65edo

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65edo

66edo →

Prime factorization	5 × 13
Step size	18.4615 ¢
Fifth	38\65 (701.538 ¢)
Semitones (A1:m2)	6:5 (110.8 ¢ : 92.31 ¢)
Consistency limit	5
Distinct consistency limit	5

The 65 equal divisions of the octave (65edo), or 65(-tone) equal temperament (65tet, 65et) when viewed from a regular temperament perspective, divides the octave into 65 equal parts of about 18.5 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, 31/16 and 47/32 well, so that it does a good job representing the 2.3.5.11.19.23.31.47 just intonation subgroup. To this one may want to add 17/16, 29/16 and 43/32, giving the 47-limit no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of schismic/Nestoria that focuses on the very primes that 53edo neglects (which instead elegantly connects primes 7, 13, 37 and 41 to nestoria). 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.

Intervals

Degree	Cents	Approximate Ratios *	Ups and Downs Notation
0	0.00	1/1	P1	D
1	18.46	81/80, 100/99, 121/120, 88/87, 93/92, 94/93, 95/94, 96/95, 115/114, 116/115, 125/124	^1	^D
2	36.92	128/125, 45/44, 46/45, 47/46, 48/47, 55/54	^^1	^^D
3	55.38	33/32, 34/33, 30/29, 32/31, 31/30	vvm2	vvEb
4	73.85	25/24, 24/23, 23/22, 47/45	vm2	vEb
5	92.31	135/128, 256/243, 18/17, 19/18, 20/19, 58/55	m2	Eb
6	110.77	16/15, 17/16, 33/31	A1/^m2	D#/^Eb
7	129.23	27/25, 14/13, 55/51	v~2	^^Eb
8	147.69	12/11, 25/23	~2	vvvE
9	166.15	11/10, 32/29	^~2	vvE
10	184.62	10/9	vM2	vE
11	203.08	9/8, 19/17, 64/57	M2	E
12	221.54	25/22, 17/15, 33/29, 58/51	^M2	^E
13	240.00	55/48, 38/33, 23/20, 31/27, 54/47	^^M2	^^E
14	258.46	64/55, 22/19, 29/25, 36/31	vvm3	vvF
15	276.92	75/64, 20/17, 27/23, 34/29	vm3	vF
16	295.38	32/27, 19/16	m3	F
17	313.85	6/5, 55/46	^m3	^F
18	332.31	40/33, 17/14, 23/19	v~3	^^F
19	350.77	11/9, 27/22, 38/31	~3	^^^F
20	369.23	26/21, 68/55, 47/38	^~3	vvF#
21	387.69	5/4, 64/51	vM3	vF#
22	406.15	81/64, 24/19, 19/15, 34/27, 29/23	M3	F#
23	424.62	32/25, 23/18	^M3	^F#
24	443.08	128/99, 22/17, 31/24, 40/31	^^M3	^^F#
25	461.54	72/55, 30/23, 47/36	vv4	vvG
26	480.00	33/25, 29/22, 62/47	v4	vG
27	498.46	4/3	P4	G
28	516.92	27/20, 23/17, 31/23	^4	^G
29	535.38	15/11, 34/25, 64/47	v~4	^^G
30	553.85	11/8, 40/29, 62/45	~4	^^^G
31	572.31	25/18, 32/23	^~4/vd5	vvG#/vAb
32	590.77	45/32, 24/17, 38/27, 31/22	vA4/d5	vG#/Ab
33	609.23	64/45, 17/12, 27/19, 44/31	A4/^d5	G#/^Ab
34	627.69	36/25, 23/16	^A4/v~5	^G#/^^Ab
35	646.15	16/11, 29/20, 45/31	~5	vvvA
36	664.62	22/15, 25/17, 47/32	^~5	vvA
37	683.08	40/27, 23/17, 31/23	v5	vA
38	701.54	3/2	P5	A
39	720.00	50/33, 44/29, 47/31	^5	^A
40	738.46	55/36, 23/15, 72/47	^^5	^^A
41	756.92	99/64, 17/11, 48/31, 31/20	vvm6	vvBb
42	775.38	25/16, 36/23	vm6	vBb
43	793.85	128/81, 19/12, 30/19, 27/17, 46/29	m6	Bb
44	812.31	8/5, 51/32	^m6	^Bb
45	830.77	21/13, 55/34, 76/47	v~6	^^Bb
46	849.23	18/11, 44/27, 31/19	~6	vvvB
47	867.69	33/20, 28/17, 38/23	^~6	vvB
48	886.15	5/3, 92/55	vM6	vB
49	904.62	27/16, 32/19	M6	B
50	923.08	128/75, 17/10, 46/27, 29/17	^M6	^B
51	941.54	55/32, 19/11, 50/29, 31/18	^^M6	^^B
52	960.00	96/55, 33/19, 40/23, 54/31, 47/27	vvm7	vvC
53	978.46	44/25, 30/17, 58/33, 51/29	vm7	vC
54	996.92	16/9, 34/19	m7	C
55	1015.38	9/5	^m7	^C
56	1033.85	20/11, 29/16	v~7	^^C
57	1052.31	11/6, 46/25	~7	^^^C
58	1070.77	50/27, 13/7, 102/55	^~7	vvC#
59	1089.23	15/8, 32/17, 62/33	vM7	vC#
60	1107.69	243/128, 256/135, 17/9, 36/19, 19/10, 55/29	M7	C#
61	1126.15	48/25, 23/12, 44/23, 90/47	^M7	^C#
62	1144.62	64/33, 33/17, 29/15, 31/16, 60/31	^^M7	^^C#
63	1163.08	125/64, 88/45, 45/23, 47/24, 92/47, 108/55	vv8	vvD
64	1181.54	160/81, 99/50, 240/121, 87/55, 184/93, 93/47, 188/95, 95/48, 228/115, 115/58, 248/125	v8	vD
65	1200.00	2/1	P8	D

* based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-103 65⟩	[⟨65 103]]	+0.131	0.131	0.71
2.3.5	32805/32768, 78732/78125	[⟨65 103 151]]	-0.110	0.358	1.94
2.3.5.11	243/242, 4000/3993, 5632/5625	[⟨65 103 151 225]]	-0.266	0.410	2.22

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	3\65	55.38	33/32	Escapade
1	9\65	166.15	11/10	Squirrel etc.
1	12\65	221.54	25/22	Hemisensi
1	19\65	350.77	11/9	Karadeniz
1	21\65	387.69	5/4	Würschmidt
1	24\65	443.08	162/125	Sensipent
1	27\65	498.46	4/3	Helmholtz / photia
1	28\65	516.92	27/20	Gravity
5	20\65 (6\65)	369.23 (110.77)	10125/8192 (16/15)	Qintosec
5	27\65 (1\65)	498.46 (18.46)	4/3 (81/80)	Pental
5	30\65 (4\65)	553.85 (73.85)	11/8 (25/24)	Trisedodge / countdown

Scales

• Photia7
• Photia12