Würschmidt comma: Difference between revisions

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 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|S9/S10]] or [[4000/3993|S10/S11]]) 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.