Tonnetz

The Tonnetz is a geometric representation of musical pitches, specifically focusing on the relationships between chords. Invented by Leonhard Euler in the 18th century, it arranges pitches in a triangular grid. The basic Tonnetz consists of three axes, typically representing the intervals of a perfect fifth, a major third, and a minor third. Adjacent notes along these axes are related by these intervals. For example, starting from C, moving up a perfect fifth leads to G, and moving up a major third leads to E. Moving up a minor third from C leads to E flat.

The power of the Tonnetz lies in its ability to visualize harmonic relationships. Triads, such as major and minor chords, can be represented as triangles on the grid. For instance, a C major triad (C-E-G) forms a triangle where C, E, and G are connected by the respective axes. Different voicings and inversions of the same chord will occupy different positions but maintain their triangular form. This visual structure allows for the exploration of chord progressions and modulations. Common progressions, like the circle of fifths, appear as straight lines on the Tonnetz. Variations and extensions of the basic Tonnetz exist to incorporate more complex harmonies and chromatic relationships. It remains a valuable tool for understanding tonal harmony and has influenced various music theorists and composers.