The poset of monotonic Fokker blocks

Still need to respond to the other posts on here, but while this idea is
fresh in my head, might as well write it down.

Pick any group of musical intervals that you like. This can be a JI
subgroup, a group of tempered intervals, the group of a scale with no
particular mapping, etc. Anything will do, but make sure your group has a
specific tuning map, so that we can speak meaningfully about which scales
in it are monotonic.

Take the set of all of the Fokker blocks in this group which are also
monotonic under the given tuning map. This set is naturally partially
ordered under inclusion, so that one block is <= another if it's a subset
of that block.

The open question: does this poset have any interesting properties by which
we can characterize it?

For a rank-2 group, there's the obviously nice thing that the partial order
becomes a total order, so we get the MOS "series" for a certain
period/generator ratio. But what about for other ranks? Clearly we won't
get a total order, but does anything interesting happen at all?

In case it isn't clear, this is supposed to be a generalization of the
concept of an MOS series, but for Fokker blocks of arbitrary rank.

Mike

On Thu, Aug 15, 2013 at 4:18 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> The open question: does this poset have any interesting properties by which we can characterize it?

Conjecture: this poset has the structure of a lattice.

If it's a lattice, that means that

a) for every pair of monotonic Fokker blocks, there's a unique
greatest lower bound - a unique largest monotonic Fokker block that's
a subset of both.
b) for every pair of monotonic Fokker blocks, there's a unique least
upper bound - a unique smallest monotonic Fokker block that contains
both of them.

For a to be true, the following two things must hold:
1) Every intersection of two monotonic Fokker blocks contains, as a
subset, at least one monotonic Fokker block, called a "subblock" of
the two blocks.
2) Exactly one of these subblocks is unique in that it is a superset
of all other monotonic subblocks in the intersection.

For b to be true, the following two things must hold:
1) Every union of two monotonic Fokker blocks is contained, as a
subset, by at least one other monotonic Fokker block, called a
"superblock" of the two blocks.
2) Exactly one of these Fokker superblocks is unique in that it is a
subset of all other monotonic superblocks containing the union.

If only a is true, then it's only a meet-semilattice. If only b is
true, then it's only a join-semilattice.

Does anyone have any insight into ANY of these conjectures? I believe
that #1 is true in both cases, but #2 is trickier. For instance, could
it be the case that two Fokker blocks have multiple common which are
incomparable, but no "least" common extension that extends to the
others? (And likewise with common restrictions)

Mike