2964edo: Difference between revisions

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Revision as of 23:43, 8 November 2025

← 2963edo 2964edo 2965edo →
Prime factorization 22 × 3 × 13 × 19
Step size 0.404858 ¢ 
Fifth 1734\2964 (702.024 ¢) (→ 289\494)
Semitones (A1:m2) 282:222 (114.2 ¢ : 89.88 ¢)
Consistency limit 7
Distinct consistency limit 7

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.

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

Approximation of prime harmonics in 2964edo
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
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