72edo: Difference between revisions

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72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].

~~72edo (with the 71.94 [[generalized patent val]]) has the highest [[path-based goodness]] out of all generalized patent vals from 1 to 140.~~

=== Prime harmonics ===

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=== Octave stretch ===

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies~~. [[Path-based goodness]] recommends an octave stretch of 0.93 cents, which lies between the two options above but is closer to 186ed6~~.

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

=== Subsets and supersets ===

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 edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].
-edo (with the 71.94 [[generalized patent val]]) has the highest [[path-based goodness]] out of all generalized patent vals from 1 to 140.
 === Prime harmonics ===
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 === Octave stretch ===
-edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies. [[Path-based goodness]] recommends an octave stretch of 0.93 cents, which lies between the two options above but is closer to 186ed6.
+edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.
 === Subsets and supersets ===