Augmented (temperament) - Revision history

FloraC at 10:18, 9 February 2026

2026-02-09T10:18:27Z

← Older revision		Revision as of 10:18, 9 February 2026
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	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Augmented		\| Title = Augmented
	\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11		\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11
	\| Comma basis = [[128/125]] (5-limit);<br>[[64/63]], [[126/125]] (7-limit);<br>[[56/55]], [[64/63]], [[100/99]] (11-limit)		\| Comma basis = [[128/125]] (5-limit); <br>[[64/63]], [[126/125]] (7-limit); <br>[[56/55]], [[64/63]], [[100/99]] (11-limit)
	\| Edo join 1 = 12 \| Edo join 2 = 15		\| Edo join 1 = 12 \| Edo join 2 = 15
	\| Mapping = 3; 1 0 -2 -2		\| Mapping = 3; 1 0 -2 -2

FloraC: Cleanup on infobox

2026-02-09T08:55:09Z

Cleanup on infobox

← Older revision		Revision as of 08:55, 9 February 2026
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	\| Edo join 1 = 12 \| Edo join 2 = 15		\| Edo join 1 = 12 \| Edo join 2 = 15
	\| Mapping = 3; 1 0 -2 -2		\| Mapping = 3; 1 0 -2 -2
	\| ~~Generator~~ = 3/2		\| Generators = 3/2
	\| ~~Generator~~ tuning = 711.6		\| Generators tuning = 711.6
	\| Optimization method = CWE		\| Optimization method = CWE
	\| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]		\| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]

Overthink: /* Tuning spectrum */ fix typo

2026-01-29T06:59:37Z

Tuning spectrum: fix typo

← Older revision		Revision as of 06:59, 29 January 2026
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	\|		\|
	\| 715.789		\| 715.789
	\| ~~53bce~~ val		\| 57bce val
	\|-		\|-
	\|		\|

Overthink: change infobox odd limits

2026-01-29T05:02:17Z

change infobox odd limits

← Older revision		Revision as of 05:02, 29 January 2026
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	\| Pergen = (P8/3, P5)		\| Pergen = (P8/3, P5)
	\| Odd limit 1 = 7 \| Mistuning 1 = 13.7 \| Complexity 1 = 12		\| Odd limit 1 = 7 \| Mistuning 1 = 13.7 \| Complexity 1 = 12
	\| Odd limit 2 = ~~(7-limit) 21~~ \| Mistuning 2 = 20.5 \| Complexity 2 = 15		\| Odd limit 2 = 11 \| Mistuning 2 = 22.4 \| Complexity 2 = 15
	}}		}}
	'''Augmented''' is a [[regular temperament\|temperament]] that sets [[5/4]] to one third of an [[2/1\|octave]], [[tempering out]] the diesis, [[128/125]], and has a [[generator]] of a [[3/2\|perfect fifth]]. The fifth can be tuned to [[12edo]], but a sharper tuning is often perferred for it to blend with the sharp 5/4. This gives rise to the natural [[7-limit]] [[extension]] (known as '''augene''') that tempers out [[64/63]] and [[126/125]], where the whole tone stands in for [[8/7]][[~]][[9/8]].		'''Augmented''' is a [[regular temperament\|temperament]] that sets [[5/4]] to one third of an [[2/1\|octave]], [[tempering out]] the diesis, [[128/125]], and has a [[generator]] of a [[3/2\|perfect fifth]]. The fifth can be tuned to [[12edo]], but a sharper tuning is often perferred for it to blend with the sharp 5/4. This gives rise to the natural [[7-limit]] [[extension]] (known as '''augene''') that tempers out [[64/63]] and [[126/125]], where the whole tone stands in for [[8/7]][[~]][[9/8]].

FloraC: /* Tunings */ + 5-limit norm-based tunings

2026-01-28T09:19:42Z

Tunings: + 5-limit norm-based tunings

@@ Line 96: / Line 96: @@
 == Tunings ==
 {| class="wikitable mw-collapsible mw-collapsed"
 |+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings

FloraC: /* Interval chain */ adopt CWE. Chain of fifths

2026-01-28T09:13:52Z

Interval chain: adopt CWE. Chain of fifths

← Older revision		Revision as of 09:13, 28 January 2026
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	\| 0		\| 0
	\| 0.0		\| 0.0
	\| 1/1		\| '''1/1'''
	\| 400.0		\| 400.0
	\| '''5/4''', 14/11		\| '''5/4''', 14/11
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	\|-		\|-
	\| 1		\| 1
	\| ~~1113~~.1		\| 711.6
			\| '''3/2'''
			\| 1111.6
	\| 15/8, 40/21		\| 15/8, 40/21
	\| ~~313~~.1		\| 311.6
	\| 6/5		\| 6/5
	~~\| 713.1~~
	~~\| '''3/2'''~~
	\|-		\|-
	\| 2		\| 2
	\| ~~1026~~.3		\| 223.2
	~~\| 9/5, 20/11~~
	~~\| 226.3~~
	\| '''8/7''', '''9/8'''		\| '''8/7''', '''9/8'''
	\| ~~626~~.3		\| 623.2
	\| 10/7, '''16/11'''		\| 10/7, '''16/11'''
			\| 1023.2
			\| 9/5, 20/11
	\|-		\|-
	\| 3		\| 3
	\| ~~939~~.4		\| 934.8
	\| 12/7		\| 12/7
	\| ~~139~~.4		\| 134.8
	\| 12/11, 15/14		\| 12/11, 15/14
	\| ~~539~~.4		\| 534.8
	\| 15/11		\| 15/11
	\|-		\|-
	\| 4		\| 4
	\| ~~852~~.6		\| 446.4
			\| 9/7
			\| 846.4
	\| 18/11		\| 18/11
	\| 52.6		\| 46.4
	\| 45/44		\| 45/44
	~~\| 452.6~~
	~~\| 9/7~~
	\|-		\|-
	\| 5		\| 5
	\| ~~765~~.7		\| 1158.0
	~~\| 54/35~~
	~~\| 1165.7~~
	\| 27/14, 96/49		\| 27/14, 96/49
	\| ~~365~~.7		\| 358.0
	\| 27/22		\| 27/22
			\| 758.0
			\| 54/35
	\|}		\|}
	<nowiki/>* In 11~~-odd~~-limit ~~minimax~~ tuning, octave reduced		<nowiki/>* In 11-limit CWE tuning, octave reduced

	== Scales ==		== Scales ==

FloraC: + link to scales

2026-01-28T09:05:16Z

+ link to scales

← Older revision		Revision as of 09:05, 28 January 2026
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	\|}		\|}
	<nowiki/>* In 11-odd-limit minimax tuning, octave reduced		<nowiki/>* In 11-odd-limit minimax tuning, octave reduced

			== Scales ==
			=== Scala files ===
			* [[12-27]] – 27edo tuning

	== Tunings ==		== Tunings ==

FloraC: Recategorize

2026-01-28T08:30:15Z

Recategorize

← Older revision		Revision as of 08:30, 28 January 2026
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	* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] – in Augene[12] tuned to 27edo		* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] – in Augene[12] tuned to 27edo

	[[Category:~~Augene~~\| ]] <!-- Main article -->		[[Category:Augmented\| ]] <!-- Main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
	[[Category:Augmented family]]		[[Category:Augmented family]]
	[[Category:Archytas clan]]		[[Category:Archytas clan]]
	[[Category:Starling temperaments]]		[[Category:Starling temperaments]]

FloraC: FloraC moved page Augene to Augmented (temperament): Canonicalization followup

2026-01-28T08:25:58Z

FloraC moved page Augene to Augmented (temperament): Canonicalization followup

← Older revision	Revision as of 08:25, 28 January 2026
(No difference)

FloraC: Rework for augmented as a single 5- to 11-limit temp

2026-01-28T08:24:53Z

Rework for augmented as a single 5- to 11-limit temp

← Older revision		Revision as of 08:24, 28 January 2026
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	{{Infobox Regtemp		{{Infobox Regtemp
	\| Title = ~~Augene~~		\| Title = Augmented
	\| Subgroups = 2.3.5.7		\| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11
	\| Comma basis = [[64/63]], [[126/125]]		\| Comma basis = [[128/125]] (5-limit);<br>[[64/63]], [[126/125]] (7-limit);<br>[[56/55]], [[64/63]], [[100/99]] (11-limit)
	\| Edo join 1 = 12 \| Edo join 2 = 15		\| Edo join 1 = 12 \| Edo join 2 = 15
	\| Mapping = 3; 1 0 -2		\| Mapping = 3; 1 0 -2 -2
	\| Generator = 3/2		\| Generator = 3/2
	\| Generator tuning = ~~709~~.3		\| Generator tuning = 711.6
	\| Optimization method = CWE		\| Optimization method = CWE
	\| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]		\| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
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	\| Odd limit 2 = (7-limit) 21 \| Mistuning 2 = 20.5 \| Complexity 2 = 15		\| Odd limit 2 = (7-limit) 21 \| Mistuning 2 = 20.5 \| Complexity 2 = 15
	}}		}}
	'''~~Augene~~''' is a [[regular temperament]] ~~of the~~ [[~~augmented family~~]]~~, which means that~~ [[~~128~~/~~125~~]] is [[tempering out~~\|tempered out~~]] and ~~three~~ [[~~5/4~~]]~~'s make~~ a [[2/1]]. ~~Augene~~ is ~~distinguished from its relative~~ [[~~august~~]] ~~by tempering~~ out [[64/63]], ~~which means that tunings~~ in ~~which the fifth is larger than~~ 7~~\12 (700 cents) are optimal. Therefore the~~ [[~~optimal ET sequence~~]] ~~of augene goes 12, 15, 27, … in constrast to august's which goes~~ 9~~, 12, 21, …~~.		'''Augmented''' is a [[regular temperament\|temperament]] that sets [[5/4]] to one third of an [[2/1\|octave]], [[tempering out]] the diesis, [[128/125]], and has a [[generator]] of a [[3/2\|perfect fifth]]. The fifth can be tuned to [[12edo]], but a sharper tuning is often perferred for it to blend with the sharp 5/4. This gives rise to the natural [[7-limit]] [[extension]] (known as '''augene''') that tempers out [[64/63]] and [[126/125]], where the whole tone stands in for [[8/7]][[~]][[9/8]].

	~~It is also~~ the ~~unique~~ [[7-limit]] [[~~regular temperament~~]] ~~that tempers out the~~ [[~~essential~~ tempering commas~~]] of the~~ [[~~5-odd-limit\|5-~~]] ([[~~128/125~~]]~~) and [[7-odd~~-limit~~]] (~~[[64/63]], [[~~126~~/~~125~~]]).		Further extension to the [[11-limit]] is available by noticing 128/125 = ([[56/55]])⋅([[176/175]]), and tempering out both commas means [[14/11]] is equated to the 1/3 octave as well. This does imply more [[damage]] in the 11-limit however, as the [[11/8]] is now conflated with [[7/5]].

	The first few ~~augene~~ [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support ~~augene~~ are [[12edo]], [[15edo]], and ~~especially~~ [[27edo]].		The first few augmented [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably [[support]] it are [[12edo]], [[15edo]], and [[27edo]].

			Alternative 7-limit extensions of augmented include [[august]] (9 & 12), which tempers out [[36/35]] and favors a flat-of-just fifth.
			Therefore the [[optimal ET sequence]] of august goes 9, 12, 21, … in constrast to augmented's which goes 12, 15, 27, ….

			See [[Augmented family #Augmented]] and [[Augmented family #Septimal augmented (augene)\|#Septimal augmented (augene)]] for technical data. See [[Augmented extensions]] for a discussion on 13-limit extensions.

	~~See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions.~~
	~~__TOC__~~
	~~{{Clear}}~~
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics 1–11 are in '''bold'''.		In the following table, odd harmonics 1–11 are in '''bold'''.