P Parallel
Hold the root and the fifth; let the third fall a semitone. C major becomes C minor — the same house, the lights turned low.
C·E·G → C·E♭·G · one voice, 1 semitone
Euler, 1739 · Speculum musicum · the tone-net
Tonnetz — harmony has a shape.
Leonhard Euler drew the twelve notes as a net: step right and you rise a perfect fifth; step up the diagonal and you rise a major third. Every triangle the net closes is a chord you already know — points up, major; points down, minor. A token is walking that net right now, and each step trades exactly one tone for a neighbour. Turn the sound on and the geometry starts to sing.
three sine voices · quiet & warm · nothing plays until you ask
NowC major
Movethe walk is gathering itself…
PathC
voices —
01
The net
The Tonnetz is a map of pitch, not of instruments. Twelve pitch classes repeat forever across a triangular lattice: the horizontal axis climbs by fifths (C · G · D · A…), the diagonal climbs by major thirds (C · E · G♯…), and the third direction falls out for free — a minor third, a fifth minus a major third. Any three mutually adjacent notes form a triangle, and every triangle is a triad: ▲ major when it points up, ▽ minor when it points down.
The payoff is that distance on the net is harmonic distance. C major and A minor sit cheek to cheek because they share two of their three tones. C major and F♯ major sit far apart because your ear has to cross the whole net to get there.
→ fifths (+7 semitones)
↗ major thirds (+4)
↘ minor thirds (+3)
▲ major triad
▽ minor triad
02
Three moves
Hugo Riemann's heirs boiled chord motion down to three reflections, each flipping a triangle across one of its own edges. Two tones hold still; one slides — never further than a whole step. These are the edges the token walks.
R Relative
Hold the root and the third; push the fifth up a whole step. C major becomes A minor — the key signature never blinks, but the centre of gravity moves.
C·E·G → C·E·A · one voice, 2 semitones
L Leading-tone
Hold the third and the fifth; let the root slip down a semitone onto its leading tone. C major becomes E minor — the gentlest exchange the net owns.
C·E·G → B·E·G · one voice, 1 semitone
03
One voice at a time
Late-Romantic harmony — Wagner, Liszt, Franck — slides between chords that no single key explains. On the net they stop being mysteries and become neighbours: every P, L or R is a reflection across a shared edge, two common tones held like held breath while one voice steps. When the sound is on, that is literally what you hear — three sine voices, and on each landing only one of them moves.
Watch the path readout: alternate P and L and after six moves you come home — C, c, A♭, a♭, E, e, C — a hexatonic cycle that visits three keys a major third apart and never once takes a voice further than a semitone. The token indulges in one now and then. Geometry, doing the voice-leading for you.