Order in Chaos

J.S. Bach's Toccata and Fugue in D minor is arguably Mankind's finest musical achievement. It is a combination of elaborate complexity superimposed on a simple underlying structure thus creating a symphony of sound that has captivated listeners for centuries.

This remarkable YouTube video allows you to see the Fugue's beautiful mathematical structure in its complexity and in its simplicity and may help explain why it is so compelling - perhaps because it mirrors, in musical form, the most fundamental design pattern found in nature: elaborate, seemingly chaotic complexity that is actually highly-ordered patterns built on simple underlying structures.

The Toccata and Fugue in D minor

To better understand how complexity and simplicity function in nature to create beauty, a look at fractal geometry may be of use.

Fractals are geometric shapes that are based on equations that undergo iteration, a method of feedback based on recursion (a process of repetition). These simple equations create remarkably beautiful structures when the results of iteration are graphically represented. One characteristic of the geometric shapes is that they are self-similar, meaning that each part is a replica of the entire structure. This Mandelbrot fractal is a classic example of fractal geometry and very clearly illustrates the phenomenon of self-similarity:

Each part is a microcosm of the whole

Depending on the base equation, the process of iteration creates an incredible variety of structures. Many of them seem to replicate recognizable natural objects and give some insight into how natural forms emerge.

On a small scale:



On a much smaller scale:

Neurons in a mouse's cerebral cortex

On a medium scale:




And even on a cosmic scale:



So perhaps it is clear now why the Fugue is so appealing. The question of how Bach composed this masterpiece (assuming he did, as there are questions about its origins) must be left for another time. Perhaps he was merely channeling the creative energy of the Cosmos...

Until then, enjoy the Fugue in all its glory.