Inversion (music)

 ( Melody ) 

In music theory, an inversion is a rearrangement of the top-to-bottom elements in an interval, a chord, a melody, or a group of contrapuntal lines of music. In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in musical set theory.

1. 'stencil = ##f

  \clef treble \time 4/4
  \new Voice \relative c' {
     \stemUp c2 c' c, c' c, c' c, c'
     }
  \new Voice \relative c' {
     \stemDown c2 c d d e e f f
     }
  \addlyrics { "P1" -- "P8" "M2" -- "m7" "M3" -- "m6" "P4" -- "P5" }
     

The tables to the right show the changes in interval quality and interval number under inversion. Thus, perfect intervals remain perfect, major intervals become minor and vice versa, and augmented intervals become diminished and vice versa. (Doubly diminished intervals become doubly augmented intervals, and vice versa.).

Traditional interval numbers add up to nine: seconds become sevenths and vice versa, thirds become sixths and vice versa, and so on. 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).

The term inversion often categorically refers to the different possibilities, though it may also be restricted to only those chords where the lowest note is not also the root of the chord. Texts that follow this restriction may use the term position instead, to refer to all of the possibilities as a category.

The following C-major triads are in root position, since the lowest note is the root. 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 – the first voicing is close voicing, while the second is open.

In an inverted chord, the root is the lowest note. The inversions are numbered in the order their lowest notes appear in a close root-position chord (from bottom to top).

As shown above, a C-major triad (or any chord with three notes) has two inversions:

1. In the first inversion, the lowest note 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.
2. In the second inversion, the lowest note 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.

Chords with four notes (such as ) work in a similar way, except that they have three inversions, instead of just two. The three inversions of a G dominant seventh chord are:

    \override Score.TimeSignature #'stencil = ##f
     \new PianoStaff <<
       \new Staff <<
           \relative c' {
               \clef treble \time 3/4
               4  
               }
           >>
       \new Staff <<
          \relative c {
               \clef bass \time 3/4
               c4 e g
               }
 \figures {
   < _ >4 <6> <6 4>
 }
           >>
   >> }
     

   {
    \override Score.TimeSignature #'stencil = ##f
     \new PianoStaff <<
       \new Staff <<
           \relative c' {
               \clef treble \time 4/4
               4   
               }
           >>
       \new Staff <<
          \relative c {
               \clef bass \time 4/4
               g4 b d f
               }
 \figures {
   <7>4 <6 5> <4 3> <4 2>
 }
           >>
   >> }
     

Figured bass is a notation in which chord inversions are indicated by Arabic numerals (the figures) either above or below the , 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 . If this triad were in first inversion (e.g., E–G–C), the figure would apply, due to the intervals of a third and a sixth appearing above the bass note E.

Certain conventional abbreviations exist in the use of figured bass. For instance, root-position triads appear without symbols (the is understood), and first-inversion triads are customarily abbreviated as just , rather than . The table to the right displays these conventions.

Figured-bass numerals express distinct intervals in a chord only as they relate to the bass note. They make no reference to the key of the progression (unlike Roman-numeral harmonic analysis), they do not express intervals pairs of upper voices themselves – for example, in a C–E–G triad, the figured bass does not signify the interval relationship between E–G, and they do not express notes 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 " chord" for a second inversion triad. Similarly, in harmonic analysis the term I⁶ refers to a tonic triad in first inversion.

In J.S. Bach's The 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 from J.S. Bach's The Well-Tempered Clavier, Book 2, both of which contain invertible counterpoint at the octave, tenth, and twelfth.

When this passage returns in bars 25–35 these lines are exchanged: J.S. 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 3–4. When this passage is repeated a few bars later in bars 7–9, the three parts are interchanged:

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 examples of invertible counterpoint occurs in the finale of Mozart's Jupiter Symphony. Here, no less than five themes are heard together:

The whole passage brings the symphony to a conclusion in a blaze of brilliant orchestral writing. According to Tom Service:

According to The Harvard Dictionary of Music, "The intervals between successive pitches may remain exact or, more often in tonal music, they may be the equivalents in the diatonic scale. Hence c'–d–e' may become c'–b–a (where the first descent is by a semitone rather than by a whole tone) instead of c'–b–a."

In set theory, the inverse operation is sometimes designated as $T\_nI$, where $I$ means "invert" and $T\_n$ means "transpose by some interval $n$" measured in number of . Thus, inversion is a combination of an inversion followed by a transposition. To apply the inversion operation $I$, you subtract the pitch class, in integer notation, from 12 (by convention, inversion is around pitch class 0). Then we apply the transposition operation $T\_n$ by adding $n$. For example, to calculate $T\_5I(3)$, first subtract 3 from 12 (giving 9) and then add 5 (giving 14, which is equivalent to 2). Thus, $T\_5I(3)=2$.

Sets are said to be inversionally symmetrical if they map onto themselves under inversion. The pitch that the sets must be inverted around is said to be the axis of symmetry (or center). An axis may either be at a specific pitch or halfway between two pitches (assuming that microtones are not used). For example, the set C–E–E–F–G–B has an axis at F, and an axis, a tritone away, at B if the set is listed as F–G–B–C–E–E. As another example, the set C–E–F–F–G–B has an axis at the dyad F/F and an axis at B/C if it is listed as F–G–B–C–E–F.

The "pitch axis" works in the context of the compound operation transpositional inversion, where transposition is carried out after inversion. However, unlike in set theory, the transposition may be a 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.

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.

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

• Chord Inversions and Exercises for Jazz Guitar