68edo: Difference between revisions

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68edo's step 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 [[support]]s [[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 [[2048/2025]], [[245/243]], [[4000/3969]], [[15625/15552]], [[3136/3125]], [[6144/6125]] and [[2401/2400]]. It [[support]]s [[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 3rd, 5th and 7th harmonics all sharp.

The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]], which also acts as the [[quartkeenlig]] temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament.

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=== Prime harmonics ===

=== Subsets and supersets ===

Since 68 factors into {{factorization|68}}, 68edo has subset edos {{EDOs| 2, 4, 17, and 34 }}.

== Intervals ==

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 edo's step 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 [[support]]s [[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 [[2048/2025]], [[245/243]], [[4000/3969]], [[15625/15552]], [[3136/3125]], [[6144/6125]] and [[2401/2400]]. It [[support]]s [[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 3rd, 5th and 7th harmonics all sharp.
 The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]], which also acts as the [[quartkeenlig]] temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament.
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 === Prime harmonics ===
 {{Harmonics in equal|68}}
+=== Subsets and supersets ===
+Since 68 factors into {{factorization|68}}, 68edo has subset edos {{EDOs| 2, 4, 17, and 34 }}.
 == Intervals ==