Würschmidt family: Difference between revisions

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The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. Its [[monzo]] is {{monzo| 17 1 -8 }}, and flipping that yields {{multival| 8 1 17 }} for the wedgie. This tells us the [[generator]] is a classic major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.

Similar to [[meantone]], würschmidt 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. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]]. [[65edo]] is the point where it is combined 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 [[243/242]], [[8019/8000]] and [[4000/3993]] (which is natural because [[243/242|S9/S11]] = [[8019/8000|S9/S10]] * [[4000/3993|S10/S11]]) and prime 19 through nestoria, among others. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].

Similar to [[meantone]], würschmidt 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. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]]. [[65edo]] is the point where it is combined 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 [[243/242]], [[8019/8000]] and [[4000/3993]] (which is natural because [[243/242|S9/S11]] = [[8019/8000|S9/S10]] * [[4000/3993|S10/S11]]; note that tempering all three without tempering the würschmidt comma yields [[gravity]]) and prime 19 through nestoria, among others. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].

Another tuning solution is to sharpen the major third by 1/8 of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]].

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 The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. Its [[monzo]] is {{monzo| 17 1 -8 }}, and flipping that yields {{multival| 8 1 17 }} for the wedgie. This tells us the [[generator]] is a classic major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6.
-Similar to [[meantone]], würschmidt 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. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]]. [[65edo]] is the point where it is combined 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 [[243/242]], [[8019/8000]] and [[4000/3993]] (which is natural because [[243/242|S9/S11]] = [[8019/8000|S9/S10]] * [[4000/3993|S10/S11]]) and prime 19 through nestoria, among others. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].
+Similar to [[meantone]], würschmidt 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. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]]. [[65edo]] is the point where it is combined 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 [[243/242]], [[8019/8000]] and [[4000/3993]] (which is natural because [[243/242|S9/S11]] = [[8019/8000|S9/S10]] * [[4000/3993|S10/S11]]; note that tempering all three without tempering the würschmidt comma yields [[gravity]]) and prime 19 through nestoria, among others. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].
 Another tuning solution is to sharpen the major third by 1/8 of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]].