# 196edo

 ← 195edo 196edo 197edo →
Prime factorization 22 × 72
Step size 6.12245¢
Fifth 115\196 (704.082¢)
Semitones (A1:m2) 21:13 (128.6¢ : 79.59¢)
Dual sharp fifth 115\196 (704.082¢)
Dual flat fifth 114\196 (697.959¢) (→57\98)
Dual major 2nd 33\196 (202.041¢)
Consistency limit 5
Distinct consistency limit 5

196 equal divisions of the octave (abbreviated 196edo or 196ed2), also called 196-tone equal temperament (196tet) or 196 equal temperament (196et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 196 equal parts of about 6.12 ¢ each. Each step represents a frequency ratio of 21/196, or the 196th root of 2.

## Theory

The equal temperament tempers out 9765625/9565938 (fifive comma) and [32 -7 -9 (escapade comma) in the 5-limit. Using the patent val, it tempers out 245/243, 65625/65536, and 235298/234375 in the 7-limit; 385/384, 896/891, 3388/3375, and 117649/117128 in the 11-limit; 352/351, 364/363, 625/624, 1001/1000, and 9295/9261 in the 13-limit; 289/288, 442/441, 715/714, and 1156/1155 in the 17-limit.

196edo can also treated as a 2.9.5.7.11.13.17 subgroup temperament (with the patent 9), providing a distinct flat tendency for harmonics 5, 7, 9, 11, 13, and 17. With the patent 9, it tempers out 321489/320000, 420175/419904, and 703125/702464 in the 2.9.5.7 subgroup; 441/440, 8019/8000, 41503/41472, and 9453125/9437184 in the 2.9.5.7.11 subgroup; 729/728, 1001/1000, 1575/1573, 6656/6655, and 10985/10976 in the 2.9.5.7.11.13 subgroup; 833/832, 936/935, 1089/1088, 1225/1224, 2025/2023, and 14365/14336 in the 2.9.5.7.11.13.17 subgroup.

Since it is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament.

### Odd harmonics

Approximation of odd harmonics in 196edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.13 -0.60 -1.48 -1.87 -0.30 -1.75 +1.53 -0.87 +2.49 +0.65 +2.34
Relative (%) +34.7 -9.8 -24.2 -30.5 -4.9 -28.6 +24.9 -14.3 +40.6 +10.6 +38.2
Steps
(reduced)
311
(115)
455
(63)
550
(158)
621
(33)
678
(90)
725
(137)
766
(178)
801
(17)
833
(49)
861
(77)
887
(103)

### Subsets and supersets

Since 196 factors into 22 × 72, 196edo has subset edos 2, 4, 7, 14, 28, 49, and 98.

## Intervals

"
Steps Cents Ups and downs notation
(dual flat fifth 114\196)
Ups and downs notation
(dual sharp fifth 115\196)
Approximate ratios
0 0 D D 1/1
1 6.12245 ^D, v3E♭♭ ^D, v12E♭
2 12.2449 ^^D, vvE♭♭ ^^D, v11E♭
3 18.3673 ^3D, vE♭♭ ^3D, v10E♭
4 24.4898 ^4D, E♭♭ ^4D, v9E♭ 65/64, 78/77
5 30.6122 ^5D, v13E♭ ^5D, v8E♭ 56/55
6 36.7347 ^6D, v12E♭ ^6D, v7E♭ 50/49
7 42.8571 ^7D, v11E♭ ^7D, v6E♭ 40/39
8 48.9796 ^8D, v10E♭ ^8D, v5E♭
9 55.102 ^9D, v9E♭ ^9D, v4E♭ 33/32
10 61.2245 ^10D, v8E♭ ^10D, v3E♭
11 67.3469 ^11D, v7E♭ ^11D, vvE♭ 26/25, 80/77
12 73.4694 ^12D, v6E♭ ^12D, vE♭
13 79.5918 ^13D, v5E♭ ^13D, E♭ 22/21
14 85.7143 D♯, v4E♭ ^14D, v20E 21/20
15 91.8367 ^D♯, v3E♭ ^15D, v19E
16 97.9592 ^^D♯, vvE♭ ^16D, v18E 55/52
17 104.082 ^3D♯, vE♭ ^17D, v17E 52/49
18 110.204 ^4D♯, E♭ ^18D, v16E 16/15
19 116.327 ^5D♯, v13E ^19D, v15E
20 122.449 ^6D♯, v12E ^20D, v14E
21 128.571 ^7D♯, v11E D♯, v13E 14/13
22 134.694 ^8D♯, v10E ^D♯, v12E
23 140.816 ^9D♯, v9E ^^D♯, v11E
24 146.939 ^10D♯, v8E ^3D♯, v10E
25 153.061 ^11D♯, v7E ^4D♯, v9E 12/11, 35/32
26 159.184 ^12D♯, v6E ^5D♯, v8E
27 165.306 ^13D♯, v5E ^6D♯, v7E 11/10
28 171.429 D𝄪, v4E ^7D♯, v6E
29 177.551 ^D𝄪, v3E ^8D♯, v5E
30 183.673 ^^D𝄪, vvE ^9D♯, v4E
31 189.796 ^3D𝄪, vE ^10D♯, v3E
32 195.918 E ^11D♯, vvE 28/25
33 202.041 ^E, v3F♭ ^12D♯, vE 55/49
34 208.163 ^^E, vvF♭ E 44/39
35 214.286 ^3E, vF♭ ^E, v12F
36 220.408 ^4E, F♭ ^^E, v11F 25/22
37 226.531 ^5E, v13F ^3E, v10F
38 232.653 ^6E, v12F ^4E, v9F 8/7
39 238.776 ^7E, v11F ^5E, v8F
40 244.898 ^8E, v10F ^6E, v7F
41 251.02 ^9E, v9F ^7E, v6F
42 257.143 ^10E, v8F ^8E, v5F 65/56
43 263.265 ^11E, v7F ^9E, v4F 64/55
44 269.388 ^12E, v6F ^10E, v3F
45 275.51 ^13E, v5F ^11E, vvF 75/64
46 281.633 E♯, v4F ^12E, vF
47 287.755 ^E♯, v3F F 13/11
48 293.878 ^^E♯, vvF ^F, v12G♭ 77/65
49 300 ^3E♯, vF ^^F, v11G♭ 25/21
50 306.122 F ^3F, v10G♭
51 312.245 ^F, v3G♭♭ ^4F, v9G♭
52 318.367 ^^F, vvG♭♭ ^5F, v8G♭ 77/64
53 324.49 ^3F, vG♭♭ ^6F, v7G♭
54 330.612 ^4F, G♭♭ ^7F, v6G♭ 40/33
55 336.735 ^5F, v13G♭ ^8F, v5G♭
56 342.857 ^6F, v12G♭ ^9F, v4G♭ 39/32
57 348.98 ^7F, v11G♭ ^10F, v3G♭ 49/40
58 355.102 ^8F, v10G♭ ^11F, vvG♭
59 361.224 ^9F, v9G♭ ^12F, vG♭ 16/13
60 367.347 ^10F, v8G♭ ^13F, G♭ 26/21
61 373.469 ^11F, v7G♭ ^14F, v20G
62 379.592 ^12F, v6G♭ ^15F, v19G
63 385.714 ^13F, v5G♭ ^16F, v18G 5/4
64 391.837 F♯, v4G♭ ^17F, v17G
65 397.959 ^F♯, v3G♭ ^18F, v16G 44/35
66 404.082 ^^F♯, vvG♭ ^19F, v15G
67 410.204 ^3F♯, vG♭ ^20F, v14G
68 416.327 ^4F♯, G♭ F♯, v13G 14/11
69 422.449 ^5F♯, v13G ^F♯, v12G
70 428.571 ^6F♯, v12G ^^F♯, v11G 32/25, 50/39
71 434.694 ^7F♯, v11G ^3F♯, v10G
72 440.816 ^8F♯, v10G ^4F♯, v9G
73 446.939 ^9F♯, v9G ^5F♯, v8G
74 453.061 ^10F♯, v8G ^6F♯, v7G 13/10
75 459.184 ^11F♯, v7G ^7F♯, v6G
76 465.306 ^12F♯, v6G ^8F♯, v5G 55/42
77 471.429 ^13F♯, v5G ^9F♯, v4G 21/16
78 477.551 F𝄪, v4G ^10F♯, v3G
79 483.673 ^F𝄪, v3G ^11F♯, vvG
80 489.796 ^^F𝄪, vvG ^12F♯, vG 65/49
81 495.918 ^3F𝄪, vG G 4/3
82 502.041 G ^G, v12A♭
83 508.163 ^G, v3A♭♭ ^^G, v11A♭ 75/56
84 514.286 ^^G, vvA♭♭ ^3G, v10A♭ 35/26
85 520.408 ^3G, vA♭♭ ^4G, v9A♭
86 526.531 ^4G, A♭♭ ^5G, v8A♭
87 532.653 ^5G, v13A♭ ^6G, v7A♭
88 538.776 ^6G, v12A♭ ^7G, v6A♭ 15/11
89 544.898 ^7G, v11A♭ ^8G, v5A♭
90 551.02 ^8G, v10A♭ ^9G, v4A♭ 11/8
91 557.143 ^9G, v9A♭ ^10G, v3A♭
92 563.265 ^10G, v8A♭ ^11G, vvA♭
93 569.388 ^11G, v7A♭ ^12G, vA♭
94 575.51 ^12G, v6A♭ ^13G, A♭ 39/28
95 581.633 ^13G, v5A♭ ^14G, v20A 7/5
96 587.755 G♯, v4A♭ ^15G, v19A
97 593.878 ^G♯, v3A♭ ^16G, v18A 55/39
98 600 ^^G♯, vvA♭ ^17G, v17A
99 606.122 ^3G♯, vA♭ ^18G, v16A 78/55
100 612.245 ^4G♯, A♭ ^19G, v15A
101 618.367 ^5G♯, v13A ^20G, v14A 10/7
102 624.49 ^6G♯, v12A G♯, v13A 56/39
103 630.612 ^7G♯, v11A ^G♯, v12A
104 636.735 ^8G♯, v10A ^^G♯, v11A
105 642.857 ^9G♯, v9A ^3G♯, v10A
106 648.98 ^10G♯, v8A ^4G♯, v9A 16/11
107 655.102 ^11G♯, v7A ^5G♯, v8A
108 661.224 ^12G♯, v6A ^6G♯, v7A 22/15
109 667.347 ^13G♯, v5A ^7G♯, v6A
110 673.469 G𝄪, v4A ^8G♯, v5A 65/44
111 679.592 ^G𝄪, v3A ^9G♯, v4A 77/52
112 685.714 ^^G𝄪, vvA ^10G♯, v3A 52/35
113 691.837 ^3G𝄪, vA ^11G♯, vvA
114 697.959 A ^12G♯, vA
115 704.082 ^A, v3B♭♭ A 3/2
116 710.204 ^^A, vvB♭♭ ^A, v12B♭
117 716.327 ^3A, vB♭♭ ^^A, v11B♭
118 722.449 ^4A, B♭♭ ^3A, v10B♭
119 728.571 ^5A, v13B♭ ^4A, v9B♭ 32/21
120 734.694 ^6A, v12B♭ ^5A, v8B♭
121 740.816 ^7A, v11B♭ ^6A, v7B♭
122 746.939 ^8A, v10B♭ ^7A, v6B♭ 20/13, 77/50
123 753.061 ^9A, v9B♭ ^8A, v5B♭
124 759.184 ^10A, v8B♭ ^9A, v4B♭
125 765.306 ^11A, v7B♭ ^10A, v3B♭
126 771.429 ^12A, v6B♭ ^11A, vvB♭ 25/16, 39/25
127 777.551 ^13A, v5B♭ ^12A, vB♭
128 783.673 A♯, v4B♭ ^13A, B♭ 11/7
129 789.796 ^A♯, v3B♭ ^14A, v20B
130 795.918 ^^A♯, vvB♭ ^15A, v19B
131 802.041 ^3A♯, vB♭ ^16A, v18B 35/22
132 808.163 ^4A♯, B♭ ^17A, v17B
133 814.286 ^5A♯, v13B ^18A, v16B 8/5
134 820.408 ^6A♯, v12B ^19A, v15B
135 826.531 ^7A♯, v11B ^20A, v14B
136 832.653 ^8A♯, v10B A♯, v13B 21/13
137 838.776 ^9A♯, v9B ^A♯, v12B 13/8
138 844.898 ^10A♯, v8B ^^A♯, v11B
139 851.02 ^11A♯, v7B ^3A♯, v10B 80/49
140 857.143 ^12A♯, v6B ^4A♯, v9B 64/39
141 863.265 ^13A♯, v5B ^5A♯, v8B
142 869.388 A𝄪, v4B ^6A♯, v7B 33/20
143 875.51 ^A𝄪, v3B ^7A♯, v6B
144 881.633 ^^A𝄪, vvB ^8A♯, v5B
145 887.755 ^3A𝄪, vB ^9A♯, v4B
146 893.878 B ^10A♯, v3B
147 900 ^B, v3C♭ ^11A♯, vvB 42/25
148 906.122 ^^B, vvC♭ ^12A♯, vB
149 912.245 ^3B, vC♭ B 22/13
150 918.367 ^4B, C♭ ^B, v12C
151 924.49 ^5B, v13C ^^B, v11C 75/44
152 930.612 ^6B, v12C ^3B, v10C
153 936.735 ^7B, v11C ^4B, v9C 55/32
154 942.857 ^8B, v10C ^5B, v8C
155 948.98 ^9B, v9C ^6B, v7C
156 955.102 ^10B, v8C ^7B, v6C
157 961.224 ^11B, v7C ^8B, v5C
158 967.347 ^12B, v6C ^9B, v4C 7/4
159 973.469 ^13B, v5C ^10B, v3C
160 979.592 B♯, v4C ^11B, vvC 44/25
161 985.714 ^B♯, v3C ^12B, vC
162 991.837 ^^B♯, vvC C 39/22
163 997.959 ^3B♯, vC ^C, v12D♭
164 1004.08 C ^^C, v11D♭ 25/14
165 1010.2 ^C, v3D♭♭ ^3C, v10D♭
166 1016.33 ^^C, vvD♭♭ ^4C, v9D♭
167 1022.45 ^3C, vD♭♭ ^5C, v8D♭
168 1028.57 ^4C, D♭♭ ^6C, v7D♭
169 1034.69 ^5C, v13D♭ ^7C, v6D♭ 20/11
170 1040.82 ^6C, v12D♭ ^8C, v5D♭
171 1046.94 ^7C, v11D♭ ^9C, v4D♭ 11/6, 64/35
172 1053.06 ^8C, v10D♭ ^10C, v3D♭
173 1059.18 ^9C, v9D♭ ^11C, vvD♭
174 1065.31 ^10C, v8D♭ ^12C, vD♭
175 1071.43 ^11C, v7D♭ ^13C, D♭ 13/7
176 1077.55 ^12C, v6D♭ ^14C, v20D
177 1083.67 ^13C, v5D♭ ^15C, v19D
178 1089.8 C♯, v4D♭ ^16C, v18D 15/8
179 1095.92 ^C♯, v3D♭ ^17C, v17D 49/26
180 1102.04 ^^C♯, vvD♭ ^18C, v16D
181 1108.16 ^3C♯, vD♭ ^19C, v15D
182 1114.29 ^4C♯, D♭ ^20C, v14D 40/21
183 1120.41 ^5C♯, v13D C♯, v13D 21/11
184 1126.53 ^6C♯, v12D ^C♯, v12D
185 1132.65 ^7C♯, v11D ^^C♯, v11D 25/13, 77/40
186 1138.78 ^8C♯, v10D ^3C♯, v10D
187 1144.9 ^9C♯, v9D ^4C♯, v9D 64/33
188 1151.02 ^10C♯, v8D ^5C♯, v8D
189 1157.14 ^11C♯, v7D ^6C♯, v7D 39/20
190 1163.27 ^12C♯, v6D ^7C♯, v6D 49/25
191 1169.39 ^13C♯, v5D ^8C♯, v5D 55/28
192 1175.51 C𝄪, v4D ^9C♯, v4D 77/39
193 1181.63 ^C𝄪, v3D ^10C♯, v3D
194 1187.76 ^^C𝄪, vvD ^11C♯, vvD
195 1193.88 ^3C𝄪, vD ^12C♯, vD
196 1200 D D 2/1