Würschmidt comma: Difference between revisions
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Make the stack of 5/4 the main way to pump this comma as the various relations between commas are somewhat anecdotal. Move other information to the temp section |
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The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[small comma|small]] [[5-limit]] [[comma]] of 11.4 [[cent]]s. | 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>. | ||
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 [[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 | * 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 | * 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. | |||
* | |||
== 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]]). | |||
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. | 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. | ||
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