Tonnetz

In musical tuning and harmony, the Tonnetz (German for 'tone net') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739.

The Tonnetz originally appeared in Leonhard Euler's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae.

In the twentieth century, composer-theorists such as Ben Johnston and James Tenney continued to developed theories and applications involving just-intoned Tonnetze.

This principle is especially important in analyzing the music of late-19th century composers like Wagner, who frequently avoided traditional tonal relationships.

[4] Neo-Riemannian music theorists David Lewin and Brian Hyer revived the Tonnetz to further explore properties of pitch structures.

Neo-Riemannian theorists typically assume enharmonic equivalence (in other words, A♭ = G♯), and so the two-dimensional plane of the 19th-century Tonnetz cycles in on itself in two different directions, and is mathematically isomorphic to a torus.

A modern rendering of the Tonnetz . The A minor triad is in dark blue, and the C major triad is in dark red. Interpreted as a torus, the Tonnetz has 12 nodes (pitches) and 24 triangles (triads).
Euler's Tonnetz
Neo-Riemannian music theory's PLR operations applied to a minor chord Q.
Tonnetz aligned with the Wicki–Hayden note layout .
Tonnetz showing enclosed chords. Capitalized chords ('Xx') are major; others ('xx') are minor.