Pitch space - AbsoluteAstronomy.com

Music theory

Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

, 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 under consideration, the models may be multidimensional

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

. Models of pitch space are often graphs

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

, groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, lattices

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...

, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class space

Pitch class space

In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...

s, which represent relationships between pitch class

Pitch class

In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

es. (Some of these models are discussed in the entry on modulatory space

Modulatory space

The spaces described in this article are pitch class spaces which model the relationships between pitch classes in some musical system. These models are often graphs, groups or lattices...

, though readers should be advised that the term "modulatory space" is not a standard music-theoretical term.) Chordal space

Chordal space

Music theorists have often used graphs, tilings, and geometrical spaces to represent the relationship between chords. We can describe these spaces as chord spaces or chordal spaces, though the terms are relatively recent in origin....

s model relationships between chords.

Linear and helical pitch space

The simplest pitch space model is the real line. A fundamental frequency f is mapped to a real number p according to the equation

This creates a linear space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and middle C is assigned the number 60, as it is in MIDI. 440 Hz is the standard frequency of 'concert A', which is the note 9 semitones above 'middle C'. Distance in this space corresponds to physical distance on keyboard instruments, orthographical distance in Western musical notation, and psychological distance as measured in psychological experiments and conceived by musicians. The system is flexible enough to include "microtones" not found on standard piano keyboards. For example, the pitch halfway between C (60) and C# (61) can be labeled 60.5.

One problem with linear pitch space is that it does not model the special relationship between octave-related pitches, or pitches sharing the same pitch class

Pitch class

In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

. This has led theorists such as M. W. Drobish (1855) and Roger Shepard (1982) to model pitch relations using a helix. In these models, linear pitch space is wrapped around a cylinder so that all octave-related pitches lie along a single line. Care must be taken when interpreting these models however, as it is not clear how to interpret "distance" in the three-dimensional space containing the helix; nor is it clear how to interpret points in the three-dimensional space not contained on the helix itself.

Higher-dimensional pitch spaces

Other theorists, such as 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...

(1739), Hermann von Helmholtz

Hermann von Helmholtz

Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

(1863/1885), Arthur von Oettingen (1866), 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...

(who should not be confused with the famous mathematician Bernhard Riemann

Bernhard Riemann

Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

), and Christopher Longuet-Higgins

H. Christopher Longuet-Higgins

Hugh Christopher Longuet-Higgins FRS was both a theoretical chemist and a cognitive scientist. He was born on April 11, 1923 in Kent, England and died on March 27, 2004....

(1978) have modeled pitch relationships using two-dimensional (or higher-dimensional) lattice

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

s, under the name of Tonnetz

Tonnetz

In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739....

. In these models, one dimension typically corresponds to acoustically-pure "perfect fifths" while the other corresponds to "major thirds." (Variations are possible in which one axis corresponds to acoustically pure minor thirds.) Additional dimensions can be used to represent additional intervals including—most typically—the octave.

A#3	—	E#4	—	B#4	—	FX5	—	CX6	—	GX6
>		>		>		>		>		>
F#3	—	C#4	—	G#4	—	D#5	—	A#5	—	E#6
>		>		>		>		>		>
D3	—	A3	—	E4	—	B4	—	F#5	—	C#6
>		>		>		>		>		>
Bb2	—	F3	—	C4	—	G4	—	D5	—	A5
>		>		>		>		>		>
Gb2	—	Db3	—	Ab3	—	Eb4	—	Bb4	—	F5
>		>		>		>		>		>
Ebb2	—	Bbb2	—	Fb3	—	Cb4	—	Gb4	—	Db5

All of these models attempt to capture the fact that intervals separated by acoustically pure intervals such as octaves, perfect fifths, and major thirds are thought to be perceptually closely related. However, proximity in these spaces need not represent physical proximity on musical instruments: by moving one's hands a very short distance on a violin string, one can move arbitrarily far in these multiple-dimensional models. For this reason, it is hard to assess the psychological relevance of distance as measured by these lattices.

History of pitch space

The idea of pitch space goes back at least as far as the ancient Greek music theorists known as the Harmonists. To quote one of their number, Bacchius,
"And what is a diagram? A representation of a musical system. And we use a diagram so that, for students of the subject, matters which are hard to grasp with the hearing may appear before their eyes." (Bacchius, in Franklin, Diatonic Music in Ancient Greece.) The Harmonists drew geometrical pictures so that the intervals of various scales could be compared visually; they thereby located the intervals in a pitch space.

Higher-dimensional pitch spaces were the earliest subjects of musical investigation. The use of a lattice

Lattice (group)

was proposed by Euler (1739) to model just intonation using an axis

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

of perfect fifths and another of major thirds. Similar models were the subject of intense investigation in the nineteenth century, chiefly by theorists such as Oettingen and Riemann

Hugo Riemann

(Cohn 1997). Contemporary theorists such as James Tenney

James Tenney

James Tenney was an American composer and influential music theorist.-Biography:Tenney was born in Silver City, New Mexico, and grew up in Arizona and Colorado. He attended the University of Denver, the Juilliard School of Music, Bennington College and the University of Illinois...

(1983) and W.A. Mathieu (1997) carry on this tradition.

M.W. Drobisch (1855) was the first to suggest a helix

Helix

A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

(i.e. the spiral of fifths) to represent octave equivalence and recurrence (Lerdahl, 2001), and hence to give a model of pitch space. Shepard (1982) regularizes Drobish's helix, and extends it to a double helix of two wholetone scales over a circle of fifths which he calls the "melodic map" (Lerdahl, 2001). Michael Tenzer

Michael Tenzer

Michael Tenzer is a composer, performer, educator and scholar.Tenzer was born in New York City and studied music at Yale University and University of California, Berkeley...

suggests its use for Balinese gamelan

Gamelan

A gamelan is a musical ensemble from Indonesia, typically from the islands of Bali or Java, featuring a variety of instruments such as metallophones, xylophones, drums and gongs; bamboo flutes, bowed and plucked strings. Vocalists may also be included....

music since the octave

Pseudo-octave

A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1 , that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency...

s are not 2:1 and thus there is even less octave equivalence than in western tonal music (Tenzer, 2000). See also chromatic circle

Chromatic circle

The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale...

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Linear and helical pitch space

Higher-dimensional pitch spaces

History of pitch space

See also

Further reading

External links