TOP and normed vector spaces

A (real) normed vector space is a real vector space with a norm
(duh :))

If u and v are two vectors, and ||u|| denotes the norm, then

1. ||u|| >= 0

2. ||u|| = 0 iff u = 0 (the zero vector, not the number)

3. If c is a scalar, ||c u|| = |c| ||u||

4. ||u + v|| <= ||u|| + ||v||

A normed vector space is a metric space, with metric

d(u, v) = ||u - v||

An example of a normed vector space is the p-limit Erlich space,
where the norm is

|| |u2 u3 u5 ... up> || = |u2|+log2(3)|u3|+ ... + log2(p)|up|

The p-limit intervals live inside this space and form a lattice.

A linear functional on a real vector space is a linear mapping from
the space to the real numbers. It is like a val, but its coordinates
can be any real number. An example is the size functional,

SIZE = <1 log2(3) ... log2(p)|

As Joe is fond of pointing out, this functional maps every point of
the Erlich space, not just lattice points, to a real number.

The space of linear functionals is the dual space. It has a norm
induced on it defined by

||f|| = sup |f(u)|/||u||, u not zero

We may change basis in the Erlich space by resizing the elements, so
that the norm is now

|| |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|

This is called the L1 norm. Each of the basis elements now represents
something of the same size as 2, but that should not worry us.

It is a standard fact that the dual space to L1 is the L infinity
norm, and vice-versa. This means (with this resizing of our basis
vectors) that the correct norm for linear functionals is

|| <f2 f2 ... fp| || = Max(|f2|, |f3|, ..., |fp|)

A tuning map T is a type of linear functional. If c is a comma which
T tempers out, then T(c) = 0. If we have a set of commas C which are
tempered out, then this defines a subspace Null(C) of the space of
linear functionals, such that for any T in Null(C), T(c)=0 for each
comma tempered out. For points in this this subspace, there will be a
minimum distance to SIZE, and using the same proceedure we use to get
a unique minimax we can find a unique minimal distance point TOP at
this minimum distance from SIZE (which is <1 1 1 ... 1| in the basis
we are now using.)

One neat thing about this is that it generalizes immediately to other
normed vector spaces containing complete p-limit (meaning, 2 is
included as a prime number) lattices. In particular, there is a
geometric complexity version of TOP.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> An example of a normed vector space is the p-limit Erlich space,

This is the Tenney space.

> We may change basis in the Erlich space by resizing the elements,
so
> that the norm is now
>
> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|

how are the v's defined?

> and using the same proceedure we use to get
> a unique minimax we can find a unique minimal distance point TOP at
> this minimum distance from SIZE

not following . . .

> One neat thing about this is that it generalizes immediately to
other
> normed vector spaces containing complete p-limit (meaning, 2 is
> included as a prime number) lattices. In particular, there is a
> geometric complexity version of TOP.

What's better about it? I think Tenney complexity is a better guide
to consonance, to tuning sensitivity, and even to musical complexity
of a chain of intervals.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > We may change basis in the Erlich space by resizing the elements,
> so
> > that the norm is now
> >
> > || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|
>
> how are the v's defined?

For a rational number, vp is the p-adic valuation of number q, that
is,the exponent in the factorization of q into primes. For other
points in the Tenney space it's just a coordinate.

> > and using the same proceedure we use to get
> > a unique minimax we can find a unique minimal distance point TOP
at
> > this minimum distance from SIZE
>
> not following . . .

Remember, we have a way of measuring distance between tuning maps.
Hence, given a tuning map SIZE and a subspace of tuning maps Null(C),
we can find those at a minimum distance from SIZE.

> > One neat thing about this is that it generalizes immediately to
> other
> > normed vector spaces containing complete p-limit (meaning, 2 is
> > included as a prime number) lattices. In particular, there is a
> > geometric complexity version of TOP.
>
> What's better about it?

What's better about it is that it is Euclidean, which is convenient
in many ways. For instance, the version of TOP here would simply be
an orthogonal projection.

I think Tenney complexity is a better guide
> to consonance, to tuning sensitivity, and even to musical
complexity
> of a chain of intervals.

Have you even thought for five seconds about the geometric complexity
map with 2 included before coming to this conclusion? In the 5-limit,
it would be

|| |u2 u3 u5> || = sqrt(u2^2+log2(3)^2 u3^2+
log2(5)^2 u5^2+u2u3+u2u5+log2(3)^2 u3u5))

So why is this so very, very much worse?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > We may change basis in the Erlich space by resizing the
elements,
> > so
> > > that the norm is now
> > >
> > > || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|
> >
> > how are the v's defined?
>
> For a rational number, vp is the p-adic valuation of number q, that
> is,the exponent in the factorization of q into primes. For other
> points in the Tenney space it's just a coordinate.

There are no other points in the Tenney space. Anyway, I lost the
train of thought.

> > > and using the same proceedure we use to get
> > > a unique minimax we can find a unique minimal distance point
TOP
> at
> > > this minimum distance from SIZE
> >
> > not following . . .
>
> Remember, we have a way of measuring distance between tuning maps.

In the dual space?

> Hence, given a tuning map SIZE and a subspace of tuning maps Null
(C),
> we can find those at a minimum distance from SIZE.

Hmm . . .

> > > One neat thing about this is that it generalizes immediately to
> > other
> > > normed vector spaces containing complete p-limit (meaning, 2 is
> > > included as a prime number) lattices. In particular, there is a
> > > geometric complexity version of TOP.
> >
> > What's better about it?
>
> What's better about it is that it is Euclidean, which is convenient
> in many ways.

Do we really need this convenience? Can't we work with the taxicab
metric?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > For a rational number, vp is the p-adic valuation of number q, that
> > is,the exponent in the factorization of q into primes. For other
> > points in the Tenney space it's just a coordinate.
>
> There are no other points in the Tenney space. Anyway, I lost the
> train of thought.

Then why did you correct me to "Tenney space"? Presumably, I knew what
space I wanted even if I didn't have the right name for it. There
*are* other points in my space, and what you seem to be talking about
is a lattice.

> > > > and using the same proceedure we use to get
> > > > a unique minimax we can find a unique minimal distance point
> TOP
> > at
> > > > this minimum distance from SIZE
> > >
> > > not following . . .
> >
> > Remember, we have a way of measuring distance between tuning maps.
>
> In the dual space?

Correct. Tuning maps are points in the dual space, and that is a
normed vector space, and hence a metric space, so we know the distance
between two tuning maps. One, SIZE, is the JI tuning map. We want a
tuning map in the subspace Null(C) as close as possible to SIZE.

> Do we really need this convenience? Can't we work with the taxicab
> metric?

I compared the two, and taxicab does seem to work better. The
Euclidean version thinks minor thirds are slightly better than major
thirds, for instance.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > For a rational number, vp is the p-adic valuation of number q,
that
> > > is,the exponent in the factorization of q into primes. For
other
> > > points in the Tenney space it's just a coordinate.
> >
> > There are no other points in the Tenney space. Anyway, I lost the
> > train of thought.
>
> Then why did you correct me to "Tenney space"?

What does that have to do with it? I just meant I lost where I was in
my thinking when I stopped to ask you about the v's.

> Presumably, I knew what
> space I wanted even if I didn't have the right name for it. There
> *are* other points in my space, and what you seem to be talking
about
> is a lattice.

OK, why do we need a space and not a lattice?

> > > > > and using the same proceedure we use to get
> > > > > a unique minimax we can find a unique minimal distance
point
> > TOP
> > > at
> > > > > this minimum distance from SIZE
> > > >
> > > > not following . . .
> > >
> > > Remember, we have a way of measuring distance between tuning
maps.
> >
> > In the dual space?
>
> Correct. Tuning maps are points in the dual space, and that is a
> normed vector space, and hence a metric space, so we know the
distance
> between two tuning maps. One, SIZE, is the JI tuning map. We want a
> tuning map in the subspace Null(C) as close as possible to SIZE.

OK . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> OK, why do we need a space and not a lattice?

I want a metric on the dual space of linear functionals, so I start
out with a normed vector space with which to define the dual.

Let's put it another way. If vp is the p-adic valuation and T is a
tuning map, then for a p-limit rational number q we have

q = 2^v2(q) 3^v3(q) ... p^vp(q)

A tuning map has numbers r2~2, r3~3, ... rp~p and is defined by

T(q) = r2^v2(q) r3^v3(q) ... rp^vp(q)

The norm ||T|| of the above tuning map (dual to the Tenney norm) is

||T|| = Max(|log2(r2)|, log3(r3), ..., logp(rp))||

However, we are not so much interested in the norm as in its distance
from the JI tuning map which I've called SIZE, which is

||T - SIZE|| = Max(|log2(r2/2)|, |log3(r3/3)|, ..., |logp(rp/p)|)

It is this which we want to minimize over the subspace of tuning maps
such that T(c)=1 (writing it multiplicatively) for each comma c we
are tempering out.