The topology.
The Tonnetz is a 200-year-old lattice that maps every triad as a triangle. Adjacent triangles share two notes. Moving between them — P, L, R — is the smallest possible voice leading. Click a triangle. Hear the chord. Watch one voice move by a single semitone. This is harmony as navigation.
Each triangle is a major or minor triad. The three vertices are its pitch classes. Click any triangle to hear it and make it the current chord. Colored triangles show which triads are one PLR step away. You can also navigate with arrow keys after clicking the lattice, or press P/L/R/N/S/H to apply transforms directly.
P (Parallel) flips major/minor, keeping root and fifth. L (Leading-tone) exchanges one note by semitone. R (Relative) connects C major to A minor. Each keeps two notes and moves one by the smallest interval.
N (Nebenverwandt) = RLP. S (Slide) = LPR. H (Hexatonic pole) = LPL — maximum distance with zero common tones. These chain single steps to reach more distant triads.
The Tonnetz was invented by Leonhard Euler in 1739. Hugo Riemann formalized it as a map of tonal relations. Two centuries later, neo-Riemannian theorists proved that its geometry explains voice leading parsimony — why some chord changes sound smooth and others feel like a leap.
Every PLR transform is involutory: apply it twice and you return to the start. P twice = identity. L twice = identity. This is why the Tonnetz has no preferred direction — every path can be reversed. Harmony is bidirectional.
The hexatonic pole (H = LPL) is the farthest you can go in three steps: C major to Ab minor. Zero common tones. Maximum distance. Yet it sounds not dissonant but otherworldly — film composers use it when reality shifts.
Adjacent triangles always share an edge (two pitch classes). This means any PLR move keeps two notes and changes one. Voice leading distance = 1 or 2 semitones. The ear tracks the moving voice and understands the connection.
The lattice wraps: if you follow fifths far enough you return to the start (circle of fifths). If you follow major thirds, you cycle through four augmented triads. The Tonnetz is not flat — it is a torus.