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

85 = 5 × 17, and 85edo 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 225/224, 245/243, and 875/864 in the 7-limit, so that it supports magic. It tempers out 100/99 and 385/384 in the 11-limit, supporting 11-limit magic, and 847/845, 1188/1183, and 1575/1573 in the 13-limit. It provides the optimal patent val for the 13-limit 36ce & 49f temperament tempering out 100/99, 540/539, 847/845 and 1575/1573.

Odd harmonics

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 (%)	+27.8	-36.4	+37.5	-44.4	-5.2	+46.3	-8.6	-43.4	-7.4	-34.7	+49.7
Steps (reduced)		135 (50)	197 (27)	239 (69)	269 (14)	294 (39)	315 (60)	332 (77)	347 (7)	361 (21)	373 (33)	385 (45)

Subsets and supersets

85edo contains 5edo and 17edo as subsets. 255edo, which triples it, is a notable tuning.

Interval table

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	14.1		^D, v4E♭
2	28.2		^^D, v3E♭
3	42.4	41/40	^3D, vvE♭
4	56.5	30/29, 31/30, 32/31	^4D, vE♭
5	70.6		^5D, E♭
6	84.7		v4D♯, ^E♭
7	98.8	37/35	v3D♯, ^^E♭
8	112.9	16/15, 31/29	vvD♯, ^3E♭
9	127.1	14/13	vD♯, ^4E♭
10	141.2	13/12, 38/35	D♯, v5E
11	155.3	23/21, 35/32	^D♯, v4E
12	169.4	32/29	^^D♯, v3E
13	183.5		^3D♯, vvE
14	197.6	37/33	^4D♯, vE
15	211.8	26/23, 35/31	E
16	225.9	33/29	^E, v4F
17	240		^^E, v3F
18	254.1	22/19, 37/32	^3E, vvF
19	268.2	7/6	^4E, vF
20	282.4	20/17, 33/28	F
21	296.5	19/16	^F, v4G♭
22	310.6		^^F, v3G♭
23	324.7	29/24, 35/29, 41/34	^3F, vvG♭
24	338.8	28/23	^4F, vG♭
25	352.9	38/31	^5F, G♭
26	367.1	26/21	v4F♯, ^G♭
27	381.2		v3F♯, ^^G♭
28	395.3		vvF♯, ^3G♭
29	409.4	19/15	vF♯, ^4G♭
30	423.5	23/18, 37/29	F♯, v5G
31	437.6	9/7	^F♯, v4G
32	451.8		^^F♯, v3G
33	465.9	38/29	^3F♯, vvG
34	480	29/22, 37/28	^4F♯, vG
35	494.1		G
36	508.2		^G, v4A♭
37	522.4		^^G, v3A♭
38	536.5	15/11	^3G, vvA♭
39	550.6	11/8	^4G, vA♭
40	564.7	18/13	^5G, A♭
41	578.8		v4G♯, ^A♭
42	592.9	31/22	v3G♯, ^^A♭
43	607.1		vvG♯, ^3A♭
44	621.2		vG♯, ^4A♭
45	635.3	13/9	G♯, v5A
46	649.4	16/11	^G♯, v4A
47	663.5	22/15	^^G♯, v3A
48	677.6		^3G♯, vvA
49	691.8		^4G♯, vA
50	705.9		A
51	720		^A, v4B♭
52	734.1	29/19	^^A, v3B♭
53	748.2	37/24	^3A, vvB♭
54	762.4	14/9	^4A, vB♭
55	776.5	36/23	^5A, B♭
56	790.6	30/19	v4A♯, ^B♭
57	804.7	35/22	v3A♯, ^^B♭
58	818.8		vvA♯, ^3B♭
59	832.9	21/13	vA♯, ^4B♭
60	847.1	31/19	A♯, v5B
61	861.2	23/14	^A♯, v4B
62	875.3		^^A♯, v3B
63	889.4		^3A♯, vvB
64	903.5	32/19	^4A♯, vB
65	917.6	17/10	B
66	931.8	12/7	^B, v4C
67	945.9	19/11	^^B, v3C
68	960		^3B, vvC
69	974.1		^4B, vC
70	988.2	23/13	C
71	1002.4		^C, v4D♭
72	1016.5		^^C, v3D♭
73	1030.6	29/16	^3C, vvD♭
74	1044.7		^4C, vD♭
75	1058.8	24/13, 35/19	^5C, D♭
76	1072.9	13/7	v4C♯, ^D♭
77	1087.1	15/8	v3C♯, ^^D♭
78	1101.2		vvC♯, ^3D♭
79	1115.3		vC♯, ^4D♭
80	1129.4		C♯, v5D
81	1143.5	29/15, 31/16	^C♯, v4D
82	1157.6		^^C♯, v3D
83	1171.8		^3C♯, vvD
84	1185.9		^4C♯, vD
85	1200	2/1	D

Scales

• Amulet⁠ ⁠^{[idiosyncratic term ]}, (approximated from 25edo, subset of magic): 7 3 7 7 3 7 10 7 7 3 7 10 7