220edo

← 219edo

220edo

221edo →

Prime factorization	22 × 5 × 11
Step size	5.45455 ¢
Fifth	129\220 (703.636 ¢)
Semitones (A1:m2)	23:15 (125.5 ¢ : 81.82 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

Using the patent val, it tempers out [20 -17 3⟩ (rodan comma) and [23 6 -14⟩ (vishnuzma) in the 5-limit; 6144/6125, 10976/10935, and 390625/388962 in the 7-limit; 1331/1323, 1375/1372, 2200/2187, and 16384/16335 in the 11-limit; 325/324, 352/351, 1001/1000, 1573/1568, and 2704/2695 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 220edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+1.68	+0.96	+2.08	-2.09	-0.41	-0.53	+2.64	-1.32	+2.49	-1.69	-1.00
	Relative (%)	+30.8	+17.6	+38.2	-38.4	-7.5	-9.7	+48.4	-24.2	+45.6	-31.0	-18.4
Steps (reduced)		349 (129)	511 (71)	618 (178)	697 (37)	761 (101)	814 (154)	860 (200)	899 (19)	935 (55)	966 (86)	995 (115)

Subsets and supersets

Since 220 factors into 2² × 5 × 11, 220edo has subset edos 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	5.45		^D, ^9E♭♭
2	10.91		^^D, ^10E♭♭
3	16.36		^3D, ^11E♭♭
4	21.82		^4D, v11E♭
5	27.27	65/64, 66/65	^5D, v10E♭
6	32.73	52/51	^6D, v9E♭
7	38.18	47/46	^7D, v8E♭
8	43.64	40/39, 41/40	^8D, v7E♭
9	49.09	36/35	^9D, v6E♭
10	54.55	32/31	^10D, v5E♭
11	60	29/28	^11D, v4E♭
12	65.45		v11D♯, v3E♭
13	70.91	25/24	v10D♯, vvE♭
14	76.36	23/22	v9D♯, vE♭
15	81.82	43/41, 65/62	v8D♯, E♭
16	87.27	41/39	v7D♯, ^E♭
17	92.73	19/18, 58/55	v6D♯, ^^E♭
18	98.18		v5D♯, ^3E♭
19	103.64	69/65	v4D♯, ^4E♭
20	109.09	33/31	v3D♯, ^5E♭
21	114.55	31/29, 47/44	vvD♯, ^6E♭
22	120	15/14	vD♯, ^7E♭
23	125.45	43/40	D♯, ^8E♭
24	130.91	41/38, 55/51, 69/64	^D♯, ^9E♭
25	136.36		^^D♯, ^10E♭
26	141.82	38/35, 51/47	^3D♯, ^11E♭
27	147.27	37/34, 49/45	^4D♯, v11E
28	152.73		^5D♯, v10E
29	158.18		^6D♯, v9E
30	163.64		^7D♯, v8E
31	169.09	43/39, 54/49	^8D♯, v7E
32	174.55	52/47	^9D♯, v6E
33	180		^10D♯, v5E
34	185.45	69/62	^11D♯, v4E
35	190.91	48/43	v11D𝄪, v3E
36	196.36	28/25, 65/58	v10D𝄪, vvE
37	201.82		v9D𝄪, vE
38	207.27	62/55	E
39	212.73	26/23	^E, ^9F♭
40	218.18		^^E, ^10F♭
41	223.64	33/29, 58/51, 74/65	^3E, ^11F♭
42	229.09		^4E, v11F
43	234.55		^5E, v10F
44	240		^6E, v9F
45	245.45		^7E, v8F
46	250.91	37/32	^8E, v7F
47	256.36	29/25, 51/44	^9E, v6F
48	261.82	50/43, 57/49, 64/55	^10E, v5F
49	267.27	7/6	^11E, v4F
50	272.73	48/41, 55/47	v11E♯, v3F
51	278.18		v10E♯, vvF
52	283.64	33/28	v9E♯, vF
53	289.09	13/11	F
54	294.55	51/43	^F, ^9G♭♭
55	300	44/37, 69/58	^^F, ^10G♭♭
56	305.45	31/26, 37/31	^3F, ^11G♭♭
57	310.91		^4F, v11G♭
58	316.36	6/5	^5F, v10G♭
59	321.82		^6F, v9G♭
60	327.27	29/24	^7F, v8G♭
61	332.73	40/33	^8F, v7G♭
62	338.18	62/51	^9F, v6G♭
63	343.64	50/41	^10F, v5G♭
64	349.09		^11F, v4G♭
65	354.55		v11F♯, v3G♭
66	360	16/13	v10F♯, vvG♭
67	365.45		v9F♯, vG♭
68	370.91		v8F♯, G♭
69	376.36	41/33, 46/37	v7F♯, ^G♭
70	381.82		v6F♯, ^^G♭
71	387.27	5/4	v5F♯, ^3G♭
72	392.73	64/51, 69/55	v4F♯, ^4G♭
73	398.18	39/31	v3F♯, ^5G♭
74	403.64	24/19	vvF♯, ^6G♭
75	409.09	19/15	vF♯, ^7G♭
76	414.55	47/37	F♯, ^8G♭
77	420	51/40, 65/51	^F♯, ^9G♭
78	425.45	55/43	^^F♯, ^10G♭
79	430.91	50/39	^3F♯, ^11G♭
80	436.36		^4F♯, v11G
81	441.82	40/31	^5F♯, v10G
82	447.27	22/17	^6F♯, v9G
83	452.73		^7F♯, v8G
84	458.18	43/33, 56/43	^8F♯, v7G
85	463.64	17/13	^9F♯, v6G
86	469.09		^10F♯, v5G
87	474.55	25/19	^11F♯, v4G
88	480	33/25, 62/47	v11F𝄪, v3G
89	485.45		v10F𝄪, vvG
90	490.91		v9F𝄪, vG
91	496.36		G
92	501.82		^G, ^9A♭♭
93	507.27		^^G, ^10A♭♭
94	512.73	39/29, 74/55	^3G, ^11A♭♭
95	518.18	58/43	^4G, v11A♭
96	523.64	23/17	^5G, v10A♭
97	529.09	19/14	^6G, v9A♭
98	534.55	49/36, 64/47	^7G, v8A♭
99	540	41/30, 56/41	^8G, v7A♭
100	545.45		^9G, v6A♭
101	550.91	11/8	^10G, v5A♭
102	556.36	40/29, 51/37	^11G, v4A♭
103	561.82	65/47	v11G♯, v3A♭
104	567.27	43/31	v10G♯, vvA♭
105	572.73	39/28	v9G♯, vA♭
106	578.18		v8G♯, A♭
107	583.64		v7G♯, ^A♭
108	589.09	52/37	v6G♯, ^^A♭
109	594.55	31/22, 55/39	v5G♯, ^3A♭
110	600	41/29, 58/41	v4G♯, ^4A♭
111	605.45	44/31	v3G♯, ^5A♭
112	610.91	37/26	vvG♯, ^6A♭
113	616.36		vG♯, ^7A♭
114	621.82		G♯, ^8A♭
115	627.27	56/39	^G♯, ^9A♭
116	632.73	62/43	^^G♯, ^10A♭
117	638.18		^3G♯, ^11A♭
118	643.64	29/20, 74/51	^4G♯, v11A
119	649.09	16/11	^5G♯, v10A
120	654.55		^6G♯, v9A
121	660	41/28, 60/41	^7G♯, v8A
122	665.45	47/32, 69/47, 72/49	^8G♯, v7A
123	670.91	28/19	^9G♯, v6A
124	676.36	34/23, 65/44	^10G♯, v5A
125	681.82	43/29	^11G♯, v4A
126	687.27	55/37, 58/39	v11G𝄪, v3A
127	692.73		v10G𝄪, vvA
128	698.18		v9G𝄪, vA
129	703.64		A
130	709.09		^A, ^9B♭♭
131	714.55	65/43	^^A, ^10B♭♭
132	720	47/31, 50/33	^3A, ^11B♭♭
133	725.45	38/25	^4A, v11B♭
134	730.91		^5A, v10B♭
135	736.36	26/17, 75/49	^6A, v9B♭
136	741.82	43/28, 66/43	^7A, v8B♭
137	747.27		^8A, v7B♭
138	752.73	17/11	^9A, v6B♭
139	758.18	31/20	^10A, v5B♭
140	763.64		^11A, v4B♭
141	769.09	39/25	v11A♯, v3B♭
142	774.55		v10A♯, vvB♭
143	780		v9A♯, vB♭
144	785.45	74/47	v8A♯, B♭
145	790.91	30/19	v7A♯, ^B♭
146	796.36	19/12	v6A♯, ^^B♭
147	801.82	62/39	v5A♯, ^3B♭
148	807.27	51/32	v4A♯, ^4B♭
149	812.73	8/5	v3A♯, ^5B♭
150	818.18	69/43	vvA♯, ^6B♭
151	823.64	37/23, 66/41	vA♯, ^7B♭
152	829.09		A♯, ^8B♭
153	834.55		^A♯, ^9B♭
154	840	13/8	^^A♯, ^10B♭
155	845.45		^3A♯, ^11B♭
156	850.91		^4A♯, v11B
157	856.36	41/25	^5A♯, v10B
158	861.82	51/31	^6A♯, v9B
159	867.27	33/20	^7A♯, v8B
160	872.73	48/29	^8A♯, v7B
161	878.18		^9A♯, v6B
162	883.64	5/3	^10A♯, v5B
163	889.09		^11A♯, v4B
164	894.55	52/31, 62/37	v11A𝄪, v3B
165	900	37/22	v10A𝄪, vvB
166	905.45		v9A𝄪, vB
167	910.91	22/13	B
168	916.36	56/33	^B, ^9C♭
169	921.82		^^B, ^10C♭
170	927.27	41/24	^3B, ^11C♭
171	932.73	12/7	^4B, v11C
172	938.18	43/25, 55/32	^5B, v10C
173	943.64	50/29, 69/40	^6B, v9C
174	949.09	64/37	^7B, v8C
175	954.55		^8B, v7C
176	960		^9B, v6C
177	965.45		^10B, v5C
178	970.91		^11B, v4C
179	976.36	51/29, 58/33, 65/37	v11B♯, v3C
180	981.82		v10B♯, vvC
181	987.27	23/13	v9B♯, vC
182	992.73	55/31	C
183	998.18		^C, ^9D♭♭
184	1003.64	25/14	^^C, ^10D♭♭
185	1009.09	43/24	^3C, ^11D♭♭
186	1014.55		^4C, v11D♭
187	1020		^5C, v10D♭
188	1025.45	47/26	^6C, v9D♭
189	1030.91	49/27	^7C, v8D♭
190	1036.36		^8C, v7D♭
191	1041.82		^9C, v6D♭
192	1047.27		^10C, v5D♭
193	1052.73	68/37	^11C, v4D♭
194	1058.18	35/19	v11C♯, v3D♭
195	1063.64		v10C♯, vvD♭
196	1069.09		v9C♯, vD♭
197	1074.55		v8C♯, D♭
198	1080	28/15	v7C♯, ^D♭
199	1085.45	58/31	v6C♯, ^^D♭
200	1090.91	62/33	v5C♯, ^3D♭
201	1096.36		v4C♯, ^4D♭
202	1101.82		v3C♯, ^5D♭
203	1107.27	36/19, 55/29	vvC♯, ^6D♭
204	1112.73		vC♯, ^7D♭
205	1118.18		C♯, ^8D♭
206	1123.64	44/23	^C♯, ^9D♭
207	1129.09	48/25	^^C♯, ^10D♭
208	1134.55		^3C♯, ^11D♭
209	1140	56/29	^4C♯, v11D
210	1145.45	31/16	^5C♯, v10D
211	1150.91	35/18	^6C♯, v9D
212	1156.36	39/20	^7C♯, v8D
213	1161.82		^8C♯, v7D
214	1167.27	51/26	^9C♯, v6D
215	1172.73	65/33	^10C♯, v5D
216	1178.18		^11C♯, v4D
217	1183.64		v11C𝄪, v3D
218	1189.09		v10C𝄪, vvD
219	1194.55		v9C𝄪, vD
220	1200	2/1	D

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[349 -220⟩	[⟨220 349]]	−0.5304	0.5302	9.72
2.3.5	[20 -17 3⟩, [23 6 -14⟩	[⟨220 349 511]]	−0.4912	0.4364	8.00
2.3.5.7	6144/6125, 10976/10935, 390625/388962	[⟨220 349 511 618]]	−0.5538	0.3932	7.21

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	43\220	234.55	729/640	Rodan (5-limit)
1	83\220	452.73	125/81	Maja (5-limit)
2	13\220	70.91	25/24	Vishnu (5-limit)
11	91\220 (9\220)	496.36 (49.09)	4/3 (36/35)	Hendecatonic
20	91\220 (3\220)	496.36 (16.36)	4/3 (126/125)	Degrees

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