21edf - Revision history

Overthink: + Theory

2025-10-22T23:39:33Z

+ Theory

← Older revision		Revision as of 23:39, 22 October 2025
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	Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]		Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]
			== Theory ==
			21edf acts as a stretched version of 36edo, though under most circumstances the stretch is more than ideal. If used as a fifth-based system, the chord 6:7:8(:9) may act as the fundamental chord of the system. The most important comma tempered out by this system is [[1029/1024]], and the related temperament is a fifth-based version of [[slendric]] with a 1/3-fifth period representing [[8/7]] and a generator of about a sixth-tone. One generator up or down from this period gives [[7/6]] and [[9/8]] respectively. Other edfs supporting this temperament include [[24edf]] and [[45edf]].
	== Approximations ==		== Approximations ==
	=== Harmonics ===		=== Harmonics ===

ArrowHead294: /* See also */

2025-06-25T19:13:09Z

ArrowHead294 at 13:46, 18 June 2025

2025-06-18T13:46:05Z

← Older revision		Revision as of 13:46, 18 June 2025
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	\|		\|
	\|}		\|}

			== See also ==
			* [[36edo]] – relative edo
			* [[57edt]] – relative edt
			* [[93ed6]] – relative ed6
			* [[101ed7]] – relative ed7
			* [[129ed12]] – relative ed12

	{{todo\|expand}}		{{todo\|expand}}

	[[Category:36edo]]		[[Category:36edo]]

ArrowHead294 at 13:40, 18 June 2025

2025-06-18T13:40:39Z

← Older revision		Revision as of 13:40, 18 June 2025
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	{{todo\|expand}}		{{todo\|expand}}

			[[Category:36edo]]

Dogwithabome: jus -> just

2025-02-11T22:17:45Z

jus -> just

← Older revision		Revision as of 22:17, 11 February 2025
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	{{ED intro}}		{{ED intro}}

	21EDF is related to [[36edo]], but with the 3/2 rather than the 2/1 being ~~jus~~, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.		21EDF is related to [[36edo]], but with the 3/2 rather than the 2/1 being just, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.

	Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]		Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]

ArrowHead294: /* Intervals */

2025-01-22T13:17:47Z

Intervals

← Older revision		Revision as of 13:17, 22 January 2025
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	! Additional ratios<br />of 2.3.7.13.17		! Additional ratios<br />of 2.3.7.13.17
	\|-		\|-
	\| colspan="2" 0		\| colspan="2" \| 0
	\| 1/1		\| 1/1
	\|		\|

ArrowHead294 at 17:20, 21 January 2025

2025-01-21T17:20:22Z

← Older revision		Revision as of 17:20, 21 January 2025
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	21EDF is related to [[36edo]], but with the 3/2 rather than the 2/1 being jus, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.		21EDF is related to [[36edo]], but with the 3/2 rather than the 2/1 being jus, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.

	Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]]		Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]

	== Approximations ==		== Approximations ==

ArrowHead294 at 17:17, 21 January 2025

2025-01-21T17:17:23Z

Show changes

BudjarnLambeth: Broke ultra wide table into two short tables for mobile readers

2024-12-19T08:33:27Z

Broke ultra wide table into two short tables for mobile readers

← Older revision		Revision as of 08:33, 19 December 2024
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	==Approximations==		==Approximations==
	===Harmonics===		===Harmonics===
	{{Harmonics in equal\|21\|3\|2\|prec=2\|columns=15}}		{{Harmonics in equal\|21\|3\|2\|prec=2\|columns=8}}
			{{Harmonics in equal\|21\|3\|2\|prec=2\|columns=8\|start=9\|title=contd.}}

	===3-limit (Pythagorean) approximations (same as 7edf):===		===3-limit (Pythagorean) approximations (same as 7edf):===

BudjarnLambeth: add todo:expand, correct heading levels, correct linking, put table in own subheading, make table collapsible

2024-12-19T08:32:03Z

add todo:expand, correct heading levels, correct linking, put table in own subheading, make table collapsible

← Older revision		Revision as of 08:32, 19 December 2024
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	{{Infobox ET}}		{{Infobox ET}}
	'''[[EDF\|Division of the just perfect fifth]] into 21 equal parts''' (21EDF) is related to [[36edo\|36 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 33.4264 cents. Unlike 36edo, it is only consistent up to the [[~~3-odd-limit\|4-~~integer-limit]], with discrepancy for the 5th harmonic.		'''[[EDF\|Division of the just perfect fifth]] into 21 equal parts''' (21EDF) is related to [[36edo\|36 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 33.4264 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.

	Lookalikes: [[36edo]], [[57edt]]		Lookalikes: [[36edo]], [[57edt]]

	=Approximations=		==Approximations==
	==Harmonics==		===Harmonics===
	{{Harmonics in equal\|21\|3\|2\|prec=2\|columns=15}}		{{Harmonics in equal\|21\|3\|2\|prec=2\|columns=15}}

	==3-limit (Pythagorean) approximations (same as 7edf):==		===3-limit (Pythagorean) approximations (same as 7edf):===
	2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.		2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.

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	128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.		128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.
	==7-limit approximations:==		===7-limit approximations:===
	===7 only:===		====7 only:====
	7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.		7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.

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	64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.		64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.
	===3 and 7:===		====3 and 7:====
	7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.		7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.

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	63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.		63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.

			==Intervals==
	The following table gives an overview of all degrees of 36edo.		The following table gives an overview of all degrees of 36edo.
	{\| class="wikitable"		{\| class="wikitable mw-collapsible"
			\|+ Intervals of 21edf
	\|-		\|-
	! \|Degree		! \|Degree
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	\|		\|
	\|}		\|}

			{{todo\|expand}}

← Older revision		Revision as of 19:13, 25 June 2025
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	* [[93ed6]] – relative ed6		* [[93ed6]] – relative ed6
	* [[101ed7]] – relative ed7		* [[101ed7]] – relative ed7
	* [[129ed12]] – relative ed12		* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo

	{{todo\|expand}}		{{todo\|expand}}

	[[Category:36edo]]		[[Category:36edo]]