01

The Tonnetz

The Tonnetz is a lattice of pitch classes arranged so that every horizontal step moves by a perfect fifth, every diagonal step moves by a major or minor third. Each triangle on the grid represents a major or minor triad. Adjacent triangles share two notes — meaning a single voice moves by just a half or whole step to get from one chord to the next. This is what musicians call parsimonious voice leading: maximum harmonic change with minimum physical movement.

02

Triads & Voice Leading

A triad is a three-note chord built from stacked thirds. Major triads (brighter triangles) contain a major third plus a minor third. Minor triads (darker triangles) reverse the order. In Neo-Riemannian theory, what matters is not which key you're in, but how efficiently voices can move from one chord to another. Chords that share two common tones and move one voice by a semitone are considered the closest possible harmonic neighbors — and they appear side-by-side on the Tonnetz.

03

P — Parallel

The Parallel transformation keeps the root and fifth of a chord unchanged and moves the third by a semitone, toggling between major and minor. C major → C minor. The root stays on C, G stays on G; only E drops to E♭. On the Tonnetz, P flips a triangle across its horizontal edge.

P

04

L — Leading-tone Exchange

The Leading-tone transformation keeps two notes common and moves one voice by a semitone in the opposite direction. C major → E minor. C and E are shared; G moves up a semitone to B. The name comes from the baroque concept of a leading tone — a note that "wants" to resolve upward. L flips a triangle across its upper-left edge on the Tonnetz.

L

05

R — Relative

The Relative transformation connects a major chord to its relative minor, keeping two tones and moving one by a whole step. C major → A minor. E and G are shared; C moves down a whole step to A. This is the familiar relationship between C major and A minor — the same six notes, differently emphasized. R flips a triangle across its upper-right edge.

R

06

Compound Operations

Any two chords can be connected by combining P, L, and R in sequence. Some combinations have names of their own: Slide (LPR) connects two triads that share a third but no other common tone. Nebenverwandt (RLP) links a major chord to its minor subdominant. Hexatonic Pole (LPL) produces the most dramatic leap — a major third away with no common tones — yet arrives through three tiny steps. These are the surprise modulations of Romantic-era music.

LPR RLP LPL PL PR

07

Cycles

Chaining the same operations repeatedly creates closed loops. The Hexatonic Cycle (PLPLPL) visits six chords — three major, three minor — all a major third apart, before returning to the start. Schubert and Wagner loved this loop for its dreamlike, suspended quality. The Octatonic Cycle (RPRPRP) visits eight chords spanning the diminished-seventh symmetry of the scale. Both cycles appear when you press LRLRLR in the app and in the Cycle Membership panel.

Hexatonic Octatonic

1

Click any triangle

Select a chord directly on the lattice. The root note glows, chord tones light up, and the sound plays immediately. Dashed arrows show your P, L, and R neighbors from that position.

2

Use the operation buttons

Press P, L, or R to step to the next chord algorithmically. Use compound buttons to make larger leaps. Watch the highlighted triangle travel across the grid with each move.

3

Build a progression

As you move, the Progression panel records your path with transform labels between each step. Colored lines trace the route on the grid itself. Press Play to hear the full sequence back.

4

Read the cycle panel

The Cycle Membership panel tells you which hexatonic or octatonic group your current chord belongs to — useful for planning modulations between harmonic regions.

C major

Controls Chord Info Progression