25edo: Difference between revisions
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{{Odd harmonics in edo|edo=25}} | {{Odd harmonics in edo|edo=25}} | ||
25EDO divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[blackwood temperament]], which takes the very sharp fifths of [[5edo]] as a given, 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 divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It is a good way to tune the [[Archytas_clan#7-limit|blackwood temperament]], which takes the very sharp fifths of [[5edo]] as a given, 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 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 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] 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. | 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]]. 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]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] 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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| [[Chromatic_pairs#Gariberttet|Gariberttet]] | | [[Chromatic_pairs#Gariberttet|Gariberttet]] | ||
| | | [[4L_1s]], [[4L_5s]], [[4L_9s]], [[4L_13s]], [[4L_17s]] | ||
|- | |- | ||
| 7\25 | | 7\25 | ||
| 1 | | 1 | ||
| | | | ||
| [[ | | [[Archytas_clan#Sixix|Sixix]] | ||
| | | [[4L_3s]], [[7L_4s]], [[7L_11s]] | ||
|- | |- | ||
| 8\25 | | 8\25 | ||
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| [[Magic]] | | [[Magic]] | ||
| | | | ||
| | | [[3L_4s]], [[3L_7s]], [[3L_10s]], [[3L_13s]], [[3L_16s]], [[3L_19s]] | ||
|- | |- | ||
| 9\25 | | 9\25 | ||