Augmented family - Revision history

FloraC: - interwiki (moved to Augmented (temperament))

2026-08-14T11:49:33Z

- interwiki (moved to Augmented (temperament))

← Older revision		Revision as of 11:49, 14 August 2026
Line 1:		Line 1:
	~~{{Interwiki~~
	~~\| en = Augmented family~~
	~~\| de = Übermässige Temperaturen~~
	}}
	{{Technical data page}}		{{Technical data page}}
	The '''augmented family''' of [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].		The '''augmented family''' of [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].

FloraC: Recategorize

2026-08-14T11:47:35Z

Recategorize

← Older revision		Revision as of 11:47, 14 August 2026
Line 741:		Line 741:
	* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]		* [https://web.archive.org/web/20211201070113/https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]

			[[Category:Augmented family\| ]] <!-- main article -->
	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:~~Augmented family\| ]] <!~~-~~- main article -->~~		[[Category:Catalogs of rank-2 temperaments]]
	~~[[Category:Rank~~ 2]]

FloraC: Text replacement - "octave" to "octave"

2026-07-14T14:15:43Z

Text replacement - "octave" to "octave"

← Older revision		Revision as of 14:15, 14 July 2026
Line 4:		Line 4:
	}}		}}
	{{Technical data page}}		{{Technical data page}}
	The '''augmented family''' of [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[~~2/1\|~~octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].		The '''augmented family''' of [[regular temperament\|temperaments]] [[tempering out\|tempers out]] the diesis a.k.a. augmented comma, [[128/125]], the amount by which three [[5/4]] major thirds fall short of an [[octave]], and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]].

	== Augmented ==		== Augmented ==

Overthink: /* Trisected */ + subgroup temperaments

2026-03-19T22:09:09Z

Trisected: + subgroup temperaments

← Older revision		Revision as of 22:09, 19 March 2026
Line 577:		Line 577:
	== Trisected ==		== Trisected ==
	{{Main\|Trisected}}		{{Main\|Trisected}}
			{{See also\|Subgroup temperaments #Trisect}}
	Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap\| 15 & 36 }} temperament.		Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap\| 15 & 36 }} temperament.

Overthink: /* Trisected */ explain extensions

2026-03-19T19:30:11Z

Trisected: explain extensions

← Older revision		Revision as of 19:30, 19 March 2026
Line 599:		Line 599:

	=== 11-limit ===		=== 11-limit ===
			This extension also tempers out [[4000/3993]], equating 3 intervals of [[11/10]] with [[4/3]].

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

Line 614:		Line 616:

	=== 13-limit ===		=== 13-limit ===
			This extension is natural since it equates the [[10/7]], which is 1/3 of [[3/1]] since 1029/1000 is tempered out, with [[13/9]], thus tempering out [[91/90]] and [[2197/2187]].

	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

FloraC: /* Trisected */ review

2026-03-19T09:54:57Z

Trisected: review

← Older revision		Revision as of 09:54, 19 March 2026
Line 580:		Line 580:

	It was named by [[User:Overthink\|Overthink]] in 2026, since it splits every Pythagorean interval in three.		It was named by [[User:Overthink\|Overthink]] in 2026, since it splits every Pythagorean interval in three.
	~~{{Todo\|review\|comment=Last digits in ~8/7 for WE may be wrong. Also, comma lists could potentially be simplified.}}~~

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
Line 586:		Line 585:
	[[Comma list]]: 128/125, 1029/1000		[[Comma list]]: 128/125, 1029/1000

	{{Mapping\|legend=1\| 3 3 7 9 \| 0 3 0 -1 }}		{{Mapping\|legend=1\| 3 0 7 10 \| 0 3 0 -1 }}
			: mapping generators: ~5/4, ~10/7
	: mapping generators: ~5/4, ~8/7

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~5/4 = 399.~~520~~{{c}}, ~8/7 = ~~234~~.~~195~~{{c}}		* [[WE]]: ~5/4 = 399.5199{{c}}, ~10/7 = 633.7146{{c}}
	: [[error map]]: {{val\| -1.440 -0.811 10.325 -7.342 }}		: [[error map]]: {{val\| -1.440 -0.811 +10.325 -7.342 }}
			* [[CWE]]: ~5/4 = 400.0000{{c}}, ~10/7 = 634.3393{{c}}
	* [[CWE]]: ~5/4 = 400.~~000~~{{c}}, ~8/7 = ~~234~~.~~339~~{{c}}
	: error map: {{val\| 0.000 +1.063 +13.686 -3.165 }}		: error map: {{val\| 0.000 +1.063 +13.686 -3.165 }}

Line 602:		Line 599:

	=== 11-limit ===		=== 11-limit ===

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

	Comma list: 56/55, 128/125, 1029/1000		Comma list: 56/55, 128/125, 1029/1000

	Mapping: {{Mapping\| 3 3 7 ~~9 11~~ \| 0 3 0 -1 -1 }}		Mapping: {{Mapping\| 3 0 7 10 12 \| 0 3 0 -1 -1 }}

	~~: mapping generators: ~5/4, ~8/7~~

	Optimal tunings:		Optimal tunings:
	* WE: ~5/4 = 399.~~305~~{{c}}, ~8/7 = ~~234~~.~~484~~{{c}}		* WE: ~5/4 = 399.3049{{c}}, ~10/7 = 633.7891{{c}}
			* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.7695{{c}}
	* CWE: ~5/4 = 400.~~000~~{{c}}, ~8/7 = ~~234~~.~~769~~{{c}}

	{{Optimal ET sequence\|legend=1\| 15, 36, 51ce }}		{{Optimal ET sequence\|legend=0\| 15, 36, 51ce }}

	[[Badness]] (Sintel): 1.60		Badness (Sintel): 1.60

	=== 13-limit ===		=== 13-limit ===

	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: 56/55, 91/90, 128/125, 1029/1000		Comma list: 56/55, 91/90, 128/125, 1029/1000

	Mapping: {{Mapping\| 3 ~~3 7 9 11~~ 7 \| 0 3 0 -1 -1 7 }}		Mapping: {{Mapping\| 3 0 7 10 12 0 \| 0 3 0 -1 -1 7 }}

	~~: mapping generators: ~5/4, ~8/7~~

	Optimal tunings:		Optimal tunings:
	* WE: ~5/4 = 399.~~319~~{{c}}, ~8/7 = ~~234~~.~~745~~{{c}}		* WE: ~5/4 = 399.3194{{c}}, ~10/7 = 634.0637{{c}}
			* CWE: ~5/4 = 400.0000{{c}}, ~10/7 = 634.9907{{c}}
	* CWE: ~5/4 = 400.~~000~~{{c}}, ~8/7 = ~~234~~.~~991~~{{c}}

	{{Optimal ET sequence\|legend=1\| 15, 36, 51ce }}		{{Optimal ET sequence\|legend=0\| 15, 36, 51ce }}

	[[Badness]] (Sintel): 1.70		Badness (Sintel): 1.70

	== Niner ==		== Niner ==

Overthink: /* Trisected */ remove higher limits for now

2026-03-19T05:45:22Z

Trisected: remove higher limits for now

← Older revision		Revision as of 05:45, 19 March 2026
Line 638:		Line 638:

	[[Badness]] (Sintel): 1.70		[[Badness]] (Sintel): 1.70

	~~=== 17-limit ===~~

	~~Subgroup: 2.3.5.7.11.13.17~~

	~~Comma list: 56/55, 91/90, 128/125, 136/135, 1029/1000~~

	~~Mapping: {{Mapping\| 3 3 7 9 11 7 7 \| 0 3 0 -1 -1 7 9 }}~~

	~~: mapping generators: ~5/4, ~8/7~~

	~~Optimal tunings:~~
	* WE: ~5/4 = 399.317{{c}}, ~8/7 = 234.609{{c}}

	* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}

	~~{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}~~

	~~[[Badness]] (Sintel): 1.83~~

	~~=== 19-limit ===~~

	~~Subgroup: 2.3.5.7.11.13.17.19~~

	~~Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 1029/1000~~

	~~Mapping: {{Mapping\| 3 3 7 9 11 7 7 10 \| 0 3 0 -1 -1 7 9 3 }}~~

	~~: mapping generators: ~5/4, ~8/7~~

	~~Optimal tunings:~~
	* WE: ~5/4 = 399.328{{c}}, ~8/7 = 234.614{{c}}

	* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}

	~~{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}~~

	~~[[Badness]] (Sintel): 1.62~~

	~~=== 23-limit ===~~

	~~Subgroup: 2.3.5.7.11.13.17.19.23~~

	~~Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 1029/1000~~

	~~Mapping: {{Mapping\| 3 3 7 9 11 7 7 10 13 \| 0 3 0 -1 -1 7 9 3 1 }}~~

	~~: mapping generators: ~5/4, ~8/7~~

	~~Optimal tunings:~~
	* WE: ~5/4 = 399.353{{c}}, ~8/7 = 234.601{{c}}

	* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.843{{c}}

	~~{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}~~

	~~[[Badness]] (Sintel): 1.54~~

	~~=== 29-limit ===~~

	~~Subgroup: 2.3.5.7.11.13.17.19.23.29~~

	~~Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 203/200, 1029/1000~~

	~~Mapping: {{Mapping\| 3 3 7 9 11 7 7 10 13 14 \| 0 3 0 -1 -1 7 9 3 1 1 }}~~

	~~: mapping generators: ~5/4, ~8/7~~

	~~Optimal tunings:~~
	* WE: ~5/4 = 399.388{{c}}, ~8/7 = 234.581{{c}}

	* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.819{{c}}

	~~{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}~~

	~~[[Badness]] (Sintel): 1.52~~

	== Niner ==		== Niner ==

Overthink: /* Trisected */ link article

2026-03-19T05:01:53Z

Trisected: link article

← Older revision		Revision as of 05:01, 19 March 2026
Line 576:		Line 576:

	== Trisected ==		== Trisected ==
			{{Main\|Trisected}}
	Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap\| 15 & 36 }} temperament.		Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap\| 15 & 36 }} temperament.

Overthink: /* Trisected */ + Todo:review

2026-03-19T04:01:59Z

Trisected: + Todo:review

← Older revision		Revision as of 04:01, 19 March 2026
Line 579:		Line 579:

	It was named by [[User:Overthink\|Overthink]] in 2026, since it splits every Pythagorean interval in three.		It was named by [[User:Overthink\|Overthink]] in 2026, since it splits every Pythagorean interval in three.
			{{Todo\|review\|comment=Last digits in ~8/7 for WE may be wrong. Also, comma lists could potentially be simplified.}}

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

Overthink: + a new temp (29-limit extension seems fairly natural, though maybe a bit overkill, since max monotonicity is 15)

2026-03-19T03:59:34Z

+ a new temp (29-limit extension seems fairly natural, though maybe a bit overkill, since max monotonicity is 15)

← Older revision		Revision as of 03:59, 19 March 2026
Line 574:		Line 574:

	Badness (Sintel): 2.67		Badness (Sintel): 2.67

			== Trisected ==
			Trisected tempers out [[1029/1024]] and [[1029/1000]], so it is the intersection of augmented and [[slendric]]. It can be described as the {{nowrap\| 15 & 36 }} temperament.

			It was named by [[User:Overthink\|Overthink]] in 2026, since it splits every Pythagorean interval in three.

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 128/125, 1029/1000

			{{Mapping\|legend=1\| 3 3 7 9 \| 0 3 0 -1 }}

			: mapping generators: ~5/4, ~8/7

			[[Optimal tuning]]s:
			* [[WE]]: ~5/4 = 399.520{{c}}, ~8/7 = 234.195{{c}}
			: [[error map]]: {{val\| -1.440 -0.811 10.325 -7.342 }}

			* [[CWE]]: ~5/4 = 400.000{{c}}, ~8/7 = 234.339{{c}}
			: error map: {{val\| 0.000 +1.063 +13.686 -3.165 }}

			{{Optimal ET sequence\|legend=1\| 15, 36, 51c }}

			[[Badness]] (Sintel): 2.61

			=== 11-limit ===

			Subgroup: 2.3.5.7.11

			Comma list: 56/55, 128/125, 1029/1000

			Mapping: {{Mapping\| 3 3 7 9 11 \| 0 3 0 -1 -1 }}

			: mapping generators: ~5/4, ~8/7

			Optimal tunings:
			* WE: ~5/4 = 399.305{{c}}, ~8/7 = 234.484{{c}}

			* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.769{{c}}

			{{Optimal ET sequence\|legend=1\| 15, 36, 51ce }}

			[[Badness]] (Sintel): 1.60

			=== 13-limit ===

			Subgroup: 2.3.5.7.11.13

			Comma list: 56/55, 91/90, 128/125, 1029/1000

			Mapping: {{Mapping\| 3 3 7 9 11 7 \| 0 3 0 -1 -1 7 }}

			: mapping generators: ~5/4, ~8/7

			Optimal tunings:
			* WE: ~5/4 = 399.319{{c}}, ~8/7 = 234.745{{c}}

			* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.991{{c}}

			{{Optimal ET sequence\|legend=1\| 15, 36, 51ce }}

			[[Badness]] (Sintel): 1.70

			=== 17-limit ===

			Subgroup: 2.3.5.7.11.13.17

			Comma list: 56/55, 91/90, 128/125, 136/135, 1029/1000

			Mapping: {{Mapping\| 3 3 7 9 11 7 7 \| 0 3 0 -1 -1 7 9 }}

			: mapping generators: ~5/4, ~8/7

			Optimal tunings:
			* WE: ~5/4 = 399.317{{c}}, ~8/7 = 234.609{{c}}

			* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}

			{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}

			[[Badness]] (Sintel): 1.83

			=== 19-limit ===

			Subgroup: 2.3.5.7.11.13.17.19

			Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 1029/1000

			Mapping: {{Mapping\| 3 3 7 9 11 7 7 10 \| 0 3 0 -1 -1 7 9 3 }}

			: mapping generators: ~5/4, ~8/7

			Optimal tunings:
			* WE: ~5/4 = 399.328{{c}}, ~8/7 = 234.614{{c}}

			* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.852{{c}}

			{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}

			[[Badness]] (Sintel): 1.62

			=== 23-limit ===

			Subgroup: 2.3.5.7.11.13.17.19.23

			Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 1029/1000

			Mapping: {{Mapping\| 3 3 7 9 11 7 7 10 13 \| 0 3 0 -1 -1 7 9 3 1 }}

			: mapping generators: ~5/4, ~8/7

			Optimal tunings:
			* WE: ~5/4 = 399.353{{c}}, ~8/7 = 234.601{{c}}

			* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.843{{c}}

			{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}

			[[Badness]] (Sintel): 1.54

			=== 29-limit ===

			Subgroup: 2.3.5.7.11.13.17.19.23.29

			Comma list: 56/55, 76/75, 91/90, 128/125, 136/135, 161/160, 203/200, 1029/1000

			Mapping: {{Mapping\| 3 3 7 9 11 7 7 10 13 14 \| 0 3 0 -1 -1 7 9 3 1 1 }}

			: mapping generators: ~5/4, ~8/7

			Optimal tunings:
			* WE: ~5/4 = 399.388{{c}}, ~8/7 = 234.581{{c}}

			* CWE: ~5/4 = 400.000{{c}}, ~8/7 = 234.819{{c}}

			{{Optimal ET sequence\|legend=1\| 15g, 36, 51ceg }}

			[[Badness]] (Sintel): 1.52

	== Niner ==		== Niner ==