5-odd-limit - Revision history

Overthink: make larger

2026-06-18T19:32:55Z

make larger

← Older revision		Revision as of 19:32, 18 June 2026
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	{{Odd-limit navigation\|5}}		{{Odd-limit navigation\|5}}
	[[File:5-odd-limit.png\|~~360px~~\|thumb\|right\|5-odd-limit intervals within an octave]]		[[File:5-odd-limit.png\|480px\|thumb\|right\|5-odd-limit intervals within an octave]]
	{{Odd-limit intro\|5}}		{{Odd-limit intro\|5}}

Overthink: add image

2026-06-14T01:08:05Z

add image

← Older revision		Revision as of 01:08, 14 June 2026
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	{{Odd-limit navigation\|5}}		{{Odd-limit navigation\|5}}
			[[File:5-odd-limit.png\|360px\|thumb\|right\|5-odd-limit intervals within an octave]]
	{{Odd-limit intro\|5}}		{{Odd-limit intro\|5}}

Overthink: Make link appear capitalized

2026-03-16T04:58:02Z

Make link appear capitalized

← Older revision		Revision as of 04:58, 16 March 2026
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	The one which is distinctly consistent in the same is [[9edo]].		The one which is distinctly consistent in the same is [[9edo]].

	The {{w\|~~natural~~ density\|density}} of edos consistent in the 5-odd-limit is 3/4. See below for proof.		The {{W\|Natural density\|density}} of edos consistent in the 5-odd-limit is 3/4. See below for proof.

	{{Proof		{{Proof

FloraC: Wording

2026-03-15T12:36:29Z

Wording

← Older revision		Revision as of 12:36, 15 March 2026
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	The one which is distinctly consistent in the same is [[9edo]].		The one which is distinctly consistent in the same is [[9edo]].

	The {{w\|natural density\|density}} of edos consistent in the 5-odd-limit is ~~expected to be~~ 3/4. See below for proof.		The {{w\|natural density\|density}} of edos consistent in the 5-odd-limit is 3/4. See below for proof.

	{{Proof		{{Proof

Eufalesio: Proposing separating the lines before the table to be separated in paragraphs

2025-10-08T13:32:11Z

Proposing separating the lines before the table to be separated in paragraphs

← Older revision		Revision as of 13:32, 8 October 2025
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	\| major sixth		\| major sixth
	\|}		\|}
	The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]~~; that~~ which is distinctly consistent in the same is [[9edo]]. The {{w\|natural density\|density}} of edos consistent in the 5-odd-limit is expected to be 3/4.		The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]].

			The one which is distinctly consistent in the same is [[9edo]].

			The {{w\|natural density\|density}} of edos consistent in the 5-odd-limit is expected to be 3/4. See below for proof.

	{{Proof		{{Proof

FloraC at 15:17, 23 September 2025

2025-09-23T15:17:13Z

← Older revision		Revision as of 15:17, 23 September 2025
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	{{~~odd~~-limit navigation}}		{{Odd-limit navigation\|5}}
	{{~~odd~~-limit intro\|5}}		{{Odd-limit intro\|5}}

	* [[1/1]]		* [[1/1]]

FloraC at 16:06, 15 August 2025

2025-08-15T16:06:03Z

← Older revision		Revision as of 16:06, 15 August 2025
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	\| major sixth		\| major sixth
	\|}		\|}
	The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]].		The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]]. The {{w\|natural density\|density}} of edos consistent in the 5-odd-limit is expected to be 3/4.

			{{Proof
			\| title = Proof for the density of edos consistent in the 5-odd-limit
			\| contents = Let the error of harmonic 3 be ''A''. Let the error of harmonic 5 be ''B''. By the {{w\|equidistribution theorem}}, the relative error of any individual interval is {{w\|equidistribution\|equidistributed}} from −50% to +50%, so we have the probability density functions {{nowrap\| ''f''<sub>''A''</sub>(''x'') {{=}} 1 }} if {{nowrap\| −1/2 ≤ ''x'' ≤ +1/2 }} and 0 otherwise, {{nowrap\| ''f''<sub>''B''</sub>(''y'') {{=}} 1 }} if {{nowrap\| −1/2 ≤ ''y'' ≤ +1/2 }} and 0 otherwise.

			An edo is consistent in the 5-odd-limit if the error {{nowrap\| ''C'' {{=}} ''B'' − ''A'' }} of the only compound interval – 5/3 – falls into the range from −50% to +50%.

			It is easy to show the probability density function ''f''<sub>''C''</sub>(''z'') is the {{w\|triangular distribution}} {{nowrap\| 1 − {{!}}''z''{{!}} }} if {{nowrap\| −1 ≤ ''z'' ≤ +1 }} and 0 otherwise.

			The density of edos consistent in the 5-odd-limit therefore equals ({{subsup\|∫\|−1/2\|+1/2}} ''f''<sub>''C''</sub>(''z'')''dz'')/({{subsup\|∫\|−∞\|+∞}} ''f''<sub>''C''</sub>(''z'')''dz''), which evaluates to 3/4.
			}}

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

Iwuqety at 14:13, 17 November 2023

2023-11-17T14:13:24Z

← Older revision		Revision as of 14:13, 17 November 2023
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	The smallest [[equal division of the octave]] which is [[consistent]] ~~through to~~ the 5-odd-limit is [[3edo]]; that which is distinctly consistent ~~through to~~ the same is [[9edo]].		The smallest [[equal division of the octave]] which is [[consistent]] in the 5-odd-limit is [[3edo]]; that which is distinctly consistent in the same is [[9edo]].

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

Iwuqety: Added final line stating smallest (distinctly) consistent edos

2023-11-17T14:01:49Z

Added final line stating smallest (distinctly) consistent edos

← Older revision		Revision as of 14:01, 17 November 2023
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			The smallest [[equal division of the octave]] which is [[consistent]] through to the 5-odd-limit is [[3edo]]; that which is distinctly consistent through to the same is [[9edo]].

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

FloraC: +link to diamond5

2023-10-06T11:12:39Z

+link to diamond5

← Older revision		Revision as of 11:12, 6 October 2023
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	== See also ==		== See also ==
	* [[5-limit]] ([[prime limit]])		* [[5-limit]] ([[prime limit]])
			* [[Diamond5]] – as a scale

	[[Category:5-odd-limit\| ]] <!-- main article -->		[[Category:5-odd-limit\| ]] <!-- main article -->

← Older revision		Revision as of 19:32, 18 June 2026
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	{{Odd-limit navigation\|5}}		{{Odd-limit navigation\|5}}
	[[File:5-odd-limit.png\|~~360px~~\|thumb\|right\|5-odd-limit intervals within an octave]]		[[File:5-odd-limit.png\|480px\|thumb\|right\|5-odd-limit intervals within an octave]]
	{{Odd-limit intro\|5}}		{{Odd-limit intro\|5}}

← Older revision		Revision as of 01:08, 14 June 2026
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	{{Odd-limit navigation\|5}}		{{Odd-limit navigation\|5}}
			[[File:5-odd-limit.png\|360px\|thumb\|right\|5-odd-limit intervals within an octave]]
	{{Odd-limit intro\|5}}		{{Odd-limit intro\|5}}