Math as Art: Part 1

I often bounce ideas off my husband, sometimes to get his initial gut reaction to something musical, and other times I use him to verbalize concepts I have difficulty realizing in practice.  Often with the later, I see his eyes glaze over as he just nods and smiles while I talk AT him…usually once I start referring to frequencies and algorithms and nested loops.  You see, my husband has an incredible innate gift of musicality.  He understands music theory and form intuitively.  And he is a man of patience to humor my intellectual ramblings:

“Look at these two axioms and how they interrelate when they are used recursively!”

“Oh, yeah…cool. That’s really interesting.”

“I hypothesize that the resulting melody will actually be tonal and aesthetically pleasing because we, as human beings tend to like things built recursively. Just look at the Golden Mean, and fractals, and flower petals…”

…and I’ve lost him.

Now, if I were to by-pass all the jargon and formula and just play the resulting music for him, he would just GET it.  He would understand exactly what is happening and even point out weaknesses and how I might make it stronger.  He is magic and I am a mere mortal who has to explore and research and use trial and error to figure out how to make these ideas a musical reality. So, I decided to try sparing him the process so he can savor the result.  What better sounding board to use freely with little to no guilt than a blog?  Over the course of the next few weeks, I plan to use this blog to help me through my latest obsession: Fractals.

And so it begins…

Fractal: A definition

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are “the same from near as from far”. Fractals may be exactly the same at every scale,  or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself. (Wikipedia: http://en.wikipedia.org/wiki/Fractal)

Inspired by the study of weather patterns and Strange Attractors, I decided to use 2 axioms which at first glance are similar, but have very slight differences. Once repeated recursively, given a set of rules, these differences will explode into 2 very unique sets.
As an example:

Axiom 1

{A, B, A, B, C, A, B}

 

Axiom 2

{A, B, A, B, D, A, B}

Rules:

A = AB

B = ABC

C = BC

D = AC

Step 1

Axiom 1 {A, B, A, B, C, A, B} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C}

Axiom 2 {A, B, A, B, D, A, B} =

{A, B, A, B, C, A, B, A, B, C, A, C, A, B, A, B, C}

At this point, the differences have been abstracted further by the use of recursion. It is difficult to see, and even more difficult to hear the differences between the two.

Step 2

Axiom 1 {A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C}


Axiom 2 {A, B, A, B, C, A, B, A, B, C, A, C, A, B, A, B, C} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

Now, just at the 2nd iteration, we can see the similarity grow farther apart:

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

This is a very simple example, using only 4 rules on 3 variables, but it gives the general idea behind this new piece I’m working on. I’m thinking the working title “Strange Attractors” is appropriate.

-S

One thought on “Math as Art: Part 1

  1. Pingback: math as art: part 2 « Sarah J Ritch

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s