Tonnetz 3D
About this app
Euler's tone network as a 3D spiral — chords as triangles, cadences as paths, Coltrane changes audible.
What a Tonnetz is
A Tonnetz arranges notes by their harmonic relationships. Neighbouring notes are a fifth or a third apart, and three related notes form a triangle, which is a major or minor triad. The idea goes back to Leonhard Euler and was taken up by Hugo Riemann in the 19th century. Neo-Riemannian theory uses the operations P, L and R to describe how triads turn into one another by moving a single note.
What the app shows
The twelve notes of equal temperament sit as spheres on a circle of fifths. Optionally the circle becomes a helix: twelve pure fifths do not return exactly to the starting note, and the vertical offset makes the Pythagorean comma visible. Triangles between the spheres stand for major and minor triads.
Try it yourself
Click a sphere to hear a single note, or a triangle to hear the chord. The Sequences panel plays II-V-I in C, I-vi-ii-V, the Coltrane changes and a P-L-R chain at the chosen tempo, lighting up each chord in turn. Notice how short the paths are in cadences and how far the Coltrane changes jump.
Frequently asked questions
- What is a Tonnetz in music theory?
- A Tonnetz is a lattice in which notes are arranged by fifth and third relationships. Every triangle of three neighbouring notes is a major or minor triad, and adjacent triangles share two notes. Chord progressions can then be read as paths through the lattice.
- What do P, L and R mean in neo-Riemannian theory?
- P (parallel) swaps major and minor on the same root, for example C major to C minor. L (leading-tone exchange) turns C major into E minor, and R (relative) turns C major into A minor. Each operation moves exactly one note by a semitone or whole tone while the other two stay put.
- What are Coltrane changes?
- Coltrane changes are a chord progression John Coltrane used in pieces such as “Giant Steps”. The key centres are a major third apart and divide the octave into three equal parts, for example B, G and E♭. In the Tonnetz they appear as large jumps, whereas cadences move in short steps.
- What does the helix in the app show?
- It shows the Pythagorean comma: twelve stacked pure fifths end about 23.5 cents above the starting note after seven octaves. The app draws this excess as a vertical offset, so the circle of fifths does not close and becomes a spiral.
Subject: Music theory
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