Würschmidt comma: Difference between revisions

'''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)⁸/6, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>.  

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)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]].

* 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]])²/([[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]])²/([[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.

[[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.