2964edo
2964 equal divisions of the octave (abbreviated 2964edo or 2964ed2), also called 2964-tone equal temperament (2964tet) or 2964 equal temperament (2964et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2964 equal parts of about 0.405 ¢ each. Each step represents a frequency ratio of 21/2964, or the 2964th root of 2.
| ← 2963edo | 2964edo | 2965edo → |
In the 13-limit, 2964edo shares the same patent val with 494edo except for the 7th harmonic, which is corrected to an extremely accurate result (absolute error 0.00000446 cents, relative error 0.0011%). 2964 is the denominator to a convergent to log27. Bordering 2964edo's patent val 7/1 on either side are 26edo's sharp approximation and 57edo's flat approximation of 7/1, having nearly identical 0.4048 ¢ errors; 2964edo exactly divides the octave into 26 and into 57 equal steps, splitting the difference between 160\57 and 73\26, as 2964 is expressible as 26 × 57 × 2.
A basis for the 13-limit commas consists of 4225/4224, 6656/6655, and 43940/43923.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.069 | -0.079 | +0.000 | +0.099 | -0.042 | -0.097 | +0.058 | +0.066 | -0.023 | -0.096 |
| Relative (%) | +0.0 | +17.1 | -19.5 | +0.0 | +24.5 | -10.3 | -24.0 | +14.3 | +16.2 | -5.6 | -23.8 | |
| Steps (reduced) |
2964 (0) |
4698 (1734) |
6882 (954) |
8321 (2393) |
10254 (1362) |
10968 (2076) |
12115 (259) |
12591 (735) |
13408 (1552) |
14399 (2543) |
14684 (2828) | |
Subsets and supersets
Since 2964 factors into 22 × 3 × 13 × 19, 2964edo has subset edos 2, 3, 4, 6, 12, 13, 19, 26, 38, 39, 52, 57, 76, 78, 114, 156, 228, 247, 494, 741, 988, and 1482.
Commas of the 2.3.5.7.11.13 subgroup tempered out in 2964edo patent val
Commas with numerator <= 2^32:
| Ratio | Monzo | Cents |
|---|---|---|
| 4225/4224 | [-7 -1 2 0 -1 2⟩ | 0.410 |
| 6656/6655 | [9 0 -1 0 -3 1⟩ | 0.260 |
| 43940/43923 | [2 -1 1 0 -4 3⟩ | 0.670 |
| 117649/117612 | [-2 -5 0 6 -2⟩ | 0.545 |
| 140625/140608 | [-6 2 6 0 0 -3⟩ | 0.209 |
| 196625/196608 | [-16 -1 3 0 2 1⟩ | 0.150 |
| 225000/224939 | [3 2 5 0 -3 -2⟩ | 0.469 |
| 234375/234256 | [-4 1 7 0 -4⟩ | 0.879 |
| 256000/255879 | [11 -9 3 0 0 -1⟩ | 0.818 |
| 371293/371250 | [-1 -3 -4 0 -1 5⟩ | 0.201 |
| 531674/531441 | [1 -12 0 0 2 3⟩ | 0.759 |
| 650000/649539 | [4 -10 5 0 -1 1⟩ | 1.228 |
| 1171875/1171456 | [-13 1 8 0 -1 -1⟩ | 0.619 |
| 1771561/1771470 | [-1 -11 -1 0 6⟩ | 0.089 |
| 1890625/1889568 | [-5 -10 6 0 2⟩ | 0.968 |
| 3327500/3326427 | [2 -9 4 0 3 -2⟩ | 0.558 |
| 3456000/3455881 | [10 3 3 0 -2 -4⟩ | 0.060 |
| 3764768/3764475 | [5 -4 -2 6 -1 -2⟩ | 0.135 |
| 4429568/4428675 | [8 -11 -2 0 3 1⟩ | 0.349 |
| 6470695/6469632 | [-11 -5 1 6 1 -1⟩ | 0.284 |
| 14236560/14235529 | [4 4 1 -6 -2 3⟩ | 0.125 |
| 14417920/14414517 | [18 -8 1 0 1 -3⟩ | 0.409 |
| 17850625/17842176 | [-14 -2 4 0 -2 4⟩ | 0.820 |
| 19500000/19487171 | [5 1 6 0 -7 1⟩ | 1.139 |
| 22151168/22143375 | [17 -11 -3 0 0 2⟩ | 0.609 |
| 23040000/23030293 | [12 2 4 0 -6 -1⟩ | 0.730 |
| 26214400/26198073 | [20 -9 2 0 -3⟩ | 1.079 |
| 28600000/28588707 | [6 -5 5 -6 1 1⟩ | 0.684 |
| 29246464/29229255 | [10 -12 -1 0 -1 4⟩ | 1.019 |
| 44302336/44289025 | [18 0 -2 0 -6 2⟩ | 0.520 |
| 46411625/46382688 | [-5 -2 3 0 -5 5⟩ | 1.080 |
| 48828125/48787596 | [-2 -8 11 0 -1 -2⟩ | 1.438 |
| 53094899/53084160 | [-17 -4 -1 0 1 6⟩ | 0.350 |
| 56953125/56942116 | [-2 6 7 -6 -2⟩ | 0.335 |
| 58492928/58461513 | [11 -1 0 0 -7 4⟩ | 0.930 |
| 62500000/62462907 | [5 -7 9 0 0 -4⟩ | 1.028 |
| 62748517/62726400 | [-8 -4 -2 0 -2 7⟩ | 0.610 |
| 73530625/73516014 | [-1 -2 4 6 -1 -5⟩ | 0.344 |
| 75521875/75497472 | [-23 -2 5 0 1 3⟩ | 0.559 |
| 100663296/100656875 | [25 1 -4 0 -5⟩ | 0.110 |
| 102105575/102036672 | [-6 -13 2 0 1 5⟩ | 1.169 |
| 104857600/104825259 | [22 -4 2 -6 -1⟩ | 0.534 |
| 123904000/123884397 | [13 -4 3 -6 2 -1⟩ | 0.274 |
| 126953125/126877696 | [-20 0 10 0 -2 1⟩ | 1.029 |
| 134217728/134165889 | [27 -8 0 0 -2 -2⟩ | 0.669 |
| 136088095/136048896 | [-8 -12 1 0 5 2⟩ | 0.499 |
| 171640625/171478296 | [-3 -11 7 0 -2 3⟩ | 1.638 |
| 171640625/171532242 | [-1 -6 7 -6 0 3⟩ | 1.094 |
| 184549376/184528125 | [24 -10 -5 0 1⟩ | 0.199 |
| 187404800/187388721 | [9 -8 2 0 4 -4⟩ | 0.149 |
| 188245551/188238400 | [-6 1 -2 -6 0 7⟩ | 0.066 |
| 191179625/191102976 | [-18 -6 3 6 0 1⟩ | 0.694 |
| 193072360/192913083 | [3 -13 1 0 -2 6⟩ | 1.429 |
| 195767936/195676965 | [7 -5 -1 6 -5 1⟩ | 0.805 |
| 218709491/218700000 | [-5 -7 -5 6 1 2⟩ | 0.075 |
| 228765625/228709656 | [-3 -5 6 -6 4⟩ | 0.424 |
| 257330216/257298363 | [3 -7 0 -6 4 3⟩ | 0.214 |
| 258474853/258280326 | [-1 -17 0 6 0 3⟩ | 1.303 |
| 330078125/329832448 | [-11 0 9 0 -5 2⟩ | 1.289 |
| 353894400/353829047 | [19 3 2 0 -5 -3⟩ | 0.320 |
| 428830605/428717762 | [-1 6 1 6 -8⟩ | 0.456 |
| 479773125/479756288 | [-15 10 4 0 -4 1⟩ | 0.061 |
| 481966875/481890304 | [-12 3 4 -6 0 4⟩ | 0.275 |
| 497067025/496793088 | [-9 -6 2 6 -3 2⟩ | 0.954 |
| 536870912/536832387 | [29 -3 0 -6 0 -2⟩ | 0.124 |
| 537109375/536832387 | [0 -3 11 -6 1 -2⟩ | 0.893 |
| 537109375/536870912 | [-29 0 11 0 1⟩ | 0.769 |
| 708984375/708837376 | [-22 1 9 0 2 -2⟩ | 0.359 |
| 726171875/725594112 | [-12 -11 8 0 1 2⟩ | 1.378 |
| 735306250/734875713 | [1 -3 5 6 -5 -2⟩ | 1.014 |
| 772289440/771895089 | [5 -8 1 -6 0 6⟩ | 0.884 |
| 808836875/808321761 | [0 -14 4 6 1 -2⟩ | 1.103 |
| 858203125/857435524 | [-2 0 8 0 -8 3⟩ | 1.549 |
| 865280000/864536409 | [13 -10 4 0 -4 2⟩ | 1.488 |
| 935886848/935859375 | [15 -2 -7 0 -3 4⟩ | 0.051 |
| 1076723648/1076168025 | [6 -16 -2 6 1 1⟩ | 0.894 |
| 1220703125/1219401216 | [-9 -9 13 0 -2⟩ | 1.847 |
| 1220703125/1219784832 | [-7 -4 13 -6⟩ | 1.303 |
| 1235663104/1235334375 | [8 -3 -5 0 -4 6⟩ | 0.461 |
| 1247410125/1247178944 | [-6 10 3 0 -7 2⟩ | 0.321 |
| 1250000000/1248774813 | [7 -8 10 0 -4 -1⟩ | 1.698 |
| 1253113875/1252726552 | [-3 3 3 -6 -3 5⟩ | 0.535 |
| 1286844416/1286491815 | [12 -7 -1 -6 1 4⟩ | 0.474 |
| 1292374265/1291467969 | [0 -6 1 6 -6 3⟩ | 1.214 |
| 1610510000/1610497161 | [4 -4 4 -6 5 -2⟩ | 0.014 |
| 1631461442/1630641375 | [1 -4 -3 0 -5 8⟩ | 0.870 |
| 1722499009/1721868840 | [-3 -16 -1 6 4⟩ | 0.633 |
| 1838265625/1836660096 | [-7 -15 6 6⟩ | 1.513 |
| 1838265625/1837465344 | [-8 -3 6 6 -2 -3⟩ | 0.754 |
| 1927561216/1927121625 | [14 -4 -3 6 -4 -1⟩ | 0.395 |
| 1930723600/1929229929 | [4 -2 2 0 -8 6⟩ | 1.340 |
| 2144153025/2143910912 | [-10 6 2 6 -5 -1⟩ | 0.195 |
| 2358180864/2357947691 | [10 11 0 0 -9 1⟩ | 0.171 |
| 2359296000/2357947691 | [21 2 3 0 -9⟩ | 0.990 |
| 2436053373/2435896375 | [0 8 -3 0 -7 5⟩ | 0.112 |
| 3132421875/3132286976 | [-11 6 8 -6 1 -1⟩ | 0.075 |
| 3173828125/3169966833 | [0 -9 12 0 -5 1⟩ | 2.108 |
| 3360173089/3359232000 | [-12 -8 -3 6 0 4⟩ | 0.485 |
| 3461120000/3459233547 | [15 -5 4 -6 -2 2⟩ | 0.944 |
| 3489136640/3486784401 | [19 -20 1 0 3⟩ | 1.168 |
| 3489660928/3486784401 | [28 -20 0 0 0 1⟩ | 1.428 |
| 3520000000/3518743761 | [12 -6 7 0 1 -6⟩ | 0.618 |
| 3764768000/3763454409 | [8 -2 3 6 -4 -4⟩ | 0.604 |
| 3955078125/3953527864 | [-3 4 11 0 -3 -5⟩ | 0.679 |
| 4060088955/4060086272 | [-25 7 1 0 -2 5⟩ | 0.001 |
| 4078653605/4076863488 | [-24 -5 1 0 0 8⟩ | 0.760 |
| 4135597648/4134796875 | [4 -7 -6 6 -2 3⟩ | 0.335 |