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 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