Euclidean??? distance

Dear Manuel, thanks for the new version of Scala.

The tips.par file contains (among others) the following two entries:
----------------------------------------------------------------
The Euclidean lattice distance given when SET ATTRIBUTE EUCLID is done, is
the sum of the absolute exponents for each prime in the interval ratio, except
2. So this gives the number of steps that needs to be made to go from the
origin (1/1) to the given ratio in lattice space. For instance for 6/5 it is
two, one for going in the positive direction of "3" plus one for going in the
negative direction of "5", prime 2 is ignored.
Similarly when SHOW DATA is done the Euclidean lattice dimensions given
indicate the total rectangular size of the lattice space occupied by the whole
scale, also ignoring prime 2, which is the sum of the sizes of each dimension.
~
The triangular lattice distance given when SET ATTRIBUTE TRIANGLE is done, is
the shortest path from the vector of prime exponents in the interval ratio
except 2, to the origin. The lattice is triangular like this:

5/4 - 15/8
/ \ / \
1/1 - 3/2 - 9/8
\ / \ /
6/5 - 9/5

For instance for 6/5 it is one, contrary to the Euclidean distance, which is
two. Prime 2 is ignored.

The triangular logarithmic complexity (harmonic distance) given when SET
ATTRIBUTE TR_LOG is done, is the sum of the base 2 logarithms of the highest
prime factors encountered when following the shortest distance path from a
lattice point to the origin. Prime 2 is also ignored. This concept is from
Paul Erlich. Tenney's function is another similar logarithmic function, but is
Euclidean and not triangular.
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End of tips.par quotes.

You are using "Euclidean distance" to mean "distance on a rectangular
lattice". This is completely at odds with my understanding of the term
"Euclidean distance".

Euclidean distance is just ordinary everyday straight-line distance i.e.
not keeping to any lattice. "City-block" I understand means keeping to the
lattice (whether rectangular or triangular or whatever), although it does
suggest rectangular. What's wrong with just "lattice distance" in general
and "rectangular lattice distance" and "triangular lattice distance".

e.g. on a 2D square lattice the lattice distance between points x1,y1 and
x2,y2 is |x1-x2| + |y1-y2| but Euclidean distance is still sqrt((x1-x2)^2 +
(y1-y2)^2).

What do others say?

Regards,
-- Dave Keenan
http://dkeenan.com

Dave,

Yes I can agree it's sloppy terminology. How about if I change
it to "Euclidean (shortest) path length", or "Minkowski distance"
or "Euclidean lattice distance"?

Manuel Op de Coul coul@ezh.nl

On Tue, 16 Mar 1999 manuel.op.de.coul@ezh.nl wrote:
> Yes I can agree it's sloppy terminology. How about if I change
> it to "Euclidean (shortest) path length", or "Minkowski distance"
> or "Euclidean lattice distance"?

Of those three, I prefer "Minkowski distance". Bringing Euclid into it
just confuses matters.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

NOTE: dehyphenate node to remove spamblock. <*>

Manuel wrote:
>Yes I can agree it's sloppy terminology. How about if I change
>it to "Euclidean (shortest) path length", or "Minkowski distance"
>or "Euclidean lattice distance"?

Paul Hahn wrote:
>Of those three, I prefer "Minkowski distance". Bringing Euclid into it
>just confuses matters.

I agree that bringing Euclid in confuses matters, but so does Minkowski. I
don't know how Euclid *or* Minkowski got in here at all! Both Euclidean and
Minkowski distances relate to *geometries*, i.e. continuous (or
approximately continuous) spaces. They have nothing to do with distances on
graphs or lattices (which are definitely discrete, even when the edges are
weighted).

Euclidean distance is our everyday idea of distance, generalised to any
number of dimensions. sqrt(dx^2 + dy^2 + dz^2 + ...). But it is definitely
not a synonym for "shortest distance", since in a non-Euclidean ("curved")
space (e.g. extended regions on the surface of a sphere, or in a 3D-space
near a large mass) the shortest distance is definitely *not* called the
Euclidean distance and is not amenable to the above formula.

Minkowski distance is where we include time as an extra dimension but
multiply it by i (sqrt(-1)) before including it in a Euclidean-type
distance formula. sqrt([i*dt]^2 + dx^2 + dy^2 + dz^2). This space-time
"distance" is what is approximately preserved even in relativistic (near
speed-of-light) changes of frame-of-reference, provided they are away from
large masses. It has nothing whatsoever to do with "city
block"/graph/lattice metrics.

Manuel, I seem to remember you may have similarly used "Cartesian"
inappropriately somewhere, possibly to mean "rectangular".

Just plain "shortest path length" in "rectangular" and "triangular"
varieties, would be fine. If you definitely need an adjective to
distinguish these from Euclidean (straight-line) distance, in case someone
thinks these things are geometrical (as opposed to being able to be
embedded in various geometries, in various ways, to aid our understanding),
I can only think of "city block". Does anyone know a better one?

Regards,
-- Dave Keenan
http://dkeenan.com

> Just plain "shortest path length" in "rectangular" and "triangular"
> varieties, would be fine. If you definitely need an adjective to
> distinguish these from Euclidean (straight-line) distance, in case someone
> thinks these things are geometrical (as opposed to being able to be
> embedded in various geometries, in various ways, to aid our
> understanding), I can only think of "city block". Does anyone know a
> better one?
>
> Regards,
> -- Dave Keenan

I use metrics every day as a mathematician and we do have technical names for
these things, even if they may not be appropriate in a musical context. The
most intuitive term I know to describe the "lattice distance" is "taxicab
geometry" (or "taxicab norm", coming close to the "city block" idea). The fact
that the lattices we are talking about in this list are integer valued is
irrelevant as for the notion of geometry and/or norm (distance). The technical
name for the "taxicab" norm/distance/geometry is

l_{infinity} {little ell infinity, where infinity is sub or superscript.

To understand the name, look at l_p, where p is a parameter between 1 and
infinity, and consider the simplest case of two dimensions (the more general
formulas are entirely analogous). The "l_p distance" from point (0,0) to point
(x,y) is given by the formula

(|x|^p + |y|^p)^{1/p} (for p smaller than infinity)

("^" denotes exponentiation). So we see that, for instance, l_2 is the usual
Euclidean (the only one!) distance, while the case p=infinity is obtained
through a limit, and in that case the formula becomes

sup{|x|,|y|}

which is exactly what one uses to measure distance in the taxicab geometry.

Hope this helps,
Vania.

Vania Mascioni ph : (1)-512-471.3149
Department of Mathematics fax : (1)-512-471.9038
The University of Texas at Austin e-mail : mascioni@math.utexas.edu
Austin, TX 78712, U.S.A. http://www.ma.utexas.edu/users/mascioni/

Vania wrote,

>The "l_p distance" from point (0,0) to point
>(x,y) is given by the formula

> (|x|^p + |y|^p)^{1/p} (for p smaller than infinity)

>("^" denotes exponentiation). So we see that, for instance, l_2 is the
usual
>Euclidean (the only one!) distance, while the case p=infinity is
obtained
>through a limit, and in that case the formula becomes

> sup{|x|,|y|}

>which is exactly what one uses to measure distance in the taxicab
geometry.

I don't think you're correct. The taxicab metric is |x| + |y|, the l_1
norm.

> I don't think you're correct. The taxicab metric is |x| + |y|, the l_1
> norm.

You are right, I was just giving all the examples in one paragraph and forgot
the point. The taxicab/cityblock is l_1, and l_infinity is the dual norm (very
much related to l_1).

Vania.