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]]). | |||
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. | |||
Latest revision as of 22:45, 20 February 2025
| ← 218edo | 219edo | 220edo → |
219 equal divisions of the octave (abbreviated 219edo or 219ed2), also called 219-tone equal temperament (219tet) or 219 equal temperament (219et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 219 equal parts of about 5.48 ¢ each. Each step represents a frequency ratio of 21/219, or the 219th root of 2.
Theory
219edo is inconsistent in the 5-odd-limit as well as higher odd limits. Its approximations to lower harmonics are exceptionally bad: 5, 11, and 13 are about halfway between its steps, and 19 and 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 ⟨214 347 509 615 758 810], which 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.59 | +2.73 | +1.04 | +2.11 | -2.17 | -0.85 | -1.62 | +1.86 | +0.56 | +0.17 |
| Relative (%) | +0.0 | -10.7 | +49.8 | +18.9 | +38.4 | -39.6 | -15.4 | -29.6 | +34.0 | +10.2 | +3.1 | |
| Steps (reduced) |
219 (0) |
347 (128) |
509 (71) |
615 (177) |
758 (101) |
810 (153) |
895 (19) |
930 (54) |
991 (115) |
1064 (188) |
1085 (209) | |
Subsets and supersets
Since 219edo factors into 3 × 73, 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.