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MA 279, Fall 2018: Fractals and Music

Daksh Jotwani, Daniel Atallah, Noah Talbot, Zoe Phillips

Introduction

“If music is invented to mimic a certain harmony in nature, the mountains are sounds; rivers are lyrics” - Zhi-Yuan Su and Tzuyin Wu

We have always been fascinated by the infinite number of patterns surrounding our environment. Patterns are a huge part of our daily observations; from uniformly distributed branches on a tree, to clouds colliding to form familiar objects, we seek to find structure in everything we see, hear, and feel around us.

A subset of patterns that we observe has been classified as fractals. Fractals are generally depicted visually, since we primarily rely on vision to observe our surroundings. These patterns, however, are also observed in a multitude of natural phenomena, and we seek to explore aural examples of them.

Music is a sequence of temporally structured sounds which are sonically appealing to the human ear. Since music is sequential, aural, and possesses a well-defined structure, we believe that exploring the structural patterns of music could lead to observation of some interesting fractals and applications of them. Therefore, we have chosen to explore and document some fascinating fractals which have been observed in the field of music.

Background

Fractals are infinite self-similar patterns. These patterns are generated by non-terminating sequences, and are usually defined recursively. Fractals are well-observed in nature, and hence are very familiar, as shown by the example below:

The tree above is self-similar, since a new tree can be formed by snapping a branch off the tree. The tree can also be infinitely generated using the following procedure: each branch must split into two new branches. This procedure is also recursive, and is defined concretely as a binary tree in many applications of modern mathematics and computer science.

Fractals can also be generated from abstract mathematical constructs. Two famous examples of these are the Mandelbrot and the Cantor set, which are both briefly explored below:

The Mandelbrot Set is the set of all complex numbers c such that the recursively defined sequence S_c:

In Figure 2, the Mandelbrot set is visually depicted on the complex plane in black, while the other colors depict values of c for with S_c diverges from a predetermined neighborhood n in k iterations, i.e. z_k > n. The visual format of the Mandelbrot Set sparked interest amongst many mathematicians, and hence was formally defined as a fractal.

The Cantor Comb, generated by the Cantor Set, is another famous example of a fractal generated from abstract mathematics. Surprisingly, a similar variant of this fractal is observed in various forms of music, one of which we will analyze and discover in greater detail. The Cantor comb can be generated as shown:

The Cantor Comb is a visual depiction of a Cantor set. The process involves generating a set of intervals as defined recursively below:

Each interval in I, if plotted sequentially, would generate the Cantor Comb as shown in Figure 3. The fractal above also looks very similar to temporal notation of notes in sheet music. We will explore this set in later parts of our exploration, where we analyze a section of Bach’s Cello Suite 3 and explore similar substructure by removing notes from a simple melody.

We will not require significant knowledge of music theory for this exploration. Whenever a musical term is used, we will provide a brief definition and examples to provide examples.