A Tetrahedral Graph of Tetrachordal Voice-leading Space

A tetrahedral graph models voice leading among the 29 T/I-type tetrachord classes. Transpositional combination plays a crucial role in the structure of the tetrahedron. A dipyramid, fusing two tetrahedra, models similar relations among the 43 T-type |4|-classes. The two graphs generalize to n-dimensional simplexes for relations among |n + 1|-classes, in modulo 12 and in other universes of even cardinality. A peculiarity is that the symmetrical graphs are a bit too large to hold the asymmetrical collection of abstract objects that they are designed to contain. A handful of set-class duplications serve as bubble wrap; the article devotes considerable attention to investigating their status. Submission received July 2003 1 A version of this paper was presented at a symposium on Music and Mathematics organized by Robert Peck and Judith Baxter, as part of the southeast regional meeting of the American Mathematical Society in Baton Rouge, March 2003. Joseph Straus’s unpublished graph prompted the questions that led to this research, and he generously permitted me to reproduce his graph concurrently with the publication under his name. Ian Quinn’s unpublished version of Straus’s graph suggested a central insight, and conversations with Ian have helped sharpen my understanding of many aspects of this presentation. Jack Douthett made a number of helpful suggestions and corrections to the proofs. A question from Panayotis Mavromatis stimulated the exploration undertaken in paragraphs [31] through [33]. Richard Plotkin created the graphics for this article. I also thank Clifton Callender, Robert Morris, John Roeder, Steve Soderberg, Zal Usiskin, and Jonathan Wild for suggestions and information. Julian Hook gave the entire manuscript a thorough reading and made many valuable suggestions.