In tertian harmonic theory, that is in a theory where chords can be considered stacks of thirds (e.g. in common practice tonality), the root of a chord is the note on which the thirds are stacked. For instance, the root of a triad such as C-E-G is C, independently of the order in which the three notes are presented. A triad knows three possible positions, a "root position" with the root in the bass, a first inversion, e.g. E-G-C, and a second inversion, e.g. G-C-E, but the root remains the same in all three cases. Four-note seventh chords know four positions, 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)).
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.
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 corresponing 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.
Chords that cannot be reduced to stacked thirds (e.g. chords of stacked fourths) are not amenable to the concept of root.
Identifying a chord's root
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.
Various devices have been imagined to notate inverted chords and their roots:
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. Thomas Campion, A New Way of Making Fowre Parts in Conterpoint, London, c1618, notes that when chords are in first inversions (sixths), the bass is not "a true Base", which is implicit a third lower. Campion’s "true Base" is the root of the chord. Full recognition of the relationship between the triad and its inversions is generally credited to Jean-Philippe Rameau’s Traité d’harmonie (1722). Rameau himself indeed was convinced having been the first to discover triadic inversion, 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.
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. 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.
An assumed root (also absent, or omitted root) is, "when a chord does not contain a root ([which is] not unusual),". 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]. "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]
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.
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. The diminished seventh chord affords, "singular facilities for modulation," as it may be notated four ways, to represent four different assumed roots.
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.
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, Yizhak Sadaï and Nicolas Meeùs.
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.
- 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."
- Wyatt and Schroeder (2002). Hal Leonard Pocket Music Theory, p.80. ISBN 0-634-04771-X.
- 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.
- Joel Lester, "Root-Position and Inverted Triads in Theory around 1600", Journal of the American Musicological Society 27/1 (Spring 1974), pp. 113-116.
- Joel Lester, op. cit., p. 112.
- B. Rivera, "The Seventeenth-Century Theory of Triadic Generation and Invertibility and its Application in Contemporaneous Rules of Composition", Music Theory Spectrum, p. 67.
- B. Rivera, op. cit., p. 66-67.
- 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.)
- Latarski, Don (1999). Ultimate Guitar Chords: First Chords, p.5. ISBN 978-0-7692-8522-1.
- Chapman, Charles (2004). Rhythm Guitar Tutor: An Essential Guide to Becoming the Consumate [sic] Rhythm Guitarist, p.4. ISBN 978-0-7866-2022-7.
- 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.
- Curwen, John (1881). The new How to observe harmony, p.44. Tonic Sol-Fa Agency.
- Richard Lawn, Jeffrey L. Hellmer (1996). Jazz: Theory and Practice, p.124. ISBN 0-88284-722-8.
- Adela Harriet Sophia Bagot Wodehouse (1890). A Dictionary of Music and Musicians: (A.D. 1450-1889), p.448. Macmillan and Co., Ltd.
- Schoenberg, Arnold (1983). Theory of Harmony, 197. ISBN 978-0-520-04944-4.
- Simon Sechter, Die Grundsätze der musikalischen Komposition, vol. I, Leipzig, 1853.
- A. Schoenberg, Theory of Harmony, op. cit., and Structural Functions of Harmony, ²1969, pp. 6-9 and passim.
- Y. Sadaï, Harmony in its Systemic and Phenomenological Aspects, Jerusalem, pp. 87-88.
- 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
- Russo, William (1975). Jazz Composition and Orchestration, p.28. ISBN 0-226-73213-4.