Beyond Functional Harmony
Traditional harmonic analysis (I, IV, V, vi) works well for most pop and classical music — but fails for chromatic progressions like C major → A♭ major → E major. These progressions don't follow traditional key-based logic, yet they sound compelling. Neo-Riemannian theory, introduced by Cohn (1998), provides the framework: instead of analyzing chords by their function in a key, it analyzes how they transform into each other. Three basic operations — Parallel (P), Leading-tone exchange (L), and Relative (R) — can generate any chromatic triad progression through a network of smooth voice-leading connections.
The Tonnetz: Harmony as Geometry
The Tonnetz is a geometric representation where triads are triangles on a 2D lattice. Adjacent triangles share two notes (smooth voice leading). The PLR operations correspond to flipping a triangle across one of its edges. Cohn (2012) showed that parsimonious (smooth) voice leading between triads creates the perception of harmonic magic — the feeling that the music is moving effortlessly through tonal space. Film composers exploit this constantly: the C → A♭ → E → C cycle (the "hexatonic pole" progression) appears in scores by John Williams, Howard Shore, and Hans Zimmer for its ethereal quality.
Tymoczko's Geometric Model
Tymoczko (2004, 2011) extended these ideas into a complete geometric theory of music. In his model, chords are points in a multi-dimensional space where distance corresponds to voice-leading efficiency. The smoothest progressions are the shortest paths. This explains why certain chromatic progressions feel "inevitable" even though they violate traditional harmonic rules — they're geometrically close even when functionally distant. MusicMind uses Tymoczko's geometric distance metric to compute harmonic sophistication scores.
Tonal Attraction & Resolution
Woolhouse (2009) proposed a model of tonal attraction — the gravitational pull between adjacent chords. Notes that are close in pitch (a semitone apart) create stronger attraction. This explains why a B resolves so strongly to C, and why the V–I cadence (with its leading-tone resolution) is the most powerful harmonic gesture in Western music. MusicMind computes tonal attraction scores for every chord transition, contributing to the overall tension and resolution mapping.
When a film score goes C → A♭ → E and your skin tingles, that's neo-Riemannian theory in action. MusicMind maps these geometric relationships in your harmony.