156edo

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← 155edo 156edo 157edo →
Prime factorization 22 × 3 × 13
Step size 7.69231 ¢ 
Fifth 91\156 (700 ¢) (→ 7\12)
Semitones (A1:m2) 13:13 (100 ¢ : 100 ¢)
Consistency limit 7
Distinct consistency limit 7

156 equal divisions of the octave (abbreviated 156edo or 156ed2), also called 156-tone equal temperament (156tet) or 156 equal temperament (156et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 156 equal parts of about 7.69 ¢ each. Each step represents a frequency ratio of 21/156, or the 156th root of 2.

It tempers out 531441/524288 (Pythagorean comma) and [-27 -2 13⟩ (ditonmic comma) in the 5-limit, as well as [8 14 -13⟩ (parakleisma); 225/224, 250047/250000, and 589824/588245 in the 7-limit. Using the patent val, it tempers out 441/440, 1375/1372, 4375/4356, and 65536/65219 in the 11-limit; 351/350, 364/363, 625/624, 1625/1617, and 13122/13013 in the 13-limit. Using the 156e val, it tempers out 385/384, 540/539, 1331/1323, and 78408/78125 in the 11-limit; 351/350, 625/624, 847/845, and 1001/1000 in the 13-limit. It supports compton and gives a good tuning for the 5- and 7-limit version thereof.

Prime harmonics

Approximation of prime harmonics in 156edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.00 -1.96 -1.70 +0.40 +2.53 -2.07 +2.74 +2.49 +2.49 +1.19 +1.12 +2.50 +1.71
Relative (%) +0.0 -25.4 -22.1 +5.3 +32.9 -26.9 +35.6 +32.3 +32.4 +15.5 +14.5 +32.5 +22.2
Steps
(reduced) 		156
(0) 		247
(91) 		362
(50) 		438
(126) 		540
(72) 		577
(109) 		638
(14) 		663
(39) 		706
(82) 		758
(134) 		773
(149) 		813
(33) 		836
(56)

Subsets and supersets

Sinece 156 factors into 22 × 3 × 13, 156edo has subset edos 2, 3, 4, 6, 12, 13, 26, 39, 52, and 78. It is the smallest edo to contain both 12edo and 13edo as subsets.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 7.7 ^D, ^E♭♭
2 15.4 ^^D, ^^E♭♭
3 23.1 ^3D, ^3E♭♭
4 30.8 56/55, 57/56, 58/57 ^4D, ^4E♭♭
5 38.5 47/46 ^5D, ^5E♭♭
6 46.2 38/37 ^6D, ^6E♭♭
7 53.8 32/31, 33/32 v6D♯, v6E♭
8 61.5 29/28, 57/55 v5D♯, v5E♭
9 69.2 25/24, 26/25, 51/49 v4D♯, v4E♭
10 76.9 23/22 v3D♯, v3E♭
11 84.6 21/20 vvD♯, vvE♭
12 92.3 58/55 vD♯, vE♭
13 100 D♯, E♭
14 107.7 33/31 ^D♯, ^E♭
15 115.4 31/29, 47/44 ^^D♯, ^^E♭
16 123.1 44/41 ^3D♯, ^3E♭
17 130.8 41/38, 55/51 ^4D♯, ^4E♭
18 138.5 13/12 ^5D♯, ^5E♭
19 146.2 37/34 ^6D♯, ^6E♭
20 153.8 35/32 v6D𝄪, v6E
21 161.5 56/51 v5D𝄪, v5E
22 169.2 32/29, 43/39 v4D𝄪, v4E
23 176.9 31/28, 41/37 v3D𝄪, v3E
24 184.6 vvD𝄪, vvE
25 192.3 19/17 vD𝄪, vE
26 200 46/41, 55/49 E
27 207.7 ^E, ^F♭
28 215.4 ^^E, ^^F♭
29 223.1 33/29, 58/51 ^3E, ^3F♭
30 230.8 8/7 ^4E, ^4F♭
31 238.5 ^5E, ^5F♭
32 246.2 ^6E, ^6F♭
33 253.8 22/19 v6E♯, v6F
34 261.5 50/43, 57/49 v5E♯, v5F
35 269.2 v4E♯, v4F
36 276.9 v3E♯, v3F
37 284.6 33/28 vvE♯, vvF
38 292.3 58/49 vE♯, vF
39 300 44/37 F
40 307.7 37/31, 43/36, 49/41 ^F, ^G♭♭
41 315.4 6/5 ^^F, ^^G♭♭
42 323.1 41/34 ^3F, ^3G♭♭
43 330.8 23/19 ^4F, ^4G♭♭
44 338.5 ^5F, ^5G♭♭
45 346.2 ^6F, ^6G♭♭
46 353.8 38/31 v6F♯, v6G♭
47 361.5 v5F♯, v5G♭
48 369.2 26/21, 47/38 v4F♯, v4G♭
49 376.9 41/33, 46/37, 51/41 v3F♯, v3G♭
50 384.6 vvF♯, vvG♭
51 392.3 vF♯, vG♭
52 400 29/23 F♯, G♭
53 407.7 ^F♯, ^G♭
54 415.4 47/37 ^^F♯, ^^G♭
55 423.1 37/29 ^3F♯, ^3G♭
56 430.8 50/39 ^4F♯, ^4G♭
57 438.5 ^5F♯, ^5G♭
58 446.2 22/17 ^6F♯, ^6G♭
59 453.8 13/10 v6F𝄪, v6G
60 461.5 v5F𝄪, v5G
61 469.2 38/29 v4F𝄪, v4G
62 476.9 29/22 v3F𝄪, v3G
63 484.6 41/31 vvF𝄪, vvG
64 492.3 vF𝄪, vG
65 500 G
66 507.7 55/41 ^G, ^A♭♭
67 515.4 31/23, 35/26 ^^G, ^^A♭♭
68 523.1 23/17 ^3G, ^3A♭♭
69 530.8 ^4G, ^4A♭♭
70 538.5 56/41 ^5G, ^5A♭♭
71 546.2 48/35 ^6G, ^6A♭♭
72 553.8 v6G♯, v6A♭
73 561.5 47/34 v5G♯, v5A♭
74 569.2 25/18, 57/41 v4G♯, v4A♭
75 576.9 v3G♯, v3A♭
76 584.6 vvG♯, vvA♭
77 592.3 31/22 vG♯, vA♭
78 600 41/29, 58/41 G♯, A♭
79 607.7 44/31 ^G♯, ^A♭
80 615.4 ^^G♯, ^^A♭
81 623.1 43/30 ^3G♯, ^3A♭
82 630.8 36/25 ^4G♯, ^4A♭
83 638.5 ^5G♯, ^5A♭
84 646.2 ^6G♯, ^6A♭
85 653.8 35/24 v6G𝄪, v6A
86 661.5 41/28 v5G𝄪, v5A
87 669.2 v4G𝄪, v4A
88 676.9 34/23 v3G𝄪, v3A
89 684.6 46/31, 49/33, 52/35 vvG𝄪, vvA
90 692.3 vG𝄪, vA
91 700 A
92 707.7 ^A, ^B♭♭
93 715.4 ^^A, ^^B♭♭
94 723.1 44/29 ^3A, ^3B♭♭
95 730.8 29/19 ^4A, ^4B♭♭
96 738.5 49/32 ^5A, ^5B♭♭
97 746.2 20/13 ^6A, ^6B♭♭
98 753.8 17/11 v6A♯, v6B♭
99 761.5 v5A♯, v5B♭
100 769.2 39/25 v4A♯, v4B♭
101 776.9 58/37 v3A♯, v3B♭
102 784.6 vvA♯, vvB♭
103 792.3 49/31 vA♯, vB♭
104 800 46/29 A♯, B♭
105 807.7 51/32 ^A♯, ^B♭
106 815.4 ^^A♯, ^^B♭
107 823.1 37/23 ^3A♯, ^3B♭
108 830.8 21/13 ^4A♯, ^4B♭
109 838.5 ^5A♯, ^5B♭
110 846.2 31/19 ^6A♯, ^6B♭
111 853.8 v6A𝄪, v6B
112 861.5 51/31 v5A𝄪, v5B
113 869.2 38/23 v4A𝄪, v4B
114 876.9 v3A𝄪, v3B
115 884.6 5/3 vvA𝄪, vvB
116 892.3 vA𝄪, vB
117 900 37/22 B
118 907.7 49/29 ^B, ^C♭
119 915.4 56/33 ^^B, ^^C♭
120 923.1 ^3B, ^3C♭
121 930.8 ^4B, ^4C♭
122 938.5 43/25, 55/32 ^5B, ^5C♭
123 946.2 19/11 ^6B, ^6C♭
124 953.8 v6B♯, v6C
125 961.5 v5B♯, v5C
126 969.2 7/4 v4B♯, v4C
127 976.9 51/29, 58/33 v3B♯, v3C
128 984.6 vvB♯, vvC
129 992.3 55/31 vB♯, vC
130 1000 41/23, 57/32 C
131 1007.7 34/19 ^C, ^D♭♭
132 1015.4 ^^C, ^^D♭♭
133 1023.1 56/31 ^3C, ^3D♭♭
134 1030.8 29/16 ^4C, ^4D♭♭
135 1038.5 51/28 ^5C, ^5D♭♭
136 1046.2 ^6C, ^6D♭♭
137 1053.8 57/31 v6C♯, v6D♭
138 1061.5 24/13 v5C♯, v5D♭
139 1069.2 v4C♯, v4D♭
140 1076.9 41/22 v3C♯, v3D♭
141 1084.6 58/31 vvC♯, vvD♭
142 1092.3 vC♯, vD♭
143 1100 C♯, D♭
144 1107.7 55/29 ^C♯, ^D♭
145 1115.4 40/21 ^^C♯, ^^D♭
146 1123.1 44/23 ^3C♯, ^3D♭
147 1130.8 25/13, 48/25 ^4C♯, ^4D♭
148 1138.5 56/29 ^5C♯, ^5D♭
149 1146.2 31/16 ^6C♯, ^6D♭
150 1153.8 37/19 v6C𝄪, v6D
151 1161.5 v5C𝄪, v5D
152 1169.2 55/28, 57/29 v4C𝄪, v4D
153 1176.9 v3C𝄪, v3D
154 1184.6 vvC𝄪, vvD
155 1192.3 vC𝄪, vD
156 1200 2/1 D
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