Crab Canons: The Backwards Walking Crustacean with a Twist

By Faithlin Hunter

Departments: Music & Math

SCE Advisors: Dr. McCollum & Dr. Wilson

 

When initially researching mathematical happenings within a musical context, I was pointed towards a YouTube video of Johann Sebastian Bach’s “Canon 1 a 2” from The Musical Offering (1747). The video begins by playing the original melody line of the canon and then introduces a retrograde of the line by playing the melody backwards, hence the canon’s nickname of the “crab canon.” However, in the transformation into a Mobius strip, the retrograded line is then also inverted and placed on the backside of the strip before twisting and linking the ends together. In topology, one could argue that a true Mobius strip was not created if a split in the strip was utilized before the twist, but scholars highly regard Bach’s Crab Canon as a true Mobius strip.

            My thesis explores this topological phenomenon of Bach’s Crab Canon and whether it is truly a composition on a Mobius strip. First, we establish all of the necessary musical terms for understanding Bach’s composition. These terms are used as preliminaries to guide our perception of aesthetics in musical composition and for further analysis in later sections. From here, we define a Mobius strip and begin deconstructing both sides of the argument for whether Bach’s Crab Canon can be transposed onto one. After analyzing the use of mathematical transformations in the construction of a Mobius strip, we further explore the significance of symmetries in music through these transformations. Through mathematical concepts such as surfaces and transformations, music visualization can be expanded to aid audiences in understanding the compositional choices of composers. Finally, we culminate all of our learned knowledge to analyze a brief composition that I made for the purposes of demonstrating musical transformations and surfaces.