85edo

← 84edo

85edo

86edo →

Prime factorization	5 × 17
Step size	14.1176¢
Fifth	50\85 (705.882¢) (→10\17)
Semitones (A1:m2)	10:5 (141.2¢ : 70.59¢)
Consistency limit	3
Distinct consistency limit	3

85 equal divisions of the octave (abbreviated 85edo or 85ed2), also called 85-tone equal temperament (85tet) or 85 equal temperament (85et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 85 equal parts of about 14.1 ¢ each. Each step represents a frequency ratio of 2^1/85, or the 85th root of 2.

Theory

Approximation of odd harmonics in 85edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+3.93	-5.14	+5.29	-6.26	-0.73	+6.53	-1.21	-6.13	-1.04	-4.90	+7.02
	relative (%)	+28	-36	+37	-44	-5	+46	-9	-43	-7	-35	+50
Steps (reduced)		135 (50)	197 (27)	239 (69)	269 (14)	294 (39)	315 (60)	332 (77)	347 (7)	361 (21)	373 (33)	385 (45)

85 = 5 * 17, and it shares the same 3.9 cent sharp fifth as 17, 34, and 68. The patent val tempers out 3125/3072 in the 5-limit and 875/864, 245/243 and 225/224 in the 7-limit, so that it supports magic. It tempers out 100/99 and 245/243 in the 11-limit, supporting 11-limit magic, and 1188/1183, 847/845 and 1575/1573 in the 13-limit. It provides the optimal patent val for the 13-limit temperament tempering out 100/99, 540/539, 847/845 and 1575/1573.

Interval table

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	14.1176	↑D, ↓4E♭
2	28.2353	↑↑D, ↓3E♭	65/64, 66/65
3	42.3529	↑3D, ↓↓E♭	45/44
4	56.4706	↑4D, ↓E♭	33/32
5	70.5882	↑5D, E♭	27/26
6	84.7059	↑6D, ↓9E
7	98.8235	↑7D, ↓8E	35/33, 52/49
8	112.941	↑8D, ↓7E	16/15, 77/72
9	127.059	↑9D, ↓6E	14/13
10	141.176	D♯, ↓5E	13/12
11	155.294	↑D♯, ↓4E	12/11, 35/32
12	169.412	↑↑D♯, ↓3E	11/10, 54/49
13	183.529	↑3D♯, ↓↓E
14	197.647	↑4D♯, ↓E
15	211.765	E
16	225.882	↑E, ↓4F	8/7
17	240	↑↑E, ↓3F
18	254.118	↑3E, ↓↓F	52/45, 65/56
19	268.235	↑4E, ↓F	7/6
20	282.353	F	33/28
21	296.471	↑F, ↓4G♭	77/65
22	310.588	↑↑F, ↓3G♭
23	324.706	↑3F, ↓↓G♭	77/64
24	338.824	↑4F, ↓G♭
25	352.941	↑5F, G♭
26	367.059	↑6F, ↓9G	26/21
27	381.176	↑7F, ↓8G	5/4, 56/45
28	395.294	↑8F, ↓7G	44/35, 49/39
29	409.412	↑9F, ↓6G	33/26
30	423.529	F♯, ↓5G
31	437.647	↑F♯, ↓4G	9/7
32	451.765	↑↑F♯, ↓3G
33	465.882	↑3F♯, ↓↓G
34	480	↑4F♯, ↓G
35	494.118	G	4/3
36	508.235	↑G, ↓4A♭
37	522.353	↑↑G, ↓3A♭	65/48
38	536.471	↑3G, ↓↓A♭	15/11, 49/36
39	550.588	↑4G, ↓A♭	11/8, 48/35
40	564.706	↑5G, A♭	18/13
41	578.824	↑6G, ↓9A	39/28
42	592.941	↑7G, ↓8A	45/32
43	607.059	↑8G, ↓7A	64/45
44	621.176	↑9G, ↓6A	56/39
45	635.294	G♯, ↓5A	13/9
46	649.412	↑G♯, ↓4A	16/11, 35/24
47	663.529	↑↑G♯, ↓3A	22/15, 72/49
48	677.647	↑3G♯, ↓↓A	65/44, 77/52
49	691.765	↑4G♯, ↓A
50	705.882	A	3/2
51	720	↑A, ↓4B♭
52	734.118	↑↑A, ↓3B♭
53	748.235	↑3A, ↓↓B♭
54	762.353	↑4A, ↓B♭	14/9
55	776.471	↑5A, B♭
56	790.588	↑6A, ↓9B	52/33
57	804.706	↑7A, ↓8B	35/22, 78/49
58	818.824	↑8A, ↓7B	8/5, 45/28, 77/48
59	832.941	↑9A, ↓6B	21/13
60	847.059	A♯, ↓5B
61	861.176	↑A♯, ↓4B
62	875.294	↑↑A♯, ↓3B	81/49
63	889.412	↑3A♯, ↓↓B
64	903.529	↑4A♯, ↓B
65	917.647	B	56/33, 75/44
66	931.765	↑B, ↓4C	12/7, 77/45
67	945.882	↑↑B, ↓3C	45/26
68	960	↑3B, ↓↓C
69	974.118	↑4B, ↓C	7/4
70	988.235	C
71	1002.35	↑C, ↓4D♭
72	1016.47	↑↑C, ↓3D♭
73	1030.59	↑3C, ↓↓D♭	20/11, 49/27
74	1044.71	↑4C, ↓D♭	11/6, 64/35
75	1058.82	↑5C, D♭	24/13
76	1072.94	↑6C, ↓9D	13/7
77	1087.06	↑7C, ↓8D	15/8
78	1101.18	↑8C, ↓7D	49/26, 66/35
79	1115.29	↑9C, ↓6D
80	1129.41	C♯, ↓5D	52/27
81	1143.53	↑C♯, ↓4D	64/33
82	1157.65	↑↑C♯, ↓3D
83	1171.76	↑3C♯, ↓↓D	65/33
84	1185.88	↑4C♯, ↓D
85	1200	D	2/1