31edf - Revision history

ArrowHead294 at 13:50, 18 June 2025

2025-06-18T13:50:58Z

← Older revision		Revision as of 13:50, 18 June 2025
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	== Theory ==		== Theory ==
	31edf is almost identical to [[53edo]], but with the [[3/2\|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning\|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit\|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1\|7]], the [[5/1\|5]], [[11/1\|11]], [[13/1\|13]], and [[19/1\|19]] remain flat but significantly less so than in 53edo.		31edf is almost identical to [[53edo]], but with the [[3/2\|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning\|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit\|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1\|7]], the [[5/1\|5]], [[11/1\|11]], [[13/1\|13]], and [[19/1\|19]] remain flat but significantly less so than in 53edo.

			The [[The Riemann zeta function and tuning\|local zeta peak]] around 53 is located at 52.996829, which has the octave stretched by 0.0718{{c}}; the octave of 31edf comes extremely close (differing by only {{sfrac\|1\|22}}{{c}}), thus minimizing relative error as much as possible.

	=== Harmonics ===		=== Harmonics ===
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	* [[84edt]] – relative edt		* [[84edt]] – relative edt
	* [[137ed6]] – relative ed6		* [[137ed6]] – relative ed6

			[[Category:53edo]]
			[[Category:Zeta-optimized tunings]]

FloraC: /* Theory */ expand

2025-04-18T14:22:14Z

Theory: expand

← Older revision		Revision as of 14:22, 18 April 2025
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	== Theory ==		== Theory ==
	31edf is almost identical to [[53edo]], but with the 3/2 rather than the [[2/1]] being [[just]]. The octave is [[~~octave stretching~~\|stretched]] by about 0.~~1166~~ [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit\|10-integer-limit]].		31edf is almost identical to [[53edo]], but with the [[3/2\|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning\|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit\|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1\|7]], the [[5/1\|5]], [[11/1\|11]], [[13/1\|13]], and [[19/1\|19]] remain flat but significantly less so than in 53edo.

	=== Harmonics ===		=== Harmonics ===

FloraC: /* See also */ +9ed9/8

2025-03-24T10:58:01Z

FloraC: /* Theory */ +link

2025-03-24T10:44:34Z

Theory: +link

← Older revision		Revision as of 10:44, 24 March 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	31edf is almost identical to [[53edo]], but with the 3/2 rather than the [[2/1]] being [[just]]. The octave is [[octave stretching\|stretched]] by about 0.1166 [[cents]]. Like 53edo, 31edf is consistent to the [[integer limit\|10-integer-limit]].		31edf is almost identical to [[53edo]], but with the 3/2 rather than the [[2/1]] being [[just]]. The octave is [[octave stretching\|stretched]] by about 0.1166 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit\|10-integer-limit]].

	=== Harmonics ===		=== Harmonics ===

FloraC: /* Theory */ +subsets and supersets

2025-03-24T10:43:29Z

Theory: +subsets and supersets

← Older revision		Revision as of 10:43, 24 March 2025
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	{{Harmonics in equal\|31\|3\|2\|intervals=integer}}		{{Harmonics in equal\|31\|3\|2\|intervals=integer}}
	{{Harmonics in equal\|31\|3\|2\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 31edf (continued)}}		{{Harmonics in equal\|31\|3\|2\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 31edf (continued)}}

			=== Subsets and supersets ===
			31edf is the 11th [[prime equal division\|prime edf]], following [[29edf]] and coming before [[37edf]]. It does not contain any nontrivial subset edfs.

	== See also ==		== See also ==

FloraC: Cleanup; -virtually duplicate stuff from 53edo that fails to undo octave reduction

2025-03-24T10:42:15Z

Cleanup; -virtually duplicate stuff from 53edo that fails to undo octave reduction

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''[[EDF|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The octave is [[Octave stretching|stretched]] by about 0.1166 [[cents]] and the step size is about 22.6437 cents. It is consistent to the 10-[[integer-limit]].
+{{ED intro}}
 == Theory ==
-edf provides excellent approximations for the classic [[5-limit]] just chords and scales, such as the Ptolemy-Zarlino "[[just major]]" scale.
+edf is almost identical to [[53edo]], but with the 3/2 rather than the [[2/1]] being [[just]]. The octave is [[octave stretching|stretched]] by about 0.1166 [[cents]]. Like 53edo, 31edf is consistent to the [[integer limit|10-integer-limit]].
 === Harmonics ===
-{{Harmonics in equal|31|3|2|intervals=prime}}
+{{Harmonics in equal|31|3|2|intervals=integer}}
-[[Category:Edf]]
+== See also ==
-[[Category:Edonoi]]
+* [[53edo]] – relative edo

BudjarnLambeth: Make style consistent with newer ET pages

2024-12-18T08:23:41Z

Make style consistent with newer ET pages

← Older revision		Revision as of 08:23, 18 December 2024
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	{{Infobox ET}}		{{Infobox ET}}
	'''[[EDF\|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo~~\|53 edo~~]], but with the 3/2 rather than the 2/1 being just. The octave is about 0.1166 cents ~~stretched~~ and the step size is about 22.6437 cents. It is consistent to the [[~~9-odd-limit\|10-~~integer-limit]].		'''[[EDF\|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo]], but with the [[3/2]] rather than the [[2/1]] being [[just]]. The octave is [[Octave stretching\|stretched]] by about 0.1166 [[cents]] and the step size is about 22.6437 cents. It is consistent to the 10-[[integer-limit]].

	Lookalikes: [[53edo]], [[84edt]]		Lookalikes: [[53edo]], [[84edt]]

	= ~~Just Approximation~~ =		== Theory ==
	31edf provides excellent approximations for the classic 5-limit ~~[[just~~]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.		31edf provides excellent approximations for the classic [[5-limit]] just chords and scales, such as the Ptolemy-Zarlino "[[just major]]" scale.

	{\| class="wikitable"		{\| class="wikitable"
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	\|}		\|}

	One notable property of ~~53EDO~~ is that it offers good approximations for both pure and Pythagorean major thirds.		One notable property of 31edf is that, like 53edo, it offers good approximations for both pure and [[Pythagorean tuning\|Pythagorean]] major thirds.

	The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! ~~53EDO~~ is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.		The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! Like 53edo, 31edf is practically equal to an extended Pythagorean.

			The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.

			=== Harmonics ===
			{{Harmonics in equal\|31\|3\|2\|intervals=prime}}

	[[Category:Edf]]		[[Category:Edf]]
	[[Category:Edonoi]]		[[Category:Edonoi]]

ArrowHead294 at 04:52, 16 November 2024

2024-11-16T04:52:39Z

← Older revision		Revision as of 04:52, 16 November 2024
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	Lookalikes: [[53edo]], [[84edt]]		Lookalikes: [[53edo]], [[84edt]]
	=Just Approximation=
			= Just Approximation =
	31edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.		31edf provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.

	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
	! ~~\|interval~~		! Interval
	! ~~\|ratio~~		! Ratio
	! ~~\|size~~		! Size
	! ~~\|difference~~		! Difference
	\|-		\|-
	\| ~~\|perfect~~ octave		\| Perfect octave
	\| \|2/1		\| 2/1
	\| style="text-align:center;" \|31		\| style="text-align: center;" \| 31
	\| \| +0.12 cents		\| +0.12 cents
	\|-		\|-
	\| \|major third		\| major third
	\| \|5/4		\| 5/4
	\| style="text-align:center;" \|17		\| style="text-align: center;" \| 17
	\| ~~\|−1~~.37 cents		\| −1.37 cents
	\|-		\|-
	\| \|minor third		\| minor third
	\| \|6/5		\| 6/5
	\| style="text-align:center;" \|14		\| style="text-align: center;" \| 14
	\| \| +1.37 cents		\| +1.37 cents
	\|-		\|-
	\| \|major tone		\| major tone
	\| \|9/8		\| 9/8
	\| style="text-align:center;" \|9		\| style="text-align: center;" \| 9
	\| ~~\|−0~~.12 cents		\| −0.12 cents
	\|-		\|-
	\| \|minor tone		\| minor tone
	\| \|10/9		\| 10/9
	\| style="text-align:center;" \|8		\| style="text-align: center;" \| 8
	\| ~~\|−1~~.25 cents		\| −1.25 cents
	\|-		\|-
	\| \|diat. semitone		\| diat. semitone
	\| \|16/15		\| 16/15
	\| style="text-align:center;" \|5		\| style="text-align: center;" \| 5
	\| \| +1.49 cents		\| +1.49 cents
	\|}One notable property of 53EDO is that it offers good approximations for both pure and ~~pythagorean~~ major thirds.		\|}

			One notable property of 53EDO is that it offers good approximations for both pure and Pythagorean major thirds.

	The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.		The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.85 cents away from the just ratio 7/4, so 31EDF can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.

	[[Category:Edf]]		[[Category:Edf]]
	[[Category:Edonoi]]		[[Category:Edonoi]]

Plumtree: Infobox ET added

2022-10-05T18:41:36Z

Infobox ET added

← Older revision		Revision as of 18:41, 5 October 2022
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			{{Infobox ET}}
	'''[[EDF\|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo\|53 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 0.1166 cents stretched and the step size is about 22.6437 cents. It is consistent to the [[9-odd-limit\|10-integer-limit]].		'''[[EDF\|Division of the just perfect fifth]] into 31 equal parts''' (31EDF) is almost identical to [[53edo\|53 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 0.1166 cents stretched and the step size is about 22.6437 cents. It is consistent to the [[9-odd-limit\|10-integer-limit]].

Moremajorthanmajor at 23:44, 20 February 2019

2019-02-20T23:44:05Z

@@ Line 2: / Line 2: @@
 Lookalikes: [[53edo]], [[84edt]]
 [[Category:Edf]]
 [[Category:Edonoi]]

← Older revision		Revision as of 10:58, 24 March 2025
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	== See also ==		== See also ==
			* [[9ed9/8]] – relative ed9/8
	* [[53edo]] – relative edo		* [[53edo]] – relative edo
	* [[84edt]] – relative edt		* [[84edt]] – relative edt
	* [[137ed6]] – relative ed6		* [[137ed6]] – relative ed6