# 115edo

 ← 114edo 115edo 116edo →
Prime factorization 5 × 23
Step size 10.4348¢
Fifth 67\115 (699.13¢)
Semitones (A1:m2) 9:10 (93.91¢ : 104.3¢)
Consistency limit 7
Distinct consistency limit 7

115 equal divisions of the octave (abbreviated 115edo or 115ed2), also called 115-tone equal temperament (115tet) or 115 equal temperament (115et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 115 equal parts of about 10.4 ¢ each. Each step represents a frequency ratio of 21/115, or the 115th root of 2.

## Theory

The equal temperament tempers out 1594323/1562500 (unicorn comma) and 2109375/2097152 (semicomma) in the 5-limit, as well as 43046721/41943040 (python comma) and [19 10 -15 (trisedodge comma); 225/224, 1728/1715, and 9920232/9765625 in the 7-limit. Using the patent val, it supports the newspeak temperament, tempering out 441/440, 1375/1372, and 1944/1925 in the 11-limit. The wizardharry comma, 4000/3993 is also tempered out.

### Odd harmonics

Approximation of odd harmonics in 115edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -2.82 -0.23 +1.61 +4.79 +1.73 +4.69 -3.05 -0.61 +5.10 -1.22 -2.19
Relative (%) -27.1 -2.2 +15.4 +45.9 +16.5 +44.9 -29.2 -5.8 +48.8 -11.7 -21.0
Steps
(reduced)
182
(67)
267
(37)
323
(93)
365
(20)
398
(53)
426
(81)
449
(104)
470
(10)
489
(29)
505
(45)
520
(60)

## Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 10.435 ^D, E♭♭
2 20.87 78/77 ^^D, v8E♭
3 31.304 50/49, 56/55, 64/63 ^3D, v7E♭
4 41.739 40/39 ^4D, v6E♭
5 52.174 33/32 ^5D, v5E♭
6 62.609 80/77 ^6D, v4E♭
7 73.043 25/24 ^7D, v3E♭
8 83.478 21/20, 22/21 ^8D, vvE♭
9 93.913 55/52 D♯, vE♭
10 104.348 35/33, 52/49 ^D♯, E♭
11 114.783 16/15 ^^D♯, v8E
12 125.217 14/13 ^3D♯, v7E
13 135.652 ^4D♯, v6E
14 146.087 ^5D♯, v5E
15 156.522 35/32 ^6D♯, v4E
16 166.957 11/10 ^7D♯, v3E
17 177.391 ^8D♯, vvE
18 187.826 39/35, 49/44 D𝄪, vE
19 198.261 28/25, 55/49 E
20 208.696 44/39 ^E, F♭
21 219.13 25/22 ^^E, v8F
22 229.565 8/7 ^3E, v7F
23 240 ^4E, v6F
24 250.435 ^5E, v5F
25 260.87 64/55, 65/56 ^6E, v4F
26 271.304 75/64 ^7E, v3F
27 281.739 33/28 ^8E, vvF
28 292.174 13/11, 77/65 E♯, vF
29 302.609 25/21 F
30 313.043 6/5 ^F, G♭♭
31 323.478 77/64 ^^F, v8G♭
32 333.913 40/33 ^3F, v7G♭
33 344.348 39/32 ^4F, v6G♭
34 354.783 49/40 ^5F, v5G♭
35 365.217 ^6F, v4G♭
36 375.652 ^7F, v3G♭
37 386.087 5/4 ^8F, vvG♭
38 396.522 44/35, 49/39, 63/50 F♯, vG♭
39 406.957 ^F♯, G♭
40 417.391 14/11, 80/63 ^^F♯, v8G
41 427.826 32/25, 50/39 ^3F♯, v7G
42 438.261 ^4F♯, v6G
43 448.696 ^5F♯, v5G
44 459.13 64/49 ^6F♯, v4G
45 469.565 21/16, 55/42 ^7F♯, v3G
46 480 33/25 ^8F♯, vvG
47 490.435 65/49 F𝄪, vG
48 500.87 4/3 G
49 511.304 35/26 ^G, A♭♭
50 521.739 ^^G, v8A♭
51 532.174 ^3G, v7A♭
52 542.609 ^4G, v6A♭
53 553.043 11/8 ^5G, v5A♭
54 563.478 ^6G, v4A♭
55 573.913 39/28 ^7G, v3A♭
56 584.348 7/5 ^8G, vvA♭
57 594.783 55/39 G♯, vA♭
58 605.217 78/55 ^G♯, A♭
59 615.652 10/7 ^^G♯, v8A
60 626.087 56/39 ^3G♯, v7A
61 636.522 ^4G♯, v6A
62 646.957 16/11 ^5G♯, v5A
63 657.391 ^6G♯, v4A
64 667.826 ^7G♯, v3A
65 678.261 65/44, 77/52 ^8G♯, vvA
66 688.696 52/35 G𝄪, vA
67 699.13 3/2 A
68 709.565 ^A, B♭♭
69 720 50/33 ^^A, v8B♭
70 730.435 32/21 ^3A, v7B♭
71 740.87 49/32 ^4A, v6B♭
72 751.304 77/50 ^5A, v5B♭
73 761.739 ^6A, v4B♭
74 772.174 25/16, 39/25 ^7A, v3B♭
75 782.609 11/7, 63/40 ^8A, vvB♭
76 793.043 A♯, vB♭
77 803.478 35/22, 78/49 ^A♯, B♭
78 813.913 8/5 ^^A♯, v8B
79 824.348 ^3A♯, v7B
80 834.783 ^4A♯, v6B
81 845.217 80/49 ^5A♯, v5B
82 855.652 64/39 ^6A♯, v4B
83 866.087 33/20 ^7A♯, v3B
84 876.522 ^8A♯, vvB
85 886.957 5/3 A𝄪, vB
86 897.391 42/25 B
87 907.826 22/13 ^B, C♭
88 918.261 56/33 ^^B, v8C
89 928.696 ^3B, v7C
90 939.13 55/32 ^4B, v6C
91 949.565 ^5B, v5C
92 960 ^6B, v4C
93 970.435 7/4 ^7B, v3C
94 980.87 44/25 ^8B, vvC
95 991.304 39/22 B♯, vC
96 1001.739 25/14 C
97 1012.174 70/39 ^C, D♭♭
98 1022.609 ^^C, v8D♭
99 1033.043 20/11 ^3C, v7D♭
100 1043.478 64/35 ^4C, v6D♭
101 1053.913 ^5C, v5D♭
102 1064.348 ^6C, v4D♭
103 1074.783 13/7 ^7C, v3D♭
104 1085.217 15/8 ^8C, vvD♭
105 1095.652 49/26, 66/35 C♯, vD♭
106 1106.087 ^C♯, D♭
107 1116.522 21/11, 40/21 ^^C♯, v8D
108 1126.957 48/25 ^3C♯, v7D
109 1137.391 77/40 ^4C♯, v6D
110 1147.826 64/33 ^5C♯, v5D
111 1158.261 39/20 ^6C♯, v4D
112 1168.696 49/25, 55/28, 63/32 ^7C♯, v3D
113 1179.13 77/39 ^8C♯, vvD
114 1189.565 C𝄪, vD
115 1200 2/1 D