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 Size (Cents)
0 0.0000
1 18.4615
2 36.9231
3 55.3846
4 73.8462
5 92.3077
6 110.7692
7 129.2308
8 147.6923
9 166.1538
10 184.6154
11 203.0769
12 221.5385
13 240.0000
14 258.4615
15 276.9231
16 295.3846
17 313.8462
18 332.3077
19 350.7692
20 369.2308
21 387.6923
22 406.1538
23 424.6154
24 443.0769
25 461.5385
26 480.0000
27 498.4615
28 516.9231
29 535.3846
30 553.8462
31 572.3077
32 590.7692
33 609.2308
34 627.6923
35 646.1538
36 664.6154
37 683.0769
38 701.5385
39 720.0000
40 738.4615
41 756.9231
42 775.3846
43 793.8462
44 812.3077
45 830.7692
46 849.2308
47 867.6923
48 886.1538
49 904.6154
50 923.0769
51 941.5385
52 960.0000
53 978.4615
54 996.9231
55 1015.3846
56 1033.8462
57 1052.3077
58 1070.7692
59 1089.2308
60 1107.6923
61 1126.1538
62 1144.6154
63 1163.0769
64 1181.5385
65 1200.0000

Scales

photia7

photia12