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Permutations and Combinations

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Encyclopedia of Mathematics & Society

Step and Tap Dancing

by Maria Droujkova

Category: Arts, Music, and Entertainment.

Fields of Study: Fields of Study: Communication; Geometry; Representations.

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

Step dance is the type of dance focusing on feet movements. It de-emphasizes the other two spatial dance aspects—hand and body movement—and repositions dancers relative to the ground to form movement patterns. There are forms of step dancing in several cultural traditions, such as Malambo from Argentina, Irish stepdance, African-American stepping, and traditional Cherokee dancing. Related forms include clog and tap dancing.

The movements of these styles of percussive dance may be performed by a single dancer or choreographed among several dancers. Tony Award–winning choreographer and dancer Danny Daniels noted that, while an individual dancer may improvise, groups must be coordinated. The rhythms and counts for the dances he designed or performed on Broadway could be organized and detailed using mathematically based musical notation. Dance theorist Rudolf Laban 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 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

There are several ratios related to music and choreography that 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 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 that has 40 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, single jig must be 112–120 beats per minute. 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 take basic elements and rearrange them in ways that may appear to be 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.

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 around the world 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 amount of steps that do not occur on beat).

Further Reading

1 

Apostolos, M. K., M. Littman, S. Lane, D. Handelman, and J. Gelfand. “Robot Choreography: An Artistic-Scientific Connection.” Computers & Mathematics with Applications 32, no. 1 (1996).

2 

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

3 

Sethares, William. Rhythm and Transforms. New York: Springer, 2007.

Citation Types

Type
Format
MLA 9th
Droujkova, Maria. "Step And Tap Dancing." Encyclopedia of Mathematics & Society, edited by Sarah J. Greenwald & Jill E. Thomley, Salem Press, 2011. Salem Online, online.salempress.com/articleDetails.do?articleName=Math_140636901406.
APA 7th
Droujkova, M. (2011). Step and Tap Dancing. In S. J. Greenwald & J. E. Thomley (Eds.), Encyclopedia of Mathematics & Society. Salem Press. online.salempress.com.
CMOS 17th
Droujkova, Maria. "Step And Tap Dancing." Edited by Sarah J. Greenwald & Jill E. Thomley. Encyclopedia of Mathematics & Society. Hackensack: Salem Press, 2011. Accessed December 14, 2025. online.salempress.com.