Transformations for the Twenty-First Century Symmetries, Intervals, and the Principle of Transformational Closure

This article addresses three interlocking topics. First, it defines voice-leading transformations that model the specific contrapuntal moves available to musicians. These are the intervals corresponding to the symmetries of voice-leading geometry, encompassing voice exchanges, transpositions along the chord, repeating contrapuntal patterns, and Richard Cohn’s version of the neo-Riemannian transformations. Second, it defines a class of ad hoc transformations known as contrapuntalinterchanges, which generalize voice-leading transformations using the retrograde. This allows us to reformulate neo-Riemannian theory without inversion, avoiding dualism while encompassing a wider range of progressions. Contrapuntal interchanges can also be used to form approximate sets such as “triad” or “fourth chord,” providing a more flexible alternative to standard set theory. Finally, the article uses these technical issues to explore larger questions about the relation between abstract mathematics and concrete musical practice. It suggests that contemporary theory is still in the grip of a modernist ideology that prioritizes systematicity over musical relevance; this ideology is as characteristic of Riemann as it is of Schoenberg.