Intellectual Property & Creative Research

Bach-Mandelbrot Zoom

Johann Sebastian Bach was a musical master of mathematical manipulation.  Many of the compositional techniques he used have analogs in the world of classical geometry.  At least some of his music also possesses characteristics of fractal geometry.

While there is no known mathematical connection between the music of Bach and the famous fractal known as the Mandelbrot set, there is certainly a poetic connection.  Bach's compositions represented the height of Baroque sensibilities; intricacy, nested levels of adornment, and the suggestion of infinite space were compelling structural properties in the art and architecture of his day.

When listening to his music, fans of Bach often refer to a sense of endless spirals or wave-like circles-within-circles.  Such descriptions are reminiscent of the infinite, Baroque-like detail of the Mandelbrot set.  In the following animation, it is striking to see how the sense of motion we experience presents a perfect counterpoint to the unfolding of Bach's music.

Click on the link below to view the full-length version of the Mandelbrot zoom produced by Harlan Brothers for Michael Lawrence's wonderful and inspiring documentary, Bach & friends:


Mandel-Bach Journey: A Fractal Zoom into the Mandelbrot Set

This animation was featured in the film festival at the Bridges 2010, art and mathematics conference in Pecs, Hungary. A paper from the conference proceedings on the subject can be found below.

Special thanks to the following people for their contributions:

Astronomer and musician, Jeffrey Hall arranged and produced the lovely rendition of Contrapunctus IX from Art of Fugue.

Award-winning producer/director, musician and composer Michael Lawrence provided the impetus and inspiration for this piece.  His brand-new film, Bach & Friends, features a stellar array of world-class musicians.

Engineer and mathematician, Jos Leys does brilliant mathematics and physics renderings and animations.  Several of the parameters for this zoom came from Chapter 6 of his film Dimensions.

If you are interested in simply exploring the Mandelbrot set, Paul Gentieu's QuickMan is a wonderful freeware application.  If you want to design and render your own fractal animations, Frederik Slijkerman's UltraFractal is a well-documented and extremely versatile tool.


H. J. Brothers, "Mandel-Bach Journey: A marriage of musical and visual fractals." Proceedings of Bridges Pecs, 2010; pages 475-478.

H. J. Brothers, "Intervallic scaling in the Bach cello suites." Fractals, Vol. 17, No. 4, 2009; pages 537-545.

H. J. Brothers, "Structural scaling in Bach's Cello Suite No. 3." Fractals, Vol. 15, No. 1, 2007; pages 89-95.

K. Hsu and A. Hsu, "Fractal geometry of music", Proc. Natl. Acad. Sci. USA Vol. 87, 1990; pages 938-941.