Chordal space
Encyclopedia
Music theorists have often used graphs
, tiling
s, 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.
in 1728;
he proposed placing the major and minor chords in a circular arrangement of twenty-four chords arranged according to the circle of fifths
; reading clockwise, ... F, d, C, a, G, ... (Capital letters represent major chords and small letters represent minor.) 1737, David Kellner
proposed an alternate arrangement, with the 12 major chords and 12 minor chords placed on concentric circles. Each chord was vertically aligned with its relative major or minor.
F. G. Vial and Gottfried Weber
suggested a grid graph or square lattice
model of chordal space; Weber's graph, centered on C major, is:
This was first proposed by Vial (1767) and later used by Gottfried Weber
, Hugo Riemann
, and Arnold Schoenberg
. Its advantage over Heinichen's and Kellner's models is that it represents a much richer set of chordal relationships. On the graph, every triad is related to its upper and lower neighbors by fifth-transposition
; its left and right neighbors are its parallel and relative triads. In addition, every triad is diagonally adjacent to the triad whose root is a major third a way, and which shares two of its three notes. A variety of other common-tone and voice leading relationships can be found among neighboring triads on the graph.
: every chord on the Vial/Weber chordal space can be associated with a triangle on the "Tonnetz" or two-dimensional pitch space discussed there.
The close correspondence between these properties -- shared common tones, efficient voice leading, and the two-dimensional pitch lattices -- is in some sense a lucky accident. As Richard Cohn
(1997) explained, analogous constructions depicting relationships among other types of chords do not have these properties.
Interest in common-tones and voice leading gradually led music theorists to modify Heinichen's original proposal. In the circular arrangement F - d - C - a ..., the chords F and d share two common tones, and can be linked by efficient voice leading. However, the chords d and C do not share any common tones, and cannot be linked by very efficient voice leading. By contrast in the series C - a - F - d ..., every chord shares two notes with its neighbors and can be transformed into them by moving one note by one or two semitones. The resulting pattern of chords can be generated in the Vial/Weber space, by moving upward along adjacent columns in the space.
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...
, tiling
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s, and geometrical spaces to represent the relationship between chord
Chord (music)
A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...
s. We can describe these spaces as chord spaces or chordal spaces, though the terms are relatively recent in origin.
History of chordal space
One of the earliest graphical models of chord-relationships was devised by Johann David HeinichenJohann David Heinichen
Johann David Heinichen was a German Baroque composer and music theorist who brought the musical genius of Venice to the court of Augustus the Strong in Dresden...
in 1728;
he proposed placing the major and minor chords in a circular arrangement of twenty-four chords arranged according to the circle of fifths
Circle of fifths
In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
; reading clockwise, ... F, d, C, a, G, ... (Capital letters represent major chords and small letters represent minor.) 1737, David Kellner
David Kellner
David Kellner was a German composer of the baroque period and a contemporary of Bach....
proposed an alternate arrangement, with the 12 major chords and 12 minor chords placed on concentric circles. Each chord was vertically aligned with its relative major or minor.
| F | C | G | D | A |
| d | a | e | b | f♯ |
F. G. Vial and Gottfried Weber
Gottfried Weber
Jacob Gottfried Weber , was a prominent German writer on music , composer, and jurist....
suggested a grid graph or square lattice
Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...
model of chordal space; Weber's graph, centered on C major, is:
| d♯ | F♯ | f♯ | A | a | C | c |
| g♯ | B | b | D | d | F | f |
| c♯ | E | e | G | g | B♭ | b♭ |
| f♯ | A | a | C | c | E♭ | e♭ |
| b | D | d | F | f | A♭ | a♭ |
| e | G | g | B♭ | b♭ | D♭ | d♭ |
| a | C | c | E♭ | e♭ | G♭ | g♭ |
This was first proposed by Vial (1767) and later used by Gottfried Weber
Gottfried Weber
Jacob Gottfried Weber , was a prominent German writer on music , composer, and jurist....
, 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...
, and Arnold 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...
. Its advantage over Heinichen's and Kellner's models is that it represents a much richer set of chordal relationships. On the graph, every triad is related to its upper and lower neighbors by fifth-transposition
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
; its left and right neighbors are its parallel and relative triads. In addition, every triad is diagonally adjacent to the triad whose root is a major third a way, and which shares two of its three notes. A variety of other common-tone and voice leading relationships can be found among neighboring triads on the graph.
Principles of chordal space
The Vial/Weber chordal space depicts two different sorts of relationships: shared common tones and efficient voice leading. For example, the proximity of the C major and e minor chords reflects the fact that the two chords share two common tones, E and G. Moreover, one chord can be transformed into another by moving a single note by just one semitone: to transform a C major chord into an E minor chord, one need only move C to B. Furthermore, the Vial/Weber chordal space is closely related to the two-dimensional lattices described in the article on pitch spacePitch 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...
: every chord on the Vial/Weber chordal space can be associated with a triangle on the "Tonnetz" or two-dimensional pitch space discussed there.
The close correspondence between these properties -- shared common tones, efficient voice leading, and the two-dimensional pitch lattices -- is in some sense a lucky accident. As Richard Cohn
Richard Cohn
Richard Cohn is a music theorist and Battell Professor of Music Theory at Yale. Early in his career, he specialized in the music of Béla Bartók, but more recently has written about Neo-Riemannian theory as well as metric dissonance.-External links:*...
(1997) explained, analogous constructions depicting relationships among other types of chords do not have these properties.
Interest in common-tones and voice leading gradually led music theorists to modify Heinichen's original proposal. In the circular arrangement F - d - C - a ..., the chords F and d share two common tones, and can be linked by efficient voice leading. However, the chords d and C do not share any common tones, and cannot be linked by very efficient voice leading. By contrast in the series C - a - F - d ..., every chord shares two notes with its neighbors and can be transformed into them by moving one note by one or two semitones. The resulting pattern of chords can be generated in the Vial/Weber space, by moving upward along adjacent columns in the space.
Further reading
- Lerdahl, Fred (2001). Tonal Pitch Space, pp. 42–43. Oxford: Oxford University Press. ISBN 0-19-505834-8.