Tonnetz & the P/L/R Group

Tap a triangle to pick a triad. Drag to pan. Keys: P L R transform, Space play.

Current triad

C major C - E - G · pc {0,4,7}

Voice-leading (last move)

At rest on C major. Apply P, L or R — watch exactly one voice slide.

Parsimony: P and L move one voice by a semitone; R moves one voice by a whole tone. Two common tones are held.

Sequence chain

Uppercase = major, lowercase = minor. Compose moves to walk the lattice.

Group structure — PLR ≅ D₂₄

The PLR group acts simply transitively on the 24 triads: between any two triads there is exactly one group word. Press a button above to see it live.

T / I lens (the dual group)

The transposition/inversion group is also order 24 (12 transpositions × inversion) and also acts simply transitively. It is the dual (commutant) of the PLR group. T moves a triad rigidly; I flips major↔minor about pitch-class 0.

Torus glue

The Tonnetz tiles the plane and repeats: shift 3 rows (three thirds = an octave, back to the same pitch classes) or 4 columns of fifths and the labels return. Gluing those two periods turns the sheet into a torus — only 12 pitch classes, 24 triangles, wrapping forever.