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In the 13-limit, 2964edo shares the same [[patent val]] with [[494edo]] except for the [[7/1|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 log<sub>2</sub>7. 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{{c}} 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 {{nowrap|26 × 57 × 2}}.  
In the 13-limit, 2964edo shares the same [[patent val]] with [[494edo]] except for the [[7/1|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 log<sub>2</sub>7. 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{{c}} 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 {{nowrap|26 × 57 × 2}}.  
A basis for the 13-limit commas consists of [[4225/4224]], [[6656/6655]], and [[43940/43923]].


=== Prime harmonics ===
=== Prime harmonics ===
Line 11: Line 9:
=== Subsets and supersets ===
=== Subsets and supersets ===
Since 2964 factors into {{factorization|2964}}, 2964edo has subset edos {{EDOs| 2, 3, 4, 6, 12, 13, 19, 26, 38, 39, 52, 57, 76, 78, 114, 156, 228, 247, 494, 741, 988, and 1482 }}.
Since 2964 factors into {{factorization|2964}}, 2964edo has subset edos {{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:
{| class="wikitable mw-collapsible mw-collapsed sortable"
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
|-
| [[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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