4459/4455 - Revision history

FloraC: /* Temperaments */

2026-03-22T11:07:12Z

Temperaments

← Older revision		Revision as of 11:07, 22 March 2026
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''tristanismic''' temperament, or in the 3.5.7.11.13 subgroup, the '''tristanic''' temperament.		[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''tristanismic''' temperament, or in the 3.5.7.11.13 subgroup, the rank-4 '''tristanic''' temperament.

	=== Tristanic ===		=== Tristanic ===

FloraC: Update data

2026-03-22T11:05:54Z

Update data

← Older revision		Revision as of 11:05, 22 March 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}

	'''4459/4455''', the '''tristanisma''', is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] no-twos [[comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[odd-particular]]s [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the [[superparticular ratio]] pairs ([[2080/2079]])⋅([[2401/2400]]) or ([[1001/1000]])⋅([[9801/9800]]).		'''4459/4455''', the '''tristanisma''', is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] no-twos [[comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[odd-particular]]s [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the [[superparticular ratio]] pairs ([[2080/2079]])⋅([[2401/2400]]) or ([[1001/1000]])⋅([[9801/9800]]).

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''tristanismic''' temperament, or in the 3.5.7.11.13 subgroup, the '''tristanic''' temperament.		[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''tristanismic''' temperament, or in the 3.5.7.11.13 subgroup, the '''tristanic''' temperament.

			=== Tristanic ===
			[[Subgroup]]: 3.5.7.11.13

			[[Comma list]]: 4459/4455

			{{Mapping\|legend=2\| 1 0 0 0 4 \| 0 1 0 0 1 \| 0 0 1 0 -3 \| 0 0 0 1 1 }}
			: mapping generators: ~3, ~5, ~7, ~11

			[[Optimal tuning]]s:
			* [[WE]]: ~3 = 1902.0648{{c}}, ~5/3 = 884.3078{{c}}, ~7/3 = 1466.5027{{c}}, ~11/9 = 347.3191{{c}}
			* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 884.3336{{c}}, ~7/3 = 1466.4567{{c}}, ~11/9 = 347.4207{{c}}

			[[Optimal ET sequence]]: [[13edt\|b13]], [[17edt\|b17]], [[22edt\|b22]], [[26edt\|b26]], [[36edt\|b36]], [[39edt\|b39]], [[92edt\|b92]], [[114edt\|b114]], [[131edt\|b131]], [[170edt\|b170]], [[245edt\|b245]], [[284edt\|b284]], [[415edt\|b415]], [[428edt\|b428]], [[559edt\|b559]], [[598edt\|b598]], [[712edt\|b712]], [[843edt\|b843]], [[882edt\|b882]], [[1725edt\|b1725]], [[4332edt\|b4332ddf]]

			[[Badness]] (Sintel): 0.0885

	=== Tristanismic ===		=== Tristanismic ===
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| ]]
	\|}		\|}
			: mapping generators: ~2, ~3, ~5, ~7, ~11

	[[Optimal tuning]] ([[CWE]]): ~2, ~3 = ~~1902~~.~~065~~{{c}}, ~5 = ~~2786~~.~~373~~{{c}}, ~7 = ~~3368~~.~~568~~{{c}}, ~11 = ~~4151~~.~~449~~{{c}}		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0648{{c}}, ~5/4 = 386.3726{{c}}, ~7/4 = 968.5675{{c}}, ~11/8 = 551.4487{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0648{{c}}, ~5/4 = 386.3726{{c}}, ~7/4 = 968.5675{{c}}, ~11/8 = 551.4487{{c}}

	{{Optimal ET sequence\|legend=1\| ~~8d, 9, 12, 14cf,~~ 17c, 19e, 22, 31f, 41, 58, 72, 118, 121, 130, 190, 198, 270, 581, 711, 981, 1171, 1441, 1752, 2022, 2463d, 2733d }}		{{Optimal ET sequence\|legend=1\| 17c, 19e, 22, 31f, 41, 58, 72, 118, 121, 130, 190, 198, 270, 581, 711, 981, 1171, 1441, 1752, 2022, 2463d, 2733d }}

	[[Badness]] (Sintel): 0.963		[[Badness]] (Sintel): 0.963

	~~=== Tristanic ===~~
	~~[[Subgroup]]: 3.5.7.11.13~~

	~~[[Comma list]]: 4459/4455~~

	~~{{Mapping\|legend=2\| 1 0 0 0 4 \| 0 1 0 0 1 \| 0 0 1 0 -3 \| 0 0 0 1 1 }}~~

	~~[[Optimal tuning]] ([[CWE]]): ~3, ~5 = 2786.289{{c}}, ~7 = 3368.412{{c}}, ~11 = 4151.331{{c}}~~

	[[Optimal ET sequence]]: [[9edt\|b9]], [[13edt\|b13]], [[17edt\|b17]], [[22edt\|b22]], [[26edt\|b26]], [[36edt\|b36]], [[39edt\|b39]], [[92edt\|b92]], [[114edt\|b114]], [[131edt\|b131]], [[170edt\|b170]], [[245edt\|b245]], [[284edt\|b284]], [[415edt\|b415]], [[428edt\|b428]], [[559edt\|b559]], [[598edt\|b598]], [[712edt\|b712]], [[843edt\|b843]], [[882edt\|b882]], [[1725edt\|b1725]], [[4332edt\|b4332ddf]]

	~~[[Badness]] (Sintel): 0.089~~

	== Etymology ==		== Etymology ==

FloraC: Cleanup. + categories

2026-03-22T10:59:27Z

Cleanup. + categories

← Older revision		Revision as of 10:59, 22 March 2026
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	}}		}}

	'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[~~unnoticeable~~ comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[~~Odd~~-particular~~\|odd-particulars~~]] [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]).		'''4459/4455''', the '''tristanisma''', is an [[unnoticeable comma\|unnoticeable]] [[13-limit]] no-twos [[comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[odd-particular]]s [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the [[superparticular ratio]] pairs ([[2080/2079]])⋅([[2401/2400]]) or ([[1001/1000]])⋅([[9801/9800]]).

	== ~~Etymology~~ ==		== Temperaments ==
	~~This comma was named by~~ [[~~User:Tristanbay\|Tristan Bay~~]] in ~~early May 2024 after himself~~.		[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''tristanismic''' temperament, or in the 3.5.7.11.13 subgroup, the '''tristanic''' temperament.

	~~== Temperaments ==~~
	=== Tristanismic ===		=== Tristanismic ===
	~~Tempering out 4459/4455 in the full 13-limit leads to this rank-5 temperament.~~

	[[Subgroup]]: 2.3.5.7.11.13		[[Subgroup]]: 2.3.5.7.11.13

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	[[Mapping]]:<br>		[[Mapping]]:<br>
	{~~{val~~\| 1 0 0 0 0 0 ~~}}<br>~~		{\| class="right-all"
	~~{{val~~\| 0 1 0 0 0 4 ~~}}<br>~~		\|-
	~~{{val~~\| 0 0 1 0 0 1 ~~}}<br>~~		\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| ],
	~~{{val~~\| 0 0 0 1 0 -3 ~~}}<br>~~		\|-
	~~{{val~~\| 0 0 0 0 1 1 }}		\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 4 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| -3 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| ]]
			\|}

	[[Optimal tuning]] ([[CWE]]): 2, ~3 = 1902.065, ~5 = 2786.373, ~7 = 3368.568, ~11 = 4151.449		[[Optimal tuning]] ([[CWE]]): ~2, ~3 = 1902.065{{c}}, ~5 = 2786.373{{c}}, ~7 = 3368.568{{c}}, ~11 = 4151.449{{c}}

	{{Optimal ET sequence\|legend=1\| 8d, 9, 12, 14cf, 17c, 19e, 22, 31f, 41, 58, 72, 118, 121, 130, 190, 198, 270, 581, 711, 981, 1171, 1441, 1752, 2022, 2463d, 2733d }}		{{Optimal ET sequence\|legend=1\| 8d, 9, 12, 14cf, 17c, 19e, 22, 31f, 41, 58, 72, 118, 121, 130, 190, 198, 270, 581, 711, 981, 1171, 1441, 1752, 2022, 2463d, 2733d }}
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	=== Tristanic ===		=== Tristanic ===
	~~This is the main temperament that tempers out 4459/4455 in the no-twos 13-limit.~~

	[[Subgroup]]: 3.5.7.11.13		[[Subgroup]]: 3.5.7.11.13

	[[Comma list]]: 4459/4455		[[Comma list]]: 4459/4455

	~~[[Mapping]]:<br>~~		{{Mapping\|legend=2\| 1 0 0 0 4 \| 0 1 0 0 1 \| 0 0 1 0 -3 \| 0 0 0 1 1 }}
	{{~~val~~\|1 0 0 0 4 ~~}}<br>~~
	~~{{val~~\|0 1 0 0 1 ~~}}<br>~~
	~~{{val~~\|0 0 1 0 -3 ~~}}<br>~~
	~~{{val~~\|0 0 0 1 1 }}

	[[Optimal tuning]] ([[CWE]]): 3, ~5 = 2786.289, ~7 = 3368.412, ~11 = 4151.331		[[Optimal tuning]] ([[CWE]]): ~3, ~5 = 2786.289{{c}}, ~7 = 3368.412{{c}}, ~11 = 4151.331{{c}}

	[[Optimal ET sequence]]: [[9edt\|b9]], [[13edt\|b13]], [[17edt\|b17]], [[22edt\|b22]], [[26edt\|b26]], [[36edt\|b36]], [[39edt\|b39]], [[92edt\|b92]], [[114edt\|b114]], [[131edt\|b131]], [[170edt\|b170]], [[245edt\|b245]], [[284edt\|b284]], [[415edt\|b415]], [[428edt\|b428]], [[559edt\|b559]], [[598edt\|b598]], [[712edt\|b712]], [[843edt\|b843]], [[882edt\|b882]], [[1725edt\|b1725]], [[4332edt\|b4332ddf]]		[[Optimal ET sequence]]: [[9edt\|b9]], [[13edt\|b13]], [[17edt\|b17]], [[22edt\|b22]], [[26edt\|b26]], [[36edt\|b36]], [[39edt\|b39]], [[92edt\|b92]], [[114edt\|b114]], [[131edt\|b131]], [[170edt\|b170]], [[245edt\|b245]], [[284edt\|b284]], [[415edt\|b415]], [[428edt\|b428]], [[559edt\|b559]], [[598edt\|b598]], [[712edt\|b712]], [[843edt\|b843]], [[882edt\|b882]], [[1725edt\|b1725]], [[4332edt\|b4332ddf]]

	[[Badness]] (Sintel): 0.089		[[Badness]] (Sintel): 0.089

			== Etymology ==
			This comma was named by [[Tristan Bay]] in early May 2024 after himself.

	[[Category:Tristanismic]]		[[Category:Tristanismic]]
			[[Category:Commas named after composers]]
			[[Category:Commas named after music theorists]]
			[[Category:Commas someone named after themselves]]

Tristanbay: /* Tristanic */Added optimal ET sequence, courtesy of FloraC

2025-11-10T17:18:56Z

Tristanic: Added optimal ET sequence, courtesy of FloraC

← Older revision		Revision as of 17:18, 10 November 2025
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	[[Optimal tuning]] ([[CWE]]): 3, ~5 = 2786.289, ~7 = 3368.412, ~11 = 4151.331		[[Optimal tuning]] ([[CWE]]): 3, ~5 = 2786.289, ~7 = 3368.412, ~11 = 4151.331

			[[Optimal ET sequence]]: [[9edt\|b9]], [[13edt\|b13]], [[17edt\|b17]], [[22edt\|b22]], [[26edt\|b26]], [[36edt\|b36]], [[39edt\|b39]], [[92edt\|b92]], [[114edt\|b114]], [[131edt\|b131]], [[170edt\|b170]], [[245edt\|b245]], [[284edt\|b284]], [[415edt\|b415]], [[428edt\|b428]], [[559edt\|b559]], [[598edt\|b598]], [[712edt\|b712]], [[843edt\|b843]], [[882edt\|b882]], [[1725edt\|b1725]], [[4332edt\|b4332ddf]]

	[[Badness]] (Sintel): 0.089		[[Badness]] (Sintel): 0.089

	[[Category:Tristanismic]]		[[Category:Tristanismic]]

Tristanbay: /* Temperaments */ Added no-twos temperament

2025-11-09T01:09:56Z

Temperaments: Added no-twos temperament

← Older revision		Revision as of 01:09, 9 November 2025
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	[[Badness]] (Sintel): 0.963		[[Badness]] (Sintel): 0.963

			=== Tristanic ===
			This is the main temperament that tempers out 4459/4455 in the no-twos 13-limit.

			[[Subgroup]]: 3.5.7.11.13

			[[Comma list]]: 4459/4455

			[[Mapping]]:<br>
			{{val\|1 0 0 0 4 }}<br>
			{{val\|0 1 0 0 1 }}<br>
			{{val\|0 0 1 0 -3 }}<br>
			{{val\|0 0 0 1 1 }}

			[[Optimal tuning]] ([[CWE]]): 3, ~5 = 2786.289, ~7 = 3368.412, ~11 = 4151.331

			[[Badness]] (Sintel): 0.089

	[[Category:Tristanismic]]		[[Category:Tristanismic]]

Tristanbay: Moved the temperament into its own section

2025-11-07T08:05:37Z

Moved the temperament into its own section

@@ Line 5: / Line 5: @@
 }}
-'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[Odd-particular|odd-particulars]] [[245/243]] and [[275/273]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[EDO|edos]] including {{EDOs | 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.
+'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[Odd-particular|odd-particulars]] [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]).
 == Etymology ==
 This comma was named by [[User:Tristanbay|Tristan Bay]] in early May 2024 after himself.
 [[Category:Tristanismic]]

Tristanbay: Formatting

2025-09-28T20:34:26Z

Formatting

← Older revision		Revision as of 20:34, 28 September 2025
Line 5:		Line 5:
	}}		}}

	'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[Odd-particular]]~~<nowiki/>s~~ [[245/243]] and [[275/273]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[EDO\|edos]] including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.		'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[Odd-particular\|odd-particulars]] [[245/243]] and [[275/273]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[EDO\|edos]] including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.

	== Etymology ==		== Etymology ==

Abnormality: I added the relationship between the tristanisma and 245/243 and 275/273.

2024-10-19T06:13:57Z

I added the relationship between the tristanisma and 245/243 and 275/273.

← Older revision		Revision as of 06:13, 19 October 2024
Line 5:		Line 5:
	}}		}}

	'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[EDO\|edos]] including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.		'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[Odd-particular]]<nowiki/>s [[245/243]] and [[275/273]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[EDO\|edos]] including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.

	== Etymology ==		== Etymology ==

Tristanbay: Fixed link redirect for EDOs and moved the "s" into the link

2024-05-08T11:11:11Z

Fixed link redirect for EDOs and moved the "s" into the link

← Older revision		Revision as of 11:11, 8 May 2024
Line 5:		Line 5:
	}}		}}

	'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[~~edo~~]]s including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.		'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[EDO\|edos]] including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.

	== Etymology ==		== Etymology ==

Tristanbay: Added another pair of superparticular factors

2024-05-07T10:35:03Z

Added another pair of superparticular factors

← Older revision		Revision as of 10:35, 7 May 2024
Line 5:		Line 5:
	}}		}}

	'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It factors into ~~simpler commas as~~ ([[2080/2079]])([[2401/2400]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[edo]]s including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.		'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]). Tempering it out and adding the octave back in yields the rank-5 [[tristanismic]] temperament, which is supported by [[edo]]s including {{EDOs \| 9, 12, 22, 27, 41, 72, 130, 190, 198, 270, and 311}}.

	== Etymology ==		== Etymology ==