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## 25 tone equal temperament

25EDO divides the octave in 25 equal steps of exact size 48 cents 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 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 subgroup tuning. Looking just at 2, 5, and 7, it equates five 8/7s 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.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the 2*25 subgroup 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

## Music

Study in Fivesby Paul RapoportFantasy for Piano in 25 Note per Octave Tuning

playby Chris VaisvilFlat fourth bluesby Fabrizio Fulvio Fausto Fiale