Mandelbrot Parameters

The Mandelbrot fractal set is obtained by applying recursively the formula

f(z) = z2 + c

where c is a complex point in the rectangle (-2, -2) - (2, 2), or -2-2i to 2+2i, if you prefer.

For a given c, the sequence

c, f(c), f(f(c)), f(f(f(c))), ..., fk(c), ...

is called the orbit of c.

For some values of c, their orbits diverge, which means that the elements of the orbit go farther and farther from the center and eventually leave the rectangle (-2, -2) - (2, 2). To generate an image of the Mandelbrot set, computers usually calculate how many iterations it takes for a point to diverge, and according to that, they assign a different color to each point. If an orbit doesn't diverge, the point is given a default color (usually black).
 

The FMG program extracts information from the orbits and also from the "color" of the points in a path (which is only a straight line). When you select "Mandelbrot" as the fractal type for a microsequence, you get the following view:


 

In the right side is a picture of the selected region in the complex plane, which initially is the rectangle  (-2, -2) - (2, 2). You can view different parts of this region by zooming into the fractal.

For example, use the "contract" button  to make the zooming region smaller.

You'll see something like this:


 

The white rectangle defines the zooming region. Press the Zoom button to select the new region. Or you can also move the zooming rectangle with the arrow buttons. For example, make a smaller zooming rectangle and move it a few steps up:

Now press the Zoom button, and you'll see something like this:


 

You can do lots of zooming before you find a "blank" area. This is another property of fractal sets: they are infinitely complex. When you hit a blank area is just because the computer doesn't have enough precision to distinguish all the points in that area.
 
 

Selecting orbits

If you want to extract microsequence information from a Mandelbrot orbit, first select the region where the first point of the orbit will be and make a zoom to that region. Then select "Orbit" and click on the "Fix Point(s)" button. The current orbit will appear and you can use the arrow buttons to move the initial point. All the points of the region will also be plotted (if they're inside the visible region). When you've found an orbit you like, click on "Fix Point" once more to select the orbit, and then click on "Save current" to save any changes to the microsequence. Here's an example of an orbit:


 

The microsequence is constructed by taking the distance from the points in the orbit to the initial point; thus the first number in the microsequence is zero (which, if mapped to a melody, corresponds to the root note of the global scale). Orbits can have many different forms. They're good for melody lines and arpeggios since they are usally well-structured and repeatitive. But remember that when you move the initial point far from the center of the complex plane, the orbits will turn chaotic and lead to strange melodies.
 
 

Selecting straigth paths

You can also extract information from a picture of the Mandelbrot fractal on the complex plane. As you zoom into the Mandelbrot set, you will see some areas with smooth color changes and other areas with abrupt color changes. You can draw a straight line and convert the color information of the points below the line in numbers. If the colors change smoothly across the line, so will the numbers in the microsequence. If the colors change abruptly, you'll get randomish sequences.

Select a "Line" path and click on "Fix Point(s)". Move the first point with the arrow buttons and click on
"Fix Point #1". Then move the other extreme and click on "Fix Point #2". As you move the second extreme, you can make the line pass through smooth or rough regions. For example, the following line goes through the smooth upper regions, to the rough middle regions and into another smooth region. Those changes will reflect in the melody or dynamics of the blocks.


 

Remember that the Mandelbrot fractal is infinitely complex, so the smoothness of a microsequence generated by one of these straight paths depends on how many points of the line we take to construct the sequence. The line is divided in a number of points equal to the period of the microsequence, so it is possible that a sequence with a short period turns into an almost constant sequence even if the line crosses a rough region. My suggestion is to experiment with different periodicities when using this kind of sequences.
 

< Back to Microsequences