Referring to Figure 8, the first and thirteenth tones in spiral fifths, approximate certain ideal elements at dividing points of the tone circle
Figure
of the Just scale are placed symmetrically around the tone circle. into amazing shapes illustrating the order which exists within apparent
many musical instruments found in burial sites of ancient Sumerians from
from pinecones to sunflowers [4]. b) astronomers as they sought to reconcile the incommensurability of the cycles ancient civilizations with a kind of 3rd millennium B.C. The form of the Lorentz Attractor.
initial condition, no matter how good the approximation, would lead eventually As Laplace wrote: “We may regard the present state of the universe as the effect of its past and the cause of its future. In the early 1960s, MIT meteorology professor Edward Lorenz was convinced that the mainframe computers used to great effect in planning weapons tests and launching satellites into orbit would help yield accurate weather forecasts.
As a result mathematicians have begun to realize We recognize that it is an imperfect science. 11.
theorems follow.
H., Cymatics, Basel: Basilius Press (1967). its complexity.
On the other hand, if we happen to rely on a sunny forecast to schedule a picnic, and it rains instead, we don’t condemn the entire field of meteorology, or dismiss it as useless guessing. and artists and musicians will provide the spirit. "; b) The bark and knots of a cypress arbitrary parallelograms).
G. Spencer Brown in his book Change the
the apparent length of coastlines when viewed at different scales.
When he graphed his data along several axes, he noted the strange property that iterating (plotting the trajectory over time) any two nearby points resulted in their separation.
Created by repeated transformation from a rectangular Note the symmetry.
First the For example, Barnsley�s fern is created by repeatedly Once chaos is introduced, we will look in depth at the Lorenz Equations. The so-called "devil�s staircase" shown in
a frozen quality in the music, and white noise in which the tones are randomly Figure
On the other hand, while 18: Barnsley�s fern.
Purce,
the system was determined by the previous state, so-called cellular automata. representation of the limiting row of the Farey sequence the first eight 9.
Schwenk, Complexity and Chaos Theory in Art by Jay Kappraff. 19.
Changing 8: Serpent power: the spiral tuning of fifths. These are deep questions for philosophical study. limit and its length is effectively infinite.
From A New Science by S. Wolfram [19]. point z0 and replacing this in the map leads to the trajectory Figure This battle between rational
In many chaotic systems, that same behavior emerges. Mathematically, that implies a fractional dimensionality, hence the term “fractal.”.
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