Didacus: Difference between revisions

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| Odd limit 2 = 11 | Mistuning 2 = 4.13 | Complexity 2 = 19

}}

'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic ~~ratios~~. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).

'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).

[[31edo]] is a very good tuning of didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS scale]]s of didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[slendric]].

@@ Line 10: / Line 10: @@
 | Odd limit 2 = 11 | Mistuning 2 = 4.13 | Complexity 2 = 19
 }}
-'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic ratios. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).
+'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).
 [[31edo]] is a very good tuning of didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS scale]]s of didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[slendric]].