"most compact" periodicity block

Is it clear what I mean when I say "most compact" periodicity block
for a particular equivalence class of bases? Won't it, in general, be
delimited by a hexagon in the 5-limit, and a rhombic dodecahedron in
the 7-limit? Could we use the resulting three or six unison vectors
as a more complete characterization of a given system, rather than an
LLL or some such reduced basis, which has some element of
arbitrariness because some "second best" reduction might be nearly as
good?

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

> Is it clear what I mean when I say "most compact" periodicity block
> for a particular equivalence class of bases? Won't it, in general,
be
> delimited by a hexagon in the 5-limit, and a rhombic dodecahedron
in
> the 7-limit?

It seems to me that depends on how you define your distance.
Incidentally, I suspect that instead of using hexagonal blocks, or
rrhombic dodecahedra, ellipsoids would work. I don't think you need a
tiling.

Could we use the resulting three or six unison vectors
> as a more complete characterization of a given system, rather than
an
> LLL or some such reduced basis, which has some element of
> arbitrariness because some "second best" reduction might be nearly
as
> good?

What's the point? I thought you wanted to produce temperaments, not
PBs.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Is it clear what I mean when I say "most compact" periodicity
block
> > for a particular equivalence class of bases? Won't it, in
general,
> be
> > delimited by a hexagon in the 5-limit, and a rhombic dodecahedron
> in
> > the 7-limit?
>
> It seems to me that depends on how you define your distance.

You know how I define distance.

> Incidentally, I suspect that instead of using hexagonal blocks, or
> rrhombic dodecahedra, ellipsoids would work.

No, because then you'd have more than one note within some
equivalence class, or no notes within some equivalence class.

> I don't think you need a
> tiling.

You automatically get a tiling if you choose one and only one note
from each equivalence class!

> Could we use the resulting three or six unison vectors
> > as a more complete characterization of a given system, rather
than
> an
> > LLL or some such reduced basis, which has some element of
> > arbitrariness because some "second best" reduction might be
nearly
> as
> > good?
>
> What's the point? I thought you wanted to produce temperaments, not
> PBs.

Right, but I think any rule characterizing which unison vectors we
should use, including relationships they may have with one another,
should be evaluated with respect to some sort of non-arbitrary
reduced basis, don't you think?

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

> You know how I define distance.

It doesn't work when you want to define blocks, though.

> > Incidentally, I suspect that instead of using hexagonal blocks,
or
> > rrhombic dodecahedra, ellipsoids would work.
>
> No, because then you'd have more than one note within some
> equivalence class, or no notes within some equivalence class.

I was thinking of using the minimal diameter definition.

> > I don't think you need a
> > tiling.
>
> You automatically get a tiling if you choose one and only one note
> from each equivalence class!

You bet, which is exaclty why you don't need to require that the
convex figures which result from taking everything less than or equal
to a certain distance produce a tiling.
>
> > Could we use the resulting three or six unison vectors
> > > as a more complete characterization of a given system, rather
> than
> > an
> > > LLL or some such reduced basis, which has some element of
> > > arbitrariness because some "second best" reduction might be
> nearly
> > as
> > > good?
> >
> > What's the point? I thought you wanted to produce temperaments,
not
> > PBs.
>
> Right, but I think any rule characterizing which unison vectors we
> should use, including relationships they may have with one another,
> should be evaluated with respect to some sort of non-arbitrary
> reduced basis, don't you think?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > You know how I define distance.
>
> It doesn't work when you want to define blocks, though.

What do you mean?
>
> > > Incidentally, I suspect that instead of using hexagonal blocks,
> or
> > > rrhombic dodecahedra, ellipsoids would work.
> >
> > No, because then you'd have more than one note within some
> > equivalence class, or no notes within some equivalence class.
>
> I was thinking of using the minimal diameter definition.

I could tell.
>
> > > I don't think you need a
> > > tiling.
> >
> > You automatically get a tiling if you choose one and only one
note
> > from each equivalence class!
>
> You bet, which is exaclty why you don't need to require that the
> convex figures which result from taking everything less than or
equal
> to a certain distance produce a tiling.

??? I don't want to simply produce convex figures which result from
taking everything less than or equal to certain distance! I want to
define equivalence relations (i.e. a kernel), and _then_ use a block
(possibly not unique up to reflections and such small changes) which,
given that there's one and only one note from each equivalence class,
is as compact as possible. Then, there should generally be three or
six operative unison vectors, right?

> >
> > > Could we use the resulting three or six unison vectors
> > > > as a more complete characterization of a given system, rather
> > than
> > > an
> > > > LLL or some such reduced basis, which has some element of
> > > > arbitrariness because some "second best" reduction might be
> > nearly
> > > as
> > > > good?
> > >
> > > What's the point? I thought you wanted to produce temperaments,
> not
> > > PBs.
> >
> > Right, but I think any rule characterizing which unison vectors
we
> > should use, including relationships they may have with one
another,
> > should be evaluated with respect to some sort of non-arbitrary
> > reduced basis, don't you think?

You didn't answer.