Chord (music)

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This article describes pitch simultaneity and harmony in traditional Western styles. For information on non-Western styles, consult the articles specific to those styles.
Typical fingering for a second inversion C major chord on a guitar.

In music and music theory a chord (from Greek χορδή: string) is a set of three or more different notes from a specific key that sound simultaneously. Most often, in European-influenced music, chords are tertian sonorities that can be constructed as stacks of thirds relative to some underlying scale. Two-note combinations are typically referred to as dyads or intervals. A succession of chords is called a chord progression.

Four ways of notating or representing chords are often used: roman numerals, figured bass, macro symbols, and popular music symbols (Benward & Saker 2003, p.77).


[edit] History

The word chord comes from cord which is a Middle English shortening of accord. In the Middle Ages, Western harmony featured the perfect intervals of a fourth, a fifth, and an octave. In the 15th and 16th centuries, the major and minor triads (see below) became increasingly common, and were soon established as the default sonority for Western music. Four-note "seventh chords" were then widely adopted from the 17th century. The harmony of many contemporary popular Western genres continues to be founded in the use of triads and seventh chords, though far from universally. Notable exceptions include: modern jazz (especially circa 1960), in which chords often include at least five notes, with seven (and occasionally more) being quite common; and atonal or post-tonal contemporary classical music (including the music of some film scores), whose chords can be far more complex, rooted in such disparate harmonic philosophies that traditional terms such as triad are rarely useful.

Chords are so well-established in Western music that sonorities of two pitches, or even monophonic melodies, are often interpreted by listeners (musicians and non-musicians alike) as "implying" chords. This psychoacoustic phenomenon occurs as a result of a lifetime of exposure to the conventional harmonies of music, with the result that the brain "supplies" the complete expected chord in its absence. [1]

Composers can and do take advantage of this tendency to surprise the listener, by deliberately avoiding certain defining tones. For instance, a composition may be predominantly composed in the pentatonic minor scale, implying common Aeolian mode to the listener, before deliberately including a more uncommon tone in a melodic progression or chord, such as a major VI (signalling Dorian mode) or a flattened II (signalling Phrygian mode).

[edit] Constructing and naming chords

Instruments playing different notes create chords.

Every chord has certain characteristics, which include:

  • the number of chromas used in constructing the chord (or the number of distinct pitch classes from which the chord takes its notes)
  • the general type of intervals it contains: for example seconds, thirds, or fourths
  • its precise intervallic construction, sometimes called "chord quality": for example, if the chord is a triad, is the triad a major, minor, augmented or diminished?
  • the scale degree of the root note
  • whether the chord is inverted in register

[edit] Number of notes

One way of classifying chords is according to the number of distinct pitch classes used in their construction, a pitch class being identified by a degree of the scale (that is, a certain musical note, such as A, B, C, D, etc.) without regard to which octave it occurs in. Chords using three pitch classes are called triads. Chords using four notes are known as tetrads. Those using five are called pentads, and those using six are hexads. They are sometimes called trichords, tetrachords, pentachords and hexachords, however these terms more usually refer to contiguous pitch classes in some scale, usually spanning a perfect fourth, and not generally played simultaneously.

Theorists differ as to whether chords consist of at least three pitches. Otto Karolyi ([cite this quote], p.63), disagrees, writing that, "two or more notes sounded simultaneously are known as a chord. The vertical combination of three sounds: fundamental note, third and fifth, gives us a chord known as a triad." In contrast, Andrew Surmani (2004, p.72), writes that, "when three or more notes are sounded together, the combination is called a chord," and George T. Jones explains (1994, p.43) "two tones sounding together are usually termed an interval, while three or mores tones are called a chord." According to Monath (1984, p.37) "A chord is a combination of three or more tones sounded simultaneously for which the distances (called intervals) between the tones are based on a particular formula. (Two notes sounded simultaneously are not considered to be chords and are simply called intervals.)"

[edit] Type of interval

Many chords can be arranged as a series whose elements are separated by intervals that are all roughly the same size. For example, a C major triad contains the notes C, E, and G. These notes can be arranged in the series C-E-G, in which the first interval (C-E) is a major third, while the second interval (E-G) is a minor third. Any chord that can be arranged as a series of (major or minor) thirds is called a tertian chord. A chord such as C-D-E is a series of seconds, containing a major second (C-D) and a minor second (D-E). Such chords are called secundal. The chord C-F-B, which consists of a perfect fourth C-F and an augmented fourth (tritone) F-B is called quartal. Most Western music uses tertian chords.

On closer examination, however, the terms "secundal", "tertian" and "quartal" can become ambiguous. The terms "second," "third," and "fourth" (and so on) are often understood relative to a scale, but it is not always clear which scale they refer to. For example, consider the pentatonic scale G-A-C-D-F. Relative to the pentatonic scale, the intervals G-C and C-F are "thirds," since there is one note between them. Relative to the chromatic scale, however, the intervals G-C and C-F are "fourths" since they are five semitones wide. For this reason the chord G-C-F might be described both as "tertian" and "quartal," depending on whether one is measuring intervals relative to the pentatonic or chromatic scales.

The use of accidentals complicates the picture. The chord B-E-A is notated as a series of diminished fourths (B-E) and (E-A). However, the chord is enharmonically equivalent to (and sonically indistinguishable from) C-E-G, which is a series of major thirds (C-E) and (E-G). Notationally, then, B-E-A is a "fourth chord," even though it sounds identical to the tertian chord C-E-G. In some circumstances it is useful to talk about how a chord is notated, while in others it is useful to talk about how it sounds. Terms such as "tertian" and "quartal" can be used in either sense, and it is important to be clear about which is intended.

[edit] Scale degree

Chords are also distinguished and notated by the scale degree of their root note or bass note.

For example, since the first scale degree of the C major scale is the note C, a triad built on top of the note C would be called the one chord, which might be notated 1, I, or even C, in which case the assumption would be made that the key signature of the particular piece of music in question would indicate to the musician what function a C major triad was fulfilling, and that any special role of the chord outside of its normal diatonic function would be inferred from the context.

When taking any major (Ionian) scale and building a triad with a base in the scale, the second, third, and sixth intervals, when used as a root, will form a minor triad. The root, fourth, and fifth form a major triad, whereas the seventh will form a diminished triad. When in minor modes, building a triad upon the tonic, fourth and fifth degrees of the scale will result in a minor chord. Building upon scale degree two will result in a diminished chord, while building a triad upon scale degrees three, six and seven will yield major chords.

Roman numerals indicate the root of the chord as a scale degree within a particular major key as follows:

Roman numeral I ii iii IV V vi viio
Scale degree tonic supertonic mediant subdominant dominant submediant leading tone/subtonic

Many analysts use lower-case Roman numerals to indicate minor triads and upper-case for major ones, with degree and plus signs (o and +) to indicate diminished and augmented triads, respectively. When they are not used, all the numerals are capital, and the qualities of the chords are inferred from the other scale degrees that chord contains; for example, a chord built on VI in C major would contain the notes A, C, and E, and would therefore be a minor triad. Chords that are not on the scale can be indicated by placing a flat/sharp sign before the chord — for example, the chord of E flat major in the key of C major is represented by III.

The scale to whose scale degrees the Roman numerals refer may be indicated to the left (e.g. F:), but may also be understood from the key signature or other contextual clues.

Unlike pop chord symbols, which are used as a guide to players, Roman numerals are used primarily as analytical tools, and so indications of inversions or added tones are sometimes omitted if they are not relevant to the analysis being performed.

[edit] Inversion

When the bass is not the same as the root, the chord is inverted.

The number of inversions that a chord can have is one fewer than the number of constituent notes. Triads, for example, (having three constituent notes) can have three positions, two of which are inversions:

  • Root position
The root note is in the bass, and above that are the third and the fifth. A triad built on the first scale degree, for example, is marked 'I'
  • First inversion
The third is in the bass, and above it are the fifth and the root. This creates an interval of a sixth and a third above the bass note, and so is marked in figured Roman notation as '6/3'. This is commonly abbreviated to 'I6' (or 'Ib') since the sixth is the characteristic interval of the inversion, and so always implies '6/3'.
  • Second inversion
The fifth is in the bass, and above it are the root and the third. This creates an interval of a sixth and a fourth above the bass note, and so is marked as 'I6/4' or 'Ic'. Second inversion is the most unstable chord position.

[edit] Types of chords

[edit] Tertian triads

Tertian triads can be described as a series of three notes. The first element is called the root note of the chord, the second note is called the "third" of the chord, and the last note is called the "fifth" of the chord. These are described below:

Chord name Component intervals Example Chord symbol Audio
major triad major third perfect fifth C-E-G C, CM, Cma, Cmaj, CΔ Major triad on C.mid play
minor triad minor third perfect fifth C-E-G Cm, Cmi, Cmin Minor triad on C.mid play
augmented triad major third augmented fifth C-E-G C+, C+, Caug Augmented triad on C.mid play
diminished triad minor third diminished fifth C-E-G Cm(5), Cº, Cdim Diminished triad on C.mid play

As an example, consider an octave of the C major scale, consisting of the notes C D E F G A B C.

C major scale C major scale.mid play
The C major triad consists of the notes C, E and G C major triad.mid play

The major triad formed using the C note as the root would consist of C (the root note of the scale), E (the third note of the scale) and G (the fifth note of the scale). This triad is major because the interval from C to E is a major third.

The D minor triad consists of the notes D, F and A D minor triad.mid play

Using the same scale (and thus, implicitly, the key of C major) a minor chord may be constructed using the D as the root note. This would be D (root), F (third note), A (fifth note).

Examination at the piano keyboard will reveal that there are four semitones between the root and third of the chord on C, but only 3 semitones between the root and third of the chord on D (while the outer notes are still a perfect fifth apart). Thus the C triad is major while the D triad is minor.

A triad can be constructed on any note of the C major scale. These will all be either minor or major, with the exception of the triad on B, the leading-tone (the last note of the scale before returning to a C, in this case), which is diminished. For more detail see the article on the mathematics of the Western music scale.

[edit] Seventh chords

Main article: Seventh chord.

Seventh chords may be thought of as the next natural step in composing tertian chords after triads. Seventh chords are constructed by adding a fourth note to a triad, at the interval of a third above the fifth of the chord. This creates the interval of a seventh above the root of the chord. There are various types of seventh chords depending on the quality of the original chord and the quality of the seventh added.

Five common types of seventh chords have standard symbols. The chord quality indications are sometimes superscripted and sometimes not (e.g. Dm7, Dm7, and Dm7 are all identical). The last three chords are not used commonly except in jazz.

Chord name Component notes (intervals) Chord symbol Audio
diminished seventh minor third diminished fifth diminished seventh Co7, Cdim7 Diminished seventh chord on C.mid Play
half-diminished seventh minor third diminished fifth minor seventh Cø7, Cm75, C-7(5) Half-diminished seventh chord on C.mid Play
minor seventh minor third perfect fifth minor seventh Cm7, C−7, C−7 Minor seventh chord on C.mid Play
minor major seventh minor third perfect fifth major seventh Cm(Maj7), C−(j7), Cm7, C−Δ7, C−maj7 Minor major seventh chord on C.mid Play
dominant seventh major third perfect fifth minor seventh C7, C7, Cdom7 Dominant seventh chord on C.mid Play
major seventh major third perfect fifth major seventh CMaj7, CMA7, CM7, CΔ7, Cj7, C+7 Major seventh chord on C.mid Play
augmented seventh major third augmented fifth minor seventh C+7, C7+, C7+5, C75 Augmented seventh chord on C.mid Play
augmented major seventh major third augmented fifth major seventh C+(Maj7), C+MA7, CMaj7+5, CMaj75, C+j7, CΔ+7 Augmented major seventh chord on C.mid Play
harmonic seventh just major third just perfect fifth harmonic seventh (approx.) C7, C7 Harmonic seventh chord on C.mid Play

When a dominant seventh chord (a major minor seventh in its most common function) is borrowed from another key, the Roman numeral corresponding with that key is shown after a slash. For example, V/V indicates the dominant of the dominant. In the key of C major, where the dominant (V) chord is G major, this secondary dominant is the chord on the fifth degree of the G major scale, i.e. D major. Note that while the chord built on D (ii) in the key of C major would normally be a minor chord, the V/V chord, also built on D, is major.

[edit] Extended chords

Main article: Extended chord.

Extended chords are tertian chords (built from thirds) or triads with notes extended, or added, beyond the seventh. Thus ninth, eleventh, and thirteenth chords are extended chords. After the thirteenth, any notes added in thirds duplicate notes elsewhere in the chord, so there are no fifteenth chords, seventeenth chords, and so on in tonal music theory, though such chords may be constructed with up to all twelve unique tones (this would obviously imply that the chord lies outside the diatonic seven-note scale) and doublings; such a chord may be of arbitrarily large size, though large names such as "23rd chord" and "25th chord" are not particularly useful.

Chord name Component notes (chord and interval) Chord symbol Audio
Dominant ninth dominant seventh major ninth - - C9 Dominant ninth chord on C.mid Play
Dominant eleventh dominant seventh
the third is usually omitted
major ninth perfect eleventh - C11 Dominant eleventh chord on C.mid Play
Dominant thirteenth dominant seventh
the eleventh is usually omitted
major ninth perfect eleventh major thirteenth C13 Dominant thirteenth chord on C.mid Play

Other extended chords follow the logic of the rules shown above.

Thus Maj9, Maj11 and Maj13 chords are the extended chords shown above with major sevenths rather than minor sevenths. Similarly, m9, m11 and m13 have minor thirds and minor sevenths.

[edit] Chromatic alterations

Although the third and seventh of the chord are always determined by the symbols shown above, the fifth, as well as the extended intervals 9, 11, and 13, may be altered through the use of accidentals. These are indicated along with the corresponding number of the element to be altered.

Accidentals are most often used in conjunction with dominant seventh chords. For example:

Chord name Component notes Chord symbol Audio
Seventh augmented fifth dominant seventh augmented fifth C7+5, C75 Seventh augmented fifth chord on C.mid Play
Seventh flat nine dominant seventh minor ninth C7-9, C79 Seventh flat nine chord on C.mid Play
Seventh sharp nine dominant seventh augmented ninth C7+9, C79 Seventh sharp nine chord on C.mid Play
Seventh augmented eleventh dominant seventh augmented eleventh C7+11, C711 Seventh augmented eleventh chord on C.mid Play
Seventh flat thirteenth dominant seventh minor thirteenth C7-13, C713 Seventh flat thirteenth chord on C.mid Play
Half-diminished seventh minor seventh diminished fifth Cø, Cm75 Half-diminished seventh chord on C.mid Play

"Altered" dominant seventh chords (C7alt) have a flat ninth, a sharp ninth, a diminished fifth and an augmented fifth (see Levine's Jazz Theory). Some write this as C7+9, which assumes also the flat ninth, diminished fifth and augmented fifth (see Aebersold's Scale Syllabus).

The augmented ninth is often referred to as a blue note, being enharmonically equivalent to the flatted third or tenth, and is used as such, particularly in blues and other jazz standards.

When superscripted numerals are used, the different numbers may be listed horizontally (as shown), or vertically.

[edit] Added tone chords

Main article: Added tone chord.

An added tone chord is a traditional chord with an extra "added" note, such as the commonly added sixth (see following section). This includes chords with an added second (ninth) or fourth (eleventh), or a combination of the three. These chords do not include "intervening" thirds as in an extended chord.

Chord name Component notes (chord and interval) Chord symbol Audio
Add nine major triad major ninth - C2, Cadd9 Add9 chord on C.mid Play
Major 4th major triad perfect fourth - C4, Cadd11 Add11 chord on C.mid Play
Major sixth major triad major sixth - C6 Add6 chord on C.mid Play
Six-nine major triad major sixth major ninth C6/9

Added chords can also have variations. Thus madd9, m4 and m6 are minor triads with extended notes.

[edit] Sixth chords

Sixth chords are chords that contain any of the various intervals of a sixth as a defining characteristic. They can be considered as belonging to either of two separate groups:

[edit] Group 1

Chords that contain a sixth chord member, i.e., a note separated by the interval of a sixth from the chord's root, such as:

  1. The major sixth chord (also called, sixth or added sixth with chord notation: 6, e.g., 'C6')
    This is by far the most common type of sixth chord of this group, and comprises a major chord plus a note forming the interval of a major sixth above the root. For example, the chord C6 contains the notes C-E-G-A.
  2. The minor sixth chord (with chord notation: min 6 or m6, e.g., Cm6)
    This is a minor chord plus a note forming the interval of a major sixth above the root. For example, the chord Cmin6 contains the notes C-E-G-A
    In chord notation, the sixth of either chord is always assumed to be a major sixth rather than a minor sixth. Minor versions exist, and in chord notation this is indicated as, e.g., Cmin (min6), or Cmin (aeolian). Such chords, however, are very rare, as the minor sixth chord member is considered an "avoid tone" due to the semitone clash between it and the chord's fifth.
  3. The augmented sixth chord (usually appearing in chord notation as an enharmonically equivalent seventh chord)
    An augmented sixth chord is a chord which contains two notes that are separated by the interval of an augmented sixth (or, by inversion, a diminished third - though this inversion is rare in compositional practice). The augmented sixth is generally used as a dissonant interval which resolves by both notes moving outward to an octave.

In Western music, the most common use of augmented sixth chords is to resolve to a dominant chord in root position (that is, a dominant triad with the root doubled to create the octave to which the augmented sixth chord resolves), or to a tonic chord in second inversion (a tonic triad with the fifth doubled for the same purpose). In this case, the tonic note of the key is included in the chord, sometimes along with an optional fourth note, to create one of the following (illustrated here in the key of C major):

  • Italian augmented sixth: A, C, F
  • French augmented sixth: A, C, D, F
  • German augmented sixth: A, C, E, F

The augmented sixth family of chords exhibits certain peculiarities. Since they are not triad-based, as are seventh chords and other sixth chords, they are not generally regarded as having roots (nor, therefore, inversions), although one re-voicing of the notes is common (with the namesake interval inverted so as to create a diminished third).

[edit] Group 2

Inverted chords, in which the interval of a sixth appears above the bass note rather than the root; inversions, traditionally, being so named from their characteristic interval of a sixth from the bass.

  1. Inverted major and minor chords
    Inverted major and minor chords may be called sixth chords. More specifically, their first and second inversions may be called six-three (6/3)and six-four (6/4) chords respectively, to indicate the intervals that the upper notes form with the bass note. Nowadays, however, this is mostly done for purposes of academic study or analysis. (see figured bass)
  2. The neapolitan sixth chord
    This chord is a major triad with the lowered supertonic scale degree as its root. The chord is referred to as a "sixth" because it is almost always found in first inversion Though a technically accurate Roman numeral analysis would be ♭II, it is generally labelled N6. In C major, the chord is spelled (assuming root position) D, F, A.
    Because it uses lowered altered tones, this chord is often grouped with the borrowed chords. However, the chord is not borrowed from the parallel major or minor, and may appear in both major and minor keys.

[edit] Suspended chords

Main article: Suspended chord.

A suspended chord, or "sus chord" (sometimes improperly called sustained chord), is a chord in which the third has been displaced by either of its dissonant neighbouring notes, forming intervals of a major second or (more commonly) a perfect fourth with the root. This results in two distinct chord types: the suspended second (sus2) and the suspended fourth (sus4). The chords, Csus2 and Csus4, for example, consist of the notes C D G and C F G, respectively. Extended versions are also possible, such as the seventh suspended fourth, for example, which, with root C, contains the notes C F G B and is notated as C7sus4 7sus4 chord on C.mid play . Csus4 is sometimes written Csus since the sus4 is more common than the sus2.

The name suspended derives from an early voice leading technique developed during the common practice period of composition, in which an anticipated stepwise melodic progression to a harmonically stable note in any particular part (voice) was often momentarily delayed or suspended simply by extending the duration of the previous note. The resulting unexpected dissonance could then be all the more satisfyingly resolved by the eventual appearance of the displaced note.

In modern usage, without regard to such considerations of voice leading, the term suspended is restricted to those chords involving the displacement of the third only, and the dissonant second or fourth no longer needs to be prepared from the previous chord. Neither is it now obligatory for the displaced note to make an appearance at all. However, in the majority of occurrences of suspended chords, the conventional stepwise resolution to the third is still observed.

Note that, in traditional music theory, the inclusion of the third in either the suspended second or suspended fourth chords negates the effect of suspension, and such chords are properly called added ninth and added eleventh chords rather than suspended chords.

A notable exception to this analysis of suspended chords occurs in jazz theory. In post-bop and modal jazz compositions and improvisations, suspended seventh chords are often used in nontraditional ways. In these contexts, they often do not function as V chords, and do not resolve the fourth to the third; the lack of resolution gives the chord an ambiguous, static quality. Indeed, the third is often played on top of a sus4 chord; in jazz theory, this doesn't negate the quality of the chord as a suspended chord. A good example is the jazz standard, Maiden Voyage.

[edit] Borrowed chords

Main article: Borrowed chord.

Borrowed chords are chords borrowed from the parallel minor or major. If the root of the borrowed chord is not in the original key, then they are named by the accidental. For instance, in major, a chord built on the parallel minor's sixth degree is a "flat six chord", written VI. Borrowed chords are an example of mode mixture.

If a chord is borrowed from the parallel key, this is usually indicated directly (e.g. IV (minor)) or explained in a footnote or accompanying text. If there is no mention of tonality upper case can be taken as the major and lower case as minor.

[edit] Polychords

Polychords are two or more chords superimposed on top of one another. See also altered chord, secundal chord, Quartal and quintal harmony and Tristan chord.

[edit] Pop chord notation

Main article: Chord notation.

Chords can be notated in abbreviated manner using pop chord symbols, usually written above the given lyrics or staff. Although these symbols are used occasionally in classical music as well, they are most common for lead sheets and fake books in jazz and other popular music. Pop chord symbols are given under the Chord symbol column in the tables of this article.

[edit] Chord sequence

Chords are commonly played in sequence, much as notes are played in sequence to form melodies. Chord sequences can be conceptualised either in a simplistic way, in which the root notes of the chords play simple melodies while tension is created and relieved by increasing and decreasing dissonance, or full attention can be paid to each note in every chord, in which case chord sequences can be regarded as multi-part harmony of unlimited complexity.

[edit] Nonchord tones and dissonance

A nonchord tone is a dissonant or unstable tone which is not a part of the chord that is currently playing and in most cases quickly resolves to a chord tone.

[edit] Simultaneity

A chord is only the harmonic function of a group of notes, and it is unnecessary for all the notes to be played together. For example, broken chords and arpeggios are ways of playing notes in succession so that they form chords. One of the most familiar broken chord figures is Alberti bass.

Since simultaneity is not a required feature of chords, there has been some academic discussion regarding the point at which a group of notes can be called a chord. Jean-Jacques Nattiez (1990, p.218) explains that, "we can encounter 'pure chords' in a musical work," such as in the "Promenade" of Modest Mussorgsky's Pictures at an Exhibition.

Mussorgsky's Pictures at an Exhibition "Promenade"

However, "often, we must go from a textual given to a more abstract representation of the chords being used," as in Claude Debussy's Première Arabesque. The chords on the second stave shown here are abstracted from the notes in the actual piece, shown on the first. "For a sound configuration to be recognized as a chord, it must have a certain duration."

Upper stave: Claude Debussy's Première Arabesque

Goldman (1965, p.26) elaborates: "the sense of harmonic relation, change, or effect depends on speed (or tempo) as well as on the relative duration of single notes or triadic units. Both absolute time (measurable length and speed) and relative time (proportion and division) must at all times be taken into account in harmonic thinking or analysis."

[edit] References

  1. ^ Schellenberg, E. Glenn; Bigand, Emmanuel; Poulin-Charronnat, Benedicte; Garnier, Cecilia; Stevens, Catherine (Nov.), "Children's implicit knowledge of harmony in Western music", Developmental Science 8 (8): 551-566 
  • Benward & Saker (2003). Music: In Theory and Practice, Vol. I. Seventh Edition. ISBN 978-0-07-294262-0.
  • Dahlhaus, Carl. Gjerdingen, Robert O. trans. (1990). Studies in the Origin of Harmonic Tonality, p.67. Princeton University Press. ISBN 0-691-09135-8.
  • Goldman (1965). Cited in Nattiez (1990).
  • Jones, George T. (1994). HarperCollins College Outline Music Theory. ISBN 0-06-467168-2.
  • Károlyi, Otto ({{}}). Introducing Music. England: Penguin Books.
  • Nattiez, Jean-Jacques (1990). Music and Discourse: Toward a Semiology of Music (Musicologie générale et sémiologue, 1987). Translated by Carolyn Abbate (1990). ISBN 0-691-02714-5.
  • Norman Monath, Norman (1984). How To Play Popular Piano In 10 Easy Lessons. Fireside Books. ISBN 0-671-53067-4.
  • Stanley Sadie and John Tyrrell, eds. (2001). The New Grove Dictionary of Music and Musicians. ISBN 1-56159-239-0.
  • Surmani, Andrew (2004). Essentials of Music Theory: A Complete Self-Study Course for All Musicians. ISBN 0-7390-3635-1.

[edit] Further reading

[edit] See also

[edit] External links

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