In
music theoryMusic theory is the field of study that deals with how music works. It examines the language and notation of music. It identifies patterns that govern composers' techniques. In a grand sense, music theory distills and analyzes the parameters or elements of music – rhythm, harmony , melody,...
, the word
inversion has several meanings. There are inverted
chords, inverted
melodies, inverted
intervals, and (in
counterpointIn music, counterpoint is the relationship between two or more voices that are independent in contour and rhythm and are harmonically interdependent. It has been most commonly identified in Western music, developing strongly during the Renaissance and in much of the common practice period,...
) inverted
voices. The concept of inversion also plays a role in musical set theory.
Inverted intervals
An
intervalIn music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....
is inverted by raising or lowering either of the notes the necessary number of
octaveIn music, an octave , is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...
s, so that both retain their names (
pitch classIn music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
) and the one which was higher is now lower and vice versa, changing the perspective or relation between the pitch classes. For example, the inversion of an interval consisting of a C with an E above it is an E with a C above it - to work this out, the C may be moved up, the E may be lowered, or both may be moved.
Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse. (Double diminished intervals become double augmented intervals, and the reverse.) Traditional interval names sum to nine: seconds become sevenths and the reverse, thirds become sixes and the reverse, and fourths become fifths and the reverse. Thus a perfect fourth becomes a perfect fifth, an augmented fourth becomes a diminished fifth, and a simple interval (that is, one that is narrower than an octave) and its inversion, when added together, will equal an octave. See also
complement (music)In traditional music theory a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th...
.
Inverted chords
A chord's
inversion describes the relationship of its bass to the other tones in the chord. For instance, a C major
triadIn music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:*the Root...
contains the tones C, E and G; its inversion is determined by which of these tones is used as the bottom note in the chord.
The term
inversion is often used to categorically refer to the different possibilities, although it may also be restricted to only those chords where the bass note is not also the root of the chord (see root position below). In texts that make this restriction, the term
position may be used instead to refer to all of the possibilities as a category.
Root position
A
root-positionIn music, the root of a chord is the note or pitch upon which such a chord is built or hierarchically centered. This centricity is present in Western music, and its verbal labeling is a basic skill for the musically trained....
chord is sometimes known as the
parent chord of its inversions. For example, C is the root of a C major triad and is in the bass when the triad is in root position; the 3rd and the 5th of the triad are sounded above the bass. Thus, a root-position chord is also known as a chord.
The following chord is also a C major triad in root position, since the root is still in the bass. The rearrangement of the notes above the bass into different octaves (here, the note E) and the doubling of notes (here, G), is known as
voicingIn music composition and arranging, a voicing is the instrumentation and vertical spacing and ordering of the pitches in a chord...
.
Inversions
In an inverted
chordIn music and music theory a chord is a set of three or more different notes from a specific key that sound simultaneously. Chords constructed of three notes are described as triads and consist of two intervals. The technical name for triad chords is tertian sonorities and is understood to be chords...
, the
rootIn music, the root of a chord is the note or pitch upon which such a chord is built or hierarchically centered. This centricity is present in Western music, and its verbal labeling is a basic skill for the musically trained....
is not in the
bassThe bass note of a chord or sonority is the lowest note played or notated. While the bass note is often the root or fundamental of the chord, it does not have to be, and sometimes one of the other pitches of the chord will be the root....
(i.e., is not the lowest note). The inversions are numbered in the order their bass tones would appear in a closed root position chord (from bottom to top).
In the first inversion of a C major triad, the bass is E—the 3rd of the triad—with the 5th and the root stacked above it (the root now shifted an octave higher), forming the intervals of a 3rd and a 6th above the inverted bass of E, respectively. A first-inversion triad is also known as a chord.
In the second inversion, the bass is G—the 5th of the triad—with the root and the 3rd above it (both again shifted an octave higher), forming a 4th and a 6th above the (inverted) bass of G, respectively. A second-inversion triad is also known as a chord. This inversion can be either
consonant or dissonantIn music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance — considered unstable...
, and analytical notation sometimes treats it differently depending on the harmonic and voice-leading context in which it occurs (
e.g. see Cadential six-four chord below).
Third inversions exist only for chords of four or more tones, such as 7th chords. In a third-inversion chord, the 7th of the chord is in the bass position. For example, a C major 7th chord in third inversion consists of B in the bass position, with C, E and G above it— being intervals of a 2nd, 4th and 6th above the (inverted) bass of B, respectively.
Figured bass
In figured bass,
Arabic numeralsThe Arabic numerals are the ten digits . They are descended from Indian numerals and the Hindu-Arabic numeral system developed by Indian mathematicians, by which a sequence of digits such as "975" is read as a whole number...
(figures) are written below each bass note. These figures refer to intervals above the bass (usually assuming octave equivalence). In a root-position triad, the intervals above the root are a 5th and a 3rd, giving the figures . Normally, however, this is abbreviated by assuming that any bass note given without symbols indicates a chord by default. Similarly, the full figuring of the first inversion ( ) is abbreviated to just ; the full figuring of the second inversion ( ) has no abbreviation.
Figured bass is also applied to
7th chordA 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 major triad with a flat seventh...
s, which have four tones. A root-position dominant-7th chord contains a 7th, 5th, and 3rd. The full figuring of 7 5 3 is usually abbreviated to just ; the full figuring of the first inversion (6 5 3) is usually rendered as just , the second inversion (6 4 3) as , and the third inversion (6 4 2) as .
The figures are often used on their own (without the bass) in music theory simply to specify a chord's inversion. This is the basis for the terms given above such as " chord"; similarly, in
harmonic analysisA diatonic function, in tonal music theory, is the specific, recognized role of each of the 7 notes and their chords in relation to the key...
the term refers to a tonic triad in first inversion.
Popular-music notation
A notation for chord inversion often used in
popularPopular music belongs to any of a number of musical genres, and stands in contrast to art music, and traditional music which was disseminated orally...
music is to write the name of a chord followed by a forward slash and then the name of the bass note. For example, the C chord above, in first inversion (i.e., with E in the bass) may be notated as
C/E. This notation works even when a note not present in a triad is the bass; for example,
F/G is a way of notating a particular approach to voicing a G11th chord (G–F–A–C). (This is quite different from analytical notations of
function; e.g., the use of
IV/V or
S/D to represent the subdominant of the dominant).
Lower-case letters
Lower-case letters may be placed after a chord symbol to indicate root position or inversion. Hence, in the key of C major, the C major chord below in first inversion may be notated as
Ib, indicating
chord I, first inversion. (Less commonly, the root of the chord is named, followed by a lower-case letter:
Cb). If no letter is added, the chord is assumed to be in root inversion, as though
a had been inserted.
Arabic numerals
A less common notation is to place the number
1,
2 or
3 etc. after a chord to indicate that it is in first, second, or third inversion respectively. The C chord above in root position is notated as
C, and in first inversion as
C1. (This notation is quite different from the Arabic numerals placed after note names to indicate the octave of a tone, typically used in acoustical contexts; for example,
C4 is often used to mean the single tone
middle CC or Do is the first note of the fixed-Do solfège.In Western music, the expression "Middle C" refers to the note "C" located exactly between the two staves of the grand staff and near the top and bottom, respectively, of the bass and soprano voices...
, and
C3 the tone an octave below it.)
Cadential six-four chord
The cadential (Figure 3) is a common harmonic phenomenon that is analyzed in two different ways: the first labels it as a second-inversion chord; the second treats it instead as part of a horizontal progression involving
voice leadingIn musical composition, voice leading is the term used to refer to a decision-making consideration when arranging voices , namely, how each voice should move in advancing from each chord to the next.- Details :...
above a stationary bass.
- In the first option, the cadential chord is considered a second inversion tonic triad because of the tones it contains. Under this designation, the progression is labeled: . Unlike the alternative analysis (see below), this label does not indicate any difference between a cadential and other uses of chords. Most older harmony textbooks use this label, and it can be traced back to the early 19th century.
- In the second option, this chord is not considered an inversion of a tonic triad but as a dissonance resolving to a consonant dominant harmony. This is notated as , in which the is not the inversion of the chord, but a dissonance that resolves to (that is, ). This function is very similar to the resolution of a 4–3 suspension. Several modern textbooks prefer this conception of the cadential , which can also be traced back to the early 19th century.
Counterpoint
Contrapuntal inversion requires that two
melodiesA melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...
, having accompanied each other once, do it again with the melody that had been in the high voice now in the low, and vice versa. Also called "double counterpoint" (if two voices are involved) or "triple counterpoint" (if three), themes that can be developed in this way are said to involve themselves in "invertible counterpoint." The action of changing the voices is called "textural inversion".
Invertible counterpoint can occur at various intervals, usually the octave (8va), less often at the 10th or 12th. To calculate the interval of inversion, add the intervals by which each voice has moved and subtract one. For example: If motive A in the high voice moves down a 6th, and motive B in the low voice moves up a 5th, in such a way as to result in A and B having exchanged registers, then the two are in double counterpoint at the 10th ((6+5)–1 = 10).
Invertible counterpoint achieves its highest expression in the four canons of JS Bach's Art of Fugue, with the first canon at the octave, the second canon at the 10th, the third canon at the 12th, and the fourth canon in augmentation and contrary motion. Other exemplars can be found in the fugues in
G minor and
B-flat major [external Shockwave movies] from Book II of Bach's
Well-Tempered ClavierThe Well-Tempered Clavier , BWV 846–893, is a collection of solo keyboard music composed by Johann Sebastian Bach...
, both of which contain invertible counterpoint at the octave, 10th, and 12th.
Inverted melodies
When applied to
melodiesA melody , also tune, voice, or line, is a linear succession of musical tones which is perceived as a single entity...
, the
inversion of a given melody is the melody turned upside-down. For instance, if the original melody has a rising major third (see
intervalIn music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....
), the inverted melody has a falling major third (or perhaps more likely, in tonal music, a falling
minor third, or even some other falling interval). Similarly, in
twelve-tone techniqueTwelve-tone technique is a method of musical composition devised by Arnold Schoenberg...
, the
inversion of the
tone rowIn music, a tone row or note row , also series and set, refers to a non-repetitive ordering of the twelve notes of the chromatic scale. Tone rows are the basis of Arnold Schoenberg's twelve-tone technique and most types of serial music...
is the so-called
prime series turned upside-down.
Inversional equivalency
Inversional equivalency or
inversional symmetry is the concept that
intervalIn music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....
s,
chordIn music and music theory a chord is a set of three or more different notes from a specific key that sound simultaneously. Chords constructed of three notes are described as triads and consist of two intervals. The technical name for triad chords is tertian sonorities and is understood to be chords...
s, and other sets of pitches are the same when inverted. It is similar to enharmonic equivalency and octave equivalency and even transpositional equivalency. Inversional equivalency is used little in
tonalTonal may refer to:* Tonal , a concept appearing in the belief systems and traditions of Mesoamerican cultures, involving a spiritual link between a person and an animal...
theory, though it is assumed a set which may be inverted onto another are remotely in common. However, taking them to be identical or near-identical is only assumed in musical set theory.
All sets of pitches with inversional symmetry have a
center or
axis of inversion. For example, the set C–E–F–F♯–G–B has one center at the dyad F and F♯ and another at the tritone, B/C, if listed F♯–G–B–C–E–F. For C–E♭–E–F♯–G–B♭ the center is F and B if listed F♯–G–B♭–C–E♭–E. (Wilson 1992, p.10-11)
Musical set theory
In musical set theory inversion may be usefully thought of as the compound operation
transpositional inversion, which is the same sense of inversion as in the Inverted melodies section above, with transposition carried out after inversion. Pitch inversion by an ordered pitch interval may be defined as:
which equals
First invert the pitch or pitches,
x = −
x, then transpose, −
x +
n.
Pitch classIn music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
inversion by a pitch class interval may be defined as:
History
In the theories of
RameauJean-Philippe Rameau was one of the most important French composers and music theorists of the Baroque era...
(1722), chords in different positions were considered functionally equivalent. However, theories of counterpoint before Rameau spoke of different intervals in different ways, such as the
regola delle terze e seste ("rule of sixths and thirds") which required the resolution of imperfect consonances to perfect ones, and would not propose a similarity between and sonorities, for instance.