5th-octave temperaments - Revision history

FloraC: Update and request review & cleanup

2026-06-28T11:39:25Z

Update and request review & cleanup

← Older revision		Revision as of 11:39, 28 June 2026
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	{{Infobox fractional-octave\|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].		{{Infobox fractional-octave\|5}}[[5edo]] is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of [[12edo]].

	The most notable 5th-octave family is [[~~limmic temperaments~~]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy ~~clan~~\|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.		The most notable 5th-octave family is [[blackwood family]] – [[tempering out]] [[256/243]] and associates 3\5 to [[3/2]] as well as 1\5 to [[9/8]], producing temperaments like [[blackwood]]. Equally notable among small equal divisions are the [[Cloudy comma #Temperaments\|cloudy temperaments]] – identifying [[8/7]] with one step of 5edo.

	Other families of 5-limit 5th-octave commas are:		Other families of 5-limit 5th-octave commas are:
	* [[Quintile family~~\|Quintile temperaments~~]] - tempers out the {{monzo\|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.		* [[Quintile family]] – tempers out the {{monzo\|-28 25 -5}} comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
	* [[Quintosec family~~\|Quintosec temperaments~~]]		* [[Quintosec family]]
	* [[Trisedodge family~~\|Trisedodge temperaments~~]]		* [[Trisedodge family]]

	== Slendroschismic ==		== Slendroschismic ==
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	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}
			{{Todo\| review \| cleanup }}

Overthink: /* Quint */ move to very low accuracy temperaments

2026-02-27T05:26:49Z

Quint: move to very low accuracy temperaments

← Older revision		Revision as of 05:26, 27 February 2026
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	Supporting ETs: {{Optimal ET sequence\|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}		Supporting ETs: {{Optimal ET sequence\|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}

	~~== Quint ==~~
	Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

	~~[[Subgroup]]: 2.3.5.7~~

	~~[[Comma list]]: 16/15, 27/25~~

	~~{{Mapping\|legend=1\| 5 8 12 0 \| 0 0 0 1 }}~~

	~~: Mapping generators: ~9/8, ~7~~

	~~[[Optimal tuning]]s:~~
	* [[POTE]]: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)
	* [[CTE]]: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)

	~~{{Optimal ET sequence\|legend=1\| 5, 15ccd }}~~

	~~[[Badness]]: 0.048312~~

	== Obscenity ==		== Obscenity ==

Xenllium at 01:35, 21 February 2026

2026-02-21T01:35:13Z

Show changes

BudjarnLambeth: Undo revision 220135 by BudjarnLambeth (talk)

2025-12-28T00:50:09Z

Undo revision 220135 by BudjarnLambeth (talk)

← Older revision		Revision as of 00:50, 28 December 2025
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	[[Support]]ing [[ET]]s: {{EDOs\|5, 65d, 70, 75, 80, 85, 90, 95}}		[[Support]]ing [[ET]]s: {{EDOs\|5, 65d, 70, 75, 80, 85, 90, 95}}

	~~''See also: [[Obsceniton]]''~~

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

BudjarnLambeth: /* Obscenity */ link to orphan page

2025-12-28T00:44:56Z

Obscenity: link to orphan page

← Older revision		Revision as of 00:44, 28 December 2025
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	[[Support]]ing [[ET]]s: {{EDOs\|5, 65d, 70, 75, 80, 85, 90, 95}}		[[Support]]ing [[ET]]s: {{EDOs\|5, 65d, 70, 75, 80, 85, 90, 95}}

			''See also: [[Obsceniton]]''

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

VectorGraphics at 05:00, 13 November 2025

2025-11-13T05:00:06Z

← Older revision		Revision as of 05:00, 13 November 2025
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	== Obscenity ==		== Obscenity ==
	Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum#Absurdity (5-limit)\|absurdity]] as a kind of septal (2.3.7) analog to it.		Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum#Absurdity (5-limit)\|absurdity]] as a kind of septal (2.3.7) analog to it. It tempers out the obsceniton, 4194304/4084101.

	[[Subgroup]]: 2.3.7		[[Subgroup]]: 2.3.7

Overthink: /* Obscenity */ fixed link

2025-10-05T23:50:16Z

Obscenity: fixed link

← Older revision		Revision as of 23:50, 5 October 2025
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	== Obscenity ==		== Obscenity ==
	Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum #Absurdity\|absurdity]] as a kind of septal (2.3.7) analog to it.		Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum#Absurdity (5-limit)\|absurdity]] as a kind of septal (2.3.7) analog to it.

	[[Subgroup]]: 2.3.7		[[Subgroup]]: 2.3.7

FloraC: Septimal blackwood is now canon

2025-09-11T15:16:46Z

Septimal blackwood is now canon

← Older revision		Revision as of 15:16, 11 September 2025
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	== Quint ==		== Quint ==
	Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[~~blacksmith~~]] does.		Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Tristanbay: /* Quint */Added obscenity temperament

2025-07-10T09:09:05Z

Quint: Added obscenity temperament

← Older revision		Revision as of 09:09, 10 July 2025
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	[[Badness]]: 0.048312		[[Badness]]: 0.048312

			== Obscenity ==
			Obscenity maps [[32/21]] to 3\5 and was named in subtle reference to [[Syntonic–chromatic equivalence continuum #Absurdity\|absurdity]] as a kind of septal (2.3.7) analog to it.

			[[Subgroup]]: 2.3.7

			[[Comma list]]: 4194304/4084101

			{{Mapping\|legend=1\| 5 0 22 \| 0 1 -1 }}

			: Mapping generators: ~512/441, ~3

			[[Support]]ing [[ET]]s: {{EDOs\|5, 65d, 70, 75, 80, 85, 90, 95}}

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

VectorGraphics at 05:06, 12 June 2025

2025-06-12T05:06:13Z

← Older revision		Revision as of 05:06, 12 June 2025
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	: Mapping generators: ~9/8, ~7		: Mapping generators: ~9/8, ~7

	~~{{Multival\|legend=1\| 0 0 5 0 8 12 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s: