Homometric sets for rank-2 with cardinality less than 6

It's true, they don't exist, an obvious case is the rank-1 Z-relation

0,1,4,6 and 0,4,6,7 which in rank-1 are <111111>, which cannot
be extended to rank-2 because the first has a span of 7 fifths and the
second has a span of 6 fifths, so they are not the same in rank-2.

Just my own way of proving rank-2 homometric sets cannot exist for
cardinality less than 6. Now I just need proof as to why they are always
in pairs, and there are not trios, quads, etc. like there are in rank-1,

I think the reason, is that most pairs have just a few notes of variance.

For example, the one cited here had three "tritones" 06, 5.11, and 6.12
with 6 used twice that were fixed across homopairs and just two
notes varying.

pgh

> Now I just need proof as to why they are always
> in pairs, and there are not trios, quads, etc. like there are in rank-1,

--the generating function approach in the Lemka-Skiena-Smith 1990 paper
shows that homometric sets in the integers always come in
(2^k)-tuplets, i.e. pairs=2, 4,8, 16 but never 3,5.

But if we are not in the integers, but instead
in some modular world like mod-31 arithmetic, or in some
weird field without unique factorization, then
this argument forcing power-of-2 no longer works.