Inversion (music)

For other uses, see Inversion (disambiguation).

Inversion example from Bach's The Well-Tempered Clavier . The melody on the first line starts on A, while the melody on the second line is identical except that it starts on E and when the first melody goes up the second goes down an equal number of diatonic steps, and when the first goes down the second goes up an equal number of steps.

Prime, retrograde, (bottom-left) inverse, and retrograde-inverse.

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.

Intervals

Interval complementation: P4 + P5 = P8

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

Interval inversions

Under inversion, perfect intervals remain perfect, major intervals become minor and vice versa, augmented intervals become diminished and vice versa. (Double diminished intervals become double augmented intervals, and vice versa.) Traditional interval names add together to make nine: seconds become sevenths and vice versa, thirds become sixes and vice versa, and fourths become fifths and vice versa. 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, equal an octave. See also complement (music).

Interval quality under inversion

Perfect	Perfect
Major	Minor
Augmented	Diminished

Interval name under inversion

Unison	Octave
Second	Seventh
Third	Sixth
Fourth	Fifth

Chords

Figure 1: the closing phrase of the hymn-setting Rustington by the English composer Hubert Parry (1848–1918), showing all three positions of the C major chord. See figured bass below for a description of the numerical symbols.

A chord's inversion describes the relationship of its bass to the other tones in the chord. For instance, a C major triad contains the tones C, E and G; its inversion is determined by which of these tones is the bottom note in the chord.

The term inversion often categorically refers to the different possibilities, though 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). Texts that follow this restriction may use the term position instead, to refer to all of the possibilities as a category.

Root position

Main article: Root (chord)

A root-position 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 third and the fifth of the triad are sounded above the bass.

C major, root position

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 voicing.

Same, different voicing

Inversions

Main articles: Root position, First inversion, and Second inversion

Root position, first inversion, and second inversion C major chords , , or . Chord roots (all the same) in red.

Root position, first inversion, and second inversion chords over C bass , , or . Chord roots in red.

In an inverted chord, the root is not in the bass (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 third of the triad – with the fifth and the root stacked above it (the root now shifted an octave higher), forming the intervals of a minor third and a minor sixth above the inverted bass of E, respectively.

F major chord

First inversion (F⁶).

Second inversion (F⁶
₄).

Third inversion F⁷ chord (F⁴
₂) .

In the second inversion , the bass is G – the fifth of the triad – with the root and the third above it (both again shifted an octave higher), forming a fourth and a sixth above the (inverted) bass of G, respectively. This inversion can be either consonant or dissonant , 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). For more details, look at Second inversion

Third inversions exist only for chords of four or more tones, such as seventh chords. In a third-inversion chord , the seventh of the chord is in the bass position. For example, a G⁷ chord in third inversion consists of F in the bass position, with G, B and D above it – being intervals of a major 2nd, augmented 4th and perfect 6th above the (inverted) bass of F, respectively. ( G⁶
₅, G⁴
₃, G⁴
₂)

Notating root position and inversions

Figured bass

Figure 2: the common chord positions and their corresponding figured-bass notation in abbreviated form

Main article: Figured bass

Figured bass is a notation convention that specifies chord inversion with Arabic numerals (figures), placed vertically, in descending numerical order) above the bass note of each chord in performing notation (or below the note in analytical indications), indicating a harmonic progression. Each numeral expresses the interval that results from the voices above it (usually assuming octave equivalence).

For example, in root-position triad C–E–G, the intervals above bass note C are a third and a fifth, giving the figures ⁵
₃. If this triad were inverted (e.g., E–G–C), the figures would apply, due to the intervals of a third and a sixth appearing above the bass note E. Figured bass is similarly applied to seventh chords, which have four tones.

Certain arbitrary conventions of abbreviation (and sometimes non-abbreviation) exist in the use of figured bass. In chords whose bass notes appear without symbols, the ⁵
₃ position is understood by default. First-inversion triads (⁶
₃) are customarily abbreviated as ⁶, i.e., presence of the third is understood. Second-inversion triads (⁶
₄) are not abbreviated. Root-position seventh chords, i.e., 75
3 , are abbreviated as ⁷. First inversion seventh chords 65
3 , are abbreviated as ⁶
₅. Second inversion seventh chords 64
3 , are abbreviated as ⁴
₃. Third inversion seventh chords 64
2  are abbreviated as either ⁴
₂ or simply ².

Figured-bass numerals express distinct intervals in a chord only as they relate to the bass voice. They make no reference to the key of the progression (unlike Roman-numeral harmonic analysis); They do not express intervals between pairs of upper voices themselves (for example, in a C–E–G triad, figured bass is unconcerned with the interval relationship E–G). They do not express tones in upper voices that double, or are unison with, the bass note. However, 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 "⁶
₄ chord"; similarly, in harmonic analysis the term I⁶ refers to a tonic triad in first inversion.

Popular-music notation

Main article: Slash chord

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 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 Fsus⁹ 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 in root inversion, as though a had been inserted.

Hindu-Arabic numerals

A less common notation is to place the number "1", "2" or "3" (and so on) 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 can be ambiguous because it clashes with the Hindu-Arabic numerals placed after note names to indicate the octave of a tone, typically used in acoustical contexts; for example, "C₄" often means the single tone middle C, and "C₃" the tone an octave below it.)

Cadential six-four chord (or Appoggiatura six-four chord)

"Six-four" redirects here. For the time signature, see Sextuple metre.

Figure 3: a cadential ⁶
₄ progression

The cadential ⁶
₄ (Figure 3) is a common harmonic device that can be analyzed in two contrasting ways: the first labels it as a second-inversion chord; the second treats it instead as part of a horizontal progression involving voice leading above a stationary bass.

1. In the first designation, the cadential ⁶
   ₄ chord features the progression: I⁶
   ₄, V, I. Most older harmony textbooks use this label, and it can be traced back to the early 19th century.
2. In the second designation, 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 V^6–5
   _4–3, I, in which the ⁶
   ₄ is not the inversion of the V chord, but a dissonance that resolves to V⁵
   ₃ (that is, V⁶
   ₄, V). This function is very similar to the resolution of a 4–3 suspension. Several modern textbooks prefer this conception of the cadential ⁶
   ₄, which can also be traced back to the early 19th century.

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". The inversion in two-part invertible counterpoint is also known as "rivolgimento."

For example in the keyboard prelude in A♭ major from Bach’s Well-Tempered Clavier, Book 1, the following passage, from bars 9–18, involves two “lines”, one in each hand:

Bach's prelude in A♭ from WTC1 bars 9–18

Bach's Prelude in A♭ from WTC1 bars 9–18

When this passage returns in bars 26–35 these lines are exchanged:

Bach's Prelude in A♭ from WTC1 bars 25–36

Bach's Prelude in A♭ from WTC1 bars 25–35

Bach’s three-part Invention in F minor, BWV 795 involves exploring the combination of three themes. Two of these are announced in the opening two bars. A third idea joins them in bars 2–4. When this passage is repeated a few bars later in bars 7–9, the three parts are interchanged:

Bach's three-part Invention (Sinfonia) in F minor BWV 795, bars 1–9

Bach's three-part Invention (Sinfonia) BWV 795, bars 1–9

The piece goes on to explore four of the six possible permutations of how these three lines can be combined in counterpoint.

One of the most spectacular illustrations of the workings of invertible counterpoint occurs in the finale of Mozart’s Symphony No. 41 in C major. Here, no less than five themes are heard together:

Mozart Symphony No. 41 Finale, bars 389–396

Mozart Symphony No. 41 Finale, bars 389–396

The whole passage brings the symphony to a conclusion in a blaze of brilliant orchestral writing. According to Tom Service, “Mozart’s composition of the finale of the Jupiter Symphony is a palimpsest on music history as well as his own. As a musical achievement, its most obvious predecessor is really the fugal finale of his G major String Quartet K. 387, but this symphonic finale trumps even that piece in its scale and ambition. If the story of that operatic tune first movement is to turn instinctive emotion into contrapuntal experience, the finale does exactly the reverse, transmuting the most complex arts of compositional craft into pure, exhilarating feeling. Its models in Michael and Joseph Haydn are unquestionable, but Mozart simultaneously pays homage to them – and transcends them. Now that’s what I call real originality.” Listen.

Invertible counterpoint can occur at various intervals, usually the octave, less often at the tenth or twelfth. 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 sixth, and motive B in the low voice moves up a fifth, in such a way as to result in A and B having exchanged registers, then the two are in double counterpoint at the tenth (6 + 5 – 1 = 10).

In J.S. Bach's Art of Fugue, the first canon is at the octave, the second canon at the tenth, the third canon at the twelfth, and the fourth canon in augmentation and contrary motion. Other exemplars can be found in the fugues in G minor and B♭ major [external Shockwave movies] from Book II of Bach's Well-Tempered Clavier, both of which contain invertible counterpoint at the octave, tenth, and twelfth.

Melodies

Figure 4: Inversion of the melody in Rachmaninoff's Rhapsody on a Theme by Paganini

When applied to melodies, the inversion of a given melody is the melody turned upside-down. For instance, if the original melody has a rising major third, 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). See bar 24 of Bach's C♯ minor fugue [external Shockwave movie], Well-Tempered Clavier Book 2 for an example of the subject in its melodic inversion.

Similarly, in twelve-tone technique, the inversion of the tone row is the so-called prime series turned upside-down, and is designated T_nI.

Given a certain prime set, with general element p_i,j; under the inversion operation, p_i,j→I(p_{i, 12 − j}); that is, each element of the prime set is mapped into an element with identical order number but with set number the complement (mod.12) [sic] of the original set number.

— Babbitt 1992, 16

each element p of [a given set] P is associated with one and only one inverse element s equals p' in [the universal set] S.

— Forte 1964, 144

For each u and v in S (v may possibly equal u), we shall define an operation I^v/u, which we shall call 'u/v inversion.'...
...[W]e conceive any sample s and its inversion I(s) [...] as balanced about the given u and v in a certain intervallic proportion. I(s) bears to v an intervallic relationship which is the inverse of the relation that s bears to u.

— Lewin 1987, 50

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 that sets that can be inverted into each other are remotely in common. However, they are only assumed identical or nearly identical 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.

Musical set theory

Main article: set (music)

Pitch class inversion

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:

• $T_{n}^{p}I(x)=-x+n$

which equals

• $T_{n}^{p}I(x)=n-x$

First invert the pitch or pitches, x = −x, then transpose, −x + n.

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

• $T_{n}I(x)=-x+n{\pmod {12}}\,$

Inversion about a pitch axis is a compound operation much like set theory's transpositional inversion, however in pitch axis inversion the transposition may be chromatic or diatonic transposition.

Pitch axis

This article is about the central pitch of an inverted melody. For the most important pitch of certain types of chord progressions, see pitch axis theory.

Pitch axis inversions of "Twinkle, Twinkle Little Star" about C and A .

In jazz theory, a pitch axis is the center around which a melody is inverted.

The "pitch axis" works in the context of the compound operation transpositional inversion, where transposition is carried out after inversion, however unlike musical set theory the transposition may be chromatic or diatonic transposition. Thus if D-A-G (P5 up, M2 down) is inverted to D-G-A (P5 down, M2 up) the "pitch axis" is D. However, if it is inverted to C-F-G the pitch axis is G while if the pitch axis is A, the melody inverts to E-A-B.

Note that the notation of octave position may determine how many lines and spaces appear to share the axis. The pitch axis of D-A-G and its inversion A-D-E either appear to be between C/B♮ or the single pitch F.

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 ⁶
₄ and ⁵
₃ sonorities, for instance.

See also

• Voicing (music)
• Pitch axis theory
• Retrograde inversion