The Logic Off Topic list

Thoughts from the mind of Tim.Dylla@..., 11-07-2002:

>If you really want to know more about fractals, I think best would be
>visiting your favourite Mathematics-Professor ;o). Maybe Hendrik van
>Veenstra from the LUG knows more about it ... (if I remember right, he's
>math-teacher)

Correct, about the math-teacher :).  Actually, the very 1st software 
I ever wrote (on the Atari) were fractal generating programs.  Some 
pictures took more than 24 hours to render (in B&W) -- now just one 
or two seconds on a G4, in millions of colours :-))  That's progress 
for you...

A "fractal as image" is basically a picture that has a lot of 
self-similarity.  I.e if you zoom in on a part of the picture, it 
will look very much like the entire picture.  If you zoom in even 
more... ditto -- up to any level of zooming.
This is contrary to "classical" mathematical objects: if you zoom in 
enough on the edge of a circle it will become a straight line and not 
a circle at all.

The best known example of a fractal is a coastline: seen from outer 
space, a coastline looks like a wiggly line.  Get closer and see just 
1 km of the coast: still a wiggly line.  Etc, until you reach the 
level of individual pebbles and rocks: still a wiggly line.

As a mathematical object, most fractals consist of just very few (2 
or so), mostly very simple functions. The core idea is to iterate the 
functions multiple times and observe what happens.  I.e. fill in 
start values in the functions, calculate the result, then fill in 
this result in the functions, calculate the next result, and keep 
doing this for maybe 1000 times.  What will now happen to the end 
result?  Will the function-outputs become very large? Very small? 
Will they cycle through some fixed values?
Even for very simple functions, the observed behaviour is extremely 
complicated.  Filling in a certain start value like 1.2340 may lead 
to a huge end-result (like a few billion), while filling in a start 
value that is only slightly different (1.2341) leads to an end result 
of almost zero.

The nice coloured pictures that everyone has come to know as fractals 
are nothing but maps of this behaviour.  Each point in the picture is 
considered to be a point in the Cartesian plane -- i.e. a simple 
(x,y)-coordinate pair like we all know from highschool.  X and Y are 
treated as described above (i.e. they're both filled in in functions 
that take 2 arguments, x and y, and produce 2 outputs, the "next" x 
and y).  Depending on the behaviour after multiple fill-in, 
calculate, fill-in, calculate loops, the point gets a colour.  Do the 
values become huge?  Colour it red.  Do they move towards zero? 
Colour it black.  Etc.

 From a philospohical point of view, fractal math (and chaos math in 
general) is extremely interesting.  For some 2500 years, we (western) 
humans have tried to explain nature in terms of Euclidian geometry: 
straight lines, circles, squares.  Looking at just a single leaf from 
a tree it's painfully obvious how inadequate circles and lines are to 
explain its shape. It was therefore thought that the "laws of nature" 
would be extremely complex since the shapes of nature are so complex.
Fractals as mathematical objects had been known since the early 20th 
century at least, but due to the huge amount of (simple) calculations 
involved, no-one had ever seen a fractal image until somewhere in the 
60-ies, when the 1st fractal images were generated on computers. 
Then all of a sudden images appeared resembling seahorse-tails, 
leaves, ferns, wiggly coastlines and the like.

So apparently achieving "natural complexity" _was_ possible without 
the need for terribly complex formulas.  In fact, the formulas used 
were so simple, that nowadays you could even let a 15-year old with a 
computer make fractals.  I'll leave it to you to figure out the 
impact this had (and still has) on our thinking about natural laws, 
the complexity of (human?) nature and the like.

MUSIC -- to keep somewhat on topic.  I don't know much about 
fractalmusic programs.  Tried many of them (even tried to write 
some), and didn't like most of what I heard.  I think it's obvious 
that once you have simple programs generate complex shapes, people 
will want to try to adapt that to other areas of human interest, such 
as music.  Wouldn't it be possible, using similar algorithms, to 
create complex sounds?  Up to a point the answer is probably: yes. 
You can create "fractal waveforms" and use those instead of plain saw 
or sine waves -- and they needn't sound that bad at all.
However, I also tried my share of fractal composing programs -- i.e. 
software that not just produces a single 10ms fractal wave, but a 
complete "composition" based on fractal principles -- and up till now 
they've all been rather disappointing.
That's not amazing, I think: music in general is highly structured 
and non-chaotic.  Fractals, by their very nature, _are_ chaotic, and 
so don't lend themselves that well to "automated composition".  All 
imo of course.

Think about it: how would you translate the "graph" of a coastline 
into music?  Would it be interesting music?  The human eye is rather 
good at seeing structure on different scales simultaneously -- you 
can see structures that are just a few millimeters big and at the 
same time see structure on the 2-meter-canvas scale.  Add to that the 
fact that visual perception is an "all at once" thing, and not 
something that develops lineair in time.
I think the average human ear is less capable in that respect. 
Structures on the millisecond scale are hardly perceived (as 
_structures_), and a structure on the 2-hour scale must be _very_ 
obvious before you'll notice it as such.  I think we're best at 
medium-sized structures, at the multiple seconds/few minutes level 
(which probably has to do with our lineair-in-time perception of 
music).  Since fractals mainly deal with "similar structures at 
various sizes", one can wonder if sounds lends itself very well to a 
fractal-treatment.

Still that doesn't mean fractal principles are totally unsuited for 
making music.  Even in music there's an amount of chaos present -- 
chorusses repeat but are not exact copies, each performance of a 
piece is slightly different, etc, etc.  Plenty of "structured chaos" 
there.  Maybe we just have to find the right perspective before being 
able to apply fractals in music in a meaningful way.

OK, end of lecture for today :-).

-- 
Hendrik Jan Veenstra  <h@...>
Omega Art: http://www.ision.nl/users/h/index.html