68edo: Difference between revisions
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The '''68 equal temperament''', often abbreviated '''68-tET''', '''68-EDO''', or '''68-ET''', is the scale derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents; this is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well for in [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly. | The '''68 equal temperament''', often abbreviated '''68-tET''', '''68-EDO''', or '''68-ET''', is the scale derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents; this is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well for in [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly. | ||
As a 7-limit system it tempers out 2048/2025, 245/243, 4000/3969, 15625/15552, 3136/3125, 6144/6125 and 2401/2400. It supports [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]]. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp. | As a 7-limit system it tempers out [[Diaschisma|2048/2025]], [[245/243]], 4000/3969, [[15625/15552]], [[3136/3125]], [[6144/6125]] and [[2401/2400]]. It supports [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]]. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp. | ||
=Diatonic scales= | =Diatonic scales= | ||
Revision as of 19:39, 15 May 2021
The 68 equal temperament, often abbreviated 68-tET, 68-EDO, or 68-ET, is the scale derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents; this is half of the step size of 34edo, which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of 17edo, which does well in the 3-limit, but not so well in the 5-limit. The luck continues: 68 is a strong 7-limit system, but does not do as well for in 11-limit; though it's certainly usable for that purpose, it does not represent the 11-limit diamond consistently.
As a 7-limit system it tempers out 2048/2025, 245/243, 4000/3969, 15625/15552, 3136/3125, 6144/6125 and 2401/2400. It supports octacot, shrutar, hemiwürschmidt, hemikleismic, clyde and neptune temperaments, and supplies the optimal patent val for 11-limit hemikleismic. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp.
Diatonic scales
Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C5)
Superpyth: 12 12 4 12 12 12 4
Superpyth quarter octave: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1
Flattone: 10 10 9 10 10 10 9
Inverse: 8 8 14 8 8 8 14
Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7