213edo: Difference between revisions

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213edo is the equal division of the octave into 213 parts of 5.6338 cents each. It is (uniquely) consistent through the [[7-limit|7-odd-limit]] and tempers out the following commas 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]]. The patent val for 213-EDO is <213 338 495 598|. The general approximations to pure 3- and 5-limit intervals are quite bad, but 7-limit intervals are slightly better tuned. However, intervals involving a factor of 5/3 or 3/5 are quite well approximated. Thus it makes sense to view this as a 2.5/3.7 subgroup temperament.

[[~~Category~~:~~Equal divisions of~~ the ~~octave~~|~~###~~]]

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 ===

=== Subsets and supersets ===

Since 213 factors into {{factorization|213}}, 213edo contains [[3edo]] and [[71edo]] as its subsets.

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 {{Infobox ET}}
-edo is the equal division of the octave into 213 parts of 5.6338 cents each. It is (uniquely) consistent through the [[7-limit|7-odd-limit]] and tempers out the following commas 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]]. The patent val for 213-EDO is &lt;213 338 495 598|. The general approximations to pure 3- and 5-limit intervals are quite bad, but 7-limit intervals are slightly better tuned. However, intervals involving a factor of 5/3 or 3/5 are quite well approximated. Thus it makes sense to view this as a 2.5/3.7 subgroup temperament.
+{{ED intro}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+edo 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.