[BROOK]

Fractal Geometry and F Sharp.....

Starting with F

Studying the frequency of natural catastrophes, one of us
(K.J.H.) came to realize an inverse log-log linear relation
between the frequency (F) and a parameter expressing the
intensity of the events (M), be they earthquakes, landslides,
floods, or meteorite impacts (2), and the relation can be stated
by the simple equation

F= c/MD.

Only later did we realize that this relation has been called
fractal by Mandelbrot (3), where c is a constant of proportionality
and D is the fractal dimension. Fractal relations have
commonly a lower and an upper limit. In the case of earthquakes,
for example, Eq. 2 holds only for the interval 3 -AM
c 9, because the smallest earthquakes are not represented by
significant statistics, nor is the energy release of large earthquakes
infinite.
Mandelbrot (3) put together certain geometric shapes
whose "monstrous" forms were very irregular and fragmented;
he coined the term fractal to denote them. Those
"monsters" were considered irrelevant to nature, akin to
modern atonal music (4), until Mandelbrot suggested that the
fractal relation could be the central conceptual tool to understand
the harmony of nature.
We have been searching for a meaning of melody. Is it
tradition or convention, or is it an instinctive expression of a
natural law? Could we find a mathematical relation to describe
a melody? Could the music of Bach be mathematically
distinguished from that of Stockhausen? Could we use mathematics
to describe the evolution of music from the primitive
folk's music to the atonal music of today? If music is an
expression of nature's harmony, could music have a fractal
geometry? Which, the atonal or the classical

http://www.pnas.org/content/87/3/938.full.pdf