Tonnetz Reference ManualNeo-Riemannian transformationsPage 1 of 4Thomas Posen, Ph.D.

The Tonnetz and the three basic moves

The Tonnetz (German for “tone network”) lays the twelve pitch classes out on a grid of triangles. Every major and minor triad is one triangle, and closely related triads sit side by side. Leonhard Euler drew the first version in 1739, and since the 1980s neo-Riemannian theory has used it to map how one chord moves to the next.

The six neighbors of C, in semitones. P5 = perfect fifth, M3 = major third, m3 = minor third.

Figure 1. C major (outlined) and the three triads one move away. Each shares an edge with C major, so it shares two of its notes. The colored edge is the one the triangle flips across.

Reading the grid

Circles are pitch classes. The grid repeats in every direction, so each note appears many times. Glued edge to edge, it closes into a torus.
Triangles are triads. ▲ is major: root lower left, fifth lower right, third on top. ▼ is minor: root upper left, fifth upper right, third at the bottom.
Directions are intervals. → is up a perfect fifth (+7 semitones), ↗ up a major third (+4), ↘ up a minor third (+3). The opposite way goes down.
Neighbors share notes. Triangles with an edge in common share two notes. Triangles that touch at one point share one.

P, L and R

Each keeps two notes and moves the third by step, flipping the triangle across the edge it keeps. So every move changes the mode, and the same move twice brings you back.

P

Parallel

Same root, opposite modeC ↔ c

Keeps: the fifth, C–G (the → edge)
Major: C E G → C E♭ G
Moves: the third, a half step
Minor: C E♭ G → C E G

L

Leading-tone exchange

Riemann’s LeittonwechselC ↔ e

Keeps: the minor third, E–G (the ↘ edge)
Major: C E G → B E G
Moves: a half step: the root of a major triad falls, the fifth of a minor triad rises
Minor: E G B → E G C

R

Relative

Relative keys share a key signatureC ↔ a

Keeps: the major third, C–E (the ↗ edge)
Major: C E G → C E A
Moves: a whole step: the fifth of a major triad rises, the root of a minor triad falls
Minor: A C E → G C E

Tonnetz Reference ManualNeo-Riemannian transformationsPage 2 of 4Thomas Posen, Ph.D.

All six transformations

S, N and H are compound moves: chains of basic moves treated as one step. Each panel starts on C major (dashed) and lands on the shaded triangle. Ringed notes stay put, and the letters on the dotted line spell out the chain.

Basic moves · one note moves

Compound moves · two or three notes move

Each compound move has two chains: S = LPR or RPL, N = RLP or PLR, H = LPL or PLP. David Lewin named the slide, and “hexatonic pole” is Richard Cohn’s term. N comes from Riemann’s theory and links major tonic to minor subdominant (I → iv) or minor tonic to major dominant (i → V).

Every triad, every move

Find the starting triad on the left and read across. Capitals are major, lower case is minor.

Each table is the other read backwards, since every move undoes itself. Enharmonic names are the same triad (c♯ = d♭, g♯ = a♭, F♯ = G♭).

Tonnetz Reference ManualNeo-Riemannian transformationsPage 3 of 4Thomas Posen, Ph.D.

Combining moves

Chains of P, L and R reach every major and minor triad. Read a chain left to right: PR means P, then R. On the grid a chain is a walk from triangle to triangle, crossing one edge per move.

Two moves

Two flips bring the mode back, so every two-move chain lands on a triad of the same mode.

Figure 2. From C major, the six two-move chains reach the six major triads that surround it.

Chromatic mediants are same-mode triads with roots a third apart that share one note. Their sudden shift of color is common in 19th-century music and film scores.

Cycles

Alternate two moves and you eventually return to the start. On the flat grid the walk runs along a strip and ends on another copy of the starting triad.

PL Hexatonic cycle

Six triads, each sharing two notes with the next. Together they use only six pitch classes, the hexatonic scale C E♭ E G A♭ B (half step and minor third, alternating). There are four such cycles. Each triad’s hexatonic pole (H) sits halfway round.

PR Octatonic cycle

Eight triads; the major roots rise by minor thirds (C, E♭, F♯, A). Together they fill the octatonic scale C D♭ E♭ E F♯ G A B♭ (half step and whole step, alternating). Three such cycles share out all 24 triads.

LR LR cycle: all 24 triads

Alternating L and R visits every major and minor triad once. The major triads climb the circle of fifths (C, G, D, A, …), and each minor triad in between shares two notes with the triads on either side. The names switch from sharps to flats halfway round (d♯ = e♭).

How the moves fit together

The moves mirror with the mode. R raises a major triad’s fifth by a whole step but lowers a minor triad’s root by a whole step. The letter names a relationship, so C → a and a → C are both R.
Smoothness. P and L move one voice a half step (1 semitone in all). R moves one voice a whole step, and S and N move two voices a half step each (2 in all). H moves all three voices (3).
Every triad is reachable. P, L and R together reach all 24 major and minor triads. They generate a group of 24 operations, the dihedral group of order 24, which mirrors the transpositions and inversions of pitch-class set theory.
Watch the German names. In German, Parallele means the relative key, and the same-root key is the Variante. The English letters follow Anglo-American usage: P for parallel (C ↔ c), R for relative (C ↔ a).

Tonnetz Reference ManualNeo-Riemannian transformationsPage 4 of 4Thomas Posen, Ph.D.

The triad network

This page turns the Tonnetz inside out. Each triad becomes a single point, and an arrow joins two triads that are one move apart. Every triad has exactly three arrows, one each for P, L and R, so the points link into a honeycomb. Jack Douthett and Peter Steinbach (1998) called this layout the chicken-wire torus.

Figure 3. The triad network around C major (outlined). Shaded circles with capitals are major triads, and white circles with lower case are minor. Arrows point both ways because every move undoes itself.

Same map, inside out. A triangle on the Tonnetz is a point here, an edge two triangles share is an arrow, and a note is a hexagon of the six triads that contain it.

Direction tells you the move. From a major triad, P points down, L up and to the right, and R up and to the left. From a minor triad each one points the opposite way.

Count arrows for distance. The shortest path between two triads is the fewest moves between them. From C, three triads are one arrow away and six are two. None is more than five; b♭ minor is the farthest.

Patterns to look for

One note, one hexagon

The six triads that contain C ring a hexagon. Walking round it alternates R, L and P: C, a, F, f, A♭, c, and back to C. All six keep C as a common tone.

SNH Compound jumps

S and N cross a hexagon to its opposite corner, so the two triads share only the note that hexagon stands for: E for C → c♯, C for C → f. H crosses three arrows and shares nothing.

Cycles run straight

Alternating two moves walks a straight zigzag (shaded). LR runs side to side through all 24 triads. PL runs on the ↗ diagonal and PR on the ↘ diagonal, through six and eight triads before the pattern repeats.