NeoRiemannian

Neo-Riemannian theory is a modern approach to harmonic analysis that focuses on the relationships among triads (major and minor chords) through simple, non-functional transformations. Developed chiefly by David Lewin in the late 20th century, it reframes tonal shifts as moves within a network of triads connected by three operations, rather than as progressions governed by functional harmony. The theory extends ideas from the 19th-century theorist Hugo Riemann about triadic relations, hence the name neo-Riemannian.

The three basic operations are P, L, and R. P, or parallel, keeps the root fixed and changes the triad's quality: for example C major becomes C minor and vice versa. L, or leading-tone exchange, moves to a triad with two common tones preserved, typically mapping C major to A minor, as E and C stay in common. R, or relative, also preserves two common tones and moves to a triad a major third away, such as C major to E minor.

Representations and concepts: The triads can be plotted on a Tonnetz, a lattice where adjacent nodes are related by P, L, or R. The three operations generate a connected graph and encode smooth voice-leading where two voices move by small intervals. The theory also emphasizes parsimonious voice-leading, i.e., minimal movement between triads.

Applications and scope: Neo-Riemannian analysis has been used to study late Romantic repertoire (Wagner, Liszt, Debussy, Scriabin) and has influenced pedagogical approaches to chromatic harmony. It has extensions to seventh chords and to larger chord collections, and researchers continue to develop its algebraic and geometric formulations.