EDOs and their friends

43-EDO can be defined over rank-3 interval space by a pure octave and tempering out the acute unison, justly tuned to 81/80, and the Grdddddd4, justly tuned to 50331648/48828125. 

If we write these interval in rank-3 prime harmonic coordinates they are [[1, 0, 0], [-4, 4, -1], [24, 1, -11]]. We might call this an EDO basis.

This matrix isn't unimodular: it has determinant -43, from which we can infer that it defines 43 EDO.

What comma would we have to use in place of the octave to get a unimodular basis? There are tons of options. If we restrict ourself to intervals with just tunings between 1 and 2, we still have tons of options. Here are some options sorted by increasing numerator size:

32805/32768 _ [-15, 8, 1]

393216/390625 _ [17, 1, -8]

531441/524288 _ [-19, 12, 0]

1990656/1953125 _ [13, 5, -9]

10077696/9765625 _ [9, 9, -10]

43046721/41943040 _ [-23, 16, -1]

51018336/48828125 _ [5, 13, -11]

258280326/244140625 _ [1, 17, -12]

3486784401/3355443200 _ [-27, 20, -2]

10460353203/9765625000 _ [-3, 21, -13]

847288609443/781250000000 _ [-7, 25, -14]

68630377364883/62500000000000 _ [-11, 29, -15]

2567836929097728/2384185791015625 _ [29, 14, -22]

12999674453557248/11920928955078125 _ [25, 18, -23]

65810851921133568/59604644775390625 _ [21, 22, -24]

333167437850738688/298023223876953125 _ [17, 26, -25]

1686660154119364608/1490116119384765625 _ [13, 30, -26]

It seems obvious to me to use the simplest one in terms of frequency ratio. That interval, [-15, 8, 1], justly tuned to 32805/32768, is affectionately called the "schisma" in music theory. If we combine it with our other two commas we get a unimodular matrix:

    [[-15, 8, 1], [-4, 4, -1], [24, 1, -11]]

which we could use to define any rank-3 interval in integer coordinates. The inverse of this unimodular matrix is:

    [-43, 89, -12], 

    [-68, 141, -19], 

    [-100, 207, -28]

The columns of this matrix, up to a factor of -1, are the tunings of  [P8, P12, M17] in [43, 89, and 12]-EDO. And those are justly tuned to the first three prime harmonics.

There's more structure here though. 

    12-EDO tempers out the schisma and Ac1, but tuns [24, 1, -11] to -1 step.

    43-EDO tempers out Ac1 and  [24, 1, -11] but tunes the schisma to -1 step.

    89-EDO tempers out the schisma and  [24, 1, -11], but tunes Ac1 to 1 step.

So ... are these three special friends? I don't know. I kind of doubt it, but let's find some other relations like these and see.

Here are some EDO bases with the octave replaced with whatever simple interval will produce a unimodular matrix:

"12-EDO": [[2, 9, -7], [-4, 4, -1], [7, 0, -3]],

"13-EDO": [[1, -2, 1], [-3, -1, 2], [9, -7, 1]],

"14-EDO": [[-4, 4, -1], [0, 3, -2], [11, -4, -2]],

"15-EDO": [[11, -4, -2], [7, 0, -3], [1, -5, 3]],

"16-EDO": [[-4, 4, -1], [-7, 3, 1], [3, 4, -4]],

"17-EDO": [[-4, 4, -1], [-3, -1, 2], [12, -9, 1]],

"18-EDO": [[1, -5, 3], [7, 0, -3], [5, -6, 2]],

"19-EDO": [[-3, -1, 2], [-4, 4, -1], [-10, -1, 5]],

"21-EDO": [[0, 3, -2], [7, 0, -3], [-11, 7, 0]],

"22-EDO": [[-3, -1, 2], [1, -5, 3], [11, -4, -2]],

"23-EDO": [[-4, 4, -1], [-7, 3, 1], [-1, 8, -5]],

"25-EDO": [[1, -5, 3], [8, -5, 0], [-10, -1, 5]],

"26-EDO": [[-3, -1, 2], [-4, 4, -1], [-13, -2, 7]],

"27-EDO": [[11, -4, -2], [7, 0, -3], [5, -9, 4]],

"28-EDO": [[7, 0, -3], [3, 4, -4], [-11, 7, 0]],

"29-EDO": [[-10, -1, 5], [1, -5, 3], [-14, 3, 4]],

"31-EDO": [[-15, 8, 1], [-4, 4, -1], [17, 1, -8]],

.

Let's take the unimodular matrices, invert, transpose, remove negative signs, and interpret as tunings of prime harmonics.

Here's the transpose of the inverse:

12-EDO [[-12, -19, -28], [27, 43, 63], [19, 30, 44]]

13-EDO [[13, 21, 30], [-5, -8, -11], [-3, -5, -7]]

14-EDO [[-14, -22, -33], [12, 19, 28], [-5, -8, -12]]

15-EDO [[-15, -24, -35], [22, 35, 51], [12, 19, 28]]

16-EDO [[-16, -25, -37], [12, 19, 28], [7, 11, 16]]

17-EDO [[17, 27, 39], [5, 8, 12], [7, 11, 16]]

18-EDO [[-18, -29, -42], [-8, -13, -19], [15, 24, 35]]

19-EDO [[19, 30, 44], [3, 5, 7], [-7, -11, -16]]

21-EDO [[21, 33, 49], [-14, -22, -33], [-9, -14, -21]]

22-EDO [[22, 35, 51], [-10, -16, -23], [7, 11, 16]]

23-EDO [[-23, -36, -53], [12, 19, 28], [7, 11, 16]]

25-EDO [[-25, -40, -58], [22, 35, 51], [15, 24, 35]]

26-EDO [[26, 41, 60], [3, 5, 7], [-7, -11, -16]]

27-EDO [[-27, -43, -63], [34, 54, 79], [12, 19, 28]]

28-EDO [[28, 44, 65], [-21, -33, -49], [12, 19, 28]]

29-EDO [[-29, -46, -67], [19, 30, 44], [22, 35, 51]]

31-EDO [[-31, -49, -72], [65, 103, 151], [-12, -19, -28]]

From which we learn that

12-EDO is friends with: 27 and 19 EDO

13-EDO is friends with: 5 and 3 EDO

14-EDO is friends with: 12 and 5 EDO

15-EDO is friends with: 22 and 12 EDO

16-EDO is friends with: 12 and 7 EDO

17-EDO is friends with: 5 and 7 EDO

18-EDO is friends with: 8 and 15 EDO

19-EDO is friends with: 3 and 7 EDO

21-EDO is friends with: 14 and 9 EDO

22-EDO is friends with: 10 and 7 EDO

23-EDO is friends with: 12 and 7 EDO

25-EDO is friends with: 22 and 15 EDO

26-EDO is friends with: 3 and 7 EDO

27-EDO is friends with: 34 and 12 EDO

28-EDO is friends with: 21 and 12 EDO

29-EDO is friends with: 19 and 22 EDO

31-EDO is friends with: 65 and 12 EDO

.

From this it seems that 12-EDO is friends with at least [5, 7, 12, 14, 15, 16, 19, 21, 22, 23, 27, 28, 31, 34, and 65]-EDO.

I think it's pretty clear that these little EDO triples are just alternative descriptions of the comma triplets, and not really specific to the EDO. Each EDO has lots of intervals that it tempers out, and lots of intervals that it tunes to -1 or 1 step, and within that large space there are ...overlaps? Like every pair of EDOs defined on rank-3 interval space has a shared comma. That doesn't mean that they have a special relationship; everyone has that relationship with everyone else. Now we're just saying, "These two EDOs have a shared tempered comma and also each one tunes one of the others' tempered commas to 1 or -1 steps." I don't know, man. It feels like there should be something deeper here, but I'm not seeing it.

How about this: It seems that in rank-n interval space, there's a unimodular matrix of {n} commas associated with every set of {n} distinct EDOs, such that each EDO tempers out {n - 1} of the commas and also tunes the remaining comma to -1 or 1 step.

And um....maybe we can use this to find EDO bases, the regular non-unimodular ones that include the octave? If you want a rank-7 basis for 87-EDO, you take 7-EDOs, including 87-EDO, and look at how they tune the intervals justly associated with the first 7 primes, and then you invert, and uh, all but one of those will be tempered commas of 87 EDO, and you can replace the last one with the octave for an EDO basis, and if the commas aren't to your liking, then roll the dice with a different set of EDOs? It's not a great procedure. I'll probably just stick with my own usual methods for finding EDO bases. But maybe it's something to keep considering.

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 bases 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.

...

First up, here are commas from Graham Breed at various interval ranks / prime limits. His code uses the Tenney-Euclidean norm to measure vector length, and like Sintel he uses this within LLL reduction to find a small basis. The tempered commas below are not all sufficient to define their associated EDOs, and they couldn't be, because e.g. his program will give two rank-3 tempered 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. 

I guess here's an example right away: 

Graham Breed's rank-5 EDO basis for 20-EDO: [Sbm2, A1, AsAsSpGrGrd1, SpGr1] :: [28/27, 25/24, 968/945, 64/63]

My rank-5 EDO basis for 20-EDO: 20-EDO: (A1 → 25/24, Sbm2 → 28/27, SbSbAcm2 → 49/48, AsAsGrd1 → 121/120)

Mine has much smaller frequency ratios and somewhat simpler interval names. I'll have to check if his ever do better than mine on those terms.

-

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]

.

Fake National Dishes

I looked up a list of national dishes and grabbed a few that I hadn't heard of before. 

Now I'm going to guess what's in them.

Bulgaria: Shopska salad.

Finland: Karjalanpaisti.

Gabon: Poulet nyembwe.

Kuwait: Machboos laham.

Latvia: Sklandrausis.

Myanmar: Lahpet thoke.

Niger: Dambou.

Singapore: Hokkien mee.

South Africa: Bobotie.

Tajikistan: Osh palov.

Uruguay: Chivito.

Now I'm going to guess within them. If you want to play too, now's a good time to pause reading and come up with your own recipes.

Bulgaria: Shopska salad. ??? Sorrel, apples, black olives, fennel, anchovies, pickled garlic cloves

Finland: Karjalanpaisti. ??? Horse shoulder, rutabaga, carrot stew. Thickened with sourdough bread. Usually served with dandelion greens and lingonberry brandy.

Gabon: Poulet nyembwe. ??? Chicken and eggplant in a peanut sauce. Served with rice and wilted cassava leaves. Usually followed by a dessert of candied plantains.

Kuwait: Machboos laham. ??? A lamb-milk and spinach lhassi, spiced with cardamom and saffron.

Latvia: Sklandrausis. ??? Cookie something like hamentashen, but made with barley flour and filled with pureed flax seed.

Myanmar: Lahpet thoke. ??? Dry rice and lentils, pureed with water, cooked to reduce in a pan, sliced into cubes, fried in oil, garnished with chili oil, strips of fish jerky, and a local herb you've never heard of, don't pretend.

Niger: Dambou. ??? Goat and sorghum curry with lots of raw red onion, thickened with palm fruit puree. Can use scotch bonnet or habenero peppers.

Singapore: Hokkien mee. ??? Squid head, reishi mushrooms, bok choy, in a fermented garlic broth over wheat noodles. Garnished with soft boiled egg halves.

South Africa: Bobotie. ??? Braised impala or sprinbok with black pepper, brown sugar, coriander, ginger, cloves. Served with saffron rice and collards that are cooked with soumbala and vinegar.

Tajikistan: Osh palov. ??? A pan fried dumpling stuffed with potatoes, topped with a mix of ground pistachios and cream cheese.

Uruguay: Chivito. ??? A burger with slices of chayote, served with a molasses, vinegar, feijoa sauce for dipping.