BP in Fokker lattice

This message goes to Paul Erlich, Kees van Prooijen, Manuel op de Coul and
everybody else interested in Fokker lattices. Stimulated by a recent exchange
of notes between Paul and Kees, I have been playing around a little with the
representation of BP scales as Fokker lattices. The result is now on the web
under
http://members.aol.com/bpsite/BPlattice.html
Since this is my first encounter with lattices at all, please let me know if
you find anything wrong with it.

Heinz

Hello Heinz,

in response to http://members.aol.com/bpsite/BPlattice.html:

>The product of the matrix (sorry, not easily reproducible in html)
resulting from >these vectors is indeed 13.

You mean "determinant", not product.

>And thus the hunt is on to find suitable unison vectors for other
"diatonic" BP >scales. . . . Two of these semitones are soon eliminated by
requesting that the >matrix products of each pair that these vectors can
form result in either 9 (for >"diatonic" scales) or 13 (for the "chromatic"
scale). 27/25 [ 0 -2 ] and 49/45 [ 2 >1 ] cannot produce these numbers in
combination with any other small interval.

I don't get it. First you say you want to find "other" scales, then you
eliminate any possibility of getting anything "other" than what you already
have . . . ?

>[ 2 3 ] * [ -3 2 ] (already used for the "chromatic" scale), [ 2 3 ] * [ -5
-1 ], >[ 2 3 ] * [ -1 5 ],
>[ -5 -1 ] * [ -3 2 ], [ -3 2 ] * [ -1 5 ],
>but only three times 9-step combinations:
>[ 2 3 ] * [ -1 3 ] (already used for the Lambda scale), [ 2 3 ] * [ -3 0 ],
[ -3 0 >] * [ -1 3 ].

Your notation indicates a product, but again you are really looking at the
determinant.

>Dur II is a symmetric scale; thus no second scale can be defined by the
same >combination of unison vectors.

On the contrary, if you shift the pattern over by about half a cell, you'll
get a different scale.

Thanks, Paul, ...

...for the "determinant". I am going to correct that; one forgets these
things when not using them for so many years.

...for pointing out unclear expressions. When I said "other" I did not mean
that I was looking for new BP scales; there are enough unexplored ones
already. I just wanted to present the other existing ones in lattice format,
too.

...for pointing out misleading notation. I'm looking for a way to replace
that.

That shifting of the pattern by half a cell, though, gets you to different
scales all right. However, they have a tendency to suffer from intervals only
a diesis wide, until you make a full cell's step.

Heinz Bohlen wrote,

>That shifting of the pattern by half a cell, though, gets you to different
>scales all right. However, they have a tendency to suffer from intervals
only
>a diesis wide, until you make a full cell's step.

If you do this correctly, there will never be two notes separated by a
unison vector within the periodicity block. By "diesis", did you mean a
different interval from the two you used as unison vectors?