Vals and Tuning Maps

Paul keeps saying stuff like this on XA:

"It's absolutely not the same space. In one case, the scalar that
comes out is the size of the interval in cents. [tuning map] In the
other, it's ... ? [val] Anyway, it certainly makes no sense to put
both these types of objects in the same space. For one thing, their
relationship would change entirely just by changing your unit of
interval measurement."

I was taught that "vals" and "tuning maps" are both objects in the
same space, which is the space of covectors acting on interval space.
Is this wrong? Can someone clarify?

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> I was taught that "vals" and "tuning maps" are both
> objects in the same space, which is the space of
> covectors acting on interval space. Is this wrong? Can
> someone clarify?

If a space includes the metric (and it does) then it's
reasonable that different metrics mean different spaces.
With vals, the inner product measures complexity and you
can also measure optimal error as an angle in the space.
With tuning maps, the metric measures the tuning. They
look different.

Graham

On Sat, Mar 3, 2012 at 5:38 PM, Graham Breed <gbreed@gmail.com> wrote:
>
> Mike Battaglia <battaglia01@gmail.com> wrote:
>
> > I was taught that "vals" and "tuning maps" are both
> > objects in the same space, which is the space of
> > covectors acting on interval space. Is this wrong? Can
> > someone clarify?
>
> If a space includes the metric (and it does) then it's
> reasonable that different metrics mean different spaces.
> With vals, the inner product measures complexity and you
> can also measure optimal error as an angle in the space.
> With tuning maps, the metric measures the tuning. They
> look different.
>
> Graham

So then there seems to be some disagreement on this. I asked this
question here and was told that both were in "tuning space"

/tuning-math/message/20095

Also, the wiki article has been called "Vals and Tuning Space" for some time

http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space

But, Paul's insisting that "Tuning Space" is the name referring to the
space of tuning maps.

I thought these were both the same space, since they're the space of
covectors acting on interval space. And, you can put the L2 norm on
either the space of vals and the space of tuning maps and you end up
in both cases with an L2-normed vector space that's dual to the
L2-normed interval space.

And I thought the name for this space in general was "Tenney-Euclidean
Tuning Space," and that these were different ways to interpret points
in it.

If this isn't the case, then for the sake of being able to
communicate, what are the mathematically and terminologically correct
names for these two spaces? Do they exist? If not, I suggest that the
space of vals be called "Temperament Space," and the space of tuning
maps be called "Tuning Space."

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> But, Paul's insisting that "Tuning Space" is the name
> referring to the space of tuning maps.

You could think of it like that. It doesn't really matter.

> And I thought the name for this space in general was
> "Tenney-Euclidean Tuning Space," and that these were
> different ways to interpret points in it.

You could think of it like that. It doesn't really matter.

> If this isn't the case, then for the sake of being able to
> communicate, what are the mathematically and
> terminologically correct names for these two spaces? Do
> they exist? If not, I suggest that the space of vals be
> called "Temperament Space," and the space of tuning maps
> be called "Tuning Space."

The lattice of vals could be called "the val lattice" or
"the complexity lattice" because it measures complexity.

Graham

On Sat, Mar 3, 2012 at 6:14 PM, Graham Breed <gbreed@gmail.com> wrote:
>
> Mike Battaglia <battaglia01@gmail.com> wrote:
>
> > But, Paul's insisting that "Tuning Space" is the name
> > referring to the space of tuning maps.
>
> You could think of it like that. It doesn't really matter.
//snip
> You could think of it like that. It doesn't really matter.

It matters to me because I'm tired of getting sucked into debates over
terminology. I started a well-planned one over the definition of
"MOS," but I didn't factor on that turning into a debate over the
definition of "tuning space." And we just had a bunch of debates over
the definition of "temperament" and "ratio." I don't care what we call
things. I just would like to know what the mathematically correct "big
picture" is, and then know what Paul's stance on it vs Gene's stance
on it is, so I can make sure I'm saying the things I mean when I use
words.

> The lattice of vals could be called "the val lattice" or
> "the complexity lattice" because it measures complexity.

I think there's some sense to be made of vals that don't form lattice.
One thing that Ryan and I have been talking a lot about recently is
analyzing subgroup vals as "fractional" versions of full-limit vals,
e.g. so that the 2.9.5-limit val <1 1 1| becomes the 2.3.5-limit val
<1 0.5 1|. That poses a few challenges but looks promising.

-Mike

On Sat, Mar 3, 2012 at 6:29 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> It matters to me because I'm tired of getting sucked into debates over
> terminology. I started a well-planned one over the definition of
> "MOS," but I didn't factor on that turning into a debate over the
> definition of "tuning space." And we just had a bunch of debates over
> the definition of "temperament" and "ratio." I don't care what we call
> things. I just would like to know what the mathematically correct "big
> picture" is, and then know what Paul's stance on it vs Gene's stance
> on it is, so I can make sure I'm saying the things I mean when I use
> words.

And now we have another catastrophic failure of terminology here:

Paul Erlich: "TE error is the Tenney-weighted sum of the squares of
the deviations of the primes from JI. Thus it is Euclidean distance in
the tuning space I'm trying to talk about. It's not an angle."

I don't think this is right. Isn't TE error supposed to be the same
thing as "TOP-RMS error?" Which of these definitions is correct?

1) TE error is the Tenney-weighted RMS prime error for =some tuning
of= some temperament
2) TE error is the Tenney-weighted RMS prime error for =the optimal
tuning of= that temperament

I thought 2. Paul is saying 1. Someone please just say "1" or "2" and
tell me what the right words are to use.

-Mike