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. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.
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
Euler's Tonnetz, pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a major third above F). Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst, discusses the relationships between keys, presenting them in a network analogous to Euler's Tonnetz, but showing keys rather than notes. The Tonnetz itself was rediscovered in 1858 by Ernst Naumann in his Harmoniesystem in dualer Entwickelung., and was disseminated in an 1866 treatise of Arthur von Oettingen. Oettingen and the influential musicologist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann) explored the capacity of the space to chart harmonic modulation between chords and motion between keys. Similar understandings of the Tonnetz appeared in the work of many late-19th century German music theorists.
The appeal of the Tonnetz to 19th-century German theorists was that it allows spatial representations of tonal distance and tonal relationships. For example, looking at the dark blue A minor triad in the graphic at the beginning of the article, its parallel major triad (A-C♯-E) is the triangle right below, sharing the vertices A and E. The relative major of A minor, C major (C-E-G) is the upper-right adjacent triangle, sharing the C and the E vertices. The dominant triad of A minor, E major (E-G♯-B) is diagonally across the E vertex, and shares no other vertices. One important point is that every shared vertex between a pair of triangles is a shared pitch between chords - the more shared vertices, the more shared pitches the chord will have. This provides a visualization of the principle of parsimonious voice-leading, in which motions between chords are considered smoother when fewer pitches change. This principle is especially important in analyzing the music of late-19th century composers like Wagner, who frequently avoided traditional tonal relationships.
https://en.wikipedia.org/wiki/Tonnetz
Harmonic Trajectories in the Tonnetz
Introduction
The Neo-Riemannian Tonnetz is a way of visualizing musical relationships between chords. It was developed by music theorists to help understand how chords can transition smoothly from one to another.
Imagine a grid or network of interconnected points. Each point represents a different chord. The horizontal lines connect chords that are closely related, while the vertical lines connect chords that share similar tonal qualities.
The Tonnetz is based on the idea that chords can be transformed or changed into one another through small movements. These transformations are represented by diagonal lines on the Tonnetz. For example, a chord can be transformed into another chord by changing one note at a time, moving in a specific direction on the grid.
By studying the Tonnetz, musicians and theorists can analyze chord progressions and see how different chords are related to each other. It provides a visual representation of the harmonic possibilities and helps to explain the underlying structure of music.
In simple terms, the Neo-Riemannian Tonnetz is a grid that shows how chords in music are connected and can be transformed smoothly from one to another. It helps musicians and theorists understand how chords fit together and how they can create pleasing transitions in music.