In musical tuning and harmony, the Tonnetz (German for "tone-network") is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. The Tonnetz shows a two-dimensional pitch space created by the network of relationships between musical pitches in just intonation. The space was rediscovered in 1866 by Arthur von Oettingen.
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- In musical tuning and harmony, the Tonnetz (German for "tone-network") is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. The Tonnetz shows a two-dimensional pitch space created by the network of relationships between musical pitches in just intonation. The space was rediscovered in 1866 by Arthur von Oettingen. The influential musicologist Hugo Riemann explored the capacity of the space to chart harmonic motion between chords and modulation between keys. Neo-Riemannian music theory's P, L, and R operations are easier to demonstrate on a tonnetz than to explain using a standard piano keyboard. Recent research (by music psychologist Carol Krumhansl, music theorist David Lewin, and others) substitutes equal temperament and enharmonic equivalence for just intonation, and explores the group-theoretic and topological aspects of the space. The harmonic table note layout is a recently developed musical interface which utilises a note layout topologically equivalent to the tonnetz. A Tonnetz of the syntonic temperament can be derived from a given isomorphic keyboard by connecting lines of successive perfect fifths, lines of successive major thirds, and lines of successive minor thirds. Such a Tonnetz, like the isomorphic keyboard itself, is tuning invariant. 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. The topology of the syntonic temperament's tonnetz is generally cylindrical. Any syntonic "equal" tuning (i.e. , a tempered width of the perfect fifth which divides the octave into a number of equally-wide intervals) snaps this cylinder into a torus (the shape of a ring doughnut, a hula hoop or an inflated tire), showing that it has a topology equivalent to S×S.
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- In musical tuning and harmony, the Tonnetz (German for "tone-network") is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. The Tonnetz shows a two-dimensional pitch space created by the network of relationships between musical pitches in just intonation. The space was rediscovered in 1866 by Arthur von Oettingen.
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