Tonnetz

In musical tuning and harmony, the Tonnetz (German: tone-network) 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.

The Tonnetz originally appeared in Euler's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae. 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). The space was rediscovered in 1858 by Ernst Naumann, 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 motion between chords and modulation between keys. Similar understandings of the Tonnetz appeared in the work of many late-19th century German music theorists.

Oettingen and Riemann both conceived of the relationships in the chart being defined through just intonation, which uses pure intervals. One can extend out one of the horizontal rows of the Tonnetz indefinitely, to form a never-ending sequence of perfect fifths: F-C-G-D-A-E-B-F#-C#(Db)-Ab-Eb-Bb-F-C- (etc.) Starting with F, after 12 perfect fifths, one reaches another F. However, perfect fifths in just intonation are slightly larger than the fifths used in equal temperament tuning systems more common in the present. This means that the F one arrives at will not be a whole number of octaves above the F we started with. Oettingen and Riemann's Tonnetz thus extended on infinitely in every direction without actually repeating any pitches.

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.

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