Tonnetz

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The Tonnetz (German for "tone-network") is a conceptual lattice diagram invented by Leonhard Euler in 1739 that shows a two-dimensional tonal pitch space created by the network of relationships between musical pitches in just intonation. The space was rediscovered in 1866 by Artur von Oettingen. The influential musicologist Hugo Riemann explored the capacity of the space to chart harmonic motion between chords and modulation between keys. 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 equal temperament Tonnetz can be rolled into the shape of a torus (the shape of a ring doughnut, a hula hoop or an inflated tire), showing that it has a topology equivalent to S¹×S¹.

[edit] See also

• Just intonation
• Pitch space
• Musical tuning
• Tuning theory
• Neo-Riemannian music theory, including the study of the PLR group

[edit] External links

• Music harmony and donuts by Paul Dysart
• Charting Enharmonicism on the Just-Intonation Tonnetz by Robert T. Kelley
• Visualization of low-dimensional structure in tonal pitch space (.pdf format)