Higher Rank EDO Generators One More Goddamn Time

Some EDOs can be defined by tempering out a single rank-2 comma and keeping octaves pure. Lots of them. Gobs. And it's dirt simple to know what the comma is, in so far as the EDO can be defined by one such comma. 

First, any EDO of {n} divisions will tune P8 to {n}-steps and P12 to 

    round(n * log_2(3))

steps. If we call this number of steps for P12 {m}, then, in so far as the EDO can be defined with a pure octave and a tempered comma, the comma will have the form

    [-m, n] or [m, -n]

Both such intervals will be tempered, either can be used for the definition, and they're inverses of each other, so they're basically the same thing. They're justly tuned to reciprocal frequency ratios and one will be larger than 1/1 and one will be smaller.

It's only when {n} and {m} are coprime that the EDO can be defined that way. Otherwise you'll need a higher interval space to define the EDO. Like, you could try it for 24-EDO, but you'd find that you had the same definition you previously found for 12-EDO, and when you tried to tune rank-2 interval space, elements of the space would only be tuned to the 12-EDO subspace.

Below 100 divisions, you can do most EDOs this way:

    rank-2 definable EDOs: [5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 26, 27, 29, 31, 32, 33, 37, 39, 40, 41, 42, 43, 45, 46, 47, 49, 50, 53, 55, 56, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 80, 81, 83, 88, 89, 90, 91, 94, 95, 97, 98, 99]-EDO

But I'm not going to devote much more to them because I've done that work in the past and also it's all very very easy to derive.

Below 100-divisions, we also have these EDOs that require higher-rank interval spaces for full definition / interpretation:

    rank-3: [10, 14, 15, 21, 25, 28, 34, 35, 48, 51, 52, 54, 58, 60, 72, 78, 84, 85, 87, 96]-EDO

    rank-4: [6, 24, 36, 38, 57, 66, 68, 76, 86, 100]-EDO

    rank-5: [20, 93]-EDO

    rank-6: [30, 44, 62, 82]-EDO

    rank-8: [92]-EDO

When you name a set of tempered commas, along with the octave, that are sufficient to define an EDO, I've sometimes heard that called a "comma basis". I don't love this, because I often use basis which are entirely made of commas, instead of commas and the octave. So I usually just call the octave and the commas the "EDO generator" but this has problems for other people. But whatever you call it, I'm going to be looking at commas that define EDOs with the addition of a pure octave in this post.

...

So, I've found sets of commas in the past that are sufficient to define all the EDOs up to 100, and indeed commas that have very small frequency ratios, in a way that I considered optimal in some sense at the time, but I never did a good job of explaining my notion of optimality. I've recently found calculators by Sintel and Graham Breed for finding the same things, but with different notions of optimality, and I'm going to put our three sets of definitions together as best as I can and really characterize EDOs. Also I think I'm going to try to show Fokker periodicity blocks for each EDO at different ranks or prime limits. I've never done that before, and I never thought Fokker periodicity blocks were very interesting, but they're growing on me a little, and I'm going to see if it's worth playing with.

...

First up, here are commas from Graham Breed at various interval ranks / prime limits. They're not all sufficient to define their associated EDOs, and they couldn't be, because e.g. his program will give two rank-3 commas to define 24-EDO, which are justly tuned to [81/80, 128/125], even though you need three rank-4 commas to do it. I'm also not not positive that I love his notion of minimality, but it's often very good, and I need to look at it more to figure out when and why I think it's inferior to what I've used before.

3-EDO, rank-6 : [M2, ReAsM2, m2, Sbm3, Asm2] :: [10/9, 44/39, 16/15, 7/6, 11/10]

3-EDO, rank-5 : [Asm2, SbAcm2, M2, m2] :: [11/10, 21/20, 10/9, 16/15]

3-EDO, rank-4 : [SbAcm2, Sbm3, m2] :: [21/20, 7/6, 16/15]

3-EDO, rank-3 : [M2, m2] :: [10/9, 16/15]


4-EDO, rank-6 : [AcM2, ReAsA1, A1, SpA1, DeAcM2] :: [9/8, 55/52, 25/24, 15/14, 12/11]

4-EDO, rank-5 : [DeAcM3, SpA1, Acm2, AcM2] :: [27/22, 15/14, 27/25, 9/8]

4-EDO, rank-4 : [SpA1, AcM2, Sp1] :: [15/14, 9/8, 36/35]

4-EDO, rank-3 : [A1, AcM2] :: [25/24, 9/8]


5-EDO, rank-6 : [ReAcA1, AsAsGrGrd3, Acm2, AsSpGr1, Asm2] :: [27/26, 484/405, 27/25, 22/21, 11/10]

5-EDO, rank-5 : [AsSpGr1, SbAcm2, Sbm2, m2] :: [22/21, 21/20, 28/27, 16/15]

5-EDO, rank-4 : [SbAcm2, m2, Sp1] :: [21/20, 16/15, 36/35]

5-EDO, rank-3 : [m2, Acm2] :: [16/15, 27/25]


6-EDO, rank-6 : [M2, m2, PrDeSp1, SbSbd3, Prm2] :: [10/9, 16/15, 78/77, 49/45, 13/12]

6-EDO, rank-5 : [AsSpGr1, M2, m2, AsSbm2] :: [22/21, 10/9, 16/15, 77/72]

6-EDO, rank-4 : [SpSpGrA0, m2, M2] :: [50/49, 16/15, 10/9]

6-EDO, rank-3 : [M2, m2] :: [10/9, 16/15]


7-EDO, rank-6 : [ReAsGr1, Ac1, A1, SpA1, DeAcA1] :: [352/351, 81/80, 25/24, 15/14, 45/44]

7-EDO, rank-5 : [AsSpGr1, SpGr1, Sp1, SpA1] :: [22/21, 64/63, 36/35, 15/14]

7-EDO, rank-4 : [SpA1, Sp1, Ac1] :: [15/14, 36/35, 81/80]

7-EDO, rank-3 : [A1, Ac1] :: [25/24, 81/80]


8-EDO, rank-6 : [Prd2, ReAs1, AsGr1, SbAcm2, m2] :: [26/25, 66/65, 55/54, 21/20, 16/15]

8-EDO, rank-5 : [Sbd3, DeSbAcM2, m2, AsGr1] :: [28/25, 35/33, 16/15, 55/54]

8-EDO, rank-4 : [SbAcm2, m2, SbM2] :: [21/20, 16/15, 175/162]

8-EDO, rank-3 : [m2, GrA1] :: [16/15, 250/243]


9-EDO, rank-6 : [ReAcA1, DeAcA1, Sp1, SbAcm2, As1] :: [27/26, 45/44, 36/35, 21/20, 33/32]

9-EDO, rank-5 : [SbAcm2, DeAcA1, AsSp1, Sp1] :: [21/20, 45/44, 297/280, 36/35]

9-EDO, rank-4 : [SbAcm2, Sp1, SpA0] :: [21/20, 36/35, 225/224]

9-EDO, rank-3 : [d2, Acm2] :: [128/125, 27/25]


10-EDO, rank-6 : [PrDem2, ReAsGr1, AsGr1, Sbm2, DeAcA1] :: [104/99, 352/351, 55/54, 28/27, 45/44]

10-EDO, rank-5 : [DeSbAcM2, Sbm2, SpGrA1, A1] :: [35/33, 28/27, 200/189, 25/24]

10-EDO, rank-4 : [SpSpGrA0, A1, Sbm2] :: [50/49, 25/24, 28/27]

10-EDO, rank-3 : [A1, Grm2] :: [25/24, 256/243]


11-EDO, rank-6 : [PrDeAcm2, Acm3, ReSbAcA1, SbAcm2, As1] :: [117/110, 243/200, 105/104, 21/20, 33/32]

11-EDO, rank-5 : [As1, SbAcm2, Spm2, DeAcA1] :: [33/32, 21/20, 192/175, 45/44]

11-EDO, rank-4 : [SbAcm2, SpA0, SpAcM2] :: [21/20, 225/224, 81/70]

11-EDO, rank-3 : [d3, AcA1] :: [144/125, 135/128]


12-EDO, rank-6 : [PrDem2, Ac1, Sp1, SpA0, DeA1] :: [104/99, 81/80, 36/35, 225/224, 100/99]

12-EDO, rank-5 : [AsSpSpGrM0, DeAcA1, Sp1, DeA1] :: [99/98, 45/44, 36/35, 100/99]

12-EDO, rank-4 : [SpSpGrA0, SpGr1, Sp1] :: [50/49, 64/63, 36/35]

12-EDO, rank-3 : [Ac1, d2] :: [81/80, 128/125]


13-EDO, rank-6 : [PrDem2, A1, SpGr1, SbSbd3, DeSbAcM2] :: [104/99, 25/24, 64/63, 49/45, 35/33]

13-EDO, rank-5 : [DeSbAcm2, A1, AsSbm2, SpGr1] :: [56/55, 25/24, 77/72, 64/63]

13-EDO, rank-4 : [SbSbd3, SpGrA1, A1] :: [49/45, 200/189, 25/24]

13-EDO, rank-3 : [A1, GrGrm3] :: [25/24, 2560/2187]


14-EDO, rank-6 : [ReAsSbm2, ReAsSpA0, Prd2, SbAcm2, Sp1] :: [616/585, 1485/1456, 26/25, 21/20, 36/35]

14-EDO, rank-5 : [Sp1, DeSbAcm2, SbAcm2, AsGrd2] :: [36/35, 56/55, 21/20, 704/675]

14-EDO, rank-4 : [SbAcm2, Sp1, SbGrdd3] :: [21/20, 36/35, 3584/3375]

14-EDO, rank-3 : [Acm2, Grd2] :: [27/25, 2048/2025]


15-EDO, rank-6 : [Sbm2, ReA1, SpGr1, DeA1, AsGr1] :: [28/27, 40/39, 64/63, 100/99, 55/54]

15-EDO, rank-5 : [SbSbSbAcdd3, DeSbAcm2, SbSbAcm2, SpGr1] :: [1029/1000, 56/55, 49/48, 64/63]

15-EDO, rank-4 : [Sbm2, SpGr1, SbAcd2] :: [28/27, 64/63, 126/125]

15-EDO, rank-3 : [GrA1, d2] :: [250/243, 128/125]


16-EDO, rank-6 : [As1, ReAs1, Sp1, SbAcd2, DeAcA1] :: [33/32, 66/65, 36/35, 126/125, 45/44]

16-EDO, rank-5 : [DeAcA1, AsSp1, SpSpGrA0, Sp1] :: [45/44, 297/280, 50/49, 36/35]

16-EDO, rank-4 : [SpSpGrA0, Sp1, AcA1] :: [50/49, 36/35, 135/128]

16-EDO, rank-3 : [Acd2, AcA1] :: [648/625, 135/128]


17-EDO, rank-6 : [A1, SpGr1, ReA1, DeAcA1, AsSpSpGrM0] :: [25/24, 64/63, 40/39, 45/44, 99/98]

17-EDO, rank-5 : [DeSbm2, SpGr1, DeSpA1, A1] :: [896/891, 64/63, 80/77, 25/24]

17-EDO, rank-4 : [SpGrA1, SpSpA0, A1] :: [200/189, 405/392, 25/24]

17-EDO, rank-3 : [A1, GrGrm2] :: [25/24, 20480/19683]


18-EDO, rank-6 : [ReAs1, Dem2, SpGrA1, ReA1, Sbm2] :: [66/65, 512/495, 200/189, 40/39, 28/27]

18-EDO, rank-5 : [AsSpGr1, Sbm2, SpGrA1, AsSb1] :: [22/21, 28/27, 200/189, 385/384]

18-EDO, rank-4 : [SpSpGrA0, Sbm2, d2] :: [50/49, 28/27, 128/125]

18-EDO, rank-3 : [d2, GrM2] :: [128/125, 800/729]


19-EDO, rank-6 : [ReSbSbAcm2, DeA1, SpA0, DeAcA1, SbSbAcm2] :: [196/195, 100/99, 225/224, 45/44, 49/48]

19-EDO, rank-5 : [Ac1, SpA0, DeAcA1, SbSbAcm2] :: [81/80, 225/224, 45/44, 49/48]

19-EDO, rank-4 : [SbSbSbAcdd3, SpA0, SbSbAcm2] :: [1029/1000, 225/224, 49/48]

19-EDO, rank-3 : [Ac1, AA0] :: [81/80, 3125/3072]


20-EDO, rank-6 : [ReA1, A1, DeDeSbAcM2, SbSbAcm2, Sbm2] :: [40/39, 25/24, 1120/1089, 49/48, 28/27]

20-EDO, rank-5 : [Sbm2, A1, AsAsSpGrGrd1, SpGr1] :: [28/27, 25/24, 968/945, 64/63]

20-EDO, rank-4 : [SpSpGrA0, A1, Sbm2] :: [50/49, 25/24, 28/27]

20-EDO, rank-3 : [A1, Grm2] :: [25/24, 256/243]


21-EDO, rank-6 : [ReAs1, d2, DeAcA1, Sp1, PrSbAcd2] :: [66/65, 128/125, 45/44, 36/35, 819/800]

21-EDO, rank-5 : [AsSpSpGrM0, DeAcA1, Sp1, AsSbAcd2] :: [99/98, 45/44, 36/35, 2079/2000]

21-EDO, rank-4 : [SbSbSbAcdd3, SpA0, Sp1] :: [1029/1000, 225/224, 36/35]

21-EDO, rank-3 : [AcAcm2, d2] :: [2187/2000, 128/125]


22-EDO, rank-6 : [AsGr1, PrSpGr1, AsSpGrd1, PrSbd2, SpGr1] :: [55/54, 65/63, 176/175, 91/90, 64/63]

22-EDO, rank-5 : [SpA0, AsSpGrd1, AsGr1, SpGr1] :: [225/224, 176/175, 55/54, 64/63]

22-EDO, rank-4 : [SpSpGrA0, SpGr1, SbSbm2] :: [50/49, 64/63, 245/243]

22-EDO, rank-3 : [GrA1, Grd2] :: [250/243, 2048/2025]


23-EDO, rank-6 : [ReAcA1, Sp1, DeA1, AcA1, DeSbAcAcm2] :: [27/26, 36/35, 100/99, 135/128, 567/550]

23-EDO, rank-5 : [DeSpA1, Sp1, AsSpSpGrA0, AcA1] :: [80/77, 36/35, 825/784, 135/128]

23-EDO, rank-4 : [Sp1, SpSpGrAA0, AcA1] :: [36/35, 625/588, 135/128]

23-EDO, rank-3 : [AcA1, AcAcd2] :: [135/128, 6561/6250]


24-EDO, rank-6 : [ReAs1, DeSbAcm2, PrSbAcd2, SbSbAcm2, AsSbd2] :: [66/65, 56/55, 819/800, 49/48, 77/75]

24-EDO, rank-5 : [DeSbAcm2, Ac1, AsSpGrd1, SbSbAcm2] :: [56/55, 81/80, 176/175, 49/48]

24-EDO, rank-4 : [Ac1, SbSbdd3, SbSbAcm2] :: [81/80, 392/375, 49/48]

24-EDO, rank-3 : [Ac1, d2] :: [81/80, 128/125]


25-EDO, rank-6 : [ReAs1, Sbm2, PrAsSpGrGr1, ReA1, ReSbAcA1] :: [66/65, 28/27, 3575/3402, 40/39, 105/104]

25-EDO, rank-5 : [Grm2, DeA1, Sbm2, AsSpGrA0] :: [256/243, 100/99, 28/27, 1375/1344]

25-EDO, rank-4 : [SpGr1, Sbm2, SpGrAA0] :: [64/63, 28/27, 3125/3024]

25-EDO, rank-3 : [GrAA1, Grm2] :: [3125/2916, 256/243]


26-EDO, rank-6 : [AsSpSpGrM0, PrDeSp1, Ac1, DeAcA1, Pr1] :: [99/98, 78/77, 81/80, 45/44, 65/64]

26-EDO, rank-5 : [AsSpSpGrM0, DeAcA1, DeA1, AsSb1] :: [99/98, 45/44, 100/99, 385/384]

26-EDO, rank-4 : [SpSpGrA0, SbAcA1, Ac1] :: [50/49, 525/512, 81/80]

26-EDO, rank-3 : [Ac1, AAA0] :: [81/80, 78125/73728]


27-EDO, rank-6 : [ReAs1, PrGrd2, AsSbd2, AsGr1, AsSpSpGrM0] :: [66/65, 416/405, 77/75, 55/54, 99/98]

27-EDO, rank-5 : [AsSpSpGrM0, AsGr1, d2, SpGr1] :: [99/98, 55/54, 128/125, 64/63]

27-EDO, rank-4 : [SbSbSbdd3, SbAcd2, SpGr1] :: [686/675, 126/125, 64/63]

27-EDO, rank-3 : [d2, GrGrA1] :: [128/125, 20000/19683]


28-EDO, rank-6 : [ReAs1, Sp1, PrAsd1, SbAcd2, DeAcA1] :: [66/65, 36/35, 1287/1280, 126/125, 45/44]

28-EDO, rank-5 : [DeSbAcAcm2, DeAcA1, Sp1, AsAc1] :: [567/550, 45/44, 36/35, 2673/2560]

28-EDO, rank-4 : [SpSpGrA0, Sp1, AcAcA1] :: [50/49, 36/35, 2187/2048]

28-EDO, rank-3 : [Acd2, AcAcA1] :: [648/625, 2187/2048]


29-EDO, rank-6 : [ReAsGr1, SpA0, PrDeSpGr1, ReSbAcA1, AsGr1] :: [352/351, 225/224, 2080/2079, 105/104, 55/54]

29-EDO, rank-5 : [AsGr1, SpA0, DeSbSbAcM2, SbSbAcm2] :: [55/54, 225/224, 1225/1188, 49/48]

29-EDO, rank-4 : [SpSpGrGrA0, SpA0, SbSbAcm2] :: [4000/3969, 225/224, 49/48]

29-EDO, rank-3 : [GrA1, AcAA0] :: [250/243, 16875/16384]


30-EDO, rank-6 : [SpGr1, DeSbAcm2, DeA1, Sbm2, PrPrSpGrd1] :: [64/63, 56/55, 100/99, 28/27, 169/168]

30-EDO, rank-5 : [SbSbSbAcdd3, DeSbAcm2, SbSbAcm2, SpGr1] :: [1029/1000, 56/55, 49/48, 64/63]

30-EDO, rank-4 : [Sbm2, SpGr1, SbAcd2] :: [28/27, 64/63, 126/125]

30-EDO, rank-3 : [GrA1, d2] :: [250/243, 128/125]


31-EDO, rank-6 : [ReAsGr1, ReSbAcA1, Ac1, SpA0, ReDeAcA1] :: [352/351, 105/104, 81/80, 225/224, 144/143]

31-EDO, rank-5 : [DeSbSbAcAcm2, SpA0, Ac1, AsSb1] :: [441/440, 225/224, 81/80, 385/384]

31-EDO, rank-4 : [SbAcd2, Ac1, SpSpSpGrM0] :: [126/125, 81/80, 1728/1715]

31-EDO, rank-3 : [Ac1, Grdddd3] :: [81/80, 393216/390625]


32-EDO, rank-6 : [AsGr1, PrGrd2, PrSpGr1, SpGr1, DeSbM2] :: [55/54, 416/405, 65/63, 64/63, 2800/2673]

32-EDO, rank-5 : [AsGr1, SpSpGrA0, DeSbSbAcM2, SpGr1] :: [55/54, 50/49, 1225/1188, 64/63]

32-EDO, rank-4 : [SpSpGrA0, SpGr1, SbSbM2] :: [50/49, 64/63, 6125/5832]

32-EDO, rank-3 : [GrAA1, Grd2] :: [3125/2916, 2048/2025]


33-EDO, rank-6 : [PrDeSp1, PrAsSpGrd1, SpA0, Sp1, DeSbSbAcAcm2] :: [78/77, 143/140, 225/224, 36/35, 441/440]

33-EDO, rank-5 : [SpA0, DeSbSbAcAcm2, Sp1, AsAsGrd1] :: [225/224, 441/440, 36/35, 121/120]

33-EDO, rank-4 : [Sp1, SpA0, SbSbSbAcAcdd3] :: [36/35, 225/224, 83349/80000]

33-EDO, rank-3 : [d2, AcAcAcm2] :: [128/125, 177147/160000]


34-EDO, rank-6 : [ReSbAcA1, DeSbAcm2, DeA1, ReAsGr1, SbSbAcm2] :: [105/104, 56/55, 100/99, 352/351, 49/48]

34-EDO, rank-5 : [DeSbAcm2, DeSbAcAcA1, SbA1, SbSbAcm2] :: [56/55, 2835/2816, 875/864, 49/48]

34-EDO, rank-4 : [SbSbAcm2, SbAcd2, SbGrm2] :: [49/48, 126/125, 2240/2187]

34-EDO, rank-3 : [GrGrA1, Grd2] :: [20000/19683, 2048/2025]


35-EDO, rank-6 : [ReAcA1, DeSbAcAcm2, AsSpSpGrM0, DeAcA1, AsSb1] :: [27/26, 567/550, 99/98, 45/44, 385/384]

35-EDO, rank-5 : [AsSpSpGrM0, DeAcA1, SbAcd2, AsSb1] :: [99/98, 45/44, 126/125, 385/384]

35-EDO, rank-4 : [SbAcd2, SpSpA0, SbAcA1] :: [126/125, 405/392, 525/512]

35-EDO, rank-3 : [AcAcd2, AcAcA1] :: [6561/6250, 2187/2048]


36-EDO, rank-6 : [PrDeSp1, ReSpAcA0, AsSpGrd1, Ac1, DeSbAcm2] :: [78/77, 729/728, 176/175, 81/80, 56/55]

36-EDO, rank-5 : [DeSpSpAcA0, DeSbAcm2, AsSpGrd1, Ac1] :: [2187/2156, 56/55, 176/175, 81/80]

36-EDO, rank-4 : [SbSbSbAcdd3, Ac1, d2] :: [1029/1000, 81/80, 128/125]

36-EDO, rank-3 : [Ac1, d2] :: [81/80, 128/125]


37-EDO, rank-6 : [PrAsGrGrd2, GrA1, AsGr1, PrSbd2, ReSbSbAcm2] :: [2288/2187, 250/243, 55/54, 91/90, 196/195]

37-EDO, rank-5 : [AsGr1, SbSbSbdd3, DeA1, SpGr1] :: [55/54, 686/675, 100/99, 64/63]

37-EDO, rank-4 : [SbSbGrdd3, SpGr1, GrA1] :: [6272/6075, 64/63, 250/243]

37-EDO, rank-3 : [GrA1, GrGrddd3] :: [250/243, 262144/253125]


38-EDO, rank-6 : [DeDeAcAcA1, ReAs1, AsSpGrd1, SpSpGrA0, Ac1] :: [243/242, 66/65, 176/175, 50/49, 81/80]

38-EDO, rank-5 : [Ac1, SpSpGrA0, AsSpGrd1, AsSb1] :: [81/80, 50/49, 176/175, 385/384]

38-EDO, rank-4 : [SpSpGrA0, Ac1, SpSpGrd1] :: [50/49, 81/80, 6144/6125]

38-EDO, rank-3 : [Ac1, AA0] :: [81/80, 3125/3072]


39-EDO, rank-6 : [ReAsSbSbm2, PrDeSbm2, AsSbd2, SbSbAcm2, DeSbAcm2] :: [1078/1053, 910/891, 77/75, 49/48, 56/55]

39-EDO, rank-5 : [DeSbAcm2, AsSpGrd1, SbSbAcm2, SbGrm2] :: [56/55, 176/175, 49/48, 2240/2187]

39-EDO, rank-4 : [SbGrm2, SbSbdd3, SbSbAcm2] :: [2240/2187, 392/375, 49/48]

39-EDO, rank-3 : [d2, GrGrGrA1] :: [128/125, 1600000/1594323]


40-EDO, rank-6 : [ReAs1, Acd2, AsSb1, AsSpSpGrM0, ReSbAcA1] :: [66/65, 648/625, 385/384, 99/98, 105/104]

40-EDO, rank-5 : [AsSpSpGrM0, SpA0, AsSb1, DeSbAcAcm2] :: [99/98, 225/224, 385/384, 567/550]

40-EDO, rank-4 : [SbSbSbAcdd3, SpA0, Acd2] :: [1029/1000, 225/224, 648/625]

40-EDO, rank-3 : [Acd2, AcAcAA0] :: [648/625, 273375/262144]


41-EDO, rank-6 : [DeA1, ReSbAcA1, ReDeAcA1, DeSbSbAcAcm2, SpA0] :: [100/99, 105/104, 144/143, 441/440, 225/224]

41-EDO, rank-5 : [DeDeSbSbAcAcM2, SbSbm2, SpA0, DeA1] :: [245/242, 245/243, 225/224, 100/99]

41-EDO, rank-4 : [SpA0, SpSpGrGrA0, SbSbm2] :: [225/224, 4000/3969, 245/243]

41-EDO, rank-3 : [AA0, GrGrA1] :: [3125/3072, 20000/19683]


42-EDO, rank-6 : [AsGr1, PrPrGrd2, AsSbd2, PrAsSpGrd1, SpGr1] :: [55/54, 169/162, 77/75, 143/140, 64/63]

42-EDO, rank-5 : [DeDeSbSbSbAcd3, AsGrd2, SpGr1, AsGr1] :: [5488/5445, 704/675, 64/63, 55/54]

42-EDO, rank-4 : [SpGr1, SbAcd2, SbSbSbGrd3] :: [64/63, 126/125, 6860/6561]

42-EDO, rank-3 : [d2, GrGrGrM2] :: [128/125, 5120000/4782969]


43-EDO, rank-6 : [SpA0, AsAsSbSbAcdd2, PrDeDeSbAcm2, AsSpGrd1, Ac1] :: [225/224, 160083/160000, 364/363, 176/175, 81/80]

43-EDO, rank-5 : [AsSpSpGrM0, SpA0, Ac1, DeDeSpAcA1] :: [99/98, 225/224, 81/80, 864/847]

43-EDO, rank-4 : [SbAcd2, Ac1, SpSpSpSpGrM0] :: [126/125, 81/80, 62208/60025]

43-EDO, rank-3 : [Ac1, Grdddddd4] :: [81/80, 50331648/48828125]


44-EDO, rank-6 : [AsGr1, ReReA1, DeSbm2, SpSpGrA0, SpGr1] :: [55/54, 512/507, 896/891, 50/49, 64/63]

44-EDO, rank-5 : [SpA0, AsSpGrd1, AsGr1, SpGr1] :: [225/224, 176/175, 55/54, 64/63]

44-EDO, rank-4 : [SpSpGrA0, SpGr1, SbSbm2] :: [50/49, 64/63, 245/243]

44-EDO, rank-3 : [GrA1, Grd2] :: [250/243, 2048/2025]


45-EDO, rank-6 : [DeAcA1, ReAsSpGrA0, ReSbSbAcAcm2, Ac1, AsSb1] :: [45/44, 275/273, 1323/1300, 81/80, 385/384]

45-EDO, rank-5 : [DeAcA1, Ac1, AsSpSpSpGrAA-1, AsSb1] :: [45/44, 81/80, 1375/1372, 385/384]

45-EDO, rank-4 : [SbA1, Ac1, SpSpSpAAA-1] :: [875/864, 81/80, 5625/5488]

45-EDO, rank-3 : [Ac1, AcAAAAAA-1] :: [81/80, 244140625/226492416]


46-EDO, rank-6 : [PrPrDeGrd2, AsGrGr1, SbAcd2, SbSbm2, PrSbd2] :: [2704/2673, 2200/2187, 126/125, 245/243, 91/90]

46-EDO, rank-5 : [SbSbSbdd3, SbAcd2, DeSbSbAcAcm2, DeSbm2] :: [686/675, 126/125, 441/440, 896/891]

46-EDO, rank-4 : [SbAcd2, SbSbm2, Grd2] :: [126/125, 245/243, 2048/2025]

46-EDO, rank-3 : [AcAcdd2, Grd2] :: [78732/78125, 2048/2025]


47-EDO, rank-6 : [DeAcA1, SbAcd2, AsSpSpGrM0, ReSbAcA1, ReSp1] :: [45/44, 126/125, 99/98, 105/104, 2304/2275]

47-EDO, rank-5 : [AsSpSpGrM0, DeAcA1, SbAcd2, AsSbAc1] :: [99/98, 45/44, 126/125, 2079/2048]

47-EDO, rank-4 : [SbAcd2, SpSpA0, SbSbSbAcAcm2] :: [126/125, 405/392, 1029/1024]

47-EDO, rank-3 : [AcAcd2, AcAA0] :: [6561/6250, 16875/16384]


48-EDO, rank-6 : [ReReA1, AsSb1, DeA1, SpSpGrA0, ReDeAcA1] :: [512/507, 385/384, 100/99, 50/49, 144/143]

48-EDO, rank-5 : [AsSpSpGrM0, AsGrGr1, DeA1, SbA1] :: [99/98, 2200/2187, 100/99, 875/864]

48-EDO, rank-4 : [SpSpGrA0, SbA1, SpAcA0] :: [50/49, 875/864, 3645/3584]

48-EDO, rank-3 : [GrGrA1, AcAA0] :: [20000/19683, 16875/16384]


49-EDO, rank-6 : [PrAsGrdd2, AsSpGrGrA0, SbSbm2, PrSbd2, DeA1] :: [1144/1125, 6875/6804, 245/243, 91/90, 100/99]

49-EDO, rank-5 : [SpGr1, AsAsSpGrGrM0, DeA1, DeSpSpA0] :: [64/63, 3025/3024, 100/99, 540/539]

49-EDO, rank-4 : [SpGr1, SbSbddd3, SbSbm2] :: [64/63, 3136/3125, 245/243]

49-EDO, rank-3 : [GrAA0, GrGrm2] :: [15625/15552, 20480/19683]


50-EDO, rank-6 : [SpA0, PrDeDeSbAcAcm2, ReSbSbAcAcm2, AsSb1, Ac1] :: [225/224, 2457/2420, 1323/1300, 385/384, 81/80]

50-EDO, rank-5 : [DeDeSbSbAcAcM2, AsSbSbAcd2, SbAcd2, Ac1] :: [245/242, 1617/1600, 126/125, 81/80]

50-EDO, rank-4 : [SbAcd2, Ac1, SbSbSbSbSbAcd3] :: [126/125, 81/80, 84035/82944]

50-EDO, rank-3 : [Ac1, AcAAAAAAA-2] :: [81/80, 1220703125/1207959552]


51-EDO, rank-6 : [PrDeSpGr1, AsGr1, ReSbSbAcm2, DeA1, ReSpSpGrA0] :: [2080/2079, 55/54, 196/195, 100/99, 640/637]

51-EDO, rank-5 : [SpA0, DeA1, AsGr1, SbSbSbAcAcm2] :: [225/224, 100/99, 55/54, 1029/1024]

51-EDO, rank-4 : [GrA1, SpA0, SbSbSbAcAcm2] :: [250/243, 225/224, 1029/1024]

51-EDO, rank-3 : [GrA1, AcAcAcAAAA-1] :: [250/243, 34171875/33554432]


52-EDO, rank-6 : [AsSpSpGrM0, PrPrGrdd2, PrDeSp1, SpA0, DeSbAcAcm2] :: [99/98, 676/675, 78/77, 225/224, 567/550]

52-EDO, rank-5 : [AsSpSpGrM0, DeSbAcAcm2, SpA0, AsSbSbAcd2] :: [99/98, 567/550, 225/224, 1617/1600]

52-EDO, rank-4 : [SpAc1, SpA0, SbSbSbAcAcm2] :: [729/700, 225/224, 1029/1024]

52-EDO, rank-3 : [Acd2, AcAcAcAA0] :: [648/625, 4428675/4194304]


53-EDO, rank-6 : [ReSpAcA0, ReAsSpGrA0, PrDeSpSpGrA0, AsSpSpGrM0, SpA0] :: [729/728, 275/273, 1625/1617, 99/98, 225/224]

53-EDO, rank-5 : [AsSpSpGrM0, AsGrGr1, SpA0, DeSpSpA0] :: [99/98, 2200/2187, 225/224, 540/539]

53-EDO, rank-4 : [SpSpGrGrA0, SpA0, SpSpSpGrM0] :: [4000/3969, 225/224, 1728/1715]

53-EDO, rank-3 : [GrAA0, AcAcA0] :: [15625/15552, 32805/32768]


54-EDO, rank-6 : [AsGr1, PrDeA1, SpA0, ReDeSbAcAA1, SpGr1] :: [55/54, 1625/1584, 225/224, 875/858, 64/63]

54-EDO, rank-5 : [SpA0, SpGr1, AsGr1, DeDeSbSbAcA2] :: [225/224, 64/63, 55/54, 30625/29403]

54-EDO, rank-4 : [SpSpGrA0, SpGr1, SbSbGrA2] :: [50/49, 64/63, 765625/708588]

54-EDO, rank-3 : [GrGrAAA1, Grd2] :: [390625/354294, 2048/2025]


55-EDO, rank-6 : [DeDeAcAcA1, ReAs1, DeSbSbAcdd3, AsSpGrd1, Ac1] :: [243/242, 66/65, 7056/6875, 176/175, 81/80]

55-EDO, rank-5 : [Ac1, AsSpGrd1, DeSbSbAcdd3, AsSb1] :: [81/80, 176/175, 7056/6875, 385/384]

55-EDO, rank-4 : [SbSbSbAcdd3, Ac1, SpSpGrd1] :: [1029/1000, 81/80, 6144/6125]

55-EDO, rank-3 : [Ac1, GrGrdddddddd5] :: [81/80, 6442450944/6103515625]


56-EDO, rank-6 : [DeA1, PrDed2, ReAsGr1, PrSbd2, DeSbSbAcAcm2] :: [100/99, 832/825, 352/351, 91/90, 441/440]

56-EDO, rank-5 : [AsSb1, DeSbSbAcAcm2, DeA1, AsSbGrd2] :: [385/384, 441/440, 100/99, 1232/1215]

56-EDO, rank-4 : [SbSbSbdd3, SbGrm2, SbA1] :: [686/675, 2240/2187, 875/864]

56-EDO, rank-3 : [Grd2, GrGrAAA0] :: [2048/2025, 1953125/1889568]


57-EDO, rank-6 : [ReAsSpGrA0, ReSbAcA1, Ac1, AsSb1, DeSpAA0] :: [275/273, 105/104, 81/80, 385/384, 625/616]

57-EDO, rank-5 : [AsSpSpGrM0, Ac1, AsSb1, DeSpAA0] :: [99/98, 81/80, 385/384, 625/616]

57-EDO, rank-4 : [SpSpSpGrM0, Ac1, SpSpSpAAA-1] :: [1728/1715, 81/80, 5625/5488]

57-EDO, rank-3 : [Ac1, AA0] :: [81/80, 3125/3072]


58-EDO, rank-6 : [ReAsAsd1, PrPrGrdd2, PrSpd1, SbAcd2, AsSpGrd1] :: [3267/3250, 676/675, 351/350, 126/125, 176/175]

58-EDO, rank-5 : [DeDeAcAcA1, SbAcd2, AsSpGrd1, DeSbm2] :: [243/242, 126/125, 176/175, 896/891]

58-EDO, rank-4 : [SbAcd2, SpSpSpGrM0, Grd2] :: [126/125, 1728/1715, 2048/2025]

58-EDO, rank-3 : [Grd2, AcAcAcdd2] :: [2048/2025, 1594323/1562500]


59-EDO, rank-6 : [PrDed2, PrSbd2, DeA1, AsGr1, ReSbSbSbdd3] :: [832/825, 91/90, 100/99, 55/54, 43904/43875]

59-EDO, rank-5 : [AsSb1, DeA1, SpGr1, SbSbSbSbdddd4] :: [385/384, 100/99, 64/63, 153664/151875]

59-EDO, rank-4 : [SpGr1, GrA1, SbSbSbSbdddd4] :: [64/63, 250/243, 153664/151875]

59-EDO, rank-3 : [GrA1, GrGrGrddddd4] :: [250/243, 536870912/512578125]


60-EDO, rank-6 : [SpA0, ReSbAcA1, DeA1, PrDeSbSbAcAcm2, AsSb1] :: [225/224, 105/104, 100/99, 5733/5632, 385/384]

60-EDO, rank-5 : [DeA1, SpA0, DeDeSbSbAcAcAcM2, AsSb1] :: [100/99, 225/224, 3969/3872, 385/384]

60-EDO, rank-4 : [SpA0, SbSbSbSbAcAcdd3, SbSbm2] :: [225/224, 64827/64000, 245/243]

60-EDO, rank-3 : [AA0, GrGrGrA1] :: [3125/3072, 1600000/1594323]


61-EDO, rank-6 : [PrAsGrdd2, SbAcd2, PrSbd2, AsSb1, ReDeSbSbAcM2] :: [1144/1125, 126/125, 91/90, 385/384, 3920/3861]

61-EDO, rank-5 : [AsSpGrd1, SbAcd2, AsSb1, SbGrm2] :: [176/175, 126/125, 385/384, 2240/2187]

61-EDO, rank-4 : [SbSbSbAcAcm2, SbAcd2, SbGrm2] :: [1029/1024, 126/125, 2240/2187]

61-EDO, rank-3 : [GrGrA1, GrGrddd3] :: [20000/19683, 262144/253125]


62-EDO, rank-6 : [SpA0, AsSpGrd1, Ac1, AsSb1, PrPrSpGrd1] :: [225/224, 176/175, 81/80, 385/384, 169/168]

62-EDO, rank-5 : [DeSbSbAcAcm2, SpA0, Ac1, AsSb1] :: [441/440, 225/224, 81/80, 385/384]

62-EDO, rank-4 : [SbAcd2, Ac1, SpSpSpGrM0] :: [126/125, 81/80, 1728/1715]

62-EDO, rank-3 : [Ac1, Grdddd3] :: [81/80, 393216/390625]


63-EDO, rank-6 : [DeA1, PrAsGrGrdd2, ReAsGr1, SpA0, DeSpSpA0] :: [100/99, 18304/18225, 352/351, 225/224, 540/539]

63-EDO, rank-5 : [DeA1, SpA0, AsSb1, AsAsSpSpGrGrGrM0] :: [100/99, 225/224, 385/384, 12100/11907]

63-EDO, rank-4 : [SpA0, SpSpSpGrGrGrA0, SbSbm2] :: [225/224, 256000/250047, 245/243]

63-EDO, rank-3 : [AA0, GrGrGrm2] :: [3125/3072, 1638400/1594323]


64-EDO, rank-6 : [ReAs1, Acd2, SpA0, ReDeAcA1, DeSbSbAcAcm2] :: [66/65, 648/625, 225/224, 144/143, 441/440]

64-EDO, rank-5 : [SpA0, DeSbSbAcAcm2, AsAsGrd1, SpAc1] :: [225/224, 441/440, 121/120, 729/700]

64-EDO, rank-4 : [SpA0, SbSbSbSbAcAcdd3, SpAc1] :: [225/224, 64827/64000, 729/700]

64-EDO, rank-3 : [Acd2, AcAcAcAcAA0] :: [648/625, 71744535/67108864]


65-EDO, rank-6 : [ReSbAcA1, PrDeDeDeSbSbAcAcAcM2, AsAsSbGrdd2, ReAsGr1, PrDeSbSbAcm2] :: [105/104, 85995/85184, 3388/3375, 352/351, 3185/3168]

65-EDO, rank-5 : [DeDeSbSbAcAcM2, DeSbSbdd3, SbAcd2, AsSb1] :: [245/242, 12544/12375, 126/125, 385/384]

65-EDO, rank-4 : [SbSbSbdd3, SbAcd2, SbSbAcAcA1] :: [686/675, 126/125, 33075/32768]

65-EDO, rank-3 : [AcAcdd2, AcAcA0] :: [78732/78125, 32805/32768]


66-EDO, rank-6 : [DeA1, ReAsSpGrA0, ReSbSbAcm2, AsGr1, Grd2] :: [100/99, 275/273, 196/195, 55/54, 2048/2025]

66-EDO, rank-5 : [AsGr1, Grd2, SbSbSbdd3, DeA1] :: [55/54, 2048/2025, 686/675, 100/99]

66-EDO, rank-4 : [SbSbSbdd3, GrA1, Grd2] :: [686/675, 250/243, 2048/2025]

66-EDO, rank-3 : [GrA1, Grd2] :: [250/243, 2048/2025]


67-EDO, rank-6 : [SbSbSbAcAcm2, PrPrDeSbAcd2, AsSpGrd1, ReSbSbAcm2, Ac1] :: [1029/1024, 3549/3520, 176/175, 196/195, 81/80]

67-EDO, rank-5 : [DeDeDeSbAcAcM2, SpSpSpGrM0, Ac1, AsSpGrd1] :: [1344/1331, 1728/1715, 81/80, 176/175]

67-EDO, rank-4 : [Ac1, SbSbSbSbAcdddd4, SpSpSpGrM0] :: [81/80, 9604/9375, 1728/1715]

67-EDO, rank-3 : [Ac1, Grddddddddd6] :: [81/80, 4174708211712/3814697265625]


68-EDO, rank-6 : [ReReA1, PrAsAsGrGrd1, DeSbm2, ReSbSbAcm2, ReAsSpGrA0] :: [512/507, 7865/7776, 896/891, 196/195, 275/273]

68-EDO, rank-5 : [DeSbm2, SpSpGrGrA0, AsSpGrd1, AsSb1] :: [896/891, 4000/3969, 176/175, 385/384]

68-EDO, rank-4 : [SpSpGrGrA0, SbSbm2, Grd2] :: [4000/3969, 245/243, 2048/2025]

68-EDO, rank-3 : [GrGrA1, Grd2] :: [20000/19683, 2048/2025]


69-EDO, rank-6 : [PrPrPrDeDeAcd2, ReAcAA0, PrDeDeSbAcm2, PrDeSp1, Ac1] :: [19773/19360, 3375/3328, 364/363, 78/77, 81/80]

69-EDO, rank-5 : [Ac1, AsSpSpGrM0, DeSpAA0, DeDeSpAcA1] :: [81/80, 99/98, 625/616, 864/847]

69-EDO, rank-4 : [Ac1, SpSpSpAAA-1, SpSpGrd1] :: [81/80, 5625/5488, 6144/6125]

69-EDO, rank-3 : [Ac1, AcAcAAAAAAAAAA-3] :: [81/80, 3814697265625/3710851743744]


70-EDO, rank-6 : [ReAsAsAsGrGrd1, PrAsGrdd2, SbAcd2, DeSpSpA0, ReAsGr1] :: [5324/5265, 1144/1125, 126/125, 540/539, 352/351]

70-EDO, rank-5 : [AsSpSpGrM0, SbAcd2, DeSpSpA0, Grd2] :: [99/98, 126/125, 540/539, 2048/2025]

70-EDO, rank-4 : [SbAcd2, SpSpSpM0, Grd2] :: [126/125, 8748/8575, 2048/2025]

70-EDO, rank-3 : [Grd2, AcAcdddd3] :: [2048/2025, 51018336/48828125]


71-EDO, rank-6 : [ReAsAsGrGr1, PrDeSbm2, DeSbSbAcAcm2, DeA1, SbA1] :: [9680/9477, 910/891, 441/440, 100/99, 875/864]

71-EDO, rank-5 : [DeA1, SbSbGrdd3, DeSbSbAcAcm2, AsSb1] :: [100/99, 6272/6075, 441/440, 385/384]

71-EDO, rank-4 : [SbSbSbGrd3, SbA1, SpSpGrGrA0] :: [6860/6561, 875/864, 4000/3969]

71-EDO, rank-3 : [GrGrAAA0, GrGrm2] :: [1953125/1889568, 20480/19683]


72-EDO, rank-6 : [DeDeAcAcA1, PrPrGrdd2, DeSbSbAcAcm2, PrGr1, PrSpd1] :: [243/242, 676/675, 441/440, 325/324, 351/350]

72-EDO, rank-5 : [SpA0, DeSbSbAcAcm2, AsSb1, AsAcd1] :: [225/224, 441/440, 385/384, 8019/8000]

72-EDO, rank-4 : [SpA0, SbSbSbAcAcm2, SbGrA1] :: [225/224, 1029/1024, 4375/4374]

72-EDO, rank-3 : [GrAA0, AcAcAcA0] :: [15625/15552, 531441/524288]


73-EDO, rank-6 : [PrSpd1, DeSbSbAcAcm2, PrSbd2, SbAcd2, DeDeSpA1] :: [351/350, 441/440, 91/90, 126/125, 2560/2541]

73-EDO, rank-5 : [AsGrGr1, SbAcd2, DeSbSbAcAcm2, DeDeAcm2] :: [2200/2187, 126/125, 441/440, 3072/3025]

73-EDO, rank-4 : [SbSbSbdd3, SbAcd2, SpGrGrd2] :: [686/675, 126/125, 131072/127575]

73-EDO, rank-3 : [AcAcdd2, GrGrddd3] :: [78732/78125, 262144/253125]


74-EDO, rank-6 : [ReDeAcAcA1, SbAcd2, DeSbSbAcAcm2, Ac1, PrPrDeSpSpGrd1] :: [729/715, 126/125, 441/440, 81/80, 2704/2695]

74-EDO, rank-5 : [AsSpSpGrM0, SbAcd2, Ac1, DeDeDeSpSpAcA1] :: [99/98, 126/125, 81/80, 331776/326095]

74-EDO, rank-4 : [SbAcd2, Ac1, SpSpSpSpSpSpSpGrGrA-1] :: [126/125, 81/80, 21233664/20588575]

74-EDO, rank-3 : [Ac1, GrGrdddddddddd6] :: [81/80, 19791209299968/19073486328125]


75-EDO, rank-6 : [ReReAsAsGr1, ReGrA1, AsSpGrd1, SpA0, PrGr1] :: [7744/7605, 3200/3159, 176/175, 225/224, 325/324]

75-EDO, rank-5 : [AsSpSpGrM0, SpA0, AsSb1, AsSpSpGrGrGrA0] :: [99/98, 225/224, 385/384, 110000/107163]

75-EDO, rank-4 : [SpA0, SpSpSpGrGrGrA0, SpSpSpGrM0] :: [225/224, 256000/250047, 1728/1715]

75-EDO, rank-3 : [GrGrA1, AcAcAAAA-1] :: [20000/19683, 2109375/2097152]


76-EDO, rank-6 : [PrSpd1, Ac1, DeDeSbSbAcAcM2, ReSpSpSpAAA-1, ReDeAcA1] :: [351/350, 81/80, 245/242, 4500/4459, 144/143]

76-EDO, rank-5 : [DeDeSbSbAcAcM2, DeSpAA0, Ac1, AsSb1] :: [245/242, 625/616, 81/80, 385/384]

76-EDO, rank-4 : [Ac1, SbSbSbSbAcdd3, AA0] :: [81/80, 2401/2400, 3125/3072]

76-EDO, rank-3 : [Ac1, AA0] :: [81/80, 3125/3072]


77-EDO, rank-6 : [PrSpd1, DeSbSbAcAcm2, SbAcd2, AsSb1, ReSpAcA0] :: [351/350, 441/440, 126/125, 385/384, 729/728]

77-EDO, rank-5 : [DeSbSbAcAcm2, SbAcd2, AsSb1, SbSbSbGrdd3] :: [441/440, 126/125, 385/384, 10976/10935]

77-EDO, rank-4 : [SpSpAcM0, SbAcd2, SbSbSbAcAcm2] :: [19683/19600, 126/125, 1029/1024]

77-EDO, rank-3 : [AcAcAcdd2, AcAcA0] :: [1594323/1562500, 32805/32768]


78-EDO, rank-6 : [PrPrDeGrd2, ReAsSpGrA0, AsSbGrd2, DeA1, AsSb1] :: [2704/2673, 275/273, 1232/1215, 100/99, 385/384]

78-EDO, rank-5 : [AsSpSpSpGrAA-1, DeA1, AsSbGrd2, AsSb1] :: [1375/1372, 100/99, 1232/1215, 385/384]

78-EDO, rank-4 : [SpSpSpGrAAA-1, SbGrm2, SbA1] :: [3125/3087, 2240/2187, 875/864]

78-EDO, rank-3 : [Grd2, GrGrGrAAAA0] :: [2048/2025, 244140625/229582512]


79-EDO, rank-6 : [ReReAsAsA0, ReAcAA0, ReAsSpGrA0, PrPrSpGrd1, AsSb1] :: [5445/5408, 3375/3328, 275/273, 169/168, 385/384]

79-EDO, rank-5 : [AsSpSpGrM0, DeDeAcAcA1, AsSb1, DeSpAA0] :: [99/98, 243/242, 385/384, 625/616]

79-EDO, rank-4 : [SpSpGrGrA0, SpAcA0, SpSpSpGrM0] :: [4000/3969, 3645/3584, 1728/1715]

79-EDO, rank-3 : [AA0, AcAcAcAcd1] :: [3125/3072, 129140163/128000000]


80-EDO, rank-6 : [PrPrDeDeAcd2, PrAsSpGrGrd1, PrSpd1, ReAsGr1, DeSpSpA0] :: [3042/3025, 572/567, 351/350, 352/351, 540/539]

80-EDO, rank-5 : [SbSbSbGrdd3, AsSpGrd1, DeSpSpA0, AsGrGr1] :: [10976/10935, 176/175, 540/539, 2200/2187]

80-EDO, rank-4 : [SpSpSpGrM0, SbSbddd3, SpSpGrGrA0] :: [1728/1715, 3136/3125, 4000/3969]

80-EDO, rank-3 : [Grd2, GrGrGrGrAAA0] :: [2048/2025, 390625000/387420489]


81-EDO, rank-6 : [PrSpd1, Ac1, SbAcd2, ReDeAcA1, ReAsAsSpSpSpSpGrA-1] :: [351/350, 81/80, 126/125, 144/143, 156816/156065]

81-EDO, rank-5 : [Ac1, SpA0, AsSb1, DeDeDeSbSbSbAcAcAcA2] :: [81/80, 225/224, 385/384, 42875/42592]

81-EDO, rank-4 : [SpA0, Ac1, SbSbSbSbSbSbSbSbAcAcdd4] :: [225/224, 81/80, 144120025/143327232]

81-EDO, rank-3 : [Ac1, AcAcAAAAAAAAAAA-4] :: [81/80, 476837158203125/474989023199232]


82-EDO, rank-6 : [SpA0, DeA1, PrPrDeSbAcd2, DeSbSbAcAcm2, AsSb1] :: [225/224, 100/99, 3549/3520, 441/440, 385/384]

82-EDO, rank-5 : [DeDeSbSbAcAcM2, SbSbm2, SpA0, DeA1] :: [245/242, 245/243, 225/224, 100/99]

82-EDO, rank-4 : [SpA0, SpSpGrGrA0, SbSbm2] :: [225/224, 4000/3969, 245/243]

82-EDO, rank-3 : [AA0, GrGrA1] :: [3125/3072, 20000/19683]


83-EDO, rank-6 : [PrSbd2, SpSpGrGrA0, ReAsSbGrA1, AsSpGrd1, ReSbSbAcm2] :: [91/90, 4000/3969, 9625/9477, 176/175, 196/195]

83-EDO, rank-5 : [AsSb1, SbGrm2, AsSpGrd1, SbSbSbdd3] :: [385/384, 2240/2187, 176/175, 686/675]

83-EDO, rank-4 : [SpSpGrGrA0, SbSbSbdd3, SpSpGrd1] :: [4000/3969, 686/675, 6144/6125]

83-EDO, rank-3 : [GrAA0, GrGrGrGrdd3] :: [15625/15552, 8388608/7971615]


84-EDO, rank-6 : [ReDeAcAcAcAcAA0, PrDeDeAcA1, PrSpd1, SpA0, DeSbAcAcA1] :: [4782969/4685824, 975/968, 351/350, 225/224, 2835/2816]

84-EDO, rank-5 : [DeSbSbAcAcm2, AsAcd1, DeSpAc1, SpA0] :: [441/440, 8019/8000, 1944/1925, 225/224]

84-EDO, rank-4 : [SpA0, SpSpSpGrM0, SbSbAcAcAcd2] :: [225/224, 1728/1715, 321489/320000]

84-EDO, rank-3 : [AcAcdd2, AcAcAcA0] :: [78732/78125, 531441/524288]


85-EDO, rank-6 : [ReAsAsSpGrGrA0, PrSpSpGrGrA0, SbSbm2, DeA1, SpA0] :: [15125/14742, 8125/7938, 245/243, 100/99, 225/224]

85-EDO, rank-5 : [SpA0, DeA1, AsSb1, AsAsSpSpSpGrGrGrGrM0] :: [225/224, 100/99, 385/384, 774400/750141]

85-EDO, rank-4 : [SpA0, SbSbm2, SpSpSpSpGrGrGrGrA0] :: [225/224, 245/243, 16384000/15752961]

85-EDO, rank-3 : [AA0, GrGrGrGrM2] :: [3125/3072, 409600000/387420489]


86-EDO, rank-6 : [AsAsAcM0, PrDeDeAcA1, PrAsd1, Ac1, AsSbSbAcd2] :: [264627/262144, 975/968, 1287/1280, 81/80, 1617/1600]

86-EDO, rank-5 : [DeDeSbSbAcAcM2, Ac1, DeSpSpA0, DeSbSbAcdd3] :: [245/242, 81/80, 540/539, 7056/6875]

86-EDO, rank-4 : [Ac1, SbSbSbSbAcdddd4, SpSpGrd1] :: [81/80, 9604/9375, 6144/6125]

86-EDO, rank-3 : [Ac1, Grdddddd4] :: [81/80, 50331648/48828125]


87-EDO, rank-6 : [ReSbSbAcm2, PrDeDeSbAcm2, DeSbSbAcAcm2, ReAA0, PrGr1] :: [196/195, 364/363, 441/440, 625/624, 325/324]

87-EDO, rank-5 : [SbSbm2, AsSb1, DeSbSbAcAcm2, AsAsSbGrdd2] :: [245/243, 385/384, 441/440, 3388/3375]

87-EDO, rank-4 : [SbSbSbAcAcm2, SbSbddd3, SbSbm2] :: [1029/1024, 3136/3125, 245/243]

87-EDO, rank-3 : [GrAA0, GrGrGrGrddd3] :: [15625/15552, 67108864/66430125]


88-EDO, rank-6 : [DeSbSbAcAcm2, ReDeAcA1, Ac1, ReSbAcA1, PrDeDeSpAcAA0] :: [441/440, 144/143, 81/80, 105/104, 219375/216832]

88-EDO, rank-5 : [AsSpSpGrM0, Ac1, AsSb1, DeSpSpAcAAAA-1] :: [99/98, 81/80, 385/384, 140625/137984]

88-EDO, rank-4 : [Ac1, SpSpSpGrM0, SpSpSpSpGrAAAAA-2] :: [81/80, 1728/1715, 78125/76832]

88-EDO, rank-3 : [Ac1, Grm-358] :: [81/80, 11920928955078125/11399736556781568]


89-EDO, rank-6 : [SbAcd2, PrDeDeSbAcm2, AsSpGrd1, PrAsd1, DeSpSpA0] :: [126/125, 364/363, 176/175, 1287/1280, 540/539]

89-EDO, rank-5 : [SbAcd2, DeSpSpA0, AsSpGrd1, AsSpAcM0] :: [126/125, 540/539, 176/175, 72171/71680]

89-EDO, rank-4 : [SbAcd2, SpSpSpAcAcAA-1, SpSpSpGrM0] :: [126/125, 177147/175616, 1728/1715]

89-EDO, rank-3 : [Acdddd3, AcAcA0] :: [10077696/9765625, 32805/32768]


90-EDO, rank-6 : [PrPrDeGrd2, ReAsSpGrA0, AsSbGrd2, AsSpGrd1, DeSbm2] :: [2704/2673, 275/273, 1232/1215, 176/175, 896/891]

90-EDO, rank-5 : [DeSbm2, AsSb1, AsSpGrd1, SpSpSpGrAAA-1] :: [896/891, 385/384, 176/175, 3125/3087]

90-EDO, rank-4 : [SpSpSpSpSpGrGrAA-1, SpSpGrd1, SbSbm2] :: [51200/50421, 6144/6125, 245/243]

90-EDO, rank-3 : [Grd2, GrGrGrGrAAAA0] :: [2048/2025, 1220703125/1162261467]


91-EDO, rank-6 : [SpA0, ReSbAcA1, AsAcd1, PrDeDeSbAcm2, AsSb1] :: [225/224, 105/104, 8019/8000, 364/363, 385/384]

91-EDO, rank-5 : [DeDeSbSbAcAcM2, AsSb1, SpA0, AsAcd1] :: [245/242, 385/384, 225/224, 8019/8000]

91-EDO, rank-4 : [SpA0, SbSbSbSbAcAcdd3, SbGrA1] :: [225/224, 64827/64000, 4375/4374]

91-EDO, rank-3 : [GrAA0, AcAcAcAcA0] :: [15625/15552, 43046721/41943040]


92-EDO, rank-6 : [PrPrDeGrd2, AsGrGr1, SbAcd2, SbSbm2, PrSbd2] :: [2704/2673, 2200/2187, 126/125, 245/243, 91/90]

92-EDO, rank-5 : [SbSbSbdd3, SbAcd2, DeSbSbAcAcm2, DeSbm2] :: [686/675, 126/125, 441/440, 896/891]

92-EDO, rank-4 : [SbAcd2, SbSbm2, Grd2] :: [126/125, 245/243, 2048/2025]

92-EDO, rank-3 : [AcAcdd2, Grd2] :: [78732/78125, 2048/2025]


93-EDO, rank-6 : [PrDeDeSbAcm2, SbAcd2, Ac1, PrAsSpSpSpGrGrm0, ReDeAcA1] :: [364/363, 126/125, 81/80, 1716/1715, 144/143]

93-EDO, rank-5 : [Ac1, SpA0, DeDeDeSbSbSbAcAcAcA2, SpSpSpGrM0] :: [81/80, 225/224, 42875/42592, 1728/1715]

93-EDO, rank-4 : [SbAcd2, Ac1, SpSpSpGrM0] :: [126/125, 81/80, 1728/1715]

93-EDO, rank-3 : [Ac1, Grdddd3] :: [81/80, 393216/390625]


94-EDO, rank-6 : [ReSpAcA0, ReAsSpGrA0, SpA0, DeSpSpA0, ReDeDeSbAcAcAA1] :: [729/728, 275/273, 225/224, 540/539, 1575/1573]

94-EDO, rank-5 : [SpA0, AsAsAsSpSpGrGrGrm0, AsGrGr1, AsSb1] :: [225/224, 1331/1323, 2200/2187, 385/384]

94-EDO, rank-4 : [SpA0, SpSpGrGrA0, SbSbSbSbAcM2] :: [225/224, 4000/3969, 1500625/1492992]

94-EDO, rank-3 : [GrGrGrAAA0, AcAcA0] :: [9765625/9565938, 32805/32768]


95-EDO, rank-6 : [AsSpGrd1, PrGr1, SbSbm2, PrDed2, ReSpSpGrA0] :: [176/175, 325/324, 245/243, 832/825, 640/637]

95-EDO, rank-5 : [AsSbGrd2, AsSpGrd1, SpSpGrGrA0, AsAsSpGrGrM0] :: [1232/1215, 176/175, 4000/3969, 3025/3024]

95-EDO, rank-4 : [SpSpGrGrA0, SbSbm2, SbGrddd3] :: [4000/3969, 245/243, 28672/28125]

95-EDO, rank-3 : [GrGrA1, GrGrGrddddd4] :: [20000/19683, 536870912/512578125]


96-EDO, rank-6 : [PrDeSpGr1, DeSpSpAcA0, AsSpGrd1, PrAsSbdd2, SpA0] :: [2080/2079, 2187/2156, 176/175, 1001/1000, 225/224]

96-EDO, rank-5 : [AsSpSpGrM0, DeDeAcAcA1, SpA0, AsAsSbSbAcdd2] :: [99/98, 243/242, 225/224, 160083/160000]

96-EDO, rank-4 : [SpA0, SpSpSpSpSpGrGrAA-1, SpSpSpM0] :: [225/224, 51200/50421, 8748/8575]

96-EDO, rank-3 : [AcAcAcdd2, AcAcAcA0] :: [1594323/1562500, 531441/524288]


97-EDO, rank-6 : [PrPrGrdd2, DeSbSbAcAcm2, DeA1, ReSbSbAcm2, AsSb1] :: [676/675, 441/440, 100/99, 196/195, 385/384]

97-EDO, rank-5 : [DeDeSbSbAcAcM2, AsSb1, DeA1, AsSbSbGrGrddd3] :: [245/242, 385/384, 100/99, 275968/273375]

97-EDO, rank-4 : [SpSpGrGrA0, SbSbGrGrdd3, SbA1] :: [4000/3969, 100352/98415, 875/864]

97-EDO, rank-3 : [GrGrAAA0, GrGrGrm2] :: [1953125/1889568, 1638400/1594323]


98-EDO, rank-6 : [ReSbSbAcm2, Ac1, AsSpGrd1, PrDeDeSbSbAcdd3, ReDeAcA1] :: [196/195, 81/80, 176/175, 15288/15125, 144/143]

98-EDO, rank-5 : [AsSpd1, Ac1, DeSpSpA0, DeDeSbAcdd3] :: [891/875, 81/80, 540/539, 387072/378125]

98-EDO, rank-4 : [Ac1, SpSpSpGrM0, SbSbddddd4] :: [81/80, 1728/1715, 401408/390625]

98-EDO, rank-3 : [Ac1, AcM357] :: [81/80, 1641562064176545792/1490116119384765625]


99-EDO, rank-6 : [PrPrSpGrd1, PrDeSpGr1, ReAsGr1, PrSbSbGrddd3, PrSpd1] :: [169/168, 2080/2079, 352/351, 10192/10125, 351/350]

99-EDO, rank-5 : [AsAsGrd1, AsSpGrd1, AsSpSpSpGrAA-1, AsGrGr1] :: [121/120, 176/175, 1375/1372, 2200/2187]

99-EDO, rank-4 : [SbSbSbGrdd3, SpGrGr1, SbGrA1] :: [10976/10935, 5120/5103, 4375/4374]

99-EDO, rank-3 : [GrGrGrA1, Grdddd3] :: [1600000/1594323, 393216/390625]

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