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The Mathematical TouristBy Ivars Peterson |
September 3, 2008
Johann Sebastian Bach surely did not have fractals in mind when he composed six suites for solo cello several centuries ago. Nonetheless, at least one movement has the repeating structure on different scales that is characteristic of a fractal.
Harlan J. Brothers of The Country School in Madison, Conn., contends that the first Bourrée in Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this musical structure can be visualized as a fractal construction called the Cantor set, he says.
Brothers' findings appear in the paper "Structural Scaling in Bach's Cello Suite No. 3," published in the March 2007 issue of the journal Fractals.
Examining only the written score, Brothers focused on the phrasing in the first section of the first Bourée. Musical phrasing refers to the way certain sequences of notes are naturally associated with each other, Brothers says.
Brothers detected repeated use of the pattern AAB on different scales, where each B section lasts twice as long as each A section.
Analysis of the first 16 measures of the Bourree from Bach's Suite No. 3. Courtesy of Harlan Brothers.
For example, the piece starts off with two eighth notes and a quarter note (m1), repeats that pattern (m2), then continues with a phrase (m3) that is twice as long. The same pattern of short, short, long (s1) is repeated (s2), followed by a longer sequence (s3).
Analogously, the first eight measures are repeated, giving two "short" sections that are followed by a 20-measure "long" section.
"Interestingly, although Bach wrote the piece with a repeat symbol at the end of this 20-measure section, anecdotal evidence suggests that some cellists choose to perform it without the second repeat," Brothers noted in his paper. "Performed in this fashion, the Bourrée Part I exhibits a full four levels of structural scaling symmetry."
The structure of Bach's music resembles that of a classic type of fractal known as a Cantor set. Start with a line segment. Remove the middle third. Then remove the middle third from the remaining pieces, and so on. The result is a "Cantor comb."
Four levels in the creation of a Cantor comb. Courtesy of Harlan Brothers.
The hierarchical nesting of the AAB phrasing in the first Bourrée produces a similar pattern.
"The fact that Bach was born almost three centuries before the formal concept of fractals came into existence may well indicate that an intuitive affinity for fractal structure is, at least for some composers, an inherent motivational element in the compositional process," Brothers concluded.
Brothers has set about establishing a mathematical foundation for the classification of fractal music and correcting widespread misconceptions about fractal music. His efforts have revealed that musicians have been composing a form of fractal music for at least six centuries. One example is a type of canon in which different voices repeat the same melody or rhythmic motif simultaneously at different tempos.
Music can exhibit a wide variety of scaling behavior, Brothers says. He has himself written a number of compositions illustrating such properties. And he is keen to have others find further examples of scaling symmetry in what he describes as "the rich and vast body of musical expression."
Comments are welcome. You can reach Ivars Peterson at ipeterson@maa.org.
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