Paul blocks

I see I was misreading Paul's construction, which actually only
allows for an extra comma which is either the sum or difference of
the other commas. This is a very nice idea, and can be formalized in
the same way as the Fokker block, via a norm.

If we have a Fokker-type norm in the 5-limit, which is
1/n [hn, v1, v2], where hn is an n-et and the other are defined on
octave equivalence classes, then instead of taking the maximum of the
absolute values of these valuations, we can take instead the maximum
of the absolute value of |hn(q)| together with the median of
{|v1(q)|, |v2(q)|, |v1(q)-v2(q)|}, or else
{|v1(q), |v2(q)|, |v1(q)+v2(q)|}--in other words, we sort three
absolute values, and take the one in the middle. This also gives us a
norm, and we can then define Paul blocks in the 5-limit in the same
way as Fokker blocks. To generalize to higher dimensions, it seems we
would need to take combinations of n vals k at a time, for k from 1
to n, giving us the verticies of the n-measure polytope (=hypercube.)
We then would take a maximum over the n smallest. At least, that
seems right, but I haven't really thought it through carefully.

--- In tuning-math@y..., genewardsmith@j... wrote:
> I see I was misreading Paul's construction, which actually only
> allows for an extra comma which is either the sum or difference of
> the other commas. This is a very nice idea, and can be formalized
in
> the same way as the Fokker block, via a norm.
>
> If we have a Fokker-type norm in the 5-limit, which is
> 1/n [hn, v1, v2], where hn is an n-et and the other are defined on
> octave equivalence classes, then instead of taking the maximum of
the
> absolute values of these valuations, we can take instead the
maximum
> of the absolute value of |hn(q)| together with the median of
> {|v1(q)|, |v2(q)|, |v1(q)-v2(q)|}, or else
> {|v1(q), |v2(q)|, |v1(q)+v2(q)|}--in other words, we sort three
> absolute values, and take the one in the middle. This also gives us
a
> norm, and we can then define Paul blocks in the 5-limit in the same
> way as Fokker blocks. To generalize to higher dimensions, it seems
we
> would need to take combinations of n vals k at a time, for k from 1
> to n, giving us the verticies of the n-measure polytope
(=hypercube.)
> We then would take a maximum over the n smallest. At least, that
> seems right, but I haven't really thought it through carefully.

I wish I could understand this. Can I ask you, what were you
misreading, and what led you to a correct reading?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I wish I could understand this. Can I ask you, what were you
> misreading, and what led you to a correct reading?

It needs fixing anyway, so I will try again later. What I had thought
you were saying would have been equivalent to saying that after a
linear transformation of the parallepiped into a hypercube of measure
1, we allow ourselves to chop up the hypercube and reassemble it into
a convex body, also of measure 1 (no Banach-Tarski, please!) This
tiles the n-space, but it allows us too much latitude.

Instead, you were putting restrictions on how the square could be
chopped up and reassembled, and we should presumably have some
restrictions also if we generalize this. I'll try to work the mess I
posted out in a way which makes more sense, but I need to define the
norm by first creating a region, and then defining the norm of a
point by scaling the region and finding what scale factor makes the
point lie on the boundry.

We could try transforming back to a regular hexagon instead, but then
I still want to know how to generalize it. Would the 3-D version be
the bee-type honeycomb of rhombic dodecahedra?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I wish I could understand this. Can I ask you, what were you
> > misreading, and what led you to a correct reading?
>
> It needs fixing anyway, so I will try again later. What I had
thought
> you were saying would have been equivalent to saying that after a
> linear transformation of the parallepiped into a hypercube of
measure
> 1, we allow ourselves to chop up the hypercube and reassemble it
into
> a convex body, also of measure 1 (no Banach-Tarski, please!) This
> tiles the n-space, but it allows us too much latitude.

Too much latitude? Why?
>
> Instead, you were putting restrictions on how the square could be
> chopped up and reassembled,

I was?

> We could try transforming back to a regular hexagon instead, but
then
> I still want to know how to generalize it.

Why a regular hexagon? And in what version of the lattice? No, I see
a desirable class of 5-limit periodicity blocks defined as a hexagon
of any shape, as long as opposite sides are parallel and congruent.

> Would the 3-D version be
> the bee-type honeycomb of rhombic dodecahedra?

Yes, but they certainly don't have to be regular. Also, they can
be "degenerate", with certain faces vanishing or combining with other
faces, so that one ends up with hexagonal prisms or parallelepipeds.