25edo: Difference between revisions

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=Theory=

25EDO divides the [[Octave|octave]] in 25 equal steps of exact size 48 [[cent|cent]]s each. It is a good way to tune the [[Blackwood_temperament|Blackwood temperament]], which takes the very sharp fifths of [[5edo|5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4|5/4]]) and 7 ([[7/4|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 divides the [[Octave|octave]] in 25 equal steps of exact size 48 [[cent|cent]]s each. It is a good way to tune the [[Blackwood_temperament|Blackwood temperament]], which takes the very sharp fifths of [[5edo|5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4|5/4]]) and 7 ([[7/4|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 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent|consistent]]. It therefore makes sense to use it as a 2.5.7 [[Just_intonation_subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125|128/125]] [[diesis|diesis]] and two [[septimal_tritones|septimal tritones]] of [[7/5|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|50EDO]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[Mavila|mavila]] temperament.

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 =Theory=
-EDO divides the [[Octave|octave]] in 25 equal steps of exact size 48 [[cent|cent]]s each. It is a good way to tune the [[Blackwood_temperament|Blackwood temperament]], which takes the very sharp fifths of [[5edo|5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4|5/4]]) and 7 ([[7/4|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 divides the [[Octave|octave]] in 25 equal steps of exact size 48 [[cent|cent]]s each. It is a good way to tune the [[Blackwood_temperament|Blackwood temperament]], which takes the very sharp fifths of [[5edo|5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4|5/4]]) and 7 ([[7/4|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 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent|consistent]]. It therefore makes sense to use it as a 2.5.7 [[Just_intonation_subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125|128/125]] [[diesis|diesis]] and two [[septimal_tritones|septimal tritones]] of [[7/5|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|50EDO]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[Mavila|mavila]] temperament.