Tonnetz
Encyclopedia
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 fifth
s, lines of successive major third
s, and lines of successive minor third
s. 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 S1×S1.
and mediant
relationships are represented horizontally and diagonally, as they may be today.
------F-- --E-DbA#-
/\ --|/|/|--
----C-A-- --A-GbD#-
/\/\ --|/|/|--
--G-E-Cb- --D-B-G#-
/\/\/ --|/|/|--
D-H-Gb--- --G-E-C#-
\/\/ --|/|/|--
--E-Db--- --C-A-F#-
\/ --|/|/|--
----B---- --F-D-B--
While Euler was attempting to represent the systematic relationship between notes in what today is called just intonation
, the contemporary chart represents these relationships within equal temperament.
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
and harmony
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
, the Tonnetz (German for "tone-network") is a conceptual lattice
Lattice (music)
In musical tuning, a lattice "is a way of modeling the tuning relationships in a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio...
diagram
Diagram
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a three-dimensional visualization which is then projected onto the two-dimensional surface...
representing tonal space first described by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in 1739.
The Tonnetz shows a two-dimensional pitch space
Pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...
created by the network of relationships between musical pitches in just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
. The space was rediscovered in 1866 by Arthur von Oettingen. The influential musicologist Hugo Riemann
Hugo Riemann
Karl Wilhelm Julius Hugo Riemann was a German music theorist.-Biography:Riemann was born at Grossmehlra, Schwarzburg-Sondershausen. He was educated in theory by Frankenberger, studied the piano with Barthel and Ratzenberger, studied law, and finally philosophy and history at Berlin and Tübingen...
explored the capacity of the space to chart harmonic motion between chords and modulation between keys. Neo-Riemannian music theory's
Neo-Riemannian theory
Neo-Riemannian theory refers to a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer...
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
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
, 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
Harmonic table note layout
The Harmonic Table note-layout, or tonal array, is a key layout for musical instruments that offers interesting advantages over the traditional keyboard layout....
is a recently developed musical interface which utilises a note layout topologically equivalent to the tonnetz.
A Tonnetz of the syntonic temperament
Syntonic temperament
The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
can be derived from a given isomorphic keyboard
Isomorphic keyboard
An isomorphic keyboard is a musical input device consisting of a two-dimensional array of note-controlling elements on which any given sequence and/or combination of musical intervals has the “same shape” on the keyboard wherever it occurs – within a key, across keys, across octaves, and across...
by connecting lines of successive perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s, lines of successive major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
s, and lines of successive minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
s. Such a Tonnetz, like the isomorphic keyboard itself, is tuning invariant.
The Tonnetz is the dual graph
Dual graph
In mathematics, the dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G. The term "dual" is used because this property is symmetric, meaning that if H is a dual...
of Schoenberg
Arnold Schoenberg
Arnold Schoenberg was an Austrian composer, associated with the expressionist movement in German poetry and art, and leader of the Second Viennese School...
'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
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of the syntonic temperament
Syntonic temperament
The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
's tonnetz is generally cylindrical
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
. Any syntonic "equal
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
" tuning (i.e., a tempered width of the perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
which divides the octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
into a number of equally-wide intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
) snaps this cylinder into a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
(the shape of a ring doughnut, a hula hoop
Hula hoop
A hula hoop is a toy hoop that is twirled around the waist, limbs or neck.Although the exact origins of hula hoops are unknown, children and adults around the world have played with hoops, twirling, rolling and throwing them throughout history...
or an inflated tire), showing that it has a topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
equivalent to S1×S1.
Chart
Euler's image (above) may be familiar, with a few adjustments, to modern musicians. For example, today it is far more common to represent the tonic-dominant relationship vertically and ascending (C-G), while in contrast Euler represents it vertically but descending (G-C). SubmediantSubmediant
In music, the submediant is the sixth scale degree of the diatonic scale, the 'lower mediant' halfway between the tonic and the subdominant or 'lower dominant'...
and mediant
Mediant
In music, the mediant is the third scale degree of the diatonic scale, being the note halfway between the tonic and the dominant. Similarly, the submediant is halfway between the tonic and subdominant...
relationships are represented horizontally and diagonally, as they may be today.
------F-- --E-DbA#-
/\ --|/|/|--
----C-A-- --A-GbD#-
/\/\ --|/|/|--
--G-E-Cb- --D-B-G#-
/\/\/ --|/|/|--
D-H-Gb--- --G-E-C#-
\/\/ --|/|/|--
--E-Db--- --C-A-F#-
\/ --|/|/|--
----B---- --F-D-B--
While Euler was attempting to represent the systematic relationship between notes in what today is called just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
, the contemporary chart represents these relationships within equal temperament.
External links
- Music harmony and donuts by Paul Dysart
- Charting Enharmonicism on the Just-Intonation Tonnetz by Robert T. Kelley
- Midi-Instrument based on Tonnetz (Melodic Table) by The Shape of Music
- Midi-Instrument based on Tonnetz (Harmonic Table) by C-Thru-Music