196edo
← 195edo | 196edo | 197edo → |
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
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 (%) | +35 | -10 | -24 | -31 | -5 | -29 | +25 | -14 | +41 | +11 | +38 | |
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, v3Ebb | ^D, v12Eb | |
2 | 12.2449 | ^^D, vvEbb | ^^D, v11Eb | |
3 | 18.3673 | ^3D, vEbb | ^3D, v10Eb | |
4 | 24.4898 | ^4D, Ebb | ^4D, v9Eb | 65/64, 78/77 |
5 | 30.6122 | ^5D, v13Eb | ^5D, v8Eb | 56/55 |
6 | 36.7347 | ^6D, v12Eb | ^6D, v7Eb | 50/49 |
7 | 42.8571 | ^7D, v11Eb | ^7D, v6Eb | 40/39 |
8 | 48.9796 | ^8D, v10Eb | ^8D, v5Eb | |
9 | 55.102 | ^9D, v9Eb | ^9D, v4Eb | 33/32 |
10 | 61.2245 | ^10D, v8Eb | ^10D, v3Eb | |
11 | 67.3469 | ^11D, v7Eb | ^11D, vvEb | 26/25, 80/77 |
12 | 73.4694 | ^12D, v6Eb | ^12D, vEb | |
13 | 79.5918 | ^13D, v5Eb | ^13D, Eb | 22/21 |
14 | 85.7143 | D#, v4Eb | ^14D, v20E | 21/20 |
15 | 91.8367 | ^D#, v3Eb | ^15D, v19E | |
16 | 97.9592 | ^^D#, vvEb | ^16D, v18E | 55/52 |
17 | 104.082 | ^3D#, vEb | ^17D, v17E | 52/49 |
18 | 110.204 | ^4D#, Eb | ^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 | Dx, v4E | ^7D#, v6E | |
29 | 177.551 | ^Dx, v3E | ^8D#, v5E | |
30 | 183.673 | ^^Dx, vvE | ^9D#, v4E | |
31 | 189.796 | ^3Dx, vE | ^10D#, v3E | |
32 | 195.918 | E | ^11D#, vvE | 28/25 |
33 | 202.041 | ^E, v3Fb | ^12D#, vE | 55/49 |
34 | 208.163 | ^^E, vvFb | E | 44/39 |
35 | 214.286 | ^3E, vFb | ^E, v12F | |
36 | 220.408 | ^4E, Fb | ^^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, v12Gb | 77/65 |
49 | 300 | ^3E#, vF | ^^F, v11Gb | 25/21 |
50 | 306.122 | F | ^3F, v10Gb | |
51 | 312.245 | ^F, v3Gbb | ^4F, v9Gb | |
52 | 318.367 | ^^F, vvGbb | ^5F, v8Gb | 77/64 |
53 | 324.49 | ^3F, vGbb | ^6F, v7Gb | |
54 | 330.612 | ^4F, Gbb | ^7F, v6Gb | 40/33 |
55 | 336.735 | ^5F, v13Gb | ^8F, v5Gb | |
56 | 342.857 | ^6F, v12Gb | ^9F, v4Gb | 39/32 |
57 | 348.98 | ^7F, v11Gb | ^10F, v3Gb | 49/40 |
58 | 355.102 | ^8F, v10Gb | ^11F, vvGb | |
59 | 361.224 | ^9F, v9Gb | ^12F, vGb | 16/13 |
60 | 367.347 | ^10F, v8Gb | ^13F, Gb | 26/21 |
61 | 373.469 | ^11F, v7Gb | ^14F, v20G | |
62 | 379.592 | ^12F, v6Gb | ^15F, v19G | |
63 | 385.714 | ^13F, v5Gb | ^16F, v18G | 5/4 |
64 | 391.837 | F#, v4Gb | ^17F, v17G | |
65 | 397.959 | ^F#, v3Gb | ^18F, v16G | 44/35 |
66 | 404.082 | ^^F#, vvGb | ^19F, v15G | |
67 | 410.204 | ^3F#, vGb | ^20F, v14G | |
68 | 416.327 | ^4F#, Gb | 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 | Fx, v4G | ^10F#, v3G | |
79 | 483.673 | ^Fx, v3G | ^11F#, vvG | |
80 | 489.796 | ^^Fx, vvG | ^12F#, vG | 65/49 |
81 | 495.918 | ^3Fx, vG | G | 4/3 |
82 | 502.041 | G | ^G, v12Ab | |
83 | 508.163 | ^G, v3Abb | ^^G, v11Ab | 75/56 |
84 | 514.286 | ^^G, vvAbb | ^3G, v10Ab | 35/26 |
85 | 520.408 | ^3G, vAbb | ^4G, v9Ab | |
86 | 526.531 | ^4G, Abb | ^5G, v8Ab | |
87 | 532.653 | ^5G, v13Ab | ^6G, v7Ab | |
88 | 538.776 | ^6G, v12Ab | ^7G, v6Ab | 15/11 |
89 | 544.898 | ^7G, v11Ab | ^8G, v5Ab | |
90 | 551.02 | ^8G, v10Ab | ^9G, v4Ab | 11/8 |
91 | 557.143 | ^9G, v9Ab | ^10G, v3Ab | |
92 | 563.265 | ^10G, v8Ab | ^11G, vvAb | |
93 | 569.388 | ^11G, v7Ab | ^12G, vAb | |
94 | 575.51 | ^12G, v6Ab | ^13G, Ab | 39/28 |
95 | 581.633 | ^13G, v5Ab | ^14G, v20A | 7/5 |
96 | 587.755 | G#, v4Ab | ^15G, v19A | |
97 | 593.878 | ^G#, v3Ab | ^16G, v18A | 55/39 |
98 | 600 | ^^G#, vvAb | ^17G, v17A | |
99 | 606.122 | ^3G#, vAb | ^18G, v16A | 78/55 |
100 | 612.245 | ^4G#, Ab | ^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 | Gx, v4A | ^8G#, v5A | 65/44 |
111 | 679.592 | ^Gx, v3A | ^9G#, v4A | 77/52 |
112 | 685.714 | ^^Gx, vvA | ^10G#, v3A | 52/35 |
113 | 691.837 | ^3Gx, vA | ^11G#, vvA | |
114 | 697.959 | A | ^12G#, vA | |
115 | 704.082 | ^A, v3Bbb | A | 3/2 |
116 | 710.204 | ^^A, vvBbb | ^A, v12Bb | |
117 | 716.327 | ^3A, vBbb | ^^A, v11Bb | |
118 | 722.449 | ^4A, Bbb | ^3A, v10Bb | |
119 | 728.571 | ^5A, v13Bb | ^4A, v9Bb | 32/21 |
120 | 734.694 | ^6A, v12Bb | ^5A, v8Bb | |
121 | 740.816 | ^7A, v11Bb | ^6A, v7Bb | |
122 | 746.939 | ^8A, v10Bb | ^7A, v6Bb | 20/13, 77/50 |
123 | 753.061 | ^9A, v9Bb | ^8A, v5Bb | |
124 | 759.184 | ^10A, v8Bb | ^9A, v4Bb | |
125 | 765.306 | ^11A, v7Bb | ^10A, v3Bb | |
126 | 771.429 | ^12A, v6Bb | ^11A, vvBb | 25/16, 39/25 |
127 | 777.551 | ^13A, v5Bb | ^12A, vBb | |
128 | 783.673 | A#, v4Bb | ^13A, Bb | 11/7 |
129 | 789.796 | ^A#, v3Bb | ^14A, v20B | |
130 | 795.918 | ^^A#, vvBb | ^15A, v19B | |
131 | 802.041 | ^3A#, vBb | ^16A, v18B | 35/22 |
132 | 808.163 | ^4A#, Bb | ^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 | Ax, v4B | ^6A#, v7B | 33/20 |
143 | 875.51 | ^Ax, v3B | ^7A#, v6B | |
144 | 881.633 | ^^Ax, vvB | ^8A#, v5B | |
145 | 887.755 | ^3Ax, vB | ^9A#, v4B | |
146 | 893.878 | B | ^10A#, v3B | |
147 | 900 | ^B, v3Cb | ^11A#, vvB | 42/25 |
148 | 906.122 | ^^B, vvCb | ^12A#, vB | |
149 | 912.245 | ^3B, vCb | B | 22/13 |
150 | 918.367 | ^4B, Cb | ^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, v12Db | |
164 | 1004.08 | C | ^^C, v11Db | 25/14 |
165 | 1010.2 | ^C, v3Dbb | ^3C, v10Db | |
166 | 1016.33 | ^^C, vvDbb | ^4C, v9Db | |
167 | 1022.45 | ^3C, vDbb | ^5C, v8Db | |
168 | 1028.57 | ^4C, Dbb | ^6C, v7Db | |
169 | 1034.69 | ^5C, v13Db | ^7C, v6Db | 20/11 |
170 | 1040.82 | ^6C, v12Db | ^8C, v5Db | |
171 | 1046.94 | ^7C, v11Db | ^9C, v4Db | 11/6, 64/35 |
172 | 1053.06 | ^8C, v10Db | ^10C, v3Db | |
173 | 1059.18 | ^9C, v9Db | ^11C, vvDb | |
174 | 1065.31 | ^10C, v8Db | ^12C, vDb | |
175 | 1071.43 | ^11C, v7Db | ^13C, Db | 13/7 |
176 | 1077.55 | ^12C, v6Db | ^14C, v20D | |
177 | 1083.67 | ^13C, v5Db | ^15C, v19D | |
178 | 1089.8 | C#, v4Db | ^16C, v18D | 15/8 |
179 | 1095.92 | ^C#, v3Db | ^17C, v17D | 49/26 |
180 | 1102.04 | ^^C#, vvDb | ^18C, v16D | |
181 | 1108.16 | ^3C#, vDb | ^19C, v15D | |
182 | 1114.29 | ^4C#, Db | ^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 | Cx, v4D | ^9C#, v4D | 77/39 |
193 | 1181.63 | ^Cx, v3D | ^10C#, v3D | |
194 | 1187.76 | ^^Cx, vvD | ^11C#, vvD | |
195 | 1193.88 | ^3Cx, vD | ^12C#, vD | |
196 | 1200 | D | D | 2/1 |