Xenharmonic Wiki

400edo

Revision as of 16:19, 6 November 2023 by FloraC (talk | contribs) (Rework; cleanup; clarify the title row of the rank-2 temp table)
← 399edo 400edo 401edo →
Prime factorization 24 × 52
Step size 3 ¢ 
Fifth 234\400 (702 ¢) (→ 117\200)
Semitones (A1:m2) 38:30 (114 ¢ : 90 ¢)
Consistency limit 21
Distinct consistency limit 21

Template:EDO intro

Theory

400edo is a strong 17- and 19-limit system, distinctly consistent to the 21-odd-limit. It shares its excellent harmonic 3/1 with 200edo, which is a semiconvergent, while correcting the higher harmonics to near-just qualities.

The equal temperament tempers out the unidecma, [-7 22 -12⟩, and the quintosec comma, [47 -15 -10⟩, in the 5-limit; 2401/2400, 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, 9801/9800, 117649/117612, and 131072/130977 in the 11-limit; 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 39366/39325 in the 13-limit, supporting the decoid temperament and the quinmite temperament. It tempers out 936/935, 1156/1155, 2058/2057, 2601/2600, 4914/4913 and 24576/24565 in the 17-limit, and 969/968, 1216/1215, 1521/1520, and 1729/1728 in the 19-limit.

Prime harmonics

Approximation of prime harmonics in 400edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.00 +0.04 +0.69 +0.17 +0.68 -0.53 +0.04 -0.51 -1.27 -0.58 +0.96 +0.66 -0.06
Relative (%) +0.0 +1.5 +22.9 +5.8 +22.7 -17.6 +1.5 -17.1 -42.5 -19.2 +32.1 +21.9 -2.1
Steps
(reduced) 		400
(0) 		634
(234) 		929
(129) 		1123
(323) 		1384
(184) 		1480
(280) 		1635
(35) 		1699
(99) 		1809
(209) 		1943
(343) 		1982
(382) 		2084
(84) 		2143
(143)

Subsets and supersets

Since 400 factors into 24 × 52, 400edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200.

Interval table

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 3 ^D, ^9E♭♭
2 6 ^^D, ^10E♭♭
3 9 ^3D, ^11E♭♭
4 12 ^4D, ^12E♭♭
5 15 115/114, 116/115, 117/116 ^5D, ^13E♭♭
6 18 95/94, 96/95, 99/98 ^6D, ^14E♭♭
7 21 82/81 ^7D, ^15E♭♭
8 24 ^8D, ^16E♭♭
9 27 64/63, 65/64 ^9D, ^17E♭♭
10 30 58/57, 117/115 ^10D, ^18E♭♭
11 33 ^11D, v19E♭
12 36 48/47, 49/48 ^12D, v18E♭
13 39 45/44 ^13D, v17E♭
14 42 42/41 ^14D, v16E♭
15 45 39/38 ^15D, v15E♭
16 48 ^16D, v14E♭
17 51 ^17D, v13E♭
18 54 65/63, 98/95 ^18D, v12E♭
19 57 31/30 ^19D, v11E♭
20 60 88/85 v18D♯, v10E♭
21 63 28/27 v17D♯, v9E♭
22 66 80/77 v16D♯, v8E♭
23 69 51/49, 77/74 v15D♯, v7E♭
24 72 49/47 v14D♯, v6E♭
25 75 47/45 v13D♯, v5E♭
26 78 68/65, 91/87 v12D♯, v4E♭
27 81 22/21 v11D♯, v3E♭
28 84 21/20 v10D♯, vvE♭
29 87 41/39 v9D♯, vE♭
30 90 99/94 v8D♯, E♭
31 93 96/91 v7D♯, ^E♭
32 96 37/35, 93/88 v6D♯, ^^E♭
33 99 18/17 v5D♯, ^3E♭
34 102 35/33, 87/82 v4D♯, ^4E♭
35 105 17/16 v3D♯, ^5E♭
36 108 33/31 vvD♯, ^6E♭
37 111 vD♯, ^7E♭
38 114 47/44 D♯, ^8E♭
39 117 ^D♯, ^9E♭
40 120 119/111 ^^D♯, ^10E♭
41 123 102/95 ^3D♯, ^11E♭
42 126 100/93 ^4D♯, ^12E♭
43 129 ^5D♯, ^13E♭
44 132 41/38, 68/63 ^6D♯, ^14E♭
45 135 40/37, 93/86 ^7D♯, ^15E♭
46 138 ^8D♯, ^16E♭
47 141 51/47 ^9D♯, ^17E♭
48 144 ^10D♯, ^18E♭
49 147 49/45 ^11D♯, v19E
50 150 ^12D♯, v18E
51 153 ^13D♯, v17E
52 156 81/74, 93/85 ^14D♯, v16E
53 159 57/52 ^15D♯, v15E
54 162 56/51 ^16D♯, v14E
55 165 11/10 ^17D♯, v13E
56 168 54/49, 119/108 ^18D♯, v12E
57 171 85/77 ^19D♯, v11E
58 174 94/85, 115/104 v18D𝄪, v10E
59 177 72/65 v17D𝄪, v9E
60 180 91/82 v16D𝄪, v8E
61 183 v15D𝄪, v7E
62 186 49/44 v14D𝄪, v6E
63 189 29/26 v13D𝄪, v5E
64 192 105/94 v12D𝄪, v4E
65 195 47/42 v11D𝄪, v3E
66 198 37/33, 102/91 v10D𝄪, vvE
67 201 64/57 v9D𝄪, vE
68 204 9/8 E
69 207 62/55 ^E, ^9F♭
70 210 35/31 ^^E, ^10F♭
71 213 95/84 ^3E, ^11F♭
72 216 111/98 ^4E, ^12F♭
73 219 42/37 ^5E, ^13F♭
74 222 108/95 ^6E, ^14F♭
75 225 41/36 ^7E, ^15F♭
76 228 ^8E, ^16F♭
77 231 8/7 ^9E, ^17F♭
78 234 87/76 ^10E, ^18F♭
79 237 86/75 ^11E, v19F
80 240 54/47, 85/74 ^12E, v18F
81 243 ^13E, v17F
82 246 98/85 ^14E, v16F
83 249 ^15E, v15F
84 252 ^16E, v14F
85 255 95/82 ^17E, v13F
86 258 65/56, 94/81 ^18E, v12F
87 261 50/43, 93/80 ^19E, v11F
88 264 99/85 v18E♯, v10F
89 267 7/6 v17E♯, v9F
90 270 90/77 v16E♯, v8F
91 273 48/41 v15E♯, v7F
92 276 95/81 v14E♯, v6F
93 279 47/40, 74/63 v13E♯, v5F
94 282 v12E♯, v4F
95 285 112/95 v11E♯, v3F
96 288 111/94 v10E♯, vvF
97 291 110/93 v9E♯, vF
98 294 32/27 F
99 297 108/91 ^F, ^9G♭♭
100 300 44/37 ^^F, ^10G♭♭
101 303 56/47, 81/68 ^3F, ^11G♭♭
102 306 37/31, 105/88 ^4F, ^12G♭♭
103 309 49/41, 104/87 ^5F, ^13G♭♭
104 312 91/76 ^6F, ^14G♭♭
105 315 ^7F, ^15G♭♭
106 318 ^8F, ^16G♭♭
107 321 65/54 ^9F, ^17G♭♭
108 324 41/34 ^10F, ^18G♭♭
109 327 93/77 ^11F, v19G♭
110 330 75/62, 98/81 ^12F, v18G♭
111 333 40/33 ^13F, v17G♭
112 336 17/14 ^14F, v16G♭
113 339 45/37 ^15F, v15G♭
114 342 39/32 ^16F, v14G♭
115 345 94/77 ^17F, v13G♭
116 348 ^18F, v12G♭
117 351 49/40, 60/49 ^19F, v11G♭
118 354 v18F♯, v10G♭
119 357 v17F♯, v9G♭
120 360 v16F♯, v8G♭
121 363 37/30 v15F♯, v7G♭
122 366 21/17 v14F♯, v6G♭
123 369 99/80 v13F♯, v5G♭
124 372 31/25, 119/96 v12F♯, v4G♭
125 375 77/62 v11F♯, v3G♭
126 378 51/41 v10F♯, vvG♭
127 381 81/65 v9F♯, vG♭
128 384 v8F♯, G♭
129 387 v7F♯, ^G♭
130 390 114/91, 119/95 v6F♯, ^^G♭
131 393 64/51 v5F♯, ^3G♭
132 396 44/35, 93/74 v4F♯, ^4G♭
133 399 34/27 v3F♯, ^5G♭
134 402 82/65, 111/88 vvF♯, ^6G♭
135 405 91/72, 115/91 vF♯, ^7G♭
136 408 81/64, 119/94 F♯, ^8G♭
137 411 52/41 ^F♯, ^9G♭
138 414 47/37, 80/63 ^^F♯, ^10G♭
139 417 ^3F♯, ^11G♭
140 420 65/51, 116/91 ^4F♯, ^12G♭
141 423 60/47 ^5F♯, ^13G♭
142 426 55/43 ^6F♯, ^14G♭
143 429 41/32 ^7F♯, ^15G♭
144 432 77/60, 95/74 ^8F♯, ^16G♭
145 435 9/7 ^9F♯, ^17G♭
146 438 85/66 ^10F♯, ^18G♭
147 441 40/31 ^11F♯, v19G
148 444 84/65 ^12F♯, v18G
149 447 ^13F♯, v17G
150 450 ^14F♯, v16G
151 453 ^15F♯, v15G
152 456 82/63 ^16F♯, v14G
153 459 ^17F♯, v13G
154 462 47/36, 64/49, 111/85 ^18F♯, v12G
155 465 ^19F♯, v11G
156 468 38/29 v18F𝄪, v10G
157 471 21/16 v17F𝄪, v9G
158 474 v16F𝄪, v8G
159 477 54/41 v15F𝄪, v7G
160 480 62/47, 95/72 v14F𝄪, v6G
161 483 37/28, 115/87 v13F𝄪, v5G
162 486 49/37 v12F𝄪, v4G
163 489 65/49 v11F𝄪, v3G
164 492 93/70 v10F𝄪, vvG
165 495 v9F𝄪, vG
166 498 4/3 G
167 501 ^G, ^9A♭♭
168 504 91/68, 99/74 ^^G, ^10A♭♭
169 507 63/47 ^3G, ^11A♭♭
170 510 47/35 ^4G, ^12A♭♭
171 513 39/29 ^5G, ^13A♭♭
172 516 66/49 ^6G, ^14A♭♭
173 519 85/63 ^7G, ^15A♭♭
174 522 119/88 ^8G, ^16A♭♭
175 525 65/48 ^9G, ^17A♭♭
176 528 ^10G, ^18A♭♭
177 531 ^11G, v19A♭
178 534 49/36, 64/47 ^12G, v18A♭
179 537 15/11 ^13G, v17A♭
180 540 56/41 ^14G, v16A♭
181 543 26/19 ^15G, v15A♭
182 546 85/62 ^16G, v14A♭
183 549 ^17G, v13A♭
184 552 ^18G, v12A♭
185 555 62/45 ^19G, v11A♭
186 558 v18G♯, v10A♭
187 561 47/34, 65/47, 112/81 v17G♯, v9A♭
188 564 v16G♯, v8A♭
189 567 68/49, 111/80 v15G♯, v7A♭
190 570 57/41 v14G♯, v6A♭
191 573 v13G♯, v5A♭
192 576 v12G♯, v4A♭
193 579 88/63, 95/68 v11G♯, v3A♭
194 582 v10G♯, vvA♭
195 585 v9G♯, vA♭
196 588 66/47 v8G♯, A♭
197 591 v7G♯, ^A♭
198 594 31/22 v6G♯, ^^A♭
199 597 24/17 v5G♯, ^3A♭
200 600 99/70 v4G♯, ^4A♭
201 603 17/12 v3G♯, ^5A♭
202 606 44/31, 105/74 vvG♯, ^6A♭
203 609 91/64 vG♯, ^7A♭
204 612 47/33 G♯, ^8A♭
205 615 117/82 ^G♯, ^9A♭
206 618 ^^G♯, ^10A♭
207 621 63/44 ^3G♯, ^11A♭
208 624 ^4G♯, ^12A♭
209 627 ^5G♯, ^13A♭
210 630 82/57 ^6G♯, ^14A♭
211 633 49/34, 111/77 ^7G♯, ^15A♭
212 636 ^8G♯, ^16A♭
213 639 68/47, 81/56, 94/65 ^9G♯, ^17A♭
214 642 ^10G♯, ^18A♭
215 645 45/31, 119/82 ^11G♯, v19A
216 648 ^12G♯, v18A
217 651 ^13G♯, v17A
218 654 ^14G♯, v16A
219 657 19/13 ^15G♯, v15A
220 660 41/28 ^16G♯, v14A
221 663 22/15 ^17G♯, v13A
222 666 47/32, 72/49, 119/81 ^18G♯, v12A
223 669 ^19G♯, v11A
224 672 115/78 v18G𝄪, v10A
225 675 96/65 v17G𝄪, v9A
226 678 v16G𝄪, v8A
227 681 v15G𝄪, v7A
228 684 49/33, 95/64 v14G𝄪, v6A
229 687 58/39, 119/80 v13G𝄪, v5A
230 690 70/47 v12G𝄪, v4A
231 693 94/63 v11G𝄪, v3A
232 696 v10G𝄪, vvA
233 699 v9G𝄪, vA
234 702 3/2 A
235 705 ^A, ^9B♭♭
236 708 ^^A, ^10B♭♭
237 711 95/63, 98/65 ^3A, ^11B♭♭
238 714 74/49 ^4A, ^12B♭♭
239 717 56/37, 115/76 ^5A, ^13B♭♭
240 720 47/31 ^6A, ^14B♭♭
241 723 41/27 ^7A, ^15B♭♭
242 726 ^8A, ^16B♭♭
243 729 32/21 ^9A, ^17B♭♭
244 732 29/19 ^10A, ^18B♭♭
245 735 ^11A, v19B♭
246 738 49/32, 72/47 ^12A, v18B♭
247 741 ^13A, v17B♭
248 744 63/41 ^14A, v16B♭
249 747 117/76 ^15A, v15B♭
250 750 ^16A, v14B♭
251 753 ^17A, v13B♭
252 756 65/42 ^18A, v12B♭
253 759 31/20 ^19A, v11B♭
254 762 v18A♯, v10B♭
255 765 14/9 v17A♯, v9B♭
256 768 120/77 v16A♯, v8B♭
257 771 64/41 v15A♯, v7B♭
258 774 86/55 v14A♯, v6B♭
259 777 47/30 v13A♯, v5B♭
260 780 91/58, 102/65 v12A♯, v4B♭
261 783 v11A♯, v3B♭
262 786 63/40, 74/47 v10A♯, vvB♭
263 789 41/26 v9A♯, vB♭
264 792 v8A♯, B♭
265 795 v7A♯, ^B♭
266 798 65/41, 111/70 v6A♯, ^^B♭
267 801 27/17 v5A♯, ^3B♭
268 804 35/22 v4A♯, ^4B♭
269 807 51/32 v3A♯, ^5B♭
270 810 91/57, 99/62 vvA♯, ^6B♭
271 813 vA♯, ^7B♭
272 816 A♯, ^8B♭
273 819 ^A♯, ^9B♭
274 822 82/51, 119/74 ^^A♯, ^10B♭
275 825 ^3A♯, ^11B♭
276 828 50/31 ^4A♯, ^12B♭
277 831 ^5A♯, ^13B♭
278 834 34/21 ^6A♯, ^14B♭
279 837 60/37 ^7A♯, ^15B♭
280 840 ^8A♯, ^16B♭
281 843 ^9A♯, ^17B♭
282 846 ^10A♯, ^18B♭
283 849 49/30, 80/49 ^11A♯, v19B
284 852 ^12A♯, v18B
285 855 77/47 ^13A♯, v17B
286 858 64/39 ^14A♯, v16B
287 861 74/45 ^15A♯, v15B
288 864 28/17 ^16A♯, v14B
289 867 33/20 ^17A♯, v13B
290 870 81/49, 119/72 ^18A♯, v12B
291 873 ^19A♯, v11B
292 876 68/41 v18A𝄪, v10B
293 879 108/65 v17A𝄪, v9B
294 882 v16A𝄪, v8B
295 885 v15A𝄪, v7B
296 888 v14A𝄪, v6B
297 891 82/49, 87/52 v13A𝄪, v5B
298 894 62/37 v12A𝄪, v4B
299 897 47/28 v11A𝄪, v3B
300 900 37/22 v10A𝄪, vvB
301 903 91/54 v9A𝄪, vB
302 906 27/16 B
303 909 93/55 ^B, ^9C♭
304 912 105/62 ^^B, ^10C♭
305 915 95/56 ^3B, ^11C♭
306 918 ^4B, ^12C♭
307 921 63/37, 80/47 ^5B, ^13C♭
308 924 ^6B, ^14C♭
309 927 41/24 ^7B, ^15C♭
310 930 77/45 ^8B, ^16C♭
311 933 12/7 ^9B, ^17C♭
312 936 ^10B, ^18C♭
313 939 43/25, 117/68 ^11B, v19C
314 942 81/47, 112/65 ^12B, v18C
315 945 ^13B, v17C
316 948 ^14B, v16C
317 951 ^15B, v15C
318 954 85/49 ^16B, v14C
319 957 ^17B, v13C
320 960 47/27 ^18B, v12C
321 963 75/43 ^19B, v11C
322 966 v18B♯, v10C
323 969 7/4 v17B♯, v9C
324 972 v16B♯, v8C
325 975 72/41 v15B♯, v7C
326 978 95/54 v14B♯, v6C
327 981 37/21 v13B♯, v5C
328 984 v12B♯, v4C
329 987 v11B♯, v3C
330 990 62/35 v10B♯, vvC
331 993 55/31 v9B♯, vC
332 996 16/9 C
333 999 57/32 ^C, ^9D♭♭
334 1002 66/37, 91/51 ^^C, ^10D♭♭
335 1005 84/47 ^3C, ^11D♭♭
336 1008 111/62 ^4C, ^12D♭♭
337 1011 52/29 ^5C, ^13D♭♭
338 1014 88/49 ^6C, ^14D♭♭
339 1017 ^7C, ^15D♭♭
340 1020 ^8C, ^16D♭♭
341 1023 65/36 ^9C, ^17D♭♭
342 1026 85/47 ^10C, ^18D♭♭
343 1029 ^11C, v19D♭
344 1032 49/27 ^12C, v18D♭
345 1035 20/11 ^13C, v17D♭
346 1038 51/28 ^14C, v16D♭
347 1041 104/57 ^15C, v15D♭
348 1044 117/64 ^16C, v14D♭
349 1047 119/65 ^17C, v13D♭
350 1050 ^18C, v12D♭
351 1053 90/49 ^19C, v11D♭
352 1056 v18C♯, v10D♭
353 1059 94/51 v17C♯, v9D♭
354 1062 v16C♯, v8D♭
355 1065 37/20 v15C♯, v7D♭
356 1068 63/34, 76/41 v14C♯, v6D♭
357 1071 v13C♯, v5D♭
358 1074 93/50, 119/64 v12C♯, v4D♭
359 1077 95/51 v11C♯, v3D♭
360 1080 v10C♯, vvD♭
361 1083 v9C♯, vD♭
362 1086 88/47 v8C♯, D♭
363 1089 v7C♯, ^D♭
364 1092 62/33 v6C♯, ^^D♭
365 1095 32/17 v5C♯, ^3D♭
366 1098 66/35 v4C♯, ^4D♭
367 1101 17/9 v3C♯, ^5D♭
368 1104 70/37 vvC♯, ^6D♭
369 1107 91/48 vC♯, ^7D♭
370 1110 C♯, ^8D♭
371 1113 78/41 ^C♯, ^9D♭
372 1116 40/21 ^^C♯, ^10D♭
373 1119 21/11 ^3C♯, ^11D♭
374 1122 65/34 ^4C♯, ^12D♭
375 1125 90/47 ^5C♯, ^13D♭
376 1128 94/49 ^6C♯, ^14D♭
377 1131 98/51 ^7C♯, ^15D♭
378 1134 77/40 ^8C♯, ^16D♭
379 1137 27/14 ^9C♯, ^17D♭
380 1140 85/44 ^10C♯, ^18D♭
381 1143 60/31 ^11C♯, v19D
382 1146 95/49 ^12C♯, v18D
383 1149 ^13C♯, v17D
384 1152 ^14C♯, v16D
385 1155 76/39 ^15C♯, v15D
386 1158 41/21 ^16C♯, v14D
387 1161 88/45 ^17C♯, v13D
388 1164 47/24, 96/49 ^18C♯, v12D
389 1167 ^19C♯, v11D
390 1170 57/29 v18C𝄪, v10D
391 1173 63/32 v17C𝄪, v9D
392 1176 v16C𝄪, v8D
393 1179 81/41 v15C𝄪, v7D
394 1182 95/48 v14C𝄪, v6D
395 1185 115/58 v13C𝄪, v5D
396 1188 v12C𝄪, v4D
397 1191 v11C𝄪, v3D
398 1194 v10C𝄪, vvD
399 1197 v9C𝄪, vD
400 1200 2/1 D

Selected intervals

Step Eliora's Naming System Associated Ratio
0 unison 1/1
28 5/12-meantone semitone 6561/6250
33 small septendecimal semitone 18/17, 55/52
35 septendecimal semitone 17/16
37 diatonic semitone 16/15
99 undevicesimal minor third 19/16
100 symmetric minor third
200 symmetric tritone 99/70, 140/99
231 Gregorian leap week fifth 525/352, 3/2 / (81/80)^(5/12)
234 perfect fifth 3/2
323 harmonic seventh 7/4
372 5/12-meantone seventh 12500/6561
400 octave 2/1

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-7 22 -12⟩, [47 -15 -10⟩ [⟨400 634 929]] -0.1080 0.1331 4.44
2.3.5.7 2401/2400, 1959552/1953125, 14348907/14336000 [⟨400 634 929 1123]] -0.0965 0.1170 3.90
2.3.5.7.11 2401/2400, 5632/5625, 9801/9800, 46656/46585 [⟨400 634 929 1123 1384]] -0.1166 0.1121 3.74
2.3.5.7.11.13 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325 [⟨400 634 929 1123 1384 1480]] -0.0734 0.1407 4.69
2.3.5.7.11.13.17 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095 [⟨400 634 929 1123 1384 1480 1635]] -0.0645 0.1321 4.40
2.3.5.7.11.13.17.19 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715 [⟨400 634 929 1123 1384 1480 1635 1699]] -0.0413 0.1380 4.60
  • 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat 354 in the 17-limit, and 311 in the 19-limit; it is bettered by 422 in either subgroup.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 83\400 249.00 [-26 18 -1⟩ Monzismic
1 33\400 99.00 18/17 Gregorian leap day
1 101\400 303.00 25/21 Quinmite
1 153\400 459.00 125/96 Majvamic
1 169\400 507.00 525/352 Gregorian leap week
2 61\400 183.00 10/9 Unidecmic
5 123\400
(37\400) 		369.00
(111.00) 		1024/891
(16/15) 		Quintosec
10 83\400
(3\400) 		249.00
(9.00) 		15/13
(176/175) 		Decoid
80 166\400
(1\400) 		498.00
(3.00) 		4/3
(245/243) 		Octogintic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

Music

Eliora
Francium
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