65edo

← 64edo

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.

Prime harmonics

Approximation of prime harmonics in 65edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Error	Absolute (¢)	+0.00	-0.42	+1.38	-8.83	+2.53	+8.70	+5.81	-2.13	-0.58	+4.27	-0.42	+7.12	-4.45	+5.41	-0.89
	Relative (%)	+0.0	-2.3	+7.5	-47.8	+13.7	+47.1	+31.5	-11.5	-3.2	+23.1	-2.3	+38.6	-24.1	+29.3	-4.8
Steps (reduced)		65 (0)	103 (38)	151 (21)	182 (52)	225 (30)	241 (46)	266 (6)	276 (16)	294 (34)	316 (56)	322 (62)	339 (14)	348 (23)	353 (28)	361 (36)

Intervals

#	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 8ve	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