Xenharmonic Wiki

684edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Contribution (talk | contribs)
3,308 edits
Tags: Reverted Visual edit
- spam
Tags: Replaced Undo
Line 3: Line 3:


== Theory ==
== 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]], [[9801/9800]], [[10648/10647]], [[32805/32768]], [[41503/41472]], [[43923/43904]], [[50193/50176]], [[59535/59488]], and [[65625/65536]] 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.
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 ===
=== Prime harmonics ===
Line 83: Line 83:


* [[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
|-
| [[2401/2400]]
| [-5 -1 -2 4⟩
| 0.721
|-
| [[3025/3024]]
| [-4 -3 2 -1 2⟩
| 0.572
|-
| [[4225/4224]]
| [-7 -1 2 0 -1 2⟩
| 0.410
|-
| [[4375/4374]]
| [-1 -7 4 1⟩
| 0.396
|-
| [[9801/9800]]
| [-3 4 -2 -2 2⟩
| 0.177
|-
| [[10648/10647]]
| [3 -2 0 -1 3 -2⟩
| 0.163
|-
| [[32805/32768]]
| [-15 8 1⟩
| 1.954
|-
| [[41503/41472]]
| [-9 -4 0 3 2⟩
| 1.294
|-
| [[43923/43904]]
| [-7 1 0 -3 4⟩
| 0.749
|-
| [[50193/50176]]
| [-10 3 0 -2 1 2⟩
| 0.586
|-
| [[59535/59488]]
| [-5 5 1 2 -1 -2⟩
| 1.367
|-
| [[65625/65536]]
| [-16 1 5 1⟩
| 2.349
|-
| [[78125/78078]]
| [-1 -1 7 -1 -1 -2⟩
| 1.042
|-
| [[85750/85683]]
| [1 -1 3 3 0 -4⟩
| 1.353
|-
| [[86625/86528]]
| [-9 2 3 1 1 -2⟩
| 1.940
|-
| [[91125/91091]]
| [0 6 3 -2 -1 -2⟩
| 0.646
|-
| [[102487/102400]]
| [-12 0 -2 1 4⟩
| 1.470
|-
| [[105644/105625]]
| [2 0 -4 4 1 -2⟩
| 0.311
|-
| [[114345/114244]]
| [-2 3 1 1 2 -4⟩
| 1.530
|-
| [[117649/117612]]
| [-2 -5 0 6 -2⟩
| 0.545
|-
| [[123201/123200]]
| [-6 6 -2 -1 -1 2⟩
| 0.014
|-
| [[142884/142805]]
| [2 6 -1 2 0 -4⟩
| 0.957
|-
| [[151263/151250]]
| [-1 2 -4 5 -2⟩
| 0.149
|-
| [[160083/160000]]
| [-8 3 -4 2 2⟩
| 0.898
|-
| [[200000/199927]]
| [6 0 5 -1 0 -4⟩
| 0.632
|-
| [[216513/216320]]
| [-8 9 -1 0 1 -2⟩
| 1.544
|-
| [[225000/224939]]
| [3 2 5 0 -3 -2⟩
| 0.469
|-
| [[234375/234256]]
| [-4 1 7 0 -4⟩
| 0.879
|-
| [[250047/250000]]
| [-4 6 -6 3⟩
| 0.325
|-
| [[369754/369603]]
| [1 -7 0 5 1 -2⟩
| 0.707
|-
| [[405769/405504]]
| [-12 -2 0 4 -1 2⟩
| 1.131
|-
| [[420175/419904]]
| [-6 -8 2 5⟩
| 1.117
|-
| [[456533/456300]]
| [-2 -3 -2 3 3 -2⟩
| 0.884
|-
| [[496125/495616]]
| [-12 4 3 2 -2⟩
| 1.777
|-
| [[703125/702464]]
| [-11 2 7 -3⟩
| 1.628
|-
| [[759375/758912]]
| [-7 5 5 -2 -2⟩
| 1.056
|-
| [[819819/819200]]
| [-15 2 -2 2 1 2⟩
| 1.308
|-
| [[820125/819896]]
| [-3 8 3 -1 -4⟩
| 0.483
|-
| [[900375/899756]]
| [-2 1 3 4 -3 -2⟩
| 1.191
|-
| [[928125/927472]]
| [-4 3 5 -3 1 -2⟩
| 1.218
|-
| [[1071875/1070784]]
| [-6 -2 5 3 -1 -2⟩
| 1.763
|-
| [[1161875/1161216]]
| [-11 -4 4 -1 1 2⟩
| 0.982
|-
| [[1240029/1239040]]
| [-11 11 -1 1 -2⟩
| 1.381
|-
| [[1399680/1399489]]
| [7 7 1 -2 0 -4⟩
| 0.236
|-
| [[1449459/1449175]]
| [0 2 -2 -3 5 -2⟩
| 0.339
|-
| [[1574573/1572864]]
| [-19 -1 0 1 3 2⟩
| 1.880
|-
| [[1574640/1574573]]
| [4 9 1 -1 -3 -2⟩
| 0.074
|-
| [[1771561/1769472]]
| [-16 -3 0 0 6⟩
| 2.043
|-
| [[1771561/1771470]]
| [-1 -11 -1 0 6⟩
| 0.089
|-
| [[1771875/1771561]]
| [0 4 5 1 -6⟩
| 0.307
|-
| [[1830125/1827904]]
| [-6 0 3 0 4 -4⟩
| 2.102
|-
| [[1890625/1889568]]
| [-5 -10 6 0 2⟩
| 0.968
|-
| [[1953125/1951488]]
| [-8 -2 9 -1 -2⟩
| 1.452
|-
| [[2100875/2097152]]
| [-21 0 3 5⟩
| 3.071
|-
| [[2250423/2249390]]
| [-1 8 -1 3 -3 -2⟩
| 0.795
|-
| [[2460375/2458624]]
| [-10 9 3 -4⟩
| 1.233
|-
| [[2657205/2656192]]
| [-6 12 1 -3 -2⟩
| 0.660
|-
| [[2773155/2768896]]
| [-14 1 1 5 1 -2⟩
| 2.661
|-
| [[3188646/3188185]]
| [1 13 -1 -3 -1 -2⟩
| 0.250
|-
| [[3247695/3246152]]
| [-3 10 1 -4 1 -2⟩
| 0.823
|-
| [[3294225/3294172]]
| [-2 2 2 -7 4⟩
| 0.028
|-
| [[3327500/3326427]]
| [2 -9 4 0 3 -2⟩
| 0.558
|-
| [[3382071/3380000]]
| [-5 1 -4 1 5 -2⟩
| 1.060
|-
| [[3456000/3455881]]
| [10 3 3 0 -2 -4⟩
| 0.060
|-
| [[3457440/3455881]]
| [5 2 1 4 -2 -4⟩
| 0.781
|-
| [[3675375/3670016]]
| [-19 5 3 -1 2⟩
| 2.526
|-
| [[3720087/3718000]]
| [-4 12 -3 1 -1 -2⟩
| 0.972
|-
| [[3750705/3748096]]
| [-8 7 1 3 -4⟩
| 1.205
|-
| [[3764768/3764475]]
| [5 -4 -2 6 -1 -2⟩
| 0.135
|-
| [[4026275/4024566]]
| [-1 -5 2 -2 5 -2⟩
| 0.735
|-
| [[4117715/4112784]]
| [-4 -2 1 7 0 -4⟩
| 2.074
|-
| [[4159375/4153344]]
| [-13 -1 5 0 3 -2⟩
| 2.512
|-
| [[4198467/4194304]]
| [-22 1 0 2 0 4⟩
| 1.717
|-
| [[4829440/4826809]]
| [8 0 1 3 1 -6⟩
| 0.943
|-
| [[4831530/4826809]]
| [1 1 1 0 5 -6⟩
| 1.692
|-
| [[4851495/4845568]]
| [-12 6 1 -1 3 -2⟩
| 2.116
|-
| [[5175625/5174928]]
| [-4 -5 4 2 -3 2⟩
| 0.233
|-
| [[5250987/5242880]]
| [-20 7 -1 4⟩
| 2.675
|-
| [[5282739/5281250]]
| [-1 4 -6 2 3 -2⟩
| 0.488
|-
| [[5623475/5619712]]
| [-14 0 2 -3 3 2⟩
| 1.159
|-
| [[5764801/5760000]]
| [-10 -2 -4 8⟩
| 1.442
|-
| [[6426225/6422528]]
| [-17 2 2 -2 0 4⟩
| 0.996
|-
| [[6940323/6938624]]
| [-13 5 0 -1 -2 4⟩
| 0.424
|-
| [[7144929/7140250]]
| [-1 10 -3 0 2 -4⟩
| 1.134
|-
| [[7503125/7496192]]
| [-9 0 5 4 -4⟩
| 1.600
|-
| [[7891499/7884864]]
| [-6 -6 0 2 5 -2⟩
| 1.456
|-
| [[8521149/8518400]]
| [-8 1 -2 5 -3 2⟩
| 0.559
|-
| [[9150625/9144576]]
| [-8 -6 4 -2 4⟩
| 1.145
|-
| [[9453125/9437184]]
| [-20 -2 7 0 2⟩
| 2.922
|-
| [[9529569/9518080]]
| [-10 4 -1 6 -1 -2⟩
| 2.088
|-
| [[9801000/9796423]]
| [3 4 3 -3 2 -4⟩
| 0.809
|-
| [[14085981/14080000]]
| [-11 5 -4 3 -1 2⟩
| 0.735
|-
| [[14348907/14336000]]
| [-14 15 -3 -1⟩
| 1.558
|-
| [[14348907/14348180]]
| [-2 15 -1 -2 -4⟩
| 0.088
|-
| [[14994525/14992384]]
| [-10 1 2 1 -4 4⟩
| 0.247
|-
| [[15882615/15859712]]
| [-17 3 1 6 -2⟩
| 2.498
|-
| [[15940925/15925248]]
| [-16 -5 2 3 1 2⟩
| 1.703
|-
| [[16808715/16777216]]
| [-24 4 1 3 2⟩
| 3.247
|-
| [[17294403/17279405]]
| [0 1 -1 8 -2 -4⟩
| 1.502
|-
| [[17850625/17842176]]
| [-14 -2 4 0 -2 4⟩
| 0.820
|-
| [[17935225/17915904]]
| [-13 -7 2 2 4⟩
| 1.866
|-
| [[18302823/18279040]]
| [-7 2 -1 5 2 -4⟩
| 2.251
|-
| [[18484375/18475776]]
| [-8 -8 6 1 -1 2⟩
| 0.806
|-
| [[18528125/18507528]]
| [-3 -4 5 2 2 -4⟩
| 1.926
|-
| [[19140625/19131876]]
| [-2 -14 8 2⟩
| 0.792
|-
| [[19487171/19468800]]
| [-9 -2 -2 0 7 -2⟩
| 1.633
|-
| [[19487171/19476912]]
| [-4 -1 0 -4 7 -2⟩
| 0.912
|-
| [[21437500/21434787]]
| [2 -11 6 3 -2⟩
| 0.219
|-
| [[21484375/21464352]]
| [-5 -4 9 -2 1 -2⟩
| 1.614
|-
| [[24149664/24134045]]
| [5 4 -1 1 3 -6⟩
| 1.120
|-
| [[26796875/26763264]]
| [-13 -3 7 3 -2⟩
| 2.173
|-
| [[27217619/27216000]]
| [-7 -5 -3 -1 5 2⟩
| 0.103
|-
| [[27227340/27217619]]
| [2 4 1 5 -5 -2⟩
| 0.618
|-
| [[28588707/28561000]]
| [-3 5 -3 6 0 -4⟩
| 1.679
|-
| [[28824005/28792192]]
| [-7 0 1 8 -3 -2⟩
| 1.912
|-
| [[29046875/29042496]]
| [-6 -3 6 -5 1 2⟩
| 0.261
|-
| [[29296875/29246464]]
| [-10 1 10 0 0 -4⟩
| 2.981
|-
| [[31250000/31244213]]
| [4 0 9 -5 -1 -2⟩
| 0.321
|-
| [[32019867/31988320]]
| [-5 7 -1 -1 4 -4⟩
| 1.707
|-
| [[32019867/32000000]]
| [-11 7 -6 0 4⟩
| 1.074
|-
| [[33792000/33787663]]
| [13 1 3 -1 1 -6⟩
| 0.222
|-
| [[35156250/35153041]]
| [1 2 9 -4 -4⟩
| 0.158
|-
| [[37812500/37786203]]
| [2 -3 7 -2 2 -4⟩
| 1.204
|-
| [[38671875/38614472]]
| [-3 2 8 0 1 -6⟩
| 2.572
|-
| [[40353607/40310784]]
| [-11 -9 0 9⟩
| 1.838
|-
| [[42109375/42052608]]
| [-10 -5 7 2 1 -2⟩
| 2.335
|-
| [[43046721/43025920]]
| [-9 16 -1 -5⟩
| 0.837
|-
| [[44289025/44255232]]
| [-11 -2 2 -4 6⟩
| 1.321
|-
| [[46200375/46137344]]
| [-22 7 3 0 -1 2⟩
| 2.364
|-
| [[47265625/47258883]]
| [0 -9 8 -4 2⟩
| 0.247
|-
| [[48828125/48771072]]
| [-12 -5 11 -2⟩
| 2.024
|-
| [[48828125/48787596]]
| [-2 -8 11 0 -1 -2⟩
| 1.438
|-
| [[48828125/48807528]]
| [-3 -1 11 -5 -2⟩
| 0.730
|-
| [[50014503/50000000]]
| [-7 10 -8 1 2⟩
| 0.502
|-
| [[52734375/52706752]]
| [-6 3 9 -7⟩
| 0.907
|-
| [[53094899/53084160]]
| [-17 -4 -1 0 1 6⟩
| 0.350
|-
| [[53144100/53094899]]
| [2 12 2 0 -1 -6⟩
| 1.604
|-
| [[53156250/53094899]]
| [1 5 6 1 -1 -6⟩
| 1.999
|-
| [[54687500/54660177]]
| [2 -5 9 1 -3 -2⟩
| 0.865
|-
| [[56723625/56689952]]
| [-5 3 3 5 -6⟩
| 1.028
|-
| [[56953125/56942116]]
| [-2 6 7 -6 -2⟩
| 0.335
|-
| [[62500000/62462907]]
| [5 -7 9 0 0 -4⟩
| 1.028
|-
| [[63412811/63375000]]
| [-3 -1 -6 8 1 -2⟩
| 1.033
|-
| [[64304361/64225280]]
| [-18 12 -1 -2 2⟩
| 2.130
|-
| [[66015625/66012408]]
| [-3 -7 8 -3 -1 2⟩
| 0.084
|-
| [[72471784/72402135]]
| [3 -1 -1 7 1 -6⟩
| 1.665
|-
| [[78125000/78121827]]
| [3 -13 10 -2⟩
| 0.070
|-
| [[79893275/79834248]]
| [-3 -10 2 4 3 -2⟩
| 1.280
|-
| [[89282088/89253125]]
| [3 13 -5 1 0 -4⟩
| 0.562
|-
| [[92421875/92274688]]
| [-23 0 7 1 -1 2⟩
| 2.759
|-
| [[95703125/95551488]]
| [-17 -6 9 2⟩
| 2.745
|-
| [[95703125/95664294]]
| [-1 -3 9 2 -6⟩
| 0.703
|-
| [[96059601/96040000]]
| [-6 8 -4 -4 4⟩
| 0.353
|-
| [[99648703/99532800]]
| [-14 -5 -2 7 2⟩
| 2.015
|-
| [[102515625/102362624]]
| [-9 8 6 -1 0 -4⟩
| 2.586
|-
| [[102942875/102795264]]
| [-11 -3 3 7 -1 -2⟩
| 2.484
|-
| [[103984375/103876864]]
| [-8 0 7 -4 3 -2⟩
| 1.791
|-
| [[109772091/109760000]]
| [-9 10 -4 -3 1 2⟩
| 0.191
|-
| [[110716875/110588192]]
| [-5 11 4 0 -2 -4⟩
| 2.013
|-
| [[110937519/110755840]]
| [-17 5 -1 3 3 -2⟩
| 2.838
|-
| [[113379904/113358609]]
| [6 -4 0 -2 6 -4⟩
| 0.325
|-
| [[120955835/120932352]]
| [-11 -10 1 1 2 4⟩
| 0.336
|-
| [[121060821/121000000]]
| [-6 1 -6 9 -2⟩
| 0.870
|-
| [[128119761/128000000]]
| [-13 2 -6 6 2⟩
| 1.619
|-
| [[130438000/130323843]]
| [4 -3 3 2 3 -6⟩
| 1.516
|-
| [[135320625/135150652]]
| [-2 9 4 -1 1 -6⟩
| 2.176
|-
| [[136088095/136048896]]
| [-8 -12 1 0 5 2⟩
| 0.499
|-
| [[137259375/137149922]]
| [-1 1 5 -4 4 -4⟩
| 1.381
|-
| [[139178767/139148064]]
| [-5 -3 0 7 -5 2⟩
| 0.382
|-
| [[139218750/139178767]]
| [1 4 7 -7 1 -2⟩
| 0.497
|-
| [[141776649/141724880]]
| [-4 10 -1 4 -6⟩
| 0.632
|-
| [[143496441/143360000]]
| [-15 4 -4 -1 6⟩
| 1.647
|-
| [[160699616/160655625]]
| [5 -2 -4 3 4 -4⟩
| 0.474
|-
| [[163980531/163840000]]
| [-18 6 -4 0 3 2⟩
| 1.484
|-
| [[173282571/173056000]]
| [-13 8 -3 4 1 -2⟩
| 2.265
|-
| [[173612978/173508075]]
| [1 -5 -2 2 6 -4⟩
| 1.046
|-
| [[175765205/175478784]]
| [-11 -1 1 4 4 -4⟩
| 2.823
|-
| [[181575625/181398528]]
| [-10 -11 4 4 2⟩
| 1.689
|-
| [[184528125/184473632]]
| [-5 10 5 -8⟩
| 0.511
|-
| [[187404800/187388721]]
| [9 -8 2 0 4 -4⟩
| 0.149
|-
| [[187578125/187388721]]
| [0 -8 7 4 0 -4⟩
| 1.749
|-
| [[199297406/199290375]]
| [1 -13 -3 7 2⟩
| 0.061
|-
| [[200120949/200000000]]
| [-9 5 -8 7⟩
| 1.047
|-
| [[201768035/201326592]]
| [-26 -1 1 9⟩
| 3.792
|-
| [[214358881/214207500]]
| [-2 -1 -4 0 8 -4⟩
| 1.223
|-
| [[214358881/214326000]]
| [-4 -7 -3 -2 8⟩
| 0.266
|-
| [[214375000/214358881]]
| [3 0 7 3 -8⟩
| 0.130
|-
| [[218709491/218700000]]
| [-5 -7 -5 6 1 2⟩
| 0.075
|-
| [[221484375/221176384]]
| [-6 4 8 1 -2 -4⟩
| 2.409
|-
| [[228765625/228709656]]
| [-3 -5 6 -6 4⟩
| 0.424
|-
| [[234365481/233971712]]
| [-13 14 0 2 0 -4⟩
| 2.911
|-
| [[234365481/234256000]]
| [-7 14 -3 2 -4⟩
| 0.809
|-
| [[236328125/236027904]]
| [-15 -1 9 -4 2⟩
| 2.201
|-
| [[244140625/243662848]]
| [-17 0 12 0 -1 -2⟩
| 3.391
|-
| [[245000000/244958571]]
| [6 -2 7 2 -5 -2⟩
| 0.293
|-
| [[246071287/245760000]]
| [-17 -1 -4 5 4⟩
| 2.191
|-
| [[246071287/246037500]]
| [-2 -9 -5 5 4⟩
| 0.238
|-
| [[256134879/256000000]]
| [-14 9 -6 1 1 2⟩
| 0.912
|-
| [[265531392/265474495]]
| [12 3 -1 4 -1 -6⟩
| 0.371
|-
| [[269028375/268435456]]
| [-28 1 3 2 4⟩
| 3.820
|-
| [[269500000/269440587]]
| [5 -13 6 2 1 -2⟩
| 0.382
|-
| [[275653125/275365888]]
| [-14 6 5 -5 2⟩
| 1.805
|-
| [[280000000/279926361]]
| [9 -4 7 1 -2 -4⟩
| 0.455
|-
| [[282475249/282268800]]
| [-7 -6 -2 10 -2⟩
| 1.266
|-
| [[282475249/282335625]]
| [0 -5 -4 10 -1 -2⟩
| 0.856
|-
| [[288240050/288178803]]
| [1 -9 2 8 -4⟩
| 0.368
|-
| [[329417200/329316273]]
| [4 -11 2 7 -1 -2⟩
| 0.530
|-
| [[330078125/329832448]]
| [-11 0 9 0 -5 2⟩
| 1.289
|-
| [[341796875/341545248]]
| [-5 -6 11 1 -4⟩
| 1.275
|-
| [[342956592/342874805]]
| [4 11 -1 -4 2 -4⟩
| 0.413
|-
| [[349274835/348913664]]
| [-18 10 1 1 -3 2⟩
| 1.791
|-
| [[355117455/354418688]]
| [-21 2 1 2 5 -2⟩
| 3.410
|-
| [[358722675/358269184]]
| [-8 15 2 -2 0 -4⟩
| 2.190
|-
| [[363862125/363569024]]
| [-7 7 3 -5 3 -2⟩
| 1.395
|-
| [[386683451/386144720]]
| [-4 0 -1 4 5 -6⟩
| 2.414
|-
| [[387420489/387058672]]
| [-4 18 0 -1 -2 -4⟩
| 1.618
|-
| [[387420489/387200000]]
| [-10 18 -5 0 -2⟩
| 0.986
|-
| [[399466375/398721024]]
| [-18 -2 3 4 3 -2⟩
| 3.233
|-
| [[411778125/411041792]]
| [-23 2 5 -2 4⟩
| 3.099
|-
| [[418161601/418037760]]
| [-14 -6 -1 -1 4 4⟩
| 0.513
|-
| [[421760625/421654016]]
| [-9 1 4 -7 3 2⟩
| 0.438
|-
| [[425329947/424759192]]
| [-3 11 0 4 -1 -6⟩
| 2.325
|-
| [[428830605/428717762]]
| [-1 6 1 6 -8⟩
| 0.456
|-
| [[430489323/430259200]]
| [-10 5 -2 -5 6⟩
| 0.926
|-
| [[432081216/431985125]]
| [6 9 -3 3 -2 -4⟩
| 0.385
|-
| [[443889677/443023360]]
| [-19 0 -1 9 1 -2⟩
| 3.382
|-
| [[443889677/443523600]]
| [-4 -8 -2 9 1 -2⟩
| 1.428
|-
| [[446265625/445906944]]
| [-18 -5 6 -1 0 4⟩
| 1.392
|-
| [[446265625/446240256]]
| [-9 -1 6 -4 -2 4⟩
| 0.098
|-
| [[446433372/446265625]]
| [2 2 -6 1 6 -4⟩
| 0.651
|-
| [[468838125/467943424]]
| [-14 7 4 3 0 -4⟩
| 3.307
|-
| [[470559375/469762048]]
| [-26 4 5 -1 1 2⟩
| 2.936
|-
| [[473513931/473027282]]
| [-1 16 0 -2 1 -6⟩
| 1.780
|-
| [[473513931/473200000]]
| [-7 16 -5 -1 1 -2⟩
| 1.148
|-
| [[478515625/477854091]]
| [0 -2 10 2 -1 -6⟩
| 2.395
|-
| [[480024727/479756288]]
| [-15 0 0 5 -4 4⟩
| 0.968
|-
| [[480024727/480000000]]
| [-11 -1 -7 5 0 4⟩
| 0.089
|-
| [[480298005/480024727]]
| [0 8 1 -5 4 -4⟩
| 0.985
|-
| [[481966875/481890304]]
| [-12 3 4 -6 0 4⟩
| 0.275
|-
| [[491941593/491724800]]
| [-13 7 -2 -4 3 2⟩
| 0.763
|-
| [[497067025/496793088]]
| [-9 -6 2 6 -3 2⟩
| 0.954
|-
| [[514714375/514434888]]
| [-3 -12 4 7 -2⟩
| 0.940
|-
| [[520613632/520524225]]
| [8 -6 -2 5 2 -4⟩
| 0.297
|-
| [[537109375/536832387]]
| [0 -3 11 -6 1 -2⟩
| 0.893
|-
| [[540078539/540000000]]
| [-8 -3 -7 4 3 2⟩
| 0.252
|-
| [[543547125/542703616]]
| [-16 3 3 -2 5 -2⟩
| 2.689
|-
| [[562166163/561971200]]
| [-16 9 -2 -3 0 4⟩
| 0.601
|-
| [[562448656/562166163]]
| [4 -9 0 4 4 -4⟩
| 0.870
|-
| [[600362847/599695360]]
| [-13 6 -1 7 -4⟩
| 1.926
|-
| [[603542016/603351125]]
| [9 7 -3 2 1 -6⟩
| 0.548
|-
| [[603993159/603351125]]
| [0 3 -3 5 3 -6⟩
| 1.841
|-
| [[604779175/603979776]]
| [-26 -2 2 1 2 4⟩
| 2.290
|-
| [[607645423/607500000]]
| [-5 -5 -7 3 6⟩
| 0.414
|-
| [[618866325/617831552]]
| [-7 8 2 3 1 -6⟩
| 2.897
|-
| [[643076643/642252800]]
| [-19 1 -2 -2 8⟩
| 2.219
|-
| [[645700815/645657712]]
| [-4 17 1 -9⟩
| 0.116
|-
| [[656128473/655360000]]
| [-20 1 -4 6 1 2⟩
| 2.029
|-
| [[672417207/671088640]]
| [-27 8 -1 1 4⟩
| 3.424
|-
| [[680440475/679477248]]
| [-23 -4 2 0 5 2⟩
| 2.452
|-
| [[698709375/697827328]]
| [-19 3 5 2 -3 2⟩
| 2.187
|-
| [[703096443/702934375]]
| [0 15 -5 2 -3 -2⟩
| 0.399
|-
| [[717482205/716538368]]
| [-9 4 1 -2 6 -4⟩
| 2.279
|-
| [[720600125/719634432]]
| [-14 -1 3 8 -4⟩
| 2.322
|-
| [[726171875/725594112]]
| [-12 -11 8 0 1 2⟩
| 1.378
|-
| [[726364926/726171875]]
| [1 2 -8 9 -1 -2⟩
| 0.460
|-
| [[734875713/734003200]]
| [-22 3 -2 -1 5 2⟩
| 2.057
|-
| [[735306250/734875713]]
| [1 -3 5 6 -5 -2⟩
| 1.014
|-
| [[781258401/781250000]]
| [-4 2 -11 2 6⟩
| 0.019
|-
| [[782954095/781943058]]
| [-1 -4 1 6 3 -6⟩
| 2.237
|-
| [[793510263/792985600]]
| [-18 4 -2 3 -2 4⟩
| 1.145
|-
| [[808321761/807403520]]
| [-21 14 -1 -1 -1 2⟩
| 1.968
|-
| [[808836875/808321761]]
| [0 -14 4 6 1 -2⟩
| 1.103
|-
| [[816293376/815730721]]
| [9 13 0 0 0 -8⟩
| 1.194
|-
| [[816480000/815730721]]
| [8 6 4 1 0 -8⟩
| 1.589
|-
| [[816750000/815730721]]
| [4 3 6 0 2 -8⟩
| 2.162
|-
| [[816820200/815730721]]
| [3 5 2 5 0 -8⟩
| 2.311
|-
| [[816903549/815730721]]
| [0 9 0 3 2 -8⟩
| 2.487
|-
| [[837019575/836323202]]
| [-1 14 2 1 -4 -4⟩
| 1.441
|-
| [[839779083/838860800]]
| [-25 5 -2 0 2 4⟩
| 1.894
|-
| [[840350000/839779083]]
| [4 -5 5 5 -2 -4⟩
| 1.177
|-
| [[850854375/849518384]]
| [-4 4 4 5 -1 -6⟩
| 2.720
|-
| [[854296875/852819968]]
| [-16 7 8 -1 -1 -2⟩
| 2.996
|-
| [[907878125/905969664]]
| [-25 -3 5 4 2⟩
| 3.643
|-
| [[922640625/921350144]]
| [-12 10 6 0 -3 -2⟩
| 2.423
|-
| [[937890625/935886848]]
| [-15 0 8 4 0 -4⟩
| 3.703
|-
| [[964565415/963780608]]
| [-13 13 1 -6 2⟩
| 1.409
|-
| [[974251369/973209600]]
| [-17 -3 -2 8 -1 2⟩
| 1.852
|-
| [[974308500/974251369]]
| [2 11 3 -8 1 -2⟩
| 0.102
|-
| [[974358550/974251369]]
| [1 0 2 -8 7 -2⟩
| 0.190
|-
| [[990234375/989069312]]
| [-18 1 9 -3 -1 2⟩
| 2.038
|-
| [[992436543/991232000]]
| [-16 10 -3 5 -2⟩
| 2.103
|-
| [[1050304563/1048576000]]
| [-23 11 -3 2 2⟩
| 2.852
|-
| [[1056655611/1056250000]]
| [-4 8 -8 0 5 -2⟩
| 0.665
|-
| [[1069453125/1068548096]]
| [-14 4 7 -2 -3 2⟩
| 1.466
|-
| [[1071794405/1071385056]]
| [-5 -14 1 -1 8⟩
| 0.661
|-
| [[1076168025/1073741824]]
| [-30 16 2⟩
| 3.907
|-
| [[1089547389/1088704760]]
| [-3 3 -1 9 -5 -2⟩
| 1.339
|-
| [[1096135733/1095120000]]
| [-7 -4 -4 7 3 -2⟩
| 1.605
|-
| [[1107225625/1106841792]]
| [-6 -1 4 -8 6⟩
| 0.600
|-
| [[1155009375/1154413568]]
| [-10 7 5 -1 -5 2⟩
| 0.893
|-
| [[1162261467/1161875000]]
| [-3 19 -7 0 -1 -2⟩
| 0.576
|-
| [[1162261467/1162084000]]
| [-5 19 -3 -4 -2⟩
| 0.264
|-
| [[1182568205/1182449664]]
| [-14 -8 1 2 -1 6⟩
| 0.174
|-
| [[1220703125/1219401216]]
| [-9 -9 13 0 -2⟩
| 1.847
|-
| [[1220703125/1219784832]]
| [-7 -4 13 -6⟩
| 1.303
|-
| [[1220703125/1220312709]]
| [0 -5 13 -3 -4⟩
| 0.554
|-
| [[1224552875/1224440064]]
| [-8 -14 3 3 0 4⟩
| 0.159
|-
| [[1238015625/1235663104]]
| [-8 1 6 4 1 -6⟩
| 3.293
|-
| [[1247410125/1247178944]]
| [-6 10 3 0 -7 2⟩
| 0.321
|-
| [[1254484803/1254400000]]
| [-13 1 -5 -2 4 4⟩
| 0.117
|-
| [[1291467969/1291315424]]
| [-5 6 0 -9 6⟩
| 0.205
|-
| [[1311721047/1310720000]]
| [-21 8 -4 1 0 4⟩
| 1.322
|-
| [[1313046875/1311844248]]
| [-3 -6 7 5 -3 -2⟩
| 1.586
|-
| [[1377751375/1377495072]]
| [-5 -16 3 2 3 2⟩
| 0.322
|-
| [[1420541793/1417674752]]
| [-23 17 0 0 1 -2⟩
| 3.498
|-
| [[1420541793/1420191500]]
| [-2 17 -3 -5 1 -2⟩
| 0.427
|-
| [[1433600000/1433562273]]
| [16 -3 5 1 -1 -6⟩
| 0.046
|-
| [[1440894015/1438646272]]
| [-22 9 1 -3 4⟩
| 2.703
|-
| [[1475824779/1475789056]]
| [-8 8 0 -8 3 2⟩
| 0.042
|-
| [[1479028005/1476395008]]
| [-27 6 1 4 -1 2⟩
| 3.085
|-
| [[1500512167/1499109768]]
| [-3 -8 0 1 8 -4⟩
| 1.619
|-
| [[1500512167/1500000000]]
| [-8 -1 -9 1 8⟩
| 0.591
|-
| [[1530966437/1528823808]]
| [-21 -6 0 7 1 2⟩
| 2.425
|-
| [[1530966437/1530550080]]
| [-6 -14 -1 7 1 2⟩
| 0.471
|-
| [[1550115875/1549681956]]
| [-2 -18 3 1 6⟩
| 0.485
|-
| [[1610510000/1610497161]]
| [4 -4 4 -6 5 -2⟩
| 0.014
|-
| [[1634180625/1631461442]]
| [-1 2 4 4 2 -8⟩
| 2.883
|-
| [[1640558367/1638400000]]
| [-19 14 -5 3⟩
| 2.279
|-
| [[1646581365/1644167168]]
| [-25 11 1 -2 1 2⟩
| 2.540
|-
| [[1650478599/1650390625]]
| [0 11 -10 1 3 -2⟩
| 0.092
|-
| [[1655976960/1655595487]]
| [10 5 1 -3 3 -6⟩
| 0.399
|-
| [[1674421875/1672646404]]
| [-2 7 6 2 -4 -4⟩
| 1.837
|-
| [[1686616064/1686498489]]
| [11 -10 0 7 0 -4⟩
| 0.121
|-
| [[1714374025/1712848896]]
| [-19 -3 2 4 -2 4⟩
| 1.541
|-
| [[1714871048/1714374025]]
| [3 0 -2 -4 8 -4⟩
| 0.502
|-
| [[1722499009/1719926784]]
| [-18 -8 0 6 4⟩
| 2.587
|-
| [[1722499009/1721868840]]
| [-3 -16 -1 6 4⟩
| 0.633
|-
| [[1775239375/1773674496]]
| [-13 -9 4 5 -1 2⟩
| 1.527
|-
| [[1779441125/1776722688]]
| [-8 -5 3 6 2 -4⟩
| 2.647
|-
| [[1801088541/1799512000]]
| [-6 7 -3 7 -3 -2⟩
| 1.516
|-
| [[1815912315/1814078464]]
| [-10 2 1 9 -6⟩
| 1.749
|-
| [[1838265625/1836660096]]
| [-7 -15 6 6⟩
| 1.513
|-
| [[1845703125/1842700288]]
| [-13 3 10 1 -3 -2⟩
| 2.819
|-
| [[1929229929/1927561216]]
| [-14 2 0 -6 8⟩
| 1.498
|-
| [[1953045675/1949302784]]
| [-20 13 2 2 -1 -2⟩
| 3.321
|-
| [[1977326743/1974136320]]
| [-9 -3 -1 11 0 -4⟩
| 2.796
|-
| [[1977326743/1975897638]]
| [-1 -12 0 11 -1 -2⟩
| 1.252
|-
| [[1977326743/1976535000]]
| [-3 -3 -4 11 -4⟩
| 0.693
|-
| [[2091059712/2090808005]]
| [9 5 -1 5 -4 -4⟩
| 0.208
|-
| [[2122043913/2119936000]]
| [-11 13 -3 -2 3 -2⟩
| 1.721
|-
| [[2129600000/2128622769]]
| [9 -2 5 -2 3 -6⟩
| 0.795
|-
| [[2152024875/2147483648]]
| [-31 3 3 3 1 2⟩
| 3.657
|-
| [[2152446615/2147483648]]
| [-31 5 1 0 6⟩
| 3.996
|-
| [[2152828125/2147483648]]
| [-31 9 6 1⟩
| 4.303
|-
| [[2159925625/2157969408]]
| [-21 -1 4 -3 2 4⟩
| 1.569
|-
| [[2197265625/2194398752]]
| [-5 2 12 -4 0 -4⟩
| 2.260
|-
| [[2204627139/2202927104]]
| [-17 4 0 -5 5 2⟩
| 1.336
|-
| [[2343775203/2339717120]]
| [-14 3 -1 2 6 -4⟩
| 3.000
|-
| [[2357947691/2354946048]]
| [-13 -5 0 -1 9 -2⟩
| 2.205
|-
| [[2373046875/2370734366]]
| [-1 5 10 -3 -2 -4⟩
| 1.688
|-
| [[2435896375/2432711736]]
| [-3 -2 3 -1 7 -6⟩
| 2.265
|-
| [[2450000000/2447192163]]
| [7 -1 8 2 0 -8⟩
| 1.985
|-
| [[2495390625/2494357888]]
| [-7 3 7 1 -7 2⟩
| 0.717
|-
| [[2516421875/2514778812]]
| [-2 -12 6 -1 5 -2⟩
| 1.131
|-
| [[2519337249/2517630976]]
| [-20 6 0 -4 2 4⟩
| 1.173
|-
| [[2542277241/2537553920]]
| [-22 2 -1 10 -2⟩
| 3.219
|-
| [[2573571875/2569273344]]
| [-18 -4 5 7 -2⟩
| 2.894
|-
| [[2573571875/2572306572]]
| [-2 -1 5 7 -8⟩
| 0.851
|-
| [[2662250409/2662000000]]
| [-7 8 -6 4 -3 2⟩
| 0.163
|-
| [[2690514981/2684354560]]
| [-29 3 -1 7 2⟩
| 3.969
|-
| [[2706784157/2704000000]]
| [-10 0 -6 5 5 -2⟩
| 1.782
|-
| [[2841733125/2835349504]]
| [-24 10 4 1 1 -2⟩
| 3.893
|-
| [[2929687500/2927131207]]
| [2 1 12 -1 -4 -4⟩
| 1.511
|-
| [[2977309629/2974400000]]
| [-9 11 -5 5 -1 -2⟩
| 1.693
|-
| [[2989355625/2984869888]]
| [-15 14 4 -2 -1 -2⟩
| 2.600
|-
| [[3228504075/3224725504]]
| [-11 17 2 -1 -3 -2⟩
| 2.027
|-
| [[3268642167/3262922884]]
| [-2 4 0 9 0 -8⟩
| 3.032
|-
| [[3293331899/3288334336]]
| [-26 0 0 -2 7 2⟩
| 2.629
|-
| [[3293331899/3292047360]]
| [-11 -8 -1 -2 7 2⟩
| 0.675
|-
| [[3293331899/3292800000]]
| [-10 -1 -5 -3 7 2⟩
| 0.280
|-
| [[3294172000/3293331899]]
| [5 0 3 7 -7 -2⟩
| 0.442
|-
| [[3349609375/3345292808]]
| [-3 0 10 3 -4 -4⟩
| 2.232
|-
| [[3360173089/3355443200]]
| [-27 0 -2 6 0 4⟩
| 2.439
|-
| [[3360173089/3359232000]]
| [-12 -8 -3 6 0 4⟩
| 0.485
|-
| [[3402639576/3402202375]]
| [3 11 -3 4 -5 -2⟩
| 0.222
|-
| [[3410254925/3406261248]]
| [-10 -9 2 1 7 -2⟩
| 2.029
|-
| [[3465028125/3461742592]]
| [-17 8 5 -4 -1 2⟩
| 1.642
|-
| [[3486784401/3483855232]]
| [-7 20 0 0 -5 -2⟩
| 1.455
|-
| [[3487704605/3486784401]]
| [0 -20 1 8 2⟩
| 0.457
|-
| [[3514671875/3511517184]]
| [-15 -7 6 -2 3 2⟩
| 1.555
|-
| [[3520000000/3518743761]]
| [12 -6 7 0 1 -6⟩
| 0.618
|-
| [[3522990625/3518743761]]
| [0 -6 5 1 5 -6⟩
| 2.088
|-
| [[3544416225/3538822144]]
| [-10 10 2 4 -2 -4⟩
| 2.735
|-
| [[3630859375/3623878656]]
| [-27 -3 9 0 1 2⟩
| 3.332
|-
| [[3660938127/3655808000]]
| [-10 6 -3 3 4 -4⟩
| 2.428
|-
| [[3742230375/3739918336]]
| [-13 11 3 -3 -3 2⟩
| 1.070
|-
| [[3751087725/3743547392]]
| [-17 11 2 1 2 -4⟩
| 3.484
|-
| [[3763454409/3758096384]]
| [-29 2 0 -1 4 4⟩
| 2.467
|-
| [[3764768000/3763454409]]
| [8 -2 3 6 -4 -4⟩
| 0.604
|-
| [[3780549773/3774873600]]
| [-24 -2 -2 5 3 2⟩
| 2.601
|-
| [[3780549773/3779136000]]
| [-9 -10 -3 5 3 2⟩
| 0.648
|-
| [[3906984375/3898605568]]
| [-21 6 6 3 -1 -2⟩
| 3.717
|-
| [[3955078125/3954653486]]
| [-1 4 11 -11⟩
| 0.186
|-
| [[4000989993/4000000000]]
| [-11 1 -9 2 5 2⟩
| 0.428
|-
| [[4041608805/4040447488]]
| [-9 14 1 -2 -5 2⟩
| 0.498
|-
| [[4044184375/4037050368]]
| [-15 -6 5 6 1 -2⟩
| 3.057
|-
| [[4078653605/4076863488]]
| [-24 -5 1 0 0 8⟩
| 0.760
|-
| [[4079910912/4078653605]]
| [15 1 -1 3 2 -8⟩
| 0.534
|-
| [[4081676544/4078653605]]
| [8 2 -1 0 6 -8⟩
| 1.283
|-
| [[4202539929/4194304000]]
| [-25 6 -3 8⟩
| 3.396
|-
| [[4226622444/4223457875]]
| [2 8 -3 -1 5 -6⟩
| 1.297
|-
| [[4227952113/4225000000]]
| [-6 3 -8 6 3 -2⟩
| 1.209
|-
| [[4253517961/4246732800]]
| [-21 -4 -2 4 6⟩
| 2.764
|-
| [[4253517961/4251528000]]
| [-6 -12 -3 4 6⟩
| 0.810
|}
Retrieved from "https://en.xen.wiki/w/684edo"