Music Mathematics Part V. Timeless Music Theory: more composing with fractals on purpose

Read Parts 1 through 4 first  

The way the quartet used a single simple pattern in Part 4 to build to more complicated patterns is similar to the best known example of a simple fractal pattern, the Koch Snowflake. To make the snowflake, you take a equilateral triangle, and then place a triangle of one third the size in the middle of each edge: this makes a hexagram, a.k.a. the star of David. Do this again, with each of the six edges feature the addition new one third size triangles, and the pattern looks like a snowflake. Repeat ad infinitum or into you get tired of doing this ...

As the scales of the new triangles become smaller, tinier rulers are needed to measure the perimeter of the snowflake. The best way determine the length of the edge of the snowflake, like the length of the coastline, is to use the trick of a fractal or Hausdorff dimension between 1 and 2.

We have the same kind of challenge with describing the pattern of the fractal Schoenberg piece. A stopwatch measures the music at 1 minute and 41 seconds long, but the patterns are in four time scales, and if we had more instruments, it would be at still more.

The quartet piece is generated from 4 time scales of 5 notes, but more time scales or notes make things richer, but so complicated that for live performers, you need to take some of the music away, like making a sculpture by removing everything that doesn’t look like a horse.

Here’s a richer fractal composition, Fractal on the Name of Haydn, played by Walter Hilse on the organ, made from multiple time scales of a short phrase from Maurice Ravel’s Minuet on the Name of Haydn.

Here is another fractal piece, Fractal on the Name of Bach, played by Steve Beck, with multiple time scales based on the way musicians often spell Bach’s last name; Bb A C B natural.