196edo

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← 195edo196edo197edo →
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 Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 114\196) 		Ups and Downs Notation
(Dual Sharp Fifth 115\196)
0 0 1/1 D D
1 6.122 ^D, v3E♭♭ ^D, v12E♭
2 12.245 ^^D, vvE♭♭ ^^D, v11E♭
3 18.367 ^3D, vE♭♭ ^3D, v10E♭
4 24.49 65/64, 78/77 ^4D, E♭♭ ^4D, v9E♭
5 30.612 56/55 ^5D, v13E♭ ^5D, v8E♭
6 36.735 50/49 ^6D, v12E♭ ^6D, v7E♭
7 42.857 40/39 ^7D, v11E♭ ^7D, v6E♭
8 48.98 ^8D, v10E♭ ^8D, v5E♭
9 55.102 33/32 ^9D, v9E♭ ^9D, v4E♭
10 61.224 ^10D, v8E♭ ^10D, v3E♭
11 67.347 26/25, 80/77 ^11D, v7E♭ ^11D, vvE♭
12 73.469 ^12D, v6E♭ ^12D, vE♭
13 79.592 22/21 ^13D, v5E♭ ^13D, E♭
14 85.714 21/20 D♯, v4E♭ ^14D, v20E
15 91.837 ^D♯, v3E♭ ^15D, v19E
16 97.959 55/52 ^^D♯, vvE♭ ^16D, v18E
17 104.082 52/49 ^3D♯, vE♭ ^17D, v17E
18 110.204 16/15 ^4D♯, E♭ ^18D, v16E
19 116.327 ^5D♯, v13E ^19D, v15E
20 122.449 ^6D♯, v12E ^20D, v14E
21 128.571 14/13 ^7D♯, v11E D♯, v13E
22 134.694 ^8D♯, v10E ^D♯, v12E
23 140.816 ^9D♯, v9E ^^D♯, v11E
24 146.939 ^10D♯, v8E ^3D♯, v10E
25 153.061 12/11, 35/32 ^11D♯, v7E ^4D♯, v9E
26 159.184 ^12D♯, v6E ^5D♯, v8E
27 165.306 11/10 ^13D♯, v5E ^6D♯, v7E
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 28/25 E ^11D♯, vvE
33 202.041 55/49 ^E, v3F♭ ^12D♯, vE
34 208.163 44/39 ^^E, vvF♭ E
35 214.286 ^3E, vF♭ ^E, v12F
36 220.408 25/22 ^4E, F♭ ^^E, v11F
37 226.531 ^5E, v13F ^3E, v10F
38 232.653 8/7 ^6E, v12F ^4E, v9F
39 238.776 ^7E, v11F ^5E, v8F
40 244.898 ^8E, v10F ^6E, v7F
41 251.02 ^9E, v9F ^7E, v6F
42 257.143 65/56 ^10E, v8F ^8E, v5F
43 263.265 64/55 ^11E, v7F ^9E, v4F
44 269.388 ^12E, v6F ^10E, v3F
45 275.51 75/64 ^13E, v5F ^11E, vvF
46 281.633 E♯, v4F ^12E, vF
47 287.755 13/11 ^E♯, v3F F
48 293.878 77/65 ^^E♯, vvF ^F, v12G♭
49 300 25/21 ^3E♯, vF ^^F, v11G♭
50 306.122 F ^3F, v10G♭
51 312.245 ^F, v3G♭♭ ^4F, v9G♭
52 318.367 77/64 ^^F, vvG♭♭ ^5F, v8G♭
53 324.49 ^3F, vG♭♭ ^6F, v7G♭
54 330.612 40/33 ^4F, G♭♭ ^7F, v6G♭
55 336.735 ^5F, v13G♭ ^8F, v5G♭
56 342.857 39/32 ^6F, v12G♭ ^9F, v4G♭
57 348.98 49/40 ^7F, v11G♭ ^10F, v3G♭
58 355.102 ^8F, v10G♭ ^11F, vvG♭
59 361.224 16/13 ^9F, v9G♭ ^12F, vG♭
60 367.347 26/21 ^10F, v8G♭ ^13F, G♭
61 373.469 ^11F, v7G♭ ^14F, v20G
62 379.592 ^12F, v6G♭ ^15F, v19G
63 385.714 5/4 ^13F, v5G♭ ^16F, v18G
64 391.837 F♯, v4G♭ ^17F, v17G
65 397.959 44/35 ^F♯, v3G♭ ^18F, v16G
66 404.082 ^^F♯, vvG♭ ^19F, v15G
67 410.204 ^3F♯, vG♭ ^20F, v14G
68 416.327 14/11 ^4F♯, G♭ F♯, v13G
69 422.449 ^5F♯, v13G ^F♯, v12G
70 428.571 32/25, 50/39 ^6F♯, v12G ^^F♯, v11G
71 434.694 ^7F♯, v11G ^3F♯, v10G
72 440.816 ^8F♯, v10G ^4F♯, v9G
73 446.939 ^9F♯, v9G ^5F♯, v8G
74 453.061 13/10 ^10F♯, v8G ^6F♯, v7G
75 459.184 ^11F♯, v7G ^7F♯, v6G
76 465.306 55/42 ^12F♯, v6G ^8F♯, v5G
77 471.429 21/16 ^13F♯, v5G ^9F♯, v4G
78 477.551 F𝄪, v4G ^10F♯, v3G
79 483.673 ^F𝄪, v3G ^11F♯, vvG
80 489.796 65/49 ^^F𝄪, vvG ^12F♯, vG
81 495.918 4/3 ^3F𝄪, vG G
82 502.041 G ^G, v12A♭
83 508.163 75/56 ^G, v3A♭♭ ^^G, v11A♭
84 514.286 35/26 ^^G, vvA♭♭ ^3G, v10A♭
85 520.408 ^3G, vA♭♭ ^4G, v9A♭
86 526.531 ^4G, A♭♭ ^5G, v8A♭
87 532.653 ^5G, v13A♭ ^6G, v7A♭
88 538.776 15/11 ^6G, v12A♭ ^7G, v6A♭
89 544.898 ^7G, v11A♭ ^8G, v5A♭
90 551.02 11/8 ^8G, v10A♭ ^9G, v4A♭
91 557.143 ^9G, v9A♭ ^10G, v3A♭
92 563.265 ^10G, v8A♭ ^11G, vvA♭
93 569.388 ^11G, v7A♭ ^12G, vA♭
94 575.51 39/28 ^12G, v6A♭ ^13G, A♭
95 581.633 7/5 ^13G, v5A♭ ^14G, v20A
96 587.755 G♯, v4A♭ ^15G, v19A
97 593.878 55/39 ^G♯, v3A♭ ^16G, v18A
98 600 ^^G♯, vvA♭ ^17G, v17A
99 606.122 78/55 ^3G♯, vA♭ ^18G, v16A
100 612.245 ^4G♯, A♭ ^19G, v15A
101 618.367 10/7 ^5G♯, v13A ^20G, v14A
102 624.49 56/39 ^6G♯, v12A G♯, v13A
103 630.612 ^7G♯, v11A ^G♯, v12A
104 636.735 ^8G♯, v10A ^^G♯, v11A
105 642.857 ^9G♯, v9A ^3G♯, v10A
106 648.98 16/11 ^10G♯, v8A ^4G♯, v9A
107 655.102 ^11G♯, v7A ^5G♯, v8A
108 661.224 22/15 ^12G♯, v6A ^6G♯, v7A
109 667.347 ^13G♯, v5A ^7G♯, v6A
110 673.469 65/44 G𝄪, v4A ^8G♯, v5A
111 679.592 77/52 ^G𝄪, v3A ^9G♯, v4A
112 685.714 52/35 ^^G𝄪, vvA ^10G♯, v3A
113 691.837 ^3G𝄪, vA ^11G♯, vvA
114 697.959 A ^12G♯, vA
115 704.082 3/2 ^A, v3B♭♭ A
116 710.204 ^^A, vvB♭♭ ^A, v12B♭
117 716.327 ^3A, vB♭♭ ^^A, v11B♭
118 722.449 ^4A, B♭♭ ^3A, v10B♭
119 728.571 32/21 ^5A, v13B♭ ^4A, v9B♭
120 734.694 ^6A, v12B♭ ^5A, v8B♭
121 740.816 ^7A, v11B♭ ^6A, v7B♭
122 746.939 20/13, 77/50 ^8A, v10B♭ ^7A, v6B♭
123 753.061 ^9A, v9B♭ ^8A, v5B♭
124 759.184 ^10A, v8B♭ ^9A, v4B♭
125 765.306 ^11A, v7B♭ ^10A, v3B♭
126 771.429 25/16, 39/25 ^12A, v6B♭ ^11A, vvB♭
127 777.551 ^13A, v5B♭ ^12A, vB♭
128 783.673 11/7 A♯, v4B♭ ^13A, B♭
129 789.796 ^A♯, v3B♭ ^14A, v20B
130 795.918 ^^A♯, vvB♭ ^15A, v19B
131 802.041 35/22 ^3A♯, vB♭ ^16A, v18B
132 808.163 ^4A♯, B♭ ^17A, v17B
133 814.286 8/5 ^5A♯, v13B ^18A, v16B
134 820.408 ^6A♯, v12B ^19A, v15B
135 826.531 ^7A♯, v11B ^20A, v14B
136 832.653 21/13 ^8A♯, v10B A♯, v13B
137 838.776 13/8 ^9A♯, v9B ^A♯, v12B
138 844.898 ^10A♯, v8B ^^A♯, v11B
139 851.02 80/49 ^11A♯, v7B ^3A♯, v10B
140 857.143 64/39 ^12A♯, v6B ^4A♯, v9B
141 863.265 ^13A♯, v5B ^5A♯, v8B
142 869.388 33/20 A𝄪, v4B ^6A♯, v7B
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 42/25 ^B, v3C♭ ^11A♯, vvB
148 906.122 ^^B, vvC♭ ^12A♯, vB
149 912.245 22/13 ^3B, vC♭ B
150 918.367 ^4B, C♭ ^B, v12C
151 924.49 75/44 ^5B, v13C ^^B, v11C
152 930.612 ^6B, v12C ^3B, v10C
153 936.735 55/32 ^7B, v11C ^4B, v9C
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 7/4 ^12B, v6C ^9B, v4C
159 973.469 ^13B, v5C ^10B, v3C
160 979.592 44/25 B♯, v4C ^11B, vvC
161 985.714 ^B♯, v3C ^12B, vC
162 991.837 39/22 ^^B♯, vvC C
163 997.959 ^3B♯, vC ^C, v12D♭
164 1004.082 25/14 C ^^C, v11D♭
165 1010.204 ^C, v3D♭♭ ^3C, v10D♭
166 1016.327 ^^C, vvD♭♭ ^4C, v9D♭
167 1022.449 ^3C, vD♭♭ ^5C, v8D♭
168 1028.571 ^4C, D♭♭ ^6C, v7D♭
169 1034.694 20/11 ^5C, v13D♭ ^7C, v6D♭
170 1040.816 ^6C, v12D♭ ^8C, v5D♭
171 1046.939 11/6, 64/35 ^7C, v11D♭ ^9C, v4D♭
172 1053.061 ^8C, v10D♭ ^10C, v3D♭
173 1059.184 ^9C, v9D♭ ^11C, vvD♭
174 1065.306 ^10C, v8D♭ ^12C, vD♭
175 1071.429 13/7 ^11C, v7D♭ ^13C, D♭
176 1077.551 ^12C, v6D♭ ^14C, v20D
177 1083.673 ^13C, v5D♭ ^15C, v19D
178 1089.796 15/8 C♯, v4D♭ ^16C, v18D
179 1095.918 49/26 ^C♯, v3D♭ ^17C, v17D
180 1102.041 ^^C♯, vvD♭ ^18C, v16D
181 1108.163 ^3C♯, vD♭ ^19C, v15D
182 1114.286 40/21 ^4C♯, D♭ ^20C, v14D
183 1120.408 21/11 ^5C♯, v13D C♯, v13D
184 1126.531 ^6C♯, v12D ^C♯, v12D
185 1132.653 25/13, 77/40 ^7C♯, v11D ^^C♯, v11D
186 1138.776 ^8C♯, v10D ^3C♯, v10D
187 1144.898 64/33 ^9C♯, v9D ^4C♯, v9D
188 1151.02 ^10C♯, v8D ^5C♯, v8D
189 1157.143 39/20 ^11C♯, v7D ^6C♯, v7D
190 1163.265 49/25 ^12C♯, v6D ^7C♯, v6D
191 1169.388 55/28 ^13C♯, v5D ^8C♯, v5D
192 1175.51 77/39 C𝄪, v4D ^9C♯, v4D
193 1181.633 ^C𝄪, v3D ^10C♯, v3D
194 1187.755 ^^C𝄪, vvD ^11C♯, vvD
195 1193.878 ^3C𝄪, vD ^12C♯, vD
196 1200 2/1 D D
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