68edo: Difference between revisions

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== Theory ==

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 in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series.

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 in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly since [[11/9]] is not mapped to its best approximation. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series.

As a 7-limit system, 68et [[tempering out|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]].

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=== Subsets and supersets ===

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

Since 68 factors into {{factorization|68}}, 68edo has subset edos {{EDOs| 2, 4, 17, and 34 }}. Important supersets include [[612edo]], a 5-limit record-holder, and [[680edo]].

== Intervals ==

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=== Interval mappings ===

{{Q-odd-limit intervals|68.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 68e val mapping}}

~~=== Zeta peak index ===~~

{{~~ZPI~~

~~| zpi = 354~~

| ~~steps =~~ 68.~~0493056282519~~

~~| step size = 17.6342725163943~~

~~| tempered height = 7.666604~~

~~| pure height = 6.147517~~

~~| integral =~~ 1~~.254592~~

| ~~gap~~ = ~~17.034505~~

| ~~octave~~ = ~~1199.13053111481~~

| ~~consistent~~ = 10

| ~~distinct~~ = 10

}}

== Regular temperament properties ==

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* 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

* Quartkeenlig[23] (Stretched 23edo): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2

== Instruments ==

* [[Lumatone mapping for 68edo]]

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/CO1AslAu9E0 ''microtonal improvisation in 68edo''] (2025)

; [[The Mercury Tree]]

* [https://m.youtube.com/watch?v=ZRFumaIM02E Grown Apart] from ''Self Similar'' (2023)

@@ Line 3: / Line 3: @@
 == Theory ==
-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 in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series.
+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 in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly since [[11/9]] is not mapped to its best approximation. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series.
 As a 7-limit system, 68et [[tempering out|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]].
@@ Line 15: / Line 15: @@
 === Subsets and supersets ===
-Since 68 factors into {{factorization|68}}, 68edo has subset edos {{EDOs| 2, 4, 17, and 34 }}.
+Since 68 factors into {{factorization|68}}, 68edo has subset edos {{EDOs| 2, 4, 17, and 34 }}. Important supersets include [[612edo]], a 5-limit record-holder, and [[680edo]].
 == Intervals ==
@@ Line 354: / Line 354: @@
 === Interval mappings ===
 {{Q-odd-limit intervals}}
+{{Q-odd-limit intervals|68.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 68e val mapping}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 354
-| steps = 68.0493056282519
-| step size = 17.6342725163943
-| tempered height = 7.666604
-| pure height = 6.147517
-| integral = 1.254592
-| gap = 17.034505
-| octave = 1199.13053111481
-| consistent = 10
-| distinct = 10
-}}
 == Regular temperament properties ==
@@ Line 425: / Line 412: @@
 * 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
 * Quartkeenlig[23] (Stretched 23edo): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2
+== Instruments ==
+* [[Lumatone mapping for 68edo]]
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/CO1AslAu9E0 ''microtonal improvisation in 68edo''] (2025)
 ; [[The Mercury Tree]]
 * [https://m.youtube.com/watch?v=ZRFumaIM02E Grown Apart] from ''Self Similar'' (2023)