Pitch space

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Pitch space

In music theory

Music theory

Music theory is the field of study that deals with how music works. It examines the language and notation of music. It identifies patterns that govern composer techniques....
, 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 mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
. 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 Binary operation that combines any two of its element to form a third element....
, lattices

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
, or geometrical figures such as helixes.

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In music theory

Music theory

Dimension

Graph (mathematics)

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
, lattices

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
, 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 that results when we ignore the difference between octave-related pitches. Mathematically, it is a quotient space that results from identifying or "gluing together" pitches sharing the same pitch class....
s, which represent relationships between pitch class

Pitch class

In music, a pitch class is a set of all Pitch 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 graph , group or lattice ....
, 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 graph , tessellations, and geometrical spaces to represent the relationship between Chord s. 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 Pitch 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 Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany.Euler made important discoveries in fields as diverse as calculus and graph theory....
(1739), Hermann von Helmholtz

Hermann von Helmholtz

Hermann Ludwig Ferdinand von Helmholtz was a Germany physician and physicist who made significant contributions to several widely varied areas of modern science....
(1863/1885), Arthur von Oettingen

Arthur von Oettingen

Arthur Joachim von Oettingen was a Baltic German physicist and musical theory who was born at the Ludenhof Estate in Dorpat, Livonia. He was the brother of theologian Alexander von Oettingen and ophthalmologist Georg von Oettingen ....
(1866), Hugo Riemann

Hugo Riemann

Karl Wilhelm Julius Hugo Riemann was a Germany music theory. He should not be confused with the mathematician Bernhard Riemann.Riemann was born at Grossmehlra, near Sondershausen....
(who should not be confused with the famous mathematician Bernhard Riemann

Bernhard Riemann

Georg Friedrich Bernhard Riemann was a Germany mathematics who made important contributions to mathematical analysis and differential geometry, some of them paving the way for the later development of general relativity....
), and Christopher Longuet-Higgins

H. Christopher Longuet-Higgins

Hugh Christopher Longuet-Higgins Fellow of the Royal Society was both a theoretical chemistry and a Cognitive science. 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 linear span the real number vector space Rn....
s, under the name of Tonnetz

Tonnetz

The Tonnetz 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....
. 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)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
was proposed by Euler (1739) to model just intonation using an axis

Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
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

Karl Wilhelm Julius Hugo Riemann was a Germany music theory. He should not be confused with the mathematician Bernhard Riemann.Riemann was born at Grossmehlra, near Sondershausen....
(Cohn 1997). Contemporary theorists such as James Tenney

James Tenney

James Tenney was an United States composer and influential music theory....
(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 special kind of space curve, i.e. a Differentiable manifold curve in three-space. As a mental image of a helix one may take the spring ....
(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. He studied music at Yale University and University of California, Berkeley . After teaching at Yale from 1986-96, he moved to University of British Columbia where he teaches ethnomusicology, musical composition, music theory and gamelan performance, co-directs the doctoral prog...
suggests its use for Balinese gamelan

Gamelan

File:Javanese Gamelan.jpgA 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....
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....
.

Pitch space

Linear and helical pitch space

Higher-dimensional pitch spaces

History of pitch space

See also

Further reading

External links