Waltz of the Cycloids

Mathematical vocabulary isn't something that I normally associate with classical music, so I was caught by surprise recently when an announcer on a local radio station gamely tried to explain the term "cycloid."

Why? The announcer was introducing a piece of music called the Cycloids Waltz (Cycloiden Walzer), written by Johann Strauss Jr (1825-1899). Strauss had dedicated his composition to the "Gentlemen Technical Students at Vienna University" and conducted its debut at their ball in the Sofienbad-Saal on Feb. 10, 1858. The apt title not only came directly out of mathematical vocabulary that would be familiar to the students but also hinted at the whirling movements of a waltz.

A cycloid is the path traced by a point on the rim of a circle that rolls (without slipping) along a straight line.


Galileo Galilei (1564–1642) studied the cycloid around 1599 and gave the curve its name, using the Greek word for "circle" as its main element.

A segment of a cycloid also represents the shape of the curve along which a bead sliding from rest and accelerated by gravity will slip from one point to another in the least time, a problem originally posed in 1696 by Johann Bernoulli (see the brachistochrone problem). Moreover, a bead sliding on a cycloid will exhibit simple harmonic motion, with a period independent of the starting point.

Galileo suggested that the cycloid would be the strongest possible arch for a bridge, and many concrete viaducts do have cycloidal arches. Cogwheels often have cycloidal sides to reduce friction as gears mesh.

In a chapter on the cycloid in Martin Gardner's 6th Book of Mathematical Diversions from Scientific American, Gardner notes that the cycloid has been called the "Helen of geometry," not only because of its beautiful properties but also because it has been the object of so many historic quarrels between eminent mathematicians.

The version of Cycloiden, Walzer, Op. 207, that I heard on the radio was recorded for the Marco Polo label by the Slovak State Philharmonic Orchestra, Kosice. The author of the accompanying album notes remarks that the title page design for the first piano edition of the piece featured a circle, encompassing the name of the work and its composer and the dedication to the students. The designer then surrounded this feature with representations of various tools of the technician (set square, compasses, theodolite, and so on) and depictions of several engineering achievements, such as the steamship, steam engine, blast furnace, and plough. But no cycloid.

Strauss' composition isn't the only one in the musical literature that refers to the cycloid. The Library of Congress collection of American sheet music includes Cycloid Polka, written by Charles Kinkel and published in 1873. Unfortunately, I can't play the piano or find a recording, so I have no idea what this piece sounds like. But I do know there's usually a lot of lively circling when you dance a polka.

A Fractal in Bach's Cello Suite

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 Bourrée 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."