311edo
| ← 310edo | 311edo | 312edo → |
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
| 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
| 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
One possible notation using ups and downs notation uses ^ and v (ups and down) to stand for 5 edosteps and / and \ (lifts and drops) to stand for 1 edostep. Double is abbreviated as "dub-".
0\311 = P1 = perfect unison
1\311 = /1 = lift unison
2\311 = //1 = dublift unison
3\311 = ^\\1 = up-dubdrop unison
4\311 = ^\1 = updrop unison
5\311 = ^1 = up unison
6\311 = ^/1 = uplift unison
7\311 = ^//1 = up-dublift unison
8\311 = ^^\\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.
| 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
| 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: |
Reduced to mode 234: 234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2: |
Ringer 311[+61, −67]
|
Scale as chord: 936:940:941:943:944:948:950:952:954:956:958:960:962: |
Reduced to mode 234: 234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2: |
Music
- Etude in C, Op. 1, No. 1 (2022)
- Baoyu（𨰻𨰻） (2023) – for electric organs tuned in 311edo
External links
Notes
- ↑ Gene is the interval size measure for 311edo, named after Gene Ward Smith
- ↑ Odd harmonics and subharmonics are in bold and linked, inconsistent intervals in italics, all 23-limit intervals linked)