Fwd: Re: deriving classical major/minor scales

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

?