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.