Second inversion

The second inversion of a chord is the voicing of a triad, seventh chord, or ninth chord in which the fifth of the chord is the bass note. In this inversion, the bass note and the root of the chord are a fourth apart which traditionally qualifies as a dissonance. There is therefore a tendency for movement and resolution. In notation form, it is referred to with a c following the chord position (For e.g., Ic. Vc or IVc). In figured bass, a second-inversion triad is a ⁶
₄ chord (as in I⁶
₄), while a second-inversion seventh chord is a ⁴
₃ chord.

Inversions are not restricted to the same number of tones as the original chord, nor to any fixed order of tones except with regard to the interval between the root, or its octave, and the bass note, hence, great variety results.[1]

${\override Score.TimeSignature#'stencil = ##f \relative c' { \clef treble \time 4/4 \key c \major <d g b>1 } }$

A G-major triad in second inversion

F major chord

Root position (F) Play

First inversion (A⁶) Play

Second inversion (C⁶
₄) Play

Third inversion of F7 chord (E^♭⁴
₂) Play

Note that any voicing above the bass is allowed. A second inversion chord must have the fifth chord factor in the bass, but it may have any arrangement of the root and third above that, including doubled notes, compound intervals, and omission (G-C-E, G-C-E-G', G-E-G-C'-E', etc.)

Examples

In the second inversion of a C-major triad, the bass is G — the fifth of the triad — with the root and third stacked above it, forming the intervals of a fourth and a sixth above the inverted bass of G, respectively.

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

In the second inversion of a G dominant seventh chord, the bass note is D, the fifth of the seventh chord.

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

Types

There are four types of second-inversion chords: cadential, passing, auxiliary, and bass arpeggiation.

Cadential

Cadential second-inversion chords are typically used in the authentic cadence I⁶
₄-V-I, or one of its variation, like I⁶
₄-V⁷-I. In this form, the chord is sometimes referred to as a cadential ⁶
₄ chord. The chord preceding I⁶
₄ is most often a chord that would introduce V as a weak to strong progression, for example, making -II-V into II-I⁶
₄-V or making IV-V into IV-I⁶
₄-V.

${ \new PianoStaff << \new Staff << \new Voice \relative c'' { \stemUp \clef treble \key c \major \time 4/4 c1 b c } \new Voice \relative c'' { \stemDown g1 g g } >> \new Staff << \new Voice \relative c' { \stemUp \clef bass \key c \major \time 4/4 e1 d e } \new Voice \relative c' { \stemDown g1 g c, \bar "||" } >> >> }$

The cadential ⁶
₄ can be analyzed in two ways: the first labels it as a second-inversion chord, while the second treats it instead as part of a horizontal progression involving voice leading above a stationary bass.

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[3] but as a dissonance resolving to a consonant dominant harmony.[4] 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 double appoggiatura on the V that resolves down by step 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.[5]

Passing

In a progression with a passing second-inversion chord, the bass passes between two tones a third apart (usually of the same harmonic function[6]). When moving from I to I⁶, the passing chord V⁶
₄ is placed between them – though some prefer VII⁶ to V⁶
₄ – creating stepwise motion in the bass (scale degrees – – ). It can also be used in the reverse direction: I⁶-V⁶
₄-I. The important point is that the V⁶
₄ chord functions as a passing chord between the two more stable chords. It occurs on the weaker beat between these two chords.[6] The upper voices usually move in step (or remain stationary) in this progression.

${ \new PianoStaff << \new Staff << \new Voice \relative c'' { \stemUp \clef treble \key c \major \time 4/4 g1 g g } \new Voice \relative c' { \stemDown e1 d c } >> \new Staff << \new Voice \relative c' { \stemUp \clef bass \key c \major \time 4/4 c1 b c } \new Voice \relative c { \stemDown c1 d e \bar "||" } >> >> }$

Auxiliary (or pedal)

In a progression with an auxiliary (or pedal) second-inversion chord, the IV⁶
₄ chord functions as the harmonization of a neighbor note in the progression, I-IV⁶
₄-I. In this progression, the third and fifth rise a step each and then fall back, creating a harmonization for the scale degrees – – in the top voice.

${ \new PianoStaff << \new Staff << \new Voice \relative c'' { \stemUp \clef treble \key c \major \time 4/4 g1 a g } \new Voice \relative c' { \stemDown e1 f e } >> \new Staff << \new Voice \relative c' { \stemUp \clef bass \key c \major \time 4/4 c1 c c } \new Voice \relative c { \stemDown c1 c c \bar "||" } >> >> }$

Bass arpeggiation

In this progression, the bass arpeggiates the root, third, and fifth of the chord. This is just a florid movement but since the fifth is present in the bass, it is referred to as a bass arpeggiation flavour of the second inversion.

${ \new PianoStaff << \new Staff << \new Voice \relative c'' { \stemUp \clef treble \key c \major \time 4/4 g1 g g } \new Voice \relative c' { \stemDown e1 e e } >> \new Staff << \new Voice \relative c' { \stemUp \clef bass \key c \major \time 4/4 c1 c c } \new Voice \relative c { \stemDown c1 e g \bar "||" } >> >> }$

References

Hubbard, William Lines (1908). The American History and Encyclopedia of Music, Vol. 10: Musical Dictionary, p.103. Irving Squire: London. [ISBN unspecified]. Also at the HathiTrust Digital Library
Weber, Theory of musical composition, p. 350, quoted in Beach, D (1967) "The functions of the six-four chord in tonal music", Journal of Music Theory, 11(1), p. 8
Aldwell, Edward; Schachter, Carl (1989), Harmony and Voice Leading (2nd ed.), San Diego, Toronto: Harcourt Brace Jovanovich, p. 263, ISBN 0-15-531519-6, OCLC 19029983, The chord does not act as an inversion of I ⁵
₃; it serves neither to extend it nor to substitute for it. LCC MT50 A444 1989.
Forte, Allen (1974), Tonal Harmony in Concept and Practice (2nd ed.), NY: Holt, Rinehart and Winston, p. 68, ISBN 0-03-077495-0.
Arnold, F.T. The art of accompaniment from a thorough-bass, Vol. 1, p. 314. ISBN 0-486-43188-6. quoted in Beach, David (1967). "The functions of the six-four chord in tonal music", p.7, Journal of Music Theory, 11(1).
Gauldin, Robert (1997). Harmonic Practice in Tonal Music New York: W.W. Norton & Company, pg 273. ISBN 0-393-97666-1