219edo: Difference between revisions

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219edo is ~~the equal division of the octave into 214 parts of 5.4795 cents each. It is incosistent~~ in the [[3-odd-limit]] as well as higher odd-limits ~~and tempers out the following commas 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. The patent val for 219-EDO is <214 347 509 615~~|~~. Its approximations to lower harmonics~~ are ~~<i>exceptionally bad</i> with at least~~ a ~~10% error (relative to the~~ step ~~size) up to the 29th harmonic and just below 5% for the 31st harmonic~~. If anything, it can be considered as a 2.17/3.~~23/11~~.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 ~~an extension~~ of [[53edo]]) or [[217edo]] (being ~~an extension~~ of [[31edo]]).

== 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]]).

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.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 219edo factors into {{factorization|219}}, 219edo contains [[3edo]] and [[73edo]] as its subsets. [[438edo]], which doubles it, provides a strong correction to the 5th harmonic and improves on the 11th and 13th.

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-edo is the equal division of the octave into 214 parts of 5.4795 cents each. It is incosistent in the [[3-odd-limit]] as well as higher odd-limits and tempers out the following commas 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. The patent val for 219-EDO is <214 347 509 615|. Its approximations to lower harmonics are <i>exceptionally bad</i> with at least a 10% error (relative to the step size) up to the 29th harmonic and just below 5% for the 31st harmonic. If anything, it can be considered as a 2.17/3.23/11.31 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 an extension of [[53edo]]) or [[217edo]] (being an extension of [[31edo]]).
+{{Infobox ET}}
+{{ED intro}}
+== 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]]).
+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.
+=== Prime harmonics ===
+{{Harmonics in equal|219}}
+=== Subsets and supersets ===
+Since 219edo factors into {{factorization|219}}, 219edo contains [[3edo]] and [[73edo]] as its subsets. [[438edo]], which doubles it, provides a strong correction to the 5th harmonic and improves on the 11th and 13th.