View source for Inversion (music)

{{Short description|Musical term}}
{{other uses|Inversion (disambiguation)}}

{{Image frame|content=<score sound="1">
{
#(set-global-staff-size 16)
      \new PianoStaff <<
        \new Staff <<
            \relative c'' {
                \set Score.currentBarNumber = #21
                \set Score.proportionalNotationDuration = #(ly:make-moment 1/8)

                \bar ""
                \clef treble \key d \minor \time 3/4
                \once \override TextScript.script-priority = #-100 a4~^\mordent^\markup { \sharp } a16 g! f e g f e d
                \override NoteHead.color = #red \stemUp e8 e' d cis b d 
                \override NoteHead.color = #black cis16
                }
            >>
        \new Staff <<
           \clef bass \key d \minor \time 3/4
           \new Voice \relative c' {
                \override NoteHead.color = #red a8 a, b cis d b
                \override NoteHead.color = #black cis16 a gis a f'4-. d\trill a'8
                }
           \new Voice \relative c' {
                \stemUp \override NoteHead.color = #red a4 r r
                }
            >>
    >> }
</score>|width=440|align=right|caption=An example of [[#Melodies|melodic inversion]] from the [[fugue]] in D minor from [[J. S. Bach]]'s ''[[The Well-Tempered Clavier]]'', Book 1.<ref>Schuijer (2008), p. 66.{{incomplete short citation|date=November 2020}}</ref> Though they start on different pitches (A and E), the second highlighted melody is the upside-down version of the first highlighted melody. That is, when the first goes {{em|up}}, the second goes {{em|down}} the same number of [[diatonic]] [[step (music)|steps]] (with some [[chromatic]] alteration); and when the first goes {{em|down}}, the second goes {{em|up}} the same number of steps.}}
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.<ref>{{Cite web |title=Inversion {{!}} Jazz, Improvisation & Harmony {{!}} Britannica |url=https://www.britannica.com/art/inversion-music |access-date=2024-01-26 |website=www.britannica.com |language=en}}</ref> In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in [[set theory (music)|musical set theory]].

==Intervals==
{{anchor|Inversion (interval)}}

An [[interval (music)|interval]] is inverted by raising or lowering either of the notes by one or more [[octave]]s so that the higher note becomes the lower note and vice versa. For example, the inversion of an interval consisting of a C with an E above it (the third measure below) 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.

:<score lang="lilypond">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
\set Score.proportionalNotationDuration = #(ly:make-moment 1/4)
\new Staff <<
   \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" }
>>
}
</score>
{| class="wikitable" style="display:inline-table;float:right;"
! colspan="3" |[[Interval number]]<br>under inversion
|-
|Unison
|↔
|Octave
|-
|Second
|↔
|Seventh
|-
|Third
|↔
|Sixth
|-
|Fourth
|↔
|Fifth
|}

{| style="display: inline-table;", style="float:right"
|}

{| class="wikitable" style="display:inline-table;float:right;"
! colspan="3" |[[Interval quality]]<br>under inversion
|-
|Perfect
|↔
|Perfect
|-
|Major
|↔
|Minor
|-
|Augmented
|↔
|Diminished
|-
|Accute
|↔
|Grave
|}

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 [[Interval (music)#Simple and compound|simple interval]] (that is, one that is narrower than an octave) and its inversion, when added together, equal an octave. See also [[complement (music)]].

{{Clear}}

==Chords==
{{anchor|Inverted chords}}{{Image frame|content=<score sound="1" override_midi="Rustington3.mid">
    {
\override Score.SpacingSpanner.strict-note-spacing = ##t
\set Score.proportionalNotationDuration = #(ly:make-moment 1/8)
      \new PianoStaff <<
        \new Staff <<
            \new Voice \relative c' {
                \clef treble \time 4/4
                \once \override NoteHead.color = #red <c g'>4 <c f> \once \override NoteHead.color = #red <c e> c 
                \stemDown c4 b \once \override NoteHead.color = #red c2
                }
               \addlyrics { To His heav'n -- ly pal -- ace gate. }
            \new Voice \relative c' {
                s1 \stemUp d4. d8 \once \override NoteHead.color = #red c2
                }
            >>
        \new Staff <<
            \new Voice \relative c, {
                \clef bass \time 4/4
                \once \override NoteHead.color = #red <e g'>4  <f a'> \once \override NoteHead.color = #red <g g'> <a e'> 
                <f a'> \stemDown g \once \override NoteHead.color = #red c2
                }
            \new Voice \relative c' {
                s1 s4 \stemUp g8 f \once \override NoteHead.color = #red e2
                }
  \figures {
    <6>2 <6 4>2 <6 5>4. <7>8
  }
            >>
    >> }
</score>|width=360|caption=The closing phrase of the hymn-setting ''Rustington'' by the English composer [[Hubert Parry]] (1897),<ref>Adapted from Measures 14–16, Parry H (1897) "Rustington". In: ''The Australian hymn book: harmony edition'', 1977, p. 492. [[Christopher Wordsworth]]'s "See, the Conqueror Mounts in Triumph".</ref> showing all three positions of the C major chord.{{efn|The root-position triad at the end has no fifth above the root. This is common at [[cadence]]s as a consequence of the [[voice leading]].}} See [[#Figured bass|figured bass]] below for a description of the numerical symbols.}}A [[Chord (music)|chord]]'s inversion describes the relationship of its lowest notes to the other notes in the chord. For instance, a C major [[Triad (music)|triad]] contains the tones C, E and G; its inversion is determined by which of these tones is the lowest note (or [[bass 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 lowest note is not also the [[root (chord)|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.

=== Root position and inverted chords ===
{{Main|1=Root position|2=First inversion|3=Second inversion|4=Third inversion|l1=|l2=1st|l3=2nd|l4=3rd inversions}}

A chord is in root position if its [[root (chord)|root]] is the lowest note. This is sometimes known as the ''parent chord'' of its inversions. For example, the root of a C-major triad is C, so a C-major triad will be in root position if C is the lowest note and its [[Third (chord)|third]] and [[fifth (chord)|fifth]] (E and G, respectively) are above it – or, on occasion, do not sound at all.

The following C-major triads are {{em|both}} 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 (music)|''voicing'']] – the first voicing is [[Close and open harmony|close]] voicing, while the second is [[Close and open harmony|open]].

{{block indent|<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
\set Score.proportionalNotationDuration = #(ly:make-moment 1/4)
\time 4/4 
\relative c' { 
      <c e g>1^\markup { \column { "Root" "position" } }
   <c g' e' g>^\markup { \column { "Root" "position" } }
   }
}
</score>}}

In an inverted chord, the root is {{em|not}} the lowest note. The inversions are numbered in the order their lowest notes appear in a close root-position chord (from bottom to top).

{{block indent|<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
\set Score.proportionalNotationDuration = #(ly:make-moment 1/4)
\time 4/4 
\relative c' { 
   <c e g>1^\markup { \column { "Root" "position" } }
   <e g c>1^\markup { \column { "First" "inversion" } }
   <g c e>1^\markup { \column { "Second" "inversion" } }
   }
}
</score>}}

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

# In the [[first inversion]], the lowest note is E – the [[Third (chord)|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.
# In the [[second inversion]], the lowest note is G – the [[fifth (chord)|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 [[seventh chord]]s) 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:

{{block indent|<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
\set Score.proportionalNotationDuration = #(ly:make-moment 1/4)
\time 4/4 
\relative c' { 
   <g b d f>1^\markup { \column { "Root" "position" } }
   <b d f g>1^\markup { \column { "First" "inversion" } }
   <d f g b>1^\markup { \column { "Second" "inversion" } }
   <f g b d>1^\markup { \column { "Third" "inversion" } }
   }
}
</score>}}

====Notating root position and inversions====

=====Figured bass=====

{| class="wikitable floatright"
|+Common Conventional Symbols for Figured Bass
! colspan="4" style="background: Silver" |Triads
|-
! align="center" | Inversion
! align="center" | Intervals<br>above bass
! align="center" | Symbol
! align="center" | Example
|-
|Root position
| align="center" | {{su|p=5|b=3}}
| align="center" | None
| rowspan="3" align="center" |<score lang="lilypond">
{
     \override Score.TimeSignature #'stencil = ##f
      \new PianoStaff <<
        \new Staff <<
            \relative c' {
                \clef treble \time 3/4
                <e g c>4 <c g' c> <c e g>
                }
            >>
        \new Staff <<
           \relative c {
                \clef bass \time 3/4
                c4 e g
                }
  \figures {
    < _ >4 <6> <6 4>
  }
            >>
    >> }
</score>
|-
|1st inversion
| align="center" | {{su|p=6|b=3}}
| align="center" | {{music|6 chord}}
|-
|2nd inversion
| align="center" | {{su|p=6|b=4}}
| align="center" | {{music|64 chord}}
|-
! colspan="4" style="background: Silver" |Seventh chords
|-
! align="center" | Inversion
! align="center" | Intervals<br>above bass
! align="center" | Symbol
! align="center" | Example
|-
|Root position
| align="center" | {{overset|7|{{SubSup||3|5}}}}
| align="center" | {{music|7 chord}}
| rowspan="4" align="center" |<score lang="lilypond">
    {
     \override Score.TimeSignature #'stencil = ##f
      \new PianoStaff <<
        \new Staff <<
            \relative c' {
                \clef treble \time 4/4
                <b d f>4 <g d' f> <b f' g > <b d g>
                }
            >>
        \new Staff <<
           \relative c {
                \clef bass \time 4/4
                g4 b d f
                }
  \figures {
    <7>4 <6 5> <4 3> <4 2>
  }
            >>
    >> }
</score>
|-
|1st inversion
| align="center" | {{overset|6|{{SubSup||3|5}}}}
| align="center" | {{music|65 chord}}
|-
|2nd inversion
| align="center" | {{overset|6|{{SubSup||3|4}}}}
| align="center" | {{music|43 chord}}
|-
|3rd inversion
| align="center" | {{overset|6|{{SubSup||2|4}}}}
| align="center" | {{music|42 chord}} or {{music|2 chord}}
|}

[[Figured bass]] is a notation in which chord inversions are indicated by [[Arabic numerals]] (the ''figures'') either above or below the [[bass note]]s, indicating a [[chord progression|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 {{su|p=5|b=3}}. If this triad were in first inversion (e.g., E–G–C), the figure {{su|p=6|b=3}} 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 {{su|p=5|b=3}} is understood), and first-inversion triads are customarily abbreviated as just {{music|6 chord}}, rather than {{su|p=6|b=3}}. 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 analysis|Roman-numeral harmonic analysis]]), they do not express intervals {{em|between}} 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 "[[64 chord|{{su|p=6|b=4}} chord]]" for a second inversion triad. Similarly, in [[Diatonic function|harmonic analysis]] the term I<sup>6</sup> refers to a tonic triad in first inversion.

{{Clear}}

=====Popular-music notation=====
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.<ref>{{cite book |last1=Wyatt |first1=Keith |last2=Schroeder |first2=Carl |title=Harmony and Theory: A Comprehensive Source for All Musicians |year=1998 |publisher=[[Hal Leonard Corporation]] |isbn=978-0-7935-7991-4 |page=74}}</ref> This is called a ''[[slash chord]]''. For example, a C-major chord in first inversion (i.e., with E in the bass) would be notated as "C/E". This notation works even when a note not present in a triad is the bass; for example, F/G<ref>[https://www.chordsology.com/guitar-chord-chart/F-bass-G/ F/G chord chart - Chordsology]</ref> is a way of notating a particular approach to voicing an Fadd<sup>9</sup> chord (G–F–A–C). This is quite different from analytical notations of ''[[Diatonic functionality|function]]''; e.g., the notation "IV/V" represents the [[subdominant]] of the [[Dominant (music)|dominant]].

=====Lower-case letters=====
Lower-case letters may be placed after a chord symbol to indicate root position or inversion.<ref>{{Citation | last = Lovelock | first = William | author-link = William Lovelock | year = 1981 | title = The Rudiments of Music | place = London | publisher = Bell & Hyman | isbn = 0-7135-0744-6 |page={{page needed|date=January 2021}}}}.</ref>{{page needed|date=August 2019}} Hence, in the key of C major, a C-major chord 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.

=== History ===
In [[Jean-Philippe Rameau]]'s ''[[Treatise on Harmony]]'' (1722), chords in different inversions are considered [[Function (music)|functionally]] equivalent and he has been credited as being the first person to recognise their underlying similarity.<ref>Christensen, Thomas. 1994. ''Rameau and musical thought in the Enlightenment,'' pp. 51–61. Cambridge</ref><ref>{{cite book |date=1954 |last=Scholes |first=Percy A. |author-link=Percy Scholes |title=The Listener's History of Music |volume=1 |edition=7th |location=London |publisher=Oxford University Press |page=95}}</ref> Earlier theorists spoke of different intervals using alternative descriptions, such as the ''regola delle terze e seste'' ("rule of sixths and thirds"). This required the resolution of [[imperfect consonances]] to [[Perfect consonance|perfect]] ones and would not propose, for example, a resemblance between {{sup sub|6|4}} and {{sup sub|5|3}} chords.

==Counterpoint==
{{Main|Counterpoint}}
{{anchor|Invertible counterpoint}}

{{Image frame|content=<score sound="1">
{
      \new PianoStaff <<
        \new Staff <<
            \relative c' {
                \clef treble \key a \minor \time 4/4
                \set Score.currentBarNumber = #18
                \bar ""
                r16 \override NoteHead.color = #red e a c b e, b' d 
                \override NoteHead.color = #blue c8 a gis e  \override NoteHead.color = #black 
                a16
                }
            >>
        \new Staff <<
           \relative c' {
                \clef bass \key a \minor \time 4/4
                \override NoteHead.color = #blue c8 a gis e 
                \override NoteHead.color = #black a16 \override NoteHead.color = #red e a c b e, b' d  
                \override NoteHead.color = #black c
                }
            >>
    >> }
</score>|width=400|align=right|caption=An example of contrapuntal inversion in one measure of [[J.S. Bach]]'s [[Inventions and Sinfonias (Bach)|Invention No. 13 in A minor, BWV 784]].}}

In [[Counterpoint|contrapuntal]] inversion, two [[melody|melodies]], having previously accompanied each other once, accompany each other again but with the melody that had been in the high voice now in the low, and vice versa. The action of changing the voices is called ''textural inversion''. This is called ''double counterpoint'' when two voices are involved and ''triple counterpoint'' when three are involved. The inversion in two-part invertible counterpoint is also known as ''rivolgimento''.<ref>{{Cite encyclopedia|title=Rivolgimento (It.)|encyclopedia=[[Grove Music Online]]|date=2001 |doi=10.1093/gmo/9781561592630.article.23544|quote=The inversion of the parts in two-part Invertible counterpoint.}}</ref>

=== Invertible counterpoint ===
[[Theme (music)|Themes]] that {{em|can}} be developed in this way without violating the rules of [[counterpoint]] are said to be in ''invertible counterpoint''. Invertible counterpoint can occur at various intervals, usually the [[octave]], less often at the tenth or [[Twelfth (interval)|twelfth]]. To calculate the interval of inversion,{{Clarify|reason=|date=October 2018}} add the intervals by which each voice has moved and subtract one. For example: If [[Motif (music)|motif]] A in the high voice moves down a sixth, and motif 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&nbsp;+&nbsp;5&nbsp;–&nbsp;1&nbsp;=&nbsp;10).

In [[J. S. Bach|J.S. Bach]]'s ''[[The Art of Fugue]]'', the first [[Canon (music)|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 [http://www2.nau.edu/tas3/wtc/ii16.html#movie G minor] {{Webarchive|url=https://web.archive.org/web/20100327103142/http://www2.nau.edu/tas3/wtc/ii16.html#movie |date=2010-03-27 }} and [https://web.archive.org/web/20100220110325/http://www2.nau.edu/tas3/wtc/ii21.html#movie B{{music|flat}} major] [external Shockwave movies] from J.S. Bach's ''[[The Well-Tempered Clavier]],'' Book 2, both of which contain invertible counterpoint at the octave, tenth, and twelfth.

=== Examples ===

For example, in the keyboard prelude in A{{music|flat}} major from J.S. Bach's ''The Well-Tempered Clavier'', Book 1, the following passage, from bars 9–18, involves two lines, one in each hand:[[File:Bach prelude in A flat from WTC1 bars 9-18.wav|thumb|Bach's prelude in A{{music|flat}} from WTC1 bars 9–18]] [[File:Bach Prelude in A flat from WTC1 bars 25-37.png|thumb|center|500px|Bach's Prelude in A{{music|flat}} from WTC1 bars 9–18]]

When this passage returns in bars 25–35 these lines are exchanged:[[File:Bach Prelude in A flat from WTC1 bars 25-36.wav|thumb|Bach's Prelude in A{{music|flat}} from WTC1 bars 25–36]]
[[File:Bach Prelude in A flat from WTC1 bars 25-35.png|thumb|center|500px|Bach's Prelude in A{{music|flat}} from WTC1 bars 25–35]]
J.S. Bach's [[Inventions and Sinfonias (Bach)|Three-Part Invention in F minor, BWV&nbsp;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:[[File:Bach Prelude in A flat from WTC1 bars 25-38.wav|thumb|Bach's three-part Invention (Sinfonia) in F minor BWV&nbsp;795, bars 1–9]][[File:Bach 3-part Invention BWV 795, bars 1-9.png|thumb|center|600px|Bach's three-part Invention (Sinfonia) BWV&nbsp;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 examples of invertible counterpoint occurs in the finale of [[Wolfgang Amadeus Mozart|Mozart]]'s [[Symphony No. 41 (Mozart)|''Jupiter Symphony'']]. Here, no less than five themes are heard together: [[File:Mozart Jupiter Finale final section, bars 389-397 all five themes together.wav|thumb|Mozart Symphony No.&nbsp;41 Finale, bars 389–396]][[File:Mozart Jupiter Finale final section, bars 389-396 all five themes together.png|thumb|center|600px|Mozart Symphony No.&nbsp;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]]:

{{Quote|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 [[String Quartet No. 14 (Mozart)|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 Haydn|Michael]] and [[Joseph Haydn]] are unquestionable, but Mozart simultaneously pays homage to them – and transcends them. Now that's what I call real originality.<ref>[[Tom Service|Service, Tom]]. (2014) "Symphony Guide: Mozart's 41st (''Jupiter'')", ''[[The Guardian]]'', 27 May.</ref>}}

==Melodies==
{{anchor|Melodic inversion}}

{{Image frame|content=<score sound="1"> {
#(set-global-staff-size 18)
\set Score.currentBarNumber = #1
\bar ""
\key g \major \time 6/8 
\relative c'' { 
   \clef treble
   g8 a16 g fis g a8 b16 a g a
   b8 a g d c'4
   b8 a g fis e'4
} }
</score>

<score sound="1"> {
#(set-global-staff-size 18)
\set Score.currentBarNumber = #28
\bar ""
\key g \major \time 6/8
\relative c { 
   \clef bass
   d8 c16 d e d c8 b16 c d c
   b8 c d g a,4
   b8 c d e fis,4
} }
</score>|width=450|align=right|caption=Two lines from the [[fugue]] in G major from [[J. S. Bach]]'s ''[[The Well-Tempered Clavier]]'', Book 1. The lowest voice in mm. 28–30 is an inversion of the opening melody in mm. 1–3.}}

A [[melody]] is inverted by flipping it "upside-down", reversing the melody's [[Contour (music)|contour]]. For instance, if the original melody has a rising [[major third]], then the inverted melody has a falling major third (or, especially in [[Tonality|tonal]] music, perhaps a falling [[minor third|{{em|minor}} third]]).

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{{music|flat}}–a{{music|flat}}."<ref>{{Cite book|url=https://archive.org/details/harvarddictionar0004unse/page/418|title=The Harvard Dictionary of Music|date=2003|publisher=Belknap Press of Harvard University Press|isbn=0674011635|editor-last=Randel|editor-first=Don Michael|editor-link=Don Michael Randel|edition=fourth|location=Cambridge, Massachusetts|pages=[https://archive.org/details/harvarddictionar0004unse/page/418 418]|oclc=52623743}}</ref> Moreover, the inversion {{em|may}} start on the same pitch as the original melody, but it does not have to, as illustrated by the example to the right.

=== Twelve-tone music ===
In [[twelve-tone technique]], the inversion of a [[tone row]] is one of its four traditional [[Permutation (music)|permutations]] (the others being the [[Prime form (music)|prime form]], the [[Retrograde (music)|retrograde]], and the [[retrograde inversion]]). These four permutations (labeled ''p''rime, ''r''etrograde, ''i''nversion, and ''r''etrograde ''i''nversion) for the tone row used in [[Arnold Schoenberg]]'s [[Variations for Orchestra (Schoenberg)|Variations for Orchestra, Op. 31]] are shown below.

{{block indent|<score>
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/1)
\new StaffGroup <<
   \new Staff 
      \relative c'' {
         \time 12/1
         bes1^\markup { P } e, fis dis f a d cis g aes b c
         c^\markup { R }  b aes g cis d a f dis fis e bes'
      }
   \new Staff {
      \relative c'' { 
         bes1^\markup { I } e d f dis b fis g cis c a aes
         aes^\markup { RI } a c cis g fis b dis f d e bis
      } }
>> 
}
</score>}}

In [[Set theory (music)|set theory]], the inverse operation is sometimes designated as <math> T_nI </math>, where <math> I </math> means "invert" and <math> T_n </math> means "transpose by some interval <math> n </math>" measured in number of [[semitone]]s. Thus, inversion is a combination of an inversion followed by a [[Transposition (music)|transposition]]. To apply the inversion operation <math> I </math>, you subtract the [[pitch class]], in [[integer notation]], from 12 (by convention, inversion is around pitch class 0). Then we apply the transposition operation <math> T_n </math> by adding <math> n </math>. For example, to calculate <math> T_5I(3) </math>, first subtract 3 from 12 (giving 9) and then add 5 (giving 14, which is equivalent to 2). Thus, <math> T_5I(3)=2 </math>.<ref>{{Cite book|title=Introduction to Post-Tonal Theory|last=Straus|first=Joseph N.|publisher=Prentice Hall|year=1990|isbn=0136866921|location=Englewood Cliffs, New Jersey|pages=34–35|oclc=20012239}}</ref> To invert a set of pitches, simply invert each pitch in the set in turn.{{sfn|Straus|1990|p=36}}

== {{anchor|Inversional equivalency}} Inversional equivalency and symmetry ==

=== Set theory ===
In set theory, ''inversional equivalency'' is the concept that [[interval (music)|interval]]s, [[Chord (music)|chords]], and other [[Pitch set|sets of pitches]] are the same when inverted.{{Citation needed|date=October 2018}} It is similar to [[enharmonic equivalency]], [[octave equivalency]] and even [[transpositional equivalency]]. Inversional equivalency is used little in [[tonality|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.

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{{music|flat}}–E–F{{music|sharp}}–G–B{{music|flat}} has an axis at F, and an axis, a tritone away, at B if the set is listed as F{{music|sharp}}–G–B{{music|flat}}–C–E{{music|flat}}–E. As another example, the set C–E–F–F{{music|sharp}}–G–B has an axis at the [[Dyad (music)|dyad]] F/F{{music|sharp}} and an axis at B/C if it is listed as F{{music|sharp}}–G–B–C–E–F.<ref>{{Citation|last=Wilson|first=Paul|title=The Music of Béla Bartók|pages=10–11|year=1992|publisher=Yale University Press |isbn=0-300-05111-5|author-link=Paul Wilson (music theorist)}}</ref>

=== Jazz theory{{anchor|Pitch axis}} ===
{{about|the central pitch of an inverted melody|the most important pitch of certain types of chord progressions|pitch axis theory}}<!--[[Pitch axis]] redirects directly here.-->{{Image frame|content=<score> { #(set-global-staff-size 15)
\set Score.tempoHideNote = ##t \tempo 4 = 120
\key c \major \time 4/4 
\set Score.proportionalNotationDuration = #(ly:make-moment 1/2)
\relative c' { 
   \clef treble
   \once \override NoteHead.color = #red c4^\markup { Melody } \once \override NoteHead.color = #red c g' g \once \override NoteHead.color = #red a \once \override NoteHead.color = #red a g2 f4 f e e d d c2 \bar "||"
} }
</score>

<score> { #(set-global-staff-size 15)
\set Score.tempoHideNote = ##t \tempo 4 = 120
\key c \major \time 4/4 
\set Score.proportionalNotationDuration = #(ly:make-moment 1/2)
\relative c' { 
   \clef bass
   \once \override NoteHead.color = #red c4^\markup { Diatonic inversion about pitch axis C } \once \override NoteHead.color = #red c f, f e e f2 g4 g a a b b c2 \bar "||"
} }
</score>

<score> { #(set-global-staff-size 15)
\set Score.tempoHideNote = ##t \tempo 4 = 120
\key c \major \time 4/4 
\set Score.proportionalNotationDuration = #(ly:make-moment 1/2)
\relative c' { 
   \clef bass
   \once \override NoteHead.color = #red c4^\markup { Chromatic inversion about pitch axis C } \once \override NoteHead.color = #red c f, f es es f2 g4 g aes aes bes bes c2 \bar "||"
} }
</score>

<score> { #(set-global-staff-size 15)
\set Score.tempoHideNote = ##t \tempo 4 = 120
\key c \major \time 4/4 
\set Score.proportionalNotationDuration = #(ly:make-moment 1/2)
\relative c'' { 
   \clef treble
   fis4^\markup { Chromatic inversion about pitch axis A } fis b, b \once \override NoteHead.color = #red a \once \override NoteHead.color = #red a b2 cis4 cis d d e e fis2 \bar "||"
} }
</score>

<score> { #(set-global-staff-size 15)
\set Score.tempoHideNote = ##t \tempo 4 = 120
\key c \major \time 4/4 
\set Score.proportionalNotationDuration = #(ly:make-moment 1/2)
\relative c'' { 
   \clef treble
   f4^\markup { Diatonic inversion about pitch axis A } f b, b \once \override NoteHead.color = #red a \once \override NoteHead.color = #red a b2 c4 c d d e e f2 \bar "||"
} }
</score>|width=350|align=right|caption=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.<ref>Pease, Ted (2003). ''Jazz Composition: Theory and Practice'', p.152. {{ISBN|0-87639-001-7}}.</ref>

The "pitch axis" works in the context of the compound operation transpositional inversion, where [[transposition (music)|transposition]] is carried out after inversion. However, unlike in set theory, the transposition may be a [[Diatonic and chromatic|chromatic]] or [[Diatonic and chromatic|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{{music|natural}} or the single pitch F.

==See also==
* [[Voicing (music)]]
* [[Pitch axis theory]]
* [[Retrograde inversion]]

==Notes==
{{notelist}}

==References==
{{reflist}}

==External links==
* [https://www.jazzguitar.be/blog/chord-inversions/ Chord Inversions and Exercises for Jazz Guitar]

{{Twelve-tone technique}}
{{Voicing (music)}}

{{DEFAULTSORT:Inversion (Music)}}
[[Category:Melody]]
[[Category:Musical symmetry]]
[[Category:Harmony]]
[[Category:Voicing (music)]]