65edo

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65 tone equal temperament

65edo divides the octave into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the wuerschmidt 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 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo 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 wuerschmidt temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 just intonation subgroup. To this one may want to add 13/8 and 17/16, giving the 19-limit no-sevens subgroup 2.3.5.11.13.17.19. 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 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.

Intervals

Degree Cents Ups and Downs
0 0.0000 P1 D
1 18.4615 ^1 ^D
2 36.9231 ^^1 ^^D
3 55.3846 vvm2 vvEb
4 73.84615 vm2 vEb
5 92.3077 m2 Eb
6 110.7692 A1/^m2 D#/^Eb
7 129.2308 v~2 ^^Eb
8 147.6923 ~2 vvvE
9 166.15385 ^~2 vvE
10 184.6154 vM2 vE
11 203.0769 M2 E
12 221.5385 ^M2 ^E
13 240 ^^M2 ^^E
14 258.4615 vvm3 vvF
15 276.9231 vm3 vF
16 295.3846 m3 F
17 313.84615 ^m3 ^F
18 332.3077 v~3 ^^F
19 350.7692 ~3 ^^^F
20 369.2308 ^~3 vvF#
21 387.6923 vM3 vF#
22 406.15385 M3 F#
23 424.6154 ^M3 ^F#
24 443.0769 ^^M3 ^^F#
25 461.5385 vv4 vvG
26 480 v4 vG
27 498.4615 P4 G
28 516.9231 ^4 ^G
29 535.3846 v~4 ^^G
30 553.84615 ~4 ^^^G
31 572.3077 ^~4/vd5 vvG#/vAb
32 590.7692 vA4/d5 vG#/Ab
33 609.2308 A4/^d5 G#/^Ab
34 627.6923 ^A4/v~5 ^G#/^^Ab
35 646.1538 ~5 vvvA
36 664.6154 ^~5 vvA
37 683.0769 v5 vA
38 701.5385 P5 A
39 720 ^5 ^A
40 738.4615 ^^5 ^^A
41 756.9231 vvm6 vvBb
42 775.3846 vm6 vBb
43 793.84615 m6 Bb
44 812.3077 ^m6 ^Bb
45 830.7692 v~6 ^^Bb
46 849.2308 ~6 vvvB
47 867.6923 ^~6 vvB
48 886.15385 vM6 vB
49 904.6154 M6 B
50 923.0769 ^M6 ^B
51 941.5385 ^^M6 ^^B
52 960 vvm7 vvC
53 978.4615 vm7 vC
54 996.9231 m7 C
55 1015.3846 ^m7 ^C
56 1033.84615 v~7 ^^C
57 1052.3077 ~7 ^^^C
58 1070.7692 ^~7 vvC#
59 1089.2308 vM7 vC#
60 1107.6923 M7 C#
61 1126.15385 ^M7 ^C#
62 1144.6154 ^^M7 ^^C#
63 1163.0769 vv8 vvD
64 1181.5385 v8 vD
65 1200 P8 D

Scales

photia7

photia12