311edo

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311edo

312edo →

Prime factorization	311 (prime)
Step size	3.85852 ¢
Fifth	182\311 (702.251 ¢)
Semitones (A1:m2)	30:23 (115.8 ¢ : 88.75 ¢)
Consistency limit	41
Distinct consistency limit	23

311edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size.

Theory

311edo is consistent through the 41-odd-limit and distinctly consistent through the 23-odd-limit, and is a zeta gap edo and a zeta peak integer edo. It achieves this since all harmonics up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 is tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently – and thus with a maximum error of ~1.929¢. This means 311edo is an extremely efficient temperament for approximating the harmonic series consistently and simply, given how much harmonic content it approximates/represents for its size.

311edo is valuable from a psychoacoustic perspective as its step is also conincidentally close enough to the just-noticeable difference, which only affirms its efficiency of interval representation.

Some 41-limit commas it tempers out are 595/594, 625/624, 697/696, 703/702, 714/713, 760/759, 784/783, 820/819, 833/832, 875/874, 900/899, 925/924, 931/930, 962/961, 969/968, 1000/999, 1015/1014, 1024/1023, 1025/1024, 1036/1035, 1045/1044, 1054/1053, 1105/1104, 1148/1147, 1156/1155, 1184/1183, 1189/1188, 1190/1189, 1197/1196, 1210/1209, 1216/1215, 1225/1224, 1275/1274, 1288/1287, 1312/1311, 1332/1331, 1353/1352, 1365/1364, 1369/1368, 1444/1443, 1445/1444, 1450/1449, 1480/1479, 1496/1495, 1519/1518, 1520/1519, 1540/1539, 1596/1595, 1600/1599, 1625/1624, 1665/1664, 1666/1665, 1681/1680, 1683/1682, 1702/1701, 1729/1728, 1768/1767, 1805/1804, 1860/1859, 1886/1885, 1887/1886, 1925/1924, 2002/2001, 2016/2015, 2025/2024, 2058/2057, 2080/2079, 2091/2090, 2109/2108, 2146/2145, 2176/2175, 2185/2184, 2205/2204, 2233/2232, 2255/2254, 2295/2294, 2296/2295, 2300/2299, 2401/2400, 2431/2430, 2432/2431, 2465/2464, 2500/2499, 2542/2541, 2553/2552, 2584/2583, 2601/2600, 2625/2624, 2640/2639, 2646/2645, 2665/2664, 2737/2736, 2738/2737, 2755/2754, 2784/2783, 2850/2849, 2926/2925, and 2945/2944.

Prime harmonics

Approximation of prime harmonics in 311edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43
Error	Absolute (¢)	+0.000	+0.296	-0.462	-0.337	+0.451	+0.630	-0.775	-0.407	+0.665	+0.648	+0.945	-0.540	-0.767	+1.666
	Relative (%)	+0.0	+7.7	-12.0	-8.7	+11.7	+16.3	-20.1	-10.5	+17.2	+16.8	+24.5	-14.0	-19.9	+43.2
Steps (reduced)		311 (0)	493 (182)	722 (100)	873 (251)	1076 (143)	1151 (218)	1271 (27)	1321 (77)	1407 (163)	1511 (267)	1541 (297)	1620 (65)	1666 (111)	1688 (133)

Subsets and supersets

311edo is the 64th prime edo.

As an interval size measure, one step of 311edo is called gene, named after Gene Ward Smith.

Intervals

Table of 311edo intervals

genes	intervals of 41-prime-limit add-73 add-89 add-101 add-109 add-113 (/2^n and 2^n/ in bold & linked, inconsistent in italics, 23-limit linked)
0	1/1, S41, S40, S38, S37, S35 = S49*S50, S34, S32, S30, S28, S25,
1	S39, S36, S33, S31, S29, S27, S26 = S13/S15, S24, S23, S22, S21, S20, S19, S17
2	S18, S16, S15, S14, 170/169
3	S13 = 169/168, S12 = 144/143, 171/170
4	124/123, 121/120, 120/119, 117/116, 116/115, 115/114, 114/113, 113/112, 112/111, 111/110, 110/109, 109/108, 105/104, 102/101, 100/99
5	101/100, 99/98, 96/95, 93/92, 92/91, 91/90, 90/89, 89/88, 88/87, 85/84, 82/81
6	81/80, 78/77, 77/76, 76/75, 75/74, 74/73, 73/72, 70/69
7	69/68, 66/65, 65/64, 64/63, 63/62, 123/121, 119/117
8	121/119, 117/115, 58/57, 115/113, 57/56, 113/111, 56/55, 111/109, 55/54
9	52/51, 51/50, 101/99, 50/49, 49/48, 95/93
10	93/91, 46/45, 91/89, 45/44, 89/87
11	87/85, 42/41, 124/121, 41/40, 40/39, 119/116
12	39/38, 116/113, 77/75, 115/112, 38/37, 113/110, 75/73, 112/109, 37/36
13	36/35, 35/34, 104/101, 34/33
14	101/98, 33/32, 98/95, 65/63, 32/31, 95/92
15	31/30, 123/119, 92/89, 91/88, 121/117, 30/29, 119/115
16	88/85, 117/113, 29/28, 115/111, 57/55, 85/82, 113/109, 28/27
17	109/105, 27/26, 80/77, 105/101
18	26/25, 77/74, 128/123, 51/49, 76/73, 126/121, 25/24
19	124/119, 99/95, 73/70, 121/116, 24/23, 119/114, 95/91
20	117/112, 93/89, 116/111, 23/22, 114/109, 91/87, 68/65, 113/108
21	89/85, 22/21, 109/104, 65/62, 85/81
22	21/20, 104/99, 41/39
23	81/77, 101/96, 121/115, 20/19, 119/113, 98/93
24	39/37, 58/55, 77/73, 96/91, 115/109, 19/18
25	93/88, 130/123, 37/35, 92/87, 55/52, 128/121, 73/69
26	18/17, 89/84, 124/117, 123/116, 35/33
27	87/82, 52/49, 121/114, 69/65, 120/113, 17/16
28	101/95, 117/110, 116/109, 33/31, 115/108, 82/77, 49/46
29	81/76, 16/15, 111/104, 95/89
30	78/73, 109/102, 31/29, 108/101, 77/72, 123/115
31	91/85, 121/113, 15/14, 119/111, 74/69
32	44/41, 117/109, 73/68, 102/95, 29/27, 130/121, 100/93
33	128/119, 99/92, 113/105, 14/13
34	69/64, 124/115, 55/51, 96/89, 41/38, 109/101, 68/63, 95/88
35	27/25, 121/112, 40/37, 119/110
36	92/85, 13/12
37	89/82, 38/35, 101/93, 63/58, 88/81, 113/104, 25/23
38	87/80, 62/57, 99/91, 37/34, 123/113, 49/45, 110/101, 85/78
39	109/100, 121/111, 12/11, 119/109, 95/87
40	130/119, 82/75, 35/32, 128/117
41	93/85, 81/74, 104/95, 23/21, 126/115, 80/73, 57/52, 34/31
42	124/113, 45/41, 101/92, 56/51, 123/112, 89/81, 100/91, 111/101
43	11/10, 120/109, 109/99, 98/89, 76/69, 119/108
44	54/49, 75/68, 32/29, 85/77
45	116/105, 21/19, 136/123, 115/104, 73/66
46	31/28, 72/65, 113/102, 41/37, 51/46, 112/101
47	132/119, 81/73, 91/82, 101/91, 111/100, 121/109, 10/9
48	109/98, 99/89, 89/80, 69/62, 128/115, 49/44
49	39/35, 126/113, 29/26, 77/69
50	124/111, 19/17, 123/110, 104/93, 85/76, 113/101
51	28/25, 121/108, 65/58, 102/91, 37/33
52	46/41, 101/90, 55/49, 64/57, 73/65, 82/73, 91/81, 100/89, 136/121
53	9/8, 98/87
54	62/55, 115/102, 44/39, 123/109, 114/101, 35/31
55	96/85, 87/77, 113/100, 26/23, 95/84, 112/99
56	77/68, 111/98, 128/113, 17/15
57	93/82, 101/89, 42/37, 109/96, 92/81, 25/22
58	108/95, 58/51, 91/80, 124/109, 33/29, 140/123, 74/65, 115/101, 41/36
59	57/50, 65/57, 138/121, 73/64, 89/78, 105/92, 113/99
60	8/7, 119/104
61	87/76, 63/55, 55/48, 102/89
62	39/34, 109/95, 101/88, 132/115, 31/27, 116/101, 85/74, 100/87
63	23/20, 130/113, 84/73, 38/33
64	121/105, 98/85, 113/98, 128/111, 15/13
65	52/45, 89/77, 126/109, 37/32, 140/121
66	81/70, 22/19, 117/101, 95/82, 73/63, 51/44, 80/69
67	138/119, 29/25, 65/56, 101/87, 36/31, 115/99, 136/117
68	93/80, 57/49, 121/104, 64/55, 85/73
69	99/85, 7/6
70	132/113, 111/95, 104/89, 90/77, 76/65
71	117/100, 48/41, 89/76, 130/111, 41/35, 116/99, 75/64, 109/93, 34/29, 95/81
72	88/75, 115/98, 27/23, 128/109, 74/63
73	87/74, 20/17, 113/96, 73/62, 119/101
74	33/28, 112/95, 46/39, 105/89, 85/72
75	124/105, 13/11, 136/115, 123/104, 110/93
76	58/49, 45/38, 77/65, 109/92, 32/27
77	121/102, 89/75, 108/91, 146/123, 19/16, 120/101, 82/69
78	101/85, 44/37, 113/95, 69/58, 119/100, 144/121, 25/21
79	81/68, 87/73, 31/26, 130/109, 68/57, 105/88, 37/31
80	117/98, 92/77, 49/41, 104/87, 55/46, 140/117
81	91/76, 109/91, 115/96, 121/101
82	6/5, 119/99
83	101/84, 89/74, 77/64, 148/123, 136/113, 65/54, 112/93
84	88/73, 41/34, 76/63, 111/92, 146/121, 35/29
85	99/82, 93/77, 29/24, 110/91, 75/62, 98/81
86	121/100, 144/119, 23/19, 132/109, 109/90, 63/52, 40/33
87	91/75, 108/89, 17/14, 113/93
88	62/51, 45/37, 73/60, 28/23, 123/101, 95/78
89	39/32, 128/105, 89/73, 50/41, 111/91
90	116/95, 138/113, 11/9, 148/121
91	104/85, 93/76, 60/49, 109/89, 49/40, 136/111, 38/31
92	92/75, 146/119, 27/22, 124/101, 70/57, 113/92
93	91/74, 123/100, 16/13, 85/69
94	117/95, 101/82, 69/56, 90/73, 37/30, 95/77, 100/81
95	121/98, 21/17, 152/123, 110/89, 89/72, 68/55, 115/93
96	99/80, 26/21, 109/88, 140/113, 57/46, 119/96, 150/121
97	31/25, 36/29, 113/91, 77/62, 41/33
98	87/70, 46/37, 148/119, 51/41, 56/45
99	81/65, 91/73, 96/77, 101/81, 111/89, 116/93, 126/101, 136/109, 146/117
100	5/4
101	154/123, 144/115, 124/99, 119/95, 114/91, 109/87
102	69/55, 64/51, 123/98, 113/90, 152/121, 49/39
103	93/74, 44/35, 39/31, 112/89, 73/58, 34/27
104	63/50, 92/73, 121/96, 150/119, 29/23, 140/111, 111/88, 82/65
105	101/80, 24/19, 115/91, 91/72, 110/87, 148/117
106	62/49, 81/64, 138/109, 19/15, 128/101
107	52/41, 33/26, 146/115, 113/89, 80/63
108	108/85, 89/70, 117/92, 14/11
109	121/95, 93/73, 144/113, 65/51, 116/91, 51/40, 88/69, 37/29
110	152/119, 23/18, 119/93
111	87/68, 32/25, 105/82, 73/57, 114/89, 41/32, 50/39
112	109/85, 77/60, 95/74, 104/81, 113/88, 140/109
113	9/7, 148/115, 130/101, 112/87, 85/66
114	58/45, 156/121, 49/38, 89/69, 40/31
115	31/24, 146/113, 115/89, 84/65, 128/99, 75/58, 119/92
116	22/17, 123/95, 101/78, 136/105, 57/44, 35/27
117	48/37, 109/84, 74/57, 100/77, 113/87, 152/117
118	13/10, 160/123, 121/93, 95/73, 82/63
119	99/76, 116/89, 73/56, 30/23
120	124/95, 111/85, 64/49, 81/62, 98/75, 115/88, 132/101, 17/13
121	89/68, 72/55, 55/42, 148/113, 38/29
122	156/119, 101/77, 21/16, 130/99
123	46/35, 117/89, 96/73, 121/92, 146/111, 25/19, 154/117
124	54/41, 112/85, 29/22, 120/91, 91/69, 95/72
125	33/25, 144/109, 37/28, 152/115, 115/87, 119/90, 160/121, 41/31
126	45/34, 49/37, 102/77
127	126/95, 65/49, 69/52, 73/55, 150/113, 77/58, 85/64
128	93/70, 101/76, 109/82, 113/85, 117/88, 121/91
129	4/3
130	123/92, 119/89
131	99/74, 91/68, 87/65, 162/121, 154/115, 75/56, 146/109
132	114/85, 55/41, 51/38, 98/73
133	121/90, 160/119, 39/29, 152/113, 113/84, 74/55, 109/81, 35/26, 136/101
134	101/75, 66/49, 128/95, 31/23, 120/89, 89/66, 85/63
135	27/20, 104/77, 77/57, 50/37, 123/91, 73/54, 119/88
136	23/17, 111/82, 88/65, 65/48, 42/31, 164/121
137	99/73, 156/115, 19/14, 148/109, 110/81
138	87/64, 121/89, 34/25, 49/36
139	162/119, 109/80, 124/91, 154/113, 15/11
140	116/85, 101/74, 56/41, 138/101, 41/30, 160/117, 119/87
141	93/68, 26/19, 115/84, 89/65, 152/111, 63/46, 100/73, 37/27, 85/62
142	48/35, 70/51, 136/99
143	11/8, 150/109, 128/93, 95/69
144	117/85, 62/45, 113/82, 164/119, 51/37, 91/66, 40/29
145	69/50, 156/113, 29/21, 105/76, 76/55, 123/89, 170/123, 112/81
146	101/73, 18/13, 140/101
147	104/75, 154/111, 111/80, 68/49, 168/121, 25/18
148	132/95, 57/41, 146/105, 89/64, 121/87, 32/23
149	39/28, 124/89, 46/33, 152/109, 113/81
150	81/58, 88/63, 95/68, 102/73, 109/78, 123/88, 130/93
151	7/5, 164/117
152	115/82, 108/77, 101/72, 87/62, 80/57, 73/52, 170/121
153	52/37, 45/32, 128/91, 38/27
154	69/49, 162/115, 31/22, 148/105, 55/39
155	24/17, 113/80, 89/63, 154/109, 65/46, 41/29, 140/99
156	99/70, 58/41, 92/65, 109/77, 126/89, 160/113, 17/12
157	78/55, 105/74, 44/31, 115/81, 98/69
158	27/19, 91/64, 64/45, 37/26
159	121/85, 104/73, 57/40, 124/87, 144/101, 77/54, 164/115
160	117/82, 10/7
161	93/65, 176/123, 156/109, 73/51, 136/95, 63/44, 116/81
162	162/113, 109/76, 33/23, 89/62, 56/39
163	23/16, 174/121, 128/89, 105/73, 82/57, 95/66
164	36/25, 121/84, 49/34, 160/111, 111/77, 75/52
165	101/70, 13/9, 146/101
166	81/56, 123/85, 178/123, 55/38, 152/105, 42/29, 113/78, 100/69
167	29/20, 132/91, 74/51, 119/82, 164/113, 45/31, 170/117
168	138/95, 93/64, 109/75, 16/11
169	99/68, 51/35, 35/24
170	124/85, 54/37, 73/50, 92/63, 111/76, 130/89, 168/115, 19/13, 136/93
171	174/119, 117/80, 60/41, 101/69, 41/28, 148/101, 85/58
172	22/15, 113/77, 91/62, 160/109, 119/81
173	72/49, 25/17, 178/121, 128/87
174	81/55, 109/74, 28/19, 115/78, 146/99
175	121/82, 31/21, 96/65, 65/44, 164/111, 34/23
176	176/119, 108/73, 182/123, 37/25, 114/77, 77/52, 40/27
177	126/85, 132/89, 89/60, 46/31, 95/64, 49/33, 150/101
178	101/68, 52/35, 162/109, 55/37, 168/113, 113/76, 58/39, 119/80, 180/121
179	73/49, 76/51, 82/55, 85/57
180	109/73, 112/75, 115/77, 121/81, 130/87, 136/91, 148/99
181	178/119, 184/123
182	3/2
183	182/121, 176/117, 170/113, 164/109, 152/101, 140/93
184	128/85, 116/77, 113/75, 110/73, 104/69, 98/65, 95/63
185	77/51, 74/49, 68/45
186	62/41, 121/80, 180/119, 174/115, 115/76, 56/37, 109/72, 50/33
187	144/95, 138/91, 91/60, 44/29, 85/56, 41/27
188	117/77, 38/25, 111/73, 184/121, 73/48, 178/117, 35/23
189	99/65, 32/21, 154/101, 119/78
190	29/19, 113/74, 84/55, 55/36, 136/89
191	26/17, 101/66, 176/115, 75/49, 124/81, 49/32, 170/111, 95/62
192	23/15, 112/73, 89/58, 152/99
193	63/41, 146/95, 186/121, 123/80, 20/13
194	117/76, 174/113, 77/50, 57/37, 168/109, 37/24
195	54/35, 88/57, 105/68, 156/101, 190/123, 17/11
196	184/119, 116/75, 99/64, 65/42, 178/115, 113/73, 48/31
197	31/20, 138/89, 76/49, 121/78, 45/29
198	132/85, 87/56, 101/65, 115/74, 14/9
199	109/70, 176/113, 81/52, 148/95, 120/77, 170/109
200	39/25, 64/41, 89/57, 114/73, 164/105, 25/16, 136/87
201	186/119, 36/23, 119/76
202	58/37, 69/44, 80/51, 91/58, 102/65, 113/72, 146/93, 190/121
203	11/7, 184/117, 140/89, 85/54
204	63/40, 178/113, 115/73, 52/33, 41/26
205	101/64, 30/19, 109/69, 128/81, 49/31
206	117/74, 87/55, 144/91, 182/115, 19/12, 160/101
207	65/41, 176/111, 111/70, 46/29, 119/75, 192/121, 73/46, 100/63
208	27/17, 116/73, 89/56, 62/39, 35/22, 148/93
209	78/49, 121/76, 180/113, 196/123, 51/32, 110/69
210	174/109, 91/57, 190/119, 99/62, 115/72, 123/77
211	8/5
212	117/73, 109/68, 101/63, 93/58, 178/111, 162/101, 77/48, 146/91, 130/81
213	45/28, 82/51, 119/74, 37/23, 140/87
214	66/41, 124/77, 182/113, 29/18, 50/31
215	121/75, 192/119, 92/57, 113/70, 176/109, 21/13, 160/99
216	186/115, 55/34, 144/89, 89/55, 123/76, 34/21, 196/121
217	81/50, 154/95, 60/37, 73/45, 112/69, 164/101, 190/117
218	138/85, 13/8, 200/123, 148/91
219	184/113, 57/35, 101/62, 44/27, 119/73, 75/46
220	31/19, 111/68, 80/49, 178/109, 49/30, 152/93, 85/52
221	121/74, 18/11, 113/69, 95/58
222	182/111, 41/25, 146/89, 105/64, 64/39
223	156/95, 202/123, 23/14, 120/73, 74/45, 51/31
224	186/113, 28/17, 89/54, 150/91
225	33/20, 104/63, 180/109, 109/66, 38/23, 119/72, 200/121
226	81/49, 124/75, 91/55, 48/29, 154/93, 164/99
227	58/35, 121/73, 184/111, 63/38, 68/41, 73/44
228	93/56, 108/65, 113/68, 123/74, 128/77, 148/89, 168/101
229	198/119, 5/3
230	202/121, 192/115, 182/109, 152/91
231	117/70, 92/55, 87/52, 82/49, 77/46, 196/117
232	62/37, 176/105, 57/34, 109/65, 52/31, 146/87, 136/81
233	42/25, 121/72, 200/119, 116/69, 190/113, 37/22, 170/101
234	69/41, 101/60, 32/19, 123/73, 91/54, 150/89, 204/121
235	27/16, 184/109, 130/77, 76/45, 49/29
236	93/55, 208/123, 115/68, 22/13, 105/62
237	144/85, 178/105, 39/23, 95/56, 56/33
238	202/119, 124/73, 192/113, 17/10, 148/87
239	63/37, 109/64, 46/27, 196/115, 75/44
240	162/95, 29/17, 186/109, 128/75, 99/58, 70/41, 111/65, 152/89, 41/24, 200/117
241	65/38, 77/45, 89/52, 190/111, 113/66
242	12/7, 170/99
243	146/85, 55/32, 208/121, 98/57, 160/93
244	117/68, 198/115, 31/18, 174/101, 112/65, 50/29, 119/69
245	69/40, 88/51, 126/73, 164/95, 202/117, 19/11, 140/81
246	121/70, 64/37, 109/63, 154/89, 45/26
247	26/15, 111/64, 196/113, 85/49, 210/121
248	33/19, 73/42, 113/65, 40/23
249	87/50, 148/85, 101/58, 54/31, 115/66, 176/101, 190/109, 68/39
250	89/51, 96/55, 110/63, 152/87
251	208/119, 7/4
252	198/113, 184/105, 156/89, 128/73, 121/69, 114/65, 100/57
253	72/41, 202/115, 65/37, 123/70, 58/33, 109/62, 160/91, 51/29, 95/54
254	44/25, 81/46, 192/109, 37/21, 178/101, 164/93
255	30/17, 113/64, 196/111, 136/77
256	99/56, 168/95, 23/13, 200/113, 154/87, 85/48
257	62/35, 101/57, 218/123, 39/22, 204/115, 55/31
258	87/49, 16/9
259	121/68, 89/50, 162/91, 73/41, 130/73, 57/32, 98/55, 180/101, 41/23
260	66/37, 91/51, 116/65, 216/121, 25/14
261	202/113, 152/85, 93/52, 220/123, 34/19, 111/62
262	138/77, 52/29, 113/63, 70/39
263	88/49, 115/64, 124/69, 160/89, 178/99, 196/109
264	9/5, 218/121, 200/111, 182/101, 164/91, 146/81, 119/66
265	101/56, 92/51, 74/41, 204/113, 65/36, 56/31
266	132/73, 208/115, 123/68, 38/21, 105/58
267	154/85, 29/16, 136/75, 49/27
268	216/119, 69/38, 89/49, 198/109, 109/60, 20/11
269	202/111, 91/50, 162/89, 224/123, 51/28, 184/101, 82/45, 113/62
270	31/17, 104/57, 73/40, 115/63, 42/23, 95/52, 148/81, 170/93
271	117/64, 64/35, 75/41, 119/65
272	174/95, 218/119, 11/6, 222/121, 200/109
273	156/85, 101/55, 90/49, 226/123, 68/37, 182/99, 57/31, 160/87
274	46/25, 208/113, 81/44, 116/63, 186/101, 35/19, 164/89
275	24/13, 85/46
276	220/119, 37/20, 224/121, 50/27
277	176/95, 63/34, 202/109, 76/41, 89/48, 102/55, 115/62, 128/69
278	13/7, 210/113, 184/99, 119/64
279	93/50, 121/65, 54/29, 95/51, 136/73, 218/117, 41/22
280	69/37, 222/119, 28/15, 226/121, 170/91
281	230/123, 144/77, 101/54, 58/31, 204/109, 73/39
282	178/95, 208/111, 15/8, 152/81
283	92/49, 77/41, 216/115, 62/33, 109/58, 220/117, 190/101
284	32/17, 113/60, 130/69, 228/121, 49/26, 164/87
285	66/35, 232/123, 117/62, 168/89, 17/9
286	138/73, 121/64, 104/55, 87/46, 70/37, 123/65, 176/93
287	36/19, 218/115, 91/48, 146/77, 55/29, 74/39
288	93/49, 226/119, 19/10, 230/121, 192/101, 154/81
289	78/41, 99/52, 40/21
290	162/85, 124/65, 208/109, 21/11, 170/89
291	216/113, 65/34, 174/91, 109/57, 44/23, 111/58, 178/93, 224/117
292	182/95, 228/119, 23/12, 232/121, 140/73, 190/99, 119/62
293	48/25, 121/63, 73/38, 98/51, 123/64, 148/77, 25/13
294	202/105, 77/40, 52/27, 210/109
295	27/14, 218/113, 164/85, 110/57, 222/115, 56/29, 226/117, 85/44
296	230/119, 29/15, 234/121, 176/91, 89/46, 238/123, 60/31
297	184/95, 31/16, 126/65, 95/49, 64/33, 196/101
298	33/17, 101/52, 68/35, 35/18
299	72/37, 109/56, 146/75, 220/113, 37/19, 224/115, 150/77, 113/58, 76/39
300	232/119, 39/20, 80/41, 121/62, 41/21, 170/87
301	174/89, 88/45, 178/91, 45/23, 182/93
302	186/95, 96/49, 49/25, 198/101, 100/51, 51/26
303	108/55, 218/111, 55/28, 222/113, 112/57, 226/115, 57/29, 230/117, 238/121
304	234/119, 242/123, 124/63, 63/32, 128/65, 65/33, 136/69
305	69/35, 144/73, 73/37, 148/75, 75/38, 152/77, 77/39, 160/81
306	81/41, 168/85, 87/44, 176/89, 89/45, 180/91, 91/46, 184/93, 95/48, 196/99, 200/101
307	99/50, 101/51, 208/105, 216/109, 109/55, 220/111, 111/56, 224/113, 113/57, 228/115, 115/58, 232/117, 119/60, 240/121, 123/62
308
309
310
311	2/1

Notation

Sagittal

Sagittal in textual form.

Steps	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Symbol	|(	)|(	)~|	~|(	~~|	/|	|)	|\	(|	(|(	~|\	//|	/|)	/|\	)/|\
Steps	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Symbol	(|)	(|\	)||(	)~||	~||(	)||~	/||	||)	||\	~||)	(||(	~||\	//||	/||)	/||\

Syntonic-rastmic subchroma notation

Syntonic-rastmic subchroma notation in textual form.

Steps	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Symbol	>	/	/>	↑\	↑<	↑	↑>	↑/	↑/>	↑↑\	↑↑<	↑↑	↑↑>	t<	t
Steps	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Symbol	t>	#↓↓<	#↓↓	#↓↓>	#↓↓/	#↓\<	#↓\	#↓<	#↓	#↓>	#↓/	#\<	#\	#<	#

Ups and downs notation

One possible notation uses / and \ (lifts and drops) to stand for 5 edosteps. Double is abbreviated as "dub-":

0\311 = P1 = perfect unison

1\311 = ^1 = up unison

2\311 = ^^1 = dup unison

3\311 = vv/1 = dudlift unison

4\311 = v/1 = downlift unison

5\311 = /1 = lift unison

6\311 = ^/1 = uplift unison

7\311 = ^^/1 = duplift unison

8\311 = vv//1 = dud-dublift unison

9\311 = v//1 = down-dublift unison

10\311 = //1 = dublift unison

11\311 = ^//1 = up-dublift unison = vv\\m2 = dud-dubdropminor second

12\311 = ^^//1 = dup-dublift unison = v\\m2 = down-dubdropminor second

13\311 = \\m2 =  dubdropminor second

14\311 = ^\\m2 =  up-dubdropminor second

15\311 = ^^\\m2 =  dup-dubdropminor second

16\311 = vv\m2 =  duddropminor second

17\311 = v\m2 =  downdropminor second

18\311 = \m2 =  dropminor second

19\311 = ^\m2 =  updropminor second

20\311 = ^^\m2 =  dupdropminor second

21\311 = vvm2 =  dudminor second

22\311 = vm2 =  downminor second

23\311 = m2 =  minor second

24\311 = ^m2 = upminor second

25\311 = ^^m2 =  dupminor second

26\311 = vv/m2 =  dudliftminor second

27\311 = v/m2 =  downliftminor second

28\311 = /m2 =  liftminor second

29\311 = ^/m2 =  upliftminor second

30\311 = ^^/m2 =  dupliftminor second

31\311 = vv\~2 =  duddropmid second

32\311 = v\~2 =  downdropmid second

33\311 = \~2 =  dropmid second

34\311 = ^\~2 = updropmid second

35\311 = ^^\~2 = dupdropmid second

36\311 = vv~2 =  dudmid second

37\311 = v~2 =  downmid second

38\311 = ~2 =  mid second

etc.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[493 -311⟩	[⟨311 493]]	-0.0933	0.0933	2.42
2.3.5	1600000/1594323, [-59 5 22⟩	[⟨311 493 722]]	+0.0040	0.1573	4.08
2.3.5.7	2401/2400, 65625/65536, 1600000/1594323	[⟨311 493 722 873]]	+0.0331	0.1453	3.76
2.3.5.7.11	2401/2400, 3025/3024, 4000/3993, 19712/19683	[⟨311 493 722 873 1076]]	+0.0004	0.1454	3.77
2.3.5.7.11.13	625/624, 1575/1573, 2080/2079, 2200/2197, 2401/2400	[⟨311 493 722 873 1076 1151]]	-0.0280	0.1472	3.81
2.3.5.7.11.13.17	595/594, 625/624, 833/832, 1156/1155, 1575/1573, 2200/2197	[⟨311 493 722 873 1076 1151 1271]]	+0.0031	0.1561	4.05
2.3.5.7.11.13.17.19	595/594, 625/624, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573	[⟨311 493 722 873 1076 1151 1271 1321]]	+0.0146	0.1492	3.87
2.3.5.7.11.13.17.19.23	595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155	[⟨311 493 722 873 1076 1151 1271 1321 1407]]	-0.0033	0.1496	3.88

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	10\311	38.59	45/44	Hemitert
1	11\311	42.44	40/39	Humorous
1	17\311	65.59	27/26	Luminal
1	20\311	77.17	256/245, 23/22	Tertiaseptal / tertiaseptia
1	22\311	84.89	21/20	Amicable / amical / amorous
1	29\311	111.90	16/15	Vavoom
1	35\311	135.05	27/25	Superlimmal
1	43\311	165.92	11/10	Satin
1	67\311	258.52	[-32 13 5⟩	Lafa
1	88\311	339.55	243/200	Paramity
1	91\311	351.13	49/40	Newt
1	108\311	416.72	14/11	Unthirds
1	129\311	497.75	4/3	Gary
1	133\311	513.18	35/26	Trinity
1	142\311	547.92	48/35	Calamity
1	143\311	551.77	11/8	Emkay
1	155\311	598.08	572/405	Vydubychi

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

Detemperaments

Ringer scales

There are two known Ringer scales based on 311edo. Both consistently map the complete mode 234 of the harmonic series using non-patent vals of 311edo, which is believed to be the highest possible complete harmonic series mode mapped by a 311-form.

Ringer 311[+61]

Scale as chord:

936:940:941:943:944:948:950:952:954:956:958:960:962:
964:966:968:970:972:974:976:980:982:984:986:988:990:
992:994:996:1000:1002:1004:1006:1008:1010:1012:1016:1018:1020:
1022:1024:1026:1028:1030:1032:1036:1038:1040:1042:1044:1048:1050:
1052:1054:1056:1060:1063:1064:1066:1068:1070:1072:1076:1078:1080:
1082:1084:1088:1090:1092:1096:1097:1100:1102:1104:1108:1110:1112:
1114:1116:1120:1122:1124:1128:1130:1132:1134:1136:1140:1142:1144:
1148:1150:1152:1156:1158:1160:1162:1164:1168:1170:1172:1176:1178:
1180:1184:1186:1188:1192:1194:1196:1200:1202:1204:1208:1210:1212:
1216:1218:1220:1224:1226:1228:1232:1234:1236:1240:1244:1246:1248:
1252:1254:1256:1260:1264:1266:1268:1272:1274:1276:1280:1282:1284:
1288:1292:1294:1296:1300:1304:1306:1308:1312:1316:1318:1320:1324:
1326:1328:1332:1336:1338:1340:1344:1348:1350:1352:1356:1360:1362:
1364:1368:1372:1374:1376:1380:1384:1388:1390:1392:1396:1400:1402:
1404:1408:1412:1414:1416:1420:1424:1428:1432:1434:1436:1440:1444:
1448:1450:1452:1456:1460:1462:1464:1468:1472:1476:1480:1484:1486:
1488:1492:1496:1500:1504:1506:1508:1512:1516:1520:1524:1526:1528:
1532:1536:1540:1544:1546:1548:1552:1556:1560:1564:1568:1572:1576:
1580:1582:1584:1588:1592:1596:1600:1604:1606:1608:1612:1616:1620:
1624:1628:1632:1636:1640:1644:1646:1648:1652:1656:1660:1664:1668:
1672:1676:1680:1684:1688:1692:1696:1700:1702:1704:1708:1712:1716:
1720:1724:1728:1732:1736:1740:1744:1748:1752:1756:1760:1764:1768:
1772:1776:1780:1784:1788:1792:1796:1800:1804:1808:1812:1816:1820:
1824:1828:1832:1836:1840:1844:1848:1852:1856:1860:1864:1868:1872

Reduced to mode 234:

234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2:
241:483/2:242:485/2:243:487/2:244:245:491/2:246:493/2:247:495/2:
248:497/2:249:250:501/2:251:503/2:252:505/2:253:254:509/2:255:
511/2:256:513/2:257:515/2:258:259:519/2:260:521/2:261:262:525/2:
263:527/2:264:265:1063/4:266:533/2:267:535/2:268:269:539/2:270:
541/2:271:272:545/2:273:274:1097/4:275:551/2:276:277:555/2:278:
557/2:279:280:561/2:281:282:565/2:283:567/2:284:285:571/2:286:
287:575/2:288:289:579/2:290:581/2:291:292:585/2:293:294:589/2:
295:296:593/2:297:298:597/2:299:300:601/2:301:302:605/2:303:
304:609/2:305:306:613/2:307:308:617/2:309:310:311:623/2:312:
313:627/2:314:315:316:633/2:317:318:637/2:319:320:641/2:321:
322:323:647/2:324:325:326:653/2:327:328:329:659/2:330:331:
663/2:332:333:334:669/2:335:336:337:675/2:338:339:340:681/2:
341:342:343:687/2:344:345:346:347:695/2:348:349:350:701/2:
351:352:353:707/2:354:355:356:357:358:717/2:359:360:361:
362:725/2:363:364:365:731/2:366:367:368:369:370:371:743/2:
372:373:374:375:376:753/2:377:378:379:380:381:763/2:382:
383:384:385:386:773/2:387:388:389:390:391:392:393:394:
395:791/2:396:397:398:399:400:401:803/2:402:403:404:405:
406:407:408:409:410:411:823/2:412:413:414:415:416:417:
418:419:420:421:422:423:424:425:851/2:426:427:428:429:
430:431:432:433:434:435:436:437:438:439:440:441:442:
443:444:445:446:447:448:449:450:451:452:453:454:455:
456:457:458:459:460:461:462:463:464:465:466:467:468

Ringer 311[+61, −67]

Scale as chord:

936:940:941:943:944:948:950:952:954:956:958:960:962:
964:966:968:970:972:974:976:980:982:984:986:988:990:
992:994:996:1000:1002:1004:1006:1008:1010:1012:1016:1018:1020:
1022:1024:1026:1028:1030:1032:1036:1038:1040:1042:1044:1048:1050:
1052:1054:1056:1060:1061:1064:1066:1068:1072:1074:1076:1078:1080:
1082:1084:1088:1090:1092:1096:1097:1100:1102:1104:1108:1110:1112:
1114:1116:1120:1122:1124:1128:1130:1132:1134:1136:1140:1142:1144:
1148:1150:1152:1156:1158:1160:1162:1164:1168:1170:1172:1176:1178:
1180:1184:1186:1188:1192:1194:1196:1200:1202:1204:1208:1210:1212:
1216:1218:1220:1224:1226:1228:1232:1234:1236:1240:1244:1246:1248:
1252:1254:1256:1260:1264:1266:1268:1272:1274:1276:1280:1282:1284:
1288:1292:1294:1296:1300:1304:1306:1308:1312:1316:1318:1320:1324:
1326:1328:1332:1336:1340:1341:1344:1348:1350:1352:1356:1360:1362:
1364:1368:1372:1374:1376:1380:1384:1388:1390:1392:1396:1400:1402:
1404:1408:1412:1414:1416:1420:1424:1428:1432:1434:1436:1440:1444:
1448:1450:1452:1456:1460:1462:1464:1468:1472:1476:1480:1484:1486:
1488:1492:1496:1500:1504:1506:1508:1512:1516:1520:1524:1526:1528:
1532:1536:1540:1544:1546:1548:1552:1556:1560:1564:1568:1572:1576:
1580:1582:1584:1588:1592:1596:1600:1604:1608:1610:1612:1616:1620:
1624:1628:1632:1636:1640:1644:1646:1648:1652:1656:1660:1664:1668:
1672:1676:1680:1684:1688:1692:1696:1700:1702:1704:1708:1712:1716:
1720:1724:1728:1732:1736:1740:1744:1748:1752:1756:1760:1764:1768:
1772:1776:1780:1784:1788:1792:1796:1800:1804:1808:1812:1816:1820:
1824:1828:1832:1836:1840:1844:1848:1852:1856:1860:1864:1868:1872

Reduced to mode 234:

234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2:
241:483/2:242:485/2:243:487/2:244:245:491/2:246:493/2:247:495/2:
248:497/2:249:250:501/2:251:503/2:252:505/2:253:254:509/2:255:
511/2:256:513/2:257:515/2:258:259:519/2:260:521/2:261:262:525/2:
263:527/2:264:265:1061/4:266:533/2:267:268:537/2:269:539/2:270:
541/2:271:272:545/2:273:274:1097/4:275:551/2:276:277:555/2:278:
557/2:279:280:561/2:281:282:565/2:283:567/2:284:285:571/2:286:
287:575/2:288:289:579/2:290:581/2:291:292:585/2:293:294:589/2:
295:296:593/2:297:298:597/2:299:300:601/2:301:302:605/2:303:
304:609/2:305:306:613/2:307:308:617/2:309:310:311:623/2:312:
313:627/2:314:315:316:633/2:317:318:637/2:319:320:641/2:321:
322:323:647/2:324:325:326:653/2:327:328:329:659/2:330:331:
663/2:332:333:334:335:1341/4:336:337:675/2:338:339:340:681/2:
341:342:343:687/2:344:345:346:347:695/2:348:349:350:701/2:
351:352:353:707/2:354:355:356:357:358:717/2:359:360:361:
362:725/2:363:364:365:731/2:366:367:368:369:370:371:743/2:
372:373:374:375:376:753/2:377:378:379:380:381:763/2:382:
383:384:385:386:773/2:387:388:389:390:391:392:393:394:
395:791/2:396:397:398:399:400:401:402:805/2:403:404:405:
406:407:408:409:410:411:823/2:412:413:414:415:416:417:
418:419:420:421:422:423:424:425:851/2:426:427:428:429:
430:431:432:433:434:435:436:437:438:439:440:441:442:
443:444:445:446:447:448:449:450:451:452:453:454:455:
456:457:458:459:460:461:462:463:464:465:466:467:468:

Music

Eliora

• Etude in C, Op. 1, No. 1 (2022)

Tee Teck Wei

• Baoyu（𨰻𨰻） (2023) – for electric organs tuned in 311edo