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7-limit symmetric lattice Minkowski reductions

🔗Gene Ward Smith <gwsmith@svpal.org>

4/23/2004 1:15:39 AM

For some important 7-limit ets, I give below the TM basis, and then
the Minkowski reduced basis using symmetric lattice distance instead
of Tenney distance; to break any ties however I do use Tenney. The
final number is a diameter measure; it is the square of the symmetric
lattice distance from the origin of the most distant of the three
commas making up the basis. It can be seen from this than some of
these ets (eg 41, 68, 171) have a more compact and more nearly cubical
shape than others (eg 72) which is an advantage.

12: [36/35, 50/49, 64/63]; [36/35, 50/49, 64/63]; 7

15: [28/27, 49/48, 126/125]; [49/48, 28/27, 126/125]; 7

19: [49/48, 81/80, 126/125]; [49/48, 126/125, 225/224]; 9

22: [50/49, 64/63, 245/243]; [50/49, 64/63, 875/864]; 10

27: [64/63, 126/125, 245/243]; [64/63, 126/125, 1728/1715]; 10

31: [81/80, 126/125, 1029/1024]; [126/125, 225/224, 1728/1715]; 10

41: [225/224, 245/243, 1029/1024]; [225/224, 875/864, 2401/2400]; 11

53: [225/224, 1728/1715, 3125/3087]; [225/224, 1728/1715, 3125/3087]; 19

68: [245/243, 2048/2025, 2401/2400]; [2401/2400, 6144/6125, 245/243]; 17

72: [225/224, 1029/1024, 4375/4374]; [225/224, 1029/1024, 4375/4374]; 35

99: [3136/3125, 2401/2400, 4375/4374]; [2401/2400, 6144/6125,
4375/4374]; 35

130: [2401/2400, 3136/3125, 19683/19600]; [2401/2400, 6144/6125,
19683/19600]; 57

140: [2401/2400, 5120/5103, 15625/15552]; [2401/2400, 15625/15552,
5120/5103]; 37

171: [2401/2400, 4375/4374, 32805/32768]; [2401/2400, 4375/4374,
65625/65536]; 38