Re: Ums

Hi Gene,

> That is not possible; the distances along diagonals do not correspond
> to the distances along sides. If you take a box with corners 1, 2, 3,
> 6, 1/5, 2/5, 3/5 and 6/5 you get 6/5 in the opposite corner from 1.
> The Euclidean distance is sqrt(1+log2(3)^2+log2(5)^2) = 2.98, but the
> taxicab distance is log2(30) = 4.91. Obviously, this isn't a normal
> kind of box.

All you are saying here is that the metric on the Tenney lattice
isn't the same as the metric on the ratios.

But that's okay. You can have two different metrics on the
same space.

Here, one of them gives the Tenney lattice as a lattice within euclidean
space. The other metric on the same lattice points is the
taxicab metric and gives the sizes of the intervals
(e.g. in cents or whatever).

You can study taxicab geometries within
a space defined using a euclidean metrics
simply by having two metrics. That is how
they are studied in geometrical research.
In fact that is how the taxicab metric
is often introduced - by drawing a diagram
in euclidean space, then defining the
metric on it using geometric notions.
There is an interesting interplay
between tilings and taxicab metrics
which I explore some in one of my
unpublished papers to get some interesting
results on discrete approximations to
the euclidean metric.

The euclidean metric here is useful for geometric
constructions such as drawing lines between
points and measuring angles and distances
in geometrical representations of the lattice.
That could be just for display purposes but
it might have some other relevance too - one
can't rule out that possibility.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> The euclidean metric here is useful for geometric
> constructions such as drawing lines between
> points and measuring angles and distances
> in geometrical representations of the lattice.

The Tenney metric gives you the same affine and topological structure,
but the angles are Euclidean. That really becomes relevant for
lattices of octave classes, where symmetry can be involved.

> That could be just for display purposes but
> it might have some other relevance too - one
> can't rule out that possibility.

See above.

>> That is not possible; the distances along diagonals do not correspond
>> to the distances along sides. If you take a box with corners 1, 2, 3,
>> 6, 1/5, 2/5, 3/5 and 6/5 you get 6/5 in the opposite corner from 1.
>> The Euclidean distance is sqrt(1+log2(3)^2+log2(5)^2) = 2.98, but the
>> taxicab distance is log2(30) = 4.91. Obviously, this isn't a normal
>> kind of box.
>
>All you are saying here is that the metric on the Tenney lattice
>isn't the same as the metric on the ratios.
>
>But that's okay. You can have two different metrics on the
>same space.

Indeed; this objection seemed ridiculous to me.

>You can study taxicab geometries within
>a space defined using a euclidean metrics
>simply by having two metrics. That is how
>they are studied in geometrical research.
>In fact that is how the taxicab metric
>is often introduced - by drawing a diagram
>in euclidean space, then defining the
>metric on it using geometric notions.
>There is an interesting interplay
>between tilings and taxicab metrics
>which I explore some in one of my
>unpublished papers to get some interesting
>results on discrete approximations to
>the euclidean metric.

This sounds interesting. Gene has demonstrated that
taxicab bounds often enclose very similar scales as
Euclidean bounds.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> That is not possible; the distances along diagonals do not correspond
> >> to the distances along sides. If you take a box with corners 1, 2, 3,
> >> 6, 1/5, 2/5, 3/5 and 6/5 you get 6/5 in the opposite corner from 1.
> >> The Euclidean distance is sqrt(1+log2(3)^2+log2(5)^2) = 2.98, but the
> >> taxicab distance is log2(30) = 4.91. Obviously, this isn't a normal
> >> kind of box.
> >
> >All you are saying here is that the metric on the Tenney lattice
> >isn't the same as the metric on the ratios.
> >
> >But that's okay. You can have two different metrics on the
> >same space.
>
> Indeed; this objection seemed ridiculous to me.

You were claiming the Tenney lattice fits inside Euclidean space, but
it seems to me what you are really doing is imagining a Euclidean
space with a Euclidean metric, and then using it to measure taxicab
distances between lattice points. This is hardly the same as saying
the Tenney lattice fits into Euclidean space. What really results if
you make that idea explicit is that both Euclidean space and Tenney
space fit into the same toplogical vector space, but neither fits
inside the other, which is impossible to accomplish. In fact in
infinite dimensions even the topology can be different, a fact which
is somtimes useful. But it does not involve stuffing one space inside
another somehow, but taking the same space and putting two different
structures on it.

> >There is an interesting interplay
> >between tilings and taxicab metrics
> >which I explore some in one of my
> >unpublished papers to get some interesting
> >results on discrete approximations to
> >the euclidean metric.
>
> This sounds interesting. Gene has demonstrated that
> taxicab bounds often enclose very similar scales as
> Euclidean bounds.

A convex polyhedron can be arbitarily close to a sphere, so these must
exist. What's dicier is trying to get a good Euclidean approximation
to something like the Tenney norm; there's only so far you can take
it. However, the Hahn metric and the symmetrical metric on 7-limit
note classes can look a lot alike in practice.

>> >> That is not possible; the distances along diagonals do not correspond
>> >> to the distances along sides. If you take a box with corners 1, 2, 3,
>> >> 6, 1/5, 2/5, 3/5 and 6/5 you get 6/5 in the opposite corner from 1.
>> >> The Euclidean distance is sqrt(1+log2(3)^2+log2(5)^2) = 2.98, but the
>> >> taxicab distance is log2(30) = 4.91. Obviously, this isn't a normal
>> >> kind of box.
>> >
>> >All you are saying here is that the metric on the Tenney lattice
>> >isn't the same as the metric on the ratios.
>> >
>> >But that's okay. You can have two different metrics on the
>> >same space.
>>
>> Indeed; this objection seemed ridiculous to me.
>
>You were claiming the Tenney lattice fits inside Euclidean space, but
>it seems to me what you are really doing is imagining a Euclidean
>space with a Euclidean metric, and then using it to measure taxicab
>distances between lattice points.

I haven't talked about measuring in this thread. In fact I thought
we agreed we did not need a norm to describe unison vectors.

>This is hardly the same as saying the Tenney lattice fits into
>Euclidean space. What really results if you make that idea explicit
>is that both Euclidean space and Tenney space fit into the same
>toplogical vector space,
//
>But it does not involve stuffing one space inside
>another somehow,

How'd you get from a lattice fitting inside a space (first sentence)
to a space in a space (second & third sentences)?

>it seems to me what you are really doing is imagining a Euclidean
>space with a Euclidean metric, and then using it to measure taxicab
>distances between lattice points.

Yes, that sounds right.

>> >There is an interesting interplay
>> >between tilings and taxicab metrics
>> >which I explore some in one of my
>> >unpublished papers to get some interesting
>> >results on discrete approximations to
>> >the euclidean metric.
>>
>> This sounds interesting. Gene has demonstrated that
>> taxicab bounds often enclose very similar scales as
>> Euclidean bounds.
>
>A convex polyhedron can be arbitarily close to a sphere, so these must
>exist.

Good reasoning.

>What's dicier is trying to get a good Euclidean approximation
>to something like the Tenney norm; there's only so far you can take
>it. However, the Hahn metric and the symmetrical metric on 7-limit
>note classes can look a lot alike in practice.

I forget what the symmetrical metric is -- the first reference I
have dates from January, when you were already using it as if it had
been defined. Later you claim it's the Euclidean distance is the
symmetrical lattice, where I'm not clear on what the "symmetrical
lattice" is.

-Carl