new graphics for Erlich's "Periodicity Block" intro

I've added color to the illustrations in Paul Erlich's
"Gentle Introduction to Fokker Periodicity Blocks"
http://www.ixpres.com/interval/td/erlich/intropblock1.htm

which should make the transformational mathematics Paul
discusses a *lot* easier to understand.

You'll have to go to the "next installment" (Part 2), and
particularly the one after that ("An Excursion"), to see
the new graphics.

-monz
http://www.monz.org
"All roads lead to n^0"

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Oops... I need to publicly thank Jon Szanto for his help
with those new graphics. Thanks, Jon!

-monz
http://www.monz.org
"All roads lead to n^0"

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Joe,

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Oops... I need to publicly thank Jon Szanto for his help
> with those new graphics. Thanks, Jon!

Youbetcha! Sure wish I understood what the hell I was looking at...

<g>

Cheers,
Jon

> ----- Original Message -----
> From: Jon Szanto <JSZANTO@ADNC.COM>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, June 16, 2001 1:02 PM
> Subject: [tuning] Re: new graphics for Erlich's "Periodicity Block" intro
>
>
> Youbetcha! Sure wish I understood what the hell I was looking at...

Ask away, Jon. I'd be glad to help you understand this stuff.

The new graphics I added should make it very clear.
The green lines are the unison vectors.

If you can see that one green line is angled so that it would
connect 10/9 and 9/8, that's the syntonic comma unison vector,
which has a vector notation of (4 -1). That simply indicates
the prime-factorization of the syntonic comma, whose ratio
is 81/80, which is (3^4)*(5^-1) in "octave"-equivalent terms
(that is, ignoring all powers of 2).

The other green line shows the other unison vector, which
in Paul's diagrams takes several different forms. The first
one he illustrates at
http://www.ixpres.com/interval/td/erlich/intropblock2.htm

is (-1 2), which designates (3^-1)*(5^2), or the ratio 25/24,
which is a 5-limit chromatic semitone. This unison vector,
paired with the syntonic comma unison vector, gives the 7-tone
diatonic scales which appear within the green parallelogram
on the lattices.

Further down that page, Paul combines the syntonic comma
unison vector with the unison vector of the diesis, which
he writes as (0 3) in vector notation. The direction of
the exponents doesn't matter in constructing periodicity
blocks, so this works. But the diesis really should be
written (0 -3), designating (3^0)*(5^-3), or the ratio
128/125. This pair of unison vectors (syntonic comma
and diesis) yields a 12-tone chromatic scale.

Then on the "Excursion":
http://www.ixpres.com/interval/td/erlich/intropblockex.htm

Paul shows how to take segments (which I've drawn in red)
from within the parallelogram (enclosed in green) and
transpose them to other parts of the lattice.

Transposing those pieces changes the shape of the periodicity
block - in these examples, from a parallelogram to a hexagon
- but it doesn't alter the ability of these shapes to periodically
tile the plane of the 2-dimensional lattice. So the hexagon
is equally valid as a periodicity block, even tho the unison
vectors are "chopped up".

Hope that helps.

-monz
http://www.monz.org
"All roads lead to n^0"

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Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> I've added color to the illustrations in Paul Erlich's
> "Gentle Introduction to Fokker Periodicity Blocks"
> http://www.ixpres.com/interval/td/erlich/intropblock1.htm
>
> which should make the transformational mathematics Paul
> discusses a *lot* easier to understand.
>
> You'll have to go to the "next installment" (Part 2), and
> particularly the one after that ("An Excursion"), to see
> the new graphics.

Thanks Monz, great job!

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_25245.html#25245

>
> I've added color to the illustrations in Paul Erlich's
> "Gentle Introduction to Fokker Periodicity Blocks"
> http://www.ixpres.com/interval/td/erlich/intropblock1.htm
>
> which should make the transformational mathematics Paul
> discusses a *lot* easier to understand.
>
> You'll have to go to the "next installment" (Part 2), and
> particularly the one after that ("An Excursion"), to see
> the new graphics.
>
>
Hi Monz!

Thanks for this coloration. It makes the parallelogram charts *so*
much more appealing, rather than just lines running everyplace!

Well, I reread the "Gentle Introduction" again, and got, again, more
out of it. Everytime I read it this happens.

I *still* get "stuck" about 3/4 of the way through... but I just
don't have the math background to complete it all, but I get the
*general idea...*

Thanks again!

_______ ______ _______
Joseph Pehrson