Definition
The Tonnetz ("tone network," in German) is a two-dimensional lattice in which notes are arranged according to three consonant intervals: the perfect fifth, the major third, and the minor third. It was proposed by Leonhard Euler in 1739 (Tentamen novae theoriae musicae), taken up in the nineteenth century by von Oettingen and Riemann, and revived by neo-Riemannian theory in the 1980s–90s.
How it is built
- Each horizontal row is a chain of fifths: … F – C – G – D – A …
- The row above is shifted up by a major third and half a cell to the right: … A – E – B – F# …
- The row below is shifted down by a major third and half a cell to the left: … Ab – Eb – Bb – F …
A portion around C (imagine the rows staggered):
- upper row: A, E, B, F#
- middle row: C, G, D, A
- lower row: Ab, Eb, Bb, F
Along the diagonal, going up to the right you find major thirds (C → E), going up to the left minor thirds (C → A).
Triads are triangles
- The triangle C – G – E (pointing up) is C major.
- The triangle C – G – Eb (pointing down) is C minor.
- A – E – C is A minor; E – B – G is E minor.
Two triangles that share an edge share two notes. Moving from one to the other is one of the basic transformations:
| Shared edge | From C to | Transformation |
|---|---|---|
| C – G (fifth) | Cm | P |
| C – E (major third) | Am | R |
| E – G (minor third) | Em | L |
Typical paths
- Straight line along the fifths (C → G → D…, alternating triangles): progressions by fifths and LR cycles.
- Vertical zigzag along the axis of major thirds (C → Cm → Ab → Abm → E → Em): the hexatonic PL cycle, closed on itself because three major thirds make an octave.
- Octatonic path PR (C → Cm → Eb → Ebm → Gb …): moves by minor thirds.
- Chords that are "far" in key (C → E, C → Ab) are often close on the map: this is why chromatic mediants sound fluid.
The torus
With equal temperament G# = Ab: the map repeats and can be rolled up into a torus (a doughnut). All 24 major and minor triads sit on the surface. Interactive Tonnetz visualizations (several are available online) show the triangles lighting up as you play.
Limits and developments
Dmitri Tymoczko (A Geometry of Music, 2011) showed that the Tonnetz represents triads well but not all chords or all voice leadings; he proposes more general geometric spaces ("orbifolds") in which distance truly measures how many half steps are moved. For the composer, though, the Tonnetz remains the most intuitive map for inventing coherent chromatic progressions.
Analysis example
Creep (Radiohead, in G): G → B → C → Cm. On the Tonnetz: from G you step down to Bm (L) and change quality to B (P): two steps; B → C is not a single step, it is a leap (you hear it as the "strangeness" of the song); C → Cm is a single step (P), and the return to G closes the circle.
Exercise for the producer
- Draw a piece of the Tonnetz on paper (5 rows × 6 notes).
- Start from Am and make a path of 8 adjacent triangles, without passing through the same triangle again.
- Play the sequence with a pad (e.g. Jup-8 V or Prophet-5 V), keeping the common tones fixed.
- Repeat in another direction: different paths = different emotions (along the fifths = sunny, along the major thirds = suspended and magical).
Related topics
Neo-Riemannian theory, chromatic mediants, chromatic voice leading, circle of fifths and closely related keys.