Gene, don't know if you saw this on the main list, or for

that matter, if it showed up. Am re-posting here.

>Date: Sun, 31 Mar 2002 17:24:45 -0800

>To: tuning@yahoogroups.com

>From: Carl Lumma <carl@lumma.org>

>Subject: Re: deriving classical major/minor scales

>

>>The periodicity block business, in suitable generality, ends

>>up saying that scales tend to be convex, which seems reasonable.

>

>Gene,

>

>Will this suitable generality be covered in your paper? Or

>would you care to explain it on these lists?

>

>-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> >Will this suitable generality be covered in your paper? Or

> >would you care to explain it on these lists?

The metric on a vector space defined from a norm

d(x,y) = ||x-y||

has the property (not shared by metrics in general) of convexity--the open and closed balls defined by the metric are convex. On the other hand, given a convex closed set S containing the origin in its interior, we can define a norm such that S is a unit ball. If we define a block as a set of octave-equivalence lattice points which are epimorphic and convex, from the above we can see this ends up equivalent to a definition in terms of normed vector spaces, where a block is an epimorphic set of lattice points of minimal diameter. The Fokker block is simply then the special case of a linearlly transformed L-infinity norm. I don't know if any of this needs to be expounded.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> I don't know if any of this needs to be expounded.

i'd like to better understand the structure of the balls of various

diameters, given the tenney norm, and extensions to octave-equivalent

lattices . . . can you help?

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> i'd like to better understand the structure of the balls of various

> diameters, given the tenney norm, and extensions to octave-equivalent

> lattices . . . can you help?

You keep bringing up the Tenney norm in this context; it isn't priviledged. Fokker blocks have paralleopiped balls and are of

L-infinity type; the Tenney norm is of L1 type and would have

cross polytope (generalized octahedron) type balls. Other norms are possible, and given a convex set you can cook up the corresponding norm, and vice-versa.

Let's look at an example. Using the sxymmetrical lattice norm on octave classes in the 7-limit, so that

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

we find that the hexany 1-15/14-5/4-10/7-3/2-12/7 is contained in a spherical (L2) ball centered at [1/2, 1/3, -1/3]. Is it a block? We set up the five equations in four unknowns [a,b,c,d] which define the val, if there is one, such that the nth degree is given value n; solving this system of equations gives the unique solution

{a=6,b=10,c=14,d=17}, so that the val in question exists and is h6; hence the hexany is a block. The L2 norm is not the only norm which would define the hexany as a block; one can always use the convec hull to generate a ball--in this case we would get an L1 norm in that way.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> we find that the hexany 1-15/14-5/4-10/7-3/2-12/7 is contained in a spherical (L2) ball centered at [1/2, 1/3, -1/3].

Sorry, this should be [1/2,1/2,-1/2].

Perhaps what follows is sufficiently correlated to your interest in this thread.

I failed until date to generalize something that seemed very simple in 3D for I believed that the

parrallelopiped was the elementary shape of periodicity blocks in which the fundamental domain

around the unison (corresponding to a set of steps) was decomposable. I was wrong.

I found the generalization way using vectorial drawing. I don't want to use neither words nor math

formalism to explain that now. I leave you to discover that by intuition with a simple 4D example.

I used animated images. The image fd4.swf shows the decomposition with 4 periodicity blocks of

the 4D fundamental domain associated with 4 steps. For comparison only, the image fd3.swf shows

the decomposition with 3 periodicity blocks of the 3D fundamental domain having only 3 steps.

(I already showed similar hexagonal shapes decomposed with 3 parallelograms.)

I guess that Gene will see immediately the link with norms and unit balls.

I could show later how matrices are nicely correlated with that. Particularly, I have now a new tool,

the TS-matrix, which is a variant of the S-matrix. (S means srutis - TS means steps and srutis)

permitting to look anew, geometrically, at the correlation between steps and unison vectors.

Don't forget to press "enter" or use "play" to start the animation.

http://www.aei.ca/~plamothe/tuning/fd4.swf

http://www.aei.ca/~plamothe/tuning/fd3.swf

Pierre

I wrote:

Don't forget to press "enter" or use "play" to start the animation.

It would have been useful with attached image but it's useless with http since the plug-in start the

animation and don't show a menu.

Pierre

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

>

> > i'd like to better understand the structure of the balls of

various

> > diameters, given the tenney norm, and extensions to octave-

equivalent

> > lattices . . . can you help?

>

> You keep bringing up the Tenney norm in this context; it isn't

>priviledged. Fokker blocks

i didn't mean for this to be in the context of fokker blocks. i've

changed the subject line to prevent further confusion on this.

> the Tenney norm is of L1 type and would have

> cross polytope (generalized octahedron) type balls.

this is what i'd like to see more on!!

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> Perhaps what follows is sufficiently correlated to your interest in

this thread.

>

> I failed until date to generalize something that seemed very simple

in 3D for I believed that the

> parrallelopiped was the elementary shape of periodicity blocks in

which the fundamental domain

> around the unison (corresponding to a set of steps) was

decomposable. I was wrong.

>

> I found the generalization way using vectorial drawing. I don't

want to use neither words nor math

> formalism to explain that now. I leave you to discover that by

intuition with a simple 4D example.

>

> I used animated images. The image fd4.swf shows the decomposition

with 4 periodicity blocks of

> the 4D fundamental domain associated with 4 steps. For comparison

only, the image fd3.swf shows

> the decomposition with 3 periodicity blocks of the 3D fundamental

domain having only 3 steps.

>

> (I already showed similar hexagonal shapes decomposed with 3

parallelograms.)

>

> I guess that Gene will see immediately the link with norms and unit

balls.

>

> I could show later how matrices are nicely correlated with that.

Particularly, I have now a new tool,

> the TS-matrix, which is a variant of the S-matrix. (S means srutis -

TS means steps and srutis)

> permitting to look anew, geometrically, at the correlation between

steps and unison vectors.

>

>

> Don't forget to press "enter" or use "play" to start the animation.

>

> http://www.aei.ca/~plamothe/tuning/fd4.swf

> http://www.aei.ca/~plamothe/tuning/fd3.swf

>

> Pierre

i'd like to understand what pierre is saying here.

gene, can you help?

does this have something to do with

http://www.ixpres.com/interval/td/erlich/intropblockex.htm

?