Augmented family - Revision history

Overthink: /* Trisected */ + subgroup temperaments

2026-03-19T22:09:09Z

Trisected: + subgroup temperaments

← Older revision		Revision as of 22:09, 19 March 2026
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	== 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
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	=== 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

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	=== 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
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	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
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	[[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 }}

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	=== 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
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	[[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
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	== 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
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	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
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	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 ==

FloraC: Review

2026-02-22T06:48:47Z

Review

← Older revision		Revision as of 06:48, 22 February 2026
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	Mapping: {{Mapping\| 3 1 7 6 8 \| 0 3 0 2 2 }}		Mapping: {{Mapping\| 3 1 7 6 8 \| 0 3 0 2 2 }}

	~~: mapping generators: ~5/4, ~15/11~~

	Optimal tunings:		Optimal tunings:
	* WE: ~5/4 = 397.~~495~~{{c}}, ~15/11 = 499.~~096~~{{c}} (~12/11 = 101.~~601~~{{c}})		* WE: ~5/4 = 397.4954{{c}}, ~15/11 = 499.0962{{c}} (~15/14 = 101.6008{{c}})
	* CWE: ~5/4 = 400.~~000~~{{c}}, ~15/11 = 500.~~775~~{{c}} (~12/11 = 100.~~775~~{{c}})		* CWE: ~5/4 = 400.0000{{c}}, ~15/11 = 500.7745{{c}} (~15/14 = 100.7745{{c}})

	{{Optimal ET sequence\|legend=0\| 3b, 9bde, 12 }}		{{Optimal ET sequence\|legend=0\| 3b, 9bde, 12 }}
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	== Subgroup extensions ==		== Subgroup extensions ==
	=== 2.3.5.19 subgroup ~~(3 & 12~~) ===		=== Augmented (2.3.5.19 subgroup) ===
	Augmented ~~temperament~~ works on the 2.3.5.19 subgroup very well, where the 1/3-octave period stands in for ~5/4, ~19/15, and ~24/19.		Augmented works on the [[2.3.5.19 subgroup]] very well, where the 1/3-octave period stands in for ~5/4, ~19/15, and ~24/19.

	[[Subgroup]]: 2.3.5.19		[[Subgroup]]: 2.3.5.19
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	{{Mapping\|legend=2\| 3 0 7 8 \| 0 1 0 1 }}		{{Mapping\|legend=2\| 3 0 7 8 \| 0 1 0 1 }}

	~~: sval mapping generators: ~5/4, ~3~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~5/4 = 399.~~075~~{{c}}, ~3/2 = 705.~~062~~{{c}} (~19/16 = ~~305~~.~~986~~{{c}})		* [[WE]]: ~5/4 = 399.0754{{c}}, ~3/2 = 705.0615{{c}} (~19/18 = 93.0893{{c}})
	* [[CWE]]: ~5/4 = 400.~~000~~{{c}}, ~3/2 = 705.~~107~~{{c}} (~19/16 = ~~305~~.~~107~~{{c}})		* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.1072{{c}} (~19/18 = 94.8928{{c}})

	{{Optimal ET sequence\|legend=1\| 3, 9, 12, 27, 39 }}		{{Optimal ET sequence\|legend=1\| 3, 9, 12, 27, 39, 51c, 90cch }}

	[[Badness]] (Sintel): 0.264		[[Badness]] (Sintel): 0.264

Xenllium: /* Subgroup extensions */

2026-02-22T03:37:26Z

Subgroup extensions

← Older revision		Revision as of 03:37, 22 February 2026
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	== Subgroup extensions ==		== Subgroup extensions ==
	=== 2.3.5.19 subgroup (3 & 12) ===		=== 2.3.5.19 subgroup (3 & 12) ===
	Augmented temperament works on the 2.3.5.19 subgroup very well, ~~tempering out both [[76~~/~~75]]~~ and ~~[[96~~/~~95]], alongside the augmented comma~~.		Augmented temperament works on the 2.3.5.19 subgroup very well, where the 1/3-octave period stands in for ~5/4, ~19/15, and ~24/19.

	[[Subgroup]]: 2.3.5.19		[[Subgroup]]: 2.3.5.19

Xenllium at 23:43, 21 February 2026

2026-02-21T23:43:31Z

← Older revision		Revision as of 23:43, 21 February 2026
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	{{Optimal ET sequence\|legend=1\| 3b, 9bd, 12 }}		{{Optimal ET sequence\|legend=1\| 3b, 9bd, 12 }}

			[[Badness]] (Sintel): 3.50

	=== 11-limit ===		=== 11-limit ===
	~~{{Todo\|unify precision}}~~
	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

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	Optimal tunings:		Optimal tunings:
	* WE: ~5/4 = 397.495{{c}}, ~15/11 = 499.096{{c}} (~15/14 = 101.601{{c}})		* WE: ~5/4 = 397.495{{c}}, ~15/11 = 499.096{{c}} (~12/11 = 101.601{{c}})
	* CWE: ~5/4 = 400.000{{c}}, ~15/11 = 500.775{{c}} (~15/14 = 100.775{{c}})		* CWE: ~5/4 = 400.000{{c}}, ~15/11 = 500.775{{c}} (~12/11 = 100.775{{c}})

			{{Optimal ET sequence\|legend=0\| 3b, 9bde, 12 }}

			Badness (Sintel): 2.60

			== Subgroup extensions ==
			=== 2.3.5.19 subgroup (3 & 12) ===
			Augmented temperament works on the 2.3.5.19 subgroup very well, tempering out both [[76/75]] and [[96/95]], alongside the augmented comma.

			[[Subgroup]]: 2.3.5.19

			[[Comma list]]: 76/75, 96/95

			{{Mapping\|legend=2\| 3 0 7 8 \| 0 1 0 1 }}

			: sval mapping generators: ~5/4, ~3

			[[Optimal tuning]]s:
			* [[WE]]: ~5/4 = 399.075{{c}}, ~3/2 = 705.062{{c}} (~19/16 = 305.986{{c}})
			* [[CWE]]: ~5/4 = 400.000{{c}}, ~3/2 = 705.107{{c}} (~19/16 = 305.107{{c}})

	{{Optimal ET sequence\|legend=1\| 3b, ~~9bde~~, 12 }}		{{Optimal ET sequence\|legend=1\| 3, 9, 12, 27, 39 }}

	[[Badness]] (Sintel): 2.60		[[Badness]] (Sintel): 0.264

	== External links ==		== External links ==