Root (chord)

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Root, in red, of a C major chord (
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). Note that the root is doubled at the octave. Root of a major chord on C.png
Root, in red, of a C major chord ( Loudspeaker.svg Play  ). Note that the root is doubled at the octave.
Root notes (blue) and bass notes (red, both=purple) from an 18th century Chorale
Play (help*info) Leading-tone triad and secondary leading-tone triad in Chorale Gotte der Vater, wohn' uns bei colored roots and bass.png
Root notes (blue) and bass notes (red, both=purple) from an 18th century Chorale Loudspeaker.svg Play  

In music theory, the concept of root is the idea that a chord can be represented and named by one of its notes. It is linked to harmonic thinking— the idea that vertical aggregates of notes can form a single unit, a chord. It is in this sense that one speaks of a "C chord" or a "chord on C"—a chord built from "C" and of which the note (or pitch) "C" is the root. When a chord is referred to in Classical music or popular music without a reference to what type of chord it is (either major or minor, in most cases), it is assumed a major triad, which for C contains the notes C, E and G. The root need not be the bass note, the lowest note of the chord: the concept of root is linked to that of the inversion of chords, which is derived from the notion of invertible counterpoint. In this concept, chords can be inverted while still retaining their root.

Music theory considers the practices and possibilities of music

Music theory is the study of the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory":

The first is what is otherwise called 'rudiments', currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, and so on. [...] The second is the study of writings about music from ancient times onwards. [...] The third is an area of current musicological study that seeks to define processes and general principles in music — a sphere of research that can be distinguished from analysis in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built.

Chord (music) harmonic set of three or more notes

A chord, in music, is any harmonic set of pitches consisting of three or more notes that are heard as if sounding simultaneously.

Classical music broad tradition of Western art music

Classical music is art music produced or rooted in the traditions of Western culture, including both liturgical (religious) and secular music. While a more precise term is also used to refer to the period from 1750 to 1820, this article is about the broad span of time from before the 6th century AD to the present day, which includes the Classical period and various other periods. The central norms of this tradition became codified between 1550 and 1900, which is known as the common-practice period. The major time divisions of Western art music are as follows:

Contents

In tertian harmonic theory, that is in a theory where chords can be considered stacks of third intervals (e.g. in common practice tonality), the root of a chord is the note on which the subsequent thirds are stacked. For instance, the root of a triad such as C Major is C, independently of the vertical order in which the three notes (C, E and G) are presented. A triad can be in three possible positions, a "root position" with the root in the bass (i.e., with the root as the lowest note, thus C, E, G or C, G, E, from lowest to highest notes), a first inversion, e.g. E, C, G or E, G, C (i.e., with the note which is a third interval above the root, E, as the lowest note) and a second inversion, e.g. G, C, E or G, E, C, in which the note that is a fifth interval above the root (G ) is the lowest note.

Tertian

In music theory, tertian describes any piece, chord, counterpoint etc. constructed from the intervals of thirds. An interval such as that between the notes A and C encompasses 3 semitone intervals and is termed a minor third while one such as that between C and E encompasses 4 semitones and is called a major third. Tertian harmony principally uses chords based on thirds; the term is typically used to contrast with quartal and quintal harmony which uses chords based on fourths or fifths.

In the history of European art music, the common practice period is the era between the formation and the decline of the tonal system. Though there are no exact dates for this phenomenon, most features of the common-practice period persisted from the mid- to late baroque period, through the Classical, Romantic and Impressionist periods, or roughly from around 1650 to 1900. While certain prevailing patterns and conventions characterize the music of this period, the time period also saw considerable stylistic evolution. Some conventions evolved during this period that were rarely employed at other times during what may still be labeled "common practice". Thus, the dates 1650–1900 are necessarily nebulous and arbitrary borders that depend on context. The most important unifying feature through this time period concerns a harmonic language to which modern music theorists can apply Roman numeral analysis.

Tonality arranges pitches or chords to induce a hierarchy of perceived relations, stabilities, and attractions

Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is called the tonic. The root of the tonic chord forms the name given to the key; so in the key of C major, the note C is both the tonic of the scale and the root of the tonic chord. Simple folk music songs often start and end with the tonic note. The most common use of the term "is to designate the arrangement of musical phenomena around a referential tonic in European music from about 1600 to about 1910". Contemporary classical music from 1910 to the 2000s may practice or avoid any sort of tonality—but harmony in almost all Western popular music remains tonal. Harmony in jazz includes many but not all tonal characteristics of the European common practice period, sometimes known as "classical music".

Regardless of whether a chord is in root position or in an inversion, the root remains the same in all three cases. Four-note seventh chords have four possible positions. That is, the chord can be played with the root as the bass note, the note a third above the root as the bass note (first inversion), the note a fifth above the root as the bass note (second inversion), or the note a seventh above the root as the bass note (third inversion). Five-note ninth chords know five positions, etc., but the root position always is that of the stack of thirds, and the root is the lowest note of this stack (see also Factor (chord)).

Factor (chord) member or component of a chord

In music, a factor or chord factor is a member or component of a chord. These are named root, third, fifth, sixth, seventh, ninth, eleventh, thirteenth, and so on, for their generic interval above the root. In harmony, the consonance and dissonance of a chord factor and a nonchord tone are distinguished, respectively.

Root position, first inversion, and second inversion C major chords
Play root position C major chord (help*info)
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Play first inversion C major chord (help*info)
, or
Play second inversion C major chord (help*info)
. Chord roots (all the same) in red. Root position, first inversion, and second inversion C major chords.png
Root position, first inversion, and second inversion C major chords Loudspeaker.svg Play root position C major chord  , Loudspeaker.svg Play first inversion C major chord  , or Loudspeaker.svg Play second inversion C major chord  . Chord roots (all the same) in red.
Root position, first inversion, and second inversion chords over C bass
Play root position C major chord (help*info)
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Play first inversion A minor chord (help*info)
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Play second inversion F major chord (help*info)
. Chord roots in red. Root position, first inversion, and second inversion chords over C bass.png
Root position, first inversion, and second inversion chords over C bass Loudspeaker.svg Play root position C major chord  , Loudspeaker.svg Play first inversion A minor chord  , or Loudspeaker.svg Play second inversion F major chord  . Chord roots in red.

Identifying a chord's root

Determining chord root from inversion
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. "Revoicing inverted triads to root position". Determining chord root from inversion.png
Determining chord root from inversion Loudspeaker.svg Play  . "Revoicing inverted triads to root position".

Although the safest way to recognize a chord’s root is, after having reduced the chord to close spacing, to rearrange it as a stack of thirds, there are shortcuts to this: in inverted triads, the root is directly above the interval of a fourth, in inverted sevenths, it is directly above the interval of a second. [1] With chord types, such as chords with added sixths or chords over pedal points, more than one possible chordal analysis may be possible. For example, in a tonal piece of music, the notes C, E, G, A, sounded as a chord, could be analyzed as a C major sixth chord in root position (a major triad – C, E, G – with an added sixth – A – above the root) or as a first inversion A minor seventh chord (the A minor seventh chord contains the notes A, C, E and G, but in this example, the C note, the third of the A minor chord, is in the bass). Deciding which note is the root of this chord could be determined by considering context. If the chord spelled C, E, G, A occurs immediately before a D7 chord (spelled D, F, A, C), most theorists and musicians would consider the first chord a minor seventh chord in first inversion, because the progression ii7–V7 is a standard chord movement.

Various devices have been imagined to notate inverted chords and their roots:

Chord names and symbols (popular music) system

Musicians use various kinds of chord names and symbols in different contexts, to represent musical chords. In most genres of popular music, including jazz, pop, and rock, a chord name and the corresponding symbol are typically composed of one or more of the following parts:

  1. The root note (e.g., C).
  2. The chord quality (e.g., minor or lowercase m, or the symbols ° or + for diminished and augmented chords; quality is usually omitted for major chords).
  3. The number of an interval (e.g., seventh, or 7), or less often, its full name or symbol (e.g., major seventh, maj7, or M7).
  4. The altered fifth (e.g., sharp five, or 5).
  5. An additional interval number (e.g., add 2 or add2), in added tone chords. For instance, the name C augmented seventh, and the corresponding symbol Caug7, or C+7, are both composed of parts 1, 2, and 3.
  6. Often, a bass note other than the root is indicated after a forward slash, following or slightly under the rest of the chord notation--for example, "GM7/B" would indicate a G major seventh chord, with B, not G, as the bass or bottom note.

In music, Roman numeral analysis uses Roman numerals to represent chords. The Roman numerals denote scale degrees ; used to represent a chord, they denote the root note on which the chord is built. For instance, III denotes the third degree of a scale or the chord built on it. Generally, uppercase Roman numerals represent major chords while lowercase Roman numerals represent minor chords ; elsewhere, upper-case Roman numerals are used for all chords. In Western classical music in the 2000s, Roman numeral analysis is used by music students and music theorists to analyze the harmony of a song or piece.

Slash chord

In music, especially modern popular music a slash chord or slashed chord, also compound chord, is a chord whose bass note or inversion is indicated by the addition of a slash and the letter of the bass note after the root note letter. It does not indicate "or". For example, a C major chord (C) in second inversion is written C/G or C/G bass, which reads "C slash G", "C over G" or "C over a G bass". If E were the bass it would be written C/E or C/E bass, which is read "C slash E", "C over E" or C/E bass. Some chords may not otherwise be notated, such as A/A. Thus, a slash chord may also indicate the chord form or shape and an additional bass note.

The concept of root has been extended for the description of intervals of two notes: the interval can either be analyzed as formed from stacked thirds (with the inner notes missing): third, fifth, seventh, etc., (i.e., intervals corresponding to odd numerals), and its low note considered as the root; or as an inversion of the same: second (inversion of a seventh), fourth (inversion of a fifth), sixth (inversion of a third), etc., (intervals corresponding to even numerals) in which cases the upper note is the root. See Interval.

Some theories of common-practice tonal music admit the sixth as a possible interval above the root and consider in some cases that 6
5
chords nevertheless are in root position – this is the case particularly in Riemannian theory. Chords that cannot be reduced to stacked thirds (e.g. chords of stacked fourths) may not be amenable to the concept of root, although in practice, in a lead sheet, the composer may specify that a quartal chord has a certain root (e.g., a fake book chart that indicates that a song uses an Asus4(add7) chord, which would use the notes A, D, G. Even though this is a quartal chord, the composer has indicated that it has a root of A.)

A major scale contains seven unique pitch classes, each of which might serve as the root of a chord:

Root position triads from C major scale
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. Root position triads from C major scale.svg
Root position triads from C major scale Loudspeaker.svg Play  .

Chords in atonal music are often of indeterminate root, as are equal-interval chords and mixed-interval chords; such chords are often best characterized by their interval content. [3]

History

The first mentions of the relation of inversion between triads appears in Otto Sigfried Harnish’s Artis musicae (1608), which describes perfect triads in which the lower note of the fifth is expressed in its own position, and imperfect ones, in which the base (i.e., root) of the chord appears only higher. Johannes Lippius, in his Disputatio musica tertia (1610) and Synopsis musicae novae (1612), is the first to use the term "triad" (trias harmonica); he also uses the term "root" (radix), but in a slightly different meaning. [4] Thomas Campion, A New Way of Making Fowre Parts in Conterpoint, London, ca. 1618, notes that when chords are in first inversions (sixths), the bass is not "...a true base,", which is implicitly a third lower. Campion’s "true base" is the root of the chord. [5]

Full recognition of the relationship between the triad and its inversions is generally credited to Jean-Philippe Rameau and his Traité d’harmonie (1722). Rameau was not the first to discover triadic inversion, [6] but his main achievement is to have recognized the importance of the succession of roots (or of chords identified by their roots) for the construction of tonality (see below, Root progressions).

Possible mathematical and scientific basis

The concept of root has some basis in the physical properties of harmonic sounds. When two notes or more notes from the harmonic series are played at the same time, people sometimes perceive the fundamental note of the series, even if that note is not present (see Missing fundamental). This property has been used in organ building for the production of low notes by resultant tones. Andreas Werckmeister’s Harmonologia (1702) describes the major triad in root position and in first inversion in terms of the harmonic series, but this description cannot be extended to the minor triad. [7]

Hindemith, who described the chromatic scale as resulting from "the juxtaposition of vibrating units in the proportions of the simple numbers from 1 to 6", i.e. from the intervals corresponding to harmonic partials 1 to 6, called the fundamental of this harmonic series the "root" of the scale. [8] From this root, he then derived a series of notes in diminishing degree of relationship, which he called Series 1 and on which he built a system of composition. This system however has been criticized for being based generically in theory derived rules and not on perception of specific instances. [3]

Assumed root

Assumed root, Am /B: A minor ninth chord without root and with B in the bass.
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Am /B, Am , then full Am9. Assumed root A minor ninth chord.png
Assumed root, Am /B: A minor ninth chord without root and with B in the bass. Loudspeaker.svg Play   Am /B, Am , then full Am9.

An assumed root (also absent, or omitted root) is, "when a chord does not contain a root ([which is] not unusual),". [10] In any context, it is the unperformed root of a performed chord. This 'assumption' may be established by the interaction of physics and perception, or by pure convention. "We only interpret a chord as having its root omitted when the habits of the ear make it absolutely necessary for us to think of the absent root in such a place."[emphasis original]. [11] "We do not acknowledge omitted Roots except in cases where the mind is necessarily conscious of them...There are also cases in instrumental accompaniment in which the root having been struck at the commencement of a measure, the ear feels it through the rest of the measure."[emphasis original] [12]

In guitar tablature, this may be indicated, "to show you where the root would be," and to assist one with, "align[ing] the chord shape at the appropriate fret," with an assumed root in grey, other notes in white, and a sounded root in black. [9]

A comparison of the diminished 7th
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and dominant 7th (9)
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chords. Diminished7thandMinor9thComparison.png
A comparison of the diminished 7th Loudspeaker.svg Play   and dominant 7th (9) Loudspeaker.svg Play   chords.
Diminished seventh chord's use in modulation: each assumed root, in parenthesis, may be used as a dominant, tonic, or supertonic.
Play ninth chords (help*info)
Thus C, taken as dominant, would modulate to F. Diminished seventh modulation.png
Diminished seventh chord's use in modulation: each assumed root, in parenthesis, may be used as a dominant, tonic, or supertonic. Loudspeaker.svg Play ninth chords   Thus C, taken as dominant, would modulate to F.

An example of an assumed root is the diminished seventh chord, of which a note a major third below the chord is often assumed to be the absent root, making it a ninth chord. [15] The diminished seventh chord affords, "singular facilities for modulation," as it may be notated four ways, to represent four different assumed roots. [14]

In jazz

In jazz and jazz fusion, roots are often omitted from chords when chord-playing musicians (e.g., electric guitar, piano, Hammond organ) are improvising chords in an ensemble that includes a bass player (either double bass, electric bass, or other bass instruments), because the bass player plays the root. For example, if a band is playing a tune in the key of C major, if there is a dominant seventh chord played on the dominant chord (i.e., G7), the chord-playing musicians typically do not play the G note in their voicing of the chord, as they expect the bass player to play the root. The chord playing musicians usually play a voicing that includes the third, seventh, and additional extensions (often the ninth and thirteenth, even if they are not specified in the chord chart). Thus a typical voicing by a chord-playing musician for a G7 chord would be the notes B and F (the third and flat seventh of the chord), along with the notes A and E (the ninth and thirteenth of the G7 chord). One possible voicing for this G7 chord would be the notes B, E, F, A (the third, thirteenth, seventh and ninth of the G7 chord). [Note: the thirteenth interval is the same "pitch class" as the sixth, except that it is one octave higher; the ninth is the same "pitch class" as the second interval, except that it is one octave higher).]

Root progressions in music

The fundamental bass (basse fondamentale) is a concept proposed by Jean-Philippe Rameau, derived from the thoroughbass, to notate what would today be called the progression of chord roots rather than the actual lowest note found in the music, the bassline. From this Rameau formed rules for the progression of chords based on the intervals between their roots. Subsequently, the Viennese theory of tonal music has typically treated chordal roots as the defining feature of harmony. [16]

Roman numeral analysis may be said to derive from the theory of the fundamental bass, although it does not particularly theorize the succession of roots. The theory of the fundamental bass properly speaking has been revived in the 20th century by Arnold Schoenberg, [17] Yizhak Sadaï [18] and Nicolas Meeùs. [19]

It may be noted in passing that the expression "fundamental bass" is somewhat improper in English, and is used here as a literal translation of the French basse fondamentale. Indeed, English makes a relative distinction between the music-theoretic concept of "root" and the acoustic concept of "fundamental", a distinction that does not exist in other languages: the links to Wikipedia articles corresponding to this one in other languages link to articles titled Grundton in German or the equivalent in other Germanic languages, or (Basse) Fondamentale in French or the equivalent in other Roman languages. The literal translations of "root" as Wurzel (German) or racine (French), etc., are not common in music theory, unless in texts translated from English.

Why is it so important to know the root of the chord? Because the roots of the chords will sound whether we want them to or not, whether or not the alphabetical symbol is correct. The root progression which emerges may not coincide with what we think we have written; it may be better or it may be worse; but art does not permit chance. The root progression supports the work. The total root progression is heard as a substantive element, almost like another melody, and it determines the tonal basis of the music. And the tonal basis of a piece is very important to the construction of themes and to the orchestration. [20]

See also

Related Research Articles

Harmony aspect of music

In music, harmony considers the process by which the composition of individual sounds, or superpositions of sounds, is analysed by hearing. Usually, this means simultaneously occurring frequencies, pitches, or chords.

Seventh chord

A seventh chord is a chord consisting of a triad plus a note forming an interval of a seventh above the chord's root. When not otherwise specified, a "seventh chord" usually means a dominant seventh chord: a major triad together with a minor seventh. However, a variety of sevenths may be added to a variety of triads, resulting in many different types of seventh chords.

Major chord chord having a root, a major third, and a perfect fifth; e.g. C–E–G or F–A–C

In music theory, a major chord is a chord that has a root, major third, and perfect fifth. When a chord has these three notes alone, 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, 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.

The term sixth chord refers to two different kinds of chord, the first in classical music and the second in modern popular music.

Thirteenth musical interval

In music or music theory, a thirteenth is the interval between the sixth and first scale degrees when the sixth is transposed up an octave, creating a compound sixth, or thirteenth. The thirteenth is most commonly major Play  or minor Play .

In music theory, a ninth chord is a chord that encompasses the interval of a ninth when arranged in close position with the root in the bass.

Function, in music, is the term used to denote the relationship of a chord or a scale degree to a tonal centre. Two main theories of tonal functions exist today:

In music, a triad is a set of three notes that can be stacked vertically in thirds. The term "harmonic triad" was coined by Johannes Lippius in his Synopsis musicae novae (1612).

Guitar chord

In music, a guitar chord is a set of notes played on a guitar. A chord's notes are often played simultaneously, but they can be played sequentially in an arpeggio. The implementation of guitar chords depends on the guitar tuning. Most guitars used in popular music have six strings with the "standard" tuning of the Spanish classical-guitar, namely E-A-D-G-B-E' ; in standard tuning, the intervals present among adjacent strings are perfect fourths except for the major third (G,B). Standard tuning requires four chord-shapes for the major triads.

The second inversion of a chord is the voicing of a triad, seventh chord, or ninth chord in which the fifth of the chord is the bass note. In this inversion, the bass note and the root of the chord are a fourth apart which traditionally qualifies as a dissonance. There is therefore a tendency for movement and resolution. In notation form, it is referred to with a c following the chord position or as a 6
4
chord.

The first inversion of a chord is the voicing of a triad, seventh chord, or ninth chord in which the third of the chord is the bass note and the root a sixth above it. In the first inversion of a C-major triad, the bass is E — the third of the triad — with the fifth and the root stacked above it, forming the intervals of a third and a sixth above the inverted bass of E, respectively.

In music theory, the word inversion has distinct, but related, meanings when applied to intervals, chords, voices, and melodies. The concept of inversion also plays an important role in musical set theory.

Dominant (music) fifth scale degree of the diatonic scale, between the subdominant and the submediant

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 dominant is sung as so in solfege. The dominant function has the role of creating instability that requires the tonic for resolution.

In very much conventionally tonal music, harmonic analysis will reveal a broad prevalence of the primary harmonies: tonic, dominant, and subdominant, and especially the first two of these.

The scheme I-x-V-I symbolizes, though naturally in a very summarizing way, the harmonic course of any composition of the Classical period. This x, usually appearing as a progression of chords, as a whole series, constitutes, as it were, the actual "music" within the scheme, which through the annexed formula V-I, is made into a unit, a group, or even a whole piece.

Third (chord) third factor of a chord is the note or pitch two scale degrees above the root or tonal center

In music, the third factor of a chord is the note or pitch two scale degrees above the root or tonal center. When the third is the bass note, or lowest note, of the expressed triad, the chord is in first inversion Play .

Seventh (chord) musical interval spanning six staff positions; either a major seventh (11 semitones) or a minor seventh (10 semitones) in a diatonic scale

In music, the seventh factor of a chord is the note or pitch seven scale degrees above the root or tonal center. When the seventh is the bass note, or lowest note, of the expressed chord, the chord is in third inversion Play .

References

  1. 1 2 Wyatt and Schroeder (2002). Hal Leonard Pocket Music Theory, p.80. ISBN   0-634-04771-X.
  2. Palmer, Manus, and Lethco (1994). The Complete Book of Scales, Chords, Arpeggios and Cadences, p.6. ISBN   0-7390-0368-2. "The root is the note from which the triad gets its name. The root of a C triad is C."
  3. 1 2 Reisberg, Horace (1975). "The Vertical Dimension in Twentieth-Century Music", Aspects of Twentieth-Century Music, p.362-72. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall. ISBN   0-13-049346-5.
  4. Joel Lester, "Root-Position and Inverted Triads in Theory around 1600", Journal of the American Musicological Society 27/1 (Spring 1974), pp. 113-116.
  5. Joel Lester, op. cit., p. 112.
  6. B. Rivera, "The Seventeenth-Century Theory of Triadic Generation and Invertibility and its Application in Contemporaneous Rules of Composition", Music Theory Spectrum, p. 67.
  7. B. Rivera, op. cit., p. 66-67.
  8. P. Hindemith, Craft of Musical Composition, A. Mendel transl., New York, 1942, p. 53. (Ein einziger Ton die Wurzel der zu ihm gehörenden Tonleiter, Unterweisung im Tonsatz, new edition, Mainz, 1940, p. 73.)
  9. 1 2 Latarski, Don (1999). Ultimate Guitar Chords: First Chords, p.5. ISBN   978-0-7692-8522-1.
  10. Chapman, Charles (2004). Rhythm Guitar Tutor: An Essential Guide to Becoming the Consumate[ sic ] Rhythm Guitarist, p.4. ISBN   978-0-7866-2022-7.
  11. John Curwen (1872). The Standard Course of Lessons and Exercises in the Tonic Sol-Fa Method of Teaching Music, p.27. Londong: Tonic Sol-Fa Agency, 8, Warwick Lane, Paternoster Row, E.C.
  12. Curwen, John (1881). The new How to observe harmony, p.44. Tonic Sol-Fa Agency.
  13. Richard Lawn, Jeffrey L. Hellmer (1996). Jazz: Theory and Practice, p.124. ISBN   0-88284-722-8.
  14. 1 2 Adela Harriet Sophia Bagot Wodehouse (1890). A Dictionary of Music and Musicians: (A.D. 1450-1889), p.448. Macmillan and Co., Ltd.
  15. Schoenberg, Arnold (1983). Theory of Harmony, 197. ISBN   978-0-520-04944-4.
  16. Simon Sechter, Die Grundsätze der musikalischen Komposition, vol. I, Leipzig, 1853.
  17. A. Schoenberg, Theory of Harmony, op. cit., and Structural Functions of Harmony, ²1969, pp. 6-9 and passim.
  18. Y. Sadaï, Harmony in its Systemic and Phenomenological Aspects, Jerusalem, pp. 87-88.
  19. N. Meeùs, “Toward a Post-Schoenbergian Grammar of Tonal and Pre-tonal Harmonic Progressions”, Music Theory Online 6/1 (2000), http://www.mtosmt.org/issues/mto.00.6.1/mto.00.6.1.meeus.html. See also http://nicolas.meeus.free.fr/NMVecteurs.html
  20. Russo, William (1975). Jazz Composition and Orchestration, p.28. ISBN   0-226-73213-4.