12edo: Difference between revisions

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The commas it tempers out include the Pythagorean comma, 3<sup>12</sup>/2<sup>19</sup>, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. ~~As of right now, it~~ is also the only known EDO that is both [[The Riemann zeta function and tuning|~~full~~ zeta]] and highly composite, and the only ~~one~~ with a step size larger than the just noticeable difference (~3-4 cents).

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. 12edo is also the only known EDO that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite, and the only strict zeta EDO with a step size larger than the [[just-noticeable difference]] (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

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 The commas it tempers out include the Pythagorean comma, 3<sup>12</sup>/2<sup>19</sup>, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
-edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).
+edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. 12edo is also the only known EDO that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite, and the only strict zeta EDO with a step size larger than the [[just-noticeable difference]] (~3-4 cents).
 edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.