Würschmidt comma - Revision history

VectorGraphics at 03:07, 9 September 2025

2025-09-09T03:07:27Z

← Older revision		Revision as of 03:07, 9 September 2025
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	}}		}}

	'''Würschmidt's comma''' (or '''würschmidt comma''')({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.		'''Würschmidt's comma''' (or '''würschmidt comma''') ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.

	It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to [[Father–3 equivalence continuum\|the family of commas that denote a specific ratio between those two intervals]]. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].		It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to [[Father–3 equivalence continuum\|the family of commas that denote a specific ratio between those two intervals]]. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].

VectorGraphics at 03:01, 9 September 2025

2025-09-09T03:01:53Z

← Older revision		Revision as of 03:01, 9 September 2025
Line 6:		Line 6:
	}}		}}

	'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.		'''Würschmidt's comma''' (or '''würschmidt comma''')({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.

	It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to [[Father–3 equivalence continuum\|the family of commas that denote a specific ratio between those two intervals]]. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].		It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to [[Father–3 equivalence continuum\|the family of commas that denote a specific ratio between those two intervals]]. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].

Lucius Chiaraviglio: Fix typo

2025-05-27T09:14:29Z

Fix typo

← Older revision		Revision as of 09:14, 27 May 2025
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	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads to 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.
	* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.

FloraC: +etymology

2025-04-17T06:18:55Z

+etymology

← Older revision		Revision as of 06:18, 17 April 2025
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	[[Magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt. This motivation and others lead to the formulation of the [[augmented–dicot equivalence continuum]], which is up to a change of basis equivalent to the [[Father–3 equivalence continuum]] focused on making certain structures more evident.		[[Magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt. This motivation and others lead to the formulation of the [[augmented–dicot equivalence continuum]], which is up to a change of basis equivalent to the [[Father–3 equivalence continuum]] focused on making certain structures more evident.

			== Etymology ==
			This comma was known as ''Würschmidt's comma'' no later than 2001, when the corresponding temperament was named<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2064.html Yahoo! Tuning Group \| Kleismic & co]</ref>.

			== Notes ==

	[[Category:Würschmidt\| ]]		[[Category:Würschmidt\| ]]
	[[Category:Commas named after composers]]		[[Category:Commas named after composers]]
	[[Category:Commas named after music theorists]]		[[Category:Commas named after music theorists]]

Godtone: /* Temperaments */ relevant to link here and to explain their equivalence

2025-03-19T22:21:43Z

Temperaments: relevant to link here and to explain their equivalence

← Older revision		Revision as of 22:21, 19 March 2025
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	Tempering out this comma leads to the [[würschmidt]] temperament and its extensions in the [[würschmidt family]]. In any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which represents a tempering of [[625/512]]~[[768/625]] and can be used to represent [[11/9]]~[[27/22]] (or more accurately [[49/40]]~[[60/49]], tempering out [[2401/2400]] instead of or in addition to [[243/242]]).		Tempering out this comma leads to the [[würschmidt]] temperament and its extensions in the [[würschmidt family]]. In any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which represents a tempering of [[625/512]]~[[768/625]] and can be used to represent [[11/9]]~[[27/22]] (or more accurately [[49/40]]~[[60/49]], tempering out [[2401/2400]] instead of or in addition to [[243/242]]).

	[[Magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt.		[[Magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt. This motivation and others lead to the formulation of the [[augmented–dicot equivalence continuum]], which is up to a change of basis equivalent to the [[Father–3 equivalence continuum]] focused on making certain structures more evident.

	[[Category:Würschmidt\| ]]		[[Category:Würschmidt\| ]]
	[[Category:Commas named after composers]]		[[Category:Commas named after composers]]
	[[Category:Commas named after music theorists]]		[[Category:Commas named after music theorists]]

Lériendil at 22:18, 19 March 2025

2025-03-19T22:18:47Z

← Older revision		Revision as of 22:18, 19 March 2025
Line 12:		Line 12:
	In terms of commas, it is the difference between:		In terms of commas, it is the difference between:
	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is ~~found in~~ found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.

Godtone: explain equivalence

2025-03-19T22:17:44Z

explain equivalence

← Older revision		Revision as of 22:17, 19 March 2025
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	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
	* ~~a [[diaschisma]]~~ and a [[~~15625/15552\|kleisma~~]]: ([[2048/2025]])/([[~~15625~~/~~15552~~]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[~~diaschismic–gothmic~~ equivalence continuum]] }}.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.
	* ~~two diaschismas~~ and a [[~~tetracot comma~~]]: ([[2048/2025]])~~<sup>2</sup>~~/([[~~20000~~/~~19683~~]]); tempering out both also leads to 34edo.		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.

	== Temperaments ==		== Temperaments ==

Godtone: the expression (16/15)^2 / (25/24)^3 corresponds to how it's found in the Father-3 equivalence continuum

2025-03-19T22:10:39Z

the expression (16/15)^2 / (25/24)^3 corresponds to how it's found in the Father-3 equivalence continuum

← Older revision		Revision as of 22:10, 19 March 2025
Line 8:		Line 8:
	'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.		'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.

	It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to a [[Father–3 equivalence continuum~~/Godtone's approach~~\|family]] of commas that denote a specific ratio between those two intervals. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].		It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to [[Father–3 equivalence continuum\|the family of commas that denote a specific ratio between those two intervals]]. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].

	In terms of commas, it is the difference between:		In terms of commas, it is the difference between:
	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]].		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* a [[~~128~~/~~125~~\|~~diesis]] and a [[magic comma~~]]: ([[128/125]])/([[~~3125~~/~~3072~~]]); tempering out both leads to the trivial tuning [[3edo]].		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* ~~two dieses and~~ a [[25/24\|~~classic chromatic semitone~~]]: ([[128/125]])~~<sup>2</sup>~~/([[25/24]]); tempering out both leads to 3edo.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo.
	* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both leads to [[34edo]].		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–gothmic equivalence continuum]] }}.
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.

Lériendil: redundancy

2025-03-19T11:52:02Z

redundancy

← Older revision		Revision as of 11:52, 19 March 2025
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	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]].		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]].
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to 3edo.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to 3edo.
	* two classic diatonic semitones and three classic chromatic semitones: ([[16/15]])<sup>2</sup>/([[25/24]])<sup>3</sup>; tempering out both leads to 3edo.
	* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both leads to [[34edo]].		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both leads to [[34edo]].
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.

Lériendil: added S-expression justification

2025-03-19T11:50:52Z

added S-expression justification

← Older revision		Revision as of 11:50, 19 March 2025
Line 8:		Line 8:
	'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.		'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.

	It is the difference between a stack of two 16/~~15s~~ and a stack of three 25/~~24s~~.		It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to a [[Father–3 equivalence continuum/Godtone's approach\|family]] of commas that denote a specific ratio between those two intervals. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].

	In terms of commas, it is the difference between:		In terms of commas, it is the difference between:
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	== Temperaments ==		== Temperaments ==
	Tempering out this comma leads to the [[würschmidt family]] ~~of temperaments~~. In any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which represents a tempering of [[625/512]]~[[768/625]] and can be used to represent [[11/9]]~[[27/22]] (or more accurately [[49/40]]~[[60/49]], tempering out [[2401/2400]] instead of or in addition to [[243/242]]).		Tempering out this comma leads to the [[würschmidt]] temperament and its extensions in the [[würschmidt family]]. In any nontrivial tuning (that is, not 3edo), there is an exact neutral third between 5/4 and 6/5, which represents a tempering of [[625/512]]~[[768/625]] and can be used to represent [[11/9]]~[[27/22]] (or more accurately [[49/40]]~[[60/49]], tempering out [[2401/2400]] instead of or in addition to [[243/242]]).

	[[Magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt.		[[Magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> which exceeds 3/1 by the magic comma, and a even simpler analogue of würschmidt is [[dicot]], where [[3/2]] is reached by ([[5/4]])<sup>2</sup>. More interesting is that there is a lower-accuracy but more complex analogue of würschmidt if we look at the pattern; the powers of [[5/4]] go 2 (dicot), 5 (magic), 8 (würschmidt), corresponding to increasingly sharp tunings of 5 where each additional three 5's represent a lowering of [[25/16]] by another [[128/125]]; finally, at ([[5/4]])<sup>11</sup> / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of würschmidt.

← Older revision		Revision as of 09:14, 27 May 2025
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	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads to 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.
	* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.

← Older revision		Revision as of 22:18, 19 March 2025
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	In terms of commas, it is the difference between:		In terms of commas, it is the difference between:
	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is ~~found in~~ found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.

← Older revision		Revision as of 22:17, 19 March 2025
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	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
	* ~~a [[diaschisma]]~~ and a [[~~15625/15552\|kleisma~~]]: ([[2048/2025]])/([[~~15625~~/~~15552~~]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[~~diaschismic–gothmic~~ equivalence continuum]] }}.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–tetracot equivalence continuum]] }}.
	* ~~two diaschismas~~ and a [[~~tetracot comma~~]]: ([[2048/2025]])~~<sup>2</sup>~~/([[~~20000~~/~~19683~~]]); tempering out both also leads to 34edo.		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both also leads to 34edo, because the kleisma is itself equal to (2048/2025)/(20000/19683), so that it's equivalent to the previous expression.

	== Temperaments ==		== Temperaments ==

← Older revision		Revision as of 22:10, 19 March 2025
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	'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.		'''Würschmidt's comma''' ({{monzo\| 17 1 -8 }} = '''393216/390625''') is a [[small comma\|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. It is the difference between an [[octave reduction\|octave-reduced]] stack of eight [[5/4\|classical major thirds]] and a [[3/2\|perfect fifth]]: (5/4)<sup>8</sup>/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.

	It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to a [[Father–3 equivalence continuum~~/Godtone's approach~~\|family]] of commas that denote a specific ratio between those two intervals. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].		It is also the difference between a stack of two [[16/15]]s and a stack of three [[25/24]]s, and therefore belongs to [[Father–3 equivalence continuum\|the family of commas that denote a specific ratio between those two intervals]]. Among these, the würschmidt comma makes a rather accurate and rather intuitive equivalence, which can be seen by writing 25/24 as 50/48 and 16/15 as 48/45 = ([[24/23\|48/46]])×([[46/45]]) where 50/48 and 48/46 differ by S24 = [[576/575]], and (46/45)<sup>2</sup> and 48/46 differ by S46<sup>2</sup>×S47 = [[12167/12150]]. Thus it can also be seen that this comma's temperament extends to the 2.3.5.23 [[subgroup]].

	In terms of commas, it is the difference between:		In terms of commas, it is the difference between:
	* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]].		* a [[syntonic comma]] and a [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]], so that this comma is found in the {{nowrap\| [[syntonic–31 equivalence continuum]] }}.
	* a [[~~128~~/~~125~~\|~~diesis]] and a [[magic comma~~]]: ([[128/125]])/([[~~3125~~/~~3072~~]]); tempering out both leads to the trivial tuning [[3edo]].		* two dieses and a [[25/24\|classic chromatic semitone]]: ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]], so that this comma is found in found in the {{nowrap\| [[augmented–dicot equivalence continuum]] }}.
	* ~~two dieses and~~ a [[25/24\|~~classic chromatic semitone~~]]: ([[128/125]])~~<sup>2</sup>~~/([[25/24]]); tempering out both leads to 3edo.		* a [[128/125\|diesis]] and a [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both also leads 3edo.
	* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both leads to [[34edo]].		* a [[diaschisma]] and a [[15625/15552\|kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both leads to [[34edo]], so that this comma is found in the {{nowrap\| [[diaschismic–gothmic equivalence continuum]] }}.
	* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.		* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.