Undim family - Revision history

FloraC: Naming of unlit and twilight

2026-03-04T10:03:52Z

Naming of unlit and twilight

← Older revision		Revision as of 10:03, 4 March 2026
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	== Unlit ==		== Unlit ==
	Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap\| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity.		Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap\| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity. It was named by [[Flora Canou]] in 2021 for it is some sort of opposite of [[#Septimal undim\|septimal undim]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Twilight ==		== Twilight ==
	Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap\| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot.		Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap\| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot. It was named by [[Flora Canou]] in 2021 for it splits the period in halves to reconcile the different mappings of [[#Septimal undim\|septimal undim]] and [[#Unlit\|unlit]].

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

FloraC: - CTE & POTE tunings

2026-01-04T16:58:37Z

- CTE & POTE tunings

← Older revision		Revision as of 16:58, 4 January 2026
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	* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}		* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}
	: error map: {{val\| 0.000 +0.670 +0.560 }}		: error map: {{val\| 0.000 +0.670 +0.560 }}
	~~<!-- * [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}~~
	~~: [[error map]]: {{val\| 0.0000 +0.7204 +0.3092 }}~~
	* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}
	~~: error map: {{val\| 0.0000 +0.6504 +0.6591 }} -->~~

	{{Optimal ET sequence\|legend=1\| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}		{{Optimal ET sequence\|legend=1\| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}
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	* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}		* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}
	: error map: {{val\| 0.000 +0.798 -0.078 +0.894 }}		: error map: {{val\| 0.000 +0.798 -0.078 +0.894 }}
	~~<!-- * [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}~~
	~~: [[error map]]: {{val\| 0.0000 +0.8398 -0.2879 +0.4308 }}~~
	* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}
	~~: error map: {{val\| 0.0000 +0.7812 +0.0051 +1.0754 }} -->~~

	{{Optimal ET sequence\|legend=1\| 12, …, 128, 140, 152, 292 }}		{{Optimal ET sequence\|legend=1\| 12, …, 128, 140, 152, 292 }}
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	* WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}		* WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}
	* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}		* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}
	~~<!-- * CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}~~
	* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12, 128e, 140, 152, 292, 444d, 596d }}		{{Optimal ET sequence\|legend=0\| 12, 128e, 140, 152, 292, 444d, 596d }}
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	* WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}		* WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}
	* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}		* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}
	~~<!-- * CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}~~
	* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}} -->

	{{Optimal ET sequence\|legend=0\| 12f, 128ef, 140, 152f, 292 }}		{{Optimal ET sequence\|legend=0\| 12f, 128ef, 140, 152f, 292 }}
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	* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}		* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}
	: error map: {{val\| 0.000 +0.617 +0.828 +0.607 }}		: error map: {{val\| 0.000 +0.617 +0.828 +0.607 }}
	~~<!-- * [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}~~
	~~: [[error map]]: {{val\| 0.0000 +0.6006 +0.9081 +0.1761 }}~~
	* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}
	~~: error map: {{val\| 0.0000 +0.6214 +0.8043 +0.7369 }} -->~~

	{{Optimal ET sequence\|legend=1\| 152, 316, 468, 620, 1088bcd, 1708bccdd }}		{{Optimal ET sequence\|legend=1\| 152, 316, 468, 620, 1088bcd, 1708bccdd }}
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	* WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}		* WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}
	* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}		* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}
	~~<!-- * CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}~~
	* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}} -->

	{{Optimal ET sequence\|legend=0\| 152, 468, 620 }}		{{Optimal ET sequence\|legend=0\| 152, 468, 620 }}
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	* WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}		* WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}
	* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}		* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}
	~~<!-- * CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}~~
	* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}} -->

	{{Optimal ET sequence\|legend=0\| 152f, 316, 468, 620f, 1088bcdf }}		{{Optimal ET sequence\|legend=0\| 152f, 316, 468, 620f, 1088bcdf }}
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	* [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}		* [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}
	: error map: {{val\| 0.000 +0.547 +1.175 +1.192 }}		: error map: {{val\| 0.000 +0.547 +1.175 +1.192 }}
	~~<!-- * [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}~~
	~~: [[error map]]: {{val\| 0.0000 +0.5215 +1.3036 +0.9865 }}~~
	* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
	~~: error map: {{val\| 0.0000 +0.5540 +1.1415 +1.2457 }} -->~~

	{{Optimal ET sequence\|legend=1\| 152, 328, 480, 1592bccddd }}		{{Optimal ET sequence\|legend=1\| 152, 328, 480, 1592bccddd }}
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	* WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}		* WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}
	* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}		* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}
	~~<!-- * CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}~~
	* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}} -->

	{{Optimal ET sequence\|legend=0\| 152, 328, 480, 1112bccddee, 1592bccdddeee }}		{{Optimal ET sequence\|legend=0\| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
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	* WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}		* WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}
	* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4670{{c}}		* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4670{{c}}
	~~<!-- * CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}~~
	* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{c}} -->

	{{Optimal ET sequence\|legend=0\| 152f, 328, 480f, 808cdeff }}		{{Optimal ET sequence\|legend=0\| 152f, 328, 480f, 808cdeff }}

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-01-04T16:57:17Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Review & expand

2026-01-04T16:36:39Z

Review & expand

← Older revision		Revision as of 16:36, 4 January 2026
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	== Undim ==		== Undim ==
	Undim features a quarter-octave period, which acts as the [[1215/1024\|ptolemaic augmented second (1215/1024)]]. That and five [[4/3\|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5. Undim equates the [[Pythagorean comma]] with a stack of four [[schisma]]s~~. This makes~~ it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap\| ''n'' {{=}} 4 }}.		Undim features a quarter-octave period, which acts as the [[1215/1024\|ptolemaic augmented second (1215/1024)]]. That and five [[4/3\|perfect fourths]] (i.e. a minor second, [[~]][[256/243]]) give the interval class of 5. Undim has a [[ploidacot]] of tetraploid monocot, and equates the [[Pythagorean comma]] with a stack of four [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap\| ''n'' {{=}} 4 }}.

	Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo\|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.

	The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)\|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group \| ''Suggested names for the unclasified temperaments'']</ref>.		The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)\|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group \| ''Suggested names for the unclasified temperaments'']</ref>.
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	=== Overview to extensions ===		=== Overview to extensions ===
	The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap\| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap\| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap\| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.		The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap\| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap\| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap\| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.

			Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious place for prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo\|1\4]].

	== Septimal undim ==		== Septimal undim ==
	Septimal undim tempers out the [[dimcomp comma]], mapping ~25/21 to the 1/4-octave period. It can be described as {{nowrap\| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)\|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.		Septimal undim tempers out the [[dimcomp comma]], mapping [[~]][[25/21]] to the 1/4-octave period. It can be described as {{nowrap\| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)\|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Unlit ==		== Unlit ==
			Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap\| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity.

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

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	== Twilight ==		== Twilight ==
			Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap\| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot.

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

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	{{Mapping\|legend=1\| 8 0 82 -79 \| 0 1 -5 8 }}		{{Mapping\|legend=1\| 8 0 82 -79 \| 0 1 -5 8 }}

	: mapping generators: ~7168/6561, ~3		: mapping generators: ~7168/6561, ~3

FloraC: /* Notes */ -> references. - redundant category

2026-01-03T14:10:26Z

Notes: -> references. - redundant category

← Older revision		Revision as of 14:10, 3 January 2026
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	Badness (Smith): 0.041365		Badness (Smith): 0.041365

	== ~~Notes~~ ==		== References ==

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Undim family\| ]] <!-- main article -->		[[Category:Undim family\| ]] <!-- main article -->
	~~[[Category:Undim\| ]] <!-- key article -->~~
	[[Category:Rank 2]]		[[Category:Rank 2]]

FloraC: Units

2026-01-03T14:09:33Z

Units

← Older revision		Revision as of 14:09, 3 January 2026
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	{{Mapping\|legend=1\| 4 0 41 \| 0 1 -5 }}		{{Mapping\|legend=1\| 4 0 41 \| 0 1 -5 }}

	: mapping generators: ~1215/1024, ~3		: mapping generators: ~1215/1024, ~3

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.6754		* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}
	: [[error map]]: {{val\| 0.0000 +0.7204 +0.3092 }}		: [[error map]]: {{val\| 0.0000 +0.7204 +0.3092 }}
	* [[POTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.6054		* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}
	: error map: {{val\| 0.0000 +0.6504 +0.6591 }}		: error map: {{val\| 0.0000 +0.6504 +0.6591 }}

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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~25/21 = 300.0000, ~3/2 = 702.7948		* [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}
	: [[error map]]: {{val\| 0.0000 +0.8398 -0.2879 +0.4308 }}		: [[error map]]: {{val\| 0.0000 +0.8398 -0.2879 +0.4308 }}
	* [[POTE]]: ~25/21 = 300.0000, ~3/2 = 702.7362		* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}
	: error map: {{val\| 0.0000 +0.7812 +0.0051 +1.0754 }}		: error map: {{val\| 0.0000 +0.7812 +0.0051 +1.0754 }}

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	Optimal tunings:		Optimal tunings:
	* CTE: ~25/21 = 300.0000, ~3/2 = 702.7433		* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}
	* POTE: ~25/21 = 300.0000, ~3/2 = 702.6886		* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}}

	{{Optimal ET sequence\|legend=0\| 140, 152, 292, 444d, 596d }}		{{Optimal ET sequence\|legend=0\| 140, 152, 292, 444d, 596d }}
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	Optimal tunings:		Optimal tunings:
	* CTE: ~25/21 = 300.0000, ~3/2 = 702.7792		* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}
	* POTE: ~25/21 = 300.0000, ~3/2 = 702.7363		* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}}

	{{Optimal ET sequence\|legend=0\| 140, 152f, 292 }}		{{Optimal ET sequence\|legend=0\| 140, 152f, 292 }}
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.5556		* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}
	: [[error map]]: {{val\| 0.0000 +0.6006 +0.9081 +0.1761 }}		: [[error map]]: {{val\| 0.0000 +0.6006 +0.9081 +0.1761 }}
	* [[POTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.5764		* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}
	: error map: {{val\| 0.0000 +0.6214 +0.8043 +0.7369 }}		: error map: {{val\| 0.0000 +0.6214 +0.8043 +0.7369 }}

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	Optimal tunings:		Optimal tunings:
	* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5582		* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}
	* POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5826		* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}}

	{{Optimal ET sequence\|legend=0\| 152, 468, 620 }}		{{Optimal ET sequence\|legend=0\| 152, 468, 620 }}
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	Optimal tunings:		Optimal tunings:
	* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5562		* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}
	* POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5741		* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}}

	{{Optimal ET sequence\|legend=0\| 152f, 316, 468, 620f, 1088bcdf }}		{{Optimal ET sequence\|legend=0\| 152f, 316, 468, 620f, 1088bcdf }}
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~7168/6561 = 150.0000, ~3/2 = 702.4765		* [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}
	: [[error map]]: {{val\| 0.0000 +0.5215 +1.3036 +0.9865 }}		: [[error map]]: {{val\| 0.0000 +0.5215 +1.3036 +0.9865 }}
	* [[POTE]]: ~7168/6561 = 150.0000, ~3/2 = 702.5090		* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
	: error map: {{val\| 0.0000 +0.5540 +1.1415 +1.2457 }}		: error map: {{val\| 0.0000 +0.5540 +1.1415 +1.2457 }}

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	Optimal tunings:		Optimal tunings:
	* CTE: ~12/11 = 150.0000, ~3/2 = 702.4692		* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}
	* POTE: ~12/11 = 150.0000, ~3/2 = 702.5090		* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}

	{{Optimal ET sequence\|legend=0\| 152, 328, 480, 1112bccddee, 1592bccdddeee }}		{{Optimal ET sequence\|legend=0\| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
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	Optimal tunings:		Optimal tunings:
	* CTE: ~12/11 = 150.0000, ~3/2 = 702.4168		* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}
	* POTE: ~12/11 = 150.0000, ~3/2 = 702.4773		* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{c}}

	{{Optimal ET sequence\|legend=0\| 152f, 328, 480f, 808cdeff }}		{{Optimal ET sequence\|legend=0\| 152f, 328, 480f, 808cdeff }}

FloraC: Rework on intro and head

2026-01-03T13:30:46Z

Rework on intro and head

← Older revision		Revision as of 13:30, 3 January 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''undim family''' of [[regular temperaments\|temperaments]] [[tempering out\|tempers out]] the [[undim comma]], {{monzo\| 41 -20 -4 }}~~, equating the [[Pythagorean comma]] with a stack of four [[schisma]]s. This makes it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap\| ''n'' {{=}} 4 }}.~~		The '''undim family''' of [[regular temperaments\|temperaments]] [[tempering out\|tempers out]] the [[undim comma]], {{monzo\| 41 -20 -4 }}.

	The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)\|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group \| ''Suggested names for the unclasified temperaments'']</ref>.

	The second comma of the [[normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap\| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap\| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap\| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.

	== Undim ==		== Undim ==
	Undim features a quarter-octave period, which acts as the [[1215/1024\|ptolemaic augmented second (1215/1024)]]. That and five [[4/3\|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5.		Undim features a quarter-octave period, which acts as the [[1215/1024\|ptolemaic augmented second (1215/1024)]]. That and five [[4/3\|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5. Undim equates the [[Pythagorean comma]] with a stack of four [[schisma]]s. This makes it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap\| ''n'' {{=}} 4 }}.

	Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo\|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.		Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo\|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.

			The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)\|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group \| ''Suggested names for the unclasified temperaments'']</ref>.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	[[Badness]] (Smith): 0.241703		[[Badness]] (Smith): 0.241703

			=== Overview to extensions ===
			The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap\| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap\| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap\| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.

	== Septimal undim ==		== Septimal undim ==

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