This whole "val space" vs "tuning space" thing

I never really got a satisfying resolution to this on XA, so I'll ask here.

Am I correct that the big picture is that there are simply two
different correct interpretations of the relationship between "vals"
and "tuning maps," and people just have different preferences as to
which one they like?

Here are the two interpretations:

1) Vals and tuning maps both form notable different structures in the
same space, which is the space of linear functionals on interval
space. Vals are a lattice that results from the embedding of a group
into a vector space. Tuning maps correspond to a fuzzy region around
the JIP. Changing units changes the location of the JIP and hence the
location of tuning maps.
2) Vals and tuning maps live in two different spaces which are
isomorphic to one another. Both spaces are dual to interval space,
which implies a trivial and canonical isomorphism sending <1 0 0| in
one space to <1.0 0.0 0.0| cents in the other space. Changing units
changes this trivial isomorphism to something like <1 0 0| -> <1.0 0.0
0.0| millioctaves instead of cents.

And so my understanding is that Paul likes #1 and Gene likes #2, but
they're really just the same thing.

Is this correct?

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> 1) Vals and tuning maps both form notable different
> structures in the same space, which is the space of
> linear functionals on interval space. Vals are a lattice
> that results from the embedding of a group into a vector
> space. Tuning maps correspond to a fuzzy region around
> the JIP. Changing units changes the location of the JIP
> and hence the location of tuning maps. 2) Vals and tuning
> maps live in two different spaces which are isomorphic to
> one another. Both spaces are dual to interval space,
> which implies a trivial and canonical isomorphism sending
> <1 0 0| in one space to <1.0 0.0 0.0| cents in the other
> space. Changing units changes this trivial isomorphism to
> something like <1 0 0| -> <1.0 0.0 0.0| millioctaves
> instead of cents.

Whatever you call the space, <1 0 0| maps to around <1200.0
0.0 0.0| cents and <1000.0 0.0 0.0| millioctaves.

Graham

On Wed, Mar 7, 2012 at 3:28 PM, Graham Breed <gbreed@gmail.com> wrote:
>
> Mike Battaglia <battaglia01@gmail.com> wrote:
>
> > 1) Vals and tuning maps both form notable different
> > structures in the same space, which is the space of
> > linear functionals on interval space. Vals are a lattice
> > that results from the embedding of a group into a vector
> > space. Tuning maps correspond to a fuzzy region around
> > the JIP. Changing units changes the location of the JIP
> > and hence the location of tuning maps. 2) Vals and tuning
> > maps live in two different spaces which are isomorphic to
> > one another. Both spaces are dual to interval space,
> > which implies a trivial and canonical isomorphism sending
> > <1 0 0| in one space to <1.0 0.0 0.0| cents in the other
> > space. Changing units changes this trivial isomorphism to
> > something like <1 0 0| -> <1.0 0.0 0.0| millioctaves
> > instead of cents.
>
> Whatever you call the space, <1 0 0| maps to around <1200.0
> 0.0 0.0| cents and <1000.0 0.0 0.0| millioctaves.
>
> Graham

Right. But, under the "two spaces" interpretation, then Gene said that
if you want both spaces to be separately dual to interval space, this
implies a canonical isomorphism from <1 0 0| to <1.0 0.0 0.0| cents.
Maybe he can clarify further.

-Mike