400edo

← 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

The 400 equal divisions of the octave (400edo), or the 400(-tone) equal temperament (400tet, 400et) when viewed from a regular temperament perspective, is the equal division of the octave into 400 parts of exact 3 cents each.

Theory

400edo is consistent in the 21-odd-limit. It tempers out the unidecma, [-7 22 -12⟩, and the qintosec 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

400 factors into 2⁴ × 5², with subset edos 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200. Notably, 200edo contains a semiconvergent approximation to 3/2.

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

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)	Cents (reduced)	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)	10125/8192 (16/15)	Qintosec (5-limit)
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

Scales

• Huntington7
• Huntington10
• Huntington17
• Monzismic[29]
• GregorianLeapWeek[71]
• ISOWeek[71]
• GregorianLeapDay[97]

Music

• Etude in Monzismic (Op. 2, No. 3) by Eliora