Würschmidt - Revision history

FloraC: /* Tunings */ tabulate minimax tuning

2026-09-09T09:34:52Z

Tunings: tabulate minimax tuning

← Older revision		Revision as of 09:34, 9 September 2026
Line 128:		Line 128:
	\|}		\|}

			=== Target tunings ===
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| [[Delta-rational chord\|DR]] and equal-beating tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| [[Delta-rational chord\|DR]] and equal-beating tunings
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	\|-		\|-
	\| 15:18:23 (+3 +5) \|\| ~5/4 = 387.9215 \|\| 4''g''<sup>7</sup> − 3''g''<sup>5</sup> − 10 = 0 \|\|		\| 15:18:23 (+3 +5) \|\| ~5/4 = 387.9215 \|\| 4''g''<sup>7</sup> − 3''g''<sup>5</sup> − 10 = 0 \|\|
			\|}

			{\| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
			\|+ style="font-size: 105%; white-space: nowrap;" \| Odd-limit-based target tunings
			! rowspan="2" \| Target
			! colspan="2" \| Minimax
			\|-
			! Generator
			! Eigenmonzo*
			\|-
			\| 5-odd-limit
			\| ~3/2 = 387.7444{{c}}
			\| 3/2
	\|}		\|}

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	\| [[3/2]]		\| [[3/2]]
	\| 387.7444		\| 387.7444
	\| 1/8-comma		\| 1/8-comma, 5-odd-limit minimax
	\|-		\|-
	\| [[294edo\|95\294]]		\| [[294edo\|95\294]]
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	=== Other tunings ===		=== Other tunings ===
	* [[DKW theory\|DKW]] ~~(2.3.5)~~: ~~~2 = 1\1,~~ ~5/4 = 387.8015		* 5-limit [[DKW theory\|DKW]]: ~5/4 = 387.8015{{c}}
	* [[5-odd-limit]] minimax: ~2 = 1\1, ~5/4 = 387.7444 ([[eigenmonzo]] 3/2 aka 1/8-comma, generator = 6<sup>1/8</sup>)

	== Music ==		== Music ==

FloraC: /* Tunings */ formatting

2026-09-09T09:00:45Z

Tunings: formatting

← Older revision		Revision as of 09:00, 9 September 2026
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	== Tunings ==		== Tunings ==
	=== ~~Optimized~~ tunings ===		=== Norm-based tunings ===
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| ~~Norm~~-based tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| 5-limit norm-based tunings
	\|-		\|-
	! rowspan="2" \| ~~Weight-skew\Order~~ !! colspan="2" \| Euclidean		! rowspan="2" \| !! colspan="3" \| Euclidean
	\|-		\|-
	! Constrained !! Destretched		! Constrained !! Constrained & skewed !! Destretched
	\|-		\|-
	! Tenney		! Tenney
	\| ~~(2.3.5)~~ CTE: ~5/4 = 387.~~734¢~~ \|\| (2.~~3.5)~~ POTE: ~5/4 = 387.~~7993¢~~		\| CTE: ~5/4 = 387.734{{c}} \|\| CWE: ~5/4 = 387.776{{c}} \|\| POTE: ~5/4 = 387.7993{{c}}
	\|-		\|-
	! ~~Weil~~		! Equilateral
	\| ~~(2.3.5) CWE~~: ~5/4 = 387.~~776¢~~ \|\|		\| CEE: ~5/4 = 387.7224{{c}}<br>(8/65-comma) \|\| \|\|
			\|}
			{\| class="wikitable mw-collapsible mw-collapsed"
			\|+ style="font-size: 105%; white-space: nowrap;" \| 5-limit add-23 norm-based tunings
			\|-
			! rowspan="2" \| !! colspan="3" \| Euclidean
	\|-		\|-
	! ~~Equilateral~~		! Constrained !! Constrained & skewed !! Destretched
	~~\| (2.3.5) CEE: ~5/4 = 387.7224¢~~
	~~(8/65-comma)~~
	\|-		\|-
	! Tenney		! Tenney
	\| ~~(2.3.5.23)~~ CTE: ~5/4 = 387.~~734¢~~ \|\| ~~(2.3.5.23) POTE~~: ~5/4 = 387.~~8051¢~~		\| CTE: ~5/4 = 387.734{{c}} \|\| CWE: ~5/4 = 387.781{{c}} \|\|POTE: ~5/4 = 387.8051{{c}}
	\|-
	~~! Weil~~
	\| ~~(2.3.5.23) CWE~~: ~5/4 = 387.~~781¢ \|\|~~
	\|}		\|}

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	=== Tuning spectrum ===		=== Tuning spectrum ===
	The below assumes the 2.3.5.11.23 subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.		The below assumes the 2.3.5.11.23-subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.

	{\| 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
	\|-		\|-
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	\|		\|
	\| 385.7143		\| 385.7143
	\| 28ei val, major thirds slightly flatter than this fall under 25&28 or [[magic]]		\| 28ei val, major thirds slightly flatter than this fall under 25 & 28 or [[magic]]
	\|-		\|-
	\|		\|

FloraC: Rework the infobox and interval table to reflect that 11 is natural

2026-09-09T08:51:27Z

Rework the infobox and interval table to reflect that 11 is natural

Show changes

FloraC: /* Extensions */ discuss these primes in increasing order, since the 11 is deemed natural

2026-09-09T08:17:53Z

Extensions: discuss these primes in increasing order, since the 11 is deemed natural

← Older revision		Revision as of 08:17, 9 September 2026
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	== Extensions ==		== Extensions ==
	~~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 [[2401/2400]] and [[6144/6125]].

	~~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-7's 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.

	~~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.		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, if one is interested in adding another prime to this temperament, 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 out [[576/575]] ({{S\|24}}) and [[12167/12150]] ([[S-expression\|S46<sup>2</sup>⋅S47]]). 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 equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]]. 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]] ([[2401/2400\|S49]]/([[25921/25920\|S161]]<sup>2</sup>)).

	== Interval chain ==		== Interval chain ==

FloraC: /* Interval chain */ reduce cent values to 1 decimal place; formatting

2026-09-09T07:57:29Z

Interval chain: reduce cent values to 1 decimal place; formatting

← Older revision		Revision as of 07:57, 9 September 2026
Line 30:		Line 30:
	In the below, octave-reduced harmonics 1–33 are indicated in '''bold'''. All intervals are in the 165-[[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-1 right-2"
	\|-		\|-
	! rowspan="2" \| &#~~35;~~ !! rowspan="2" \| Cents* !! colspan="2" \| Approximate ratios		! rowspan="2" \| # !! rowspan="2" \| Cents* !! colspan="2" \| Approximate ratios
	\|-		\|-
	! 2.3.5.23 subgroup !! Add-11 extension		! 2.3.5.23 subgroup !! Add-11 extension
	\|-		\|-
	\| 0 \|\| 0.00 \|\| '''1/1''' \|\|		\| 0 \|\| 0.0 \|\| '''1/1''' \|\|
	\|-		\|-
	\| 1 \|\| 387.~~652~~ \|\| '''5/4''', 144/115 \|\| 69/55		\| 1 \|\| 387.7 \|\| '''5/4''', 144/115 \|\| 69/55
	\|-		\|-
	\| 2 \|\| 775.~~304~~ \|\| '''25/16''', 36/23 \|\| 69/44		\| 2 \|\| 775.3 \|\| '''25/16''', 36/23 \|\| 69/44
	\|-		\|-
	\| 3 \|\| ~~1162~~.~~956~~ \|\| 45/23, 125/64 \|\| 88/45, 108/55		\| 3 \|\| 1163.0 \|\| 45/23, 125/64 \|\| 88/45, 108/55
	\|-		\|-
	\| 4 \|\| 350.~~608~~ \|\| 92/75 \|\| 11/9, 27/22		\| 4 \|\| 350.6 \|\| 92/75 \|\| 11/9, 27/22
	\|-		\|-
	\| 5 \|\| 738.~~260~~ \|\| 23/15, 192/125 \|\| 55/36, 135/88		\| 5 \|\| 738.3 \|\| 23/15, 192/125 \|\| 55/36, 135/88
	\|-		\|-
	\| 6 \|\| 1125.~~912~~ \|\| 23/12, 48/25 \|\| 44/23		\| 6 \|\| 1125.9 \|\| 23/12, 48/25 \|\| 44/23
	\|-		\|-
	\| 7 \|\| 313.~~564~~ \|\| 6/5, 115/96 \|\| 55/46		\| 7 \|\| 313.6 \|\| 6/5, 115/96 \|\| 55/46
	\|-		\|-
	\| 8 \|\| 701.~~216~~ \|\| '''3/2''' \|\| 121/81		\| 8 \|\| 701.2 \|\| '''3/2''' \|\| 121/81
	\|-		\|-
	\| 9 \|\| 1088.~~868~~ \|\| '''15/8''', 216/115 \|\|		\| 9 \|\| 1088.9 \|\| '''15/8''', 216/115 \|\|
	\|-		\|-
	\| 10 \|\| 276.~~520~~ \|\| 27/23, 75/64 \|\| 88/75		\| 10 \|\| 276.5 \|\| 27/23, 75/64 \|\| 88/75
	\|-		\|-
	\| 11 \|\| 664.~~172~~ \|\| 184/125, 135/92 \|\| 22/15, 81/55		\| 11 \|\| 664.2 \|\| 184/125, 135/92 \|\| 22/15, 81/55
	\|-		\|-
	\| 12 \|\| 1051.~~824~~ \|\| 46/25 \|\| 11/6, 81/44		\| 12 \|\| 1051.8 \|\| 46/25 \|\| 11/6, 81/44
	\|-		\|-
	\| 13 \|\| 239.~~476~~ \|\| 23/20, 144/125 \|\| 55/48, 132/115		\| 13 \|\| 239.5 \|\| 23/20, 144/125 \|\| 55/48, 132/115
	\|-		\|-
	\| 14 \|\| 627.~~128~~ \|\| '''23/16''', 36/25 \|\| 33/23		\| 14 \|\| 627.1 \|\| '''23/16''', 36/25 \|\| 33/23
	\|-		\|-
	\| 15 \|\| 1014.~~780~~ \|\| 9/5, 115/64 \|\| 165/92, 242/135		\| 15 \|\| 1014.8 \|\| 9/5, 115/64 \|\| 165/92, 242/135
	\|-		\|-
	\| 16 \|\| 202.~~432~~ \|\| '''9/8''' \|\| 121/108		\| 16 \|\| 202.4 \|\| '''9/8''' \|\| 121/108
	\|-		\|-
	\| 17 \|\| 590.~~084~~ \|\| 45/32, 162/115 \|\|		\| 17 \|\| 590.1 \|\| 45/32, 162/115 \|\|
	\|-		\|-
	\| 18 \|\| 977.~~736~~ \|\| 81/46 \|\| 44/25		\| 18 \|\| 977.7 \|\| 81/46 \|\| 44/25
	\|-		\|-
	\| 19 \|\| 165.~~388~~ \|\| 138/125 \|\| 11/10		\| 19 \|\| 165.4 \|\| 138/125 \|\| 11/10
	\|-		\|-
	\| 20 \|\| 553.~~040~~ \|\| 69/50 \|\| '''11/8'''		\| 20 \|\| 553.0 \|\| 69/50 \|\| '''11/8'''
	\|-		\|-
	\| 21 \|\| 940.~~692~~ \|\| 69/40, 216/125 \|\| 55/32		\| 21 \|\| 940.7 \|\| 69/40, 216/125 \|\| 55/32
	\|-		\|-
	\| 22 \|\| 128.~~344~~ \|\| 27/25, 69/64 \|\| 99/92		\| 22 \|\| 128.3 \|\| 27/25, 69/64 \|\| 99/92
	\|-		\|-
	\| 23 \|\| ~~515~~.~~996~~ \|\| 27/20 \|\|		\| 23 \|\| 516.0 \|\| 27/20 \|\|
	\|-		\|-
	\| 24 \|\| 903.~~648~~ \|\| '''27/16''' \|\|		\| 24 \|\| 903.6 \|\| '''27/16''' \|\|
	\|-		\|-
	\| 25 \|\| 91.~~300~~ \|\| 135/128 \|\| 132/125		\| 25 \|\| 91.3 \|\| 135/128 \|\| 132/125
	\|-		\|-
	\| 26 \|\| ~~478~~.~~952~~ \|\| \|\| 33/25		\| 26 \|\| 479.0 \|\| \|\| 33/25
	\|-		\|-
	\| 27 \|\| 866.~~604~~ \|\| 207/125 \|\| 33/20		\| 27 \|\| 866.6 \|\| 207/125 \|\| 33/20
	\|-		\|-
	\| 28 \|\| 54.~~256~~ \|\| \|\| '''33/32'''		\| 28 \|\| 54.3 \|\| \|\| '''33/32'''
	\|-		\|-
	\| 29 \|\| 441.~~908~~ \|\| 162/125 \|\| 165/128		\| 29 \|\| 441.9 \|\| 162/125 \|\| 165/128
	\|-		\|-
	\| 30 \|\| 829.~~560~~ \|\| 81/50 \|\| 121/75		\| 30 \|\| 829.6 \|\| 81/50 \|\| 121/75
	\|-		\|-
	\| 31 \|\| 17.~~212~~ \|\| 81/80 \|\| 121/120		\| 31 \|\| 17.2 \|\| 81/80 \|\| 121/120
	\|}		\|}
	<nowiki />* In 5-limit [[CTE]] tuning		<nowiki/>* In 5-limit [[CTE]] tuning

	== Tunings ==		== Tunings ==

Lucius Chiaraviglio: Fix confusing wording

2026-06-29T05:31:21Z

Fix confusing wording

← Older revision		Revision as of 05:31, 29 June 2026
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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 [[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]] has 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}}

Hstraub: Link to German

2026-03-08T21:27:17Z

Link to German

← Older revision		Revision as of 21:27, 8 March 2026
Line 1:		Line 1:
			{{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
Line 1:		Line 1:
	{{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
Line 150:		Line 150:
	\|		\|
	\| 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
Line 150:		Line 150:
	\|		\|
	\| 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