684edo: Difference between revisions

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* [[15-odd-limit|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64
* [[15-odd-limit|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64
== Commas of the 2.3.5.7.11.13 subgroup tempered out in 684edo patent val ==
Commas with numerator <= 2^32:
{| class="wikitable mw-collapsible mw-collapsed sortable"
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
|-
| [[1610510000/1610497161]]
| [4 -4 4 -6 5 -2⟩
| 0.014
|-
| [[123201/123200]]
| [-6 6 -2 -1 -1 2⟩
| 0.014
|-
| [[781258401/781250000]]
| [-4 2 -11 2 6⟩
| 0.019
|-
| [[3294225/3294172]]
| [-2 2 2 -7 4⟩
| 0.028
|-
| [[1475824779/1475789056]]
| [-8 8 0 -8 3 2⟩
| 0.042
|-
| [[1433600000/1433562273]]
| [16 -3 5 1 -1 -6⟩
| 0.046
|-
| [[3456000/3455881]]
| [10 3 3 0 -2 -4⟩
| 0.060
|-
| [[199297406/199290375]]
| [1 -13 -3 7 2⟩
| 0.061
|-
| [[78125000/78121827]]
| [3 -13 10 -2⟩
| 0.070
|-
| [[1574640/1574573]]
| [4 9 1 -1 -3 -2⟩
| 0.074
|-
| [[218709491/218700000]]
| [-5 -7 -5 6 1 2⟩
| 0.075
|-
| [[66015625/66012408]]
| [-3 -7 8 -3 -1 2⟩
| 0.084
|-
| [[14348907/14348180]]
| [-2 15 -1 -2 -4⟩
| 0.088
|-
| [[1771561/1771470]]
| [-1 -11 -1 0 6⟩
| 0.089
|-
| [[480024727/480000000]]
| [-11 -1 -7 5 0 4⟩
| 0.089
|-
| [[1650478599/1650390625]]
| [0 11 -10 1 3 -2⟩
| 0.092
|-
| [[446265625/446240256]]
| [-9 -1 6 -4 -2 4⟩
| 0.098
|-
| [[974308500/974251369]]
| [2 11 3 -8 1 -2⟩
| 0.102
|-
| [[27217619/27216000]]
| [-7 -5 -3 -1 5 2⟩
| 0.103
|-
| [[645700815/645657712]]
| [-4 17 1 -9⟩
| 0.116
|-
| [[1254484803/1254400000]]
| [-13 1 -5 -2 4 4⟩
| 0.117
|-
| [[1686616064/1686498489]]
| [11 -10 0 7 0 -4⟩
| 0.121
|-
| [[214375000/214358881]]
| [3 0 7 3 -8⟩
| 0.130
|-
| [[3764768/3764475]]
| [5 -4 -2 6 -1 -2⟩
| 0.135
|-
| [[187404800/187388721]]
| [9 -8 2 0 4 -4⟩
| 0.149
|-
| [[151263/151250]]
| [-1 2 -4 5 -2⟩
| 0.149
|-
| [[35156250/35153041]]
| [1 2 9 -4 -4⟩
| 0.158
|-
| [[1224552875/1224440064]]
| [-8 -14 3 3 0 4⟩
| 0.159
|-
| [[10648/10647]]
| [3 -2 0 -1 3 -2⟩
| 0.163
|-
| [[2662250409/2662000000]]
| [-7 8 -6 4 -3 2⟩
| 0.163
|-
| [[1182568205/1182449664]]
| [-14 -8 1 2 -1 6⟩
| 0.174
|-
| [[9801/9800]]
| [-3 4 -2 -2 2⟩
| 0.177
|-
| [[3955078125/3954653486]]
| [-1 4 11 -11⟩
| 0.186
|-
| [[974358550/974251369]]
| [1 0 2 -8 7 -2⟩
| 0.190
|-
| [[109772091/109760000]]
| [-9 10 -4 -3 1 2⟩
| 0.191
|-
| [[1291467969/1291315424]]
| [-5 6 0 -9 6⟩
| 0.205
|-
| [[2091059712/2090808005]]
| [9 5 -1 5 -4 -4⟩
| 0.208
|-
| [[21437500/21434787]]
| [2 -11 6 3 -2⟩
| 0.219
|-
| [[33792000/33787663]]
| [13 1 3 -1 1 -6⟩
| 0.222
|-
| [[3402639576/3402202375]]
| [3 11 -3 4 -5 -2⟩
| 0.222
|-
| [[5175625/5174928]]
| [-4 -5 4 2 -3 2⟩
| 0.233
|-
| [[1399680/1399489]]
| [7 7 1 -2 0 -4⟩
| 0.236
|-
| [[246071287/246037500]]
| [-2 -9 -5 5 4⟩
| 0.238
|-
| [[47265625/47258883]]
| [0 -9 8 -4 2⟩
| 0.247
|-
| [[14994525/14992384]]
| [-10 1 2 1 -4 4⟩
| 0.247
|-
| [[3188646/3188185]]
| [1 13 -1 -3 -1 -2⟩
| 0.250
|-
| [[540078539/540000000]]
| [-8 -3 -7 4 3 2⟩
| 0.252
|-
| [[29046875/29042496]]
| [-6 -3 6 -5 1 2⟩
| 0.261
|-
| [[1162261467/1162084000]]
| [-5 19 -3 -4 -2⟩
| 0.264
|-
| [[214358881/214326000]]
| [-4 -7 -3 -2 8⟩
| 0.266
|-
| [[481966875/481890304]]
| [-12 3 4 -6 0 4⟩
| 0.275
|-
| [[3293331899/3292800000]]
| [-10 -1 -5 -3 7 2⟩
| 0.280
|-
| [[245000000/244958571]]
| [6 -2 7 2 -5 -2⟩
| 0.293
|-
| [[520613632/520524225]]
| [8 -6 -2 5 2 -4⟩
| 0.297
|-
| [[1771875/1771561]]
| [0 4 5 1 -6⟩
| 0.307
|-
| [[105644/105625]]
| [2 0 -4 4 1 -2⟩
| 0.311
|-
| [[31250000/31244213]]
| [4 0 9 -5 -1 -2⟩
| 0.321
|-
| [[1247410125/1247178944]]
| [-6 10 3 0 -7 2⟩
| 0.321
|-
| [[1377751375/1377495072]]
| [-5 -16 3 2 3 2⟩
| 0.322
|-
| [[113379904/113358609]]
| [6 -4 0 -2 6 -4⟩
| 0.325
|-
| [[250047/250000]]
| [-4 6 -6 3⟩
| 0.325
|-
| [[56953125/56942116]]
| [-2 6 7 -6 -2⟩
| 0.335
|-
| [[120955835/120932352]]
| [-11 -10 1 1 2 4⟩
| 0.336
|-
| [[1449459/1449175]]
| [0 2 -2 -3 5 -2⟩
| 0.339
|-
| [[53094899/53084160]]
| [-17 -4 -1 0 1 6⟩
| 0.350
|-
| [[96059601/96040000]]
| [-6 8 -4 -4 4⟩
| 0.353
|-
| [[288240050/288178803]]
| [1 -9 2 8 -4⟩
| 0.368
|-
| [[265531392/265474495]]
| [12 3 -1 4 -1 -6⟩
| 0.371
|-
| [[269500000/269440587]]
| [5 -13 6 2 1 -2⟩
| 0.382
|-
| [[139178767/139148064]]
| [-5 -3 0 7 -5 2⟩
| 0.382
|-
| [[432081216/431985125]]
| [6 9 -3 3 -2 -4⟩
| 0.385
|-
| [[4375/4374]]
| [-1 -7 4 1⟩
| 0.396
|-
| [[1655976960/1655595487]]
| [10 5 1 -3 3 -6⟩
| 0.399
|-
| [[703096443/702934375]]
| [0 15 -5 2 -3 -2⟩
| 0.399
|-
| [[4225/4224]]
| [-7 -1 2 0 -1 2⟩
| 0.410
|-
| [[342956592/342874805]]
| [4 11 -1 -4 2 -4⟩
| 0.413
|-
| [[607645423/607500000]]
| [-5 -5 -7 3 6⟩
| 0.414
|-
| [[228765625/228709656]]
| [-3 -5 6 -6 4⟩
| 0.424
|-
| [[6940323/6938624]]
| [-13 5 0 -1 -2 4⟩
| 0.424
|-
| [[1420541793/1420191500]]
| [-2 17 -3 -5 1 -2⟩
| 0.427
|-
| [[4000989993/4000000000]]
| [-11 1 -9 2 5 2⟩
| 0.428
|-
| [[421760625/421654016]]
| [-9 1 4 -7 3 2⟩
| 0.438
|-
| [[3294172000/3293331899]]
| [5 0 3 7 -7 -2⟩
| 0.442
|-
| [[280000000/279926361]]
| [9 -4 7 1 -2 -4⟩
| 0.455
|-
| [[428830605/428717762]]
| [-1 6 1 6 -8⟩
| 0.456
|-
| [[3487704605/3486784401]]
| [0 -20 1 8 2⟩
| 0.457
|-
| [[726364926/726171875]]
| [1 2 -8 9 -1 -2⟩
| 0.460
|-
| [[225000/224939]]
| [3 2 5 0 -3 -2⟩
| 0.469
|-
| [[1530966437/1530550080]]
| [-6 -14 -1 7 1 2⟩
| 0.471
|-
| [[160699616/160655625]]
| [5 -2 -4 3 4 -4⟩
| 0.474
|-
| [[820125/819896]]
| [-3 8 3 -1 -4⟩
| 0.483
|-
| [[1550115875/1549681956]]
| [-2 -18 3 1 6⟩
| 0.485
|-
| [[3360173089/3359232000]]
| [-12 -8 -3 6 0 4⟩
| 0.485
|-
| [[5282739/5281250]]
| [-1 4 -6 2 3 -2⟩
| 0.488
|-
| [[139218750/139178767]]
| [1 4 7 -7 1 -2⟩
| 0.497
|-
| [[4041608805/4040447488]]
| [-9 14 1 -2 -5 2⟩
| 0.498
|-
| [[136088095/136048896]]
| [-8 -12 1 0 5 2⟩
| 0.499
|-
| [[1714871048/1714374025]]
| [3 0 -2 -4 8 -4⟩
| 0.502
|-
| [[50014503/50000000]]
| [-7 10 -8 1 2⟩
| 0.502
|-
| [[184528125/184473632]]
| [-5 10 5 -8⟩
| 0.511
|-
| [[418161601/418037760]]
| [-14 -6 -1 -1 4 4⟩
| 0.513
|-
| [[329417200/329316273]]
| [4 -11 2 7 -1 -2⟩
| 0.530
|-
| [[4079910912/4078653605]]
| [15 1 -1 3 2 -8⟩
| 0.534
|-
| [[117649/117612]]
| [-2 -5 0 6 -2⟩
| 0.545
|-
| [[603542016/603351125]]
| [9 7 -3 2 1 -6⟩
| 0.548
|-
| [[1220703125/1220312709]]
| [0 -5 13 -3 -4⟩
| 0.554
|-
| [[3327500/3326427]]
| [2 -9 4 0 3 -2⟩
| 0.558
|-
| [[8521149/8518400]]
| [-8 1 -2 5 -3 2⟩
| 0.559
|-
| [[89282088/89253125]]
| [3 13 -5 1 0 -4⟩
| 0.562
|-
| [[3025/3024]]
| [-4 -3 2 -1 2⟩
| 0.572
|-
| [[1162261467/1161875000]]
| [-3 19 -7 0 -1 -2⟩
| 0.576
|-
| [[50193/50176]]
| [-10 3 0 -2 1 2⟩
| 0.586
|-
| [[1500512167/1500000000]]
| [-8 -1 -9 1 8⟩
| 0.591
|-
| [[1107225625/1106841792]]
| [-6 -1 4 -8 6⟩
| 0.600
|-
| [[562166163/561971200]]
| [-16 9 -2 -3 0 4⟩
| 0.601
|-
| [[3764768000/3763454409]]
| [8 -2 3 6 -4 -4⟩
| 0.604
|-
| [[3520000000/3518743761]]
| [12 -6 7 0 1 -6⟩
| 0.618
|-
| [[27227340/27217619]]
| [2 4 1 5 -5 -2⟩
| 0.618
|-
| [[200000/199927]]
| [6 0 5 -1 0 -4⟩
| 0.632
|-
| [[141776649/141724880]]
| [-4 10 -1 4 -6⟩
| 0.632
|-
| [[1722499009/1721868840]]
| [-3 -16 -1 6 4⟩
| 0.633
|-
| [[91125/91091]]
| [0 6 3 -2 -1 -2⟩
| 0.646
|-
| [[3780549773/3779136000]]
| [-9 -10 -3 5 3 2⟩
| 0.648
|-
| [[446433372/446265625]]
| [2 2 -6 1 6 -4⟩
| 0.651
|-
| [[2657205/2656192]]
| [-6 12 1 -3 -2⟩
| 0.660
|-
| [[1071794405/1071385056]]
| [-5 -14 1 -1 8⟩
| 0.661
|-
| [[1056655611/1056250000]]
| [-4 8 -8 0 5 -2⟩
| 0.665
|-
| [[3293331899/3292047360]]
| [-11 -8 -1 -2 7 2⟩
| 0.675
|-
| [[1977326743/1976535000]]
| [-3 -3 -4 11 -4⟩
| 0.693
|-
| [[95703125/95664294]]
| [-1 -3 9 2 -6⟩
| 0.703
|-
| [[369754/369603]]
| [1 -7 0 5 1 -2⟩
| 0.707
|-
| [[2495390625/2494357888]]
| [-7 3 7 1 -7 2⟩
| 0.717
|-
| [[2401/2400]]
| [-5 -1 -2 4⟩
| 0.721
|-
| [[48828125/48807528]]
| [-3 -1 11 -5 -2⟩
| 0.730
|-
| [[4026275/4024566]]
| [-1 -5 2 -2 5 -2⟩
| 0.735
|-
| [[14085981/14080000]]
| [-11 5 -4 3 -1 2⟩
| 0.735
|-
| [[43923/43904]]
| [-7 1 0 -3 4⟩
| 0.749
|-
| [[4078653605/4076863488]]
| [-24 -5 1 0 0 8⟩
| 0.760
|-
| [[491941593/491724800]]
| [-13 7 -2 -4 3 2⟩
| 0.763
|-
| [[3457440/3455881]]
| [5 2 1 4 -2 -4⟩
| 0.781
|-
| [[19140625/19131876]]
| [-2 -14 8 2⟩
| 0.792
|-
| [[2129600000/2128622769]]
| [9 -2 5 -2 3 -6⟩
| 0.795
|-
| [[2250423/2249390]]
| [-1 8 -1 3 -3 -2⟩
| 0.795
|-
| [[18484375/18475776]]
| [-8 -8 6 1 -1 2⟩
| 0.806
|-
| [[9801000/9796423]]
| [3 4 3 -3 2 -4⟩
| 0.809
|-
| [[234365481/234256000]]
| [-7 14 -3 2 -4⟩
| 0.809
|-
| [[4253517961/4251528000]]
| [-6 -12 -3 4 6⟩
| 0.810
|-
| [[17850625/17842176]]
| [-14 -2 4 0 -2 4⟩
| 0.820
|-
| [[3247695/3246152]]
| [-3 10 1 -4 1 -2⟩
| 0.823
|-
| [[43046721/43025920]]
| [-9 16 -1 -5⟩
| 0.837
|-
| [[2573571875/2572306572]]
| [-2 -1 5 7 -8⟩
| 0.851
|-
| [[282475249/282335625]]
| [0 -5 -4 10 -1 -2⟩
| 0.856
|-
| [[54687500/54660177]]
| [2 -5 9 1 -3 -2⟩
| 0.865
|-
| [[562448656/562166163]]
| [4 -9 0 4 4 -4⟩
| 0.870
|-
| [[121060821/121000000]]
| [-6 1 -6 9 -2⟩
| 0.870
|-
| [[234375/234256]]
| [-4 1 7 0 -4⟩
| 0.879
|-
| [[456533/456300]]
| [-2 -3 -2 3 3 -2⟩
| 0.884
|-
| [[537109375/536832387]]
| [0 -3 11 -6 1 -2⟩
| 0.893
|-
| [[1155009375/1154413568]]
| [-10 7 5 -1 -5 2⟩
| 0.893
|-
| [[160083/160000]]
| [-8 3 -4 2 2⟩
| 0.898
|-
| [[52734375/52706752]]
| [-6 3 9 -7⟩
| 0.907
|-
| [[19487171/19476912]]
| [-4 -1 0 -4 7 -2⟩
| 0.912
|-
| [[256134879/256000000]]
| [-14 9 -6 1 1 2⟩
| 0.912
|-
| [[430489323/430259200]]
| [-10 5 -2 -5 6⟩
| 0.926
|-
| [[514714375/514434888]]
| [-3 -12 4 7 -2⟩
| 0.940
|-
| [[4829440/4826809]]
| [8 0 1 3 1 -6⟩
| 0.943
|-
| [[497067025/496793088]]
| [-9 -6 2 6 -3 2⟩
| 0.954
|-
| [[142884/142805]]
| [2 6 -1 2 0 -4⟩
| 0.957
|-
| [[1890625/1889568]]
| [-5 -10 6 0 2⟩
| 0.968
|-
| [[480024727/479756288]]
| [-15 0 0 5 -4 4⟩
| 0.968
|-
| [[3720087/3718000]]
| [-4 12 -3 1 -1 -2⟩
| 0.972
|-
| [[1161875/1161216]]
| [-11 -4 4 -1 1 2⟩
| 0.982
|-
| [[480298005/480024727]]
| [0 8 1 -5 4 -4⟩
| 0.985
|-
| [[387420489/387200000]]
| [-10 18 -5 0 -2⟩
| 0.986
|-
| [[6426225/6422528]]
| [-17 2 2 -2 0 4⟩
| 0.996
|-
| [[735306250/734875713]]
| [1 -3 5 6 -5 -2⟩
| 1.014
|-
| [[62500000/62462907]]
| [5 -7 9 0 0 -4⟩
| 1.028
|-
| [[56723625/56689952]]
| [-5 3 3 5 -6⟩
| 1.028
|-
| [[63412811/63375000]]
| [-3 -1 -6 8 1 -2⟩
| 1.033
|-
| [[78125/78078]]
| [-1 -1 7 -1 -1 -2⟩
| 1.042
|-
| [[173612978/173508075]]
| [1 -5 -2 2 6 -4⟩
| 1.046
|-
| [[200120949/200000000]]
| [-9 5 -8 7⟩
| 1.047
|-
| [[759375/758912]]
| [-7 5 5 -2 -2⟩
| 1.056
|-
| [[3382071/3380000]]
| [-5 1 -4 1 5 -2⟩
| 1.060
|-
| [[3742230375/3739918336]]
| [-13 11 3 -3 -3 2⟩
| 1.070
|-
| [[32019867/32000000]]
| [-11 7 -6 0 4⟩
| 1.074
|-
| [[808836875/808321761]]
| [0 -14 4 6 1 -2⟩
| 1.103
|-
| [[420175/419904]]
| [-6 -8 2 5⟩
| 1.117
|-
| [[24149664/24134045]]
| [5 4 -1 1 3 -6⟩
| 1.120
|-
| [[2516421875/2514778812]]
| [-2 -12 6 -1 5 -2⟩
| 1.131
|-
| [[405769/405504]]
| [-12 -2 0 4 -1 2⟩
| 1.131
|-
| [[7144929/7140250]]
| [-1 10 -3 0 2 -4⟩
| 1.134
|-
| [[9150625/9144576]]
| [-8 -6 4 -2 4⟩
| 1.145
|-
| [[793510263/792985600]]
| [-18 4 -2 3 -2 4⟩
| 1.145
|-
| [[473513931/473200000]]
| [-7 16 -5 -1 1 -2⟩
| 1.148
|-
| [[5623475/5619712]]
| [-14 0 2 -3 3 2⟩
| 1.159
|-
| [[2519337249/2517630976]]
| [-20 6 0 -4 2 4⟩
| 1.173
|-
| [[840350000/839779083]]
| [4 -5 5 5 -2 -4⟩
| 1.177
|-
| [[900375/899756]]
| [-2 1 3 4 -3 -2⟩
| 1.191
|-
| [[816293376/815730721]]
| [9 13 0 0 0 -8⟩
| 1.194
|-
| [[37812500/37786203]]
| [2 -3 7 -2 2 -4⟩
| 1.204
|-
| [[3750705/3748096]]
| [-8 7 1 3 -4⟩
| 1.205
|-
| [[4227952113/4225000000]]
| [-6 3 -8 6 3 -2⟩
| 1.209
|-
| [[928125/927472]]
| [-4 3 5 -3 1 -2⟩
| 1.218
|-
| [[214358881/214207500]]
| [-2 -1 -4 0 8 -4⟩
| 1.223
|-
| [[2460375/2458624]]
| [-10 9 3 -4⟩
| 1.233
|-
| [[1977326743/1975897638]]
| [-1 -12 0 11 -1 -2⟩
| 1.252
|-
| [[282475249/282268800]]
| [-7 -6 -2 10 -2⟩
| 1.266
|-
| [[341796875/341545248]]
| [-5 -6 11 1 -4⟩
| 1.275
|-
| [[79893275/79834248]]
| [-3 -10 2 4 3 -2⟩
| 1.280
|-
| [[4081676544/4078653605]]
| [8 2 -1 0 6 -8⟩
| 1.283
|-
| [[330078125/329832448]]
| [-11 0 9 0 -5 2⟩
| 1.289
|-
| [[41503/41472]]
| [-9 -4 0 3 2⟩
| 1.294
|-
| [[4226622444/4223457875]]
| [2 8 -3 -1 5 -6⟩
| 1.297
|-
| [[1220703125/1219784832]]
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| 1.303
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| 1.321
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| 1.322
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| 1.339
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| 1.353
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| 1.428
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| 1.455
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| 1.456
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| 1.498
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| 1.502
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| 2.423
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| 2.425
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| 2.428
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| 2.439
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| 2.452
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| 2.484
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| 2.512
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| 2.720
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| 2.894
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| [-26 4 5 -1 1 2⟩
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| [-10 1 10 0 0 -4⟩
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| [-14 3 -1 2 6 -4⟩
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| 3.032
|-
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| 3.057
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| [-21 0 3 5⟩
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| [-27 6 1 4 -1 2⟩
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| [-24 4 1 3 2⟩
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| 3.332
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| [-17 0 12 0 -1 -2⟩
| 3.391
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| 3.396
|-
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| [-21 2 1 2 5 -2⟩
| 3.410
|-
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| [-27 8 -1 1 4⟩
| 3.424
|-
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| [-17 11 2 1 2 -4⟩
| 3.484
|-
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| [-23 17 0 0 1 -2⟩
| 3.498
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| [-25 -3 5 4 2⟩
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| [-31 3 3 3 1 2⟩
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| 3.703
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| [-21 6 6 3 -1 -2⟩
| 3.717
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| [-26 -1 1 9⟩
| 3.792
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| [-28 1 3 2 4⟩
| 3.820
|-
| [[2841733125/2835349504]]
| [-24 10 4 1 1 -2⟩
| 3.893
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| [[1076168025/1073741824]]
| [-30 16 2⟩
| 3.907
|-
| [[2690514981/2684354560]]
| [-29 3 -1 7 2⟩
| 3.969
|-
| [[2152446615/2147483648]]
| [-31 5 1 0 6⟩
| 3.996
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| [-31 9 6 1⟩
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Revision as of 22:14, 8 November 2025

← 683edo 684edo 685edo →
Prime factorization 22 × 32 × 19
Step size 1.75439 ¢ 
Fifth 400\684 (701.754 ¢) (→ 100\171)
Semitones (A1:m2) 64:52 (112.3 ¢ : 91.23 ¢)
Consistency limit 17
Distinct consistency limit 17

684 equal divisions of the octave (abbreviated 684edo or 684ed2), also called 684-tone equal temperament (684tet) or 684 equal temperament (684et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 684 equal parts of about 1.75 ¢ each. Each step represents a frequency ratio of 21/684, or the 684th root of 2.

Theory

684edo divides the steps of 171edo into four. It is consistent to the 17-odd-limit, tempering out 2401/2400, 3025/3024, 4225/4224, 4375/4374, and 32805/32768 in the 13-limit; 1089/1088, 1225/1224, 1701/1700, 2025/2023, 2058/2057, 2500/2499, 8624/8619, and 14875/14872 in the 17-limit.

Prime harmonics

Approximation of prime harmonics in 684edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.201 -0.349 -0.405 -0.441 -0.177 +0.308 +0.733 -0.204 +0.247 +0.578
Relative (%) +0.0 -11.4 -19.9 -23.1 -25.1 -10.1 +17.5 +41.8 -11.6 +14.1 +33.0
Steps
(reduced) 		684
(0) 		1084
(400) 		1588
(220) 		1920
(552) 		2366
(314) 		2531
(479) 		2796
(60) 		2906
(170) 		3094
(358) 		3323
(587) 		3389
(653)

Subsets and supersets

Since 684 factors into 22 × 32 × 19, 684edo has subset edos 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342.

Approximation to JI

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
5818zpi 683.938935 1.754543 14.267321 7.268914 1.773752 20.109967 1200.107141 0.107141 18 18

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5.7.11.13 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768 [⟨684 1084 1588 1920 2366 2531]] +0.0994 0.0558 3.18
2.3.5.7.11.13.17 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005 [⟨684 1084 1588 1920 2366 2531 2796]] +0.0744 0.0800 4.56
  • 684et is the first equal temperament past 494 with a lower 13-limit absolute error. The next equal temperament that is better tuned is 764.

Rank-2 temperaments

Note: 11-limit temperaments supported by 342et are not shown.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
18 271\684
(5\684) 		475.44
(8.77) 		1053/800
(1287/1280) 		Semihemiennealimmal
38 151\684
(7\684) 		264.91
(12.28) 		500/429
(144/143) 		Semihemienneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

  • Diamond15: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64

Commas of the 2.3.5.7.11.13 subgroup tempered out in 684edo patent val

Commas with numerator <= 2^32:

Ratio Monzo Cents
1610510000/1610497161 [4 -4 4 -6 5 -2⟩ 0.014
123201/123200 [-6 6 -2 -1 -1 2⟩ 0.014
781258401/781250000 [-4 2 -11 2 6⟩ 0.019
3294225/3294172 [-2 2 2 -7 4⟩ 0.028
1475824779/1475789056 [-8 8 0 -8 3 2⟩ 0.042
1433600000/1433562273 [16 -3 5 1 -1 -6⟩ 0.046
3456000/3455881 [10 3 3 0 -2 -4⟩ 0.060
199297406/199290375 [1 -13 -3 7 2⟩ 0.061
78125000/78121827 [3 -13 10 -2⟩ 0.070
1574640/1574573 [4 9 1 -1 -3 -2⟩ 0.074
218709491/218700000 [-5 -7 -5 6 1 2⟩ 0.075
66015625/66012408 [-3 -7 8 -3 -1 2⟩ 0.084
14348907/14348180 [-2 15 -1 -2 -4⟩ 0.088
1771561/1771470 [-1 -11 -1 0 6⟩ 0.089
480024727/480000000 [-11 -1 -7 5 0 4⟩ 0.089
1650478599/1650390625 [0 11 -10 1 3 -2⟩ 0.092
446265625/446240256 [-9 -1 6 -4 -2 4⟩ 0.098
974308500/974251369 [2 11 3 -8 1 -2⟩ 0.102
27217619/27216000 [-7 -5 -3 -1 5 2⟩ 0.103
645700815/645657712 [-4 17 1 -9⟩ 0.116
1254484803/1254400000 [-13 1 -5 -2 4 4⟩ 0.117
1686616064/1686498489 [11 -10 0 7 0 -4⟩ 0.121
214375000/214358881 [3 0 7 3 -8⟩ 0.130
3764768/3764475 [5 -4 -2 6 -1 -2⟩ 0.135
187404800/187388721 [9 -8 2 0 4 -4⟩ 0.149
151263/151250 [-1 2 -4 5 -2⟩ 0.149
35156250/35153041 [1 2 9 -4 -4⟩ 0.158
1224552875/1224440064 [-8 -14 3 3 0 4⟩ 0.159
10648/10647 [3 -2 0 -1 3 -2⟩ 0.163
2662250409/2662000000 [-7 8 -6 4 -3 2⟩ 0.163
1182568205/1182449664 [-14 -8 1 2 -1 6⟩ 0.174
9801/9800 [-3 4 -2 -2 2⟩ 0.177
3955078125/3954653486 [-1 4 11 -11⟩ 0.186
974358550/974251369 [1 0 2 -8 7 -2⟩ 0.190
109772091/109760000 [-9 10 -4 -3 1 2⟩ 0.191
1291467969/1291315424 [-5 6 0 -9 6⟩ 0.205
2091059712/2090808005 [9 5 -1 5 -4 -4⟩ 0.208
21437500/21434787 [2 -11 6 3 -2⟩ 0.219
33792000/33787663 [13 1 3 -1 1 -6⟩ 0.222
3402639576/3402202375 [3 11 -3 4 -5 -2⟩ 0.222
5175625/5174928 [-4 -5 4 2 -3 2⟩ 0.233
1399680/1399489 [7 7 1 -2 0 -4⟩ 0.236
246071287/246037500 [-2 -9 -5 5 4⟩ 0.238
47265625/47258883 [0 -9 8 -4 2⟩ 0.247
14994525/14992384 [-10 1 2 1 -4 4⟩ 0.247
3188646/3188185 [1 13 -1 -3 -1 -2⟩ 0.250
540078539/540000000 [-8 -3 -7 4 3 2⟩ 0.252
29046875/29042496 [-6 -3 6 -5 1 2⟩ 0.261
1162261467/1162084000 [-5 19 -3 -4 -2⟩ 0.264
214358881/214326000 [-4 -7 -3 -2 8⟩ 0.266
481966875/481890304 [-12 3 4 -6 0 4⟩ 0.275
3293331899/3292800000 [-10 -1 -5 -3 7 2⟩ 0.280
245000000/244958571 [6 -2 7 2 -5 -2⟩ 0.293
520613632/520524225 [8 -6 -2 5 2 -4⟩ 0.297
1771875/1771561 [0 4 5 1 -6⟩ 0.307
105644/105625 [2 0 -4 4 1 -2⟩ 0.311
31250000/31244213 [4 0 9 -5 -1 -2⟩ 0.321
1247410125/1247178944 [-6 10 3 0 -7 2⟩ 0.321
1377751375/1377495072 [-5 -16 3 2 3 2⟩ 0.322
113379904/113358609 [6 -4 0 -2 6 -4⟩ 0.325
250047/250000 [-4 6 -6 3⟩ 0.325
56953125/56942116 [-2 6 7 -6 -2⟩ 0.335
120955835/120932352 [-11 -10 1 1 2 4⟩ 0.336
1449459/1449175 [0 2 -2 -3 5 -2⟩ 0.339
53094899/53084160 [-17 -4 -1 0 1 6⟩ 0.350
96059601/96040000 [-6 8 -4 -4 4⟩ 0.353
288240050/288178803 [1 -9 2 8 -4⟩ 0.368
265531392/265474495 [12 3 -1 4 -1 -6⟩ 0.371
269500000/269440587 [5 -13 6 2 1 -2⟩ 0.382
139178767/139148064 [-5 -3 0 7 -5 2⟩ 0.382
432081216/431985125 [6 9 -3 3 -2 -4⟩ 0.385
4375/4374 [-1 -7 4 1⟩ 0.396
1655976960/1655595487 [10 5 1 -3 3 -6⟩ 0.399
703096443/702934375 [0 15 -5 2 -3 -2⟩ 0.399
4225/4224 [-7 -1 2 0 -1 2⟩ 0.410
342956592/342874805 [4 11 -1 -4 2 -4⟩ 0.413
607645423/607500000 [-5 -5 -7 3 6⟩ 0.414
228765625/228709656 [-3 -5 6 -6 4⟩ 0.424
6940323/6938624 [-13 5 0 -1 -2 4⟩ 0.424
1420541793/1420191500 [-2 17 -3 -5 1 -2⟩ 0.427
4000989993/4000000000 [-11 1 -9 2 5 2⟩ 0.428
421760625/421654016 [-9 1 4 -7 3 2⟩ 0.438
3294172000/3293331899 [5 0 3 7 -7 -2⟩ 0.442
280000000/279926361 [9 -4 7 1 -2 -4⟩ 0.455
428830605/428717762 [-1 6 1 6 -8⟩ 0.456
3487704605/3486784401 [0 -20 1 8 2⟩ 0.457
726364926/726171875 [1 2 -8 9 -1 -2⟩ 0.460
225000/224939 [3 2 5 0 -3 -2⟩ 0.469
1530966437/1530550080 [-6 -14 -1 7 1 2⟩ 0.471
160699616/160655625 [5 -2 -4 3 4 -4⟩ 0.474
820125/819896 [-3 8 3 -1 -4⟩ 0.483
1550115875/1549681956 [-2 -18 3 1 6⟩ 0.485
3360173089/3359232000 [-12 -8 -3 6 0 4⟩ 0.485
5282739/5281250 [-1 4 -6 2 3 -2⟩ 0.488
139218750/139178767 [1 4 7 -7 1 -2⟩ 0.497
4041608805/4040447488 [-9 14 1 -2 -5 2⟩ 0.498
136088095/136048896 [-8 -12 1 0 5 2⟩ 0.499
1714871048/1714374025 [3 0 -2 -4 8 -4⟩ 0.502
50014503/50000000 [-7 10 -8 1 2⟩ 0.502
184528125/184473632 [-5 10 5 -8⟩ 0.511
418161601/418037760 [-14 -6 -1 -1 4 4⟩ 0.513
329417200/329316273 [4 -11 2 7 -1 -2⟩ 0.530
4079910912/4078653605 [15 1 -1 3 2 -8⟩ 0.534
117649/117612 [-2 -5 0 6 -2⟩ 0.545
603542016/603351125 [9 7 -3 2 1 -6⟩ 0.548
1220703125/1220312709 [0 -5 13 -3 -4⟩ 0.554
3327500/3326427 [2 -9 4 0 3 -2⟩ 0.558
8521149/8518400 [-8 1 -2 5 -3 2⟩ 0.559
89282088/89253125 [3 13 -5 1 0 -4⟩ 0.562
3025/3024 [-4 -3 2 -1 2⟩ 0.572
1162261467/1161875000 [-3 19 -7 0 -1 -2⟩ 0.576
50193/50176 [-10 3 0 -2 1 2⟩ 0.586
1500512167/1500000000 [-8 -1 -9 1 8⟩ 0.591
1107225625/1106841792 [-6 -1 4 -8 6⟩ 0.600
562166163/561971200 [-16 9 -2 -3 0 4⟩ 0.601
3764768000/3763454409 [8 -2 3 6 -4 -4⟩ 0.604
3520000000/3518743761 [12 -6 7 0 1 -6⟩ 0.618
27227340/27217619 [2 4 1 5 -5 -2⟩ 0.618
200000/199927 [6 0 5 -1 0 -4⟩ 0.632
141776649/141724880 [-4 10 -1 4 -6⟩ 0.632
1722499009/1721868840 [-3 -16 -1 6 4⟩ 0.633
91125/91091 [0 6 3 -2 -1 -2⟩ 0.646
3780549773/3779136000 [-9 -10 -3 5 3 2⟩ 0.648
446433372/446265625 [2 2 -6 1 6 -4⟩ 0.651
2657205/2656192 [-6 12 1 -3 -2⟩ 0.660
1071794405/1071385056 [-5 -14 1 -1 8⟩ 0.661
1056655611/1056250000 [-4 8 -8 0 5 -2⟩ 0.665
3293331899/3292047360 [-11 -8 -1 -2 7 2⟩ 0.675
1977326743/1976535000 [-3 -3 -4 11 -4⟩ 0.693
95703125/95664294 [-1 -3 9 2 -6⟩ 0.703
369754/369603 [1 -7 0 5 1 -2⟩ 0.707
2495390625/2494357888 [-7 3 7 1 -7 2⟩ 0.717
2401/2400 [-5 -1 -2 4⟩ 0.721
48828125/48807528 [-3 -1 11 -5 -2⟩ 0.730
4026275/4024566 [-1 -5 2 -2 5 -2⟩ 0.735
14085981/14080000 [-11 5 -4 3 -1 2⟩ 0.735
43923/43904 [-7 1 0 -3 4⟩ 0.749
4078653605/4076863488 [-24 -5 1 0 0 8⟩ 0.760
491941593/491724800 [-13 7 -2 -4 3 2⟩ 0.763
3457440/3455881 [5 2 1 4 -2 -4⟩ 0.781
19140625/19131876 [-2 -14 8 2⟩ 0.792
2129600000/2128622769 [9 -2 5 -2 3 -6⟩ 0.795
2250423/2249390 [-1 8 -1 3 -3 -2⟩ 0.795
18484375/18475776 [-8 -8 6 1 -1 2⟩ 0.806
9801000/9796423 [3 4 3 -3 2 -4⟩ 0.809
234365481/234256000 [-7 14 -3 2 -4⟩ 0.809
4253517961/4251528000 [-6 -12 -3 4 6⟩ 0.810
17850625/17842176 [-14 -2 4 0 -2 4⟩ 0.820
3247695/3246152 [-3 10 1 -4 1 -2⟩ 0.823
43046721/43025920 [-9 16 -1 -5⟩ 0.837
2573571875/2572306572 [-2 -1 5 7 -8⟩ 0.851
282475249/282335625 [0 -5 -4 10 -1 -2⟩ 0.856
54687500/54660177 [2 -5 9 1 -3 -2⟩ 0.865
562448656/562166163 [4 -9 0 4 4 -4⟩ 0.870
121060821/121000000 [-6 1 -6 9 -2⟩ 0.870
234375/234256 [-4 1 7 0 -4⟩ 0.879
456533/456300 [-2 -3 -2 3 3 -2⟩ 0.884
537109375/536832387 [0 -3 11 -6 1 -2⟩ 0.893
1155009375/1154413568 [-10 7 5 -1 -5 2⟩ 0.893
160083/160000 [-8 3 -4 2 2⟩ 0.898
52734375/52706752 [-6 3 9 -7⟩ 0.907
19487171/19476912 [-4 -1 0 -4 7 -2⟩ 0.912
256134879/256000000 [-14 9 -6 1 1 2⟩ 0.912
430489323/430259200 [-10 5 -2 -5 6⟩ 0.926
514714375/514434888 [-3 -12 4 7 -2⟩ 0.940
4829440/4826809 [8 0 1 3 1 -6⟩ 0.943
497067025/496793088 [-9 -6 2 6 -3 2⟩ 0.954
142884/142805 [2 6 -1 2 0 -4⟩ 0.957
1890625/1889568 [-5 -10 6 0 2⟩ 0.968
480024727/479756288 [-15 0 0 5 -4 4⟩ 0.968
3720087/3718000 [-4 12 -3 1 -1 -2⟩ 0.972
1161875/1161216 [-11 -4 4 -1 1 2⟩ 0.982
480298005/480024727 [0 8 1 -5 4 -4⟩ 0.985
387420489/387200000 [-10 18 -5 0 -2⟩ 0.986
6426225/6422528 [-17 2 2 -2 0 4⟩ 0.996
735306250/734875713 [1 -3 5 6 -5 -2⟩ 1.014
62500000/62462907 [5 -7 9 0 0 -4⟩ 1.028
56723625/56689952 [-5 3 3 5 -6⟩ 1.028
63412811/63375000 [-3 -1 -6 8 1 -2⟩ 1.033
78125/78078 [-1 -1 7 -1 -1 -2⟩ 1.042
173612978/173508075 [1 -5 -2 2 6 -4⟩ 1.046
200120949/200000000 [-9 5 -8 7⟩ 1.047
759375/758912 [-7 5 5 -2 -2⟩ 1.056
3382071/3380000 [-5 1 -4 1 5 -2⟩ 1.060
3742230375/3739918336 [-13 11 3 -3 -3 2⟩ 1.070
32019867/32000000 [-11 7 -6 0 4⟩ 1.074
808836875/808321761 [0 -14 4 6 1 -2⟩ 1.103
420175/419904 [-6 -8 2 5⟩ 1.117
24149664/24134045 [5 4 -1 1 3 -6⟩ 1.120
2516421875/2514778812 [-2 -12 6 -1 5 -2⟩ 1.131
405769/405504 [-12 -2 0 4 -1 2⟩ 1.131
7144929/7140250 [-1 10 -3 0 2 -4⟩ 1.134
9150625/9144576 [-8 -6 4 -2 4⟩ 1.145
793510263/792985600 [-18 4 -2 3 -2 4⟩ 1.145
473513931/473200000 [-7 16 -5 -1 1 -2⟩ 1.148
5623475/5619712 [-14 0 2 -3 3 2⟩ 1.159
2519337249/2517630976 [-20 6 0 -4 2 4⟩ 1.173
840350000/839779083 [4 -5 5 5 -2 -4⟩ 1.177
900375/899756 [-2 1 3 4 -3 -2⟩ 1.191
816293376/815730721 [9 13 0 0 0 -8⟩ 1.194
37812500/37786203 [2 -3 7 -2 2 -4⟩ 1.204
3750705/3748096 [-8 7 1 3 -4⟩ 1.205
4227952113/4225000000 [-6 3 -8 6 3 -2⟩ 1.209
928125/927472 [-4 3 5 -3 1 -2⟩ 1.218
214358881/214207500 [-2 -1 -4 0 8 -4⟩ 1.223
2460375/2458624 [-10 9 3 -4⟩ 1.233
1977326743/1975897638 [-1 -12 0 11 -1 -2⟩ 1.252
282475249/282268800 [-7 -6 -2 10 -2⟩ 1.266
341796875/341545248 [-5 -6 11 1 -4⟩ 1.275
79893275/79834248 [-3 -10 2 4 3 -2⟩ 1.280
4081676544/4078653605 [8 2 -1 0 6 -8⟩ 1.283
330078125/329832448 [-11 0 9 0 -5 2⟩ 1.289
41503/41472 [-9 -4 0 3 2⟩ 1.294
4226622444/4223457875 [2 8 -3 -1 5 -6⟩ 1.297
1220703125/1219784832 [-7 -4 13 -6⟩ 1.303
819819/819200 [-15 2 -2 2 1 2⟩ 1.308
44289025/44255232 [-11 -2 2 -4 6⟩ 1.321
1311721047/1310720000 [-21 8 -4 1 0 4⟩ 1.322
2204627139/2202927104 [-17 4 0 -5 5 2⟩ 1.336
1089547389/1088704760 [-3 3 -1 9 -5 -2⟩ 1.339
85750/85683 [1 -1 3 3 0 -4⟩ 1.353
59535/59488 [-5 5 1 2 -1 -2⟩ 1.367
726171875/725594112 [-12 -11 8 0 1 2⟩ 1.378
137259375/137149922 [-1 1 5 -4 4 -4⟩ 1.381
1240029/1239040 [-11 11 -1 1 -2⟩ 1.381
446265625/445906944 [-18 -5 6 -1 0 4⟩ 1.392
363862125/363569024 [-7 7 3 -5 3 -2⟩ 1.395
964565415/963780608 [-13 13 1 -6 2⟩ 1.409
443889677/443523600 [-4 -8 -2 9 1 -2⟩ 1.428
48828125/48787596 [-2 -8 11 0 -1 -2⟩ 1.438
837019575/836323202 [-1 14 2 1 -4 -4⟩ 1.441
5764801/5760000 [-10 -2 -4 8⟩ 1.442
1953125/1951488 [-8 -2 9 -1 -2⟩ 1.452
3486784401/3483855232 [-7 20 0 0 -5 -2⟩ 1.455
7891499/7884864 [-6 -6 0 2 5 -2⟩ 1.456
1069453125/1068548096 [-14 4 7 -2 -3 2⟩ 1.466
102487/102400 [-12 0 -2 1 4⟩ 1.470
163980531/163840000 [-18 6 -4 0 3 2⟩ 1.484
1929229929/1927561216 [-14 2 0 -6 8⟩ 1.498
17294403/17279405 [0 1 -1 8 -2 -4⟩ 1.502
2929687500/2927131207 [2 1 12 -1 -4 -4⟩ 1.511
1838265625/1836660096 [-7 -15 6 6⟩ 1.513
130438000/130323843 [4 -3 3 2 3 -6⟩ 1.516
1801088541/1799512000 [-6 7 -3 7 -3 -2⟩ 1.516
1775239375/1773674496 [-13 -9 4 5 -1 2⟩ 1.527
114345/114244 [-2 3 1 1 2 -4⟩ 1.530
1714374025/1712848896 [-19 -3 2 4 -2 4⟩ 1.541
216513/216320 [-8 9 -1 0 1 -2⟩ 1.544
3514671875/3511517184 [-15 -7 6 -2 3 2⟩ 1.555
14348907/14336000 [-14 15 -3 -1⟩ 1.558
2159925625/2157969408 [-21 -1 4 -3 2 4⟩ 1.569
1313046875/1311844248 [-3 -6 7 5 -3 -2⟩ 1.586
816480000/815730721 [8 6 4 1 0 -8⟩ 1.589
7503125/7496192 [-9 0 5 4 -4⟩ 1.600
53144100/53094899 [2 12 2 0 -1 -6⟩ 1.604
1096135733/1095120000 [-7 -4 -4 7 3 -2⟩ 1.605
21484375/21464352 [-5 -4 9 -2 1 -2⟩ 1.614
387420489/387058672 [-4 18 0 -1 -2 -4⟩ 1.618
1500512167/1499109768 [-3 -8 0 1 8 -4⟩ 1.619
128119761/128000000 [-13 2 -6 6 2⟩ 1.619
703125/702464 [-11 2 7 -3⟩ 1.628
19487171/19468800 [-9 -2 -2 0 7 -2⟩ 1.633
3465028125/3461742592 [-17 8 5 -4 -1 2⟩ 1.642
143496441/143360000 [-15 4 -4 -1 6⟩ 1.647
72471784/72402135 [3 -1 -1 7 1 -6⟩ 1.665
28588707/28561000 [-3 5 -3 6 0 -4⟩ 1.679
2373046875/2370734366 [-1 5 10 -3 -2 -4⟩ 1.688
181575625/181398528 [-10 -11 4 4 2⟩ 1.689
4831530/4826809 [1 1 1 0 5 -6⟩ 1.692
2977309629/2974400000 [-9 11 -5 5 -1 -2⟩ 1.693
15940925/15925248 [-16 -5 2 3 1 2⟩ 1.703
32019867/31988320 [-5 7 -1 -1 4 -4⟩ 1.707
4198467/4194304 [-22 1 0 2 0 4⟩ 1.717
2122043913/2119936000 [-11 13 -3 -2 3 -2⟩ 1.721
187578125/187388721 [0 -8 7 4 0 -4⟩ 1.749
1815912315/1814078464 [-10 2 1 9 -6⟩ 1.749
1071875/1070784 [-6 -2 5 3 -1 -2⟩ 1.763
496125/495616 [-12 4 3 2 -2⟩ 1.777
473513931/473027282 [-1 16 0 -2 1 -6⟩ 1.780
2706784157/2704000000 [-10 0 -6 5 5 -2⟩ 1.782
103984375/103876864 [-8 0 7 -4 3 -2⟩ 1.791
349274835/348913664 [-18 10 1 1 -3 2⟩ 1.791
275653125/275365888 [-14 6 5 -5 2⟩ 1.805
1674421875/1672646404 [-2 7 6 2 -4 -4⟩ 1.837
40353607/40310784 [-11 -9 0 9⟩ 1.838
603993159/603351125 [0 3 -3 5 3 -6⟩ 1.841
1220703125/1219401216 [-9 -9 13 0 -2⟩ 1.847
974251369/973209600 [-17 -3 -2 8 -1 2⟩ 1.852
17935225/17915904 [-13 -7 2 2 4⟩ 1.866
1574573/1572864 [-19 -1 0 1 3 2⟩ 1.880
839779083/838860800 [-25 5 -2 0 2 4⟩ 1.894
28824005/28792192 [-7 0 1 8 -3 -2⟩ 1.912
18528125/18507528 [-3 -4 5 2 2 -4⟩ 1.926
600362847/599695360 [-13 6 -1 7 -4⟩ 1.926
86625/86528 [-9 2 3 1 1 -2⟩ 1.940
32805/32768 [-15 8 1⟩ 1.954
808321761/807403520 [-21 14 -1 -1 -1 2⟩ 1.968
2450000000/2447192163 [7 -1 8 2 0 -8⟩ 1.985
53156250/53094899 [1 5 6 1 -1 -6⟩ 1.999
110716875/110588192 [-5 11 4 0 -2 -4⟩ 2.013
99648703/99532800 [-14 -5 -2 7 2⟩ 2.015
48828125/48771072 [-12 -5 11 -2⟩ 2.024
3228504075/3224725504 [-11 17 2 -1 -3 -2⟩ 2.027
3410254925/3406261248 [-10 -9 2 1 7 -2⟩ 2.029
656128473/655360000 [-20 1 -4 6 1 2⟩ 2.029
990234375/989069312 [-18 1 9 -3 -1 2⟩ 2.038
1771561/1769472 [-16 -3 0 0 6⟩ 2.043
734875713/734003200 [-22 3 -2 -1 5 2⟩ 2.057
4117715/4112784 [-4 -2 1 7 0 -4⟩ 2.074
3522990625/3518743761 [0 -6 5 1 5 -6⟩ 2.088
9529569/9518080 [-10 4 -1 6 -1 -2⟩ 2.088
1830125/1827904 [-6 0 3 0 4 -4⟩ 2.102
992436543/991232000 [-16 10 -3 5 -2⟩ 2.103
4851495/4845568 [-12 6 1 -1 3 -2⟩ 2.116
64304361/64225280 [-18 12 -1 -2 2⟩ 2.130
816750000/815730721 [4 3 6 0 2 -8⟩ 2.162
26796875/26763264 [-13 -3 7 3 -2⟩ 2.173
135320625/135150652 [-2 9 4 -1 1 -6⟩ 2.176
698709375/697827328 [-19 3 5 2 -3 2⟩ 2.187
358722675/358269184 [-8 15 2 -2 0 -4⟩ 2.190
246071287/245760000 [-17 -1 -4 5 4⟩ 2.191
236328125/236027904 [-15 -1 9 -4 2⟩ 2.201
2357947691/2354946048 [-13 -5 0 -1 9 -2⟩ 2.205
643076643/642252800 [-19 1 -2 -2 8⟩ 2.219
3349609375/3345292808 [-3 0 10 3 -4 -4⟩ 2.232
782954095/781943058 [-1 -4 1 6 3 -6⟩ 2.237
18302823/18279040 [-7 2 -1 5 2 -4⟩ 2.251
2197265625/2194398752 [-5 2 12 -4 0 -4⟩ 2.260
2435896375/2432711736 [-3 -2 3 -1 7 -6⟩ 2.265
173282571/173056000 [-13 8 -3 4 1 -2⟩ 2.265
717482205/716538368 [-9 4 1 -2 6 -4⟩ 2.279
1640558367/1638400000 [-19 14 -5 3⟩ 2.279
604779175/603979776 [-26 -2 2 1 2 4⟩ 2.290
816820200/815730721 [3 5 2 5 0 -8⟩ 2.311
720600125/719634432 [-14 -1 3 8 -4⟩ 2.322
425329947/424759192 [-3 11 0 4 -1 -6⟩ 2.325
42109375/42052608 [-10 -5 7 2 1 -2⟩ 2.335
65625/65536 [-16 1 5 1⟩ 2.349
46200375/46137344 [-22 7 3 0 -1 2⟩ 2.364
478515625/477854091 [0 -2 10 2 -1 -6⟩ 2.395
221484375/221176384 [-6 4 8 1 -2 -4⟩ 2.409
386683451/386144720 [-4 0 -1 4 5 -6⟩ 2.414
922640625/921350144 [-12 10 6 0 -3 -2⟩ 2.423
1530966437/1528823808 [-21 -6 0 7 1 2⟩ 2.425
3660938127/3655808000 [-10 6 -3 3 4 -4⟩ 2.428
3360173089/3355443200 [-27 0 -2 6 0 4⟩ 2.439
680440475/679477248 [-23 -4 2 0 5 2⟩ 2.452
3763454409/3758096384 [-29 2 0 -1 4 4⟩ 2.467
102942875/102795264 [-11 -3 3 7 -1 -2⟩ 2.484
816903549/815730721 [0 9 0 3 2 -8⟩ 2.487
15882615/15859712 [-17 3 1 6 -2⟩ 2.498
4159375/4153344 [-13 -1 5 0 3 -2⟩ 2.512
3675375/3670016 [-19 5 3 -1 2⟩ 2.526
1646581365/1644167168 [-25 11 1 -2 1 2⟩ 2.540
38671875/38614472 [-3 2 8 0 1 -6⟩ 2.572
102515625/102362624 [-9 8 6 -1 0 -4⟩ 2.586
1722499009/1719926784 [-18 -8 0 6 4⟩ 2.587
2989355625/2984869888 [-15 14 4 -2 -1 -2⟩ 2.600
3780549773/3774873600 [-24 -2 -2 5 3 2⟩ 2.601
3293331899/3288334336 [-26 0 0 -2 7 2⟩ 2.629
1779441125/1776722688 [-8 -5 3 6 2 -4⟩ 2.647
2773155/2768896 [-14 1 1 5 1 -2⟩ 2.661
5250987/5242880 [-20 7 -1 4⟩ 2.675
543547125/542703616 [-16 3 3 -2 5 -2⟩ 2.689
1440894015/1438646272 [-22 9 1 -3 4⟩ 2.703
850854375/849518384 [-4 4 4 5 -1 -6⟩ 2.720
3544416225/3538822144 [-10 10 2 4 -2 -4⟩ 2.735
95703125/95551488 [-17 -6 9 2⟩ 2.745
92421875/92274688 [-23 0 7 1 -1 2⟩ 2.759
4253517961/4246732800 [-21 -4 -2 4 6⟩ 2.764
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1845703125/1842700288 [-13 3 10 1 -3 -2⟩ 2.819
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1634180625/1631461442 [-1 2 4 4 2 -8⟩ 2.883
2573571875/2569273344 [-18 -4 5 7 -2⟩ 2.894
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9453125/9437184 [-20 -2 7 0 2⟩ 2.922
470559375/469762048 [-26 4 5 -1 1 2⟩ 2.936
29296875/29246464 [-10 1 10 0 0 -4⟩ 2.981
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2343775203/2339717120 [-14 3 -1 2 6 -4⟩ 3.000
3268642167/3262922884 [-2 4 0 9 0 -8⟩ 3.032
4044184375/4037050368 [-15 -6 5 6 1 -2⟩ 3.057
2100875/2097152 [-21 0 3 5⟩ 3.071
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411778125/411041792 [-23 2 5 -2 4⟩ 3.099
2542277241/2537553920 [-22 2 -1 10 -2⟩ 3.219
399466375/398721024 [-18 -2 3 4 3 -2⟩ 3.233
16808715/16777216 [-24 4 1 3 2⟩ 3.247
1238015625/1235663104 [-8 1 6 4 1 -6⟩ 3.293
468838125/467943424 [-14 7 4 3 0 -4⟩ 3.307
1953045675/1949302784 [-20 13 2 2 -1 -2⟩ 3.321
3630859375/3623878656 [-27 -3 9 0 1 2⟩ 3.332
443889677/443023360 [-19 0 -1 9 1 -2⟩ 3.382
244140625/243662848 [-17 0 12 0 -1 -2⟩ 3.391
4202539929/4194304000 [-25 6 -3 8⟩ 3.396
355117455/354418688 [-21 2 1 2 5 -2⟩ 3.410
672417207/671088640 [-27 8 -1 1 4⟩ 3.424
3751087725/3743547392 [-17 11 2 1 2 -4⟩ 3.484
1420541793/1417674752 [-23 17 0 0 1 -2⟩ 3.498
907878125/905969664 [-25 -3 5 4 2⟩ 3.643
2152024875/2147483648 [-31 3 3 3 1 2⟩ 3.657
937890625/935886848 [-15 0 8 4 0 -4⟩ 3.703
3906984375/3898605568 [-21 6 6 3 -1 -2⟩ 3.717
201768035/201326592 [-26 -1 1 9⟩ 3.792
269028375/268435456 [-28 1 3 2 4⟩ 3.820
2841733125/2835349504 [-24 10 4 1 1 -2⟩ 3.893
1076168025/1073741824 [-30 16 2⟩ 3.907
2690514981/2684354560 [-29 3 -1 7 2⟩ 3.969
2152446615/2147483648 [-31 5 1 0 6⟩ 3.996
2152828125/2147483648 [-31 9 6 1⟩ 4.303
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