2964edo

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2964edo

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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 2^1/2964, 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.

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
4060088955/4060086272	[-25 7 1 0 -2 5⟩	0.001
1610510000/1610497161	[4 -4 4 -6 5 -2⟩	0.014
935886848/935859375	[15 -2 -7 0 -3 4⟩	0.051
3456000/3455881	[10 3 3 0 -2 -4⟩	0.060
479773125/479756288	[-15 10 4 0 -4 1⟩	0.061
188245551/188238400	[-6 1 -2 -6 0 7⟩	0.066
3132421875/3132286976	[-11 6 8 -6 1 -1⟩	0.075
218709491/218700000	[-5 -7 -5 6 1 2⟩	0.075
1771561/1771470	[-1 -11 -1 0 6⟩	0.089
100663296/100656875	[25 1 -4 0 -5⟩	0.110
2436053373/2435896375	[0 8 -3 0 -7 5⟩	0.112
536870912/536832387	[29 -3 0 -6 0 -2⟩	0.124
14236560/14235529	[4 4 1 -6 -2 3⟩	0.125
3764768/3764475	[5 -4 -2 6 -1 -2⟩	0.135
187404800/187388721	[9 -8 2 0 4 -4⟩	0.149
196625/196608	[-16 -1 3 0 2 1⟩	0.150
2358180864/2357947691	[10 11 0 0 -9 1⟩	0.171
2144153025/2143910912	[-10 6 2 6 -5 -1⟩	0.195
184549376/184528125	[24 -10 -5 0 1⟩	0.199
371293/371250	[-1 -3 -4 0 -1 5⟩	0.201
140625/140608	[-6 2 6 0 0 -3⟩	0.209
257330216/257298363	[3 -7 0 -6 4 3⟩	0.214
6656/6655	[9 0 -1 0 -3 1⟩	0.260
123904000/123884397	[13 -4 3 -6 2 -1⟩	0.274
481966875/481890304	[-12 3 4 -6 0 4⟩	0.275
6470695/6469632	[-11 -5 1 6 1 -1⟩	0.284
353894400/353829047	[19 3 2 0 -5 -3⟩	0.320
1247410125/1247178944	[-6 10 3 0 -7 2⟩	0.321
56953125/56942116	[-2 6 7 -6 -2⟩	0.335
4135597648/4134796875	[4 -7 -6 6 -2 3⟩	0.335
73530625/73516014	[-1 -2 4 6 -1 -5⟩	0.344
4429568/4428675	[8 -11 -2 0 3 1⟩	0.349
53094899/53084160	[-17 -4 -1 0 1 6⟩	0.350
708984375/708837376	[-22 1 9 0 2 -2⟩	0.359
1927561216/1927121625	[14 -4 -3 6 -4 -1⟩	0.395
14417920/14414517	[18 -8 1 0 1 -3⟩	0.409
4225/4224	[-7 -1 2 0 -1 2⟩	0.410
228765625/228709656	[-3 -5 6 -6 4⟩	0.424
428830605/428717762	[-1 6 1 6 -8⟩	0.456
1235663104/1235334375	[8 -3 -5 0 -4 6⟩	0.461
225000/224939	[3 2 5 0 -3 -2⟩	0.469
1286844416/1286491815	[12 -7 -1 -6 1 4⟩	0.474
3360173089/3359232000	[-12 -8 -3 6 0 4⟩	0.485
136088095/136048896	[-8 -12 1 0 5 2⟩	0.499
44302336/44289025	[18 0 -2 0 -6 2⟩	0.520
104857600/104825259	[22 -4 2 -6 -1⟩	0.534
1253113875/1252726552	[-3 3 3 -6 -3 5⟩	0.535
117649/117612	[-2 -5 0 6 -2⟩	0.545
3327500/3326427	[2 -9 4 0 3 -2⟩	0.558
75521875/75497472	[-23 -2 5 0 1 3⟩	0.559
3764768000/3763454409	[8 -2 3 6 -4 -4⟩	0.604
22151168/22143375	[17 -11 -3 0 0 2⟩	0.609
62748517/62726400	[-8 -4 -2 0 -2 7⟩	0.610
3520000000/3518743761	[12 -6 7 0 1 -6⟩	0.618
1171875/1171456	[-13 1 8 0 -1 -1⟩	0.619
1722499009/1721868840	[-3 -16 -1 6 4⟩	0.633
134217728/134165889	[27 -8 0 0 -2 -2⟩	0.669
43940/43923	[2 -1 1 0 -4 3⟩	0.670
3955078125/3953527864	[-3 4 11 0 -3 -5⟩	0.679
28600000/28588707	[6 -5 5 -6 1 1⟩	0.684
191179625/191102976	[-18 -6 3 6 0 1⟩	0.694
23040000/23030293	[12 2 4 0 -6 -1⟩	0.730
1838265625/1837465344	[-8 -3 6 6 -2 -3⟩	0.754
531674/531441	[1 -12 0 0 2 3⟩	0.759
4078653605/4076863488	[-24 -5 1 0 0 8⟩	0.760
537109375/536870912	[-29 0 11 0 1⟩	0.769
195767936/195676965	[7 -5 -1 6 -5 1⟩	0.805
256000/255879	[11 -9 3 0 0 -1⟩	0.818
17850625/17842176	[-14 -2 4 0 -2 4⟩	0.820
1631461442/1630641375	[1 -4 -3 0 -5 8⟩	0.870
234375/234256	[-4 1 7 0 -4⟩	0.879
772289440/771895089	[5 -8 1 -6 0 6⟩	0.884
537109375/536832387	[0 -3 11 -6 1 -2⟩	0.893
1076723648/1076168025	[6 -16 -2 6 1 1⟩	0.894
58492928/58461513	[11 -1 0 0 -7 4⟩	0.930
3461120000/3459233547	[15 -5 4 -6 -2 2⟩	0.944
497067025/496793088	[-9 -6 2 6 -3 2⟩	0.954
1890625/1889568	[-5 -10 6 0 2⟩	0.968
2359296000/2357947691	[21 2 3 0 -9⟩	0.990
735306250/734875713	[1 -3 5 6 -5 -2⟩	1.014
29246464/29229255	[10 -12 -1 0 -1 4⟩	1.019
62500000/62462907	[5 -7 9 0 0 -4⟩	1.028
126953125/126877696	[-20 0 10 0 -2 1⟩	1.029
26214400/26198073	[20 -9 2 0 -3⟩	1.079
46411625/46382688	[-5 -2 3 0 -5 5⟩	1.080
171640625/171532242	[-1 -6 7 -6 0 3⟩	1.094
808836875/808321761	[0 -14 4 6 1 -2⟩	1.103
19500000/19487171	[5 1 6 0 -7 1⟩	1.139
3489136640/3486784401	[19 -20 1 0 3⟩	1.168
102105575/102036672	[-6 -13 2 0 1 5⟩	1.169
1292374265/1291467969	[0 -6 1 6 -6 3⟩	1.214
650000/649539	[4 -10 5 0 -1 1⟩	1.228
330078125/329832448	[-11 0 9 0 -5 2⟩	1.289
1220703125/1219784832	[-7 -4 13 -6⟩	1.303
258474853/258280326	[-1 -17 0 6 0 3⟩	1.303
1930723600/1929229929	[4 -2 2 0 -8 6⟩	1.340
726171875/725594112	[-12 -11 8 0 1 2⟩	1.378
3489660928/3486784401	[28 -20 0 0 0 1⟩	1.428
193072360/192913083	[3 -13 1 0 -2 6⟩	1.429
48828125/48787596	[-2 -8 11 0 -1 -2⟩	1.438
865280000/864536409	[13 -10 4 0 -4 2⟩	1.488
1838265625/1836660096	[-7 -15 6 6⟩	1.513
858203125/857435524	[-2 0 8 0 -8 3⟩	1.549
171640625/171478296	[-3 -11 7 0 -2 3⟩	1.638
1250000000/1248774813	[7 -8 10 0 -4 -1⟩	1.698
1220703125/1219401216	[-9 -9 13 0 -2⟩	1.847
3173828125/3169966833	[0 -9 12 0 -5 1⟩	2.108