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.

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]]~~. It is a sharp-tending system, with the 3rd, 5th and 7th harmonics all sharp~~.

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 (temperament)|neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]].

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{{c}}). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament.

The 5th degree of 68edo can be used as a generator for [[88cET]], while ~~one step~~ almost perfectly ~~approximates~~ [[50/49]], allowing for the tempering out of 2/(50/49)34.

The 5th degree of 68edo can be used as a generator for [[88cET]], while two steps almost perfectly approximate [[50/49]], allowing for the tempering out of 2/(50/49)34.

=== Prime harmonics ===

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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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== Approximation to JI ==

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

=== Interval mappings ===

{{~~ZPI~~

~~| zpi = 354~~

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

| ~~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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| 2.3.5.7

| 245/243, 2048/2025, 2401/2400

| {{~~mapping~~| 68 108 158 191 }}

| {{Mapping| 68 108 158 191 }}

| −0.983

| 0.915

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| 2.3.5.7.11

| 121/120, 176/175, 245/243, 1375/1372

| {{~~mapping~~| 68 108 158 191 235 }}

| {{Mapping| 68 108 158 191 235 }}

| −0.541

| 1.206

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| 2.3.5.7.11.13

| 121/120, 176/175, 196/195, 245/243, 275/273

| {{~~mapping~~| 68 108 158 191 235 252 }}

| {{Mapping| 68 108 158 191 235 252 }}

| −0.745

| 1.191

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| 2.3.5.7.11.13.17

| 121/120, 136/135, 154/153, 176/175, 196/195, 275/273

| {{~~mapping~~| 68 108 158 191 235 252 278 }}

| {{Mapping| 68 108 158 191 235 252 278 }}

| −0.671

| 1.118

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| 2.3.5.7.11.13.17.19

| 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273

| {{~~mapping~~| 68 108 158 191 235 252 278 289 }}

| {{Mapping| 68 108 158 191 235 252 278 289 }}

| −0.661

| 1.046

| 5.93

|}

=== Rank 2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperament

|-

| 1

| 5\68

| 88.24

| 20/19

| [[Octacot]]

|-

| 1

| 11\68

| 194.12

| 28/25

| [[Hemiwürschmidt]] (68e) / hemiwur (68)

|-

| 2

| 3\68

| 52.94

| 33/32

| [[Shrutar]]

|-

| 4

| 6\68

| 105.88

| 17/16

| [[Bidia]]

|}

<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

== Scales ==

: ''See also: [[34edo #Scales]] and [[17edo #Scales]].''

: ''See also: [[List of MOS scales in 68edo]], [[34edo #Scales]], and [[17edo #Scales]].''

* [[Deeptone]][7]: 10 10 9 10 10 10 9

* [[Hemiwur]][31]: 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2

* Inverse half octave{{clarify}}: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7

* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')

* [[Octacot]][27]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3

* [[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

* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3

* [[Superpyth]] quarter octave{{clarify}}: 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

* Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C)

== Instruments ==

* ~~Deeptone~~[7]~~: 10 10 9 10 10 10 9~~

* [[Lumatone mapping for 68edo]]

* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7

* 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~~

== 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)

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[[Category:Shrutar]]

[[Category:Quartismic]]

[[Category:Todo:add rank 2 temperaments table]]

@@ 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.
+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]]. It is a sharp-tending system, with the 3rd, 5th and 7th harmonics all sharp.
+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 (temperament)|neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]].
 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{{c}}). It also works as a [[22L&nbsp;1s]] MOS of the quartkeenlig temperament.
-The 5th degree of 68edo can be used as a generator for [[88cET]], while one step almost perfectly approximates [[50/49]], allowing for the tempering out of 2/(50/49)<sup>34</sup>.
+The 5th degree of 68edo can be used as a generator for [[88cET]], while two steps almost perfectly approximate [[50/49]], allowing for the tempering out of 2/(50/49)<sup>34</sup>.
 === Prime harmonics ===
@@ 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 352: / Line 352: @@
 == Approximation to JI ==
-=== Zeta peak index ===
+=== Interval mappings ===
-{{ZPI
+{{Q-odd-limit intervals}}
-| zpi = 354
+{{Q-odd-limit intervals|68.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 68e val mapping}}
-| 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 380: / Line 370: @@
 | 2.3.5.7
 | 245/243, 2048/2025, 2401/2400
-| {{mapping| 68 108 158 191 }}
+| {{Mapping| 68 108 158 191 }}
 | −0.983
 | 0.915
@@ Line 387: / Line 377: @@
 | 2.3.5.7.11
 | 121/120, 176/175, 245/243, 1375/1372
-| {{mapping| 68 108 158 191 235 }}
+| {{Mapping| 68 108 158 191 235 }}
 | −0.541
 | 1.206
@@ Line 394: / Line 384: @@
 | 2.3.5.7.11.13
 | 121/120, 176/175, 196/195, 245/243, 275/273
-| {{mapping| 68 108 158 191 235 252 }}
+| {{Mapping| 68 108 158 191 235 252 }}
 | −0.745
 | 1.191
@@ Line 401: / Line 391: @@
 | 2.3.5.7.11.13.17
 | 121/120, 136/135, 154/153, 176/175, 196/195, 275/273
-| {{mapping| 68 108 158 191 235 252 278 }}
+| {{Mapping| 68 108 158 191 235 252 278 }}
 | −0.671
 | 1.118
@@ Line 408: / Line 398: @@
 | 2.3.5.7.11.13.17.19
 | 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273
-| {{mapping| 68 108 158 191 235 252 278 289 }}
+| {{Mapping| 68 108 158 191 235 252 278 289 }}
 | −0.661
 | 1.046
 | 5.93
 |}
+=== Rank 2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperament
+|-
+| 1
+| 5\68
+| 88.24
+| 20/19
+| [[Octacot]]
+|-
+| 1
+| 11\68
+| 194.12
+| 28/25
+| [[Hemiwürschmidt]] (68e) / hemiwur (68)
+|-
+| 2
+| 3\68
+| 52.94
+| 33/32
+| [[Shrutar]]
+|-
+| 4
+| 6\68
+| 105.88
+| 17/16
+| [[Bidia]]
+|}
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 == Scales ==
-: ''See also: [[34edo #Scales]] and [[17edo #Scales]].''
+: ''See also: [[List of MOS scales in 68edo]], [[34edo #Scales]], and [[17edo #Scales]].''
+* [[Deeptone]][7]: 10 10 9 10 10 10 9
+* [[Hemiwur]][31]: 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2
+* Inverse half octave{{clarify}}: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
+* Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'')
+* [[Octacot]][27]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3
+* [[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
+* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3
+* [[Superpyth]] quarter octave{{clarify}}: 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
-* Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C)
+== Instruments ==
-* Deeptone[7]: 10 10 9 10 10 10 9
+* [[Lumatone mapping for 68edo]]
-* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
-* 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
 == 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 434: / Line 469: @@
 [[Category:Shrutar]]
 [[Category:Quartismic]]
+[[Category:Todo:add rank 2 temperaments table]]