Triadic Inversion and Figures

Musicians often prioritize the note that is in the bass voice of triadic harmonies, often simply called the “bass,” which is the lowest part (or voice) of a composition, regardless of what instrument or voice type is singing or playing that lowest note. Example 1 shows an A major triad with three different notes in the bass and chord symbols (above the staff):

https://musescore.com/user/32728834/scores/6839459/embed

Example 1. An A major triad in root position, first inversion, and second inversion.

An A major triad consists of three notes, the root (A), the third (C♯), and the fifth (E). When a triad is stacked in thirds (i.e. “snowperson form”), we say the triad is in root position. The bass note in root position is the root. Chords that do not have the root in the bass are said to be inverted, as summarized in Example 2:

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Example 2. A summary of triadic inversions, root, and bass.

As seen in Example 2, when the third appears in the bass, we say the triad is in first inversion, and when the fifth appears in the bass we say the triad is in second inversion.

It is important to note that the bass voice of the chord is NOT the same thing as the chord’s root. The root of an A major triad is always A, regardless of whether the triad is in root position, first inversion, or second inversion. However, the bass voice changes between these inversions, from A to C♯ to E, as seen in Example 1 and Example 2.

You might think of first inversion triads as looking like a snowperson whose feet have been moved above their head, while a second inversion triad looks like a snow person whose head has been moved to where their feet would normally appear. Example 3 demonstrates this similarity:

Sometimes musicians use chord symbols to indicate inversions, as seen in Example 1. However, musicians also use figured bass to indicate inversion. Figured bass uses Arabic numerals and some symbols which indicate intervals above the bass (NOT the root) note. These are then interpreted as chords by musicians.

Example 4 shows the full figured bass for triads underneath their chord symbols:

https://musescore.com/user/32728834/scores/6839460/embed

Example 4. The full figured bass for triadic inversions.

As you can see in Example 4, a root position triad has a third and a fifth above the bass. A first inversion triad has a third and a sixth above the bass, while a second inversion triad has a fourth and a sixth above the bass. In figured bass, the larger numerals (intervals) always appear above the smaller ones.

Many centuries ago, however, musicians abbreviated the full figured bass for triads and seventh chords in order to save time and supplies (paper and ink were very expensive before the industrial revolution). Example 5 shows the abbreviated figured bass for triads that we usually use today underneath their chord symbols:

https://musescore.com/user/32728834/scores/6839461/embed

Example 5. The abbreviated figured bass for triadic inversions.

As you can see, no figure appears for root position. First inversion triads are abbreviated with the superscript number “6,” while a second inversion triad keeps its full figures, “[latex]\begin{smallmatrix}6\\4\end{smallmatrix}[/latex],” to distinguish it from a first inversion triad.

Figured bass is not usually added to chord symbols; however, it is added to triadic shorthand notation. For example, the last measure of example 5 would be notated as “A/E” in chord symbols. Using triadic shorthand, this chord would be notated as “A[latex]\begin{smallmatrix}6\\4\end{smallmatrix}[/latex].”

When musicians turn figured bass into chords—either on paper or in performance—this is called realizing the figured bass. Example 6 shows the process of realization for several triads:

https://musescore.com/user/32728834/scores/6839462/embed

Example 6. Some triads with figured bass and their realizations.

As seen in the first measure of Example 6, an E♭ appears with no figured bass next to it. Therefore, we can assume that we are realizing an E♭ major triad in root position. This chord is realized (written out with notes) in the next measure. In measure 3, we see the number “6” below the bass note G. We can understand that notation to mean that we are realizing an E♭ major triad in first inversion. This chord is realized in the next measure. In measure 5, we see the figures “[latex]\begin{smallmatrix}6\\4\end{smallmatrix}[/latex]” below the bass note B♭. This notation means that we are realizing an E♭ major triad in second inversion, realized in the next measure.

Example 7 shows the process of realization for a triad in first inversion in more detail:

https://musescore.com/user/32728834/scores/6839464/embed

Example 7. Realizing a Gm triad in first inversion in three steps.

As seen in Example 7, we first see the bass note B♭ with the figure “6” underneath. This means that the third of the triad is in the bass. We can now find the root of the chord, which is G. In the second measure of Example 7, a G minor triad in root position has been realized in parentheses. In the third measure of Example 7, the third of the chord (the B♭) is in the bass; now the triad is in first inversion. The last measure of Example 7 is the correct “answer” or realization of this chord symbol.

Example 8 shows the process of realization for a triad in second inversion in more detail:

https://musescore.com/user/32728834/scores/6839465/embed

Example 8. Realizing a B triad in second inversion in three steps.

As seen in Example 8, we are realizing a B [latex]\begin{smallmatrix}6\\4\end{smallmatrix}[/latex] triad—a B major triad in second inversion. In the first measure, we see an F# with the figured bass “[latex]\begin{smallmatrix}6\\4\end{smallmatrix}[/latex] “underneath. This means that the fifth of the triad is in the bass, and we must find the root of the chord, which is B. In the second measure of Example 8, a B major triad in root position has been realized in parentheses. In the third measure of Example 8, the fifth of the chord (the F♯) appears in the bass; this chord is now in second inversion. The last measure of Example 8 is the correct “answer” or realization of this figure.

Identifying Triads

Triads are identified according to their root, quality, and inversion. With the addition of inversion, you can identify triads in four steps:

1. Identify and write its root.
2. Identify and write its quality.
3. Identify its inversion.
4. Write the appropriate figured bass figures if applicable.

Example 9 shows a triad in inversion (measure 1) and root position (measure 2):

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Example 9. A triad in inversion (measure 1) and root position (measure 2).

The four-step process of identification for the triad in measure 1 is as follows:

1. In measure 2, the chord has been put into root position. Now we can see the root of the triad is D.
2. This triad is minor.
3. The third of the triad is in the bass; therefore this triad is in first inversion.
4. Using figured bass, we would identify this triad as Dm⁶. Using chord symbols, we would identify this triad as Dm/F.

Example 10 shows another triad in inversion (measure 1) and root position (measure 2):

https://musescore.com/user/32728834/scores/6876514/embed

Example 10. A triad in inversion (measure 1) and root position (measure 2).

The four-step process of identification for the triad in measure 1 is as follows:

1. In measure 2, the chord has been put into root position. Now we can see the root is A.
2. This triad is major.
3. The fifth of the triad in the bass; therefore this triad is in second inversion.
4. Using figured bass, we would identify this triad is A[latex]\begin{smallmatrix}6\\4\end{smallmatrix}[/latex]. Using chord symbols, we would identify this triad as A/E.

Note that the second measure of example 9 and example 10 are in parentheses. It is recommended that you imagine the chord in root position rather than write it out in order to save time.

Seventh Chord Inversion and Figures

Like triads, seventh chords can also be inverted; in other words, their root doesn’t necessarily have to be the bass. Example 11 shows an A⁷ chord in root position, first inversion, second inversion, and third inversion:

https://musescore.com/user/32728834/scores/6843779/embed

Example 11. Seventh chord inversions.

As you can see in Example 11, when a seventh chord has its root in the bass it is in root position. When the third appears in the bass it is in first inversion, and when the fifth appears in the bass it is in second inversion. Seventh chords have one more note than triads, so they have one additional inversion. When the chordal seventh of a seventh chord is in the bass it is in third inversion. Don’t forget that the bass and the root of the chord are NOT synonymous. In Example 11 the root of the chord is always A, regardless of its inversion and bass note.

A summary of the different seventh chord inversions can be seen in Example 12:

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Example 12. A summary of seventh chord inversion, root, and bass.

We also use figured bass to indicate the inversion of seventh chords. Example 13 shows the full figured bass for seventh chords underneath their chord symbols:

https://musescore.com/user/32728834/scores/6843780/embed

Example 13. Seventh chord inversions with chord and figured bass.

As you can see in Example 13, a root position seventh chord has a third, fifth, and chordal seventh above the bass. A first inversion seventh chord has a third, fifth, and a sixth above the bass, while a second inversion seventh chord has a third, a fourth, and a sixth above the bass. Finally, a third inversion seventh chord has a second, a fourth, and a sixth above the bass.

Example 14 shows the abbreviated figured bass for seventh chords that musicians use today underneath their chord symbols:

https://musescore.com/user/32728834/scores/6843782/s/mesgHK/embed

Example 14. The abbreviated figures for seventh chord inversion.

As seen in Example 14, a root position seventh chord is abbreviated with just the number “7,” while a first inversion seventh chord is abbreviated as “[latex]^6_5[/latex].” A second inversion seventh chord is abbreviated “[latex]^4_3[/latex],” and a third inversion seventh chord is abbreviated as “[latex]^4_2[/latex].” Sometimes, in older style figured bass notation, a third inversion seventh chord is notated as “2.”

You can realize figured bass for inverted seventh chords in a similar way to how you realized them for triads. To realize an inverted seventh chord, first realize the chord in root position, then invert the chord. Example 15 shows this process for a Gmm[latex]^6_5[/latex] chord (chord symbol Gmi⁷/B♭).

https://musescore.com/user/32728834/scores/6843784/embed

Example 15. Realizing an inverted seventh chord.

As seen in example 15, we first see the bass note B♭ with the figured bass “[latex]^6_5[/latex]” underneath. This means that the third of the seventh chord is in the bass. We can now find the root of the chord, which is G. In the second measure of example 15, a Gmm7 (chord symbol Gmi⁷) seventh chord has been realized in parentheses. In the third measure of example 15, the third of the chord (B♭) is in the bass; now the triad is in first inversion.

Example 16 shows another figured bass realization with inversion:

https://musescore.com/user/32728834/scores/6876603/embed

Example 16. Realizing an inverted seventh chord.

As seen in example 16, we first see the bass note F with the figured bass “[latex]^4_3[/latex]” underneath. This means that the fifth of the seventh chord is in the bass. We can now find the root of the chord, which is B♭. In the second measure of example 16, a B♭MM⁷ (chord symbol B♭M⁷) seventh chord has been realized in parentheses. In the third measure of example 16, the fifth of the chord (F) is in the bass; now the triad is in second inversion.

Identifying Seventh Chords

Like triads, seventh chords are also identified according to their root, quality, and inversion. You can identify seventh chords in five steps:

1. Identify and write its root.
2. Identify and write its quality of triad.
3. Identify and write its quality of seventh.
4. Identify its inversion if applicable.
5. Write the appropriate figured bass figures if applicable.

Example 17 shows a seventh chord in inversion and the process of identification:

https://musescore.com/user/32728834/scores/6843787/embed

Example 17. A seventh chord in inversion (measure 1) and root position (measure 2).

The five-step process of identification for the seventh chord in measure 1 is as follows:

1. In measure 2, the seventh chord has been put into root position (measure 2). Now we can see the root of the chord is E.
2. This triad is minor.
3. The chordal seventh is also minor.
4. The original example is in first inversion, because the third is in the bass.
5. This chord is an Emm[latex]^6_5[/latex] (chord symbol Emi⁷/G) chord.

Another seventh chord in inversion is shown in Example 18, along with the process of identification:

https://musescore.com/user/32728834/scores/6876617/embed

Example 18. A seventh chord in inversion (measure 1) and root position (measure 2).

The five-step process of identification for the seventh chord in measure 1 is as follows:

1. In measure 2, the seventh chord has been put into root position (measure 2). Now we can see the root of the chord is G.
2. The triad is major.
3. The chordal seventh is minor.
4. The original example is in third inversion, because the seventh is in the bass.
5. This chord is an GMm[latex]^4_2[/latex] (chord symbol G⁷/F) chord.

Note that the second measure of Example 17 and Example 18 are in parentheses. It is recommended that you imagine the chord in root position rather than write it out in order to save time.

Other Figured Bass Symbols

In order to denote chromatic alterations to notes, musicians put accidentals (♭, ♯, ♮) before the figure that is altered. Example 19 shows a few realizations of figures with accidentals:

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Example 19. Realizations of figures with accidentals. 

Musicians also use slashes through a figure or a plus sign before a figure, in order to indicate raising the note by a half step. These symbols and their realizations are shown in Example 20:

https://musescore.com/user/32728834/scores/6878013/embed

Example 20. Realization of figures with other symbols. 

An orphaned accidental (or hanging accidental) is also common. These are accidentals, that appear by themselves, not accompanying any other figure or symbol. In these cases, the accidental is assumed to apply to the third above the bass, as seen in Example 21:

https://musescore.com/user/32728834/scores/6878023/embed

Example 21. Realization of orphaned accidentals.

Take note that if only a “7” appears below the bass, a root position seventh chord is assumed. If no figures appear below the bass, a root position triad is assumed.

• Understanding Inversions of Triads and Seventh Chords (Justin Rubin)
• Triad Inversion (musictheory.net)
• Seventh Chord Inversion (musictheory.net)
• Figured Bass Inversion Symbols (Robert Hutchinson)
• Triad Chord Inversions (musictheoryacademy.com)
• How Do Chord Inversions Work? (Hello Music Theory)
• Chord Inversions (key-notes.com)
• Triad Chord Inversion (teoria)
• Figured Bass: How to Read Chord Inversion Symbols (LANDR)
• 7th Chord inversion (Gracemusic.us)
• Figured Bass (thinkingmusic.ca)
• What are Chord Inversions? (YouTube)
• What are Chord Inversions? Piano (YouTube)
• Seventh Chord Ear Training (teoria)
• Chord Ear Training (musictheory.net)
• Chord Ear Training (Tone Savvy)

1. Triad Construction (Inversion) (.pdf)
2. Triad Construction and Identification (Inversion), p.10 (.pdf), pp.4–8 (.pdf),
3. Triad Chord Identification (Inversion) (.pdf, .pdf)
4. Seventh Chord Construction (Inversion) (.pdf)
5. Seventh Chord Identification, Major (Root Position and Inversion), pp. 2–4 (.pdf)
6. Seventh Chord Identification, Minor (Root Position and Inversion), pp. 2–4 (.pdf)

1. Triadic Inversions (.pdf, .mcsz)
2. Seventh Chord Inversions (.pdf, .mcsz)