 | This article possibly contains original research .(December 2007) |
Music theorists have often used graphs, tilings, and geometrical spaces to represent the relationship between chords. These spaces can be described as chord spaces or chordal spaces, though the terms are relatively recent in origin.[ citation needed ]
One of the earliest graphical models of chord-relationships was devised by Johann David Heinichen 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 | C | G | D | A |
| d | a | e | b | f♯ |
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:
| 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, 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 major triad is diagonally adjacent to the minor triad whose root is a major third above, and which shares two of its three notes (this is the diagonal above and to the left); every minor triad is diagonally adjacent to the major triad whose root is a third below, and which shares two of its three notes (this is the diagonal below and to the right). A variety of other common-tone and voice leading relationships can be found among neighboring triads on the graph.
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 space: 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.
The major scale is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double its frequency so that it is called a higher octave of the same note.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
In music theory, a leading-tone is a note or pitch which resolves or "leads" to a note one semitone higher or lower, being a lower and upper leading-tone, respectively. Typically, the leading tone refers to the seventh scale degree of a major scale, a major seventh above the tonic. In the movable do solfège system, the leading-tone is sung as ti.
In music, modulation is the change from one tonality to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest. Treatment of a chord as the tonic for less than a phrase is considered tonicization.
Modulation is the essential part of the art. Without it there is little music, for a piece derives its true beauty not from the large number of fixed modes which it embraces but rather from the subtle fabric of its modulation.
In music theory, the circle of fifths is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music, the sequence is: C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯ (F), C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle.
In music theory, a major chord is a chord that has a root, a major third, and a perfect fifth. When a chord comprises only these three notes, it is called a major triad. For example, the major triad built on C, called a C major triad, has pitches C–E–G:
In music theory, a diminished triad is a triad consisting of two minor thirds above the root. It is a minor triad with a lowered (flattened) fifth. When using chord symbols, it may be indicated by the symbols "dim", "o", "m♭5", or "MI(♭5)". However, in most popular-music chord books, the symbol "dim" and "o" represents a diminished seventh chord, which in some modern jazz books and music theory books is represented by the "dim7" or "o7" symbols.
In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.
In music, a triad is a set of three notes that can be stacked vertically in thirds. Triads are the most common chords in Western music.
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 under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes. Chordal spaces model relationships between chords.
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. Closely related to pitch class space is pitch space, which represents pitches rather than pitch classes, and chordal space, which models relationships between chords.
Modes of limited transposition are musical modes or scales that fulfill specific criteria relating to their symmetry and the repetition of their interval groups. These scales may be transposed to all twelve notes of the chromatic scale, but at least two of these transpositions must result in the same pitch classes, thus their transpositions are "limited". They were compiled by the French composer Olivier Messiaen, and published in his book La technique de mon langage musical.
In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.
Transformational theory is a branch of music theory developed by David Lewin in the 1980s, and formally introduced in his 1987 work, Generalized Musical Intervals and Transformations. The theory—which models musical transformations as elements of a mathematical group—can be used to analyze both tonal and atonal music.
Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.
In music theory, an inversion is a rearrangement of the top-to-bottom elements in an interval, a chord, a melody, or a group of contrapuntal lines of music. In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in musical set theory.
Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly to each other, without necessary reference to a tonic. Initially, those harmonies were major and minor triads; subsequently, neo-Riemannian theory was extended to standard dissonant sonorities as well. Harmonic proximity is characteristically gauged by efficiency of voice leading. Thus, C major and E minor triads are close by virtue of requiring only a single semitonal shift to move from one to the other. Motion between proximate harmonies is described by simple transformations. For example, motion between a C major and E minor triad, in either direction, is executed by an "L" transformation. Extended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire system of harmonic relations. Where consensus is lacking is on the question of what is most central to the theory: smooth voice leading, transformations, or the system of relations that is mapped by the geometries. The theory is often invoked when analyzing harmonic practices within the Late Romantic period characterized by a high degree of chromaticism, including work of Schubert, Liszt, Wagner and Bruckner.
The Harmonic Table note-layout, or tonal array, is a key layout for musical instruments that offers interesting advantages over the traditional keyboard layout.
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.
Among alternative tunings for guitar, a major-thirds tuning is a regular tuning in which each interval between successive open strings is a major third. Other names for major-thirds tuning include major-third tuning, M3 tuning, all-thirds tuning, and augmented tuning. By definition, a major-third interval separates two notes that differ by exactly four semitones.
