Würschmidt comma: Difference between revisions
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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 | 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> | |||
* the [[syntonic comma]] and the [[semicomma]]: ([[81/80]])/([[2109375/2097152]]); tempering out both leads to [[31edo]] | |||
* the [[syntonic comma]] and the [[semicomma]] | * the [[diesis]] and the [[magic comma]]: ([[128/125]])/([[3125/3072]]); tempering out both leads to the trivial tuning [[3edo]] | ||
* the [[diesis]] and the [[magic comma]] | * two classic diatonic semitones and three classic chromatic semitones: ([[16/15]])<sup>2</sup>/([[25/24]])<sup>3</sup> | ||
* two classic diatonic semitones and three classic chromatic semitones | * two [[diaschisma]]s and the [[tetracot comma]]: ([[2048/2025]])<sup>2</sup>/([[20000/19683]]); tempering out both leads to [[34edo]] | ||
* two [[diaschisma]]s and the [[tetracot comma]] | ** equivalently, one diaschisma and the [[kleisma]]: ([[2048/2025]])/([[15625/15552]]); tempering out both thus also corresponds to [[34edo]] | ||
** equivalently, one diaschisma and the [[kleisma]] | * 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]]. | ||
* finally, between two dieses and the just chromatic semitone | |||
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]]). | 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]]). | ||
Revision as of 17:48, 29 May 2024
| Interval information |
The 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]
- the syntonic comma and the semicomma: (81/80)/(2109375/2097152); tempering out both leads to 31edo
- the diesis and the magic comma: (128/125)/(3125/3072); tempering out both leads to the trivial tuning 3edo
- two classic diatonic semitones and three classic chromatic semitones: (16/15)2/(25/24)3
- two diaschismas and the tetracot comma: (2048/2025)2/(20000/19683); tempering out both leads to 34edo
- equivalently, one diaschisma and the kleisma: (2048/2025)/(15625/15552); tempering out both thus also corresponds to 34edo
- finally, between two dieses and the just chromatic semitone: (128/125)2/(25/24); tempering out both leads to the trivial tuning 3edo.
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)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.
Temperaments
Tempering out this comma leads to the würschmidt family of temperaments.