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.