This paper proves the existence and uniqueness of an automorphism on the Riemannian group that preserves voice-leading properties. The musical significance of this result is as follows: in Western music, there are two different (and conflicting) ways to think about how to move from one chord to another. The first way of thinking about chord progressions (popular among contemporary guitar players) is called “root-intervallic” motion, and it reflects modern inclinations of thinking about chords as single units. The second way of thinking about chord progressions (more popular among classically-trained musicians) is called “voice-leading”. This approach entails a level of detail in pitch-by-pitch specificity.
The two methods of thinking about chord progressions, root-intervallic and voice-leading, present two different ways of thinking about moving from one chord to another. In common-practice Western music, there are idiomatic rules for how both aspects of a chord progression should look like. (As a grossly simplified example, in the key of C-major, an G-major chord is not allowed to move to a F-major chord. This is the root-intervallic constraint. On the other hand, the pitch B in the G-major chord must move to a C. This is the voice-leading constraint.) These two aspects are often at odds with one another, and a solution must be found that satisfies both. Indeed, such problem-solving is the focus of most first-semester music theory courses.
The Neo-Riemannian approach to analyzing chord progressions has traditionally been split between the root-intervallic and the voice-leading approaches. The problem is that these two approaches are not mathematically equivalent. Thus, a serious logical flaw in the theory exists unless we prove that the two approaches are equivalent. This paper does this by proving the existence and uniqueness of The Voice-Leading Automorphism.
The first section of this paper includes a friendly introduction to mathematical music theory, for those who may not be initiated to the subject.