320edo

← 319edo

320edo

321edo →

Prime factorization	26 × 5
Step size	3.75 ¢
Fifth	187\320 (701.25 ¢)
Semitones (A1:m2)	29:25 (108.8 ¢ : 93.75 ¢)
Consistency limit	19
Distinct consistency limit	19

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	3.75		^D, ^5E♭♭
2	7.5		^^D, ^6E♭♭
3	11.25		^3D, ^7E♭♭
4	15		^4D, ^8E♭♭
5	18.75	91/90, 94/93, 95/94, 96/95	^5D, ^9E♭♭
6	22.5	76/75, 77/76, 78/77	^6D, ^10E♭♭
7	26.25	65/64, 66/65	^7D, ^11E♭♭
8	30		^8D, ^12E♭♭
9	33.75	51/50, 52/51	^9D, ^13E♭♭
10	37.5	93/91	^10D, ^14E♭♭
11	41.25	42/41, 43/42	^11D, v14E♭
12	45	39/38, 77/75	^12D, v13E♭
13	48.75	36/35	^13D, v12E♭
14	52.5		^14D, v11E♭
15	56.25	31/30, 94/91	v14D♯, v10E♭
16	60	88/85	v13D♯, v9E♭
17	63.75		v12D♯, v8E♭
18	67.5	26/25	v11D♯, v7E♭
19	71.25	25/24, 99/95	v10D♯, v6E♭
20	75	47/45, 95/91	v9D♯, v5E♭
21	78.75	45/43, 68/65	v8D♯, v4E♭
22	82.5	43/41	v7D♯, v3E♭
23	86.25	41/39	v6D♯, vvE♭
24	90	99/94	v5D♯, vE♭
25	93.75	19/18	v4D♯, E♭
26	97.5	55/52, 91/86	v3D♯, ^E♭
27	101.25	35/33	vvD♯, ^^E♭
28	105	17/16	vD♯, ^3E♭
29	108.75	33/31, 82/77	D♯, ^4E♭
30	112.5		^D♯, ^5E♭
31	116.25	77/72	^^D♯, ^6E♭
32	120	15/14	^3D♯, ^7E♭
33	123.75		^4D♯, ^8E♭
34	127.5		^5D♯, ^9E♭
35	131.25	41/38, 55/51	^6D♯, ^10E♭
36	135	40/37, 93/86	^7D♯, ^11E♭
37	138.75	13/12	^8D♯, ^12E♭
38	142.5	38/35	^9D♯, ^13E♭
39	146.25	37/34, 99/91	^10D♯, ^14E♭
40	150	12/11	^11D♯, v14E
41	153.75	47/43	^12D♯, v13E
42	157.5		^13D♯, v12E
43	161.25	45/41	^14D♯, v11E
44	165	11/10	v14D𝄪, v10E
45	168.75	43/39, 54/49	v13D𝄪, v9E
46	172.5	95/86	v12D𝄪, v8E
47	176.25	31/28	v11D𝄪, v7E
48	180	91/82	v10D𝄪, v6E
49	183.75		v9D𝄪, v5E
50	187.5	39/35	v8D𝄪, v4E
51	191.25	86/77	v7D𝄪, v3E
52	195	47/42	v6D𝄪, vvE
53	198.75		v5D𝄪, vE
54	202.5		E
55	206.25		^E, ^5F♭
56	210	35/31	^^E, ^6F♭
57	213.75	43/38	^3E, ^7F♭
58	217.5	93/82	^4E, ^8F♭
59	221.25	25/22	^5E, ^9F♭
60	225	41/36, 74/65	^6E, ^10F♭
61	228.75		^7E, ^11F♭
62	232.5		^8E, ^12F♭
63	236.25	47/41, 55/48	^9E, ^13F♭
64	240	54/47, 85/74	^10E, ^14F♭
65	243.75	38/33, 99/86	^11E, v14F
66	247.5	15/13	^12E, v13F
67	251.25	37/32	^13E, v12F
68	255	51/44, 95/82	^14E, v11F
69	258.75	36/31	v14E♯, v10F
70	262.5	64/55	v13E♯, v9F
71	266.25	7/6	v12E♯, v8F
72	270	90/77	v11E♯, v7F
73	273.75	41/35	v10E♯, v6F
74	277.5		v9E♯, v5F
75	281.25	20/17	v8E♯, v4F
76	285	33/28	v7E♯, v3F
77	288.75	13/11	v6E♯, vvF
78	292.5	45/38	v5E♯, vF
79	296.25		F
80	300	44/37	^F, ^5G♭♭
81	303.75	56/47	^^F, ^6G♭♭
82	307.5	43/36	^3F, ^7G♭♭
83	311.25	91/76	^4F, ^8G♭♭
84	315		^5F, ^9G♭♭
85	318.75		^6F, ^10G♭♭
86	322.5	47/39	^7F, ^11G♭♭
87	326.25	93/77, 99/82	^8F, ^12G♭♭
88	330	75/62, 98/81	^9F, ^13G♭♭
89	333.75	57/47	^10F, ^14G♭♭
90	337.5	62/51	^11F, v14G♭
91	341.25	95/78	^12F, v13G♭
92	345	94/77	^13F, v12G♭
93	348.75		^14F, v11G♭
94	352.5	38/31	v14F♯, v10G♭
95	356.25	43/35, 70/57	v13F♯, v9G♭
96	360	16/13	v12F♯, v8G♭
97	363.75	95/77	v11F♯, v7G♭
98	367.5	47/38, 68/55	v10F♯, v6G♭
99	371.25		v9F♯, v5G♭
100	375	77/62	v8F♯, v4G♭
101	378.75	56/45	v7F♯, v3G♭
102	382.5		v6F♯, vvG♭
103	386.25	5/4	v5F♯, vG♭
104	390		v4F♯, G♭
105	393.75	54/43	v3F♯, ^G♭
106	397.5	39/31	vvF♯, ^^G♭
107	401.25	29/23	vF♯, ^3G♭
108	405	24/19, 91/72	F♯, ^4G♭
109	408.75	19/15	^F♯, ^5G♭
110	412.5	33/26	^^F♯, ^6G♭
111	416.25		^3F♯, ^7G♭
112	420	51/40, 65/51	^4F♯, ^8G♭
113	423.75		^5F♯, ^9G♭
114	427.5	32/25	^6F♯, ^10G♭
115	431.25	77/60	^7F♯, ^11G♭
116	435	9/7	^8F♯, ^12G♭
117	438.75		^9F♯, ^13G♭
118	442.5	31/24	^10F♯, ^14G♭
119	446.25	22/17	^11F♯, v14G
120	450	48/37	^12F♯, v13G
121	453.75	13/10	^13F♯, v12G
122	457.5	56/43, 99/76	^14F♯, v11G
123	461.25	47/36	v14F𝄪, v10G
124	465	17/13	v13F𝄪, v9G
125	468.75		v12F𝄪, v8G
126	472.5		v11F𝄪, v7G
127	476.25	54/41	v10F𝄪, v6G
128	480	62/47, 95/72	v9F𝄪, v5G
129	483.75	41/31	v8F𝄪, v4G
130	487.5	57/43	v7F𝄪, v3G
131	491.25	85/64, 93/70	v6F𝄪, vvG
132	495		v5F𝄪, vG
133	498.75		G
134	502.5		^G, ^5A♭♭
135	506.25	75/56	^^G, ^6A♭♭
136	510	47/35, 51/38	^3G, ^7A♭♭
137	513.75	74/55	^4G, ^8A♭♭
138	517.5		^5G, ^9A♭♭
139	521.25	50/37, 77/57	^6G, ^10A♭♭
140	525	65/48, 88/65	^7G, ^11A♭♭
141	528.75	19/14	^8G, ^12A♭♭
142	532.5	34/25	^9G, ^13A♭♭
143	536.25		^10G, ^14A♭♭
144	540	56/41	^11G, v14A♭
145	543.75		^12G, v13A♭
146	547.5		^13G, v12A♭
147	551.25	11/8	^14G, v11A♭
148	555	51/37, 62/45	v14G♯, v10A♭
149	558.75		v13G♯, v9A♭
150	562.5		v12G♯, v8A♭
151	566.25	43/31	v11G♯, v7A♭
152	570	57/41	v10G♯, v6A♭
153	573.75	39/28	v9G♯, v5A♭
154	577.5		v8G♯, v4A♭
155	581.25		v7G♯, v3A♭
156	585		v6G♯, vvA♭
157	588.75	52/37	v5G♯, vA♭
158	592.5		v4G♯, A♭
159	596.25		v3G♯, ^A♭
160	600	99/70	vvG♯, ^^A♭
161	603.75		vG♯, ^3A♭
162	607.5		G♯, ^4A♭
163	611.25	37/26	^G♯, ^5A♭
164	615		^^G♯, ^6A♭
165	618.75		^3G♯, ^7A♭
166	622.5		^4G♯, ^8A♭
167	626.25	56/39	^5G♯, ^9A♭
168	630	82/57, 95/66	^6G♯, ^10A♭
169	633.75	62/43, 75/52	^7G♯, ^11A♭
170	637.5		^8G♯, ^12A♭
171	641.25		^9G♯, ^13A♭
172	645	45/31, 74/51	^10G♯, ^14A♭
173	648.75	16/11	^11G♯, v14A
174	652.5		^12G♯, v13A
175	656.25		^13G♯, v12A
176	660	41/28	^14G♯, v11A
177	663.75	91/62	v14G𝄪, v10A
178	667.5	25/17	v13G𝄪, v9A
179	671.25	28/19	v12G𝄪, v8A
180	675	65/44, 96/65	v11G𝄪, v7A
181	678.75	37/25	v10G𝄪, v6A
182	682.5		v9G𝄪, v5A
183	686.25	55/37	v8G𝄪, v4A
184	690	70/47, 76/51	v7G𝄪, v3A
185	693.75		v6G𝄪, vvA
186	697.5		v5G𝄪, vA
187	701.25		A
188	705		^A, ^5B♭♭
189	708.75		^^A, ^6B♭♭
190	712.5	86/57	^3A, ^7B♭♭
191	716.25	62/41	^4A, ^8B♭♭
192	720	47/31	^5A, ^9B♭♭
193	723.75	41/27	^6A, ^10B♭♭
194	727.5		^7A, ^11B♭♭
195	731.25		^8A, ^12B♭♭
196	735	26/17	^9A, ^13B♭♭
197	738.75	72/47, 95/62	^10A, ^14B♭♭
198	742.5	43/28	^11A, v14B♭
199	746.25	20/13	^12A, v13B♭
200	750	37/24	^13A, v12B♭
201	753.75	17/11	^14A, v11B♭
202	757.5	48/31	v14A♯, v10B♭
203	761.25		v13A♯, v9B♭
204	765	14/9	v12A♯, v8B♭
205	768.75		v11A♯, v7B♭
206	772.5	25/16	v10A♯, v6B♭
207	776.25		v9A♯, v5B♭
208	780	80/51	v8A♯, v4B♭
209	783.75		v7A♯, v3B♭
210	787.5	52/33	v6A♯, vvB♭
211	791.25	30/19	v5A♯, vB♭
212	795	19/12	v4A♯, B♭
213	798.75	46/29	v3A♯, ^B♭
214	802.5	62/39	vvA♯, ^^B♭
215	806.25	43/27	vA♯, ^3B♭
216	810	91/57, 99/62	A♯, ^4B♭
217	813.75	8/5	^A♯, ^5B♭
218	817.5		^^A♯, ^6B♭
219	821.25	45/28	^3A♯, ^7B♭
220	825		^4A♯, ^8B♭
221	828.75		^5A♯, ^9B♭
222	832.5	55/34, 76/47	^6A♯, ^10B♭
223	836.25		^7A♯, ^11B♭
224	840	13/8	^8A♯, ^12B♭
225	843.75	57/35, 70/43	^9A♯, ^13B♭
226	847.5	31/19	^10A♯, ^14B♭
227	851.25	85/52	^11A♯, v14B
228	855	77/47	^12A♯, v13B
229	858.75		^13A♯, v12B
230	862.5	51/31	^14A♯, v11B
231	866.25	94/57	v14A𝄪, v10B
232	870	81/49	v13A𝄪, v9B
233	873.75		v12A𝄪, v8B
234	877.5	78/47	v11A𝄪, v7B
235	881.25		v10A𝄪, v6B
236	885		v9A𝄪, v5B
237	888.75		v8A𝄪, v4B
238	892.5	72/43	v7A𝄪, v3B
239	896.25	47/28	v6A𝄪, vvB
240	900	37/22	v5A𝄪, vB
241	903.75	91/54	B
242	907.5	76/45	^B, ^5C♭
243	911.25	22/13	^^B, ^6C♭
244	915	56/33, 95/56	^3B, ^7C♭
245	918.75	17/10	^4B, ^8C♭
246	922.5		^5B, ^9C♭
247	926.25	70/41	^6B, ^10C♭
248	930	77/45	^7B, ^11C♭
249	933.75	12/7	^8B, ^12C♭
250	937.5	55/32	^9B, ^13C♭
251	941.25	31/18	^10B, ^14C♭
252	945	88/51	^11B, v14C
253	948.75	64/37	^12B, v13C
254	952.5	26/15	^13B, v12C
255	956.25	33/19	^14B, v11C
256	960	47/27	v14B♯, v10C
257	963.75	82/47, 96/55	v13B♯, v9C
258	967.5		v12B♯, v8C
259	971.25		v11B♯, v7C
260	975	65/37, 72/41	v10B♯, v6C
261	978.75	44/25	v9B♯, v5C
262	982.5		v8B♯, v4C
263	986.25	76/43, 99/56	v7B♯, v3C
264	990	62/35	v6B♯, vvC
265	993.75		v5B♯, vC
266	997.5		C
267	1001.25		^C, ^5D♭♭
268	1005	84/47	^^C, ^6D♭♭
269	1008.75	77/43	^3C, ^7D♭♭
270	1012.5	70/39	^4C, ^8D♭♭
271	1016.25		^5C, ^9D♭♭
272	1020		^6C, ^10D♭♭
273	1023.75	56/31	^7C, ^11D♭♭
274	1027.5		^8C, ^12D♭♭
275	1031.25	49/27, 78/43	^9C, ^13D♭♭
276	1035	20/11	^10C, ^14D♭♭
277	1038.75	82/45	^11C, v14D♭
278	1042.5		^12C, v13D♭
279	1046.25	86/47	^13C, v12D♭
280	1050	11/6	^14C, v11D♭
281	1053.75	68/37	v14C♯, v10D♭
282	1057.5	35/19	v13C♯, v9D♭
283	1061.25	24/13	v12C♯, v8D♭
284	1065	37/20	v11C♯, v7D♭
285	1068.75	76/41	v10C♯, v6D♭
286	1072.5		v9C♯, v5D♭
287	1076.25		v8C♯, v4D♭
288	1080	28/15	v7C♯, v3D♭
289	1083.75		v6C♯, vvD♭
290	1087.5		v5C♯, vD♭
291	1091.25	62/33, 77/41	v4C♯, D♭
292	1095	32/17	v3C♯, ^D♭
293	1098.75	66/35	vvC♯, ^^D♭
294	1102.5		vC♯, ^3D♭
295	1106.25	36/19	C♯, ^4D♭
296	1110		^C♯, ^5D♭
297	1113.75	78/41	^^C♯, ^6D♭
298	1117.5	82/43	^3C♯, ^7D♭
299	1121.25	65/34, 86/45	^4C♯, ^8D♭
300	1125	90/47	^5C♯, ^9D♭
301	1128.75	48/25	^6C♯, ^10D♭
302	1132.5	25/13	^7C♯, ^11D♭
303	1136.25		^8C♯, ^12D♭
304	1140	85/44	^9C♯, ^13D♭
305	1143.75	60/31, 91/47	^10C♯, ^14D♭
306	1147.5		^11C♯, v14D
307	1151.25	35/18	^12C♯, v13D
308	1155	76/39	^13C♯, v12D
309	1158.75	41/21, 84/43	^14C♯, v11D
310	1162.5		v14C𝄪, v10D
311	1166.25	51/26, 100/51	v13C𝄪, v9D
312	1170		v12C𝄪, v8D
313	1173.75	65/33	v11C𝄪, v7D
314	1177.5	75/38, 77/39	v10C𝄪, v6D
315	1181.25	93/47, 95/48	v9C𝄪, v5D
316	1185		v8C𝄪, v4D
317	1188.75		v7C𝄪, v3D
318	1192.5		v6C𝄪, vvD
319	1196.25		v5C𝄪, vD
320	1200	2/1	D

Theory

320edo is consistent in the 19-odd-limit and a fairly good tuning for the 19-limit. It has a flat tendency for most prime harmonics from 3 to 19, with the sole exception of 17.

The equal temperament tempers out 65625/65536 (horwell comma) and 420175/419904 (wizma) in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports the varuna temperament, the rank-3 temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the optimal patent val. It also provides the optimal patent val for the rank-4 werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to an extension of varuna.

Prime harmonics

Approximation of prime harmonics in 320edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.71	-0.06	-1.33	-0.07	-0.53	+0.04	-1.26	+1.73	+1.67	-1.29
	Relative (%)	+0.0	-18.8	-1.7	-35.4	-1.8	-14.1	+1.2	-33.7	+46.0	+44.6	-34.3
Steps (reduced)		320 (0)	507 (187)	743 (103)	898 (258)	1107 (147)	1184 (224)	1308 (28)	1359 (79)	1448 (168)	1555 (275)	1585 (305)

Subsets and supersets

Since 320 factors into 2⁶ × 5, 320edo has subset edos 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-507 320⟩	[⟨320 507]]	+0.2224	0.2224	5.93
2.3.5	[23 6 -14⟩, [-28 25 -5⟩	[⟨320 507 743]]	+0.1574	0.2036	5.43
2.3.5.7	65625/65536, 235298/234375, 321489/320000	[⟨320 507 743 898]]	+0.2361	0.2229	5.94
2.3.5.7.11	441/440, 8019/8000, 41503/41472, 65625/65536	[⟨320 507 743 898 1107]]	+0.1928	0.2173	5.80
2.3.5.7.11.13	441/440, 729/728, 1001/1000, 4225/4224, 6656/6655	[⟨320 507 743 898 1107 1184]]	+0.1845	0.1993	5.31
2.3.5.7.11.13.17	441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224	[⟨320 507 743 898 1107 1184 1308]]	+0.1565	0.1968	5.25
2.3.5.7.11.13.17.19	441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520	[⟨320 507 743 898 1107 1184 1308 1359]]	+0.1741	0.1899	5.06

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	7\320	26.25	[-2 13 -8⟩	Sfourth (5-limit)
1	131\320	491.25	3645/2744	Fifthplus
1	157\320	588.75	45/32	Untriton (5-limit)
1	93\320	348.75	6144/3757	Hectosaros leap week
2	19\320	71.25	25/24	Narayana
5	133\320 (5\320)	498.75 (18.75)	4/3 (81/80)	Pental
8	133\320 (9\320)	566.25 (33.75)	104/75 (55/54)	Octowerck
10	19\320 (13\320)	71.25 (48.75)	25/24 (36/35)	Decavish
10	133\320 (5\320)	498.75 (18.75)	4/3 (81/80)	Decal
20	151\320 (7\320)	566.25 (26.25)	165/119 (?)	Soviet ferris wheel
32	133\320 (3\320)	498.75 (11.25)	4/3 (?)	Bezique
80	99\320 (3\320)	371.25 (11.25)	2275/1836 (?)	Mercury
80	133\320 (1\320)	498.75 (3.75)	4/3 (245/243)	Octogintic

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