Hunting Bach’s fractals

Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Mathematician Benoit Mandelbrot invented the word “fractal” in a 1975 book on the subject, and his landmark 1982 book The Fractal Geometry of Nature, which records the geometric patterns’ prevalence, is largely credited with popularizing them.

Because nature is full of fractals, the patterns are highly familiar. Trees, rivers, beaches, mountains, clouds, seashells, hurricanes, and so forth. Abstract fractals, such as the Mandelbrot Set or the Sierpinski Triangle, may be produced by a computer repeatedly solving a simple equation.

Fractal animation of the Sierpinski Triangle ©Wikipedia

A fractal is sometimes defined as having self-similarity, which indicates that same (or nearly identical) patterns emerge whether the form is examined up close or from a distance. That is, the portion appears to be the whole, and the whole appears to be a part. Not surprisingly, music scholars have tried to investigate such patterns in compositions, despite the fact that the mathematical roots of music are widely established.

For example, physicists Richard F. Voss and John Clarke (University of California) studied numerous audio recordings of music and discovered scaling patterns in loudness fluctuations and melody fluctuations of popular music on the radio.

Identifying fractals in music, on the other hand, needs a different technique than perceiving them in images.

Harlan Brothers, a jazz guitarist, composer, and mathematician from Branford, Connecticut, discovered numerous forms of scaling when researching the function of power laws in music, including self-similarity with respect to duration, pitch, interval, theme, and structure. He has been looking for fractals in Bach’s work as well.

“Unlike a picture, which is all laid out so that you can instantly see the structure, music is fundamentally a serial phenomenon,” Brothers says. “With music, the whole piece takes shape in your mind. This makes it more challenging to identify the self-similarity.”

For much over a decade, Brothers reported on fractal structure in the phrasing of notes employed by Bach to construct his Cello Suite No. 3 in a research published in Fractals in 2007. Patterns of long and short notes within measures resurfaced inside that piece as patterns of long and short phrases at greater scales. Brothers noticed that the suite’s self-similarity bore a remarkable resemblance to the Cantor Comb, a representation of a historical fractal known as the Cantor Set. (which Georg Cantor described in 1883, a century before Mandelbrot coined the name “fractal”).

The top of this drawing shows a Cantor comb, which depicts self-similar patterns repeating at different scales on different lines. The lower diagram depicts the distribution of note durations in a 16-measure excerpt from a cello suite by Bach. The two patterns are similar. © Harlan Brothers

One of Harlan’s early discoveries was that musicians have been creating fractal music for at least six centuries. Many of the great Flemish composers who created the technique of the mensuration or prolation canons, such as Johannes Ockeghem and Josquin des Prez, were familiar with motivic scaling. This form of canon is distinguished by a melody or rhythmic theme that is repeated in multiple voices at different tempos at the same time. To be clear, not all mensuration canons are fractal; basic conditions must be satisfied in order for an item to be classified as such.

Nonetheless, a percentage of predictability is given by all these patterns. In search of a definitive conclusion, Daniel Levitin analyzed the rhythm spectra of 1,788 movements from 558 compositions of Western classical music.

“Because music has a beat and is based on repetition, it has been said that ‘what’ the next musical event will be is not always easy to guess, but ‘when’ it is likely to happen can be easily predicted.”

Distribution of rhythm spectral exponents for musical genres ordered from largest mean exponent to smallest. Larger exponents indicate correlations over longer timescales, and hence more predictable rhythms (vertical gray arrow). Circles are mean exponents, and error bars are 95%

The differences in the predictability for different genres of music and composers are more relevant than the fractal structure of the rhythms. Composers with a more diverse style, like Mozart or Joplin, wrote music with more variable spectral exponents than Beethoven or Vivaldi. Ragtime and madrigals, for example, are considerably less predictable than symphonies or scherzos. The ensuing variations in rhythmic predictability would allow them to uniquely identify their compositions and differentiate them from those of their contemporaries.

Looking at the time of musical notes, one can see that they are not as regularly spaced as one might anticipate and, in fact, have a fractal structure. The researchers discovered that musical rhythm follows a 1/f pattern, the majority of the music we listen to being in the 1/f pattern.

It is characterized as a power-law decline in the correlations of the pitch with time, and it has the perfect balance of pattern and unexpectedness, as well as being appealing to the human ear, the shape of the curve defined by 1/f music having a fractal shape. 

While the subject is being researched further, it is clear that our brains appear to be particularly well suited for processing some types of sounds and find those with a fractal structure to be particularly appealing.

Science and Culture: Hunting fractals in the music of J. S. Bach, Stephen Ornes

Published: July 2014, Proceedings of the National Academy of Sciences
DOI: 10.1073/pnas.1410330111


Musical rhythm spectra from Bach to Joplin obey a 1/f power law, Daniel J. Levitin, Parag Chordia, and Vinod Menon

Published: March 2012, Proceedings of the National Academy of Sciences
DOI: 10.1073/pnas.1113828109

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