Würschmidt comma: Difference between revisions

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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]]).

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 (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 lower-accuracy analogue of würschmidt, reaching [[3/1]] with ([[5/4]])5 (which exceeds 3/1 by the magic comma), 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 (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]].

== Temperaments ==

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 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]]).
-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 (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 lower-accuracy analogue of würschmidt, reaching [[3/1]] with ([[5/4]])<sup>5</sup> (which exceeds 3/1 by the magic comma), 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 (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]].
 == Temperaments ==