Folding music: modeling transformations between voice-leading spaces

Abstract

Geometric voice-leading spaces have been outlined by Clifton Callender, Ian Quinn, and Dmitri Tymoczko, who explain how quotient spaces, using traditional musical equivalences, can be used to model voice leading in music. They discuss how five musical equivalences, which they refer to as the "OPTIC" equivalences (octave, permutation, transposition, inversion, and cardinality), can be applied in any combination in order to highlight different musical properties. However, they do not address all of these combinations in an exhaustive manner. Nor have any articles been written about these spaces that were specifically directed to the music theory community, especially in terms of explaining the mathematical concepts that are the foundation of these spaces. This project fulfills this need, discussing the background mathematics required to understand how these spaces function, and addressing every possible combination of the OPTIC equivalences, using video graphics in order to fully explain and imagine these spaces, the way they transform from one to another, and the motion within each space. In addition, two new equivalence relations are presented, namely transpositional inversion and contour. The last section of the video provides a practical application of these spaces by examining Joseph Straus's concept of "fuzzy transposition" and the metrics by which he measures transpositional relatedness. Seeing how the spaces fold and interrelate provides a better understanding of the spaces themselves, how they function, and the musical equivalences that they embody.