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

Theory

86 = 2 × 43, and the patent val is a contorted 43edo 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 (%)	-30.7	+31.4	-43.3	+38.6	+48.9	-23.8	+0.7	+47.8	-32.2	+26.1	-2.6
Steps (reduced)		136 (50)	200 (28)	241 (69)	273 (15)	298 (40)	318 (60)	336 (78)	352 (8)	365 (21)	378 (34)	389 (45)

Subsets and supersets

86edo contains 2edo and 43edo as subsets. 258edo, which triples it, is a notable tuning.

Interval table

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	14		^D, vE♭♭
2	27.9		^^D, E♭♭
3	41.9	41/40	^3D, ^E♭♭
4	55.8	30/29, 31/30, 32/31, 33/32	vvD♯, ^^E♭♭
5	69.8		vD♯, v3E♭
6	83.7		D♯, vvE♭
7	97.7	37/35	^D♯, vE♭
8	111.6	16/15	^^D♯, E♭
9	125.6	14/13	^3D♯, ^E♭
10	139.5	13/12, 38/35	vvD𝄪, ^^E♭
11	153.5	35/32	vD𝄪, v3E
12	167.4	11/10	D𝄪, vvE
13	181.4		^D𝄪, vE
14	195.3	37/33	E
15	209.3	26/23, 35/31	^E, vF♭
16	223.3	25/22, 33/29	^^E, F♭
17	237.2		^3E, ^F♭
18	251.2	37/32	vvE♯, ^^F♭
19	265.1	7/6	vE♯, v3F
20	279.1	20/17	E♯, vvF
21	293		^E♯, vF
22	307	31/26, 37/31	F
23	320.9		^F, vG♭♭
24	334.9	40/33	^^F, G♭♭
25	348.8		^3F, ^G♭♭
26	362.8	37/30	vvF♯, ^^G♭♭
27	376.7	41/33	vF♯, v3G♭
28	390.7		F♯, vvG♭
29	404.7	24/19	^F♯, vG♭
30	418.6		^^F♯, G♭
31	432.6	9/7	^3F♯, ^G♭
32	446.5	22/17	vvF𝄪, ^^G♭
33	460.5	30/23	vF𝄪, v3G
34	474.4		F𝄪, vvG
35	488.4		^F𝄪, vG
36	502.3		G
37	516.3	31/23, 35/26	^G, vA♭♭
38	530.2	19/14, 34/25	^^G, A♭♭
39	544.2	26/19	^3G, ^A♭♭
40	558.1	40/29	vvG♯, ^^A♭♭
41	572.1	32/23, 39/28	vG♯, v3A♭
42	586		G♯, vvA♭
43	600	41/29	^G♯, vA♭
44	614		^^G♯, A♭
45	627.9	23/16, 33/23	^3G♯, ^A♭
46	641.9	29/20	vvG𝄪, ^^A♭
47	655.8	19/13, 35/24	vG𝄪, v3A
48	669.8	25/17, 28/19	G𝄪, vvA
49	683.7		^G𝄪, vA
50	697.7		A
51	711.6		^A, vB♭♭
52	725.6	35/23	^^A, B♭♭
53	739.5	23/15	^3A, ^B♭♭
54	753.5	17/11	vvA♯, ^^B♭♭
55	767.4	14/9	vA♯, v3B♭
56	781.4		A♯, vvB♭
57	795.3	19/12	^A♯, vB♭
58	809.3		^^A♯, B♭
59	823.3	37/23	^3A♯, ^B♭
60	837.2		vvA𝄪, ^^B♭
61	851.2		vA𝄪, v3B
62	865.1	33/20	A𝄪, vvB
63	879.1		^A𝄪, vB
64	893		B
65	907		^B, vC♭
66	920.9	17/10	^^B, C♭
67	934.9	12/7	^3B, ^C♭
68	948.8		vvB♯, ^^C♭
69	962.8		vB♯, v3C
70	976.7		B♯, vvC
71	990.7	23/13	^B♯, vC
72	1004.7		C
73	1018.6		^C, vD♭♭
74	1032.6	20/11	^^C, D♭♭
75	1046.5		^3C, ^D♭♭
76	1060.5	24/13, 35/19	vvC♯, ^^D♭♭
77	1074.4	13/7	vC♯, v3D♭
78	1088.4	15/8	C♯, vvD♭
79	1102.3		^C♯, vD♭
80	1116.3		^^C♯, D♭
81	1130.2		^3C♯, ^D♭
82	1144.2	29/15, 31/16	vvC𝄪, ^^D♭
83	1158.1		vC𝄪, v3D
84	1172.1		C𝄪, vvD
85	1186		^C𝄪, vD
86	1200	2/1	D