After Tymoczko's Geometry of Musical Chords (Science, 2006): chords live in quotient spaces (orbifolds), and voice leading is short motion through them. Everything makes sound — click or drag, and listen to the geometry. Master volume is capped low.
Two pitches live on a 2-torus T² = (R/12Z)² (octave equivalence wraps each axis at 12). A chord is an unordered pair, so we quotient by the swap S₂: (x,y)~(y,x). Geometrically the swap is a reflection across the diagonal.
The shaded triangle x ≤ y is a fundamental domain: one representative per chord. Its two non-diagonal edges glue to each other with a flip (arrows) — a single orientation-reversing identification. One reflection makes the surface non-orientable: the result is a Mobius strip.
The diagonal x = y is the singular boundary (unisons / doublings): the swap fixes it, so it is a mirror wall, the edge of the strip. The tritone {0,6} splits the octave evenly — the maximally even dyad — and rides the strip's center line.
Modding out transposition too, a 3-note chord is fixed by its three adjacent gaps (i₁,i₂,i₃) with i₁+i₂+i₃=12. That is a 2-simplex (triangle) — a cross-section of the full orbifold T³/S₃ (a twisted triangular prism).
The center (4,4,4) is the augmented triad, the maximally even trichord. Move outward and chords grow lopsided; on the boundary some gap hits 0, meaning two notes coincide — a doubling, a singular stratum. Major (4-3-5) and minor (3-4-5) are mirror images across the triangle's axis (chirality).
The straight segment is a schematic of the shortest path; the audio glides the exact minimal voice leading computed in pitch space.
A voice leading pairs each note of A with a note of B; its size is the taxicab (L₁) distance — total semitones moved, each voice by its shortest mod-12 route. We minimize over all pairings (and, for unequal sizes, over which note to double). In the orbifold this is a short path (a near-geodesic) between the two points.
Small numbers are the parsimony behind P / L / R (neo-Riemannian): C major -> C minor moves one voice by 1 (P), C major -> E minor by 1 (L), C major -> A minor by 2 (R). Nearly-even chords are exactly the ones with tiny voice leadings among their transpositions.