Brats for temperaments

Below I give 40 commas and the associated mapping, optimal generators
for exponents 2,3,4 and infinity (ie., minimax), the associated brats,
a reasonable choice for a brat on the basis of the above, and the brat
as a rational function of the generator. By solving for the generator,
we can find what corresponds to a particular choice of brat, such as
my "reasonable choice".

27/25 [[1, 2, 3], [0, -2, -3]]

[268.056438833948, 267.435479555972, 268.056438834223, 271.228762045052]
[7.41821754512150, 6.28707119757482, 7.41821754573784, 2026193647657.34]
b = infinity
2*(-6+5*g)/(-8+5*g^3)

135/128 [[1, 2, 1], [0, -1, 3]]

[522.862345874114, 523.070610486922, 522.862345843404, 521.089678249860]
[-.595712670824538, -.689980196352613, -.595712657415004,
.145521127181207e-11]
b = 0
(3*g^4-10)/g/(2*g^3-5)

256/243 [[5, 8, 12], [0, 0, -1]]

[84.6637865678555, 84.6637865678584, 84.6637865678614, 84.6637865678580]
[-1.50783780618484, -1.50783780618484, -1.50783780619035,
-1.50783780618484]
b = -3/2
2^(3/5)*(-3*2^(4/5)+5*g)/(-4*2^(2/5)+5*g)

25/24 [[1, 1, 2], [0, 2, 1]]

[350.977500432694, 350.977500432688, 350.977500431322, 350.977500432690]
[1.50000000000012, 1.49999999999890, 1.49999999988252, 1.50000000000012]
b = 3/2
-g*(-6+5*g)/(4*g-5)

648/625 [[4, 6, 9], [0, 1, 1]]

[94.1343573651068, 94.1343573651113, 94.1343573651289, 94.1343573651120]
[2.99323921078598, 2.99323921078598, 2.99323921077971, 2.99323921078598]
b = 3
-2*(-6+5*2^(1/4))*g/(8*g-5*2^(3/4))

16875/16384 [[1, 2, 2], [0, -4, 3]]

[126.238272015258, 126.205405491448, 126.238272048931, 126.336958999920]
[.136091735477899, .179098457309447, .136091690828154,
.780659798090136e-11]
b = 0
2*(3*g^7-5)/g^4/(4*g^3-5)

250/243 [[1, 2, 3], [0, -3, -5]]

[162.996026370546, 163.069650366763, 162.996026374613, 162.737257227031]
[12.0314476213430, 9.49916182634974, 12.0314474417337, 683526730904.000]
b = infinity
2*(-6+5*g^2)/(-8+5*g^5)

128/125 [[3, 5, 7], [0, -1, 0]]

[91.2018560670246, 91.2018560670318, 91.2018560671082, 91.2018560670318]
[.751482280984136, .751482280984136, .751482280993138, .751482280984136]
b = 3/4
1/3*(25*2^(1/3)+20*2^(2/3)+32)*(6*g-5*2^(1/3))/g

3125/3072 [[1, 0, 2], [0, 5, 1]]

[379.967949195095, 379.932993406921, 379.967950000885, 380.391000173076]
[.999388966181635, .961015450204820, .999389855671830, 1.50000000000604]
b = 1
-1/2*g*(-12+5*g^4)/(4*g-5)

20000/19683 [[1, 1, 1], [0, 4, 9]]

[176.282270436412, 176.291430912569, 176.282270741616, 176.257079318316]
[-19.5178863924243, -14.0908094624819, -19.5176398261720,
333148837511.168]
b = infinity
g^4*(3*g^5-5)/(2*g^9-5)

531441/524288 [[12, 19, 28], [0, 0, -1]]

[14.6637865678600, 14.6637865678561, 14.6637865678567, 14.6637865678560]
[-1.49915329944943, -1.49915329944943, -1.49915329944943,
-1.49915329944943]
b = -3/2
2^(7/12)*(-3*2^(3/4)+5*g)/(-4*2^(1/3)+5*g)

81/80 [[1, 2, 4], [0, -1, -4]]

[503.835154026037, 503.883744407432, 503.835153661954, 503.421571533790]
[-3.74373500460168, -3.23201443400256, -3.74373928931788,
372097502304.000]
b = infinity
2*(-12+5*g^3)/(-16+5*g^4)

2048/2025 [[2, 3, 5], [0, 1, -2]]

[105.446531009815, 105.492093791319, 105.446530999138, 105.213762333520]
[-.375378300113750, -.463999206632087, -.375378280006185, 0.]
b = 0
2^(1/2)*(-6+5*g^3)/(-4*2^(1/2)+5*g^2)

67108864/66430125 [[3, 5, 6], [0, -1, 4]]

[96.7879385616958, 96.7810140152084, 96.7879385539879, 96.8717425998888]
[-.75027233388, -.84003132898, -.75027243052, -.3e-10]
b = 0
(6*g^5-5*2^(2/3))/g/(4*g^4-5)

78732/78125 [[1, -1, -1], [0, 7, 9]]

[442.979297439105, 442.984696404155, 442.979314219411, 442.923745984982]
[4.79847558521374, 4.40299365566162, 4.79712734696446, -47754600097.7616]
b = infinity
1/2*g^7*(3*g^2-5)/(g^9-10)

393216/390625 [[1, -1, 2], [0, 8, 1]]

[387.819673068348, 387.822889697592, 387.819654171468, 387.744375108173]
[.900156573210775, .875861107340688, .900299608507285, 1.50000000004430]
b = 1
-1/4*g*(-24+5*g^7)/(4*g-5)

2109375/2097152 [[1, 0, 3], [0, 7, -3]]

[271.589599585246, 271.593697588711, 271.589601548604, 271.564128700056]
[.352856800255913, .405031315378923, .352882080754974,
-.752603975517185e-10]
b = 0
1/2*(-24+5*g^10)/(-8+5*g^3)

4294967296/4271484375 [[1, 2, 2], [0, -9, 7]]

[55.2754932571411, 55.2765687202447, 55.2754940620164, 55.2724195624650]
[.120019601606355, .160053073802616, .120049929465304,
.113835249438497e-9]
b = 0
2*(3*g^16-5)/g^9/(4*g^7-5)

15625/15552 [[1, 0, 1], [0, 6, 5]]

[317.079675185760, 317.085190264962, 317.079673835428, 316.992500144230]
[2.35749900106746, 2.44007011449707, 2.35747939320971, 1.49999999997026]
b = 2
-1/2*g^5*(-6+5*g)/(2*g^5-5)

1600000/1594323 [[1, 3, 6], [0, -5, -13]]

[339.508825625716, 339.507387256433, 339.508875042446, 339.514329702704]
[-9.00130202082944, -6.94510840078312, -9.09121293421452,
10500688224.0000]
b = infinity
4*(-24+5*g^8)/(-64+5*g^13)

1224440064/1220703125 [[1, 5, 6], [0, -13, -14]]

[315.250913337821, 315.250648792898, 315.250774706137, 315.263306152511]
[3.37478861493051, 3.30649638615107, 3.33864128093987, 5239355206.80000]
b = infinity
16*(-6+5*g)/(-64+5*g^14)

10485760000/10460353203 [[1, 0, -6], [0, 4, 21]]

[475.542233398946, 475.542500964170, 475.537866896123, 475.538748279278]
[-2.88479361344442, -2.59252603021660, 17.4229369490071,
-3465801815.30269]
b = infinity
1/2*g^4*(3*g^17-320)/(g^21-320)

6115295232/6103515625 [[2, 4, 5], [0, -7, -3]]

[71.1460635722381, 71.1449905180548, 71.1460673212410, 71.1492855906600]
[1.36364524574015, 1.32056202037312, 1.36379772876677, 1.49999999998592]
b = 3/2
-2^(1/2)*(6*g^4-5*2^(1/2))/g^4/(-4*2^(1/2)+5*g^3)

19073486328125/19042491875328 [[19, 30, 44], [0, 1, 1]]

[7.29225210195604, 7.29225210195535, 7.29225210203665, 7.29225210195600]
[3.00006419857425, 3.00006419857425, 3.00006420206449, 3.00006419857425]
b = 3
-2*(-6+5*2^(5/19))*g/(8*g-5*2^(13/19))

295578376007080078125/295147905179352825856 [[1, 0, 4], [0, 17, -18]]

[111.875426120873, 111.875408100894, 111.873024328626, 111.875465342873]
[-.288455519924944e-1, -.422903799249676e-1, -4.55207607290570,
.600359663647405e-9]
b = 0
1/2*(-48+5*g^35)/(-16+5*g^18)

68719476736000/68630377364883 [[1, 3, 16], [0, -3, -29]]

[565.988014913065, 565.987975909345, 565.997677421647, 565.989182280523]
[-2.08700529757088, -1.97604825636316, 1.81643654799886, 1099510569.80000]
b = infinity
4*(-24576+5*g^26)/(-65536+5*g^29)

32805/32768 [[1, 2, -1], [0, -1, 8]]

[498.272487171564, 498.272868834532, 498.272049258382, 498.262079222160]
[-1.04993101558579, -1.11384696853000, -.980096572950770,
.299047265951068e-10]
b = -1
1/2*(3*g^9-40)/g/(g^8-10)

2954312706550833698643/2951479051793528258560 [[1, 2, 21], [0, -1, -45]]

[498.082318148220, 498.082321198085, 498.072699083572, 498.081917469671]
[-1.60463387655409, -1.58143258093245, 1.60014916042977,
-479153673.714286]
b = infinity
2*(-1572864+5*g^44)/(-2097152+5*g^45)

274877906944/274658203125 [[1, 4, 2], [0, -15, 2]]

[193.199614933860, 193.199695616494, 193.197870548564, 193.197571352909]
[.609383934851494, .632249777329436, .930137288618768e-1,
.159458989868289e-9]
b = 0
2*(3*g^17-20)/g^15/(4*g^2-5)

50031545098999707/50000000000000000 [[1, -1, -3], [0, 17, 35]]

[182.466089137089, 182.466081068180, 182.468384070474, 182.466106110424]
[-77.9993086006430, -52.6184224616755, 1.35836621555422, 211999114.193699]
b = infinity
1/2*g^17*(3*g^18-20)/(g^35-40)

7629394531250/7625597484987 [[9, 15, 22], [0, -2, -3]]

[49.0088197863282, 49.0092660738190, 49.0088122512589, 49.0065398228312]
[7.50005927378393, 6.35414803641991, 7.52327055663510, -7526906337.33860]
b = infinity
2^(2/3)*(-3*2^(7/9)+5*g)/(-4*2^(4/9)+5*g^3)

2475880078570760549798248448/2474715001881122589111328125 [[1, 5, 1],
[0, -31, 12]]

[132.194510561451, 132.194527882796, 132.191628330792, 132.194388674406]
[.385136835303054, .435434635544873, 12.5616518433724,
-.142174950967159e-7]
b = infinity
(3*g^43-80)/g^31/(2*g^12-5)

9010162353515625/9007199254740992 [[2, 1, 6], [0, 8, -5]]

[162.741892126380, 162.741958792709, 162.744739215164, 162.741637461582]
[.214284486343732, .266546184643909, 1.61679609019013,
-.698683814530209e-9]
b = 1
1/2*2^(1/2)*(-12*2^(1/2)+5*g^13)/(-8+5*g^5)

116450459770592056836096/116415321826934814453125 [[1, 17, 14], [0,
-33, -25]]

[560.546969532518, 560.546989674304, 560.550415366598, 560.546818155593]
[2.12194817527459, 2.23544091312893, -.903028139088175, 1.50000000092280]
b = 3/2
-8192*(3*g^8-40)/g^8/(-16384+5*g^25)

444089209850062616169452667236328125/444002166576103304796646509039845376
[[1, 15, 16], [0, -51, -52]]

[315.647874693155, 315.647874725983, 315.649938234086, 315.647813194906]
[3.09000325776413, 3.08836970087491, .117431939393260, -46321886.6176000]
b = infinity
16384*(-6+5*g)/(-65536+5*g^52)

450359962737049600/450283905890997363 [[1, 2, 10], [0, -2, -37]]

[249.018447894646, 249.018446056434, 249.013842268974, 249.018548273923]
[-1.77273144947357, -1.71533485435417, 1.35084773049401,
-50081536.7142858]
b = infinity
2*(-768+5*g^35)/(-1024+5*g^37)

162285243890121480027996826171875/162259276829213363391578010288128
[[1, -1, 11], [0, 14, -47]]

[221.567865486044, 221.567867194692, 221.562607414796, 221.567889950828]
[-.659999130523142, -.599704461945995, 1.98303836002560,
-.102372548206321e-6]
b = 0
1/4*(-12288+5*g^61)/(-2048+5*g^47)

22300745198530623141535718272648361505980416/22297583945629639856633730232715606689453125
[[1, -2, 4], [0, 47, -22]]

[91.5310212152784, 91.5310188118782, 91.5331152603505, 91.5310331449356]
[.323276236582079, .383101227190776, 5.10076534940776,
-.152099898191020e-6]
b = infinity
1/8*(-192+5*g^69)/(-16+5*g^22)

381520424476945831628649898809/381469726562500000000000000000 [[1, 11,
19], [0, -35, -62]]

[322.801386591773, 322.801392385865, 322.810340953662, 322.801391711857]
[18.1874711880147, -132.546574791409, .643135139215846, 37459128.8888889]
b = infinity
1024*(-768+5*g^27)/(-524288+5*g^62)

1292469707114105741986576081359316958696581423282623291015625/1292388115393055295535123767426518253869322213731913416835072
[[2, 14, 19], [0, -43, -57]]

[151.117308517283, 151.117310901324, 151.120216709418, 151.117303265530]
[5.17241093569130, 3.67251559849278, .377067159030008, -14320613.1809524]
b = infinity
64*(-12*2^(1/2)+5*g^14)/(-512*2^(1/2)+5*g^57)