[comma]]s up to the 13-limit: [[32805/32768]] in the 5-limit; [[243/242]], [[441/440]] and [[65536/65219]] in the 11-limit; [[364/363]] in the 13-limit.