311edo: Difference between revisions

← 310edo 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

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VisualWikitext

Revision as of 23:38, 8 November 2025

311 equal divisions of the octave (abbreviated 311edo or 311ed2), also called 311-tone equal temperament (311tet) or 311 equal temperament (311et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 311 equal parts of about 3.86 ¢ each. Each step represents a frequency ratio of 2^1/311, or the 311th root of 2.

311edo is notable for its extremely high consistency limit, which provides efficient and well-tempered just interval representation relative to its size.

Theory

311edo is consistent through the 41-odd-limit and nearly distinctly consistent through the 27-odd-limit with the single exception of 25/24 ~26/25, tempering out 625/624 (S25), and is a zeta gap edo and a zeta peak integer edo. This is because all harmonics up to the 42nd, and all composite harmonics up to the 80th, are more in tune than out of tune. (Prime 73 is also unusually accurate, more so than all smaller primes.) As a result, all ratios among those harmonics are mapped consistently, with a maximum error of about 1.929 ¢. This means 311edo is a serendipitously efficient temperament for approximating the harmonic series and the 41-limit in general, consistently and simply, given how much harmonic content it approximates/represents for its size. The smallest edo that has a higher consistency limit is 17461, being consistent in the 45-odd-limit, though one may prefer 20567, as it is consistent in the 57-odd-limit.

It is also the smallest edo that is purely consistent on all the first 32 harmonics (in this case, up to the 42nd harmonic). The next edo with less relative error is 16808. The smallest edo purely consistent on the first 64 harmonics is 3159811.

Although it does not do as well as 270edo in the 13-limit, it is still very accurate in the lower limits. It tempers out the amity comma, 1600000/1594323, the lafa comma, [77 -31 -12⟩, the vavoom comma, [-68 18 17⟩ in the 5-limit; 2401/2400 (breedsma), 65625/65536 (horwell comma), and 33554432/33480783 (garischisma) in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 12005/11979, and 19712/19683 in the 11-limit; and 625/624, 1575/1573, 2080/2079, 2200/2197, 4096/4095, and 4225/4224 in the 13-limit. It allows petrmic and nicolic chords in the 15-odd-limit.

Beyond the 13-limit, primes 17 and 23 are 311edo's first notable improvements over 270edo's approximation. It tempers out 595/594, 833/832, 1156/1155, 1225/1224, 1275/1274, 2058/2057, 2431/2430 in the 17-limit; 969/968, 1216/1215, 1445/1444, 1540/1539, 1729/1728 in the 19-limit; and 760/759, 875/874, 1105/1104, 1197/1196, 1288/1287, 1496/1495 in the 23-limit.

It is valuable from a psychoacoustic perspective as its step is also coincidentally above the melodic just-noticeable difference, which only affirms its efficiency of interval representation.

Prime harmonics

Approximation of prime harmonics in 311edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41
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
Error	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
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)

Approximation of prime harmonics in 311edo (continued)
Harmonic		43	47	53	59	61	67	71	73	79	83	89	97	101
Error	Absolute (¢)	+1.666	-1.841	-1.479	+1.922	-1.772	+1.722	+1.654	-0.137	-1.836	+1.400	+0.181	+1.648	+1.143
Error	Relative (%)	+43.2	-47.7	-38.3	+49.8	-45.9	+44.6	+42.9	-3.5	-47.6	+36.3	+4.7	+42.7	+29.6
Steps (reduced)		1688 (133)	1727 (172)	1781 (226)	1830 (275)	1844 (289)	1887 (21)	1913 (47)	1925 (59)	1960 (94)	1983 (117)	2014 (148)	2053 (187)	2071 (205)

Subsets and supersets

311edo is the 64th prime edo.

As an interval size measure, one step of 311edo is called gene, named by Joseph Monzo in 2007 after Gene Ward Smith^[1].

Intervals

The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely consistently, and as aforementioned, the 77-odd-limit subset of that odd-limit is purely consistent, to which a variety of odds can be added that keep pure consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.

There are 884 interval pairs in that odd limit (the 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit), where "pairs" refers to that each interval has an octave complement with equal and opposite error. That odd limit can be described explicitly as the tonality diamond of {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}. We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.

Of those 884 interval pairs, only 42 interval pairs (< 4.8%) are inconsistent, not mapped to the nearest interval of 311edo but to the second-nearest interval. Reduced to the lower half of the octave, these intervals, from smallest to largest, are: 101/100, 100/99, 82/81, 121/119, 119/117, 95/93, 87/85, 124/119, 85/81, 101/95, 100/93, 85/78, 93/85, 119/108, 93/82, 81/70, 138/119, 136/117, 99/85, 117/100, 95/81, 119/101, 101/85, 81/68, 140/117, 119/99, 117/95, 85/69, 100/81, 108/85, 119/93, 85/66, 156/119, 93/70, 162/119, 93/68, 119/87, 85/62, 117/85, 140/101, 164/117, 170/121.

Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of 311edo of error relative to where they are mapped in 311edo by the patent val, which is to say less than 3/5 = 60% relative interval error, which is equal to 2.3 ¢. The 6 highest-error intervals mentioned instead have less than 2/3 = 67% relative interval error.

The below table was generated by a simple Python 3 script to print it in plaintext using Godtone's code to simplify certain steps.

It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the square-particulars up to S41 = (41/40)/(42/41) were added manually for completeness and reference in understanding the mapping of the 41-odd-limit by 311edo for the first three edosteps and the unison. The rest of the table is algorithmically generated.

Interval table

Table of 311edo intervals
#	Cents	Marks	Approximate Intervals^{[note 1]}
0	0.0	P1	1/1, S41, S40, S38, S37, S35 = S49S50, S34, S32, S30, S28, S25*
1	3.85		S39, S36, S33, S31, S29, S27, S26 = S13/S15, S24, S23, S22, S21 = 441/440, S20 = 400/399, S19 = 361/360, S17 = 289/288
2	7.71		S18 = 324/323, S16 = 256/255, S9/S11 = 243/242, S15 = 225/224, S14 = 196/195, 170/169
3	11.57		S13 = 169/168, S12 = 144/143, 171/170
4	15.43		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	19.29		101/100, 99/98, 96/95, 93/92, 92/91, 91/90, 90/89, 89/88, 88/87, 85/84, 82/81
6	23.15		81/80, 78/77, 77/76, 76/75, 75/74, 74/73, 73/72, 70/69
7	27.0		69/68, 66/65, 65/64, 64/63, 63/62, 123/121, 119/117
8	30.86	sd2	121/119, 117/115, 58/57, 115/113, 57/56, 113/111, 56/55, 111/109, 55/54
9	34.72		52/51, 51/50, 101/99, 50/49, 49/48, 95/93
10	38.58		93/91, 46/45, 91/89, 45/44, 89/87
11	42.44		87/85, 42/41, 124/121, 41/40, 40/39, 119/116
12	46.3		39/38, 116/113, 77/75, 115/112, 38/37, 113/110, 75/73, 112/109, 37/36
13	50.16		36/35, 35/34, 104/101, 34/33
14	54.01		101/98, 33/32, 98/95, 65/63, 32/31, 95/92
15	57.87	sA1	31/30, 123/119, 92/89, 91/88, 121/117, 30/29, 119/115
16	61.73		88/85, 117/113, 29/28, 115/111, 57/55, 85/82, 113/109, 28/27
17	65.59		109/105, 27/26, 80/77, 105/101
18	69.45		26/25, 77/74, 128/123, 51/49, 76/73, 126/121, 25/24
19	73.31		124/119, 99/95, 73/70, 121/116, 24/23, 119/114, 95/91
20	77.17		117/112, 93/89, 116/111, 23/22, 114/109, 91/87, 68/65, 113/108
21	81.02		89/85, 22/21, 109/104, 65/62, 85/81
22	84.88		21/20, 104/99, 41/39
23	88.74	m2	81/77, 101/96, 121/115, 20/19, 119/113, 98/93
24	92.6		39/37, 58/55, 77/73, 96/91, 115/109, 19/18
25	96.46		93/88, 130/123, 37/35, 92/87, 55/52, 128/121, 73/69
26	100.32		18/17, 89/84, 124/117, 123/116, 35/33
27	104.18		87/82, 52/49, 121/114, 69/65, 120/113, 17/16
28	108.03		101/95, 117/110, 116/109, 33/31, 115/108, 82/77, 49/46
29	111.89		81/76, 16/15, 111/104, 95/89
30	115.75	A1	78/73, 109/102, 31/29, 108/101, 77/72, 123/115
31	119.61		91/85, 121/113, 15/14, 119/111, 74/69
32	123.47		44/41, 117/109, 73/68, 102/95, 29/27, 130/121, 100/93
33	127.33		128/119, 99/92, 113/105, 14/13
34	131.18		69/64, 124/115, 55/51, 96/89, 41/38, 109/101, 68/63, 95/88
35	135.04		27/25, 121/112, 40/37, 119/110
36	138.9		92/85, 13/12
37	142.76		89/82, 38/35, 101/93, 63/58, 88/81, 113/104, 25/23
38	146.62	N2	87/80, 62/57, 99/91, 37/34, 123/113, 49/45, 110/101, 85/78
39	150.48		109/100, 121/111, 12/11, 119/109, 95/87
40	154.34		130/119, 82/75, 35/32, 128/117
41	158.19		93/85, 81/74, 104/95, 23/21, 126/115, 80/73, 57/52, 34/31
42	162.05		124/113, 45/41, 101/92, 56/51, 123/112, 89/81, 100/91, 111/101
43	165.91		11/10, 120/109, 109/99, 98/89, 76/69, 119/108
44	169.77		54/49, 75/68, 32/29, 85/77
45	173.63		116/105, 21/19, 136/123, 115/104, 73/66
46	177.49	d3	31/28, 72/65, 113/102, 41/37, 51/46, 112/101
47	181.35		132/119, 81/73, 91/82, 101/91, 111/100, 121/109, 10/9
48	185.2		109/98, 99/89, 89/80, 69/62, 128/115, 49/44
49	189.06		39/35, 126/113, 29/26, 77/69
50	192.92		124/111, 19/17, 123/110, 104/93, 85/76, 113/101
51	196.78		28/25, 121/108, 65/58, 102/91, 37/33
52	200.64		46/41, 101/90, 55/49, 64/57, 73/65, 82/73, 91/81, 100/89, 136/121
53	204.5	M2	9/8, 98/87
54	208.36		62/55, 115/102, 44/39, 123/109, 114/101, 35/31
55	212.21		96/85, 87/77, 113/100, 26/23, 95/84, 112/99
56	216.07		77/68, 111/98, 128/113, 17/15
57	219.93		93/82, 101/89, 42/37, 109/96, 92/81, 25/22
58	223.79		108/95, 58/51, 91/80, 124/109, 33/29, 140/123, 74/65, 115/101, 41/36
59	227.65		57/50, 65/57, 138/121, 73/64, 89/78, 105/92, 113/99
60	231.51		8/7, 119/104
61	235.36	sd3	87/76, 63/55, 55/48, 102/89
62	239.22		39/34, 109/95, 101/88, 132/115, 31/27, 116/101, 85/74, 100/87
63	243.08		23/20, 130/113, 84/73, 38/33
64	246.94		121/105, 98/85, 113/98, 128/111, 15/13
65	250.8		52/45, 89/77, 126/109, 37/32, 140/121
66	254.66		81/70, 22/19, 117/101, 95/82, 73/63, 51/44, 80/69
67	258.52		138/119, 29/25, 65/56, 101/87, 36/31, 115/99, 136/117
68	262.37	sA2	93/80, 57/49, 121/104, 64/55, 85/73
69	266.23		99/85, 7/6
70	270.09		132/113, 111/95, 104/89, 90/77, 76/65
71	273.95		117/100, 48/41, 89/76, 130/111, 41/35, 116/99, 75/64, 109/93, 34/29, 95/81
72	277.81		88/75, 115/98, 27/23, 128/109, 74/63
73	281.67		87/74, 20/17, 113/96, 73/62, 119/101
74	285.53		33/28, 112/95, 46/39, 105/89, 85/72
75	289.38		124/105, 13/11, 136/115, 123/104, 110/93
76	293.24	m3	58/49, 45/38, 77/65, 109/92, 32/27
77	297.1		121/102, 89/75, 108/91, 146/123, 19/16, 120/101, 82/69
78	300.96		101/85, 44/37, 113/95, 69/58, 119/100, 144/121, 25/21
79	304.82		81/68, 87/73, 31/26, 130/109, 68/57, 105/88, 37/31
80	308.68		117/98, 92/77, 49/41, 104/87, 55/46, 140/117
81	312.54		91/76, 109/91, 115/96, 121/101
82	316.39		6/5, 119/99
83	320.25	A2	101/84, 89/74, 77/64, 148/123, 136/113, 65/54, 112/93
84	324.11		88/73, 41/34, 76/63, 111/92, 146/121, 35/29
85	327.97		99/82, 93/77, 29/24, 110/91, 75/62, 98/81
86	331.83		121/100, 144/119, 23/19, 132/109, 109/90, 63/52, 40/33
87	335.69		91/75, 108/89, 17/14, 113/93
88	339.54		62/51, 45/37, 73/60, 28/23, 123/101, 95/78
89	343.4		39/32, 128/105, 89/73, 50/41, 111/91
90	347.26		116/95, 138/113, 11/9, 148/121
91	351.12	N3	104/85, 93/76, 60/49, 109/89, 49/40, 136/111, 38/31
92	354.98		92/75, 146/119, 27/22, 124/101, 70/57, 113/92
93	358.84		91/74, 123/100, 16/13, 85/69
94	362.7		117/95, 101/82, 69/56, 90/73, 37/30, 95/77, 100/81
95	366.55		121/98, 21/17, 152/123, 110/89, 89/72, 68/55, 115/93
96	370.41		99/80, 26/21, 109/88, 140/113, 57/46, 119/96, 150/121
97	374.27		31/25, 36/29, 113/91, 77/62, 41/33
98	378.13		87/70, 46/37, 148/119, 51/41, 56/45
99	381.99	d4	81/65, 91/73, 96/77, 101/81, 111/89, 116/93, 126/101, 136/109, 146/117
100	385.85		5/4
101	389.71		154/123, 144/115, 124/99, 119/95, 114/91, 109/87
102	393.56		69/55, 64/51, 123/98, 113/90, 152/121, 49/39
103	397.42		93/74, 44/35, 39/31, 112/89, 73/58, 34/27
104	401.28		63/50, 92/73, 121/96, 150/119, 29/23, 140/111, 111/88, 82/65
105	405.14		101/80, 24/19, 115/91, 91/72, 110/87, 148/117
106	409.0	M3	62/49, 81/64, 138/109, 19/15, 128/101
107	412.86		52/41, 33/26, 146/115, 113/89, 80/63
108	416.72		108/85, 89/70, 117/92, 14/11
109	420.57		121/95, 93/73, 144/113, 65/51, 116/91, 51/40, 88/69, 37/29
110	424.43		152/119, 23/18, 119/93
111	428.29		87/68, 32/25, 105/82, 73/57, 114/89, 41/32, 50/39
112	432.15		109/85, 77/60, 95/74, 104/81, 113/88, 140/109
113	436.01		9/7, 148/115, 130/101, 112/87, 85/66
114	439.87	sd4	58/45, 156/121, 49/38, 89/69, 40/31
115	443.72		31/24, 146/113, 115/89, 84/65, 128/99, 75/58, 119/92
116	447.58		22/17, 123/95, 101/78, 136/105, 57/44, 35/27
117	451.44		48/37, 109/84, 74/57, 100/77, 113/87, 152/117
118	455.3		13/10, 160/123, 121/93, 95/73, 82/63
119	459.16		99/76, 116/89, 73/56, 30/23
120	463.02		124/95, 111/85, 64/49, 81/62, 98/75, 115/88, 132/101, 17/13
121	466.88	sA3	89/68, 72/55, 55/42, 148/113, 38/29
122	470.73		156/119, 101/77, 21/16, 130/99
123	474.59		46/35, 117/89, 96/73, 121/92, 146/111, 25/19, 154/117
124	478.45		54/41, 112/85, 29/22, 120/91, 91/69, 95/72
125	482.31		33/25, 144/109, 37/28, 152/115, 115/87, 119/90, 160/121, 41/31
126	486.17		45/34, 49/37, 102/77
127	490.03		126/95, 65/49, 69/52, 73/55, 150/113, 77/58, 85/64
128	493.89		93/70, 101/76, 109/82, 113/85, 117/88, 121/91
129	497.74	P4	4/3
130	501.6		123/92, 119/89
131	505.46		99/74, 91/68, 87/65, 162/121, 154/115, 75/56, 146/109
132	509.32		114/85, 55/41, 51/38, 98/73
133	513.18		121/90, 160/119, 39/29, 152/113, 113/84, 74/55, 109/81, 35/26, 136/101
134	517.04		101/75, 66/49, 128/95, 31/23, 120/89, 89/66, 85/63
135	520.9		27/20, 104/77, 77/57, 50/37, 123/91, 73/54, 119/88
136	524.75	A3	23/17, 111/82, 88/65, 65/48, 42/31, 164/121
137	528.61		99/73, 156/115, 19/14, 148/109, 110/81
138	532.47		87/64, 121/89, 34/25, 49/36
139	536.33		162/119, 109/80, 124/91, 154/113, 15/11
140	540.19		116/85, 101/74, 56/41, 138/101, 41/30, 160/117, 119/87
141	544.05		93/68, 26/19, 115/84, 89/65, 152/111, 63/46, 100/73, 37/27, 85/62
142	547.9		48/35, 70/51, 136/99
143	551.76		11/8, 150/109, 128/93, 95/69
144	555.62	sA4	117/85, 62/45, 113/82, 164/119, 51/37, 91/66, 40/29
145	559.48		69/50, 156/113, 29/21, 105/76, 76/55, 123/89, 170/123, 112/81
146	563.34		101/73, 18/13, 140/101
147	567.2		104/75, 154/111, 111/80, 68/49, 168/121, 25/18
148	571.06		132/95, 57/41, 146/105, 89/64, 121/87, 32/23
149	574.91		39/28, 124/89, 46/33, 152/109, 113/81
150	578.77		81/58, 88/63, 95/68, 102/73, 109/78, 123/88, 130/93
151	582.63		7/5, 164/117
152	586.49	d5	115/82, 108/77, 101/72, 87/62, 80/57, 73/52, 170/121
153	590.35		52/37, 45/32, 128/91, 38/27
154	594.21		69/49, 162/115, 31/22, 148/105, 55/39
155	598.07		24/17, 113/80, 89/63, 154/109, 65/46, 41/29, 140/99
156	601.92		99/70, 58/41, 92/65, 109/77, 126/89, 160/113, 17/12
157	605.78		78/55, 105/74, 44/31, 115/81, 98/69
158	609.64		27/19, 91/64, 64/45, 37/26
159	613.5	A4	121/85, 104/73, 57/40, 124/87, 144/101, 77/54, 164/115
160	617.36		117/82, 10/7
161	621.22		93/65, 176/123, 156/109, 73/51, 136/95, 63/44, 116/81
162	625.08		162/113, 109/76, 33/23, 89/62, 56/39
163	628.93		23/16, 174/121, 128/89, 105/73, 82/57, 95/66
164	632.79		36/25, 121/84, 49/34, 160/111, 111/77, 75/52
165	636.65		101/70, 13/9, 146/101
166	640.51		81/56, 123/85, 178/123, 55/38, 152/105, 42/29, 113/78, 100/69
167	644.37	sd5	29/20, 132/91, 74/51, 119/82, 164/113, 45/31, 170/117
168	648.23		138/95, 93/64, 109/75, 16/11
169	652.09		99/68, 51/35, 35/24
170	655.94		124/85, 54/37, 73/50, 92/63, 111/76, 130/89, 168/115, 19/13, 136/93
171	659.8		174/119, 117/80, 60/41, 101/69, 41/28, 148/101, 85/58
172	663.66		22/15, 113/77, 91/62, 160/109, 119/81
173	667.52		72/49, 25/17, 178/121, 128/87
174	671.38		81/55, 109/74, 28/19, 115/78, 146/99
175	675.24	d6	121/82, 31/21, 96/65, 65/44, 164/111, 34/23
176	679.09		176/119, 108/73, 182/123, 37/25, 114/77, 77/52, 40/27
177	682.95		126/85, 132/89, 89/60, 46/31, 95/64, 49/33, 150/101
178	686.81		101/68, 52/35, 162/109, 55/37, 168/113, 113/76, 58/39, 119/80, 180/121
179	690.67		73/49, 76/51, 82/55, 85/57
180	694.53		109/73, 112/75, 115/77, 121/81, 130/87, 136/91, 148/99
181	698.39		178/119, 184/123
182	702.25	P5	3/2
183	706.1		182/121, 176/117, 170/113, 164/109, 152/101, 140/93
184	709.96		128/85, 116/77, 113/75, 110/73, 104/69, 98/65, 95/63
185	713.82		77/51, 74/49, 68/45
186	717.68		62/41, 121/80, 180/119, 174/115, 115/76, 56/37, 109/72, 50/33
187	721.54		144/95, 138/91, 91/60, 44/29, 85/56, 41/27
188	725.4		117/77, 38/25, 111/73, 184/121, 73/48, 178/117, 35/23
189	729.26		99/65, 32/21, 154/101, 119/78
190	733.11	sd6	29/19, 113/74, 84/55, 55/36, 136/89
191	736.97		26/17, 101/66, 176/115, 75/49, 124/81, 49/32, 170/111, 95/62
192	740.83		23/15, 112/73, 89/58, 152/99
193	744.69		63/41, 146/95, 186/121, 123/80, 20/13
194	748.55		117/76, 174/113, 77/50, 57/37, 168/109, 37/24
195	752.41		54/35, 88/57, 105/68, 156/101, 190/123, 17/11
196	756.27		184/119, 116/75, 99/64, 65/42, 178/115, 113/73, 48/31
197	760.12	sA5	31/20, 138/89, 76/49, 121/78, 45/29
198	763.98		132/85, 87/56, 101/65, 115/74, 14/9
199	767.84		109/70, 176/113, 81/52, 148/95, 120/77, 170/109
200	771.7		39/25, 64/41, 89/57, 114/73, 164/105, 25/16, 136/87
201	775.56		186/119, 36/23, 119/76
202	779.42		58/37, 69/44, 80/51, 91/58, 102/65, 113/72, 146/93, 190/121
203	783.27		11/7, 184/117, 140/89, 85/54
204	787.13		63/40, 178/113, 115/73, 52/33, 41/26
205	790.99	m6	101/64, 30/19, 109/69, 128/81, 49/31
206	794.85		117/74, 87/55, 144/91, 182/115, 19/12, 160/101
207	798.71		65/41, 176/111, 111/70, 46/29, 119/75, 192/121, 73/46, 100/63
208	802.57		27/17, 116/73, 89/56, 62/39, 35/22, 148/93
209	806.43		78/49, 121/76, 180/113, 196/123, 51/32, 110/69
210	810.28		174/109, 91/57, 190/119, 99/62, 115/72, 123/77
211	814.14		8/5
212	818.0	A5	117/73, 109/68, 101/63, 93/58, 178/111, 162/101, 77/48, 146/91, 130/81
213	821.86		45/28, 82/51, 119/74, 37/23, 140/87
214	825.72		66/41, 124/77, 182/113, 29/18, 50/31
215	829.58		121/75, 192/119, 92/57, 113/70, 176/109, 21/13, 160/99
216	833.44		186/115, 55/34, 144/89, 89/55, 123/76, 34/21, 196/121
217	837.29		81/50, 154/95, 60/37, 73/45, 112/69, 164/101, 190/117
218	841.15		138/85, 13/8, 200/123, 148/91
219	845.01		184/113, 57/35, 101/62, 44/27, 119/73, 75/46
220	848.87	N6	31/19, 111/68, 80/49, 178/109, 49/30, 152/93, 85/52
221	852.73		121/74, 18/11, 113/69, 95/58
222	856.59		182/111, 41/25, 146/89, 105/64, 64/39
223	860.45		156/95, 202/123, 23/14, 120/73, 74/45, 51/31
224	864.3		186/113, 28/17, 89/54, 150/91
225	868.16		33/20, 104/63, 180/109, 109/66, 38/23, 119/72, 200/121
226	872.02		81/49, 124/75, 91/55, 48/29, 154/93, 164/99
227	875.88		58/35, 121/73, 184/111, 63/38, 68/41, 73/44
228	879.74	d7	93/56, 108/65, 113/68, 123/74, 128/77, 148/89, 168/101
229	883.6		198/119, 5/3
230	887.45		202/121, 192/115, 182/109, 152/91
231	891.31		117/70, 92/55, 87/52, 82/49, 77/46, 196/117
232	895.17		62/37, 176/105, 57/34, 109/65, 52/31, 146/87, 136/81
233	899.03		42/25, 121/72, 200/119, 116/69, 190/113, 37/22, 170/101
234	902.89		69/41, 101/60, 32/19, 123/73, 91/54, 150/89, 204/121
235	906.75	M6	27/16, 184/109, 130/77, 76/45, 49/29
236	910.61		93/55, 208/123, 115/68, 22/13, 105/62
237	914.46		144/85, 178/105, 39/23, 95/56, 56/33
238	918.32		202/119, 124/73, 192/113, 17/10, 148/87
239	922.18		63/37, 109/64, 46/27, 196/115, 75/44
240	926.04		162/95, 29/17, 186/109, 128/75, 99/58, 70/41, 111/65, 152/89, 41/24, 200/117
241	929.9		65/38, 77/45, 89/52, 190/111, 113/66
242	933.76		12/7, 170/99
243	937.62	sd7	146/85, 55/32, 208/121, 98/57, 160/93
244	941.47		117/68, 198/115, 31/18, 174/101, 112/65, 50/29, 119/69
245	945.33		69/40, 88/51, 126/73, 164/95, 202/117, 19/11, 140/81
246	949.19		121/70, 64/37, 109/63, 154/89, 45/26
247	953.05		26/15, 111/64, 196/113, 85/49, 210/121
248	956.91		33/19, 73/42, 113/65, 40/23
249	960.77		87/50, 148/85, 101/58, 54/31, 115/66, 176/101, 190/109, 68/39
250	964.63	sA6	89/51, 96/55, 110/63, 152/87
251	968.48		208/119, 7/4
252	972.34		198/113, 184/105, 156/89, 128/73, 121/69, 114/65, 100/57
253	976.2		72/41, 202/115, 65/37, 123/70, 58/33, 109/62, 160/91, 51/29, 95/54
254	980.06		44/25, 81/46, 192/109, 37/21, 178/101, 164/93
255	983.92		30/17, 113/64, 196/111, 136/77
256	987.78		99/56, 168/95, 23/13, 200/113, 154/87, 85/48
257	991.63		62/35, 101/57, 218/123, 39/22, 204/115, 55/31
258	995.49	m7	87/49, 16/9
259	999.35		121/68, 89/50, 162/91, 73/41, 130/73, 57/32, 98/55, 180/101, 41/23
260	1003.21		66/37, 91/51, 116/65, 216/121, 25/14
261	1007.07		202/113, 152/85, 93/52, 220/123, 34/19, 111/62
262	1010.93		138/77, 52/29, 113/63, 70/39
263	1014.79		88/49, 115/64, 124/69, 160/89, 178/99, 196/109
264	1018.64		9/5, 218/121, 200/111, 182/101, 164/91, 146/81, 119/66
265	1022.5	A6	101/56, 92/51, 74/41, 204/113, 65/36, 56/31
266	1026.36		132/73, 208/115, 123/68, 38/21, 105/58
267	1030.22		154/85, 29/16, 136/75, 49/27
268	1034.08		216/119, 69/38, 89/49, 198/109, 109/60, 20/11
269	1037.94		202/111, 91/50, 162/89, 224/123, 51/28, 184/101, 82/45, 113/62
270	1041.8		31/17, 104/57, 73/40, 115/63, 42/23, 95/52, 148/81, 170/93
271	1045.65		117/64, 64/35, 75/41, 119/65
272	1049.51		174/95, 218/119, 11/6, 222/121, 200/109
273	1053.37	N7	156/85, 101/55, 90/49, 226/123, 68/37, 182/99, 57/31, 160/87
274	1057.23		46/25, 208/113, 81/44, 116/63, 186/101, 35/19, 164/89
275	1061.09		24/13, 85/46
276	1064.95		220/119, 37/20, 224/121, 50/27
277	1068.81		176/95, 63/34, 202/109, 76/41, 89/48, 102/55, 115/62, 128/69
278	1072.66		13/7, 210/113, 184/99, 119/64
279	1076.52		93/50, 121/65, 54/29, 95/51, 136/73, 218/117, 41/22
280	1080.38		69/37, 222/119, 28/15, 226/121, 170/91
281	1084.24	d8	230/123, 144/77, 101/54, 58/31, 204/109, 73/39
282	1088.1		178/95, 208/111, 15/8, 152/81
283	1091.96		92/49, 77/41, 216/115, 62/33, 109/58, 220/117, 190/101
284	1095.81		32/17, 113/60, 130/69, 228/121, 49/26, 164/87
285	1099.67		66/35, 232/123, 117/62, 168/89, 17/9
286	1103.53		138/73, 121/64, 104/55, 87/46, 70/37, 123/65, 176/93
287	1107.39		36/19, 218/115, 91/48, 146/77, 55/29, 74/39
288	1111.25	M7	93/49, 226/119, 19/10, 230/121, 192/101, 154/81
289	1115.11		78/41, 99/52, 40/21
290	1118.97		162/85, 124/65, 208/109, 21/11, 170/89
291	1122.82		216/113, 65/34, 174/91, 109/57, 44/23, 111/58, 178/93, 224/117
292	1126.68		182/95, 228/119, 23/12, 232/121, 140/73, 190/99, 119/62
293	1130.54		48/25, 121/63, 73/38, 98/51, 123/64, 148/77, 25/13
294	1134.4		202/105, 77/40, 52/27, 210/109
295	1138.26		27/14, 218/113, 164/85, 110/57, 222/115, 56/29, 226/117, 85/44
296	1142.12	sd8	230/119, 29/15, 234/121, 176/91, 89/46, 238/123, 60/31
297	1145.98		184/95, 31/16, 126/65, 95/49, 64/33, 196/101
298	1149.83		33/17, 101/52, 68/35, 35/18
299	1153.69		72/37, 109/56, 146/75, 220/113, 37/19, 224/115, 150/77, 113/58, 76/39
300	1157.55		232/119, 39/20, 80/41, 121/62, 41/21, 170/87
301	1161.41		174/89, 88/45, 178/91, 45/23, 182/93
302	1165.27		186/95, 96/49, 49/25, 198/101, 100/51, 51/26
303	1169.13	sA7	108/55, 218/111, 55/28, 222/113, 112/57, 226/115, 57/29, 230/117, 238/121
304	1172.99		234/119, 242/123, 124/63, 63/32, 128/65, 65/33, 136/69
305	1176.84		69/35, 144/73, 73/37, 148/75, 75/38, 152/77, 77/39, 160/81
306	1180.7		81/41, 168/85, 87/44, 176/89, 89/45, 180/91, 91/46, 184/93, 95/48, 196/99, 200/101
307	1184.56		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	1188.42
309	1192.28
310	1196.14
311	1200.0	P8	2/1

Notation

Sagittal notation

The Sagittal notation for 311edo uses alterations of the Promethian set. Since the apotome can be split in two, a half-sharp and a half-flat may be used.

Steps		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Symbol	Evo SZ																															
	Evo																															
	Revo																															

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	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Symbol	>	/	/>	↑\	↑<	↑	↑>	↑/	↑/>	↑↑\	↑↑<	↑↑	↑↑>	t<	t	t>	#↓↓<	#↓↓	#↓↓>	#↓↓/	#↓\<	#↓\	#↓<	#↓	#↓>	#↓/	#\<	#\	#<	#

Ups and downs notation

Ups and downs notation uses ^ and v (up and down) to stand for 1 edostep and > and < (quip and quid) to stand for 5 edosteps. The spoken names run up, dup, trup, quup/downquip, quip, upquip, etc. >> is quipquip and >>> is tripquip. Quarter-tone accidentals can also be used for 311edo.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61
Sharp symbol
Flat symbol

JI approximation

Selected just intervals

The following table shows how 41-odd-limit intervals are represented in 311edo. Prime harmonics are in bold.

As 311edo is consistent in the 41-odd-limit, the mappings by direct approximation and through the patent val are identical.

41-odd-limit intervals in 311edo
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
41/34, 68/41	0.009	0.2
29/23, 46/29	0.017	0.4
29/26, 52/29	0.018	0.5
39/31, 62/39	0.019	0.5
35/34, 68/35	0.023	0.6
41/35, 70/41	0.032	0.8
23/13, 26/23	0.035	0.9
13/9, 18/13	0.038	1.0
21/16, 32/21	0.041	1.1
19/10, 20/19	0.055	1.4
29/18, 36/29	0.056	1.5
31/27, 54/31	0.058	1.5
19/14, 28/19	0.070	1.8
23/18, 36/23	0.073	1.9
37/20, 40/37	0.079	2.0
33/23, 46/33	0.082	2.1
33/29, 58/33	0.098	2.6
33/26, 52/33	0.116	3.0
7/5, 10/7	0.124	3.2
37/19, 38/37	0.133	3.5
11/9, 18/11	0.141	3.7
25/17, 34/25	0.148	3.8
11/6, 12/11	0.155	4.0
41/25, 50/41	0.157	4.1
15/8, 16/15	0.166	4.3
15/14, 28/15	0.171	4.4
13/11, 22/13	0.179	4.6
29/22, 44/29	0.197	5.1
33/31, 62/33	0.199	5.2
37/28, 56/37	0.203	5.3
23/22, 44/23	0.214	5.5
27/23, 46/27	0.223	5.8
41/37, 74/41	0.226	5.9
37/34, 68/37	0.235	6.1
29/27, 54/29	0.240	6.2
19/15, 30/19	0.241	6.2
27/26, 52/27	0.258	6.7
37/35, 70/37	0.259	6.7
39/23, 46/39	0.261	6.8
39/29, 58/39	0.278	7.2
31/23, 46/31	0.281	7.3
3/2, 4/3	0.296	7.7
31/29, 58/31	0.297	7.7
41/40, 80/41	0.305	7.9
17/10, 20/17	0.314	8.1
31/26, 52/31	0.315	8.2
13/12, 24/13	0.334	8.7
7/4, 8/7	0.337	8.7
29/24, 48/29	0.352	9.1
31/18, 36/31	0.354	9.2
41/38, 76/41	0.360	9.3
21/19, 38/21	0.366	9.5
19/17, 34/19	0.368	9.5
23/12, 24/23	0.369	9.6
37/30, 60/37	0.374	9.7
37/25, 50/37	0.383	9.9
35/19, 38/35	0.392	10.2
19/16, 32/19	0.407	10.5
21/20, 40/21	0.420	10.9
41/28, 56/41	0.429	11.1
27/22, 44/27	0.437	11.3
17/14, 28/17	0.438	11.4
11/8, 16/11	0.451	11.7
5/4, 8/5	0.462	12.0
39/22, 44/39	0.475	12.3
21/11, 22/21	0.492	12.7
31/22, 44/31	0.495	12.8
37/21, 42/37	0.499	12.9
25/19, 38/25	0.516	13.4
37/32, 64/37	0.540	14.0
25/14, 28/25	0.586	15.2
9/8, 16/9	0.592	15.3
41/30, 60/41	0.601	15.6
17/15, 30/17	0.610	15.8
15/11, 22/15	0.616	16.0
13/8, 16/13	0.630	16.3
7/6, 12/7	0.633	16.4
29/16, 32/29	0.648	16.8
31/24, 48/31	0.649	16.8
23/16, 32/23	0.665	17.2
21/13, 26/21	0.671	17.4
29/21, 42/29	0.689	17.9
19/12, 24/19	0.703	18.2
23/21, 42/23	0.706	18.3
41/21, 42/41	0.725	18.8
21/17, 34/21	0.734	19.0
33/32, 64/33	0.746	19.3
5/3, 6/5	0.757	19.6
41/32, 64/41	0.767	19.9
17/16, 32/17	0.775	20.1
11/7, 14/11	0.788	20.4
15/13, 26/15	0.796	20.6
35/32, 64/35	0.799	20.7
29/15, 30/29	0.814	21.1
23/15, 30/23	0.830	21.5
37/24, 48/37	0.836	21.7
19/11, 22/19	0.857	22.2
25/21, 42/25	0.882	22.9
27/16, 32/27	0.887	23.0
11/10, 20/11	0.912	23.6
25/16, 32/25	0.923	23.9
39/32, 64/39	0.926	24.0
9/7, 14/9	0.929	24.1
31/16, 32/31	0.945	24.5
13/7, 14/13	0.967	25.1
29/28, 56/29	0.985	25.5
31/21, 42/31	0.986	25.6
37/22, 44/37	0.991	25.7
19/18, 36/19	0.999	25.9
23/14, 28/23	1.002	26.0
19/13, 26/19	1.037	26.9
9/5, 10/9	1.053	27.3
29/19, 38/29	1.055	27.3
41/24, 48/41	1.062	27.5
17/12, 24/17	1.071	27.8
23/19, 38/23	1.071	27.8
33/28, 56/33	1.084	28.1
13/10, 20/13	1.092	28.3
35/24, 48/35	1.095	28.4
29/20, 40/29	1.110	28.8
31/30, 60/31	1.111	28.8
23/20, 40/23	1.126	29.2
37/36, 72/37	1.132	29.3
33/19, 38/33	1.153	29.9
37/26, 52/37	1.170	30.3
37/29, 58/37	1.188	30.8
37/23, 46/37	1.205	31.2
33/20, 40/33	1.208	31.3
41/22, 44/41	1.217	31.5
25/24, 48/25	1.219	31.6
27/14, 28/27	1.225	31.7
17/11, 22/17	1.226	31.8
35/22, 44/35	1.249	32.4
39/28, 56/39	1.263	32.7
31/28, 56/31	1.282	33.2
37/33, 66/37	1.287	33.3
27/19, 38/27	1.294	33.5
39/38, 76/39	1.333	34.5
27/20, 40/27	1.349	35.0
31/19, 38/31	1.352	35.0
41/36, 72/41	1.358	35.2
17/9, 18/17	1.367	35.4
25/22, 44/25	1.374	35.6
39/20, 40/39	1.387	36.0
35/18, 36/35	1.390	36.0
41/26, 52/41	1.396	36.2
17/13, 26/17	1.405	36.4
31/20, 40/31	1.407	36.5
41/29, 58/41	1.414	36.7
29/17, 34/29	1.423	36.9
37/27, 54/37	1.428	37.0
35/26, 52/35	1.429	37.0
41/23, 46/41	1.431	37.1
23/17, 34/23	1.440	37.3
35/29, 58/35	1.447	37.5
35/23, 46/35	1.463	37.9
39/37, 74/39	1.466	38.0
37/31, 62/37	1.485	38.5
41/33, 66/41	1.513	39.2
25/18, 36/25	1.515	39.3
33/17, 34/33	1.522	39.4
35/33, 66/35	1.545	40.0
25/13, 26/25	1.553	40.3
29/25, 50/29	1.571	40.7
25/23, 46/25	1.588	41.2
41/27, 54/41	1.654	42.9
27/17, 34/27	1.663	43.1
33/25, 50/33	1.670	43.3
35/27, 54/35	1.686	43.7
41/39, 78/41	1.692	43.9
39/34, 68/39	1.701	44.1
41/31, 62/41	1.712	44.4
31/17, 34/31	1.720	44.6
39/35, 70/39	1.724	44.7
35/31, 62/35	1.744	45.2
27/25, 50/27	1.811	46.9
39/25, 50/39	1.849	47.9
31/25, 50/31	1.868	48.4

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

311et has lower relative errors than any previous equal temperaments in the 23-limit and beyond. In the 23-limit it beats 282 and is bettered by 373g in terms of absolute error, and by 581 in terms of relative error.
311et is also notable in the 17- and 19-limit, with lower absolute errors than any previous equal temperaments, beating 270 in both subgroups and is bettered by 354 in the 17-limit, and by 400 in the 19-limit.

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 distinct

Commas

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.



595/594	[-1 -3 1 1 -1 0 1⟩	2.912
625/624	[-4 -1 4 0 0 -1⟩	2.772
697/696	[-3 -1 0 0 0 0 1 0 0 -1 0 0 1⟩	2.486
703/702	[-1 -3 0 0 0 -1 0 1 0 0 0 1⟩	2.464
714/713	[1 1 0 1 0 0 1 0 -1 0 -1⟩	2.426
760/759	[3 -1 1 0 -1 0 0 1 -1⟩	2.279
784/783	[4 -3 0 2 0 0 0 0 0 -1⟩	2.210
820/819	[2 -2 1 -1 0 -1 0 0 0 0 0 0 1⟩	2.113
833/832	[-6 0 0 2 0 -1 1⟩	2.080
875/874	[-1 0 3 1 0 0 0 -1 -1⟩	1.980
900/899	[2 2 2 0 0 0 0 0 0 -1 -1⟩	1.925
925/924	[-2 -1 2 -1 -1 0 0 0 0 0 0 1⟩	1.873
931/930	[-1 -1 -1 2 0 0 0 1 0 0 -1⟩	1.861
962/961	[1 0 0 0 0 1 0 0 0 0 -2 1⟩	1.801
969/968	[-3 1 0 0 -2 0 1 1⟩	1.788
1000/999	[3 -3 3 0 0 0 0 0 0 0 0 -1⟩	1.732
1015/1014	[-1 -1 1 1 0 -2 0 0 0 1⟩	1.706
1024/1023	[10 -1 0 0 -1 0 0 0 0 0 -1⟩	1.691
1025/1023	[0 -1 2 0 -1 0 0 0 0 0 -1 0 1⟩	3.381
1025/1024	[-10 0 2 0 0 0 0 0 0 0 0 0 1⟩	1.690
1036/1035	[2 -2 -1 1 0 0 0 0 -1 0 0 1⟩	1.672
1045/1044	[-2 -2 1 0 1 0 0 1 0 -1⟩	1.657
1054/1053	[1 -4 0 0 0 -1 1 0 0 0 1⟩	1.643
1105/1104	[-4 -1 1 0 0 1 1 0 -1⟩	1.567
1148/1147	[2 0 0 1 0 0 0 0 0 0 -1 -1 1⟩	1.509
1156/1155	[2 -1 -1 -1 -1 0 2⟩	1.498
1184/1183	[5 0 0 -1 0 -2 0 0 0 0 0 1⟩	1.463
1189/1188	[-2 -3 0 0 -1 0 0 0 0 1 0 0 1⟩	1.457
1190/1189	[1 0 1 1 0 0 1 0 0 -1 0 0 -1⟩	1.455
1197/1196	[-2 2 0 1 0 -1 0 1 -1⟩	1.447
1210/1209	[1 -1 1 0 2 -1 0 0 0 0 -1⟩	1.431
1216/1215	[6 -5 -1 0 0 0 0 1⟩	1.424
1225/1224	[-3 -2 2 2 0 0 -1⟩	1.414
1275/1274	[-1 1 2 -2 0 -1 1⟩	1.358
1288/1287	[3 -2 0 1 -1 -1 0 0 1⟩	1.345
1312/1311	[5 -1 0 0 0 0 0 -1 -1 0 0 0 1⟩	1.320
1332/1331	[2 2 0 0 -3 0 0 0 0 0 0 1⟩	1.300
1353/1352	[-3 1 0 0 1 -2 0 0 0 0 0 0 1⟩	1.280
1365/1364	[-2 1 1 1 -1 1 0 0 0 0 -1⟩	1.269
1369/1368	[-3 -2 0 0 0 0 0 -1 0 0 0 2⟩	1.265
1444/1443	[2 -1 0 0 0 -1 0 2 0 0 0 -1⟩	1.199
1445/1443	[0 -1 1 0 0 -1 2 0 0 0 0 -1⟩	2.398
1445/1444	[-2 0 1 0 0 0 2 -2⟩	1.199
1450/1449	[1 -2 2 -1 0 0 0 0 -1 1⟩	1.194
1480/1479	[3 -1 1 0 0 0 -1 0 0 -1 0 1⟩	1.170
1496/1495	[3 0 -1 0 1 -1 1 0 -1⟩	1.158
1519/1518	[-1 -1 0 2 -1 0 0 0 -1 0 1⟩	1.140
1520/1519	[4 0 1 -2 0 0 0 1 0 0 -1⟩	1.139
1540/1539	[2 -4 1 1 1 0 0 -1⟩	1.125
1575/1573	[0 2 2 1 -2 -1⟩	2.200
1596/1595	[2 1 -1 1 -1 0 0 1 0 -1⟩	1.085
1600/1599	[6 -1 2 0 0 -1 0 0 0 0 0 0 -1⟩	1.082
1615/1612	[-2 0 1 0 0 -1 1 1 0 0 -1⟩	3.219
1625/1624	[-3 0 3 -1 0 1 0 0 0 -1⟩	1.066
1665/1664	[-7 2 1 0 0 -1 0 0 0 0 0 1⟩	1.040
1666/1665	[1 -2 -1 2 0 0 1 0 0 0 0 -1⟩	1.039
1681/1680	[-4 -1 -1 -1 0 0 0 0 0 0 0 0 2⟩	1.030
1683/1682	[-1 2 0 0 1 0 1 0 0 -2⟩	1.029
1702/1701	[1 -5 0 -1 0 0 0 0 1 0 0 1⟩	1.017
1729/1728	[-6 -3 0 1 0 1 0 1⟩	1.002
1768/1767	[3 -1 0 0 0 1 1 -1 0 0 -1⟩	0.979
1805/1804	[-2 0 1 0 -1 0 0 2 0 0 0 0 -1⟩	0.959
1860/1859	[2 1 1 0 -1 -2 0 0 0 0 1⟩	0.931
1862/1859	[1 0 0 2 -1 -2 0 1⟩	2.792
1886/1885	[1 0 -1 0 0 -1 0 0 1 -1 0 0 1⟩	0.918
1887/1885	[0 1 -1 0 0 -1 1 0 0 -1 0 1⟩	1.836
1887/1886	[-1 1 0 0 0 0 1 0 -1 0 0 1 -1⟩	0.918
1925/1922	[-1 0 2 1 1 0 0 0 0 0 -2⟩	2.700
1925/1924	[-2 0 2 1 1 -1 0 0 0 0 0 -1⟩	0.900
1955/1953	[0 -2 1 -1 0 0 1 0 1 0 -1⟩	1.772
2002/2001	[1 -1 0 1 1 1 0 0 -1 -1⟩	0.865
2016/2015	[5 2 -1 1 0 -1 0 0 0 0 -1⟩	0.859
2025/2024	[-3 4 2 0 -1 0 0 0 -1⟩	0.855
2058/2057	[1 1 0 3 -2 0 -1⟩	0.841
2080/2079	[5 -3 1 -1 -1 1⟩	0.833
2091/2090	[-1 1 -1 0 -1 0 1 -1 0 0 0 0 1⟩	0.828
2109/2108	[-2 1 0 0 0 0 -1 1 0 0 -1 1⟩	0.821
2146/2145	[1 -1 -1 0 -1 -1 0 0 0 1 0 1⟩	0.807
2176/2175	[7 -1 -2 0 0 0 1 0 0 -1⟩	0.796
2185/2184	[-3 -1 1 -1 0 -1 0 1 1⟩	0.793
2200/2197	[3 0 2 0 1 -3⟩	2.362
2205/2204	[-2 2 1 2 0 0 0 -1 0 -1⟩	0.785
2233/2232	[-3 -2 0 1 1 0 0 0 0 1 -1⟩	0.775
2255/2254	[-1 0 1 -2 1 0 0 0 -1 0 0 0 1⟩	0.768
2295/2294	[-1 3 1 0 0 0 1 0 0 0 -1 -1⟩	0.755
2296/2295	[3 -3 -1 1 0 0 -1 0 0 0 0 0 1⟩	0.754
2300/2299	[2 0 2 0 -2 0 0 -1 1⟩	0.753
2401/2400	[-5 -1 -2 4⟩	0.721
2431/2430	[-1 -5 -1 0 1 1 1⟩	0.712
2432/2431	[7 0 0 0 -1 -1 -1 1⟩	0.712
2465/2464	[-5 0 1 -1 -1 0 1 0 0 1⟩	0.702
2500/2499	[2 -1 4 -2 0 0 -1⟩	0.693
2516/2511	[2 -4 0 0 0 0 1 0 0 0 -1 1⟩	3.444
2527/2523	[0 -1 0 1 0 0 0 2 0 -2⟩	2.743
2542/2541	[1 -1 0 -1 -2 0 0 0 0 0 1 0 1⟩	0.681
2553/2552	[-3 1 0 0 -1 0 0 0 1 -1 0 1⟩	0.678
2584/2583	[3 -2 0 -1 0 0 1 1 0 0 0 0 -1⟩	0.670
2601/2600	[-3 2 -2 0 0 -1 2⟩	0.666
2625/2624	[-6 1 3 1 0 0 0 0 0 0 0 0 -1⟩	0.660
2640/2639	[4 1 1 -1 1 -1 0 0 0 -1⟩	0.656
2646/2645	[1 3 -1 2 0 0 0 0 -2⟩	0.654
2665/2662	[-1 0 1 0 -3 1 0 0 0 0 0 0 1⟩	1.950
2665/2664	[-3 -2 1 0 0 1 0 0 0 0 0 -1 1⟩	0.650
2695/2691	[0 -2 1 2 1 -1 0 0 -1⟩	2.571
2720/2717	[5 0 1 0 -1 -1 1 -1⟩	1.911
2737/2736	[-4 -2 0 1 0 0 1 -1 1⟩	0.633
2738/2737	[1 0 0 -1 0 0 -1 0 -1 0 0 2⟩	0.632
2755/2754	[-1 -4 1 0 0 0 -1 1 0 1⟩	0.629
2784/2783	[5 1 0 0 -2 0 0 0 -1 1⟩	0.622
2788/2783	[2 0 0 0 -2 0 1 0 -1 0 0 0 1⟩	3.108
2850/2849	[1 1 2 -1 -1 0 0 1 0 0 0 -1⟩	0.608
2873/2871	[0 -2 0 0 -1 2 1 0 0 -1⟩	1.206
2875/2871	[0 -2 3 0 -1 0 0 0 1 -1⟩	2.410
2875/2873	[0 0 3 0 0 -2 -1 0 1⟩	1.205
2888/2883	[3 -1 0 0 0 0 0 2 0 0 -2⟩	3.000
2890/2883	[1 -1 1 0 0 0 2 0 0 0 -2⟩	4.198
2926/2925	[1 -2 -2 1 1 -1 0 1⟩	0.592
2945/2944	[-7 0 1 0 0 0 0 1 -1 0 1⟩	0.588
3025/3024	[-4 -3 2 -1 2⟩	0.572
3060/3059	[2 2 1 -1 0 0 1 -1 -1⟩	0.566
3136/3135	[6 -1 -1 2 -1 0 0 -1⟩	0.552
3179/3174	[-1 -1 0 0 1 0 2 0 -2⟩	2.725
3213/3211	[0 3 0 1 0 -2 1 -1⟩	1.078
3220/3219	[2 -1 1 1 0 0 0 0 1 -1 0 -1⟩	0.538
3249/3248	[-4 2 0 -1 0 0 0 2 0 -1⟩	0.533
3250/3249	[1 -2 3 0 0 1 0 -2⟩	0.533
3256/3255	[3 -1 -1 -1 1 0 0 0 0 0 -1 1⟩	0.532
3325/3321	[0 -4 2 1 0 0 0 1 0 0 0 0 -1⟩	2.084
3367/3364	[-2 0 0 1 0 1 0 0 0 -2 0 1⟩	1.543
3367/3366	[-1 -2 0 1 -1 1 -1 0 0 0 0 1⟩	0.514
3381/3380	[-2 1 -1 2 0 -2 0 0 1⟩	0.512
3400/3393	[3 -2 2 0 0 -1 1 0 0 -1⟩	3.568
3451/3450	[-1 -1 -2 1 0 0 1 0 -1 1⟩	0.502
3510/3509	[1 3 1 0 -2 1 0 0 0 -1⟩	0.493
3515/3509	[0 0 1 0 -2 0 0 1 0 -1 0 1⟩	2.958
3520/3519	[6 -2 1 0 1 0 -1 0 -1⟩	0.492
3553/3549	[0 -1 0 -1 1 -2 1 1⟩	1.950
3553/3552	[-5 -1 0 0 1 0 1 1 0 0 0 -1⟩	0.487
3565/3564	[-2 -4 1 0 -1 0 0 0 1 0 1⟩	0.486
3567/3565	[0 1 -1 0 0 0 0 0 -1 1 -1 0 1⟩	0.971
3626/3625	[1 0 -3 2 0 0 0 0 0 -1 0 1⟩	0.478
3690/3689	[1 2 1 -1 0 0 -1 0 0 0 -1 0 1⟩	0.469
3705/3703	[0 1 1 -1 0 1 0 1 -2⟩	0.935
3731/3726	[-1 -4 0 1 0 1 0 0 -1 0 0 0 1⟩	2.322
3757/3751	[0 0 0 0 -2 1 2 0 0 0 -1⟩	2.767
3773/3770	[-1 0 -1 3 1 -1 0 0 0 -1⟩	1.377
3773/3772	[-2 0 0 3 1 0 0 0 -1 0 0 0 -1⟩	0.459
3774/3773	[1 1 0 -3 -1 0 1 0 0 0 0 1⟩	0.459
3875/3872	[-5 0 3 0 -2 0 0 0 0 0 1⟩	1.341
3876/3875	[2 1 -3 0 0 0 1 1 0 0 -1⟩	0.447
3888/3887	[4 5 0 0 0 -2 0 0 -1⟩	0.445
3895/3887	[0 0 1 0 0 -2 0 1 -1 0 0 0 1⟩	3.559
3895/3888	[-4 -5 1 0 0 0 0 1 0 0 0 0 1⟩	3.114
3969/3968	[-7 4 0 2 0 0 0 0 0 0 -1⟩	0.436
3971/3968	[-7 0 0 0 1 0 0 2 0 0 -1⟩	1.308
3971/3969	[0 -4 0 -2 1 0 0 2⟩	0.872
4000/3993	[5 -1 3 0 -3⟩	3.032
4060/4059	[2 -2 1 1 -1 0 0 0 0 1 0 0 -1⟩	0.426
4096/4095	[12 -2 -1 -1 0 -1⟩	0.423
4125/4123	[0 1 3 -1 1 0 0 -1 0 0 -1⟩	0.840
4186/4185	[1 -3 -1 1 0 1 0 0 1 0 -1⟩	0.414
4200/4199	[3 1 2 1 0 -1 -1 -1⟩	0.412
4225/4224	[-7 -1 2 0 -1 2⟩	0.410
4235/4232	[-3 0 1 1 2 0 0 0 -2⟩	1.227
4256/4255	[5 0 -1 1 0 0 0 1 -1 0 0 -1⟩	0.407
4264/4263	[3 -1 0 -2 0 1 0 0 0 -1 0 0 1⟩	0.406
4305/4301	[0 1 1 1 -1 0 -1 0 -1 0 0 0 1⟩	1.609
4352/4347	[8 -3 0 -1 0 0 1 0 -1⟩	1.990
4403/4394	[-1 0 0 1 0 -3 1 0 0 0 0 1⟩	3.542
4403/4400	[-4 0 -2 1 -1 0 1 0 0 0 0 1⟩	1.180
4437/4433	[0 2 0 0 -1 -1 1 0 0 1 -1⟩	1.561
4440/4433	[3 1 1 0 -1 -1 0 0 0 0 -1 1⟩	2.732
4459/4455	[0 -4 -1 3 -1 1⟩	1.554
4480/4477	[7 0 1 1 -2 0 0 0 0 0 0 -1⟩	1.160
4551/4550	[-1 1 -2 -1 0 -1 0 0 0 0 0 1 1⟩	0.380
4624/4617	[4 -5 0 0 0 0 2 -1⟩	2.623
4625/4617	[0 -5 3 0 0 0 0 -1 0 0 0 1⟩	2.997
4625/4624	[-4 0 3 0 0 0 -2 0 0 0 0 1⟩	0.374
4641/4640	[-5 1 -1 1 0 1 1 0 0 -1⟩	0.373
4674/4669	[1 1 0 -1 0 0 0 1 -1 -1 0 0 1⟩	1.853
4675/4669	[0 0 2 -1 1 0 1 0 -1 -1⟩	2.223
4675/4674	[-1 -1 2 0 1 0 1 -1 0 0 0 0 -1⟩	0.370
4693/4692	[-2 -1 0 0 0 1 -1 2 -1⟩	0.369
4715/4712	[-3 0 1 0 0 0 0 -1 1 0 -1 0 1⟩	1.102
4750/4743	[1 -2 3 0 0 0 -1 1 0 0 -1⟩	2.553
4785/4784	[-4 1 1 0 1 -1 0 0 -1 1⟩	0.362
4802/4797	[1 -2 0 4 0 -1 0 0 0 0 0 0 -1⟩	1.804
4807/4805	[0 0 -1 0 1 0 0 1 1 0 -2⟩	0.720
4810/4807	[1 0 1 0 -1 1 0 -1 -1 0 0 1⟩	1.080
4879/4875	[0 -1 -3 1 0 -1 1 0 0 0 0 0 1⟩	1.420
4913/4901	[0 0 0 0 0 -2 3 0 0 -1⟩	4.234
4921/4920	[-3 -1 -1 1 0 0 0 1 0 0 0 1 -1⟩	0.352
4960/4959	[5 -2 1 0 0 0 0 -1 0 -1 1⟩	0.349
4961/4959	[0 -2 0 0 2 0 0 -1 0 -1 0 0 1⟩	0.698
4961/4960	[-5 0 -1 0 2 0 0 0 0 0 -1 0 1⟩	0.349
4992/4991	[7 1 0 -1 0 1 0 0 -1 0 -1⟩	0.347
4995/4991	[0 3 1 -1 0 0 0 0 -1 0 -1 1⟩	1.387
5000/4991	[3 0 4 -1 0 0 0 0 -1 0 -1⟩	3.119
5054/5049	[1 -3 0 1 -1 0 -1 2⟩	1.714
5083/5082	[-1 -1 0 -1 -2 1 1 0 1⟩	0.341
5084/5083	[2 0 0 0 0 -1 -1 0 -1 0 1 0 1⟩	0.341
5104/5103	[4 -6 0 -1 1 0 0 0 0 1⟩	0.339
5244/5239	[2 1 0 0 0 -2 0 1 1 0 -1⟩	1.651
5248/5239	[7 0 0 0 0 -2 0 0 0 0 -1 0 1⟩	2.972
5250/5239	[1 1 3 1 0 -2 0 0 0 0 -1⟩	3.631
5291/5290	[-1 0 -1 0 1 1 0 0 -2 0 0 1⟩	0.327
5292/5291	[2 3 0 2 -1 -1 0 0 0 0 0 -1⟩	0.327
5415/5408	[-5 1 1 0 0 -2 0 2⟩	2.239
5425/5423	[0 0 2 1 -1 0 -1 0 0 -1 1⟩	0.638
5439/5434	[-1 1 0 2 -1 -1 0 -1 0 0 0 1⟩	1.592
5440/5439	[6 -1 1 -2 0 0 1 0 0 0 0 -1⟩	0.318
5453/5445	[0 -2 -1 1 -2 0 0 1 0 0 0 0 1⟩	2.542
5491/5481	[0 -3 0 -1 0 0 2 1 0 -1⟩	3.156
5491/5488	[-4 0 0 -3 0 0 2 1⟩	0.946
5600/5589	[5 -5 2 1 0 0 0 0 -1⟩	3.404
5625/5624	[-3 2 4 0 0 0 0 -1 0 0 0 -1⟩	0.308
5635/5632	[-9 0 1 2 -1 0 0 0 1⟩	0.922
5643/5642	[-1 3 0 -1 1 -1 0 1 0 0 -1⟩	0.307
5684/5681	[2 0 0 2 0 -1 0 -1 -1 1⟩	0.914
5735/5733	[0 -2 1 -2 0 -1 0 0 0 0 1 1⟩	0.604
5776/5775	[4 -1 -2 -1 -1 0 0 2⟩	0.300
5797/5796	[-2 -2 0 -1 1 0 1 0 -1 0 1⟩	0.299
5800/5797	[3 0 2 0 -1 0 -1 0 0 1 -1⟩	0.896
5824/5819	[6 0 0 1 -1 1 0 0 -2⟩	1.487
5831/5819	[0 0 0 3 -1 0 1 0 -2⟩	3.566
5863/5859	[0 -3 0 -1 1 1 0 0 0 0 -1 0 1⟩	1.182
5865/5863	[0 1 1 0 -1 -1 1 0 1 0 0 0 -1⟩	0.590
5888/5887	[8 0 0 -1 0 0 0 0 1 -2⟩	0.294
5890/5887	[1 0 1 -1 0 0 0 1 0 -2 1⟩	0.882
5916/5915	[2 1 -1 -1 0 -2 1 0 0 1⟩	0.293
5929/5928	[-3 -1 0 2 2 -1 0 -1⟩	0.292
5957/5952	[-6 -1 0 1 0 0 0 0 1 0 -1 1⟩	1.454
5985/5984	[-5 2 1 1 -1 0 -1 1⟩	0.289
6068/6061	[2 0 0 0 -1 0 0 -1 0 -1 0 1 1⟩	1.998
6069/6061	[0 1 0 1 -1 0 2 -1 0 -1⟩	2.284
6069/6068	[-2 1 0 1 0 0 2 0 0 0 0 -1 -1⟩	0.285
6076/6075	[2 -5 -2 2 0 0 0 0 0 0 1⟩	0.285
6175/6171	[0 -1 2 0 -2 1 -1 1⟩	1.122
6175/6174	[-1 -2 2 -3 0 1 0 1⟩	0.280
6250/6237	[1 -4 5 -1 -1⟩	3.605
6256/6253	[4 0 0 0 0 -2 1 0 1 0 0 -1⟩	0.830
6273/6272	[-7 2 0 -2 0 0 1 0 0 0 0 0 1⟩	0.276
6290/6279	[1 -1 1 -1 0 -1 1 0 -1 0 0 1⟩	3.030
6293/6292	[-2 0 0 1 -2 -1 0 0 0 1 1⟩	0.275
6325/6318	[-1 -5 2 0 1 -1 0 0 1⟩	1.917
6325/6324	[-2 -1 2 0 1 0 -1 0 1 0 -1⟩	0.274
6327/6325	[0 2 -2 0 -1 0 0 1 -1 0 0 1⟩	0.547
6355/6348	[-2 -1 1 0 0 0 0 0 -2 0 1 0 1⟩	1.908
6358/6355	[1 0 -1 0 1 0 2 0 0 0 -1 0 -1⟩	0.817
6422/6417	[1 -2 0 0 0 2 0 1 -1 0 -1⟩	1.348
6480/6479	[4 4 1 0 -1 0 0 -1 0 0 -1⟩	0.267
6517/6512	[-4 0 0 3 -1 0 0 1 0 0 0 -1⟩	1.329
6601/6591	[0 -1 0 1 0 -3 0 0 1 0 0 0 1⟩	2.625
6601/6600	[-3 -1 -2 1 -1 0 0 0 1 0 0 0 1⟩	0.262
6647/6642	[-1 -4 0 0 0 0 2 0 1 0 0 0 -1⟩	1.303
6650/6647	[1 0 2 1 0 0 -2 1 -1⟩	0.781
6656/6655	[9 0 -1 0 -3 1⟩	0.260
6664/6655	[3 0 -1 2 -3 0 1⟩	2.340
6670/6669	[1 -3 1 0 0 -1 0 -1 1 1⟩	0.260
6728/6727	[3 0 0 -1 0 0 0 0 0 2 -2⟩	0.257
6732/6727	[2 2 0 -1 1 0 1 0 0 0 -2⟩	1.286
6845/6831	[0 -3 1 0 -1 0 0 0 -1 0 0 2⟩	3.544
6859/6851	[0 0 0 0 0 -1 -1 3 0 0 -1⟩	2.020
6860/6851	[2 0 1 3 0 -1 -1 0 0 0 -1⟩	2.273
6860/6859	[2 0 1 3 0 0 0 -3⟩	0.252
6882/6877	[1 1 0 0 0 -1 0 0 -2 0 1 1⟩	1.258
6885/6877	[0 4 1 0 0 -1 1 0 -2⟩	2.013
6888/6877	[3 1 0 1 0 -1 0 0 -2 0 0 0 1⟩	2.767
6902/6897	[1 -1 0 1 -2 0 1 -1 0 1⟩	1.255
6919/6912	[-8 -3 0 0 1 0 1 0 0 0 0 1⟩	1.752
6919/6916	[-2 0 0 -1 1 -1 1 -1 0 0 0 1⟩	0.751
6930/6929	[1 2 1 1 1 -2 0 0 0 0 0 0 -1⟩	0.250
6936/6929	[3 1 0 0 0 -2 2 0 0 0 0 0 -1⟩	1.748
6993/6992	[-4 3 0 1 0 0 0 -1 -1 0 0 1⟩	0.248
7011/7007	[0 2 0 -2 -1 -1 0 1 0 0 0 0 1⟩	0.988
7105/7104	[-6 -1 1 2 0 0 0 0 0 1 0 -1⟩	0.244
7106/7105	[1 0 -1 -2 1 0 1 1 0 -1⟩	0.244
7163/7161	[0 -1 0 -1 -1 1 0 1 0 1 -1⟩	0.483
7168/7163	[10 0 0 1 0 -1 0 -1 0 -1⟩	1.208
7175/7163	[0 0 2 1 0 -1 0 -1 0 -1 0 0 1⟩	2.898
7203/7192	[-3 1 0 4 0 0 0 0 0 -1 -1⟩	2.646
7216/7215	[4 -1 -1 0 1 -1 0 0 0 0 0 -1 1⟩	0.240
7225/7216	[-4 0 2 0 -1 0 2 0 0 0 0 0 -1⟩	2.158
7344/7337	[4 3 0 0 -1 0 1 0 -1 -1⟩	1.651
7350/7337	[1 1 2 2 -1 0 0 0 -1 -1⟩	3.065
7425/7424	[-8 3 2 0 1 0 0 0 0 -1⟩	0.233
7429/7424	[-8 0 0 0 0 0 1 1 1 -1⟩	1.166
7429/7425	[0 -3 -2 0 -1 0 1 1 1⟩	0.932
7511/7502	[-1 0 0 1 -2 0 0 0 0 1 -1 1⟩	2.076
7514/7511	[1 0 0 -1 0 1 2 0 0 -1 0 -1⟩	0.691
7540/7533	[2 -5 1 0 0 1 0 0 0 1 -1⟩	1.608
7544/7533	[3 -5 0 0 0 0 0 0 1 0 -1 0 1⟩	2.526
7546/7533	[1 -5 0 3 1 0 0 0 0 0 -1⟩	2.985
7585/7569	[0 -2 1 0 0 0 0 0 0 -2 0 1 1⟩	3.656
7585/7581	[0 -1 1 -1 0 0 0 -2 0 0 0 1 1⟩	0.913
7616/7605	[6 -2 -1 1 0 -2 1⟩	2.502
7657/7656	[-3 -1 0 0 -1 1 0 1 0 -1 1⟩	0.226
7659/7657	[0 2 0 0 0 -1 0 -1 1 0 -1 1⟩	0.452
7667/7657	[0 0 0 0 1 -1 1 -1 0 0 -1 0 1⟩	2.260
7667/7659	[0 -2 0 0 1 0 1 0 -1 0 0 -1 1⟩	1.807
7695/7688	[-3 4 1 0 0 0 0 1 0 0 -2⟩	1.576
7696/7695	[4 -4 -1 0 0 1 0 -1 0 0 0 1⟩	0.225
7733/7728	[-4 -1 0 -1 1 0 0 1 -1 0 0 1⟩	1.120
7735/7733	[0 0 1 1 -1 1 1 -1 0 0 0 -1⟩	0.448
7749/7744	[-6 3 0 1 -2 0 0 0 0 0 0 0 1⟩	1.117
7750/7749	[1 -3 3 -1 0 0 0 0 0 0 1 0 -1⟩	0.223
7866/7865	[1 2 -1 0 -2 -1 0 1 1⟩	0.220
7872/7865	[6 1 -1 0 -2 -1 0 0 0 0 0 0 1⟩	1.540
7889/7888	[-4 0 0 3 0 0 -1 0 1 -1⟩	0.219
7905/7904	[-5 1 1 0 0 -1 1 -1 0 0 1⟩	0.219
7936/7935	[8 -1 -1 0 0 0 0 0 -2 0 1⟩	0.218
7942/7935	[1 -1 -1 0 1 0 0 2 -2⟩	1.527
8029/8019	[0 -6 0 1 -1 0 0 0 0 0 1 1⟩	2.158
8036/8019	[2 -6 0 2 -1 0 0 0 0 0 0 0 1⟩	3.666
8075/8064	[-7 -2 2 -1 0 0 1 1⟩	2.360
8092/8073	[2 -3 0 1 0 -1 2 0 -1⟩	4.070
8096/8091	[5 -2 0 0 1 0 0 0 1 -1 -1⟩	1.070
8125/8118	[-1 -2 4 0 -1 1 0 0 0 0 0 0 -1⟩	1.492
8281/8280	[-3 -2 -1 2 0 2 0 0 -1⟩	0.209
8303/8294	[-1 0 0 0 -1 -1 0 2 1 -1⟩	1.878
8323/8320	[-7 0 -1 1 0 -1 0 0 0 1 0 0 1⟩	0.624
8325/8323	[0 2 2 -1 0 0 0 0 0 -1 0 1 -1⟩	0.416
8381/8370	[-1 -3 -1 0 0 0 2 0 0 1 -1⟩	2.274
8381/8372	[-2 0 0 -1 0 -1 2 0 -1 1⟩	1.860
8381/8379	[0 -2 0 -2 0 0 2 -1 0 1⟩	0.413
8405/8398	[-1 0 1 0 0 -1 -1 -1 0 0 0 0 2⟩	1.442
8464/8463	[4 -1 0 -1 0 -1 0 0 2 0 -1⟩	0.205
8500/8487	[2 -2 3 0 0 0 1 0 -1 0 0 0 -1⟩	2.650
8526/8525	[1 1 -2 2 -1 0 0 0 0 1 -1⟩	0.203
8528/8525	[4 0 -2 0 -1 1 0 0 0 0 -1 0 1⟩	0.609
8569/8556	[-2 -1 0 0 1 0 0 1 -1 0 -1 0 1⟩	2.628
8569/8568	[-3 -2 0 -1 1 0 -1 1 0 0 0 0 1⟩	0.202
8575/8556	[-2 -1 2 3 0 0 0 0 -1 0 -1⟩	3.840
8575/8569	[0 0 2 3 -1 0 0 -1 0 0 0 0 -1⟩	1.212
8624/8619	[4 -1 0 2 1 -2 -1⟩	1.004
8625/8624	[-4 1 3 -2 -1 0 0 0 1⟩	0.201
8680/8671	[3 0 1 1 0 -1 0 0 -1 -1 1⟩	1.796
8856/8855	[3 3 -1 -1 -1 0 0 0 -1 0 0 0 1⟩	0.195
8897/8892	[-2 -2 0 1 0 -1 0 -1 0 0 1 0 1⟩	0.973
8959/8954	[-1 0 0 0 -2 0 2 0 0 0 1 -1⟩	0.966
8960/8959	[8 0 1 1 0 0 -2 0 0 0 -1⟩	0.193
8991/8990	[-1 5 -1 0 0 0 0 0 0 -1 -1 1⟩	0.193
8993/8990	[-1 0 -1 0 0 0 1 0 2 -1 -1⟩	0.578
8993/8991	[0 -5 0 0 0 0 1 0 2 0 0 -1⟩	0.385
9000/8993	[3 2 3 0 0 0 -1 0 -2⟩	1.347
9025/9009	[0 -2 2 -1 -1 -1 0 2⟩	3.072
9025/9016	[-3 0 2 -2 0 0 0 2 -1⟩	1.727
9065/9048	[-3 -1 1 2 0 -1 0 0 0 -1 0 1⟩	3.250
9065/9061	[0 0 1 2 0 -1 -1 0 0 0 0 1 -1⟩	0.764
9139/9135	[0 -2 -1 -1 0 1 0 1 0 -1 0 1⟩	0.758
9177/9176	[-3 1 0 1 0 0 0 1 1 0 -1 -1⟩	0.189
9261/9251	[0 3 0 3 -1 0 0 0 0 -2⟩	1.870
9280/9269	[6 0 1 0 0 -1 0 0 -1 1 -1⟩	2.053
9317/9315	[0 -4 -1 1 3 0 0 0 -1⟩	0.372
9361/9360	[-4 -2 -1 0 1 -1 0 0 1 0 0 1⟩	0.185
9367/9360	[-4 -2 -1 0 0 -1 1 1 0 1⟩	1.294
9367/9361	[0 0 0 0 -1 0 1 1 -1 1 0 -1⟩	1.109
9375/9361	[0 1 5 0 -1 0 0 0 -1 0 0 -1⟩	2.587
9375/9367	[0 1 5 0 0 0 -1 -1 0 -1⟩	1.478
9425/9424	[-4 0 2 0 0 1 0 -1 0 1 -1⟩	0.184
9435/9424	[-4 1 1 0 0 0 1 -1 0 0 -1 1⟩	2.020
9472/9471	[8 -1 0 -1 -1 0 0 0 0 0 0 1 -1⟩	0.183
9500/9477	[2 -6 3 0 0 -1 0 1⟩	4.196
9583/9568	[-5 0 0 1 0 -1 0 0 -1 0 0 2⟩	2.712
9583/9570	[-1 -1 -1 1 -1 0 0 0 0 -1 0 2⟩	2.350
9758/9747	[1 -3 0 1 0 0 1 -2 0 0 0 0 1⟩	1.953
9775/9768	[-3 -1 2 0 -1 0 1 0 1 0 0 -1⟩	1.240
9802/9801	[1 -4 0 0 -2 2 0 0 0 1⟩	0.177
9826/9801	[1 -4 0 0 -2 0 3⟩	4.410
9947/9936	[-4 -3 0 3 0 0 0 0 -1 1⟩	1.916
9947/9945	[0 -2 -1 3 0 -1 -1 0 0 1⟩	0.348

Scales

MOS scales

See: User:BudjarnLambeth/311edo MOS scales.

Mode 16 of the harmonic series

311edo accurately approximates the mode 16 of harmonic series.

Overtones	16	17	18	19	20	21	22	23	24
JI ratios	1/1	17/16	9/8	19/16	5/4	21/16	11/8	23/16	3/2
…in cents	0	104.955	203.910	297.513	386.314	470.781	551.318	628.274	701.955
Degrees in 311edo	0	27	53	77	100	122	143	163	182
…in cents	0	104.180	204.502	297.106	385.852	470.740	551.768	628.939	702.251

Overtones	25	26	27	28	29	30	31	32
JI ratios	25/16	13/8	27/16	7/4	29/16	15/8	31/16	2/1
…in cents	772.627	840.528	905.865	968.826	1029.577	1088.269	1145.036	1200
Degrees in 311edo	200	218	235	251	267	282	297	311
…in cents	771.704	841.158	906.752	968.489	1030.23	1088.1	1145.98	1200

The scale in adjacent steps is 27, 26, 24, 23, 22, 21, 20, 19, 18, 18, 17, 16, 16, 15, 15, 14. Three interval pairs are conflated: 25/24 ~ 26/25, 28/27 ~ 29/28, and 30/29 ~ 31/30.

Detemperaments

The most otonally simple way of detempering 311edo is a Ringer scale. See 311edo/Ringer 311 for details.

Music

Eliora

Etude in C, Op. 1, No. 1 (2022)

Francium

"From the Ground" from Scoop (2024) – Spotify | Bandcamp | YouTube
"Translator Server Error" from Naughty Girl Era (2024) – Spotify | Bandcamp | YouTube
"Vermin Supreme" from The Scallop Disco Accident (2025) – Spotify | Bandcamp | YouTube
"Love Is Just a Flying Pig Going to a Funeral." from Random Sentences (2025) – Spotify | Bandcamp | YouTube
"kumturd" from wiloliquy (2025) – Spotify | Bandcamp | YouTube
"Is That An Albino Duck?" from Questions, Vol. 2 (2025) – Spotify | Bandcamp | YouTube
"Don't Worry About Me" from Don't (2025) – Spotify | Bandcamp | YouTube

Tee Teck Wei

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

External links

gene, 311-edo on Tonalsoft Encyclopedia

Notes

↑ Odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all 23-limit intervals linked)

References

↑ Tonalsoft Encyclopedia | gene, 311-edo

[2] Odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all 23-limit intervals linked)

[1] Tonalsoft Encyclopedia | gene, 311-edo

[1]

[note 1]

311edo: Difference between revisions

Revision as of 23:38, 8 November 2025

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