25edo: Difference between revisions

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== Theory ==

25edo is a good way to tune the [[blackwood]] temperament, which closes ~~the~~ circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25edo is a good way to tune the [[blackwood]] temperament, which closes each circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.

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 == Theory ==
-edo is a good way to tune the [[blackwood]] temperament, which closes the circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
+edo is a good way to tune the [[blackwood]] temperament, which closes each circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
 edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.