Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly to each other, without necessary reference to a tonic. Initially, those harmonies were major and minor triads; subsequently, neo-Riemannian theory was extended to standard dissonant sonorities as well. Harmonic proximity is characteristically gauged by efficiency of voice leading. Thus, C major and E minor triads are close by virtue of requiring only a single semitonal shift to move from one to the other. Motion between proximate harmonies is described by simple transformations. For example, motion between a C major and E minor triad, in either direction, is executed by an "L" transformation. Extended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire system of harmonic relations. Where consensus is lacking is on the question of what is most central to the theory: smooth voice leading, transformations, or the system of relations that is mapped by the geometries. The theory is often invoked when analyzing harmonic practices within the Late Romantic period characterized by a high degree of chromaticism, including work of Schubert, Liszt, Wagner and Bruckner. Neo-Riemannian theory is named after Hugo Riemann (1849–1919), whose "dualist" system for relating triads was adapted from earlier 19th-century harmonic theorists. (The term "dualism" refers to the emphasis on the inversional relationship between major and minor, with minor triads being considered "upside down" versions of major triads; this "dualism" is what produces the change-in-direction described above. See also: Utonality) In the 1880s, Riemann proposed a system of transformations that related triads directly to each other. The revival of this aspect of Riemann's writings, independently of the dualist premises under which they were initially conceived, originated with David Lewin (1933–2003), particularly in his article "Amfortas's Prayer to Titurel and the Role of D in Parsifal" (1984) and his influential book, Generalized Musical Intervals and Transformations (1987). Subsequent development in the 1990s and 2000s has expanded the scope of neo-Riemannian theory considerably, with further mathematical systematization to its basic tenets, as well as inroads into 20th century repertoires and music psychology.
Triadic transformations and voice leading The principal transformations of neo-Riemannian triadic theory connect triads of different species (major and minor), and are their own inverses (a second application undoes the first). These transformations are purely harmonic, and do not need any particular voice leading between chords: all instances of motion from a C major to a C minor triad represent the same neo-Riemannian transformation, no matter how the voices are distributed in register.
The three transformations move one of the three notes of the triad to produce a different triad:
The P transformation exchanges a triad for its parallel: applied to a major triad, it lowers the third by a semitone (C major to C minor); conversely, applied to a minor triad, it raises the third by a semitone (C minor to C major) The R transformation exchanges a triad for its relative: applied to a major triad, it raises the fifth by a tone (C major to A minor); conversely, applied to a minor triad, it lowers the root by a tone (A minor to C major) The L transformation exchanges a triad for its mediant or submediant: applied to a major triad, it lowers the root by a semitone (C major to E minor); conversely, applied to a minor triad, it raises the fifth by a semitone (E minor to C major) Observe that P preserves the perfect fifth interval (so given say C and G there are only two candidates for the third note: E and E♭), R preserves the major third interval (given C and E our candidates are G and A), and L preserves the minor third interval (given E and G our candidates are C and B). Secondary operations can be constructed by combining these basic operations:
The N (or Nebenverwandt) relation exchanges a major triad for its minor subdominant, and a minor triad for its major dominant (C major and F minor). The "N" transformation can be obtained by applying R, L, and P successively. The S (or Slide) relation exchanges two triads that share a third (C major and C♯ minor); it can be obtained by applying L, P, and R successively in that order. The H relation (LPL) exchanges a triad for its hexatonic pole (C major and A♭ minor) Any combination of the L, P, and R transformations will act inversely on major and minor triads: for instance, R-then-P transposes C major down a minor third, to A major via A minor, whilst transposing C minor to E♭ minor up a minor 3rd via E♭ major. Initial work in neo-Riemannian theory treated these transformations in a largely harmonic manner, without explicit attention to voice leading. Later, Cohn pointed out that neo-Riemannian concepts arise naturally when thinking about certain problems in voice leading. For example, two triads (major or minor) share two common tones and can be connected by stepwise voice leading the third voice if and only if they are linked by one of the L, P, R transformations described above. (This property of stepwise voice leading in a single voice is called voice-leading parsimony.) Note that here the emphasis on inversional relationships arises naturally, as a byproduct of interest in "parsimonious" voice leading, rather than being a fundamental theoretical postulate, as it was in Riemann's work. Dmitri Tymoczko has argued that the connection between neo-Riemannian operations and voice leading is only approximate (see below). Furthermore, the formalism of neo-Riemannian theory treats voice leading in a somewhat oblique manner: "neo-Riemannian transformations," as defined above, are purely harmonic relationships that do not necessarily involve any particular mapping between the chords' notes.
Graphical representations
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