15
Oct
12

Math as Art: Part 2

Immediately following part 1 of this blog series going live, I received an assortment of cool links to and recommendations for other related projects/books/pieces. I have to say I was very happy to see the interest in this topic! What I wasn’t prepared for was the reaction I would get from my Professor{Discrete Structures, Network Theory, Cryptography, Complexity Theory}.  While excited by the project and the idea of expressing abstract mathematics through sound/music, he had issue with my choice of algorithm:

The Mandelbrot set M is defined by a family of complex quadratic polynomials

P_c:\mathbb C\to\mathbb C

given by

P_c: z\mapsto z^2 + c,

where c is a complex parameter. For each c, one considers the behavior of the sequence

(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots)

obtained by iterating P_c(z) starting at critical point z = 0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points c such that the above sequence does not escape to infinity.

A mathematician’s depiction of the Mandelbrot set M. A point c is coloured black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

More formally, if P_c^n(z) denotes the nth iterate of P_c(z) (i.e. P_c(z) composed with itself n times), the Mandelbrot set is the subset of the complex plane given by

M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \mathbb N, |P_c^n(0)| \le s \right\}.

You see, the Mandelbrot Set and fractals in general are considered “pop” math.  My complexity (and primes) loving Professor brought to light an important topic: Does pop have any place in high art, and if so what is it?

It seems like an old hat that has been worn many times, by Andy Warhol, by Roy Lichtenstein, by Michael Daugherty, and many other prominent artists and composers.  I think, for me the better question is: Where does COMPLEXITY fit into my practice?

If my goal with this whole thing is to try to find a way to express mathematics in sound, does it matter what function/algorithm I chose?  Now, I understand the desire for an elegant solution to this problem. I appreciate the beauty in abstract mathematics but I try to follow a simple programmers crede: Keep It Simple Stupid (KISS).

This doesn’t mean to limit yourself to remedial tasks but to only use the level of complexity that is necessary for the task.  I chose this algorithm and topic in mathematics as an entry point to the greater problem of expression of mathematic properties in music.  Fractals are familiar and even easy to understand by the average person. My goal is not to alienate but to share this abstract theory that math and music can co-exist in their pure forms without the degradation of either.  Perhaps I’ll gain enough trust to take my audience on the long road to primes…but not today.

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3 Responses to “Math as Art: Part 2”


  1. November 26, 2012 at 2:53 pm

    HI Sarah,

    Have you considered the colored mandelbrot set? Since the graph (when mapped to X,Y) exists on the Real numbers within the disc r=2, with the z-axis representing the values of the polynomial Pc, coloration maps the relative z-order values in ranges to various RGB colors.
    I can see this tonally being mapped either specifically to scale ABCD…. , or maybe also mapped to Hz frequencies. Maybe the “lake” of outlying values (outside the graph), and the “center” (inside the graph) being boundaries on the frequency range. It might be possible to derive a polynomial function on z : H(z) that defines the frequency for each value, maybe ranged by rounding?
    In thinking about this, since a melody is linear temporally, there needs to be a way to “linearize” in time the z-values… so maybe an iteration across the X,Y range of the graph, scanning across the graph like television electron-rays, with each iteration occupying a time-slice, T milliseconds (100ms, 200ms, other?).

    There are several variations of this on google search: mandelbrot set music

    In general, this is really really cool! This could lead to a whole album of chaos-based music : Mandelbrot, Julia, etc.

    • November 26, 2012 at 3:05 pm

      Hi Peter! You have great suggestions…I’m going to think about that! I have another piece coming up that I am working on….thinking Primes might be fun. Oh, and the Julia set is definitely a good one as well.

      • November 26, 2012 at 3:29 pm

        Ah, yes, Primes. There has been much work around graphing primes and variations, and a bit of music work as well. This is so cool, and I could totally lose myself in researching and learning this stuff.

        Hope to hear more about your work!


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