213edo: Difference between revisions
Jump to navigation
Jump to search
m Infobox ET added |
Rework |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro}} | |||
[[ | 213edo is [[consistency|(uniquely) consistent]] through the [[7-odd-limit]], but [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. Higher [[prime harmonic|prime]]s are slightly better tuned. Moreover, intervals involving a factor of 5/3 or 15 are quite well approximated. Thus it makes sense to view this as a 2.9.15.7.11.13 [[subgroup]] temperament. | ||
The full 13-limit [[patent val]] for 213edo is {{val| 213 338 495 598 737 788 }}, which [[tempering out|tempers out]] the following [[comma]]s up to the [[13-limit]]: {{monzo| 3 -10 11 }} in the [[5-limit]]; {{monzo| 6 -5 -4 4 }}, {{monzo| 10 -11 2 1 }} and [[6144/6125]] in the [[7-limit]]; [[896/891]] in the [[11-limit]]; {{monzo| 12 -7 0 1 0 -1 }}, [[325/324]], [[352/351]] and [[364/363]] in the [[13-limit]]. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|213}} | |||
=== Subsets and supersets === | |||
Since 213 factors into {{factorization|213}}, 213edo contains [[3edo]] and [[71edo]] as its subsets. | |||
Revision as of 09:00, 3 April 2024
| ← 212edo | 213edo | 214edo → |
213edo is (uniquely) consistent through the 7-odd-limit, but harmonics 3 and 5 are about halfway between its steps. Higher primes are slightly better tuned. Moreover, intervals involving a factor of 5/3 or 15 are quite well approximated. Thus it makes sense to view this as a 2.9.15.7.11.13 subgroup temperament.
The full 13-limit patent val for 213edo is ⟨213 338 495 598 737 788], which tempers out the following commas up to the 13-limit: [3 -10 11⟩ in the 5-limit; [6 -5 -4 4⟩, [10 -11 2 1⟩ and 6144/6125 in the 7-limit; 896/891 in the 11-limit; [12 -7 0 1 0 -1⟩, 325/324, 352/351 and 364/363 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.27 | +2.42 | +0.19 | -1.09 | +0.79 | -1.09 | -0.94 | +2.09 | +1.08 | +2.46 | +2.71 |
| Relative (%) | +40.3 | +42.9 | +3.3 | -19.4 | +14.1 | -19.4 | -16.8 | +37.0 | +19.1 | +43.6 | +48.1 | |
| Steps (reduced) |
338 (125) |
495 (69) |
598 (172) |
675 (36) |
737 (98) |
788 (149) |
832 (193) |
871 (19) |
905 (53) |
936 (84) |
964 (112) | |
Subsets and supersets
Since 213 factors into 3 × 71, 213edo contains 3edo and 71edo as its subsets.