20edo: Difference between revisions

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~~'''~~20 ~~equal divisions of the octave''' ('''20edo''') divides the [[octave]] into exactly 20 [[equal]] steps of 60 [[cent]]s each.~~

== Theory ==

20edo contains smaller {{EDOs|legend=1| 2, 4, 5, and 10 }} and is part of the 5''n'' family of equal divisions of the octave. It fairly approximates the harmonics 7 (from [[5edo]]), 11, 13 & 15 (from [[10edo]]), 19 & 27 (from [[4edo]]), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15. As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard. Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead. The 11\20 generator is a near-optimal tuning for both [[No-threes_subgroup_temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator) and [[Chromatic pairs#Score|Score]] temperament (which has a ~[[16/11]] generator).

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 {{Infobox ET}}
-'''20 equal divisions of the octave''' ('''20edo''') divides the [[octave]] into exactly 20 [[equal]] steps of 60 [[cent]]s each.
+{{EDO intro|20}}
 == Theory ==
 edo contains smaller {{EDOs|legend=1| 2, 4, 5, and 10 }} and is part of the 5''n'' family of equal divisions of the octave. It fairly approximates the harmonics 7 (from [[5edo]]), 11, 13 &amp; 15 (from [[10edo]]), 19 &amp; 27 (from [[4edo]]), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15. As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard. Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead. The 11\20 generator is a near-optimal tuning for both [[No-threes_subgroup_temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator)  and [[Chromatic pairs#Score|Score]] temperament (which has a ~[[16/11]] generator).