Würschmidt - Revision history

Hstraub: Link to German

2026-03-08T21:27:17Z

Link to German

← Older revision		Revision as of 21:27, 8 March 2026
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			{{interwiki
			\| de = Würschmidt
			\| en = Würschmidt
			}}
	{{Infobox regtemp		{{Infobox regtemp
	\| Title = Würschmidt		\| Title = Würschmidt

FloraC: Cleanup on infobox

2026-02-09T09:53:06Z

Cleanup on infobox

← Older revision		Revision as of 09:53, 9 February 2026
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	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Würschmidt		\| Title = Würschmidt
	\| Subgroups = 2.3.5, 2.3.5.23		\| Subgroups = 2.3.5, 2.3.5.23
	\| Comma basis = [[393216/390625]] (2.3.5); <br> [[576/575]], [[12167/12150]] (2.3.5.23)		\| Comma basis = [[393216/390625]] (2.3.5); <br>[[576/575]], [[12167/12150]] (2.3.5.23)
	\| Edo join 1 = 31 \| Edo join 2 = 34		\| Edo join 1 = 31 \| Edo join 2 = 34
	\| ~~Generator~~ = 5/4 \| ~~Generator~~ tuning = 387.~~734~~ \| Optimization method = ~~CTE~~		\| Mapping = 1; 8 1 14
			\| Generators = 5/4 \| Generators tuning = 387.8 \| Optimization method = CWE
	\| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]		\| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
	~~\| Mapping = 1; 8 1 14~~
	\| Pergen = (P8, ccP5/8)		\| Pergen = (P8, ccP5/8)
	\| Color name = Saquadbiguti		\| Color name = Saquadbiguti
	\| Odd limit 1 = 5 \| Mistuning 1 = 1.43 \| Complexity 1 = 19		\| Odd limit 1 = 5 \| Mistuning 1 = 1.43 \| Complexity 1 = 19
	\| Odd limit 2 = (2.3.5.23) 25 \| Mistuning 2 = 2.86 \| Complexity 2 = 25		\| Odd limit 2 = 2.3.5.23 25 \| Mistuning 2 = 2.86 \| Complexity 2 = 25
	}}		}}
	'''Würschmidt''' is a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo\|legend=1\| 17 1 -8 }}). It can be treated as analogous to [[schismic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator\|generated]] by a [[5/4\|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.		'''Würschmidt''' is a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo\|legend=1\| 17 1 -8 }}). It can be treated as analogous to [[schismic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator\|generated]] by a [[5/4\|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.

Fitzgerald Lee: /* Tuning spectrum */

2025-11-08T00:41:56Z

Tuning spectrum

← Older revision		Revision as of 00:41, 8 November 2025
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	\|		\|
	\| 385.7143		\| 385.7143
	\| 28ei val, major thirds flatter than this fall under 25&28 or [[magic]]		\| 28ei val, major thirds slightly flatter than this fall under 25&28 or [[magic]]
	\|-		\|-
	\|		\|

Fitzgerald Lee: /* Tuning spectrum */

2025-11-08T00:41:17Z

Tuning spectrum

← Older revision		Revision as of 00:41, 8 November 2025
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	\|		\|
	\| 385.7143		\| 385.7143
	\| 28ei val		\| 28ei val, major thirds flatter than this fall under 25&28 or [[magic]]
	\|-		\|-
	\|		\|
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	\|		\|
	\| '''400.0000'''		\| '''400.0000'''
	\| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone'''		\| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone''', major thirds slightly sharper than this fall under [[smate_family\|smate]]
	\|}		\|}
	<nowiki />* Besides the octave		<nowiki />* Besides the octave

FloraC: Text replacement - "Prime-optimized" to "Norm-based"

2025-09-12T14:08:41Z

Text replacement - "Prime-optimized" to "Norm-based"

← Older revision		Revision as of 14:08, 12 September 2025
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	=== Optimized tunings ===		=== Optimized tunings ===
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| ~~Prime~~-~~optimized~~ tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| Norm-based tunings
	\|-		\|-
	! rowspan="2" \| Weight-skew\Order !! colspan="2" \| Euclidean		! rowspan="2" \| Weight-skew\Order !! colspan="2" \| Euclidean

Lériendil: got rid of stupidly high-odd-limit intervals

2025-07-15T21:55:45Z

got rid of stupidly high-odd-limit intervals

← Older revision		Revision as of 21:55, 15 July 2025
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	== Interval chain ==		== Interval chain ==
	In the below, octave-reduced harmonics ~~1–125~~ are indicated in '''bold'''. All intervals are in the ~~625~~-[[odd limit]].		In the below, octave-reduced harmonics 1–33 are indicated in '''bold'''. All intervals are in the 165-[[odd limit]].

	{\| class="wikitable center-all right-2"		{\| class="wikitable center-all right-2"
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	\| 1 \|\| 387.652 \|\| '''5/4''', 144/115 \|\| 69/55		\| 1 \|\| 387.652 \|\| '''5/4''', 144/115 \|\| 69/55
	\|-		\|-
	\| 2 \|\| 775.304 \|\| '''25/16''', 36/23 \|\| 69/44~~, 352/225~~		\| 2 \|\| 775.304 \|\| '''25/16''', 36/23 \|\| 69/44
	\|-		\|-
	\| 3 \|\| 1162.956 \|\| 45/23, ~~'''~~125/64~~''', 736/375~~ \|\| 88/45, 108/55		\| 3 \|\| 1162.956 \|\| 45/23, 125/64 \|\| 88/45, 108/55
	\|-		\|-
	\| 4 \|\| 350.608 \|\| 92/75~~, 225/184, 625/512~~ \|\| 11/9, 27/22		\| 4 \|\| 350.608 \|\| 92/75 \|\| 11/9, 27/22
	\|-		\|-
	\| 5 \|\| 738.260 \|\| 23/15, 192/125 \|\| 55/36, 135/88		\| 5 \|\| 738.260 \|\| 23/15, 192/125 \|\| 55/36, 135/88
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	\| 8 \|\| 701.216 \|\| '''3/2''' \|\| 121/81		\| 8 \|\| 701.216 \|\| '''3/2''' \|\| 121/81
	\|-		\|-
	\| 9 \|\| 1088.868 \|\| '''15/8''', 216/115 \|\| ~~207/110, 253/135~~		\| 9 \|\| 1088.868 \|\| '''15/8''', 216/115 \|\|
	\|-		\|-
	\| 10 \|\| 276.520 \|\| 27/23, ~~'''~~75/64~~'''~~ \|\| 88/75~~, 207/176~~		\| 10 \|\| 276.520 \|\| 27/23, 75/64 \|\| 88/75
	\|-		\|-
	\| 11 \|\| 664.172 \|\| 184/125, 135/92~~, 375/256~~ \|\| 22/15, 81/55		\| 11 \|\| 664.172 \|\| 184/125, 135/92 \|\| 22/15, 81/55
	\|-		\|-
	\| 12 \|\| 1051.824 \|\| 46/25~~, 675/368~~ \|\| 11/6, 81/44		\| 12 \|\| 1051.824 \|\| 46/25 \|\| 11/6, 81/44
	\|-		\|-
	\| 13 \|\| 239.476 \|\| 23/20, 144/125 \|\| 55/48, 132/115		\| 13 \|\| 239.476 \|\| 23/20, 144/125 \|\| 55/48, 132/115
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	\| 14 \|\| 627.128 \|\| '''23/16''', 36/25 \|\| 33/23		\| 14 \|\| 627.128 \|\| '''23/16''', 36/25 \|\| 33/23
	\|-		\|-
	\| 15 \|\| 1014.780 \|\| 9/5, ~~'''~~115/64~~'''~~ \|\| 165/92, 242/135		\| 15 \|\| 1014.780 \|\| 9/5, 115/64 \|\| 165/92, 242/135
	\|-		\|-
	\| 16 \|\| 202.432 \|\| '''9/8''' \|\| 121/108		\| 16 \|\| 202.432 \|\| '''9/8''' \|\| 121/108
	\|-		\|-
	\| 17 \|\| 590.084 \|\| ~~'''~~45/32~~'''~~, 162/115 \|\| ~~253/180~~		\| 17 \|\| 590.084 \|\| 45/32, 162/115 \|\|
	\|-		\|-
	\| 18 \|\| 977.736 \|\| 81/46~~, 225/128~~ \|\| 44/25		\| 18 \|\| 977.736 \|\| 81/46 \|\| 44/25
	\|-		\|-
	\| 19 \|\| 165.388 \|\| 138/125~~, 405/368~~ \|\| 11/10~~, 243/220~~		\| 19 \|\| 165.388 \|\| 138/125 \|\| 11/10
	\|-		\|-
	\| 20 \|\| 553.040 \|\| 69/50~~, 864/625~~ \|\| '''11/8'''~~, 243/176~~		\| 20 \|\| 553.040 \|\| 69/50 \|\| '''11/8'''
	\|-		\|-
	\| 21 \|\| 940.692 \|\| 69/40, 216/125 \|\| ~~'''~~55/32~~'''~~		\| 21 \|\| 940.692 \|\| 69/40, 216/125 \|\| 55/32
	\|-		\|-
	\| 22 \|\| 128.344 \|\| 27/25, ~~'''~~69/64~~'''~~ \|\| 99/92		\| 22 \|\| 128.344 \|\| 27/25, 69/64 \|\| 99/92
	\|-		\|-
	\| 23 \|\| 515.996 \|\| 27/20~~, 345/256~~ \|\|		\| 23 \|\| 515.996 \|\| 27/20 \|\|
	\|-		\|-
	\| 24 \|\| 903.648 \|\| '''27/16''' \|\| ~~253/150~~		\| 24 \|\| 903.648 \|\| '''27/16''' \|\|
	\|-		\|-
	\| 25 \|\| 91.300 \|\| 135/128~~, 243/230~~ \|\| 132/125~~, 253/240~~		\| 25 \|\| 91.300 \|\| 135/128 \|\| 132/125
	\|-		\|-
	\| 26 \|\| 478.952 \|\| ~~243/184, 828/625~~ \|\| 33/25~~, 253/192~~		\| 26 \|\| 478.952 \|\| \|\| 33/25
	\|-		\|-
	\| 27 \|\| 866.604 \|\| 207/125 \|\| 33/20		\| 27 \|\| 866.604 \|\| 207/125 \|\| 33/20
	\|-		\|-
	\| 28 \|\| 54.256 \|\| ~~207/200, 648/625~~ \|\| '''33/32'''		\| 28 \|\| 54.256 \|\| \|\| '''33/32'''
	\|-		\|-
	\| 29 \|\| 441.908 \|\| 162/125~~, 207/160~~ \|\| 165/128		\| 29 \|\| 441.908 \|\| 162/125 \|\| 165/128
	\|-		\|-
	\| 30 \|\| 829.560 \|\| 81/50~~, 207/128~~ \|\| 121/75		\| 30 \|\| 829.560 \|\| 81/50 \|\| 121/75
	\|-		\|-
	\| 31 \|\| 17.212 \|\| 81/80 \|\| 121/120		\| 31 \|\| 17.212 \|\| 81/80 \|\| 121/120

Lériendil: /* Extensions */

2025-06-26T01:49:54Z

Extensions

← Older revision		Revision as of 01:49, 26 June 2025
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	Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as {{nowrap\|([[24/23]]) × ([[46/45]])}}, and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering {{nowrap\|S24 {{=}} [[576/575]]}} and {{nowrap\|S46<sup>2</sup> × S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. A perhaps more direct way of seeing why equating 25/24 with 24/23 is natural is that würschmidt's generator is a slightly sharpened 5/4 with a slightly flat 3/2 in an optimised tuning, so that 25/24 is sharpened and 24/n is flattened, so equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]] = {{nowrap\|[[2401/2400\|S49]] / ([[25921/25920\|S161]]<sup>2</sup>)}}.		Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as {{nowrap\|([[24/23]]) × ([[46/45]])}}, and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering {{nowrap\|S24 {{=}} [[576/575]]}} and {{nowrap\|S46<sup>2</sup> × S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. A perhaps more direct way of seeing why equating 25/24 with 24/23 is natural is that würschmidt's generator is a slightly sharpened 5/4 with a slightly flat 3/2 in an optimised tuning, so that 25/24 is sharpened and 24/n is flattened, so equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]] = {{nowrap\|[[2401/2400\|S49]] / ([[25921/25920\|S161]]<sup>2</sup>)}}.

	Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt\|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap\|S161 {{=}} [[25921/25920]]}}).		Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt\|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] and [[2401/2400]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap\|S161 {{=}} [[25921/25920]]}}).

	Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with [[45/44]] and therefore 625/512 with [[11/9]] (by tempering out [[243/242]] and [[5632/5625]]), finding the 11th harmonic at 20 generators up, is highly natural in addition to the aforementioned extension to prime 23.		Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with [[45/44]] and therefore 625/512 with [[11/9]] (by tempering out [[243/242]] and [[5632/5625]]), finding the 11th harmonic at 20 generators up, is highly natural in addition to the aforementioned extension to prime 23.

Lériendil: Text replacement - "(unchanged-interval)" to "(unchanged interval)"

2025-06-21T06:59:27Z

Text replacement - "(unchanged-interval)" to "(unchanged interval)"

← Older revision		Revision as of 06:59, 21 June 2025
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	{\| class="wikitable center-all left-4"		{\| class="wikitable center-all left-4"
	! Edo<br />generator		! Edo<br />generator
	! [[Eigenmonzo\|Eigenmonzo<br />(unchanged-interval)]]*		! [[Eigenmonzo\|Eigenmonzo<br />(unchanged interval)]]*
	! Generator (¢)		! Generator (¢)
	! Comments		! Comments

FloraC: - parent category

2025-04-28T14:50:40Z

- parent category

← Older revision		Revision as of 14:50, 28 April 2025
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	* ''Extrospection'' (2013) – [https://web.archive.org/web/20201127013550/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt{{lbrack}}16{{rbrack}}-out.mp3 play] \| [https://soundcloud.com/jdfreivald/extrospection SoundCloud] – Würschmidt[16] in 31edo tuning.		* ''Extrospection'' (2013) – [https://web.archive.org/web/20201127013550/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt{{lbrack}}16{{rbrack}}-out.mp3 play] \| [https://soundcloud.com/jdfreivald/extrospection SoundCloud] – Würschmidt[16] in 31edo tuning.

	[[Category:~~Temperaments~~]]		[[Category:Würschmidt\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
	[[Category:Würschmidt family]]		[[Category:Würschmidt family]]

Lériendil: "schismatic" to "schismic"; personal convention has updated

2025-04-22T19:43:40Z

"schismatic" to "schismic"; personal convention has updated

← Older revision		Revision as of 19:43, 22 April 2025
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	\| Odd limit 2 = (2.3.5.23) 25 \| Mistuning 2 = 2.86 \| Complexity 2 = 25		\| Odd limit 2 = (2.3.5.23) 25 \| Mistuning 2 = 2.86 \| Complexity 2 = 25
	}}		}}
	'''Würschmidt''' is a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo\|legend=1\| 17 1 -8 }}). It can be treated as analogous to [[~~schismatic~~]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator\|generated]] by a [[5/4\|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in ~~schismatic~~, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.		'''Würschmidt''' is a [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo\|legend=1\| 17 1 -8 }}). It can be treated as analogous to [[schismic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator\|generated]] by a [[5/4\|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.

	{{Tdlink\|Würschmidt family #Würschmidt}}		{{Tdlink\|Würschmidt family #Würschmidt}}