196edo: Difference between revisions

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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|196}}
{{Harmonics in equal|196}}
=== Subsets and supersets ===
Since 196 factors into {{factorization|196}}, 196edo has subset edos {{EDOs| 2, 4, 7, 14, 28, 49, and 98 }}.


== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}

Revision as of 13:37, 15 April 2024

← 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

Template:EDO intro

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 6.12 ^D, v3E♭♭ ^D, ^9E♭♭
2 12.24 ^^D, vvE♭♭ ^^D, ^10E♭♭
3 18.37 ^3D, vE♭♭ ^3D, v10E♭
4 24.49 ^4D, E♭♭ ^4D, v9E♭
5 30.61 56/55 ^5D, ^E♭♭ ^5D, v8E♭
6 36.73 47/46, 48/47 ^6D, ^^E♭♭ ^6D, v7E♭
7 42.86 40/39, 41/40 ^7D, ^3E♭♭ ^7D, v6E♭
8 48.98 v6D♯, ^4E♭♭ ^8D, v5E♭
9 55.1 32/31 v5D♯, ^5E♭♭ ^9D, v4E♭
10 61.22 29/28 v4D♯, ^6E♭♭ ^10D, v3E♭
11 67.35 26/25 v3D♯, v7E♭ v10D♯, vvE♭
12 73.47 24/23 vvD♯, v6E♭ v9D♯, vE♭
13 79.59 22/21, 45/43 vD♯, v5E♭ v8D♯, E♭
14 85.71 41/39 D♯, v4E♭ v7D♯, ^E♭
15 91.84 39/37, 58/55 ^D♯, v3E♭ v6D♯, ^^E♭
16 97.96 55/52 ^^D♯, vvE♭ v5D♯, ^3E♭
17 104.08 17/16 ^3D♯, vE♭ v4D♯, ^4E♭
18 110.2 ^4D♯, E♭ v3D♯, ^5E♭
19 116.33 31/29, 46/43 ^5D♯, ^E♭ vvD♯, ^6E♭
20 122.45 44/41 ^6D♯, ^^E♭ vD♯, ^7E♭
21 128.57 14/13 ^7D♯, ^3E♭ D♯, ^8E♭
22 134.69 40/37 v6D𝄪, ^4E♭ ^D♯, ^9E♭
23 140.82 51/47 v5D𝄪, ^5E♭ ^^D♯, ^10E♭
24 146.94 37/34 v4D𝄪, ^6E♭ ^3D♯, v10E
25 153.06 47/43 v3D𝄪, v7E ^4D♯, v9E
26 159.18 34/31 vvD𝄪, v6E ^5D♯, v8E
27 165.31 11/10 vD𝄪, v5E ^6D♯, v7E
28 171.43 32/29 D𝄪, v4E ^7D♯, v6E
29 177.55 41/37, 51/46 ^D𝄪, v3E ^8D♯, v5E
30 183.67 ^^D𝄪, vvE ^9D♯, v4E
31 189.8 29/26, 48/43 ^3D𝄪, vE ^10D♯, v3E
32 195.92 28/25 E v10D𝄪, vvE
33 202.04 ^E, v3F♭ v9D𝄪, vE
34 208.16 44/39, 62/55 ^^E, vvF♭ E
35 214.29 43/38 ^3E, vF♭ ^E, ^9F♭
36 220.41 25/22, 42/37 ^4E, F♭ ^^E, ^10F♭
37 226.53 ^5E, ^F♭ ^3E, v10F
38 232.65 ^6E, ^^F♭ ^4E, v9F
39 238.78 ^7E, ^3F♭ ^5E, v8F
40 244.9 38/33 v6E♯, ^4F♭ ^6E, v7F
41 251.02 37/32 v5E♯, ^5F♭ ^7E, v6F
42 257.14 29/25, 65/56 v4E♯, ^6F♭ ^8E, v5F
43 263.27 64/55 v3E♯, v7F ^9E, v4F
44 269.39 vvE♯, v6F ^10E, v3F
45 275.51 34/29 vE♯, v5F v10E♯, vvF
46 281.63 20/17 E♯, v4F v9E♯, vF
47 287.76 ^E♯, v3F F
48 293.88 ^^E♯, vvF ^F, ^9G♭♭
49 300 44/37 ^3E♯, vF ^^F, ^10G♭♭
50 306.12 37/31 F ^3F, v10G♭
51 312.24 ^F, v3G♭♭ ^4F, v9G♭
52 318.37 ^^F, vvG♭♭ ^5F, v8G♭
53 324.49 35/29, 41/34 ^3F, vG♭♭ ^6F, v7G♭
54 330.61 23/19 ^4F, G♭♭ ^7F, v6G♭
55 336.73 17/14 ^5F, ^G♭♭ ^8F, v5G♭
56 342.86 39/32, 50/41 ^6F, ^^G♭♭ ^9F, v4G♭
57 348.98 ^7F, ^3G♭♭ ^10F, v3G♭
58 355.1 v6F♯, ^4G♭♭ v10F♯, vvG♭
59 361.22 v5F♯, ^5G♭♭ v9F♯, vG♭
60 367.35 47/38, 68/55 v4F♯, ^6G♭♭ v8F♯, G♭
61 373.47 31/25 v3F♯, v7G♭ v7F♯, ^G♭
62 379.59 vvF♯, v6G♭ v6F♯, ^^G♭
63 385.71 5/4 vF♯, v5G♭ v5F♯, ^3G♭
64 391.84 F♯, v4G♭ v4F♯, ^4G♭
65 397.96 39/31 ^F♯, v3G♭ v3F♯, ^5G♭
66 404.08 24/19 ^^F♯, vvG♭ vvF♯, ^6G♭
67 410.2 19/15 ^3F♯, vG♭ vF♯, ^7G♭
68 416.33 ^4F♯, G♭ F♯, ^8G♭
69 422.45 37/29, 60/47 ^5F♯, ^G♭ ^F♯, ^9G♭
70 428.57 41/32 ^6F♯, ^^G♭ ^^F♯, ^10G♭
71 434.69 ^7F♯, ^3G♭ ^3F♯, v10G
72 440.82 40/31 v6F𝄪, ^4G♭ ^4F♯, v9G
73 446.94 22/17 v5F𝄪, ^5G♭ ^5F♯, v8G
74 453.06 v4F𝄪, ^6G♭ ^6F♯, v7G
75 459.18 30/23, 43/33 v3F𝄪, v7G ^7F♯, v6G
76 465.31 17/13 vvF𝄪, v6G ^8F♯, v5G
77 471.43 21/16 vF𝄪, v5G ^9F♯, v4G
78 477.55 29/22 F𝄪, v4G ^10F♯, v3G
79 483.67 41/31 ^F𝄪, v3G v10F𝄪, vvG
80 489.8 65/49 ^^F𝄪, vvG v9F𝄪, vG
81 495.92 ^3F𝄪, vG G
82 502.04 G ^G, ^9A♭♭
83 508.16 55/41, 63/47 ^G, v3A♭♭ ^^G, ^10A♭♭
84 514.29 35/26 ^^G, vvA♭♭ ^3G, v10A♭
85 520.41 50/37 ^3G, vA♭♭ ^4G, v9A♭
86 526.53 42/31 ^4G, A♭♭ ^5G, v8A♭
87 532.65 34/25 ^5G, ^A♭♭ ^6G, v7A♭
88 538.78 56/41 ^6G, ^^A♭♭ ^7G, v6A♭
89 544.9 63/46 ^7G, ^3A♭♭ ^8G, v5A♭
90 551.02 11/8 v6G♯, ^4A♭♭ ^9G, v4A♭
91 557.14 40/29 v5G♯, ^5A♭♭ ^10G, v3A♭
92 563.27 v4G♯, ^6A♭♭ v10G♯, vvA♭
93 569.39 v3G♯, v7A♭ v9G♯, vA♭
94 575.51 46/33 vvG♯, v6A♭ v8G♯, A♭
95 581.63 7/5 vG♯, v5A♭ v7G♯, ^A♭
96 587.76 66/47 G♯, v4A♭ v6G♯, ^^A♭
97 593.88 31/22 ^G♯, v3A♭ v5G♯, ^3A♭
98 600 41/29, 58/41 ^^G♯, vvA♭ v4G♯, ^4A♭
99 606.12 44/31 ^3G♯, vA♭ v3G♯, ^5A♭
100 612.24 47/33 ^4G♯, A♭ vvG♯, ^6A♭
101 618.37 10/7 ^5G♯, ^A♭ vG♯, ^7A♭
102 624.49 33/23 ^6G♯, ^^A♭ G♯, ^8A♭
103 630.61 ^7G♯, ^3A♭ ^G♯, ^9A♭
104 636.73 v6G𝄪, ^4A♭ ^^G♯, ^10A♭
105 642.86 29/20 v5G𝄪, ^5A♭ ^3G♯, v10A
106 648.98 16/11 v4G𝄪, ^6A♭ ^4G♯, v9A
107 655.1 v3G𝄪, v7A ^5G♯, v8A
108 661.22 41/28, 63/43 vvG𝄪, v6A ^6G♯, v7A
109 667.35 25/17 vG𝄪, v5A ^7G♯, v6A
110 673.47 31/21 G𝄪, v4A ^8G♯, v5A
111 679.59 37/25 ^G𝄪, v3A ^9G♯, v4A
112 685.71 52/35, 55/37 ^^G𝄪, vvA ^10G♯, v3A
113 691.84 ^3G𝄪, vA v10G𝄪, vvA
114 697.96 A v9G𝄪, vA
115 704.08 ^A, v3B♭♭ A
116 710.2 ^^A, vvB♭♭ ^A, ^9B♭♭
117 716.33 62/41 ^3A, vB♭♭ ^^A, ^10B♭♭
118 722.45 44/29 ^4A, B♭♭ ^3A, v10B♭
119 728.57 32/21 ^5A, ^B♭♭ ^4A, v9B♭
120 734.69 26/17 ^6A, ^^B♭♭ ^5A, v8B♭
121 740.82 23/15, 66/43 ^7A, ^3B♭♭ ^6A, v7B♭
122 746.94 v6A♯, ^4B♭♭ ^7A, v6B♭
123 753.06 17/11 v5A♯, ^5B♭♭ ^8A, v5B♭
124 759.18 31/20 v4A♯, ^6B♭♭ ^9A, v4B♭
125 765.31 v3A♯, v7B♭ ^10A, v3B♭
126 771.43 64/41 vvA♯, v6B♭ v10A♯, vvB♭
127 777.55 47/30, 58/37 vA♯, v5B♭ v9A♯, vB♭
128 783.67 A♯, v4B♭ v8A♯, B♭
129 789.8 30/19 ^A♯, v3B♭ v7A♯, ^B♭
130 795.92 19/12 ^^A♯, vvB♭ v6A♯, ^^B♭
131 802.04 62/39 ^3A♯, vB♭ v5A♯, ^3B♭
132 808.16 ^4A♯, B♭ v4A♯, ^4B♭
133 814.29 8/5 ^5A♯, ^B♭ v3A♯, ^5B♭
134 820.41 ^6A♯, ^^B♭ vvA♯, ^6B♭
135 826.53 50/31 ^7A♯, ^3B♭ vA♯, ^7B♭
136 832.65 55/34 v6A𝄪, ^4B♭ A♯, ^8B♭
137 838.78 v5A𝄪, ^5B♭ ^A♯, ^9B♭
138 844.9 v4A𝄪, ^6B♭ ^^A♯, ^10B♭
139 851.02 v3A𝄪, v7B ^3A♯, v10B
140 857.14 41/25, 64/39 vvA𝄪, v6B ^4A♯, v9B
141 863.27 28/17 vA𝄪, v5B ^5A♯, v8B
142 869.39 38/23 A𝄪, v4B ^6A♯, v7B
143 875.51 58/35, 63/38, 68/41 ^A𝄪, v3B ^7A♯, v6B
144 881.63 ^^A𝄪, vvB ^8A♯, v5B
145 887.76 ^3A𝄪, vB ^9A♯, v4B
146 893.88 62/37 B ^10A♯, v3B
147 900 37/22 ^B, v3C♭ v10A𝄪, vvB
148 906.12 ^^B, vvC♭ v9A𝄪, vB
149 912.24 ^3B, vC♭ B
150 918.37 17/10 ^4B, C♭ ^B, ^9C♭
151 924.49 29/17 ^5B, ^C♭ ^^B, ^10C♭
152 930.61 ^6B, ^^C♭ ^3B, v10C
153 936.73 55/32 ^7B, ^3C♭ ^4B, v9C
154 942.86 50/29 v6B♯, ^4C♭ ^5B, v8C
155 948.98 64/37 v5B♯, ^5C♭ ^6B, v7C
156 955.1 33/19 v4B♯, ^6C♭ ^7B, v6C
157 961.22 v3B♯, v7C ^8B, v5C
158 967.35 vvB♯, v6C ^9B, v4C
159 973.47 vB♯, v5C ^10B, v3C
160 979.59 37/21, 44/25 B♯, v4C v10B♯, vvC
161 985.71 ^B♯, v3C v9B♯, vC
162 991.84 39/22, 55/31 ^^B♯, vvC C
163 997.96 ^3B♯, vC ^C, ^9D♭♭
164 1004.08 25/14 C ^^C, ^10D♭♭
165 1010.2 43/24, 52/29 ^C, v3D♭♭ ^3C, v10D♭
166 1016.33 ^^C, vvD♭♭ ^4C, v9D♭
167 1022.45 ^3C, vD♭♭ ^5C, v8D♭
168 1028.57 29/16 ^4C, D♭♭ ^6C, v7D♭
169 1034.69 20/11 ^5C, ^D♭♭ ^7C, v6D♭
170 1040.82 31/17 ^6C, ^^D♭♭ ^8C, v5D♭
171 1046.94 ^7C, ^3D♭♭ ^9C, v4D♭
172 1053.06 68/37 v6C♯, ^4D♭♭ ^10C, v3D♭
173 1059.18 v5C♯, ^5D♭♭ v10C♯, vvD♭
174 1065.31 37/20 v4C♯, ^6D♭♭ v9C♯, vD♭
175 1071.43 13/7 v3C♯, v7D♭ v8C♯, D♭
176 1077.55 41/22 vvC♯, v6D♭ v7C♯, ^D♭
177 1083.67 43/23, 58/31 vC♯, v5D♭ v6C♯, ^^D♭
178 1089.8 C♯, v4D♭ v5C♯, ^3D♭
179 1095.92 32/17 ^C♯, v3D♭ v4C♯, ^4D♭
180 1102.04 ^^C♯, vvD♭ v3C♯, ^5D♭
181 1108.16 55/29 ^3C♯, vD♭ vvC♯, ^6D♭
182 1114.29 ^4C♯, D♭ vC♯, ^7D♭
183 1120.41 21/11 ^5C♯, ^D♭ C♯, ^8D♭
184 1126.53 23/12 ^6C♯, ^^D♭ ^C♯, ^9D♭
185 1132.65 25/13 ^7C♯, ^3D♭ ^^C♯, ^10D♭
186 1138.78 56/29 v6C𝄪, ^4D♭ ^3C♯, v10D
187 1144.9 31/16 v5C𝄪, ^5D♭ ^4C♯, v9D
188 1151.02 v4C𝄪, ^6D♭ ^5C♯, v8D
189 1157.14 39/20 v3C𝄪, v7D ^6C♯, v7D
190 1163.27 47/24 vvC𝄪, v6D ^7C♯, v6D
191 1169.39 55/28 vC𝄪, v5D ^8C♯, v5D
192 1175.51 C𝄪, v4D ^9C♯, v4D
193 1181.63 ^C𝄪, v3D ^10C♯, v3D
194 1187.76 ^^C𝄪, vvD v10C𝄪, vvD
195 1193.88 ^3C𝄪, vD v9C𝄪, vD
196 1200 2/1 D D
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