Inversion (music)
In
Intervals
An interval is inverted by raising or lowering either of the notes by one or more octaves 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.
Interval number under inversion | ||
---|---|---|
Unison | ↔ | Octave |
Second | ↔ | Seventh |
Third | ↔ | Sixth |
Fourth | ↔ | Fifth |
Interval quality under inversion | ||
---|---|---|
Perfect | ↔ | Perfect |
Major | ↔ | Minor |
Augmented | ↔ | Diminished |
Accute | ↔ | Grave |
The tables to the right show the changes in
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).
Chords
A chord's inversion describes the relationship of its lowest notes to the other notes 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 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 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
A chord is in root position if its 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 and fifth (E and G, respectively) are above it – or, on occasion, do not sound at all.
The following C-major triads are 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 – the first voicing is close voicing, while the second is open.
In an inverted chord, the root is 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).
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 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 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 chords) 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:
Notating root position and inversions
Figured bass
Triads | |||
---|---|---|---|
Inversion | Intervals above bass |
Symbol | Example |
Root position | 5 3 |
None | |
1st inversion | 6 3 |
6 | |
2nd inversion | 6 4 |
6 4 | |
Seventh chords | |||
Inversion | Intervals above bass |
Symbol | Example |
Root position | 7 | ||
1st inversion | 6 5 | ||
2nd inversion | 4 3 | ||
3rd inversion | 4 2 or 2 |
3. If this triad were in first inversion (e.g., E–G–C), the figure 6
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 5
3 is understood), and first-inversion triads are customarily abbreviated as just 6, rather than 6
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 harmonic analysis), they do not express intervals 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 "
Popular-music notation
A notation for chord inversion often used in
Lower-case letters
Lower-case letters may be placed after a chord symbol to indicate root position or inversion.[6][page needed] 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
4 and 5
3 chords.
Counterpoint
In contrapuntal inversion, two 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.[9]
Invertible counterpoint
In
Examples
For example, in the keyboard prelude in A♭ 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:
When this passage returns in bars 25–35 these lines are exchanged:
J.S. Bach's
The piece goes on to explore four of the six possible
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:
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.[10]
Melodies
A
According to
Twelve-tone music
In
In set theory, the inverse operation is sometimes designated as , where means "invert" and means "transpose by some interval " measured in number of semitones. Thus, inversion is a combination of an inversion followed by a transposition. To apply the inversion operation , you subtract the
Inversional equivalency and symmetry
Set theory
In set theory, inversional equivalency is the concept that
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
Jazz theory
In
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.
See also
Notes
- ^ The root-position triad at the end has no fifth above the root. This is common at cadences as a consequence of the voice leading.
References
- incomplete short citation]
- ^ "Inversion | Jazz, Improvisation & Harmony | Britannica". www.britannica.com. Retrieved 2024-01-26.
- ^ 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".
- ISBN 978-0-7935-7991-4.
- ^ F/G chord chart - Chordsology
- ISBN 0-7135-0744-6.
- ^ Christensen, Thomas. 1994. Rameau and musical thought in the Enlightenment, pp. 51–61. Cambridge
- ^ Scholes, Percy A. (1954). The Listener's History of Music. Vol. 1 (7th ed.). London: Oxford University Press. p. 95.
- .
The inversion of the parts in two-part Invertible counterpoint.
- ^ Service, Tom. (2014) "Symphony Guide: Mozart's 41st (Jupiter)", The Guardian, 27 May.
- OCLC 52623743.
- OCLC 20012239.
- ^ Straus 1990, p. 36.
- ISBN 0-300-05111-5
- ISBN 0-87639-001-7.