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{{Infobox Interval
{{Infobox Interval
| Ratio = 393216/390625
| Ratio = 393216/390625
| Name = Würschmidt comma
| Name = würschmidt comma
| Color name = sg<sup>8</sup>3, Saquadbigu comma
| Color name = sg<sup>8</sup>3, Saquadbigu comma
| Comma = yes
| Comma = yes
}}
}}


The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[small comma|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s.
'''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 the amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)<sup>8</sup>(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. (Therefore, it is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves), that is, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.)
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:
In terms of commas, it is the difference between:  
* the difference between the [[syntonic comma]] and the [[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]] }}.
* the difference between the [[diesis]] and the [[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]] }}.
* the difference between two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]].
* 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.
* equivalently, between one diaschisma and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]].  
* 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]] }}.  
* finally, between two dieses and the just chromatic semitone, ([[128/125]])<sup>2</sup>/([[25/24]]); tempering out both leads to the trivial tuning [[3edo]].  
* 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.
The last expression means tempering it out 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]]).


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, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is little reason to use magus unless you want a sharp [[5/4]] and/or want to use a temperament that happens to support it, a notable tuning of which is [[46edo]].
== Temperaments ==
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>.  


== Temperaments ==
== Notes ==
Tempering out this comma leads to the [[würschmidt family]] of temperaments.


[[Category:Würschmidt|#]] <!-- list on top of cat -->
[[Category:Würschmidt| ]]
[[Category:Commas named after composers]]
[[Category:Commas named after music theorists]]
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