86edo

← 85edo

86edo

87edo →

Prime factorization	2 × 43
Step size	13.9535¢
Fifth	50\86 (697.674¢) (→25\43)
Semitones (A1:m2)	6:8 (83.72¢ : 111.6¢)
Consistency limit	3
Distinct consistency limit	3

86 equal divisions of the octave (abbreviated 86edo or 86ed2), also called 86-tone equal temperament (86tet) or 86 equal temperament (86et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 86 equal parts of about 14 ¢ each. Each step represents a frequency ratio of 2^1/86, or the 86th root of 2.

86 = 2 × 43, and the patent val is a contorted 43 in the 5-limit. In the 7-limit the patent val tempers out 6144/6125, so that it supports mohajira temperament. In the 11-limit it tempers out 245/242, 540/539 and 4000/3993, and in the 13-limit 144/143, 196/195 and 676/675. It provides the optimal patent val for the 13-limit 9 & 86 temperament tempering out 144/143, 196/195, 245/242 and 676/675.

86edo is closely related to the delta scale, which is the equal division of the classic diatonic semitone into eight parts of 13.9664 cents each.

Odd harmonics

Approximation of odd harmonics in 86edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-4.28	+4.38	-6.04	+5.39	+6.82	-3.32	+0.10	+6.67	-4.49	+3.64	-0.37
	relative (%)	-31	+31	-43	+39	+49	-24	+1	+48	-32	+26	-3
Steps (reduced)		136 (50)	200 (28)	241 (69)	273 (15)	298 (40)	318 (60)	336 (78)	352 (8)	365 (21)	378 (34)	389 (45)

Interval table

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	13.9535	↑D, ↓E♭♭
2	27.907	↑↑D, E♭♭	65/64, 66/65
3	41.8605	↑3D, ↓5E♭	77/75
4	55.814	↑4D, ↓4E♭	33/32
5	69.7674	↑5D, ↓3E♭	80/77
6	83.7209	D♯, ↓↓E♭
7	97.6744	↑D♯, ↓E♭	35/33
8	111.628	↑↑D♯, E♭	16/15
9	125.581	↑3D♯, ↓5E	14/13
10	139.535	↑4D♯, ↓4E	13/12
11	153.488	↑5D♯, ↓3E	35/32
12	167.442	D𝄪, ↓↓E	11/10, 54/49
13	181.395	↑D𝄪, ↓E
14	195.349	E
15	209.302	↑E, ↓F♭
16	223.256	↑↑E, F♭	25/22
17	237.209	↑3E, ↓5F
18	251.163	↑4E, ↓4F	15/13, 52/45
19	265.116	↑5E, ↓3F	7/6
20	279.07	E♯, ↓↓F	75/64
21	293.023	↑E♯, ↓F	77/65
22	306.977	F
23	320.93	↑F, ↓G♭♭	77/64
24	334.884	↑↑F, G♭♭	40/33
25	348.837	↑3F, ↓5G♭
26	362.791	↑4F, ↓4G♭	16/13
27	376.744	↑5F, ↓3G♭	56/45
28	390.698	F♯, ↓↓G♭	5/4, 49/39
29	404.651	↑F♯, ↓G♭
30	418.605	↑↑F♯, G♭
31	432.558	↑3F♯, ↓5G	9/7, 77/60
32	446.512	↑4F♯, ↓4G
33	460.465	↑5F♯, ↓3G
34	474.419	F𝄪, ↓↓G
35	488.372	↑F𝄪, ↓G
36	502.326	G	4/3
37	516.279	↑G, ↓A♭♭	35/26
38	530.233	↑↑G, A♭♭	49/36, 65/48
39	544.186	↑3G, ↓5A♭	48/35
40	558.14	↑4G, ↓4A♭	18/13
41	572.093	↑5G, ↓3A♭	39/28
42	586.047	G♯, ↓↓A♭	45/32
43	600	↑G♯, ↓A♭
44	613.953	↑↑G♯, A♭	64/45
45	627.907	↑3G♯, ↓5A	56/39
46	641.86	↑4G♯, ↓4A	13/9
47	655.814	↑5G♯, ↓3A	35/24
48	669.767	G𝄪, ↓↓A	72/49
49	683.721	↑G𝄪, ↓A	52/35, 77/52
50	697.674	A	3/2
51	711.628	↑A, ↓B♭♭
52	725.581	↑↑A, B♭♭
53	739.535	↑3A, ↓5B♭
54	753.488	↑4A, ↓4B♭
55	767.442	↑5A, ↓3B♭	14/9
56	781.395	A♯, ↓↓B♭
57	795.349	↑A♯, ↓B♭
58	809.302	↑↑A♯, B♭	8/5, 78/49
59	823.256	↑3A♯, ↓5B	45/28, 77/48
60	837.209	↑4A♯, ↓4B	13/8
61	851.163	↑5A♯, ↓3B
62	865.116	A𝄪, ↓↓B	33/20, 81/49
63	879.07	↑A𝄪, ↓B
64	893.023	B
65	906.977	↑B, ↓C♭
66	920.93	↑↑B, C♭	75/44
67	934.884	↑3B, ↓5C	12/7, 77/45
68	948.837	↑4B, ↓4C	26/15, 45/26
69	962.791	↑5B, ↓3C
70	976.744	B♯, ↓↓C	44/25
71	990.698	↑B♯, ↓C
72	1004.65	C
73	1018.6	↑C, ↓D♭♭
74	1032.56	↑↑C, D♭♭	20/11, 49/27
75	1046.51	↑3C, ↓5D♭	64/35
76	1060.47	↑4C, ↓4D♭	24/13
77	1074.42	↑5C, ↓3D♭	13/7
78	1088.37	C♯, ↓↓D♭	15/8
79	1102.33	↑C♯, ↓D♭	66/35
80	1116.28	↑↑C♯, D♭
81	1130.23	↑3C♯, ↓5D	77/40
82	1144.19	↑4C♯, ↓4D	64/33
83	1158.14	↑5C♯, ↓3D
84	1172.09	C𝄪, ↓↓D	65/33
85	1186.05	↑C𝄪, ↓D
86	1200	D	2/1