65edo: Difference between revisions

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== Intervals ==
== Intervals ==


{| class="wikitable"
{| class="wikitable center-all right-2"
|-
|-
! [[Degree|Degree]]
! [[Degree]]
![[cent|Cents]]
! [[Cent]]s
! colspan="2" |[[Ups and Downs Notation]]
! colspan="2" |[[Ups and Downs Notation]]
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0.0000
| 0.00
|P1
| P1
|D
| D
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 18.4615
| 18.46
|^1
| ^1
|^D
| ^D
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" | 36.9231
| 36.92
|^^1
| ^^1
|^^D
| ^^D
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 55.3846
| 55.38
|vvm2
| vvm2
|vvEb
| vvEb
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 73.84615
| 73.85
|vm2
| vm2
|vEb
| vEb
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 92.3077
| 92.31
|m2
| m2
|Eb
| Eb
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 110.7692
| 110.77
|A1/^m2
| A1/^m2
|D#/^Eb
| D#/^Eb
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 129.2308
| 129.23
|v~2
| v~2
|^^Eb
| ^^Eb
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 147.6923
| 147.69
|~2
| ~2
|vvvE
| vvvE
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 166.15385
| 166.15
|^~2
| ^~2
|vvE
| vvE
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 184.6154
| 184.62
|vM2
| vM2
|vE
| vE
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 203.0769
| 203.08
|M2
| M2
|E
| E
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 221.5385
| 221.54
|^M2
| ^M2
|^E
| ^E
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 240
| 240.00
|^^M2
| ^^M2
|^^E
| ^^E
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:right;" | 258.4615
| 258.46
|vvm3
| vvm3
|vvF
| vvF
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:right;" | 276.9231
| 276.92
|vm3
| vm3
|vF
| vF
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:right;" | 295.3846
| 295.38
|m3
| m3
|F
| F
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:right;" | 313.84615
| 313.85
|^m3
| ^m3
|^F
| ^F
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:right;" | 332.3077
| 332.31
|v~3
| v~3
|^^F
| ^^F
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:right;" | 350.7692
| 350.77
|~3
| ~3
|^^^F
| ^^^F
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:right;" | 369.2308
| 369.23
|^~3
| ^~3
|vvF#
| vvF#
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:right;" | 387.6923
| 387.69
|vM3
| vM3
|vF#
| vF#
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:right;" | 406.15385
| 406.15
|M3
| M3
|F#
| F#
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:right;" | 424.6154
| 424.62
|^M3
| ^M3
|^F#
| ^F#
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:right;" | 443.0769
| 443.08
|^^M3
| ^^M3
|^^F#
| ^^F#
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:right;" | 461.5385
| 461.54
|vv4
| vv4
|vvG
| vvG
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:right;" | 480
| 480.00
|v4
| v4
|vG
| vG
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:right;" | 498.4615
| 498.46
|P4
| P4
|G
| G
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:right;" | 516.9231
| 516.92
|^4
| ^4
|^G
| ^G
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:right;" | 535.3846
| 535.38
|v~4
| v~4
|^^G
| ^^G
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:right;" | 553.84615
| 553.85
|~4
| ~4
|^^^G
| ^^^G
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:right;" | 572.3077
| 572.31
|^~4/vd5
| ^~4/vd5
|vvG#/vAb
| vvG#/vAb
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:right;" | 590.7692
| 590.77
|vA4/d5
| vA4/d5
|vG#/Ab
| vG#/Ab
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:right;" | 609.2308
| 609.23
|A4/^d5
| A4/^d5
|G#/^Ab
| G#/^Ab
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:right;" | 627.6923
| 627.69
|^A4/v~5
| ^A4/v~5
|^G#/^^Ab
| ^G#/^^Ab
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:right;" | 646.1538
| 646.15
|~5
| ~5
|vvvA
| vvvA
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:right;" | 664.6154
| 664.62
|^~5
| ^~5
|vvA
| vvA
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:right;" | 683.0769
| 683.08
|v5
| v5
|vA
| vA
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:right;" | 701.5385
| 701.54
|P5
| P5
|A
| A
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:right;" | 720
| 720.00
|^5
| ^5
|^A
| ^A
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:right;" | 738.4615
| 738.46
|^^5
| ^^5
|^^A
| ^^A
|-
|-
| style="text-align:center;" | 41
| 41
| style="text-align:right;" | 756.9231
| 756.92
|vvm6
| vvm6
|vvBb
| vvBb
|-
|-
| style="text-align:center;" | 42
| 42
| style="text-align:right;" | 775.3846
| 775.38
|vm6
| vm6
|vBb
| vBb
|-
|-
| style="text-align:center;" | 43
| 43
| style="text-align:right;" | 793.84615
| 793.85
|m6
| m6
|Bb
| Bb
|-
|-
| style="text-align:center;" | 44
| 44
| style="text-align:right;" | 812.3077
| 812.31
|^m6
| ^m6
|^Bb
| ^Bb
|-
|-
| style="text-align:center;" | 45
| 45
| style="text-align:right;" | 830.7692
| 830.77
|v~6
| v~6
|^^Bb
| ^^Bb
|-
|-
| style="text-align:center;" | 46
| 46
| style="text-align:right;" | 849.2308
| 849.23
|~6
| ~6
|vvvB
| vvvB
|-
|-
| style="text-align:center;" | 47
| 47
| style="text-align:right;" | 867.6923
| 867.69
|^~6
| ^~6
|vvB
| vvB
|-
|-
| style="text-align:center;" | 48
| 48
| style="text-align:right;" | 886.15385
| 886.15
|vM6
| vM6
|vB
| vB
|-
|-
| style="text-align:center;" | 49
| 49
| style="text-align:right;" | 904.6154
| 904.62
|M6
| M6
|B
| B
|-
|-
| style="text-align:center;" | 50
| 50
| style="text-align:right;" | 923.0769
| 923.08
|^M6
| ^M6
|^B
| ^B
|-
|-
| style="text-align:center;" | 51
| 51
| style="text-align:right;" | 941.5385
| 941.54
|^^M6
| ^^M6
|^^B
| ^^B
|-
|-
| style="text-align:center;" | 52
| 52
| style="text-align:right;" | 960
| 960.00
|vvm7
| vvm7
|vvC
| vvC
|-
|-
| style="text-align:center;" | 53
| 53
| style="text-align:right;" | 978.4615
| 978.46
|vm7
| vm7
|vC
| vC
|-
|-
| style="text-align:center;" | 54
| 54
| style="text-align:right;" | 996.9231
| 996.92
|m7
| m7
|C
| C
|-
|-
| style="text-align:center;" | 55
| 55
| style="text-align:right;" | 1015.3846
| 1015.38
|^m7
| ^m7
|^C
| ^C
|-
|-
| style="text-align:center;" | 56
| 56
| style="text-align:right;" | 1033.84615
| 1033.85
|v~7
| v~7
|^^C
| ^^C
|-
|-
| style="text-align:center;" | 57
| 57
| style="text-align:right;" | 1052.3077
| 1052.31
|~7
| ~7
|^^^C
| ^^^C
|-
|-
| style="text-align:center;" | 58
| 58
| style="text-align:right;" | 1070.7692
| 1070.77
|^~7
| ^~7
|vvC#
| vvC#
|-
|-
| style="text-align:center;" | 59
| 59
| style="text-align:right;" | 1089.2308
| 1089.23
|vM7
| vM7
|vC#
| vC#
|-
|-
| style="text-align:center;" | 60
| 60
| style="text-align:right;" | 1107.6923
| 1107.69
|M7
| M7
|C#
| C#
|-
|-
| style="text-align:center;" | 61
| 61
| style="text-align:right;" | 1126.15385
| 1126.15
|^M7
| ^M7
|^C#
| ^C#
|-
|-
| style="text-align:center;" | 62
| 62
| style="text-align:right;" | 1144.6154
| 1144.62
|^^M7
| ^^M7
|^^C#
| ^^C#
|-
|-
| style="text-align:center;" | 63
| 63
| style="text-align:right;" | 1163.0769
| 1163.08
|vv8
| vv8
|vvD
| vvD
|-
|-
| style="text-align:center;" | 64
| 64
| style="text-align:right;" | 1181.5385
| 1181.54
|v8
| v8
|vD
| vD
|-
|-
| style="text-align:center;" | 65
| 65
| style="text-align:right;" | 1200
| 1200.00
|P8
| P8
|D
| D
|}
|}


Revision as of 12:55, 27 July 2021

65edo divides the octave into 65 equal parts of 18.4615 cents each.

Theory

65et can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the würschmidt comma. In the 7-limit, there are two different maps; the first is ⟨65 103 151 182], tempering out 126/125, 245/243 and 686/675, so that it supports sensi temperament, and the second is ⟨65 103 151 183] (65d), tempering out 225/224, 3125/3087, 4000/3969 and 5120/5103, so that it supports garibaldi temperament. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit würschmidt temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals 3/2, 5/4, 11/8, 19/16, 23/16 and 31/16 well, so that it does a good job representing the 2.3.5.11.19.23.31 just intonation subgroup. To this one may want to add 17/16 and 29/16, giving the 31-limit no-7's no-13's subgroup 2.3.5.11.17.19.23.29.31. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the zeta edo 130edo.

65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Rubble: a Xenuke Unfolded.

Prime harmonics

Script error: No such module "primes_in_edo".

Intervals

Degree Cents Ups and Downs Notation
0 0.00 P1 D
1 18.46 ^1 ^D
2 36.92 ^^1 ^^D
3 55.38 vvm2 vvEb
4 73.85 vm2 vEb
5 92.31 m2 Eb
6 110.77 A1/^m2 D#/^Eb
7 129.23 v~2 ^^Eb
8 147.69 ~2 vvvE
9 166.15 ^~2 vvE
10 184.62 vM2 vE
11 203.08 M2 E
12 221.54 ^M2 ^E
13 240.00 ^^M2 ^^E
14 258.46 vvm3 vvF
15 276.92 vm3 vF
16 295.38 m3 F
17 313.85 ^m3 ^F
18 332.31 v~3 ^^F
19 350.77 ~3 ^^^F
20 369.23 ^~3 vvF#
21 387.69 vM3 vF#
22 406.15 M3 F#
23 424.62 ^M3 ^F#
24 443.08 ^^M3 ^^F#
25 461.54 vv4 vvG
26 480.00 v4 vG
27 498.46 P4 G
28 516.92 ^4 ^G
29 535.38 v~4 ^^G
30 553.85 ~4 ^^^G
31 572.31 ^~4/vd5 vvG#/vAb
32 590.77 vA4/d5 vG#/Ab
33 609.23 A4/^d5 G#/^Ab
34 627.69 ^A4/v~5 ^G#/^^Ab
35 646.15 ~5 vvvA
36 664.62 ^~5 vvA
37 683.08 v5 vA
38 701.54 P5 A
39 720.00 ^5 ^A
40 738.46 ^^5 ^^A
41 756.92 vvm6 vvBb
42 775.38 vm6 vBb
43 793.85 m6 Bb
44 812.31 ^m6 ^Bb
45 830.77 v~6 ^^Bb
46 849.23 ~6 vvvB
47 867.69 ^~6 vvB
48 886.15 vM6 vB
49 904.62 M6 B
50 923.08 ^M6 ^B
51 941.54 ^^M6 ^^B
52 960.00 vvm7 vvC
53 978.46 vm7 vC
54 996.92 m7 C
55 1015.38 ^m7 ^C
56 1033.85 v~7 ^^C
57 1052.31 ~7 ^^^C
58 1070.77 ^~7 vvC#
59 1089.23 vM7 vC#
60 1107.69 M7 C#
61 1126.15 ^M7 ^C#
62 1144.62 ^^M7 ^^C#
63 1163.08 vv8 vvD
64 1181.54 v8 vD
65 1200.00 P8 D

Scales

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