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. See lattice for other meanings of this term, both within and without mathematics. ... Sample flowchart diagram A diagram is a simplified and structured visual representation of concepts, ideas, constructions, relations, statistical data, anatomy etc used in all aspects of human activities to visualize and clarify the topic. ... Euler redirects here. ... In music pitch space is the modeling of pitch relationships, represented through mathematical models, most often multidimensional, describing how near or far pitches are from each other. ... In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. ... Dr. Hugo Riemann (full name: Karl Wilhelm Julius Hugo Riemann) (July 18, 1849 - July 10, 1919) was a German musicologist. ... David Lewin (July 2, 1933-May 5, 2003) was an American music theorist and composer. ...

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. A torus. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...

See also

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

In music, just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. ... In music pitch space is the modeling of pitch relationships, represented through mathematical models, most often multidimensional, describing how near or far pitches are from each other. ... In music, there are two common meanings for tuning: Tuning practice The act of tuning an instrument or voice. ...

External links

• Music harmony and donuts by Paul Dysart
• [1] a Tonnetz generation program.
• Tonnetz at the Encyclopedia of Microtonal Music Theory
• Charting Enharmonicism on the Just-Intonation Tonnetz by Robert T. Kelley
• Score Generation in Voice-Leading Orbifolds (.pdf format)
• Visualization of low-dimensional structure in tonal pitch space (.pdf format)