II. Diatonic Polyphony and Functional Harmony

# 21.1 Introduction

Chapter 13 introduced the concept of figured bass numerals as a technique for indicating chord inversion. In this chapter you will learn about various other uses of figured bass.

Figured bass comes from a Baroque compositional practice in which composers used a numerical shorthand to provide an accompanist with a harmonic blueprint. This consisted of a notated bass line coupled with a series of Arabic numerals (1, 2, 3, etc.) and various other symbols. The following excerpt provides an example:

The numerals and symbols above or below the notes on the lower staff indicated intervals to be played above the bass. The placement of the actual pitches (register, doublings, etc.) was left to the accompanist. In this way, the composer would be able to quickly specify harmonic progressions, though not the chord voicings or, for the most part, the voice-leading from one chord to the next.

For music analysts today, figured bass is useful in two ways:

1. for indicating chord inversions and
2. for representing intervals and melodic motion above a bass line.

In this chapter, we will discuss both of these applications and how they interact. Because figured bass developed as a type of shorthand, numerous abbreviations are used; our discussion will cover the most common ones.

# 21.2 Intervals above the bass

Example 21–2 shows a bass note with figures:

As explained above, the Arabic numerals indicate intervals above the bass. In other words, the 6 and the 3 specify that a sixth and a third must occur somewhere above the A. The quality of each interval (major, minor, etc.) is determined by the key signature unless otherwise specified (more on this below). In this case, a third above the A in the bass would be C# and a sixth above the bass would be an F#, as dictated by the A-major key signature. The following example shows the complete chord, an F#-minor triad in first inversion:

The figures specify the intervals to be played above the bass in a generic way. They do not specify the register of pitches forming those intervals. In other words, any interval indicated by the bass figures may be compounded by one or more octaves. Take another look at Example 21–3. The C# was placed two steps above the bass note A. It would have been equally valid to place the C# on the first ledger line above the bass staff—or in any other octave, provided the note lies somewhere above the bass:

As we will see later on, the decision to realize a bass figure as a simple or compound interval will depend on the musical context. Furthermore, bass figures do not specify anything about doublings—two or more members of the same pitch class appearing in different registers. It would therefore be equally valid to include both the C# just above the bass and the C# an octave higher:

The following examples show valid SATB voicings of the figured bass from Example 21–2:

Example 21–6 has wider spacing and doubles the bass two octaves above in the alto. Example 21–7 doubles the sixth and has the voices more tightly arranged. Despite these somewhat superficial differences, however, all of the examples above show the same chord: an F#-minor triad in first inversion (a i chord in F#-minor or a vi chord in A major).

Activity 21-1

Activity 21–1

In this activity, you will be presented with a series of single-note figured bass examples. For each exercise, indicate the pitches that must appear above the bass according to the figures. (Remember, unless otherwise specified, the quality of the interval is determined by the key signature.)

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

A and F#

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

Bb and G

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

G# and D

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

A and E

Figures under a bass line can also indicate melodic motion in the upper voices:

In Example 21–8, the figures indicate that the sixth above the bass will step down to the fifth, from F# to E. This is indicated specifically by the “6–5” figure. (The “–5” applies only to the “6” because they are found next to one another on the same line horizontally.) Simultaneous motion in several voices can also be indicated in this manner:

In some cases this kind of motion in the upper voices may end up creating an entirely new chord. We will return to this idea momentarily.

Activity 21-2

Activity 21–2

The figured bass signatures in each of the following exercises indicate the presence of melodic motion in one or more of the upper voices. For each exercise, identify the voice or voices where the melodic motion should occur. Then, indicate the pitch to which that voice should move.

### Question

In which voice will the 6–5 motion indicated by the figured bass occur?

Hint

Which voice forms a sixth (or compound sixth) with the bass?

Alto

### Follow-up question

To which pitch should the voice in question move?

Hint

Remember to make sure that your answer forms a fifth with the bass and corresponds with the key signature.

F#

### Question

In which voice will the 8–7 motion indicated by the figured bass occur?

Hint

Which voice forms an octave (or compound octave) with the bass?

Alto

### Follow-up question

To which pitch should the voice in question move?

Hint

Remember to make sure that your answer forms a fifth with the bass and corresponds with the key signature.

Eb

### Question

In this example, there is melodic motion in two voices. In which voice will the 6–5 motion indicated by the figured bass occur? In which voice will the 4–3 motion indicated by the figured bass occur?

Hint

Which voice forms a sixth (or compound sixth) with the bass? Which voice forms a fourth (or compound fourth) with the bass?

The alto voice forms a sixth above the bass. The soprano voice forms a fourth above the bass.

### Follow-up question

To which pitches should the voices in question move?

Hint

Remember to make sure that your answers form the specified intervals with the bass and that they correspond with the key signature.

The alto should move to E and the soprano should move to C#.

### Question

In which voice will the 6–5 motion indicated by the figured bass occur?

Hint

Which voice forms a sixth (or compound sixth) with the bass?

Tenor

### Follow-up question

To which pitch should the voice in question move?

Hint

Remember to make sure that your answer forms a fifth with the bass and corresponds with the key signature.

G

Activity 21-3

Activity 21–3

In this activity, you will be presented with a series of single-note figured bass examples. For each exercise, indicate the pitches that must appear above the bass according to the figures. The exercises will consist of both triads and seventh chords, and the figured bass signatures may be abbreviated. (Remember, unless otherwise specified, the quality of the interval is determined by the key signature.)

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

D, F#, and A

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

D and Ab

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

D#, F#, and A

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, the Arabic numerals indicate the intervals above the bass. The quality is determined by the key signature.

E and G

# 21.3 Chromatic alterations

As explained above, figured-bass signatures assume diatonic intervals above bass notes. In other words, the key signature determines the quality of the intervals. If a non-diatonic pitch is required, accidentals (b, #, etc.) appear next to the figured-bass signatures.

Example 21–13 shows a bass A with the figure  5/#3. The accidental next to the 3 specifies that a C# is required instead of a C. An accidental that occurs by itself is assumed to affect the third above the bass (since the third is the most the most frequently altered member of a chord):

Multiple chromatic alterations may occur simultaneously as well:

Another common convention is to indicate a raised pitch by drawing a slash or a small vertical line through the appropriate figure. The A-major triad of Example 21–15 could also be indicated by the following figured bass:

The slash through the 3 indicates that the third above the bass must be raised by a semitone (C# instead of C).

Activity 21-4

Activity 21–4

Each of the figured bass signatures in the following exercises require at least one chromatic alteration. For each exercise, indicate the pitches that must appear above the bass according to the figures. The exercises will consist of both triads and seventh chords, and the figured bass signatures may be abbreviated.

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, a slash through a numeral indicates a pitch a half step above the diatonic pitch.

F# and A

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, an accidental by itself is applied to the third above the bass.

E# and G#

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, an accidental by itself is applied to the third above the bass.

G#, B, and D

### Question

According to the figured bass signature, what pitches must appear above this bass note?

Hint

Remember, accidentals in the figured bass signature must be applied to the appropriate intervals.

Ab and Eb

# 21.4 Roman numeral analysis with figured bass

As discussed in Chapter 13 and Chapter 18, figured bass signatures can be used to indicate inversions of triads or seventh chords. They can also be combined with Roman numerals to indicate roots and positions of triads. Roman numerals indicate the scale degree of each chord’s root. Figured-bass symbols, on the other hand, are determined by the intervals above the bass, irrespective of roots. Consider the following excerpt from a Bach chorale:

In Example 21–14, a Roman numeral appears under every beat. These Roman numerals indicate the root of each harmony: the anacrusis is labeled “vi” because the root of that chord is A (scale degree $\hat6$ in C major), the downbeat of the first measure is labeled “I” because the root of that chord is C (scale degree $\hat1$ in C major), and so on. Some of the Roman numerals are accompanied by figured-bass signatures, which indicate chord inversions. The C-major chords on beats one and three of the first measure are labeled “I6 because they are in first inversion. Likewise, the chord on the downbeat of measure two is a first inversion seventh chord whose root is D (hence “ii6/5”).

As mentioned earlier, figured-bass signatures may also indicate melodic motion above the bass, as in the passing seventh in the second measure. The figures 87 indicate that one of the upper voices—in this case, the alto—first forms an octave above the bass, and then steps down to form a seventh with the bass before the next beat.

It is crucial to remember that figured-bass signatures do not always indicate chord inversions. As with the V87 in Example 21–14, they may instead indicate movement over a stationary bass. Consider the following excerpt from a Bach chorale:

On the fourth beat of m. 7 we find what appears to be iii6 chord in F major: A, C, and E with C in the bass. The bass C is doubled at the octave in the tenor while the alto and soprano have E and A respectively. Although this sonority contains all the pitches of a iii6 chord, it would be incorrect to label it as such. The soprano and tenor voices contain accented passing tones: A and C, respectively. (See Chapter 15 for a discussion of passing and other nonharmonic tones.) These passing tones resolve to G and Bb on the second eighth note, creating a root-position V7 chord. The passage should therefore be analyzed like this:

To label beat four as anything other than a V chord would undermine its important role in the underlying IV–V–vi progression, a common harmonic pattern in this style of music. It is important, in other words, that you consider the context when analyzing and labeling a chord. What appears to be an A-minor chord on beat four is actually just a byproduct of voice-leading. Such byproducts are quite common and it is important that you learn to identify them. To clarify how voice-leading can create apparent chords, consider the following example:

In Example 21–17, the entire measure consists of a C-major chord. Melodic motion above a stationary bass appears in two of the upper voices: the soprano and alto voices are each decorated with upper neighbor tones. Because the two neighbor notes together with the stationary bass coincidentally produce the pitches of an F-major chord, it is tempting to analyze passage like this:

However, that analysis is less convincing because the chord does not perform the typical function of a subdominant harmony. As we will see in the following chapters, IV chords usually lead to V chords. Such coincidentally formed chords as the IV6/4 in Example 21–18 will here be referred to as auxiliary sonorities. We will return to the somewhat tricky topics of harmonic function (Chapter 22 and Chapter 24) and auxiliary sonorities (Chapter 23) later on. For the time being it will suffice to say that a correct analysis of this progression will demonstrate that the I chord is being prolonged throughout the measure using figured-bass signatures to indicate the melodic motion:

If we compare the figures in this example with the figures in Example 21–12, we can see the different roles of figured bass. In Example 21–12, the figures indicate chord inversion. In Example 21–19, the figures indicate part movement above a stationary bass. Many sonorities—like the F-major chord in Examples 21–17 through 21–19—arise from melodic motion in one or more of the upper voices.

Making the kind of analytical decisions described here can be difficult at first and it may take some time to develop this skill. As with most skills, however, you will find that practice is crucial. With each analysis you complete you will become more familiar with common patterns and progressions. And with this familiarity you will grow more confident in the choices you make.

Activity 21-5

Activity 21–5

In this activity, you will be presented with a short passage from a chorale. Each exercise will have you analyze a single chord from the excerpt by providing a Roman numeral with figures.

### Question

What Roman numeral should appear under the chord indicated by the arrow?

Hint

Remember, the Roman numeral should be determined by the root of the chord (which may be in inversion).

V

### Follow-up question

What figures, if any, should accompany the Roman numeral?

6, 6/#, or  6/#3

### Question

What Roman numeral should appear under the chord indicated by the arrow?

Hint

Remember, the Roman numeral should be determined by the root of the chord (which may be in inversion).

V

### Follow-up question

What figures, if any, should accompany the Roman numeral?

87, 87/3,87/#3, 87/#, 87/5/3, 87/5/#3, or 87/5/#

### Question

What Roman numeral should appear under the chord indicated by the arrow?

Hint

Remember, the Roman numeral should be determined by the root of the chord (which may be in inversion).

ii

### Follow-up question

What figures, if any, should accompany the Roman numeral?

6/5 or 6/5/3

### Question

What Roman numeral should appear under the chord indicated by the arrow?

Hint

Remember, the Roman numeral should be determined by the root of the chord (which may be in inversion).

i

### Follow-up question

What figures, if any, should accompany the Roman numeral?