The bass voice of triadic harmonies, often simply called the "bass," determines inversion.

Inverted harmonies do not have the root in the bass. When the third appears in the bass, we say the chord is in first inversion, when the fifth appears in the bass we say the chord is in second inversion, and when the seventh appears in the bass we say the chord is in third inversion.

In addition to chord symbols, musicians also use figured bass to indicate inversion. Each triad and seventh chord is indicated by unique figures, which are often abbreviated.

Figured bass is not usually added to chord symbols; however, it is added to triadic shorthand notation.

It is called realizing figured bass when musicians turn figured bass into chords, either on paper or in performance.

Triads and seventh chords are identified according to their root, quality, and inversion. Inversion includes the appropriate figures if applicable.

In order to denote chromatic alterations to notes, musicians put accidentals (♭, ♯, ♮) before the figure that is altered. 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.

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):

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:

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:

Example 3. Visual similarity between snowpeople and inverted triads.

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:

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:

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, "\begin{smallmatrix}6\\4\end{smallmatrix}," 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\begin{smallmatrix}6\\4\end{smallmatrix}."

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:



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 "\begin{smallmatrix}6\\4\end{smallmatrix}" 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:

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 thir