Re: periodicity blocks algorithm

----- Original Message ----- >
> The unison vectors determine an equivalence relation over the lattice
space.
> Flipping the sign of a unison vector doesn't change the relative
properties.
> That's why I only searched half the hexagon. Half of the parallelogram
lies
> inside this hexagon. The 'faraway' triangle is equivalent with the mirror
of
> the 'inside' triangle in the mirror of the traversed half hexagon. So any
> smaller intervals therein should have been found within the current level
of
> complexity (I think)
>
Paul,

Thinking a little further I guess I can't promise there won't be any smaller
intervals in the parallelogram outside the complexity limit. I think you
make the mistake associating the parallelogram with the scale determined by
the unison vectors. What I think is the right interpretation of periodicity
blocks is, as I said, taking the unison vectors as an equivalence relation
_inside_ the complexity limit. Musically speaking that means you can pick
any scale of at most n notes, where n is the determinant of the periodicity
matrix, inside the hexagon, as long as there are not two notes in that scale
related by a unison vector. You can either have that scale JI or approached
by n ET.

Coming back to your question about leaving larger intervals in the series, I
think I did that to see if interesting matrices appeared using the first and
third smallest intervals. I always checked for the determinant not to be
zero, (meaning the intervals would be linear dependent). When you do this
you get numbers like 22 and 65, which I don't think are right (in 5-limit
context !), but you might disagree from a musical standpoint :-) So I
skipped mentioning the determinant, but left the intervals in the list to
see the development.

Kees

Kees van Prooijen wrote,

> Anyway, let me ask you this: Considering the 5-limit case for now, how do
> you know that there won't be an interval in the scale that is smaller than
a
> unison vector? Even if the unison vectors both are the smallest within the
> half-hexagonal "boundary", the parallogram defined by those vectors may
> extend beyond the boundary, and may happen to include one or more smaller
> unison vectors. (Flipping the sign of one of the unison vectors will lead
to
> a parallelogram that extends over a different part of the boundary, and
> again one or more unison vectors may lie in that region.) Isn't that so?

>The unison vectors determine an equivalence relation over the lattice space.
>Flipping the sign of a unison vector doesn't change the relative properties.

When constructed from the origin, flipping the sign of _one_ of the unison
vectors changes the parallelogram's orientation with respect to the fixed
hexagon.

>That's why I only searched half the hexagon. Half of the parallelogram lies
>inside this hexagon. The 'faraway' triangle is equivalent with the mirror of
>the 'inside' triangle in the mirror of the traversed half hexagon. So any
>smaller intervals therein should have been found within the current level of
>complexity (I think)

I would think that in general more than half the parallelogram lies inside the
hexagon. And though the outside fragment is equivalent to one inside the
hexagon, might there not be a smaller interval between a note on the outside
and one on the inside?

When I get back to the office, I'll have to actually calculate your 5-limit PDs
and see if I can find an example.

-Paul

Kees wrote,

>Paul,

>Thinking a little further I guess I can't promise there won't be any smaller
>intervals in the parallelogram outside the complexity limit. I think you
>make the mistake associating the parallelogram with the scale determined by
>the unison vectors.

I took that from Fokker (have you seen his papers?).

>What I think is the right interpretation of periodicity
>blocks is, as I said, taking the unison vectors as an equivalence relation
>_inside_ the complexity limit. Musically speaking that means you can pick
>any scale of at most n notes, where n is the determinant of the periodicity
>matrix, inside the hexagon, as long as there are not two notes in that scale
>related by a unison vector.

I'm happy with that definition. But still, you may have an interval between two
notes (say, in opposite halves of the hexagon) that is smaller than a unison
vector, since the complexity limit is defined only with respect to the note at
the origin. Isn't that right?

>Coming back to your question about leaving larger intervals in the series, I
>think I did that to see if interesting matrices appeared using the first and
>third smallest intervals. I always checked for the determinant not to be
>zero, (meaning the intervals would be linear dependent). When you do this
>you get numbers like 22 and 65, which I don't think are right (in 5-limit
>context !), but you might disagree from a musical standpoint :-) So I
>skipped mentioning the determinant, but left the intervals in the list to
>see the development.

One interesting question is whether these alternate numbers lead to periodicity
blocks with the property I'm looking for (all intervals smaller than unison
vectors).

From: <PERLICH@ACADIAN-ASSET.COM>
Sent: Monday, October 11, 1999 1:34 PM
Subject: [tuning] Re: periodicity blocks algorithm

> I took that from Fokker (have you seen his papers?).
No, I haven't :-) And I don't even need a translation. Sorry.

>
> But still, you may have an interval between two
> notes (say, in opposite halves of the hexagon) that is smaller than a
unison
> vector, since the complexity limit is defined only with respect to the
note at
> the origin. Isn't that right?
>
I guess that could happen. But I'm pretty sure it would be a linear
combination of 'known' unison vectors. So it won't spoil the 'logic' of the
interpretation inside the complexity limit. (I think)

> >Coming back to your question about leaving larger intervals in the
series, I
> >think I did that to see if interesting matrices appeared using the first
and
> >third smallest intervals. I always checked for the determinant not to be
> >zero, (meaning the intervals would be linear dependent). When you do this
> >you get numbers like 22 and 65, which I don't think are right (in 5-limit
> >context !), but you might disagree from a musical standpoint :-) So I
> >skipped mentioning the determinant, but left the intervals in the list to
> >see the development.
>
> One interesting question is whether these alternate numbers lead to
periodicity
> blocks with the property I'm looking for (all intervals smaller than
unison
> vectors).
>
I'm still not sure what you mean by 'all intervals of a periodicity block'.
Does that concept follow from the Fokker material? Do you mean all intervals
inside the parallelogram? From my interpretation a (d determinant)
periodicity block maps all possible (p-limit) intervals to d equivalence
classes. From that point of view there is not _a_ parallelogram, but lots of
possible shapes containing representatives of those classes. Can you say in
that context which intervals should be smaller than the unison vectors ?

Kees

I wrote,

>> But still, you may have an interval between two
>> notes (say, in opposite halves of the hexagon) that is smaller than a
unison
>> vector, since the complexity limit is defined only with respect to the
note at
>> the origin. Isn't that right?

Kees wrote,

>I guess that could happen. But I'm pretty sure it would be a linear
>combination of 'known' unison vectors. So it won't spoil the 'logic' of the
>interpretation inside the complexity limit. (I think)

Well, to me, interpreting a JI pitch set as "closed" seems illogical in such
a case.

>> One interesting question is whether these alternate numbers lead to
periodicity
>> blocks with the property I'm looking for (all intervals smaller than
unison
>> vectors).
>
>I'm still not sure what you mean by 'all intervals of a periodicity block'.
>Does that concept follow from the Fokker material? Do you mean all
intervals
>inside the parallelogram? From my interpretation a (d determinant)
>periodicity block maps all possible (p-limit) intervals to d equivalence
>classes. From that point of view there is not _a_ parallelogram, but lots
of
>possible shapes containing representatives of those classes. Can you say in
>that context which intervals should be smaller than the unison vectors ?

I did mention in one post that I don't like "holes" -- so I guess what I'm
looking for, out of the infinitude of possible shapes, are ones that are
fully connected in the triangular lattice. If you can find any such shape in
which all intervals are smaller than the unison vectors, then you would seem
justified in claiming that a scale of d notes can arise as a naturally
closed system in JI. I posted a couple of 3-D (3,5,7) lattice diagrams of
the parallelopiped shape for a few UV triplets. The 27-tone one did have the
property I'm looking for.

The variety of possible shapes to consider is still huge. Perhaps limiting
oneself to the parallelogram/parallelopiped is one way to make the search
reasonable -- even in that case, you usually still have a few alternatives
to consider (e.g., vertex on a lattice point, center on a lattice point,
etc.). Or if you have an alternate suggestion for a consistent
shape-formulation, I'd love to hear it.

Also, I believe your work raises a few other interesting issues that are
relevant to previous discussions on this list -- I hope you stick around!

-Paul