Harmonic Theory
Pioneered by Leonhard Euler and popularized by Hugo Riemann, the Tonnetz
(tone-network) is a slanted grid representing pitch space. Notes are connected along
three geometric axes of consonant intervals:
- Horizontal Axis: Perfect Fifths (P5, +7 semitones).
- Upward Diagonal Axis: Major Thirds (M3, +4 semitones).
- Downward Diagonal Axis: Minor Thirds (m3, +3 semitones).
Triangular cells in this grid represent triads: triangles pointing upwards are Major triads,
and triangles pointing downwards are Minor triads.
Neo-Riemannian theory
analyzes chord transformations based on note preservation (minimal voice leading distance),
focusing on three primary operations:
- P (Parallel): Flips a major triad into its parallel minor (C major
↔ C minor) by changing the 3rd. It flips the triangle across its P5 edge.
- L (Leading-tone exchange): Relates a major triad to its minor
partner a major third above (C major ↔ E minor) by moving the root a semitone.
It flips the triangle across its minor third edge.
- R (Relative): Flips a major triad into its relative minor (C major
↔ A minor) by moving the fifth. It flips the triangle across its major third
edge.
How to Use the View
Hover over the transformations panel to trigger and trace L, P, and R operations. The
grid highlights the triad shapes and draws arrows between them, visualizing voice
leading transitions. Common notes shared between chords glow to show how notes pivot
during transitions.
Pitch Class Node
Nodes represent individual pitches. Clicking a node sets it as the key signature
tonic.
Interval Axes
Lines connecting notes by fifths (horizontal), major thirds (upwards), and minor
thirds (downwards).
Triad Triangles
Major triads point upward. Minor triads point downward. Highlighting shows active
notes in your chord.
Neo-Riemannian Panel
Hover or click transformations (like L-P-R paths) to watch the triangles flip across
their shared edges.
