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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.
 
== 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 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 [[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.
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.
=== Prime harmonics ===
{{Primes in edo|68}}


== Intervals ==
== Intervals ==
Line 7: Line 13:
! Degrees
! Degrees
! Cents
! Cents
! Approximate Ratios
|-
|-
|0
| 0
|0.0000
| 0.00
| 1/1
|-
|-
|1
| 1
|17.6471
| 17.65
| 64/63, 126/125, 225/224
|-
|-
|2
| 2
|35.2941
| 35.29
| 81/80, 49/48, 50/49
|-
|-
|3
| 3
|52.9411
| 52.94
| 28/27, 36/35, 33/32
|-
|-
|4
| 4
|70.5882
| 70.59
| 25/24, ''22/21''
|-
|-
|5
| 5
|88.2353
| 88.24
| 21/20, 19/18, 20/19
|-
|-
|6
| 6
|105.8824
| 105.88
| 16/15, 17/16, 18/17
|-
|-
|7
| 7
|123.5294
| 123.53
| 15/14, 14/13
|-
|-
|8
| 8
|141.1765
| 141.18
| 13/12
|-
|-
|9
| 9
|158.8235
| 158.82
| 12/11, 11/10
|-
|-
|10
| 10
|176.4706
| 176.47
| 10/9
|-
|-
|11
| 11
|194.1176
| 194.12
| 28/25, 19/17
|-
|-
|12
| 12
|211.7647
| 211.76
| 9/8
|-
|-
|13
| 13
|229.4118
| 229.41
| 8/7
|-
|-
|14
| 14
|247.0588
| 247.06
| 15/13
|-
|-
|15
| 15
|264.7059
| 264.71
| 7/6
|-
|-
|16
| 16
|282.3529
| 282.35
| 20/17
|-
|-
|17
| 17
|300.0000
| 300.00
| ''13/11'', 19/16
|-
|-
|18
| 18
|317.6471
| 317.65
| 6/5
|-
|-
|19
| 19
|335.2941
| 335.29
| ''11/9'', 40/33, 17/14
|-
|-
|20
| 20
|352.9412
| 352.94
| 16/13, ''39/32''
|-
|-
|21
| 21
|370.5882
| 370.59
| ''27/22'', 26/21, 21/17
|-
|-
|22
| 22
|388.2353
| 388.24
| 5/4
|-
|-
|23
| 23
|405.8824
| 405.88
| 24/19, 19/15
|-
|-
|24
| 24
|423.5294
| 423.53
| 14/11
|-
|-
|25
| 25
|441.1765
| 441.18
| 9/7
|-
|-
|26
| 26
|458.8235
| 458.82
| 13/10, 17/13
|-
|-
|27
| 27
|476.4706
| 476.47
| 21/16
|-
|-
|28
| 28
|494.1176
| 494.12
| 4/3
|-
|-
|29
| 29
|511.7647
| 511.76
| 75/56
|-
|-
|30
| 30
|529.4118
| 529.41
| 27/20, 19/14
|-
|-
|31
| 31
|547.0588
| 547.06
| 11/8, ''15/11''
|-
|-
|32
| 32
|564.7059
| 564.71
| 25/18, 18/13, 26/19
|-
|-
|33
| 33
|582.3529
| 582.35
| 7/5
|-
|-
|34
| 34
|600.0000
| 600.00
| 17/12, 24/17
|-
|-
|35
| 35
|617.6471
| 617.65
| 10/7
|-
|-
|36
| 36
|635.2941
| 635.29
| 36/25, 13/9, 19/13
|-
|-
|37
| 37
|652.9412
| 652.94
| 16/11, ''22/15''
|-
|-
|38
| 38
|670.5882
| 670.59
| 40/27, 28/19
|-
|-
|39
| 39
|688.2353
| 688.24
| 112/75
|-
|-
|40
| 40
|705.8824
| 705.88
| 3/2
|-
|-
|41
| 41
|723.5294
| 723.53
| 32/21
|-
|-
|42
| 42
|741.1765
| 741.18
| 16/13, 26/17
|-
|-
|43
| 43
|758.8235
| 758.82
| 14/9
|-
|-
|44
| 44
|776.4706
| 776.47
| 11/7
|-
|-
|45
| 45
|794.1176
| 794.12
| 19/12, 30/19
|-
|-
|46
| 46
|811.7647
| 811.76
| 8/5
|-
|-
|47
| 47
|829.4118
| 829.41
| ''44/27'', 21/13, 34/21
|-
|-
|48
| 48
|847.0588
| 847.06
| 13/8, ''64/39''
|-
|-
|49
| 49
|864.7059
| 864.71
| ''18/11'', 33/20, 28/17
|-
|-
|50
| 50
|882.3529
| 882.35
| 5/3
|-
|-
|51
| 51
|900.0000
| 900.00
| ''22/13'', 32/19
|-
|-
|52
| 52
|917.6471
| 917.65
| 17/10
|-
|-
|53
| 53
|935.2941
| 935.29
| 12/7
|-
|-
|54
| 54
|952.9412
| 952.94
| 26/15
|-
|-
|55
| 55
|970.5882
| 970.59
| 7/4
|-
|-
|56
| 56
|988.2353
| 988.24
| 16/9
|-
|-
|57
| 57
|1005.8824
| 1005.88
| 25/14, 34/19
|-
|-
|58
| 58
|1023.5294
| 1023.53
| 9/5
|-
|-
|59
| 59
|1041.1765
| 1041.18
| 11/6, 20/11
|-
|-
|60
| 60
|1058.8235
| 1058.82
| 24/13
|-
|-
|61
| 61
|1076.4706
| 1076.47
| 28/15, 13/7
|-
|-
|62
| 62
|1094.1176
| 1094.12
| 15/8, 32/17, 17/9
|-
|-
|63
| 63
|1111.7647
| 1111.76
| 40/21, 36/19, 19/10
|-
|-
|64
| 64
|1129.4118
| 1129.41
| 48/25, ''21/11''
|-
|-
|65
| 65
|1147.0588
| 1147.06
| 27/14, 35/18, 64/33
|-
|-
|66
| 66
|1164.7059
| 1164.71
| 160/81, 96/49, 49/25
|-
|-
|67
| 67
|1182.3529
| 1182.35
| 63/32, 125/64, 448/225
|-
|-
|68
| 68
|1200.0000
| 1200.00
| 2/1
|}
|}


Line 231: Line 307:
Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7


[[Category:Equal divisions of the octave]]
[[Category:Clyde]]
[[Category:Clyde]]
[[Category:Equal divisions of the octave]]
[[Category:Hemikleismic]]
[[Category:Hemikleismic]]
[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Neptune]]
[[Category:Neptune]]
[[Category:Octacot]]
[[Category:Octacot]]
[[Category:Shrutar]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Shrutar]]
Retrieved from "https://en.xen.wiki/w/68edo"