Würschmidt family - Revision history

FloraC: Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "

2026-09-30T17:44:30Z

Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "

← Older revision		Revision as of 17:44, 30 September 2026
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	Comma list: 243/242, 5632/5625		Comma list: 243/242, 5632/5625

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 -1 2 -3 \| 0 8 1 20 }}		{{Mapping\|legend=2\| 1 -1 2 -3 \| 0 8 1 20 }}

	Optimal tuning:		Optimal tuning:
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	Comma list: 243/242, 276/275, 529/528		Comma list: 243/242, 276/275, 529/528

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 -1 2 -3 0 \| 0 8 1 20 14 }}		{{Mapping\|legend=2\| 1 -1 2 -3 0 \| 0 8 1 20 14 }}

	Optimal tuning:		Optimal tuning:
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	Comma list: 576/575, 12167/12150		Comma list: 576/575, 12167/12150

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 -1 2 0 \| 0 8 1 14 }}		{{Mapping\|legend=2\| 1 -1 2 0 \| 0 8 1 14 }}

	Optimal tunings:		Optimal tunings:

FloraC: Fixes

2026-09-29T08:31:47Z

Fixes

← Older revision		Revision as of 08:31, 29 September 2026
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	Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.		Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.

	2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a [[2.3.7.11-subgroup\|2.3.7.11 subgroup]] temperament.		2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a [[2.3.7.11 subgroup\|2.3.7.11-subgroup]] temperament.

	The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175\|S8/S10]], [[243/242\|S9/S11]], [[225/224\|S15]]}. Tempering out [[81/80\|S9]] or [[121/120\|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63\|S8]] or [[100/99\|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.		The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175\|S8/S10]], [[243/242\|S9/S11]], [[225/224\|S15]]}. Tempering out [[81/80\|S9]] or [[121/120\|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63\|S8]] or [[100/99\|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.
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	== Other subgroup extensions ==		== Other subgroup extensions ==
	=== Würschmidt (2.3.5.23) ===		=== Würschmidt (2.3.5.23) ===
	Extensions to harmonics [[47/1\|47]] and [[49/1\|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment\|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1\|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].		Extensions for harmonics [[47/1\|47]] and [[49/1\|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment\|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1\|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].

	Subgroup: 2.3.5.23		Subgroup: 2.3.5.23

FloraC: Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

2026-09-24T12:32:16Z

Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

← Older revision		Revision as of 12:32, 24 September 2026
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	Comma list: 99/98, 176/175, 243/242		Comma list: 99/98, 176/175, 243/242

	~~Mapping:~~ {{~~mapping~~\| 1 -1 2 -3 -3 \| 0 8 1 18 20 }}		{{Mapping\|legend=0\| 1 -1 2 -3 -3 \| 0 8 1 18 20 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 99/98, 144/143, 176/175, 275/273		Comma list: 99/98, 144/143, 176/175, 275/273

	~~Mapping:~~ {{~~mapping~~\| 1 -1 2 -3 -3 5 \| 0 8 1 18 20 -4 }}		{{Mapping\|legend=0\| 1 -1 2 -3 -3 5 \| 0 8 1 18 20 -4 }}

	Optimal tunings:		Optimal tunings:
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	Commas: 66/65, 99/98, 105/104, 243/242		Commas: 66/65, 99/98, 105/104, 243/242

	~~Mapping:~~ {{~~mapping~~\| 1 -1 2 -3 -3 -5 \| 0 8 1 18 20 27 }}		{{Mapping\|legend=0\| 1 -1 2 -3 -3 -5 \| 0 8 1 18 20 27 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 126/125, 243/242, 385/384		Comma list: 126/125, 243/242, 385/384

	~~Mapping:~~ {{~~mapping~~\| 1 -1 2 7 -3 \| 0 8 1 -13 20 }}		{{Mapping\|legend=0\| 1 -1 2 7 -3 \| 0 8 1 -13 20 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 243/242, 896/891, 4375/4356		Comma list: 243/242, 896/891, 4375/4356

	~~Mapping:~~ {{~~mapping~~\| 1 -1 2 -14 -3 \| 0 8 1 52 20 }}		{{Mapping\|legend=0\| 1 -1 2 -14 -3 \| 0 8 1 52 20 }}

	Optimal tunings:		Optimal tunings:

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

2026-09-09T09:17:40Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Move part of the intro to the first section, too

2026-09-09T07:42:08Z

Move part of the intro to the first section, too

← Older revision		Revision as of 07:42, 9 September 2026
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	{{Technical data page}}		{{Technical data page}}
	The ~~[[5-limit]] parent comma for the~~ '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt~~. The [[generator]] is a [[5/4\|classical major third]], and to get to the interval class of 3 requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6~~.		The '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') of [[regular temperament\|temperaments]] [[tempering out\|tempers out]] [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt.

	Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo\|11\34]] and especially [[65edo\|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].		== Würschmidt ==
			{{Main\| Würschmidt }}

			The [[generator]] of würschmidt is a [[5/4\|classical major third]], and to get to the interval class of the [[3/1\|3rd harmonic]] requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.

			Similar to [[meantone]], würschmidt implies that [[3/2]] will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo\|11\34]] and especially [[65edo\|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].

	[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety\|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.		[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety\|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.

	~~== Würschmidt ==~~
	~~{{Main\| Würschmidt }}~~

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

FloraC: Re-organize and clarify a few things

2026-09-09T07:38:43Z

Re-organize and clarify a few things

← Older revision		Revision as of 07:38, 9 September 2026
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	{{Technical data page}}		{{Technical data page}}
	The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a ~~classic~~ major third, and to get to the interval class of ~~fifths~~ requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.		The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a [[5/4\|classical major third]], and to get to the interval class of 3 requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.

	Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo\|11\34]] and especially [[65edo\|21\65]], which notably is the point where it is combined with [[schismic~~]]/[[nestoria~~]] and [[gravity~~]]/[[larry~~]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].		Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo\|11\34]] and especially [[65edo\|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].

	~~Another tuning solution is to sharpen the major third by 1/8 of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]].~~		[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety\|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.

	[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[~~Proper~~]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.

	== Würschmidt ==		== Würschmidt ==
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]]; furthermore, like practically any 5-limit temperament with this accuracy level of [[3/2]] available, extensions to prime 19 exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo and [[nestoria]]).		Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]].

	~~However~~, ~~as discussed in the main article, the "free" higher~~ prime ~~for würschmidt outside the 5-limit is in fact 23, via~~ tempering out ~~S24 =~~ [[~~576~~/~~575~~]] ~~and S46<sup>2</sup> × S47 =~~ [[~~12167~~/~~12150~~]]~~. Therefore~~, ~~the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions~~.		With this accuracy level of [[3/2]] available, extensions that add prime [[19/1\|19]] exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo), but they are very complex.

	~~=== 2.3.~~5~~.23 subgroup ===~~		The "free" higher prime for würschmidt outside the 5-limit is in fact [[23/1\|23]], via tempering out [[576/575]] ({{S\|24}}) and [[12167/12150]] ([[S-expression\|S46<sup>2</sup>⋅S47]]). This is considered immediately below, and the no-11 restriction thereof is considered in [[#Other subgroup extensions]].
	~~Extensions to harmonics~~ [[47/1\|47]] ~~and~~ [[49/~~1\|49~~]] ~~are also available at +11~~ and ~~+5 generator steps respectively, equalizing the 45::50~~ [[~~harmonic series segment\|segment~~]] ~~of the~~ [[~~harmonic series~~]]. ~~If we derive a mapping of prime [[7/1\|7]] from this mapping of 49~~, ~~then we get~~ the ~~weak extension~~ [[~~hemiwürschmidt~~]].

	~~Subgroup: 2.3.5.23~~

	~~Comma list: 576/575, 12167/12150~~

	~~Subgroup-val mapping: {{mapping\| 1 -1 2 0 \| 0 8 1 14 }}~~

	~~Optimal tunings:~~
	* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}}
	* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.805{{c}}

	~~{{Optimal ET sequence\|legend=0\| 3, …, 28i, 31, 34, 65, 99, 164 }}~~

	~~Badness (Smith): 0.00530~~

	==== 2.3.5.11.23 subgroup ====		==== 2.3.5.11.23 subgroup ====
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	== Septimal würschmidt ==		== Septimal würschmidt ==
	~~Würschmidt~~, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.		Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.

	2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 subgroup temperament.		2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a [[2.3.7.11-subgroup\|2.3.7.11 subgroup]] temperament.

	The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175\|S8/S10]], [[243/242\|S9/S11]], [[225/224\|S15]]}. Tempering out [[81/80\|S9]] or [[121/120\|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63\|S8]] or [[100/99\|S10]] results in [[34edo]]~~, but specifically,~~ the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by ~~equating~~ them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap\| {[[5120/5103\|S8/S9]], [[8019/8000\|S9/S10]], [[4000/3993\|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap\| {S8/9/10/11} }}*. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7 ~~no matter what mapping you use (with the sharp 7 being used for this temperament)~~; even so, it's fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, ~~65d + 31 =~~ [[96edo]] is also a reasonable choice, as it has the advantage of being [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.		The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175\|S8/S10]], [[243/242\|S9/S11]], [[225/224\|S15]]}. Tempering out [[81/80\|S9]] or [[121/120\|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63\|S8]] or [[100/99\|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.

	(<nowiki>*</nowiki> The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).)

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Worschmidt ==		== Worschmidt ==
	Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is {{val\| 127 201 295 '''356''' }} (127d) and not {{val\| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.		Worschmidt tempers out [[126/125]] rather than [[225/224]], and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is {{val\| 127 201 295 '''356''' }} (127d) and not {{val\| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	Badness (Smith): 0.058325		Badness (Smith): 0.058325

			== Other subgroup extensions ==
			=== Würschmidt (2.3.5.23) ===
			Extensions to harmonics [[47/1\|47]] and [[49/1\|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment\|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1\|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].

			Subgroup: 2.3.5.23

			Comma list: 576/575, 12167/12150

			Subgroup-val mapping: {{mapping\| 1 -1 2 0 \| 0 8 1 14 }}

			Optimal tunings:
			* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}}
			* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.805{{c}}

			{{Optimal ET sequence\|legend=0\| 3, …, 28i, 31, 34, 65, 99, 164 }}

			Badness (Smith): 0.00530

	[[Category:Temperament families]]		[[Category:Temperament families]]

FloraC: Units; misc. cleanup; recategorize

2026-09-09T07:05:56Z

Units; misc. cleanup; recategorize

← Older revision		Revision as of 07:05, 9 September 2026
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	{{Mapping\|legend=1\| 1 -1 2 \| 0 8 1 }}		{{Mapping\|legend=1\| 1 -1 2 \| 0 8 1 }}

	: mapping generators: ~2, ~5/4		: mapping generators: ~2, ~5/4

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~5/4 = 387.734		* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}}
	* [[POTE]]: ~2 = ~~1\1~~, ~5/4 = 387.799		* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.799{{c}}

	{{Optimal ET sequence\|legend=1\| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}		{{Optimal ET sequence\|legend=1\| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}
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	Comma list: 576/575, 12167/12150		Comma list: 576/575, 12167/12150

	~~Sval~~ mapping: {{mapping\| 1 -1 2 0 \| 0 8 1 14 }}		Subgroup-val mapping: {{mapping\| 1 -1 2 0 \| 0 8 1 14 }}

	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.734		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}}
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.805		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.805{{c}}

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 3, …, 28i, 31, 34, 65, 99, 164 }}		{{Optimal ET sequence\|legend=0\| 3, …, 28i, 31, 34, 65, 99, 164 }}

	Badness (Smith): 0.00530		Badness (Smith): 0.00530
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	Comma list: 243/242, 276/275, 529/528		Comma list: 243/242, 276/275, 529/528

	~~Sval~~ mapping: {{mapping\| 1 -1 2 -3 0 \| 0 8 1 20 14 }}		Subgroup-val mapping: {{mapping\| 1 -1 2 -3 0 \| 0 8 1 20 14 }}

	Optimal tuning:		Optimal tuning:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.652		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.652{{c}}
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.690		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.690{{c}}

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 31, 34, 65 }}		{{Optimal ET sequence\|legend=0\| 31, 34, 65 }}

	Badness (Smith): 0.00660		Badness (Smith): 0.00660
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~5/4 = 387.379		* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.379{{c}}
	* [[POTE]]: ~2 = ~~1\1~~, ~5/4 = 387.383		* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.383{{c}}

	{{Optimal ET sequence\|legend=1\| 31, 96, 127 }}		{{Optimal ET sequence\|legend=1\| 31, 96, 127 }}
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	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.441		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.441{{c}}
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.447		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.447{{c}}

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 31, 65d, 96, 127 }}		{{Optimal ET sequence\|legend=0\| 31, 65d, 96, 127 }}

	Badness (Smith): 0.024413		Badness (Smith): 0.024413
Line 116:		Line 115:

	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.469		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.469{{c}}
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.626		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.626{{c}}

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 31, 65d }}		{{Optimal ET sequence\|legend=0\| 31, 65d }}

	Badness (Smith): 0.023593		Badness (Smith): 0.023593
Line 131:		Line 130:

	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.179		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.179{{c}}
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.099		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.099{{c}}

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 3def, 28def, 31 }}		{{Optimal ET sequence\|legend=0\| 3def, 28def, 31 }}

	Badness (Smith): 0.034382		Badness (Smith): 0.034382
Line 148:		Line 147:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~5/4 = 387.406		* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.406{{c}}
	* [[POTE]]: ~2 = ~~1\1~~, ~5/4 = 387.392		* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.392{{c}}

	{{Optimal ET sequence\|legend=1\| 31, 96d, 127d }}		{{Optimal ET sequence\|legend=1\| 31, 96d, 127d }}
Line 163:		Line 162:

	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.472		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.472{{c}}
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.407		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.407{{c}}

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 31, 65, 96d, 127d }}		{{Optimal ET sequence\|legend=0\| 31, 65, 96d, 127d }}

	Badness (Smith): 0.033436		Badness (Smith): 0.033436
Line 180:		Line 179:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~2 = ~~1\1~~, ~5/4 = 387.853		* [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.853
	* [[POTE]]: ~2 = ~~1\1~~, ~5/4 = 387.881		* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.881

	{{Optimal ET sequence\|legend=1\| 34d, 65, 99 }}		{{Optimal ET sequence\|legend=1\| 34d, 65, 99 }}
Line 195:		Line 194:

	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = ~~1\1~~, ~5/4 = 387.829		* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.829
	* POTE: ~2 = ~~1\1~~, ~5/4 = 387.882		* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.882

	~~Optimal ET sequence:~~ {{~~optimal~~ ET sequence\| 34d, 65, 99e }}		{{Optimal ET sequence\|legend=0\| 34d, 65, 99e }}

	Badness (Smith): 0.058325		Badness (Smith): 0.058325
Line 205:		Line 204:
	[[Category:Würschmidt family\| ]] <!-- main article -->		[[Category:Würschmidt family\| ]] <!-- main article -->
	[[Category:Würschmidt\| ]] <!-- key article -->		[[Category:Würschmidt\| ]] <!-- key article -->
	[[Category:~~Rank~~ 2]]		[[Category:Catalogs of rank-2 temperaments]]

Overthink: /* 2.3.5.23 subgroup */ note further extensions

2026-06-27T16:16:28Z

2.3.5.23 subgroup: note further extensions

← Older revision		Revision as of 16:16, 27 June 2026
Line 39:		Line 39:

	=== 2.3.5.23 subgroup ===		=== 2.3.5.23 subgroup ===
			Extensions to harmonics [[47/1\|47]] and [[49/1\|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment\|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1\|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].

	Subgroup: 2.3.5.23		Subgroup: 2.3.5.23

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:17:19Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:17, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Würschmidt family\| ]] <!-- main article -->		[[Category:Würschmidt family\| ]] <!-- main article -->
	[[Category:Würschmidt\| ]] <!-- key article -->		[[Category:Würschmidt\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:34:21Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:34, 24 June 2025
Line 201:		Line 201:

	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Würschmidt family\| ]] <!-- main article -->		[[Category:Würschmidt family\| ]] <!-- main article -->
	[[Category:Würschmidt\| ]] <!-- key article -->		[[Category:Würschmidt\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]