311edo

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← 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
Special properties

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 21/311, 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.

311edo's step size is sometimes called a gene, in honor of Gene Ward Smith, when used as an interval size unit.

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 S25 (625/624), 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.

It is also the lowest edo that maintains relative interval errors of no greater than 25% on all of the first 42 harmonics of the harmonic series. The next lowest edo that approximates the 43rd harmonic while maintaining the same maximum relative errors on the 42nd and lower is 20567, and the smallest edo that maintains less than 25% relative error on the first 64 harmonics is 3159811.

It is still very accurate in the lower limits. Although it does not do as well as 270edo in the 13-limit, it makes for an interesting comparison. The equal temperament 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 conincidentally close enough to the 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 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

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 perfectly consistent, to which a variety of odds can be added that keep perfect 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. Therefore, the very beginning of the table (from 0\311 to 3\311 inclusive) is the only part that is not algorithmically generated.

Interval table

Table of 311edo intervals
Genes[note 1] Cents Marks Approximate Intervals[note 2]
0 0.0 P1 1/1, S41, S40, S38, S37, S35 = S49*S50, 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

Sagittal notation 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

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.

  • 0\311 = P1 = perfect unison
  • 1\311 = ^1 = upunison
  • 2\311 = ^^1 = dup unison
  • 3\311 = ^31 = trup unison (or ^^^1)
  • 4\311 = ^41 = quup unison (or v>, or downquip)
  • 5\311 = >1 = quip unison
  • 6\311 = ^>1 = upquip unison
  • 7\311 = ^^> = dupquip unison

TO DO: complete this


8\311 = ^3>1 = dup-dubdrop unison

9\311 = ^^\1 = dupdrop unison

10\311 = ^^1 = dup unison

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

12\311 = ^^//1 = dup-dublift unison = vv\m2 = duddropminor second

13\311 = vvm2 = dudminor second

14\311 = vv/m2 = dudliftminor second

15\311 = vv//m2 = dud-dubliftminor second

16\311 = v\\m2 = down-dubdropminor second

17\311 = v\m2 = downdropminor second

18\311 = vm2 = downminor second

19\311 = v/m2 = downliftminor second

20\311 = v//m2 = down-dubliftminor second

21\311 = \\m2 = dubdropminor second

22\311 = \m2 = dropminor second

23\311 = m2 = minor second

24\311 = /m2 = liftminor second

25\311 = //m2 = dubliftminor second

26\311 = ^\\m2 = up-dubdropminor second

27\311 = ^\m2 = updropminor second

28\311 = ^m2 = upminor second

29\311 = ^/m2 = upliftminor second

30\311 = ^//m2 = up-dubliftminor second

31\311 = ^^\\m2 = dup-dubdropminor second = v\\~2 = down-dubdropmid second

32\311 = ^^\m2 = dupdropminor second = v\~2 = downdropmid second

33\311 = ^^m2 = dupminor second = v~2 = downmid second

34\311 = ^^/m2 = dupliftminor second = v/~2 = downliftmid second

35\311 = ^^//m2 = dup-dubliftminor second = v//~2 = down-dubliftmid second

36\311 = \\~2 = dubdropmid second

37\311 = \~2 = dropmid second

38\311 = ~2 = mid second

etc.

JI approximation

Interval mappings

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
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 it is 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.

Detemperaments

Ringer scales

There are two known Ringer scales based on 311edo. Both include 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. However, if one is looking for a Ringer scale, 217edo's corresponding Ringer 217 may be preferred due to mapping all of mode 167, which is exceptional for its size (as a Ringer scale), just as 311edo is exceptional for its size (as an edo interpreted as a temperament; it's unknown whether Ringer 311 possesses any special properties that are not a direct result of 311edo's special properties as a temperament).

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
Francium
Tee Teck Wei

External links

Notes

  1. ↑ Gene is the interval size measure for 311edo, named after Gene Ward Smith
  2. ↑ Odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all 23-limit intervals linked)
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