Würschmidt comma: Difference between revisions

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* equivalently, between 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~~|S49~~]] instead of or in addition to [[243/242~~|S9/11~~]]).

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, 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]].

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, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is little 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 out this comma 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 würschmidt sharpens the 5/4 by up to 1.43{{cent}} (corresponding to 1/8-comma würschmidt, 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]] or [[4000/3993]]) and prime 19 (through nestoria), among others. In an ideal tuning of würschmidt, [[5/4]] is sharpened by about 1.4{{cent}} leading to a tuning of 5/4 of about 387.7{{cent}}. 31 + 65 = [[96edo]] is also within the range of ideal tunings, corresponding to the fifth of [[12edo]], being 12 × 8.

Tempering out this comma leads to the [[würschmidt family]] of temperaments.

[[Category:Würschmidt|#]]

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 * equivalently, between 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]])<sup>2</sup>/([[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|S49]] instead of or in addition to [[243/242|S9/11]]).
+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]])<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, 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]].
+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, which is in some sense the logical dual of magic, which tunes 5/4 flat. There is little 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 out this comma 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 würschmidt sharpens the 5/4 by up to 1.43{{cent}} (corresponding to 1/8-comma würschmidt, 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]] or [[4000/3993]]) and prime 19 (through nestoria), among others. In an ideal tuning of würschmidt, [[5/4]] is sharpened by about 1.4{{cent}} leading to a tuning of 5/4 of about 387.7{{cent}}. 31 + 65 = [[96edo]] is also within the range of ideal tunings, corresponding to the fifth of [[12edo]], being 12 × 8.
+Tempering out this comma leads to the [[würschmidt family]] of temperaments.
 [[Category:Würschmidt|#]] <!-- list on top of cat -->