213edo - Revision history

ArrowHead294 at 22:45, 20 February 2025

2025-02-20T22:45:21Z

← Older revision		Revision as of 22:45, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED 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.		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.

FloraC: Rework

2024-04-03T09:00:38Z

Rework

← Older revision		Revision as of 09:00, 3 April 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	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.		{{EDO intro}}

	[[~~Category~~:~~Equal divisions of~~ the ~~octave~~\|~~###~~]] <!-- 3-~~digit number~~ -->		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.

Plumtree: Infobox ET added

2022-10-04T21:22:38Z

Infobox ET added

← Older revision		Revision as of 21:22, 4 October 2022
Line 1:		Line 1:
			{{Infobox ET}}
	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.		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\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->

Fredg999 category edits: Categories

2022-07-03T18:33:29Z

Xenwolf: lc monzo

2021-01-17T12:36:05Z

lc monzo

← Older revision		Revision as of 12:36, 17 January 2021
Line 1:		Line 1:
	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.		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.

ThisIsOutrageous: So, I've never really gotten behind the high-level math stuff, so I don't know if I will add more stuff later.

2019-08-24T13:31:26Z

So, I've never really gotten behind the high-level math stuff, so I don't know if I will add more stuff later.

New page

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.

← Older revision		Revision as of 18:33, 3 July 2022
Line 1:		Line 1:
	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.		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\|###]] <!-- 3-digit number -->