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".