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, vE♭♭
2	27.907	^^D, E♭♭	65/64, 66/65
3	41.8605	^3D, v5E♭	77/75
4	55.814	^4D, v4E♭	33/32
5	69.7674	^5D, v3E♭	80/77
6	83.7209	D♯, vvE♭
7	97.6744	^D♯, vE♭	35/33
8	111.628	^^D♯, E♭	16/15
9	125.581	^3D♯, v5E	14/13
10	139.535	^4D♯, v4E	13/12
11	153.488	^5D♯, v3E	35/32
12	167.442	D𝄪, vvE	11/10, 54/49
13	181.395	^D𝄪, vE
14	195.349	E
15	209.302	^E, vF♭
16	223.256	^^E, F♭	25/22
17	237.209	^3E, v5F
18	251.163	^4E, v4F	15/13, 52/45
19	265.116	^5E, v3F	7/6
20	279.07	E♯, vvF	75/64
21	293.023	^E♯, vF	77/65
22	306.977	F
23	320.93	^F, vG♭♭	77/64
24	334.884	^^F, G♭♭	40/33
25	348.837	^3F, v5G♭
26	362.791	^4F, v4G♭	16/13
27	376.744	^5F, v3G♭	56/45
28	390.698	F♯, vvG♭	5/4, 49/39
29	404.651	^F♯, vG♭
30	418.605	^^F♯, G♭
31	432.558	^3F♯, v5G	9/7, 77/60
32	446.512	^4F♯, v4G
33	460.465	^5F♯, v3G
34	474.419	F𝄪, vvG
35	488.372	^F𝄪, vG
36	502.326	G	4/3
37	516.279	^G, vA♭♭	35/26
38	530.233	^^G, A♭♭	49/36, 65/48
39	544.186	^3G, v5A♭	48/35
40	558.14	^4G, v4A♭	18/13
41	572.093	^5G, v3A♭	39/28
42	586.047	G♯, vvA♭	45/32
43	600	^G♯, vA♭
44	613.953	^^G♯, A♭	64/45
45	627.907	^3G♯, v5A	56/39
46	641.86	^4G♯, v4A	13/9
47	655.814	^5G♯, v3A	35/24
48	669.767	G𝄪, vvA	72/49
49	683.721	^G𝄪, vA	52/35, 77/52
50	697.674	A	3/2
51	711.628	^A, vB♭♭
52	725.581	^^A, B♭♭
53	739.535	^3A, v5B♭
54	753.488	^4A, v4B♭
55	767.442	^5A, v3B♭	14/9
56	781.395	A♯, vvB♭
57	795.349	^A♯, vB♭
58	809.302	^^A♯, B♭	8/5, 78/49
59	823.256	^3A♯, v5B	45/28, 77/48
60	837.209	^4A♯, v4B	13/8
61	851.163	^5A♯, v3B
62	865.116	A𝄪, vvB	33/20, 81/49
63	879.07	^A𝄪, vB
64	893.023	B
65	906.977	^B, vC♭
66	920.93	^^B, C♭	75/44
67	934.884	^3B, v5C	12/7, 77/45
68	948.837	^4B, v4C	26/15, 45/26
69	962.791	^5B, v3C
70	976.744	B♯, vvC	44/25
71	990.698	^B♯, vC
72	1004.65	C
73	1018.6	^C, vD♭♭
74	1032.56	^^C, D♭♭	20/11, 49/27
75	1046.51	^3C, v5D♭	64/35
76	1060.47	^4C, v4D♭	24/13
77	1074.42	^5C, v3D♭	13/7
78	1088.37	C♯, vvD♭	15/8
79	1102.33	^C♯, vD♭	66/35
80	1116.28	^^C♯, D♭
81	1130.23	^3C♯, v5D	77/40
82	1144.19	^4C♯, v4D	64/33
83	1158.14	^5C♯, v3D
84	1172.09	C𝄪, vvD	65/33
85	1186.05	^C𝄪, vD
86	1200	D	2/1