68edo: Difference between revisions

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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.
= Intervals =
{| class="wikitable"
!Degrees
!Cents
|-
|0
|0.0000
|-
|1
|17.6471
|-
|2
|35.2941
|-
|3
|52.9411
|-
|4
|70.5882
|-
|5
|88.2353
|-
|6
|105.8824
|-
|7
|123.5294
|-
|8
|141.1765
|-
|9
|158.8235
|-
|10
|176.4706
|-
|11
|194.1176
|-
|12
|211.7647
|-
|13
|229.4118
|-
|14
|247.0588
|-
|15
|264.7059
|-
|16
|282.3529
|-
|17
|300.0000
|-
|18
|317.6471
|-
|19
|335.2941
|-
|20
|352.9412
|-
|21
|370.5882
|-
|22
|388.2353
|-
|23
|405.8824
|-
|24
|423.5294
|-
|25
|441.1765
|-
|26
|458.8235
|-
|27
|476.4706
|-
|28
|494.1176
|-
|29
|511.7647
|-
|30
|529.4118
|-
|31
|547.0588
|-
|32
|564.7059
|-
|33
|582.3529
|-
|34
|600.0000
|-
|35
|617.6471
|-
|36
|635.2941
|-
|37
|652.9412
|-
|38
|670.5882
|-
|39
|688.2353
|-
|40
|705.8824
|-
|41
|723.5294
|-
|42
|741.1765
|-
|43
|758.8235
|-
|44
|776.4706
|-
|45
|794.1176
|-
|46
|811.7647
|-
|47
|829.4118
|-
|48
|847.0588
|-
|49
|864.7059
|-
|50
|882.3529
|-
|51
|900.0000
|-
|52
|917.6471
|-
|53
|935.2941
|-
|54
|952.9412
|-
|55
|970.5882
|-
|56
|988.2353
|-
|57
|1005.8824
|-
|58
|1023.5294
|-
|59
|1041.1765
|-
|60
|1058.8235
|-
|61
|1076.4706
|-
|62
|1094.1176
|-
|63
|1111.7647
|-
|64
|1129.4118
|-
|65
|1147.0588
|-
|66
|1164.7059
|-
|67
|1182.3529
|-
|68
|1200.0000
|}


=Diatonic scales=
=Diatonic scales=

Revision as of 01:18, 28 June 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.

Intervals

Degrees Cents
0 0.0000
1 17.6471
2 35.2941
3 52.9411
4 70.5882
5 88.2353
6 105.8824
7 123.5294
8 141.1765
9 158.8235
10 176.4706
11 194.1176
12 211.7647
13 229.4118
14 247.0588
15 264.7059
16 282.3529
17 300.0000
18 317.6471
19 335.2941
20 352.9412
21 370.5882
22 388.2353
23 405.8824
24 423.5294
25 441.1765
26 458.8235
27 476.4706
28 494.1176
29 511.7647
30 529.4118
31 547.0588
32 564.7059
33 582.3529
34 600.0000
35 617.6471
36 635.2941
37 652.9412
38 670.5882
39 688.2353
40 705.8824
41 723.5294
42 741.1765
43 758.8235
44 776.4706
45 794.1176
46 811.7647
47 829.4118
48 847.0588
49 864.7059
50 882.3529
51 900.0000
52 917.6471
53 935.2941
54 952.9412
55 970.5882
56 988.2353
57 1005.8824
58 1023.5294
59 1041.1765
60 1058.8235
61 1076.4706
62 1094.1176
63 1111.7647
64 1129.4118
65 1147.0588
66 1164.7059
67 1182.3529
68 1200.0000

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

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