Pitch class 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 class space is the circular space

Space

Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

representing all the notes in a musical 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"...

.
In this space, there is no distinction between tones that are separated by an integral number of octaves. For example, C4, C5, and C6, though different pitches, are represented by the same point in pitch class space.

Since pitch class space is a circle, we return to our starting point by taking a series of steps in the same direction: beginning with C, we can move "upward" in pitch class space, through the pitch classes C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, and B, returning finally to C. By contrast, 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...

is a linear space: the more steps we take in a single direction, the further we get from our starting point.

Tonal pitch class space

Deutsch

Diana Deutsch

Diana Deutsch is a British-American perceptual and cognitive psychologist, born in London, England. She is currently Professor of Psychology at the University of California, San Diego, and is one of the most prominent researchers on the psychology of music...

and Feroe (1981), and Lerdahl

Fred Lerdahl

Alfred Whitford Lerdahl is the Fritz Reiner Professor of Musical Composition at Columbia University, and a composer and music theorist best known for his work on pitch space and cognitive constraints on compositional systems or "musical grammar[s]." He has written many orchestral and chamber...

and Jackendoff

Ray Jackendoff

Ray Jackendoff is an American linguist. He is professor of philosophy, Seth Merrin Chair in the Humanities and, with Daniel Dennett, Co-director of the Center for Cognitive Studies at Tufts University...

(1983) use a "reductional format" to represent the perception of pitch class relations in tonal contexts. These two-dimensional models resemble bar graphs, using height to represent a pitch class's degree of importance or centricity. Lerdahl's version uses five levels: the first (highest) contains only the tonic

Tonic (music)

In music, the tonic is the first scale degree of the diatonic scale and the tonal center or final resolution tone. The triad formed on the tonic note, the tonic chord, is thus the most significant chord...

, the second contains tonic and dominant

Dominant (music)

In music, the dominant is the fifth scale degree of the diatonic scale, called "dominant" because it is next in importance to the tonic,and a dominant chord is any chord built upon that pitch, using the notes of the same diatonic scale...

, the third contains tonic, 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...

, and dominant, the fourth contains all the notes of the diatonic scale

Diatonic scale

In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...

, and the fifth contains the chromatic scale

Chromatic scale

The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...

. In addition to representing centricity or importance, the individual levels are also supposed to represent "alphabets" that describe the melodic possibilities in tonal music (Lerdahl, 2001, 44–46). The model asserts that tonal melodies will be cognized in terms of one of the five levels a-e:

Level a:

Level b:

Level c:

Level d:

Level e:

D♭

E♭

F♯

A♭

B♭

(Lerdahl 1992,)

Note that Lerdahl's model is meant to be cyclical, with its right edge identical to its left. One could therefore display Lerdahl's graph as a series of five concentric circles representing the five melodic "alphabets." In this way one could unite the circular representation depicted at the beginning of this article with Lerdahl's flat two-dimensional representation depicted above.

According to David Kopp (2002, 1), "Harmonic space, or tonal space as defined by Fred Lerdahl, is the abstract nexus of possible normative harmonic connections in a system, as opposed to the actual series of temporal connections in a realized work, linear or otherwise."

Sources

Kopp David (2002). Chromatic Transformations in Nineteenth-Century Music. Cambridge University Press. ISBN 0-521-80463-9.
Lerdahl, Fred (1992). "Cognitive Constraints on Compositional Systems". Contemporary Music Review 6 (2), pp. 97–121.
Lerdahl, Fred (2001). Tonal Pitch Space. Oxford University Press. Full Text

Tonal pitch class space

Sources

Further reading

See also