In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and (in counterpoint) inverted voices. The concept of inversion also plays a role in musical set theory.

Inverted chords

An inverted chord is a chord in which the bass is not the root. For example, C is the bass of a C major triad in root position, in which the third and the fifth of the triad are above it. Thus, a root-position triad is known as a chord. (A triad in root position is also in normal form.)

In the first inversion of a C major triad, the bass is E—the third of the triad—with the root and the fifth above it, forming the intervals of a sixth and a third above the bass, respectively. Thus, a first-inversion triad is known as a chord.

In the second inversion, the bass is G—the fifth of the triad—with the root and the third above it, forming a fourth and a sixth above the bass, respectively. Thus, a second-inversion triad is known as a chord. This inversion is usually considered a dissonance, and analytical notation will often treat it differently than the other chords (see Notations for inverted chords below).

Third inversions exist only for chords that comprise four or more tones, such as seventh and ninth chords. In a third-inversion chord, the seventh of the chord is in the bass. For example, a C major seventh chord in third inversion consists of B in the bass, with C, E and G above it—a second, fourth and sixth above the bass, respectively.

Notations for inverted chords

There are at least four systems of notation for the inversions of chords.

(i) A commonly used method is figured bass. In this system, inversions are indicated by numerals written below each bass note. These numerals refer to intervals above the bass. A written note C with no digits stands for C 5 3, which is a root position C major triad (given a key signature that does not alter E or G), as G and E are a fifth and third above this note. Its first inversion would be notated E 6, which stands for E 6 3, giving E G and C. Its second inversion is G 6 4. Similarly a G dominant seventh chord would have a root figure of G 7 (fully 7 5 3), first inversion of B 6 5 (6 5 3), second of D 4 3 (6 4 3), and third of F 4 2 (6 4 2). (Chord tones which fall outside the given key signature are specified with accidentals. See Figured bass for a full explanation.)

In addition to this, the numbered figures used in figured bass are often used in music theory to simply denote a chord's inversion. Thus a 6/4 chord refers to a chord in second inversion, and is often seen with roman numeral analyses of harmonic function. For instance, a common cadential progression might be written: (see picture). Alternatively, this progression is often written as in order to signify that the chord here is a dissonance resolving to .

(ii) A notation for chord inversion often used in popular music is to write the name of a chord, followed by a forward slash, and then the name of the note that is to sound in the bass. For example, the C chord above, in first inversion (i.e. with E in the bass) may be notated as C/E. Interestingly, this notation works consistently even when a note not present in a triad or other chord is to sound in the bass, e.g. F/G is a way of notating a particular approach to voicing a G11th chord. This should not be confused with notations of the "function of function" style, for instance the subdominant of the dominant is IV/V or S/D.

(iii) The letters a, b, c, etc., may be placed after any chord symbol to indicate the root, first and second inversion respectively. Hence the C chord below, in first inversion (i.e. with E in the bass) may be notated as Cb. (If no letter is added, the chord is assumed to be in root inversion, having the same meaning as if 'a' had been added explicitly.)

(iv) A less common, but occasionally used, notation for chord inversion 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. Hence the C chord above, in first inversion (i.e. with E in the bass) may be notated as C1. No number is added in the case of a chord in root inversion. This notation should not be confused with a quite different meaning of the same notation, where a number is placed after a note name to indicate the octave in which a single note is to sound, e.g. C4 is often used simply to mean the single note middle C.

Inverted intervals

An interval is inverted by raising or lowering either of the notes the necessary number of octaves, so that both retain their names (pitch class) 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).

Inversion in counterpoint

Contrapuntal inversion requires that two melodies, 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 J. S. Bach's Art of Fugue, with the first canon at the 8va, 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 Clavier, both of which contain invertible counterpoint at the 8va, 10th, and 12th.

Inverted melodies

When applied to melodies, the inversion of a given melody is the melody turned upside-down. The notes are reflected about the center line of the staff. For instance, if the original melody has a rising major third (see interval), 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 technique, the inversion of the tone row is the so-called prime series turned upside-down.

Inversional equivalency

Inversional equivalency or inversional symmetry is the concept that intervals, chords, 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 tonal 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)

Inversion in musical set theory

Main article: set (music)

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 class inversion by a pitch class interval may be defined as:

History

In the theories of Rameau (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.

Source

• Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.