II. Diatonic Polyphony and Functional Harmony
In Chapter 25 we discussed the fundamentals of diatonic sequences and examined the most common type: those in which the harmonies descend by root motion of a fifth. In this chapter, we will discuss several other varieties of diatonic sequences.
26.2 Harmonic root motion and labeling sequences
The chords in a sequence do not usually progress in a functionally meaningful way. As we saw in Chapter 25, one encounters harmonic progressions in a sequence that do not usually appear in any other context. Roman numerals can provide a helpful summary of the changing pitch content in a sequence, but they can also be misleading since a chord in a sequence might behave in a very different manner to the same chord heard in a normal phrase.
In this chapter, we will use a supplementary line of analytical markings showing chord roots and the intervallic motion between them:
We refer to the progression in Example 26–1 as a “↓4th ↑2nd” sequence, since the root motion first descends by a fourth (as in G, the root of the I chord, moving to D, the root of the V6 chord) and then ascends by a second (as in V6to vi). This labeling system can be confusing at first since, as in the example above, the bass line may not match up with the root motion. Here, the bass line moves smoothly down by step with each new chord in the sequence despite the disjunct root motion. On the other hand, this labeling system is very useful since it can be used to highlight the similarities between sequences with superficially contrasting voice-leading.
Note: The intervals described in a label such as “↓4th ↑2nd” are abstract. They may, for example, be inverted in the actual music. A descending fourth, in other words, may be written as an ascending fifth. When determining the intervals to use in describing the root motion, select the smaller interval of the inversional pair. For example, use ” ↑2nd” instead of “↓7th.”
26.3 Sequences based on thirds
Sequences in which the harmonic units move by seconds or thirds run a greater risk of creating parallel fifths and octaves than those that move by fifths. Composers use a number of strategies to avoid such undesirable interval progressions. The following example shows a descending-third sequence in which intervening first-inversion chords break up the parallel motion:
Each step in the chain of descending thirds appears on the second beat of its measure: I–vi–IV. Between each step of the sequence, however, we find an intervening chord: there is a V6 chord between I and vi and a iii6 between vi and IV. These intermediate harmonies—along with the melodic figuration and the weak metric placement of the main chords—help obscure the parallel fifths between each step in the sequence.
Notice that the root motion here is the same as in Example 26–1: the root of the V6 chord is down a fourth from I and the root of the vi chord is a step above that of V6. The root motion heard from one chord to the next is one of the defining characteristics of a sequence and so, again, we will refer to this particular progression as a ↓4th ↑2nd sequence.
The following example shows a voice-leading reduction of the outer voices:
As this reduction shows, alternating between root position and first inversion produces a desirable effect: a stepwise descending bass line The intervening chords (V6 and iii6) break up the parallel fifths that would normally result from successive descending-third root motions. The result is a series of fifths suspended to become sixths as the bass steps down on the downbeat of each measure. As the second level of Roman numeral analysis shows, this sequence prolongs the initial tonic as I moves to I6. The descending 5–6 technique was a popular contrapuntal strategy in the Renaissance and was continually used in later music.
The following example shows another ↓4th ↑2nd sequence:
This sequence is very similar to the one shown in Example 26–2, though here there is much more melodic decoration. The descending steps in the bass occur on the beats in each measure, but here with octave leaps and passing tones. This sequence also moves through a 5–6 interval progression, but here the fifths are delayed by thirds on the downbeat of each measure. The following reduction clarifies:
In the following excerpt, from a well-known canon, the order of harmonies (I–V–vi–iii–IV–I) is virtually identical to the examples above, though here they all appear in root position:
Beginning with the initial tonic chord, this descending-third sequence continues until the next I chord on beat two of the second measure. Each step in the chain of descending thirds appears on a metrically strong beat (I–vi–IV), with intervening chords on the weak beats. Because each of the chords appears in root position, it is very easy to see the ↓4th ↑2nd root motion.
The following reduction removes the inner voices to reveal the contrapuntal framework in this sequence and clarify the voice-leading:
Between the outer voices we find tenths on the strong beats alternating with fifths on the weak beats. The intervening tenths obscure the parallel fifths. Like Example 26–2, this sequence prolongs the initial tonic harmony (as shown by the second level of Roman numeral analysis).
As we saw in Example 26–7, Johann Pachelbel’s Canon in D finds the outer voices alternating between tenths and fifths. But take note of the inner voices reintroduced below (arranged in SATB for ease of reading). What LIP is formed by the “tenor” and “alto” of this reduction?
Note the LIP appearing between the “tenor” and “alto.” On the first downbeat, the alto (A) forms a fifth above the tenor (D). The A is held as the tenor steps down to C, forming an oblique 5–6 interval progression. This pattern then repeats twice more. This inner-voice interval progression is the same one we saw in Example 26–3: a descending 5–6 pattern. Because this pattern is so recognizable, sequences such as the one found in Example 26–8 are often referred to as root-position variants of the descending 5–6 technique.
26.4 Sequences based on seconds
The following example shows a different strategy for avoiding objectionable parallels, this time in an ascending-second (and then descending-second) sequence:
In this passage, starting in m. 56, the bass note and accompanying melodic figure are each transposed up by step several times before being brought back down to their original pitches. Parallel fifths are not an issue here because the fifth of each chord has simply been omitted! There are parallel octaves, but these are obscured by the rhythm since notes sounding an octave apart never begin at the same time.
Contrapuntally, we may view the entire sequence as a series of parallel thirds prolonging the tonic harmony:
The following example shows an ascending-seconds sequence from a composition by Barbara Strozzi. What is the interval progression heard between the two voices in this sequence, and how are the parallel fifths concealed?
In determining the contrapuntal framework, look to the notes that fall on the beat.
The voices in this sequence move through a progression of parallel thirds. The parallel fifths are concealed by the rhythmic placement of the chord roots:
In the following example, first-inversion chords mediate between each step of an ascending-second sequence:
Like sequences based on thirds, ascending-second sequences often make use of intervening chords to break up parallel fifths and octaves. The IV chord on the downbeat of m. 12 initiates the sequence (IV–V–vi) with intervening chords on the weak beats. The intervening chords appear in first inversion, thereby forming stepwise motion in the bass. Again, despite the stepwise bass line, we refer to sequences of this sort as ↓3rd ↑4th, summarizing the root motion from one chord to the next. The following reduction reveals the outer-voice interval progression and how the intervening chords obscure the parallel fifths:
The ascending 5–6 motion seen in this reduction is remarkably similar to what we saw in Example 26–3. The only difference is that here the voices ascend instead of descend. (In this case, the sequence prolongs the pre-dominant harmony.) Once again, the intervening sixths obscure the parallel fifths by approaching them through oblique motion. Such interval progressions are often referred to as ascending 5–6 LIPs.
The following excerpt begins with an ascending-third sequence starting with the V7 chord in the first measure:
This ascending-third sequence features an ascending stepwise line in the uppermost voice. Again, intervening chords break up the inevitable parallel fifths. Note that while parallel octaves do appear on the downbeats between the bass and the middle voice, they quickly leap up to tenths on the second beat of each measure.
Looking at the reduction, we can see how the mediating chords break up the parallel fifths:
Instead of moving directly from one fifth to the next on the second beat of each measure, thirds intervene on the downbeats, changing the parallel motion to contrary motion. This results in the ascending stepwise motion of the entire upper line. If we consider the partially concealed inner voice, however, we find a familiar pattern:
Consider the interval progression formed by the inner voice and the upper voice. With the anacrusis to m. 2, we find the upper voice (D) a fifth above the inner voice (G). The G is held into m. 2 while the upper voice steps up to Eb forming a sixth with the inner voice. The pattern then repeats: 5–6–5–6–etc. This is the same LIP we saw in Example 26–12! In this case, however, each of the harmonies appears in root position. You can think of this pattern as a root-position variant of the ascending 5–6 technique.
Note as well that in this case, instead of prolonging a single harmony, the sequence prolongs the progression from i to ii7.
26.5 Ascending-fifth sequences
Ascending-fifth sequences are far less common than their descending-fifth counterparts. Nonetheless, they do appear with some frequency and have a decidedly different effect. Consider the following example:
The melodic figure in m. 20 is passed back and forth between the bass and uppermost voice with each change in harmony. Starting with the tonic chord in m. 20, the harmonic progression ascends by fifth in each subsequent measure: I–V–ii–vi. In m. 24, the root of the chord is again a fifth higher, but the pattern is broken by the altered melodic line in the upper voice.
Note as well that m. 24 introduces B§. That chord, initially heard as V/vi in Eb major, turns out to be an auxiliary sonority prolonging the C-minor chord of m. 23, which in light of the ensuing cadence in G minor is retroactively interpreted as iv in that key. (Changes of key such as this one will be explored in great detail in Chapter 28.)
Beginning with the I chord in m. 20 and going through the vi6 chord in m. 23, what is the LIP between the outer voices? (Hint: Look at the notes on the downbeat of each measure, ignoring, for the moment, the decorations that appear above and below.)
The primary notes of this LIP are found on the downbeats of each measure. Don’t be distracted by the decorative eighth notes that appear above and below later on in each measure.
Looking at the outer-voice-leading reduction, we again see a familiar interval progression:
Here, the root-position chords have a third above the bass while the first-inversion chords have a sixth. 3–6 interval progressions are also very common. Note that in contrast to descending-fifth sequences, in which the overall motion descends, here the overall motion ascends.
The beginning of the following excerpt, begins with a ↑5 sequence (Example 26–19 provides a reduction):
Throughout the passage, we encounter suspensions and other techniques smoothing out the ascending-fifth progressions. In m. 1, the V chord is introduced as the continuation of a bass arpeggiation of I. The suspended fourth (C on beat 3) resolves on the fourth beat as the upper voice makes a consonant leap up to the root of the triad. That voice is then suspended as a dissonant fourth into the next measure before resolving to the tenth above the bass and repeating the pattern. The basic framework of this sequence, then, is a series of alternating tenths and fifths.
Note: Neither of the ascending-fifth sequences discussed above run past the fifth chord in the sequence—to do so would introduce a tritone in the bass as viio moved to IV. Sometimes composers will skip entire steps of a sequence to avoid this situation. Consider the following example:
This ascending-fifth sequence begins normally, but breaks the pattern in mm. 449 to avoid the tritone bass motion: I–V–ii–vi–[missing: iii–viio]–IV–I.
Other interval patterns are possible (e.g. 10–5 and 10–10) with ascending-fifth sequences, depending on which chords are inverted, and on what chord member appears in the uppermost voice.
While the majority of sequences move by descending-fifth root motion, you will also encounter sequences that move by seconds or thirds. Many of these are structured by a 5–6 interval pattern and include intervening chords that offset the parallel fifths that inevitably arise. In a descending 5–6 sequence, each repetition descends by a third but, typically, intervening chords lead to a ↑4th↓2nd root motion within each step. Ascending 5–6 sequences, on the other hand, typically follow ↑3rd↓4th root motion from one chord to the next. Ascending-fifth sequences have a unique effect, but occur much less frequently than their descending-fifth counterparts.