Question for Gene

Gene, could you please comment on this:

http://www.kees.cc/tuning/lat_perbl.html

In particular, I'm assuming a city-block or taxicab metric. Is Kees
observing that in his final lattice? It looks like he isn't.

What else can you say?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> What else can you say?

Before I can say much else, I need to read up on what you two are
doing, but I can make one comment now: if you want to put a
symmetrical metric onto the note-classes defined as octave
equivalents of 3^a 5^b, the way to do it is with the quadratic form
q(3^a 5^b) = a^2 + ab + b^2. This and the bilinear form associated to
it will make 5/4 and 5/3 the same size, both being one step away from
1.

This is quite interesting when looking at automorphisms of the 5-
limit which preserve octave equivalence and also the 3-et; these can
be thought of as generalizations of the major/minor transformation.
In the same way, putting the quadratic form

q(3^a 5^b 7^c) = a^2 + b^2 + c^2 + ab + ac + bc

onto the octave-equivalence classes of the 7-limit gives a lattice
structure containing tetrahedrons and octahedrons, which is the only
semiregular 3D honeycomb; around each vertex are eight tetrahedra and
six octahedra in a pattern like the triangles and squares of a
cubeoctahedron. If we look for automorphism groups which preserve
this metric and the 4-et we get another interesting group of musical
transformations--the octahedral group applied to 7-limit music.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Gene, could you please comment on this:

It looks to me like he's trying to do something which doesn't quite
work, which is to analyze a taxicab metric in Euclidean terms.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Gene, could you please comment on this:
>
> It looks to me like he's trying to do something which doesn't quite
> work, which is to analyze a taxicab metric in Euclidean terms.

What if we forget about the Euclidean part -- can you shed any light on the "paradox" in pure taxicab terms?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> In particular, I'm assuming a city-block or taxicab metric. Is Kees
> observing that in his final lattice? It looks like he isn't.

Kees is talking about circles and transforming as if in a Euclidean
space, so you aren't on the same wavelength.

> What else can you say?

I'm not sure what your triangular lattice metric is. A taxicab needs
two lines to run along; you can make these 120 degrees to each other
but you can't get an array of equilateral triangles out of it.

By "lattice", mathematicians usually mean one of two things. The
first has to do with partial orderings and need not concern us, the
second defines a lattice as a discrete subgroup of R^n whose quotient
group is compact. I'm not always sure what people mean when they say
lattice in this neighborhood.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > In particular, I'm assuming a city-block or taxicab metric. Is
Kees
> > observing that in his final lattice? It looks like he isn't.
>
> Kees is talking about circles and transforming as if in a Euclidean
> space, so you aren't on the same wavelength.

What about the previous lattices on that page?
>
> > What else can you say?
>
> I'm not sure what your triangular lattice metric is. A taxicab
needs
> two lines to run along; you can make these 120 degrees to each
other
> but you can't get an array of equilateral triangles out of it.

Well, they're not equilateral triangles, but they are triangles. The
metric is the shortest path along the edges of this triangular graph.
Is it not correct to call that a "taxicab metric"?
>
> By "lattice", mathematicians usually mean one of two things. The
> first has to do with partial orderings and need not concern us, the
> second defines a lattice as a discrete subgroup of R^n whose
quotient
> group is compact. I'm not always sure what people mean when they
say
> lattice in this neighborhood.

Sir, there is an accepted definition of "lattice" that is used in
geometry and crystallography. Every point, and its local connections,
is congruent to every other point and its local connections . . .
something like that. We had this discussion a long time ago on the
tuning list.

In post 984 <genewardsmith@j...> wrote:

<<

> By "lattice", mathematicians usually mean one of two things. The
> first has to do with partial orderings and need not concern us, the
> second defines a lattice as a discrete subgroup of R^n whose quotient
> group is compact. I'm not always sure what people mean when they say
> lattice in this neighborhood.

and Paul respond:

<<

Sir, there is an accepted definition of "lattice" that is used in
geometry and crystallography. Every point, and its local connections,
is congruent to every other point and its local connections . . .
something like that. We had this discussion a long time ago on the tuning
list.

>>

We don't have this problem in French for we use "treillis" in the fist
sense of partial ordering where two elements have always inferior and
superior "bornes", and "réseau" in the sense of discrete Z-module.

Personally, I use both the "treillis" (in melodic representation) and the
"réseaux" (in harmonic representation). I have the problem to be understood
with the unique term "lattice".