Math as Art: Part 3

For the past 3 days, I have been obsessing over the letters: ABCD.










Every axiom I used combined with every “clever” set of rules, produced the most monotonous dribble I have ever heard. I was ready to admit defeat and go back to writing cute little melodies when I realized I had forgotten the most important factor!!!  When I was first conceiving of this project, one of the first thoughts I had was this:

How many unique pitch classes at a minimum will allow a piece of music to be interesting?

My answer? It depends.

One pitch is all that is necessary if the piece is minimalist and the goal is to explore the timbral and sonorous  shades to a single pitch.

Two pitches are all that is necessary if your goal is to exploit the relationship between the two.

However, if your goal is to write a melodic, tonal piece of music inheriting the functions of the classical western tradition….

No less than Five pitches are necessary.

I had assumed this intuitively, but had not performed any experiments to prove it for myself. Well, I can assure you that the last 3 days proved to me without a doubt that no less than 5 pitches are absolutely necessary to create a piece of music that is melodic, tonal, and following in the traditions of western classical music.  With my new assumption, I set out creating a new set of 5 pitch axioms and a set of rules which both imitated formal structures from classical music.  Remember, my goal is to combine classical mathematics and classical music theory in their pure forms to reach my hypothesis that it can be done without threatening either.

So, I chose to use a musical form that is self-referential for the axioms: Rondo




For the rules, I used the idea of building harmonic structures:

ruleA = “AE”;
ruleB = “ACE”;
ruleC = “BD”;
ruleD = “CBDAE”;
ruleE = “EA”;
What resulted was a two-voice counterpoint that has a clear tonal center, with clear melodic themes that repeat and interact without ever becoming monotonous.

I’ll let you hear a short excerpt for yourself:

Now, to “wash” it with my human hand, adding intuitive musicality such as dynamics, phrasing, rhythmic variation, and articulation.


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.