Würschmidt comma: Difference between revisions

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{{Infobox Interval

| Ratio = 393216/390625

| Name = ~~Würschmidt~~ comma

| Name = würschmidt comma

| Color name = sg83, Saquadbigu comma

| Comma = yes

}}

~~The~~ '''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)8/6, 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)8/6, 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)2 and 48/46 differ by S462×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:

* 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]])2/([[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]])~~2~~/([[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 to 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.

* two ~~classic diatonic semitones~~ and ~~three classic chromatic semitones~~: ([[16/15]])2/([[25/24]])~~3~~; tempering out both leads to ~~3edo~~.

* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])2/([[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 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.

* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])~~2~~/([[20000/19683]])~~; tempering out both also leads~~ to ~~34edo~~.

== 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 ~~usually~~ represents ~[[11/9]] (or more accurately [[49/40]], 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]])5 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]])2. 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]])11 / ([[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>.

Notice that [[magic]] is a simpler analogue of würschmidt, reaching [[3/1]] with ([[5/4]])5 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]])2. 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]])11 / ([[12/1]]), we get [[magus]], a sharp-major-third analogue of Würschmidt.

== Notes ==

[[Category:Würschmidt| ]] ~~~~

[[Category:Würschmidt| ]]

[[Category:Commas named after composers]]

[[Category:Commas named after music theorists]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = 393216/390625
-| Name = Würschmidt comma
+| Name = würschmidt comma
 | Color name = sg<sup>8</sup>3, Saquadbigu comma
 | Comma = yes
 }}
-The '''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]].
 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 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 to 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression.
-* 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.
+* 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 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.
-* two diaschismas and a [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both also leads to 34edo.
 == 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 usually represents ~[[11/9]] (or more accurately [[49/40]], 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. 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>.
-Notice that [[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.
+== Notes ==
-[[Category:Würschmidt| ]] <!-- key article -->
+[[Category:Würschmidt| ]]
+[[Category:Commas named after composers]]
+[[Category:Commas named after music theorists]]