Introduction Transformation theory is a branch of music theory that studies how musical objects, such as tonal areas, rhythm, or chords can transform from one to another. Neo-riemannian music theory, named after the music theorist Hugo Riemann, studies the transformation of chords through the so-called neo-Riemannian operators. These neo-Riemannian operators form a group, called the Riemannian group. In the early formulations of Riemannian operators, they were defined by their voice-leading properties. In other words, they were defined by how individual notes move from one chord to the next. In particular, the Riemannian group is generated by the maximally smooth voice-leading operations on the set class of major and minor triads. Unfortunately, this has the disadvantage of not easily being generalizable to arbitrary pc-set classes. Some have suggested a root-intervallic approach to Riemannian operators. Unfortunately, this has the disadvantage of abandoning the operators’ voice-leading properties. In particular, isomorphic chord progressions on different pc-set classes have different common-tone retention. In our paper, we show that there exists “dual” pc-set classes, on which isomorphic chord progressions have same common-tone retention. We do this by defining the voice-leading automorphism on the Riemannian group, which preserves common-tone retention. Music theory preliminaries We review basic standard music theory terminology, for the benefit of the musically uninitiated. The equal-tempered 12 chromatic pitches of Western music is represented by. A pitch-class set (abbreviated as “pc-set”) is simply a subset of, which could represent a chord, a melodic line, or a scale. C-major triad = {0,4,7} Next, we define two musically significant operators on pc-sets: (1) The transposition operator, which is given by (2) The inversion operator, which is given by Both of the operators defined above have the property that they preserve all of the intervals present in the pc-set, thus preserving the sound of the chord. Now define an equivalence relation on the set of all pc-sets with the same cardinality as follows: if or for some n. Such an equivalence class is called a pc-set class. For example, the set of all major and minor triads forms a pc-set class. Literature cited Cohn, Richard. Neo-riemannian operations, parsimonious trichords, and their “tonnetz” representations. Journal of Music Theory 41(1):1-66, Hook, Julian. Uniform Triadic Transformations. Journal of Music Theory 46(1/2):57-126, Hook, Julian. Cross-type transformations and the path consistency condition. Music Theory Spectrum 29(1):1-40, Hyer, Brian. Reimag(in)ing Riemann. Journal of Music Theory 39(1): , Please refer to the paper for a full list of works cited. Elementary results in Riemannian music theory Throughout this paper, γ shall denote a pc-set class of 24 elements. Also,, which is a coordinate system used to represent elements of γ, as we now define. Definition 1.1/1.2 [Operators on Coordinate Space] (1) The transposition operator is given by (2) The inversion operator is given by Definition 1.3 [Coordinate System on pc-Set Class] We choose to represent elements of γ with Γ via a bijection called the coordinate system, such that We remark that such a coordinate system is necessary, as chords in γ do not have the notion of a “root” in general. Definition [Riemannian Operators] Let, denoted as, where Theorem [The Riemannian Group], the set of all Riemannian operators, form a group under function composition. This group is isomorphic to. The Riemannian group is actually the normal subgroup of a larger class of operators, called uniform triadic transformations. Since the Riemannian operators act on Γ directly, its action on γ will vary depending on the choice of the coordinate system f. The following theorem shows that the action of a Riemannian operator on γ through a coordinate system can be preserved via an automorphism. Theorem 2.1 [Preservation Automorphism] Let f and g be coordinate systems on a pc-set class γ. Then there exists an automorphism Ω on such that for all, In other words, the following diagram commutes: In general, the preservation automorphism is an inner automorphism. The automorphism can be computed explicitly by using the modulation function, which measures the difference between two coordinate systems f and g. Please refer to the paper for details. Example We now proceed with an example to demonstrate the voice-leading automorphism. Maxx Cho Department of Mathematics/Department of Music, Swarthmore College Recent advancements I have recently begun computing dual pc-set classes for tetrachords. In particular, the set class of dominant seventh/half-diminished seventh chords and the set class containing {0,2,3,6} are dual in that they have algebraically compatible voice-leading. Acknowledgments I would like to thank the 2008 summer REU at UNCA, where this research was conducted. In particular, I would like to thank professors Sam Kaplan, Dave Peifer, and Patrick Bahls. I would also like to thank mathematical music theorists Julian Hook (University of Indiana) and Jon Kochavi (Swarthmore College) for reading my paper and providing helpful suggestions. Please contact for a copy of the paper. Algebraic compatibility of Riemannian operators’ voice-leading properties Retention operators Once a coordinate system has been chosen, this induces operators directly on the space downstairs, as the above diagram shows. We shall denote this group downstairs by. One of the most important tenets of Western music up to the 19th century is minimal voice-leading. In particular, retention notes are an important aspect of a chord progression. Definition 3.1 [Retention Value] Let γ be a pc-set class with coordinate system f. Consider some. (1) Suppose that for all. Then n is called the retention value of C and is denoted by. (2) The retention value of some relative to f is Voice-leading automorphism Consider some Riemannian operator U, which induces an operator G on γ. Now suppose that we apply the same G (that is, with the same retention value) to a different pc-set class γ’. This, in turn, induces a Riemannian operator U’. When is the mapping an automorphism? This question is important as such an automorphism would imply that isomorphic chord progressions have similar voice-leading. Definition 4.1 [Dual pc-Set Classes] Let γ and γ’ be pc-set classes of trichords. Also, fix some and. Now, suppose that and. Then γ and γ’ are dual if and. Here is the central result of this paper: Theorem 4.2 [Voice-Leading Isomorphism] Let γ and γ’ be dual pc-set classes, with corresponding groups and. Then there exists a nontrivial group isomorphism such that Corollary 4.5 [Voice-Leading Automorphim] Let γ and γ’ be dual pc-set classes with coordinate systems f and f’, respectively. Then there exists a unique automorphism such that DUAL {0,2,3,6}C7={0,4,7,10} {0,4,7} {11,4,7} ff (0,+) (11,-) {0,1,4} {5,4,1} gg (0,+) (5,-) The pc-set class containing {0,4,7} and the pc-set class containing {0,1,4} are dual. Thus, there exists an automorphism on the Riemannian group that preserves common-tone retention. In this case, the voice-leading automorphism sends the generators and. Thus,