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.

It is the amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)8(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. It is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves)~~. In other words~~, (5/4)7(393216/390625)/4 = 6/5.

It is the amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)8(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. (Therefore, it is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves), that is, (5/4)7(393216/390625)/4 = 6/5.)

In terms of commas it is the difference between the ~~lesser~~ diesis and the magic comma, (128/125)/(3125/3072).

In terms of commas, it is the difference between:

* the difference between the diesis and the magic comma, (128/125)/(3125/3072); tempering both leads to [[3edo]]

* between two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])2/([[20000/19683]]), corresponding to [[34edo]]

* or equivalently, between one [[diaschisma]] and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]), thus also corresponding to [[34edo]], and finally,

* between two dieses and the just chromatic semitone, ([[128/125]])2/([[25/24]]), corresponding to [[3edo]]

The last expression means that if you temper 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 [[2401/2400|S49]] instead of (or in addition to) [[243/242|S9/11]]).

Tempering it out leads to the [[würschmidt family]] of temperaments. ~~As in~~ [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.

Notice that [[magic]] is a lower-accuracy analogue of würschmidt, reaching [[3/1]] with ([[5/4]])5, and a trivial analogue of wurschmidt 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 (wurschmidt), 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, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is no real reason to use magus unless you want a sharp [[5/4]] and/or want to use a temperament that happens to support it, a notable tuning of which is [[46edo]].

== Temperaments ==

Tempering it out leads to the [[würschmidt family]] of temperaments. Similar to [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is ''far'' more accurate; an ideal tuning of wurschmidt sharpens the 5/4 by up to 1.43{{cent}} (corresponding to 1/8-comma wurschmidt, where 3/2's are pure). Combining it with meantone gives [[31edo]] as the first real tuning but increasingly good 5-limit edo tunings after 31 (all of which distinguish the [[syntonic comma]]) are [[34edo]] and especially [[65edo]], although 34+65 = [[99edo]] certainly makes sense if you prefer its tuning properties. [[65edo]] has the distinguishing property of being the smallest würschmidt edo with a 5/4 in the aforementioned ideal tuning range, and corresponds to combining it with [[schismic]] (especially the extension to include prime 19 called [[nestoria]]) and [[gravity]], so is a very accurate 5-limit tuning that extends naturally to prime 11 (through the aforementioned [[243/242]] or equivalently through [[8019/8000|S9/S10]] or [[4000/3993|S10/S11]]) and prime 19 (through nestoria), among others. In an ideal tuning of wurschmidt, [[5/4]] is sharpened by {{cent}}

[[Category:Würschmidt|#]]

@@ Line 8: / Line 8: @@
 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 amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)<sup>8</sup>(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. It is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves). In other words, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.
+It is the amount by which an [[octave reduction|octave-reduced]] stack of eight [[5/4|classical major thirds]] falls short of a [[3/2|perfect fifth]]: (5/4)<sup>8</sup>(393216/390625)/4 = 3/2, which comes from 5/4 being a convergent in the continued fraction of <math>\sqrt[8]{6}</math>. (Therefore, it is also equal to the difference between seven major thirds and 24/5 (i.e. 6/5 plus two octaves), that is, (5/4)<sup>7</sup>(393216/390625)/4 = 6/5.)
-In terms of commas it is the difference between the lesser diesis and the magic comma, (128/125)/(3125/3072).
+In terms of commas, it is the difference between:
+* the difference between the diesis and the magic comma, (128/125)/(3125/3072); tempering both leads to [[3edo]]
+* between two [[diaschisma]]s and the [[tetracot comma]], ([[2048/2025]])<sup>2</sup>/([[20000/19683]]), corresponding to [[34edo]]
+* or equivalently, between one [[diaschisma]] and the [[kleisma]], ([[2048/2025]])/([[15625/15552]]), thus also corresponding to [[34edo]], and finally,
+* between two dieses and the just chromatic semitone, ([[128/125]])<sup>2</sup>/([[25/24]]), corresponding to [[3edo]]
+The last expression means that if you temper 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 [[2401/2400|S49]] instead of (or in addition to) [[243/242|S9/11]]).
-Tempering it out leads to the [[würschmidt family]] of temperaments. As in [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.
+Notice that [[magic]] is a lower-accuracy analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup>, and a trivial analogue of wurschmidt 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 (wurschmidt), 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, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is no real reason to use magus unless you want a sharp [[5/4]] and/or want to use a temperament that happens to support it, a notable tuning of which is [[46edo]].
+== Temperaments ==
+Tempering it out leads to the [[würschmidt family]] of temperaments. Similar to [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is ''far'' more accurate; an ideal tuning of wurschmidt sharpens the 5/4 by up to 1.43{{cent}} (corresponding to 1/8-comma wurschmidt, where 3/2's are pure). Combining it with meantone gives [[31edo]] as the first real tuning but increasingly good 5-limit edo tunings after 31 (all of which distinguish the [[syntonic comma]]) are [[34edo]] and especially [[65edo]], although 34+65 = [[99edo]] certainly makes sense if you prefer its tuning properties. [[65edo]] has the distinguishing property of being the smallest würschmidt edo with a 5/4 in the aforementioned ideal tuning range, and corresponds to combining it with [[schismic]] (especially the extension to include prime 19 called [[nestoria]]) and [[gravity]], so is a very accurate 5-limit tuning that extends naturally to prime 11 (through the aforementioned [[243/242]] or equivalently through [[8019/8000|S9/S10]] or [[4000/3993|S10/S11]]) and prime 19 (through nestoria), among others. In an ideal tuning of wurschmidt, [[5/4]] is sharpened by {{cent}}
 [[Category:Würschmidt|#]] <!-- list on top of cat -->