Würschmidt comma: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 393216/390625 | | Ratio = 393216/390625 | ||
| Name = | | 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 | ||
}} | }} | ||
'''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>. | |||
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 [[ | * 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 to 3edo, because the magic comma is itself equal to (25/24)/(128/125), so that it's equivalent to the previous expression. | ||
* two | * 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 | * 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 == | ||
Tempering out this comma leads to the [[würschmidt family]] | [[Tempering out]] this comma leads to the [[würschmidt]] temperament and its extensions in the [[würschmidt family]]. This temperament has 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. | The comma 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 [[576/575]] ({{S|24}}), and where (46/45)<sup>2</sup> and 48/46 differ by [[12167/12150]] (S46<sup>2</sup>⋅S47). Thus it can also be seen that this comma's temperament extends to the [[2.3.5.23 subgroup]]. | ||
[[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>. | |||
== References == | |||
[[Category:Würschmidt| ]] | [[Category:Würschmidt| ]] | ||
[[Category:Commas named after composers]] | |||
[[Category:Commas named after music theorists]] | |||