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>. | |||
It is the | 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]], 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]] }}. | |||
* 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]] }}. | |||
* 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. | |||
Tempering | == 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. | |||
[[Category:Würschmidt| | == 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:Commas named after composers]] | |||
[[Category:Commas named after music theorists]] | |||
Latest revision as of 03:07, 9 September 2025
| Interval information |
Würschmidt's comma (or würschmidt comma) ([17 1 -8⟩ = 393216/390625) is a small 5-limit comma of 11.4 cents. It is the difference between an octave-reduced stack of eight classical major thirds and a perfect fifth: (5/4)8/6, which comes from 5/4 being a convergent in the continued fraction of [math]\displaystyle{ \sqrt[8]{6} }[/math].
It is also the difference between a stack of two 16/15s and a stack of three 25/24s, and therefore belongs to 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 = (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, so that this comma is found in the syntonic–31 equivalence continuum.
- two dieses and a 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 augmented–dicot equivalence continuum.
- a 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)2/(20000/19683); tempering out both leads to 34edo, so that this comma is found in the diaschismic–tetracot equivalence continuum.
- a diaschisma and a 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
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[1].