# 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 21/85, 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 14.118 ^D, v4E♭
2 28.235 65/64, 66/65 ^^D, v3E♭
3 42.353 45/44 ^3D, vvE♭
4 56.471 33/32 ^4D, vE♭
5 70.588 27/26 ^5D, E♭
6 84.706 ^6D, v9E
7 98.824 35/33, 52/49 ^7D, v8E
8 112.941 16/15, 77/72 ^8D, v7E
9 127.059 14/13 ^9D, v6E
10 141.176 13/12 D♯, v5E
11 155.294 12/11, 35/32 ^D♯, v4E
12 169.412 11/10, 54/49 ^^D♯, v3E
13 183.529 ^3D♯, vvE
14 197.647 ^4D♯, vE
15 211.765 E
16 225.882 8/7 ^E, v4F
17 240 ^^E, v3F
18 254.118 52/45, 65/56 ^3E, vvF
19 268.235 7/6 ^4E, vF
20 282.353 33/28 F
21 296.471 77/65 ^F, v4G♭
22 310.588 ^^F, v3G♭
23 324.706 77/64 ^3F, vvG♭
24 338.824 ^4F, vG♭
25 352.941 ^5F, G♭
26 367.059 26/21 ^6F, v9G
27 381.176 5/4, 56/45 ^7F, v8G
28 395.294 44/35, 49/39 ^8F, v7G
29 409.412 33/26 ^9F, v6G
30 423.529 F♯, v5G
31 437.647 9/7 ^F♯, v4G
32 451.765 ^^F♯, v3G
33 465.882 ^3F♯, vvG
34 480 ^4F♯, vG
35 494.118 4/3 G
36 508.235 ^G, v4A♭
37 522.353 65/48 ^^G, v3A♭
38 536.471 15/11, 49/36 ^3G, vvA♭
39 550.588 11/8, 48/35 ^4G, vA♭
40 564.706 18/13 ^5G, A♭
41 578.824 39/28 ^6G, v9A
42 592.941 45/32 ^7G, v8A
43 607.059 64/45 ^8G, v7A
44 621.176 56/39 ^9G, v6A
45 635.294 13/9 G♯, v5A
46 649.412 16/11, 35/24 ^G♯, v4A
47 663.529 22/15, 72/49 ^^G♯, v3A
48 677.647 65/44, 77/52 ^3G♯, vvA
49 691.765 ^4G♯, vA
50 705.882 3/2 A
51 720 ^A, v4B♭
52 734.118 ^^A, v3B♭
53 748.235 ^3A, vvB♭
54 762.353 14/9 ^4A, vB♭
55 776.471 ^5A, B♭
56 790.588 52/33 ^6A, v9B
57 804.706 35/22, 78/49 ^7A, v8B
58 818.824 8/5, 45/28, 77/48 ^8A, v7B
59 832.941 21/13 ^9A, v6B
60 847.059 A♯, v5B
61 861.176 ^A♯, v4B
62 875.294 81/49 ^^A♯, v3B
63 889.412 ^3A♯, vvB
64 903.529 ^4A♯, vB
65 917.647 56/33, 75/44 B
66 931.765 12/7, 77/45 ^B, v4C
67 945.882 45/26 ^^B, v3C
68 960 ^3B, vvC
69 974.118 7/4 ^4B, vC
70 988.235 C
71 1002.353 ^C, v4D♭
72 1016.471 ^^C, v3D♭
73 1030.588 20/11, 49/27 ^3C, vvD♭
74 1044.706 11/6, 64/35 ^4C, vD♭
75 1058.824 24/13 ^5C, D♭
76 1072.941 13/7 ^6C, v9D
77 1087.059 15/8 ^7C, v8D
78 1101.176 49/26, 66/35 ^8C, v7D
79 1115.294 ^9C, v6D
80 1129.412 52/27 C♯, v5D
81 1143.529 64/33 ^C♯, v4D
82 1157.647 ^^C♯, v3D
83 1171.765 65/33 ^3C♯, vvD
84 1185.882 ^4C♯, vD
85 1200 2/1 D