Tonnetz

In musical tuning and harmony, the (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.

History through 1900

The ''Tonnetz'' originally appeared in Leonhard Euler's 1739 . 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, , 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 ., 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.

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♯-G♯-D♯-A♯-E♯-B♯-F𝄪-C𝄪-G𝄪- (etc.) Starting with F, after 12 perfect fifths, one reaches E♯. Perfect fifths in just intonation are slightly larger than the compromised fifths used in equal temperament tuning systems more common in the present. This means that when one stacks 12 fifths starting from F, the E♯ we arrive at will not be seven octaves above the F we started with. Oettingen and Riemann's ''Tonnetz'' thus extended on infinitely in every direction without actually repeating any pitches. In the twentieth century, composer-theorists such as Ben Johnston and James Tenney continued to develop theories and applications involving just-intoned ''Tonnetze''.

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.

Twentieth-century reinterpretation

Neo-Riemannian music theorists David Lewin and Brian Hyer revived the ''Tonnetz'' to further explore properties of pitch structures. Modern music theorists generally construct the ''Tonnetz'' in equal temperament and without distinction between octave transpositions of a pitch (i.e., using pitch classes). Under equal temperament, the never-ending series of ascending fifths mentioned earlier becomes a cycle. 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.

Neo-Riemannian theorists have also used the ''Tonnetz'' to visualize non-tonal triadic relationships. For example, the diagonal going up and to the left from C in the diagram at the beginning of the article forms a division of the octave in three major thirds: C-A♭-E-C (the E is actually an F♭, and the final C a D♭♭). Richard Cohn argues that while a sequence of triads built on these three pitches (C major, A♭ major, and E major) cannot be adequately described using traditional concepts of functional harmony, this cycle has smooth voice leading and other important group properties which can be easily observed on the ''Tonnetz''.

Similarities to other graphical systems

The harmonic table note layout is a note layout that is topologically equivalent to the ''Tonnetz'', and is used on several music instruments that allow playing major and minor chords with a single finger.

The ''Tonnetz'' can be overlayed on the Wicki–Hayden note layout, where the major second can be found halfway towards the major third.

The ''Tonnetz'' is the dual graph of Schoenberg's chart of the regions, and of course ''vice versa''. Research into music cognition has demonstrated that the human brain uses a "chart of the regions" to process tonal relationships.

See also

* Chordal space
* Fokker periodicity block
* Neo-Riemannian theory
* Musical set-theory
* Riemannian theory
* Transformational theory
* Tuning theory
* '' Traité de l'harmonie réduite à ses principes naturels''

References

Further reading

*Johnston, Ben (2006). "Rational Structure in Music", ''"Maximum Clarity" and Other Writings on Music'', edited by Bob Gilmore. Urbana: University of Illinois Press. .
*Wannamaker, Robert,
The Music of James Tenney, Volume 1: Contexts and Paradigms
' (University of Illinois Press, 2021), 155-65.

External links

by Robert T. Kelley

The Tonnetz
(interactive visualization that works with any keyboard) by Corentin Guichaoua and Moreno Andreatta.

TonnetzViz (interactive visualization)
by Ondřej Cífka
a modified version
by Anton Salikhmetov

Midi-Instrument based on Tonnetz (Harmonic Table)
by C-Thru-Music

{{Riemannian theory

Diagrams
Lattice theory
Pitch space
Topology
Music theory