65edo

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