Bird's eye view of temperaments by accuracy: Difference between revisions
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* 18 for adding {9, 23, 45, 75, 115} ([[3L 16s]]) | * 18 for adding {9, 23, 45, 75, 115} ([[3L 16s]]) | ||
* 22 (or 23) for adding {11, 55(, 69)} ([[3L 19s]] or [[3L 22s]]) | * 22 (or 23) for adding {11, 55(, 69)} ([[3L 19s]] or [[3L 22s]]) | ||
[[#Generator tunings|Generator tunings]]: 21\65, 32\99, 53\164 | |||
Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)<sup>8</sup>. The rationale for this is that (5/4)<sup>3</sup> falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)<sup>2</sup> = [[25/16]] by [[128/125]] twice we get [[~]][[3/2]]. Therefore, in an optimized tuning, we can expect the fifth to be slightly flat, so that [[25/24]] is sharpened so that it makes sense to equate with a slightly flattened [[~]][[24/23]] by tempering out their difference, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|as documented in its interval chain]]. Because of dividing 6/1 into eight, it admits a neutral third at 4 generators so that an extension to prime 11 is also natural by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]] so that [[~]][[11/9]][[~]][[27/22]] is the neutral third. | Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)<sup>8</sup>. The rationale for this is that (5/4)<sup>3</sup> falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)<sup>2</sup> = [[25/16]] by [[128/125]] twice we get [[~]][[3/2]]. Therefore, in an optimized tuning, we can expect the fifth to be slightly flat, so that [[25/24]] is sharpened so that it makes sense to equate with a slightly flattened [[~]][[24/23]] by tempering out their difference, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|as documented in its interval chain]]. Because of dividing 6/1 into eight, it admits a neutral third at 4 generators so that an extension to prime 11 is also natural by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]] so that [[~]][[11/9]][[~]][[27/22]] is the neutral third. | ||