Mandelbrot

Aha. Here is a short but pretty good description of the relation of the
Julia set to the Mandelbrot set.
http://www2.vo.lu/homepages/phahn/fractals/julia.htm
--
John Starrett
"We have nothing to fear but the scary stuff."
http://www-math.cudenver.edu/~jstarret/microtone.html

There's a strong connection between the Mandelbrot/Julia fractals and
the Scale Tree. Can anyone see it?

These pictures (already published)

<http://www.aei.ca/~plamothe/pix/mandelbr.htm>

had been prepared on the basis of Robert L. Devaney's work. See

<http://math.bu.edu/DYSYS/papers.html>

--- In tuning@y..., Pierre Lamothe <plamothe@a...> wrote:
>
> These pictures (already published)
>
> <http://www.aei.ca/~plamothe/pix/mandelbr.htm>

Hi Pierre -- the pictures don't show up on my browser. Are the
picture files still in the correct locations? Or is it just my
browser?

Paul!
Was the quality you saw in the scale tree sharing with the Mandelbrot the fact that it looks
the same whether you look at it afar or close up?

PERLICH@ACADIAN-ASSET.COM wrote:

>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> Was the quality you saw in the scale tree sharing with the
Mandelbrot the fact that it looks
> the same whether you look at it afar or close up?

Yes, they both have these fractal-like qualities . . . but I was
thinking of a couple of things a little more specific than that. I
think Pierre is on to one of those things (but it would be great if
we could see those pictures)!

He left the links pointing to his local disk.
You can address them directly though:
http://www.aei.ca/~plamothe/pix/poire_blues.gif
http://www.aei.ca/~plamothe/pix/poire_musique.gif

----- Original Message -----
From: <PERLICH@ACADIAN-ASSET.COM>
To: <tuning@yahoogroups.com>
Sent: Monday, April 16, 2001 7:02 PM
Subject: [tuning] Re: Mandelbrot

> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > Paul!
> > Was the quality you saw in the scale tree sharing with the
> Mandelbrot the fact that it looks
> > the same whether you look at it afar or close up?
>
> Yes, they both have these fractal-like qualities . . . but I was
> thinking of a couple of things a little more specific than that. I
> think Pierre is on to one of those things (but it would be great if
> we could see those pictures)!
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold
for the tuning group.
> tuning-digest@yahoogroups.com - change your subscription to daily digest
mode.
> tuning-normal@yahoogroups.com - change your subscription to individual
emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

Oups . . . Now ok

<http://www.aei.ca/~plamothe/pix/mandelbr.htm>

--- In tuning@y..., Pierre Lamothe <plamothe@a...> wrote:
>
>
> Oups . . . Now ok
>
> <http://www.aei.ca/~plamothe/pix/mandelbr.htm>

Cool, now interpret those fractions logarithmically, as fractions of
an octave, rather than as frequency-ratios, and you have the scale
tree.

What Pierre didn't show, but undoubtedly knows, is how the fractions
simply read out the angle (as a fraction of a full circle) relative
to the point 0.25 + 0*i.

But the really interesting part is when you actually look at the
stuctures near the "bud" jutting out from the point corresponding to
a particular rational angle (as a fraction of a full circle).

Look at the little "pinwheels" and the gradation of sizes of the
various "spokes". Does it ring any musical bells?

Paul Erlich wrote:

> > <http://www.aei.ca/~plamothe/pix/mandelbr.htm>
>
> Cool, now interpret those fractions logarithmically, as fractions of
> an octave, rather than as frequency-ratios, and you have the scale
> tree.
>
> What Pierre didn't show, but undoubtedly knows, is how the fractions
> simply read out the angle (as a fraction of a full circle) relative
> to the point 0.25 + 0*i.
>
> But the really interesting part is when you actually look at the
> stuctures near the "bud" jutting out from the point corresponding to
> a particular rational angle (as a fraction of a full circle).
>
> Look at the little "pinwheels" and the gradation of sizes of the
> various "spokes". Does it ring any musical bells?

I'm sorry but I must confess that your post and Pierre's "Page normale sans titre" ("Normal page with no
title"?) diagrams are almost completely opaque to me. I can't identify anything that is so specifically
*a* "bud," "pinwheel," or "spoke" that I'm certain it's the same thing you mean. There are self-similar
buds everywhere in sizes that decrease to the infinitesimal (or would be if the image were high res). I
remember seeing pinwheels in fractal diagrams, esp. Julia set diagrams, but I don't see them here. I do
clearly recognize Pierre's diagrams as outlines of a plotted Mandelbrot set. Are the ratio plottings
somewhat arbitrary or do they relate directly to the formula used to plot the diagram? The seeds, maybe?

You're not comparing this to the infinite, self-similar division of intervals in the Golden horagrams?
That's similar in a way but the scaling is different. Self-similarity in the Mandelbrot set is on a scale
of about 10:1 if I remember rightly, as compared with phi for the Golden horagrams. Plus, it's regular,
not fractal.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@y..., "David J. Finnamore" <daeron@b...> wrote:
>
> I'm sorry but I must confess that your post and Pierre's "Page
normale sans titre" ("Normal page with no
> title"?) diagrams are almost completely opaque to me. I can't
identify anything that is so specifically
> *a* "bud," "pinwheel," or "spoke" that I'm certain it's the same
thing you mean. There are self-similar
> buds everywhere in sizes that decrease to the infinitesimal (or
would be if the image were high res).

Yup, those are the buds.

> I
> remember seeing pinwheels in fractal diagrams, esp. Julia set
diagrams, but I don't see them here.

No, you won't see them in Pierre's small diagram. You'll have to
investigate further, such as in the other links Pierre provided.

> I do
> clearly recognize Pierre's diagrams as outlines of a plotted
Mandelbrot set. Are the ratio plottings
> somewhat arbitrary or do they relate directly to the formula used
to plot the diagram? The seeds, maybe?

Here was my explanation of that. Let me know if it isn't clear:

> What Pierre didn't show, but undoubtedly knows, is how the fractions
> simply read out the angle (as a fraction of a full circle) relative
> to the point 0.25 + 0*i.

>
> You're not comparing this to the infinite, self-similar division of
> intervals in the Golden horagrams?

Nope!

> That's similar in a way but the scaling is different. Self-
>similarity in the Mandelbrot set is on a scale
> of about 10:1 if I remember rightly,

I don't think there's any single ratio, but if you have information
to the contrary, please let me know.

Without looking at underlying maths I had just seen analogy with
Stern-Brocot tree at moment I saw this site

http://math.bu.edu/DYSYS/papers.html

where we find How to count. I made immediately my pictures and post them to
Robert L. Devaney who replied

<< From your pictures, it appears that you are familiar
with the Farey tree, which is fairly well known to be
represented in the Mandelbrot set. >>

The red numbers associated with the biggest holes in poire_musique.gif are
the original ones

0 -- 1/4 -- 1/3 -- 2/5
\
\
1/2
/
/
1 -- 3/4 -- 2/3 -- 3/5

Adding 1 to each numbers, what means interpreting the sequence as an octave
of ratios, rather than a log (octave) in base 2, gives the numbers found in

http://www.aei.ca/~plamothe/pix/poire_blues.gif

1 -- 5/4 -- 4/3 -- 7/5
\
\
3/2
/
/
2 -- 7/4 -- 5/3 -- 8/5

Now, if we consider the symmetry in the figure we see that the symmetric of
a ratio a/b in original numbers is (b-a)/b. There exist an alternate manner
to obtain ratios with the same counts : the use of b/a as symmetric of a/b.

So starting with the original red numbers but with an alternate
interpretation for the limits (1 and 1/2)

1/4 -- 1/3 -- 2/5 -- 1/2
/
/
1
\
\
3/4 -- 2/3 -- 3/5 -- 1/2

and then using the multiplicative symmetry for upper numbers rather than
the additive one it gives what is shown in

http://www.aei.ca/~plamothe/pix/poire_musique.gif

4/3 -- 3/2 -- 5/3 -- 2/1
/
/
1
\
\
3/4 -- 2/3 -- 3/5 -- 1/2

We have now two octaves rather than one and the other octaves are
represented in the prolongation of subsequent bulbs.

I don't have taken time to do more than to obtain global coherent mappings
of rationals in holes of Mandelbrot set.

Pierre Lamothe

--- In tuning@y..., PERLICH@A... wrote:
> Here was my explanation of that. Let me know if it isn't clear:
>
> > What Pierre didn't show, but undoubtedly knows, is how the
fractions
> > simply read out the angle (as a fraction of a full circle)
relative
> > to the point 0.25 + 0*i.

Oh, I think I see what that means now. Before I was interpreting that
as an angle of an imaginary circle rather than a point plotted in the
set. But Pierre's spokes don't point at the same point?!

So. Ringing musical bells. The diagram is shaped like a bell. 8-)

Let's see. There seems to be a hint of otonal/utonal but that does't
correlate with the buds.

The C spokes point at the bases of buds, while the Gs an Fs point at
the ends of buds, and the A and Eb point at the sides of buds.

Boy, I'm having to restrain myself from the bud and spoke puns!

David F

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

/tuning/topicId_21184.html#21186

>
> These pictures (already published)
>
> <http://www.aei.ca/~plamothe/pix/mandelbr.htm>
>
> had been prepared on the basis of Robert L. Devaney's work. See
>
> <http://math.bu.edu/DYSYS/papers.html>

Lots of interesting interactive stuff in here, and Pierre is even
seeing pitches in the Mandelbrot! This has NOTHING to do with
April 1...

________ _____ ________ _
Joseph Pehrson