321edo: Difference between revisions

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== Theory ==
== Theory ==
321edo is in[[consistent]] in the [[5-odd-limit]]. The [[patent val]] [[Tempering out|tempers out]] [[2401/2400]], [[5120/5103]] and [[10976/10935]] in the 7-limit, supporting [[hemififths]]. In the 11-limit it tempers out [[385/384]] and 1375/1372, and in the 13-limit [[325/324]], [[352/351]], [[847/845]] and [[2080/2079]], providing the [[optimal patent val]] for 11- and 13-limit [[akea]] temperament.
321edo is in[[consistent]] in the [[5-odd-limit]]. The [[patent val]] [[Tempering out|tempers out]] [[2401/2400]], [[5120/5103]] and [[10976/10935]] in the 7-limit, supporting [[hemififths]]. In the 11-limit it tempers out [[385/384]] and 1375/1372, and in the 13-limit [[325/324]], [[352/351]], [[847/845]], [[2080/2079]] and [[4096/4095]], providing the [[optimal patent val]] for 11- and 13-limit [[akea]] temperament.


=== Prime harmonics ===
=== Prime harmonics ===
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== Music ==
== Music ==
; [[JUMBLE]]
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=4HT5onKQaz0 ''Sun Through The Window''] (2023)
* [https://www.youtube.com/watch?v=4HT5onKQaz0 ''Sun Through The Window''] (2023) – ambient
* [https://www.youtube.com/watch?v=cQoWVBbKnuA ''BLASTOFF!''] (2023)
* [https://www.youtube.com/watch?v=cQoWVBbKnuA ''BLASTOFF!''] (2023) – synthwave, [[9L 8s]] scale
 
[[Category:Akea]]

Revision as of 04:27, 15 September 2023

← 320edo 321edo 322edo →
Prime factorization 3 × 107
Step size 3.73832 ¢ 
Fifth 188\321 (702.804 ¢)
Semitones (A1:m2) 32:23 (119.6 ¢ : 85.98 ¢)
Consistency limit 3
Distinct consistency limit 3

Template:EDO intro

Theory

321edo is inconsistent in the 5-odd-limit. The patent val tempers out 2401/2400, 5120/5103 and 10976/10935 in the 7-limit, supporting hemififths. In the 11-limit it tempers out 385/384 and 1375/1372, and in the 13-limit 325/324, 352/351, 847/845, 2080/2079 and 4096/4095, providing the optimal patent val for 11- and 13-limit akea temperament.

Prime harmonics

Approximation of prime harmonics in 321edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.85 -1.27 -0.60 -1.79 +0.59 -0.28 +1.55 -0.24 -1.54 -1.11
Relative (%) +0.0 +22.7 -33.9 -16.1 -47.8 +15.9 -7.6 +41.5 -6.3 -41.2 -29.7
Steps
(reduced)
321
(0)
509
(188)
745
(103)
901
(259)
1110
(147)
1188
(225)
1312
(28)
1364
(80)
1452
(168)
1559
(275)
1590
(306)

Interval table

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 3.74 ^D, ^10E♭♭
2 7.48 ^^D, ^11E♭♭
3 11.21 ^3D, ^12E♭♭
4 14.95 ^4D, ^13E♭♭
5 18.69 91/90, 92/91, 93/92, 94/93, 95/94, 96/95 ^5D, ^14E♭♭
6 22.43 ^6D, ^15E♭♭
7 26.17 66/65 ^7D, v16E♭
8 29.91 ^8D, v15E♭
9 33.64 52/51 ^9D, v14E♭
10 37.38 47/46, 93/91, 95/93 ^10D, v13E♭
11 41.12 42/41, 43/42 ^11D, v12E♭
12 44.86 39/38 ^12D, v11E♭
13 48.6 ^13D, v10E♭
14 52.34 ^14D, v9E♭
15 56.07 94/91, 95/92 ^15D, v8E♭
16 59.81 88/85 ^16D, v7E♭
17 63.55 v15D♯, v6E♭
18 67.29 v14D♯, v5E♭
19 71.03 99/95 v13D♯, v4E♭
20 74.77 47/45, 95/91 v12D♯, v3E♭
21 78.5 45/43, 68/65 v11D♯, vvE♭
22 82.24 43/41, 65/62 v10D♯, vE♭
23 85.98 41/39 v9D♯, E♭
24 89.72 99/94 v8D♯, ^E♭
25 93.46 19/18 v7D♯, ^^E♭
26 97.2 92/87 v6D♯, ^3E♭
27 100.93 v5D♯, ^4E♭
28 104.67 17/16 v4D♯, ^5E♭
29 108.41 33/31 v3D♯, ^6E♭
30 112.15 16/15 vvD♯, ^7E♭
31 115.89 31/29 vD♯, ^8E♭
32 119.63 15/14 D♯, ^9E♭
33 123.36 ^D♯, ^10E♭
34 127.1 99/92 ^^D♯, ^11E♭
35 130.84 69/64 ^3D♯, ^12E♭
36 134.58 40/37, 94/87 ^4D♯, ^13E♭
37 138.32 13/12 ^5D♯, ^14E♭
38 142.06 ^6D♯, ^15E♭
39 145.79 37/34, 87/80, 99/91 ^7D♯, v16E
40 149.53 ^8D♯, v15E
41 153.27 ^9D♯, v14E
42 157.01 23/21 ^10D♯, v13E
43 160.75 45/41 ^11D♯, v12E
44 164.49 11/10 ^12D♯, v11E
45 168.22 ^13D♯, v10E
46 171.96 95/86 ^14D♯, v9E
47 175.7 31/28 ^15D♯, v8E
48 179.44 ^16D♯, v7E
49 183.18 v15D𝄪, v6E
50 186.92 49/44 v14D𝄪, v5E
51 190.65 48/43 v13D𝄪, v4E
52 194.39 47/42 v12D𝄪, v3E
53 198.13 37/33 v11D𝄪, vvE
54 201.87 v10D𝄪, vE
55 205.61 98/87 E
56 209.35 ^E, ^10F♭
57 213.08 95/84 ^^E, ^11F♭
58 216.82 17/15 ^3E, ^12F♭
59 220.56 ^4E, ^13F♭
60 224.3 33/29, 74/65 ^5E, ^14F♭
61 228.04 ^6E, ^15F♭
62 231.78 8/7 ^7E, v16F
63 235.51 ^8E, v15F
64 239.25 ^9E, v14F
65 242.99 ^10E, v13F
66 246.73 98/85 ^11E, v12F
67 250.47 52/45 ^12E, v11F
68 254.21 95/82 ^13E, v10F
69 257.94 65/56 ^14E, v9F
70 261.68 ^15E, v8F
71 265.42 ^16E, v7F
72 269.16 v15E♯, v6F
73 272.9 48/41 v14E♯, v5F
74 276.64 88/75 v13E♯, v4F
75 280.37 87/74 v12E♯, v3F
76 284.11 33/28 v11E♯, vvF
77 287.85 v10E♯, vF
78 291.59 58/49 F
79 295.33 51/43 ^F, ^10G♭♭
80 299.07 82/69 ^^F, ^11G♭♭
81 302.8 56/47 ^3F, ^12G♭♭
82 306.54 37/31 ^4F, ^13G♭♭
83 310.28 ^5F, ^14G♭♭
84 314.02 ^6F, ^15G♭♭
85 317.76 ^7F, v16G♭
86 321.5 ^8F, v15G♭
87 325.23 35/29, 76/63 ^9F, v14G♭
88 328.97 52/43, 75/62 ^10F, v13G♭
89 332.71 40/33, 63/52 ^11F, v12G♭
90 336.45 17/14 ^12F, v11G♭
91 340.19 28/23 ^13F, v10G♭
92 343.93 ^14F, v9G♭
93 347.66 ^15F, v8G♭
94 351.4 49/40 ^16F, v7G♭
95 355.14 v15F♯, v6G♭
96 358.88 16/13 v14F♯, v5G♭
97 362.62 37/30 v13F♯, v4G♭
98 366.36 21/17 v12F♯, v3G♭
99 370.09 26/21 v11F♯, vvG♭
100 373.83 v10F♯, vG♭
101 377.57 51/41 v9F♯, G♭
102 381.31 86/69 v8F♯, ^G♭
103 385.05 v7F♯, ^^G♭
104 388.79 v6F♯, ^3G♭
105 392.52 64/51 v5F♯, ^4G♭
106 396.26 44/35, 93/74 v4F♯, ^5G♭
107 400 v3F♯, ^6G♭
108 403.74 vvF♯, ^7G♭
109 407.48 62/49 vF♯, ^8G♭
110 411.21 52/41 F♯, ^9G♭
111 414.95 ^F♯, ^10G♭
112 418.69 ^^F♯, ^11G♭
113 422.43 60/47 ^3F♯, ^12G♭
114 426.17 87/68 ^4F♯, ^13G♭
115 429.91 ^5F♯, ^14G♭
116 433.64 ^6F♯, ^15G♭
117 437.38 85/66 ^7F♯, v16G
118 441.12 40/31 ^8F♯, v15G
119 444.86 75/58 ^9F♯, v14G
120 448.6 ^10F♯, v13G
121 452.34 100/77 ^11F♯, v12G
122 456.07 82/63 ^12F♯, v11G
123 459.81 30/23 ^13F♯, v10G
124 463.55 98/75 ^14F♯, v9G
125 467.29 ^15F♯, v8G
126 471.03 21/16 ^16F♯, v7G
127 474.77 v15F𝄪, v6G
128 478.5 29/22, 91/69 v14F𝄪, v5G
129 482.24 37/28 v13F𝄪, v4G
130 485.98 49/37 v12F𝄪, v3G
131 489.72 65/49, 69/52 v11F𝄪, vvG
132 493.46 v10F𝄪, vG
133 497.2 G
134 500.93 ^G, ^10A♭♭
135 504.67 87/65, 91/68 ^^G, ^11A♭♭
136 508.41 ^3G, ^12A♭♭
137 512.15 43/32 ^4G, ^13A♭♭
138 515.89 66/49 ^5G, ^14A♭♭
139 519.63 ^6G, ^15A♭♭
140 523.36 23/17 ^7G, v16A♭
141 527.1 ^8G, v15A♭
142 530.84 ^9G, v14A♭
143 534.58 64/47 ^10G, v13A♭
144 538.32 86/63 ^11G, v12A♭
145 542.06 93/68 ^12G, v11A♭
146 545.79 85/62 ^13G, v10A♭
147 549.53 ^14G, v9A♭
148 553.27 95/69 ^15G, v8A♭
149 557.01 40/29 ^16G, v7A♭
150 560.75 47/34, 65/47 v15G♯, v6A♭
151 564.49 v14G♯, v5A♭
152 568.22 v13G♯, v4A♭
153 571.96 32/23 v12G♯, v3A♭
154 575.7 v11G♯, vvA♭
155 579.44 95/68 v10G♯, vA♭
156 583.18 v9G♯, A♭
157 586.92 87/62 v8G♯, ^A♭
158 590.65 45/32 v7G♯, ^^A♭
159 594.39 v6G♯, ^3A♭
160 598.13 65/46 v5G♯, ^4A♭
161 601.87 92/65 v4G♯, ^5A♭
162 605.61 v3G♯, ^6A♭
163 609.35 64/45, 91/64 vvG♯, ^7A♭
164 613.08 vG♯, ^8A♭
165 616.82 G♯, ^9A♭
166 620.56 93/65 ^G♯, ^10A♭
167 624.3 ^^G♯, ^11A♭
168 628.04 23/16 ^3G♯, ^12A♭
169 631.78 ^4G♯, ^13A♭
170 635.51 ^5G♯, ^14A♭
171 639.25 68/47, 94/65 ^6G♯, ^15A♭
172 642.99 29/20 ^7G♯, v16A
173 646.73 93/64 ^8G♯, v15A
174 650.47 99/68 ^9G♯, v14A
175 654.21 ^10G♯, v13A
176 657.94 ^11G♯, v12A
177 661.68 63/43, 85/58 ^12G♯, v11A
178 665.42 47/32 ^13G♯, v10A
179 669.16 ^14G♯, v9A
180 672.9 ^15G♯, v8A
181 676.64 34/23 ^16G♯, v7A
182 680.37 v15G𝄪, v6A
183 684.11 49/33, 95/64 v14G𝄪, v5A
184 687.85 64/43 v13G𝄪, v4A
185 691.59 v12G𝄪, v3A
186 695.33 v11G𝄪, vvA
187 699.07 v10G𝄪, vA
188 702.8 A
189 706.54 ^A, ^10B♭♭
190 710.28 98/65 ^^A, ^11B♭♭
191 714.02 74/49 ^3A, ^12B♭♭
192 717.76 56/37 ^4A, ^13B♭♭
193 721.5 44/29, 91/60 ^5A, ^14B♭♭
194 725.23 ^6A, ^15B♭♭
195 728.97 32/21, 99/65 ^7A, v16B♭
196 732.71 ^8A, v15B♭
197 736.45 75/49 ^9A, v14B♭
198 740.19 23/15 ^10A, v13B♭
199 743.93 63/41 ^11A, v12B♭
200 747.66 77/50 ^12A, v11B♭
201 751.4 ^13A, v10B♭
202 755.14 99/64 ^14A, v9B♭
203 758.88 31/20 ^15A, v8B♭
204 762.62 87/56 ^16A, v7B♭
205 766.36 v15A♯, v6B♭
206 770.09 v14A♯, v5B♭
207 773.83 v13A♯, v4B♭
208 777.57 47/30 v12A♯, v3B♭
209 781.31 v11A♯, vvB♭
210 785.05 v10A♯, vB♭
211 788.79 41/26 v9A♯, B♭
212 792.52 49/31 v8A♯, ^B♭
213 796.26 v7A♯, ^^B♭
214 800 v6A♯, ^3B♭
215 803.74 35/22 v5A♯, ^4B♭
216 807.48 51/32 v4A♯, ^5B♭
217 811.21 v3A♯, ^6B♭
218 814.95 vvA♯, ^7B♭
219 818.69 69/43 vA♯, ^8B♭
220 822.43 82/51 A♯, ^9B♭
221 826.17 ^A♯, ^10B♭
222 829.91 21/13 ^^A♯, ^11B♭
223 833.64 34/21 ^3A♯, ^12B♭
224 837.38 60/37 ^4A♯, ^13B♭
225 841.12 13/8 ^5A♯, ^14B♭
226 844.86 ^6A♯, ^15B♭
227 848.6 80/49 ^7A♯, v16B
228 852.34 ^8A♯, v15B
229 856.07 ^9A♯, v14B
230 859.81 23/14 ^10A♯, v13B
231 863.55 28/17 ^11A♯, v12B
232 867.29 33/20 ^12A♯, v11B
233 871.03 43/26 ^13A♯, v10B
234 874.77 58/35, 63/38 ^14A♯, v9B
235 878.5 93/56 ^15A♯, v8B
236 882.24 ^16A♯, v7B
237 885.98 v15A𝄪, v6B
238 889.72 v14A𝄪, v5B
239 893.46 62/37 v13A𝄪, v4B
240 897.2 47/28 v12A𝄪, v3B
241 900.93 69/41 v11A𝄪, vvB
242 904.67 86/51 v10A𝄪, vB
243 908.41 49/29 B
244 912.15 ^B, ^10C♭
245 915.89 56/33 ^^B, ^11C♭
246 919.63 ^3B, ^12C♭
247 923.36 75/44 ^4B, ^13C♭
248 927.1 41/24 ^5B, ^14C♭
249 930.84 ^6B, ^15C♭
250 934.58 ^7B, v16C
251 938.32 ^8B, v15C
252 942.06 ^9B, v14C
253 945.79 ^10B, v13C
254 949.53 45/26 ^11B, v12C
255 953.27 85/49 ^12B, v11C
256 957.01 ^13B, v10C
257 960.75 ^14B, v9C
258 964.49 ^15B, v8C
259 968.22 7/4 ^16B, v7C
260 971.96 v15B♯, v6C
261 975.7 58/33, 65/37 v14B♯, v5C
262 979.44 v13B♯, v4C
263 983.18 30/17 v12B♯, v3C
264 986.92 99/56 v11B♯, vvC
265 990.65 v10B♯, vC
266 994.39 87/49 C
267 998.13 ^C, ^10D♭♭
268 1001.87 66/37, 91/51 ^^C, ^11D♭♭
269 1005.61 84/47 ^3C, ^12D♭♭
270 1009.35 43/24 ^4C, ^13D♭♭
271 1013.08 88/49 ^5C, ^14D♭♭
272 1016.82 ^6C, ^15D♭♭
273 1020.56 ^7C, v16D♭
274 1024.3 56/31 ^8C, v15D♭
275 1028.04 ^9C, v14D♭
276 1031.78 ^10C, v13D♭
277 1035.51 20/11 ^11C, v12D♭
278 1039.25 82/45 ^12C, v11D♭
279 1042.99 42/23, 95/52 ^13C, v10D♭
280 1046.73 ^14C, v9D♭
281 1050.47 ^15C, v8D♭
282 1054.21 68/37 ^16C, v7D♭
283 1057.94 v15C♯, v6D♭
284 1061.68 24/13 v14C♯, v5D♭
285 1065.42 37/20, 87/47 v13C♯, v4D♭
286 1069.16 v12C♯, v3D♭
287 1072.9 v11C♯, vvD♭
288 1076.64 95/51 v10C♯, vD♭
289 1080.37 28/15 v9C♯, D♭
290 1084.11 58/31 v8C♯, ^D♭
291 1087.85 15/8 v7C♯, ^^D♭
292 1091.59 62/33 v6C♯, ^3D♭
293 1095.33 32/17 v5C♯, ^4D♭
294 1099.07 v4C♯, ^5D♭
295 1102.8 87/46 v3C♯, ^6D♭
296 1106.54 36/19 vvC♯, ^7D♭
297 1110.28 vC♯, ^8D♭
298 1114.02 78/41 C♯, ^9D♭
299 1117.76 82/43 ^C♯, ^10D♭
300 1121.5 65/34, 86/45 ^^C♯, ^11D♭
301 1125.23 90/47 ^3C♯, ^12D♭
302 1128.97 ^4C♯, ^13D♭
303 1132.71 ^5C♯, ^14D♭
304 1136.45 ^6C♯, ^15D♭
305 1140.19 85/44 ^7C♯, v16D
306 1143.93 91/47 ^8C♯, v15D
307 1147.66 ^9C♯, v14D
308 1151.4 ^10C♯, v13D
309 1155.14 76/39 ^11C♯, v12D
310 1158.88 41/21, 84/43 ^12C♯, v11D
311 1162.62 92/47 ^13C♯, v10D
312 1166.36 51/26 ^14C♯, v9D
313 1170.09 ^15C♯, v8D
314 1173.83 65/33 ^16C♯, v7D
315 1177.57 v15C𝄪, v6D
316 1181.31 91/46, 93/47, 95/48 v14C𝄪, v5D
317 1185.05 v13C𝄪, v4D
318 1188.79 v12C𝄪, v3D
319 1192.52 v11C𝄪, vvD
320 1196.26 v10C𝄪, vD
321 1200 2/1 D

Music

JUMBLE