Würschmidt family - Revision history

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
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	=== 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
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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]]

VectorGraphics at 05:06, 12 June 2025

2025-06-12T05:06:03Z

← Older revision		Revision as of 05:06, 12 June 2025
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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 ~~{{Multival\| 8 1 18 20 … }}~~ 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.		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 temperament.

VectorGraphics at 04:53, 12 June 2025

2025-06-12T04:53:23Z

← Older revision		Revision as of 04:53, 12 June 2025
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	{{Mapping\|legend=1\| 1 -1 2 -3 \| 0 8 1 18 }}		{{Mapping\|legend=1\| 1 -1 2 -3 \| 0 8 1 18 }}

	~~{{Multival\|legend=1\| 8 1 18 -17 6 39 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 1 -1 2 7 \| 0 8 1 -13 }}		{{Mapping\|legend=1\| 1 -1 2 7 \| 0 8 1 -13 }}

	~~{{Multival\|legend=1\| 8 1 -13 -17 -43 -33 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 1 -1 2 -14 \| 0 8 1 52 }}		{{Mapping\|legend=1\| 1 -1 2 -14 \| 0 8 1 52 }}

	~~{{Multival\|legend=1\| 8 1 52 -17 60 118 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:42:22Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

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	{{Technical data page}}~~<br><br>~~		{{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 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.

BudjarnLambeth: {{Technical data page}}

2025-04-13T05:42:26Z

{{Technical data page}}<br><br>

← Older revision		Revision as of 05:42, 13 April 2025
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			{{Technical data page}}<br><br>
	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 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.

Godtone: /* Septimal würschmidt */ note 96edo as the smallest patent tuning above 31edo, move explanation of shortened form to avoid diverting focus

2025-03-09T22:21:10Z

Septimal würschmidt: note 96edo as the smallest patent tuning above 31edo, move explanation of shortened form to avoid diverting focus

← Older revision		Revision as of 22:21, 9 March 2025
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	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 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} }}. (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).). 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.		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.

			(<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

Godtone: /* Septimal würschmidt */ add nowrap, an explanation of the shortened form, and discuss structural advantages of 65edo as a wurschmidt tuning

2025-03-09T22:16:34Z

Septimal würschmidt: add nowrap, an explanation of the shortened form, and discuss structural advantages of 65edo as a wurschmidt tuning

← Older revision		Revision as of 22:16, 9 March 2025
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	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 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 {[[5120/5103\|S8/S9]], [[8019/8000\|S9/S10]], [[4000/3993\|S10/S11]]}, which may be expressed in shortened form as {S8/9/10/11}. The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied by reading them off as ratios, like 8/10 and 9/11, in ~~the shortened form~~.		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} }}. (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).). 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.

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

Godtone: /* Septimal würschmidt */ add S-expression-based comma list

2025-03-09T22:07:25Z

Septimal würschmidt: add S-expression-based comma list

← Older revision		Revision as of 22:07, 9 March 2025
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	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 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 {[[5120/5103\|S8/S9]], [[8019/8000\|S9/S10]], [[4000/3993\|S10/S11]]}, which may be expressed in shortened form as {S8/9/10/11}. The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied by reading them off as ratios, like 8/10 and 9/11, in the shortened form.

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