23.1 Introduction

Techniques of harmonic expansion play a crucial role in tonal Western art music. By prolonging harmonies over time, composers are able to build or sustain musical tension and expand musical works to large dimensions. We have already learned about melodic embellishment as a means of decorating individual tones (see Chapter 15). These techniques can be broadened, through combination with basic interval progressions, to embrace entire harmonies.

In this chapter you will apply your knowledge of basic interval progressions to expand a single harmony. We will begin by expanding a chord in root position with a neighboring auxiliary sonority. From there we will expand a harmony as it changes from root position to first inversion through a passing auxiliary sonority. In each case, the chord undergoing expansion will be called a reference sonority. Examples from actual compositions will be provided throughout.

23.2 Neighboring auxiliary sonorities

Let us begin by considering the expansion of a single triad:

In this example, a root-position tonic triad will serve as the reference sonority. The soprano voice may be decorated through a simple melodic embellishment, a lower neighbor tone (N):

This neighbor note in the soprano can be accompanied in the bass, creating parallel thirds between the outer voices.

The example above may be left alone as a simple expansion of tonic harmony, with melodic embellishments in the outer voices. However, we can take the embellishments a step further. In the following example, the tenor is given an upper neighbor while the alto sustains a G:

Here we have the tenor moving away from C to its neighbor note, D, so that it is consonant with the neighbor notes in the soprano and bass. Both of the inner voices form valid interval progressions: the alto creates a 6–5–6 with the soprano and a 5–6–5 with the bass, while the tenor forms a 3–1–3 with the soprano and a 1–3–1 with the bass. Coincidentally, the multiple neighbor notes on the second beat of Example 23–4 produce the pitches of a first-inversion V chord. We will refer to such four-part expansions as auxiliary sonorities.

Note that a perfect fourth appears between the tenor and alto on the second beat. This fourth is a resultant interval and does not present a problem:

1. because it is not formed with the bass and, additionally,
2. because the auxiliary (neighboring) sonority is merely transitory, subordinate to the tonic undergoing expansion.

This type of expansion occurs frequently in tonal music. Consider the following examples:

Both of these excerpts—the first in major, the second in minor—show the expansion of the opening tonic triad with an auxiliary sonority. In each case, the bass is embellished with a lower neighbor tone while one of the upper voices is held and the other two are decorated with nonharmonic tones. In each case, the auxiliary sonority resembles a dominant triad in first inversion.

The examples above are by no means the only way to expand a chord. What if, for example, the bass were given an upper neighbor? The outer voices would form a 3–1–3 progression:

This expansion may be “filled out” in the same manner as above. Note that the tenor embellishment has been changed to a lower neighbor in order to avoid parallel octaves with the bass:

Here we have an interesting situation. In Chapter 12, it was stated that perfect fourths are considered dissonant in two-voice textures. That rule was qualified in Chapter 14 to permit perfect fourths as resultant intervals for voice pairs not involving the bass. In this example, however, we see a perfect fourth formed between the alto and the bass. This type of situation requires a further qualification regarding how composers use perfect fourths.

Perfect fourths may occur with the bass when they are part of an auxiliary sonority associated with a reference sonority. In Example 23–9, the perfect fourth between the alto and the bass occurs during the expansion of a tonic triad in a sonority that coincidentally produces the pitches of a V6/4 chord.

A similar type of neighboring auxiliary sonority occurs when the bass is held. Consider the following example where the soprano of the same reference chord is given an upper neighbor note.

This F in the soprano forms a dissonance (again, a perfect fourth) with the bass, but is permissible since it is merely a melodic embellishment. The following example shows the alto accompanying the soprano in parallel sixths with an upper neighbor note of its own:

The tenor and bass may hold C throughout, as the following example indicates:

This progression illustrates again how an auxiliary sonority may coincidentally produce the pitches of an inverted chord. In this case, however, the auxiliary sonority resembles a IV chord in second inversion. The interval progressions formed by the upper voices against the bass are also shown in Example 23–12. The alto creates a 5–6–5 progression with the bass while the soprano creates a 3–4–3 progression. As was the case with the auxiliary V6, the perfect fourth between the bass and soprano is permissible here because of the inherent instability of a neighboring auxiliary sonority.

The following excerpt shows an example of this sort of neighboring auxiliary sonority (compare Examples 23–12 and 23–13):

Here, the neighboring motion is seen clearly in the arpeggios of the left hand. The Es and Gs of mm. 1–2 and 5 are embellished with the upper–neighbor Fs and As in mm. 3–4 above a sustained C.

23.3 Passing auxiliary sonorities

Expansions with auxiliary sonorities are not limited to neighboring motion. Consider the following example of the common I–I6 progression:

What do you notice about the outer voices? The soprano moves from E to C, while the bass does the opposite, moving from C to E. When two voices swap pitch classes like this, it is known as a voice exchange. (This interval progression is discussed in Chapter 12.) In this case, the voice exchange creates a 10–6 interval progression:

Very frequently, leaps in the outer voices are filled in with passing tones. These pitches are not consonant with the reference sonority (the I chord, in this case). They are instead understood as melodic embellishments of the outer voices:

In the above example, the outer voices form a 10–8–6 progression. Just as before, we may enhance this expansion by embellishing the inner voices so that they harmonize with the passing Ds:

In this case, the sonority resulting from the multiple embellishments coincidentally produces the pitches of a vii^o6 chord. Chapter 16 discussed the special case of this diminished triad. The triad built on the seventh scale degree of the major scale will have a diminished fifth between the root and fifth. This highly dissonant interval is permitted only when treated in specific ways. By presenting the vii^o chord in first inversion, all upper voices are consonant with the bass and the dissonant tritone is hidden in the inner voices. In Example 23–17, the diminished fifth appears between the tenor B and the alto F. The tritone resolves as both inner voices ascend by step to form a perfect fifth, the tenor creating the basic interval progression 6–6 with the bass, and the alto 10–10 with the bass.

This type of expansion, like the neighboring auxiliary sonorities outlined above, occurs frequently in tonal music. Consider the following examples:

Examples 23–18 through 23–19 each show the expansion of the reference tonic triad with a passing auxiliary harmony coincidentally producing the pitches of a vii^o6 chord. In each case, the bass moves stepwise through a third while the upper voices either follow suit or decorate a single pitch with a neighbor tone. Take note of the interval patterns between the various voice pairs as well as the prominent voice exchanges in each example. (Example 23–20 has two voice exchanges: one between the bass and alto, the other between the tenor and soprano!)

23.4 “Root position” auxiliary sonorities

The auxiliary sonorities we have examined so far have all resembled inverted triads. However, root-position triads may also serve as auxiliary sonorities. Consider the following excerpt (Example 23–21b shows a simplified reduction of Example 23–21a):

The auxiliary sonority found in m. 2 of Example 23–21 is of the exact same type shown in Example 23–13. Over a sustained bass, the third and fifth of the tonic triad (G and Bb, respectively) step to their upper neighbors in the first half of m. 2 and then return. The result is a sonority that resembles a IV chord in second inversion. A few bars later we find an analogous progression when the singer joins the piano. In this case, however, the root of the IV chord (Ab) is in the bass, as shown in Example 23–21b. As you can see, the leaps up to Ab and then back down to Eb are more disjunct than the voice leading in the introduction. Nevertheless, the auxiliary function of the IV chord is clear from the surrounding reference sonority and the embellishing neighbor-note patterns in the upper voices.

The following excerpt shows a similar situation:

Like Example 23–21, Example 23–22 shows an expansion of a tonic triad with a root-position auxiliary sonority on a strong beat. Again, the bass leaps up a perfect fourth and then back down to the tonic. The upper voices either sustain the tonic or step to upper neighbors and back.

Similarly, a root-position dominant chord may serve as an auxiliary sonority to a tonic harmony. Consider the following example:

Here we see a tonic harmony being expanded with a root-position auxiliary dominant chord—in this case a V7—in mm. 8–10. As before, note the neighbor-note patterns, this time in the upper and lower parts of the right-hand piano chords. In this case the same auxiliary sonority appears again in m. 11 and is then inverted in m. 12 before returning once again to the tonic.

It is important to remember that auxiliary sonorities serve a different purpose than the functional harmonies made up of the same pitches. In Example 23–23, for instance, the auxiliary sonority in m. 10 and 12–13 does not act as a functional V chord; instead, it expands the tonic harmony. The V chord in m. 16 is different. It ends the phrase with a half cadence and is therefore a functional dominant.

This particular pattern of root position sonorities—I–(V)–I—commonly appears at the end of a piece, where tonic is strongly reinforced. In codas, for instance, after the cadential I chord has arrived, we frequently encounter such expanded reinforcements of tonic harmony. Consider the following excerpt:

The V chord in m. 24 is heard as a functional dominant since it participates in the cadence that concludes the main melody. The V chords in the last two measures, on the other hand, do not function in this manner. In such concluding passages, where the music routinely alternates between tonic and dominant chords, the tonics are clearly primary, the dominants subsidiary. They are auxiliary sonorities used to expand the referential tonic.

While expansions of the I chord with an auxiliary dominant chord in root position are common at the ends of pieces, they are by no means limited to such sections. This type of expansion may occur wherever tonic harmony is stressed. Consider the following example where the opening measures consist of tonic harmony prolonged through several auxiliary root-position dominants:

Root-position auxiliary chords are also sometimes used to expand inverted chords. In the following example, a I chord in first inversion is expanded with a root-position auxiliary ii chord:

Note the usual auxiliary sonority voice-leading patterns in this excerpt. In this case, every voice is embellished with a neighbor tone.

23.5 Expansion of other chords

Expansions of tonic chords are very common in tonal Western art music. Other chords may be similarly expanded but this happens far less frequently. When we do find expansions of non-tonic chords, we see that composers tend to follow the same principles shown in the examples above.

Consider, for example, the following passage from a string quartet (note that despite the four sharps in the key signature, this passage is in B major):

Here, an auxiliary sonority coincidentally containing the pitches of a I6 chord is used to expand a ii6 chord. This is very similar to the I–vii^o–I model described above, where the apparent root of the auxiliary sonority lies a step below that of the reference sonority.

The same model can be used to expand a dominant chord. Consider the beginning of the second phrase from this chorale:

In Example 23–28, m. 3 is all tonic: beat 1 is a tonic chord that concludes the first phrase of the chorale and beat 4 continues that tonic. M. 4 begins with a V6 on beat 1. The second beat of m. 4 is a neighboring auxiliary IV6 which leads right back to V6, which in turn resolves back to tonic.

Non-tonic harmonies may be expanded by root-position auxiliary sonorities as well. In the following example, the reference harmony V is expanded by an auxiliary tonic.

After several I and V chords in various positions, we find a V6 on the third beat of the first full measure. With the exception of the F in the right hand of the piano part, all of the notes are decorated with upper neighbors or passing tones. These pitches outline a tonic harmony, but it would be a mistake to hear tonic as the reference sonority. The V is the active chord at this point and the I is performing an auxiliary role.

The following example shows another possibility for expanding a V chord. In this case a root-position vi chord serves as the auxiliary sonority:

Example 23–32 shows another expansion of a chord that is not the tonic or dominant:

In this passage, a subdominant (IV) chord is expanded with an auxiliary I6/4.

23.6 Common-tone fully-diminished seventh chords

Fully-diminished seventh chords also appear as auxiliary sonorities, as in the following example:

In m. 5 of this piece by Felix Mendelssohn, we find a chromatic sonority consisting of Fx, A#, C#, and E. The chord is flanked on either side by tonic triads. Compared to all of the other leading-tone seventh chords discussed in this chapter, the way this sonority resolves is unique for it shares one of its members with the chord that follows. Looking at the motion in each individual voice it becomes clear that this is a neighboring auxiliary sonority and not a functional dominant. Scale degrees # [latex]\hat2[/latex] (Fx) and # [latex]\hat4[/latex] (A#) are chromatic neighbors to [latex]\hat3[/latex] and [latex]\hat5[/latex], and [latex]\hat6[/latex] (C#) is a diatonic neighbor of [latex]\hat5[/latex]. The seventh of the chord (E) is the same as the root of the framing tonic triads and is held throughout.

The following example shows a very similar progression, this time in F major:

Once again, the auxiliary fully-diminished seventh chord appears between two tonic triads. And like Example 23–17, the seventh of the chord is the same as the root of the framing tonics.

The following examples incorporate auxiliary fully-diminished seventh chords, but here they play slightly different roles:

In Example 23–35 we see an auxiliary fully-diminished chord expanding not just a single harmony but rather the motion from one harmony (V7) to another (I). It does not have any tones in common with the chord that comes before it. Example 23–36 includes three auxiliary fully-diminished chords, the second of which resolves to a submediant harmony. Note that because the vi chord has a seventh, there are two common tones with the preceding auxiliary chord. (It is also possible to think of the Eb as a nonharmonic tone: a suspension.)

23.7 The cadential 6/4 chord

One final auxiliary sonority deserves special attention. This particular sonority appears primarily at cadences and coincidentally contains the pitches of a I chord in second inversion. Consider the following example:

Following a IV chord in m. 18, the chord on the downbeat of m. 19 contains the pitches C and E above a G in the bass. Subdominant chords can and sometimes do move directly to tonic chords, so you may be tempted to label this sonority I6/4. To do so, however, would contradict the manner in which this chord functions. The G in the bass in m. 19 is the root of the dominant at the end of the phrase. Therefore, the sonority on the downbeat of m. 19 is not a functional tonic chord, but rather an embellishment and intensification of the dominant.

Look again at the C and G. Both are nonharmonic tones: the C is a 4–3 suspension from the IV chord and the G is an accented passing tone. Both notes resolve down by step to form a root position V chord at the end of the measure, setting up the perfect authentic cadence in m. 20. The sonority on the downbeat of m. 19 is referred to as a cadential 6/4 chord. (“Cadential” because it typically appears as part of a cadence, and “6/4” because of the intervals formed above the bass.) The progression heard in m. 19 is analyzed as V65/43 to reflect the underlying importance of the V chord in the cadential progression.

This progression is ubiquitous in tonal Western art music: the cadential 6/4 chord typically appears after a ii or IV chord—a pre-dominant harmony, to be discussed in Chapter 24—and resolves to a root-position V chord as part of a structurally important cadence. The excerpts shown in the following examples show cadential 6/4 chords appearing after ii chords:

Cadential 6/4 chords usually resolve to a root-position V chord in the manner demonstrated by the examples above. Sometimes they resolve to a V7 chord, as in the following example:

In m. 15 (and again in m. 19) of Example 23–40, we find a cadential 6/4 chord resolving as expected to a root position V. In this case, however, the bass note (E) is doubled in the right hand of the piano part. The E then steps down to D, adding a seventh to the dominant triad.

A cadential 6/4 is a type of accented 6/4 chord. An accented 6/4 is a strong-beat chord, dominant or otherwise, that is delayed by a sixth and a fourth above the bass. These nonharmonic tones may be introduced as suspensions, accented passing notes, or appoggiaturas. The following excerpt contains two such sonorities:

The V7 chord at the end of m. 1 leads to a vi chord that is delayed until beat two with a fourth (G) and a sixth (B) suspended above the bass D. Even though the pitches on the downbeat of m. 2 seem to imply the presence of a ii6/4 chord, the progression V7–ii6/4 would be nonsensical. V7–vi65/4–3 is the correct way to analyze this progression. The arrival of the V chord in m. 4 is similarly delayed, in this case with a cadential 6/4 chord.

23.8 Summary

In this chapter we have seen how common interval progressions, coupled with melodic embellishments, may be used to expand harmonies. These individual embellishments combine to form full, four-voiced sonorities. Often, these sonorities coincidentally contain the pitches of familiar chords in first or second inversion. However, even though auxiliary sonorities may resemble those chords, it is important to remember that they do not carry the functional meaning of those chords. In other words, the V in a I–(V6)–I expansion does not have the dominant function it would in a I–IV–V–I.

Instead, we refer to these as “auxiliary sonorities.” Accordingly, we have either placed the auxiliary sonority in parentheses (i.e. I–(vii^o6)–I6) or refrained from labeling it altogether (i.e. “I          ”). Although root-position triads are permissible, most auxiliary sonorities appear as inverted triads for the sake of smooth voice-leading.

Although the majority of this chapter demonstrated the expansion of a I chord, we also saw that these progressions may serve as models for expanding other reference harmonies. In other words, the I–(vii^o6)–I6 expansion can be used as a model for expanding a ii harmony: ii–(I6)–ii6.

Some sonorities appear above a passing note in the bass. These are referred to as “passing auxiliary sonorities.” Others appear above neighboring motion in the bass and are correspondingly referred to as “neighboring auxiliary sonorities.” Among these, we saw several that resembled triads in second inversion: I-(IV6/4)-I and IV6-(I6/4)-IV, for example. Fully-diminished seventh chords are used as auxiliary sonorities too, sometimes sharing an unusual common tone with the reference sonority. The most common auxiliary sonority, however, is the cadential 6/4 chord which delays the upper voices of the dominant at important cadences.