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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