Mathematics of step and tap dancing

SUMMARY: Step and tap dancing each involve rhythms and combinations that can be analyzed mathematically.

Step dance is a type of dance focusing on foot movements. It de-emphasizes other spatial dance aspects—hand and body movements—and repositions dancers relative to the ground to form movement patterns. There are forms of step dancing in several cultural traditions. Argentinian Malambo folk dancing, often associated with nomadic horsemen and cowhands called gauchos, is not formally choreographed. It is often performed as a competition between men who use brushing, scrubbing, and stomping motions in rhythmic patterns to create complex performances. The tradition of stepping in Black American culture, tapping and clogging in Irish culture, and various traditional dances performed in Cherokee cultural events also present important examples of mathematics in dance.

The movements of these percussive dance styles may be performed by a single dancer or choreographed among several dancers. Tony Award-winning choreographer and dancer Danny Daniels (1924-2017) noted that while an individual dancer may improvise, groups must be coordinated. The rhythms and counts for the dances he designed and performed on Broadway could be organized and detailed using mathematically based musical notation. Dance theorist Rudolf Laban (1879-1958) used ideas from various fields, including crystallography, when he modeled dance dynamics.

Scientists and dancers continue to develop notation and models to express human movement in tap and other dances. Dance algorithms may help create natural robotic movement. Dancer Gregory Hines (1946-2003) said, “My style is part choreography, part improvisation. That gives me a chance to show people the possibilities of tap dancing, which, at its heart, is mathematics with endless possibilities.”

Ratio and Proportion

Several ratios related to music and choreography determine movement in step dancing. Music time signature is written as a fraction, with the denominator signifying the size of the notes used and the numerator signifying the total length—in such notes—of a bar, which is the unit of music. For example, traditional music for Irish slip-jig has a 9/8 time signature in the note pattern: quarter, eighth, quarter, eighth, dotted quarter (three-eighth). The five notes in the time signature correspond to two-and-a-half dance steps per bar, with long, graceful slides between the steps.

The formula for a dance includes the number of bars in each repeating cycle (sometimes performed symmetrically), first for one starting foot and then the other. For example, a song with forty bars may be choreographed to include five step cycles, each spanning eight bars. Another ratio important for step dancing is the tempo of music, measured in beats per minute (bpm). Dancing competitions specify the tempo range for each type of dance. For example, a single jig must be 112–120 bpm. Tap dancers of the past used their signature “time steps” (particular combinations of taps) to communicate the tempo to the accompanying band.

Patterns and Improvisation

In step dances, themes are expressed using sequences of the basic elements or steps. For example, common elements in tap dancing include shuffles, flaps, pullbacks, wings, and stomps. These sequences may be strictly choreographed from beginning to end, sometimes with repeating patterns or permutations of shorter elements, which can be repeated by any dancer who has learned the sequence. Improvisation allows the dancer to rearrange basic elements in ways that may appear random to the casual observer.

Some step dance music has built-in departures from the standard bar structures. For example, Irish stepdance “crooked tunes” may have seven-and-one-half bar parts in addition to eight bar parts. Step dance patterns have multiple levels: steps within a bar, combinations of steps spanning multiple bars, and patterns of these step combinations. Order and perceived randomness can be manifested at all levels. These elements of dance support the neurological functions needed to excel in many fields, like engineering, computer science, mathematical modeling, logistics, and data science.

Dance-Dance Revolution

Dance-Dance Revolution (DDR) is a step dancing video game. The goal of the game is to match the pattern of steps on the screen and their rhythm on the special gaming pad with four or eight foot positions. The combination of visual, audio, and kinesthetic representations of the same rhythm have kept versions of the game popular worldwide since its release in 1998.

Later versions of DDR use a mathematical visualization of multi-dimensional data, called radar diagrams, to rate the difficulty of individual dances. The variables describe different characteristics of the dance, such as steam (the density of steps) and chaos (the number of steps that do not occur on beat). These games are sold in formats that are compatible with many modern gaming platforms like Xbox, PlayStation, and virtual reality programs.

Dance and Education

Dance is also a useful teaching tool in mathematics, patterns, puzzle solving, and strengthening memorization in adolescent students. For example, programs like Dance Equations use dance's time, space, force, and body elements to allow students to explore patterns and learn mathematical operations using repetitive gestures. Malke Rosenfeld is a well-known math and dance instructor who designs curriculums that harness the power of math and dance to improve students' understanding and abilities.

Bibliography

Apostolos, M. K., et al. “Robot Choreography: An Artistic-Scientific Connection.” Computers & Mathematics with Applications, vol. 32, no. 1, 1996, pp. 1–4, doi.org/10.1016/0898-1221(96)00081-8.

Becker, Kelly Mancini. Learning through Movement in the K-6 Classroom: Integrating Theater and Dance to Achieve Educational Equity. Taylor & Francis Group, 2023.

Farmer, Melanie A. "An Engineer Tap Dances His Way to Tackling Big Data." Columbia University, 9 May 2017, news.columbia.edu/news/engineer-tap-dances-his-way-tackling-big-data. Accessed 1 Oct. 2024.

Maletic, Vera. Dance Dynamics: Effort and Phrasing. Grade A Notes, 2005.

"Mathematics in Dance." Dance Teaching Ideas, 12 Apr. 2021, danceteachingideas.com/mathematics-in-dance. Accessed 1 Oct. 2024.

Sethares, William. Rhythm and Transforms. Springer, 2007.