The Tonnetz (German for "tone-network") is a conceptual lattice
diagraminvented by Leonhard Eulerin 1739 that shows a two-dimensional tonal pitch spacecreated 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 Riemannexplored 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 topologyequivalent to S1×S1.
Neo-Riemannian music theory, including the study of the PLR group
* [http://members.cox.net/dysartp/ Music harmony and donuts] by Paul Dysart
* [http://www.robertkelleyphd.com/justtonnetz.htm Charting Enharmonicism on the Just-Intonation Tonnetz] by Robert T. Kelley
* [http://www.cis.upenn.edu/~lsaul/papers/visual_icmc05.pdf Visualization of low-dimensional structure in tonal pitch space] (.pdf format)
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