To shed light on the nature of sequences, we will begin by looking at some familiar patterns, starting with the one in figure 3.2 (A typ-ical ascending-fifth sequence). This particular phrase has two main components: a sequence and a cadence. The former consists of pairs of fifth-related harmonies that are transposed down a third: C–G,
Figure 3.1 (continued).
A–E, F–C, D–A. The latter, meanwhile, consists of a typical progression II5^–V–I. Innocuous as these observations may seem, they actually suggest that each component encapsulates a quite dif-ferent principle of voice leading. These differences are most obvi-ous in the soprano and alto voices. Indeed, figure 3.2 clearly shows that, while the two upper voices move by parallel thirds during the sequence, they ultimately converge on the tonic at the cadence.
This point has important ramifications; among other things, it con-firms the intuition that parallel motion creates a sense of openness, whereas convergence creates a sense of closure. We might also note that the soprano and alto lines make contrapuntal sense on their own; the other voices could be omitted and the phrase would still define the tonic C. This point is supported by the fact the two upper voices can potentially be supported by a long tonic pedal in the bass. We will return to this connection between sequences and pedals later in the chapter. But for the time being, we can simply conclude that sequences are ultimately contrapuntal, rather than harmonic, in origin and that they might be generated from parallel voice leading, rather than from the outer voice counterpoint.
This observation allows us to generate the pattern in the manner shown in figure 3.3 (Deriving ascending-fifth sequences).
Figure 3.3a begins by presuming that the sequence occurs within the context of a single phrase, in this case one that descends by step
3–2–1 and ends with the cadential progression II5^–V8–7–I. Since this derivation is context-dependent, we can avoid ‘The Top-Down/Bottom-Up Problem.’ Next, the soprano tone
3 is transferred up an octave and the intervening space is filled by a stepwise descentFigure 3.2. A typical ascending-fifth sequence.
(see figure 3.3b). In figure 3.3c, this descent is harmonized by a string of parallel thirds. Finally, figure 3.3d harmonizes this string of thirds according to two principles: each third must be supported by a root triad and each triad should not produce parallel perfect octaves and fifths with its neighbors. Significantly, the laws of tonal
Figure 3.3. Deriving ascending-fifth sequences.
voice leading dictate that only one such harmonization is possible from a given starting chord. For example, although the third D/B can belong to triads on G or B, the latter creates parallel perfect octaves and fifths with the initial tonic chord. This leaves G as the only option. Similarly, when the third D/B descends to A/C, the latter can be harmonized by triads whose roots are F and A. But since the triad on F creates parallel perfect octaves and fifths with the preceding G chord, the triad on A must be picked. In other words, we can resolve ‘The Parallel Problem’ by generating the sequence from the upper voices and not the bass.
The same strategy can be used with suitable adjustments to derive other sequences. Perhaps the easiest way to change the pattern in figure 3.3 is to invert the voices contrapuntally, thereby transforming the chain of parallel thirds into a chain of parallel sixths. This process is shown in figure 3.4 (Restacking ascending-fifth sequences). In figure 3.4a, the soprano spans an octave
8 to 1. The final stepwise descent 2–1 is harmonized by a cadential progression II5^–V–I. Next, figure 3.4b supports the stepwise descent of the soprano with a string of parallel sixths from E to E in the tenor voice. Finally, these parallel sixths are harmonized by root triads (see figure 3.4c). As with figure 3.3, the starting chord and laws of counterpoint dictate that only one such harmoniza-tion is possible.With further adjustments, we can generate still more sequences (see figure 3.5, Deriving ascending-third sequences). Figure 3.5a starts again with a phrase model consisting of a descent
3–2–1 inthe soprano supported by a simple cadential progression I–II5^–
V8–7–I. As before, the soprano tone
3 is transferred up an octave.The intervening octave space is then filled by step (see figure 3.5b) and harmonized by a string of parallel thirds in the alto voice (fig-ure 3.5c). But in fig(fig-ure 3.5d, the soprano E from the first third E/C is displaced over the second alto note B to create the pattern 3–4–3. Similar displacements occur when C/A descends to B/D, when A/F descends to G/E, and when F/D descends to E/C. Instead of creating a chain of parallel thirds, this process produces the suc-cession 3–4–3–3–4–3–3–4–3–3–4–3. Although figures 3.5a–d set the soprano and alto voices against a single tonic pedal, figure 3.5e harmonizes the descent 4–3 as a passing motion; E/B–D/B are supported by a triad on E, C/G–B/G by a triad on C, A/E–G/E by
a triad on A, and F/C–E/C by a triad on F. For variety, several chromaticisms have been added; these introduce applied chords that tonicize the triads on VI, IV, and II.
Whereas the sequence in figure 3.5 elaborates the chain of paral-lel thirds by displacing the lines to create the pattern 3–4–3, other sequences can be created by inserting intermediary tones (see figure 3.6, Deriving descending-fifth sequences). This sequence is, of course, analogous to the one in figure 3.1. Figure 3.6a begins with a stepwise descent
5–4–3– 2– 1 in the soprano. This motion is supported by aFigure 3.4. Restacking ascending-fifth sequences.
Figure 3.5. Deriving ascending-third sequences.
simple cadential progression I–II5^–V8–7–I. Next, the soprano part is elaborated with escape tones (see figure 3.6b) and the new line is har-monized by a string of parallel thirds (see figure 3.6c). Figure 3.6d then harmonizes the soprano and alto lines exclusively with root triads.
Once again, there is only one possible harmonization. Although the
Figure 3.6. Deriving descending-fifth sequences.
third A/F belongs to triads on D and F, the former is impossible because it produces parallel perfect octaves and fifths with the open-ing tonic chord. Similarly, when the third A/F leaps to F/D, the latter can be harmonized only by a triad on B because a triad on D creates parallel perfect octaves and fifths with the preceding F harmony.
Figure 3.4 modifies the sequence in figure 3.3 by inverting the lines contrapuntally; the soprano line in figure 3.4 was originally the alto line in figure 3.3, and the tenor line in figure 3.4 was origi-nally the soprano line in figure 3.3. But, if the alto line in figure 3.3 is placed in the bass, below the tenor voice, then this no longer the case; the resulting pattern is now reclassified as a descending 5–6 sequence. Figure 3.7 (Deriving descending 5–6 sequences) shows how to generate such a configuration. Figure 3.7a starts with a descent
3–2–1 in the soprano, harmonized by the cadential pro-gression I–II5^–V8–7–I. In figure 3.7b, 3 in the soprano line is trans-ferred up an octave and the intervening space is filled by a stepwise descent. Figure 3.7c then supports the octave descent with a string of parallel tenths in the bass. Finally, figure 3.7d adds inner voices to this parallel string. Although these tenths can be harmonized in other ways—perhaps by a string of parallel 6/3 sonorities—figure 3.7d follows the same scheme as figure 3.3, thereby underscoring the close connections between these two forms of sequence.In the same vein, we can develop further transformations of our model. One such transformation is shown in figure 3.8 (Deriving alternative descending-fifth sequences). Figure 3.8a starts with a descent
3–2–1 in the soprano, harmonized by the cadential pro-gression I–II5^–V8–7–I. In figure 3.8b, the alto voice G moves up by step through A and B to C, with each of these notes repeated. This motion from an inner voice is then harmonized in parallel thirds (see figure 3.8c). Finally, figure 3.8d harmonizes the chain of thirds with root chords. Given the starting chord and the laws of counter-point, only one harmonization is possible. Notice how the tenor part rises by step from C to G before descending back to E.A similar strategy is shown in figure 3.9 (Deriving ascending 5–6 sequences). Once again, figure 3.9a starts with a descent
3–2–1in the soprano, harmonized by the cadential progression I–II5^–V8–7–I. In figure 3.9b, the alto moves up by step from G to C, and in figure 3.9c it is supported by parallel thirds with the tenor.
Figure 3.7. Deriving descending 5–6 sequences.
Next, figure 3.9d repeats each new member of the alto and tenor voice; it also displaces them so as to create the pattern 3–4–3–4–3–4–3. Finally, in figure 3.9e, this pattern is then harmo-nized by alternating root and first inversion triads. As in figure 3.5, a few chromaticisms have been added; these allow us to treat the 6/3 sonorities as applied dominants.
Figure 3.8. Deriving alternative descending-fifth sequences.
Figure 3.9. Deriving ascending 5–6 sequences.
We have already shown in figures 3.5 and 3.9 that sequences often allow composers to introduce chromaticisms; in both cases, applied chords were added to tonicize particular harmonies. But there is another important source of chromaticism in tonal con-texts, namely the principle of mixture; figure 3.10 (Simple mixture)
Figure 3.10. Simple mixture.
and figure 3.11 (Double mixture) derive two chromatic sequences.
Figure 3.10 presents a variant of the descending-fifth sequence that we derived in figure 3.6. Figure 3.10a begins with a stepwise descent in the soprano from
5 to 1, though the third is mixed 5–4–3–2–1.This motion is supported by a simple cadential progression V8–7–I.
Figure 3.11. Double mixture.
Next, the soprano part is elaborated with escape tones (see figure 3.10b) and the resulting line is harmonized by a string of parallel thirds (see figure 3.10c). Figure 3.10d then harmonizes the soprano and alto lines exclusively with root chords. Again, this harmoni-zation is the only one possible. Although the third A/F in m. 2 belongs to triads on D and F, the former is impossible because it pro-duces parallel perfect octaves and fifths with the opening tonic chord. Similarly, when the third A/F leaps to F/D, the latter can be harmonized only by a triad on B because a triad on D creates paral-lel perfect octaves and fifths with the preceding F harmony. Notice how the D harmony II serves as a pre-dominant at the cadence.
Figure 3.11 follows the same strategy, though the double mixture is created by changing the quality of the triads on B, E, and A from major to minor.
To sum up, figures 3.3–3.11 resolve the problems of generating sequences in two quite different ways. On the one hand, they overcome the ‘The Top-Down/Bottom-Up Problem’ by deriving sequences within the context of a phrase. This move guarantees that the goal of the sequence is always specified before its surface features are completely worked out. On the other hand, figures 3.3–3.11 sidestep ‘The Parallel Problem’ by deriving the sequence from parallel motion in the upper voices and not necessarily from the counterpoint between the outer voices. Not only does this approach contrast with most conventional accounts of sequences, but it also confirms the notion that sequences are basically contra-puntal, rather than harmonic in nature. Indeed, by showing that the bass motion is ultimately controlled by the upper-voice coun-terpoint, figures 3.3–3.11 also imply that harmonic function in a Riemannian sense emerges from contrapuntal motion. This point is evident both inside the sequence, where functionally related har-monies derive from parallel step motion, and at the cadence, where the penultimate dominant chord converges on the final tonic, with the soprano descending
2–1, the alto ascending 7–1, and the pre-dominant chord converges on the pre-dominant chord, with 4 and 6both moving to
5. These derivations even suggest that interesting connections can be found between sequences and pedals. But this contrapuntal view of the sequence begs its own set of questions: to what extent can we find precursors of sequences in traditionalcontrapuntal theory, especially Fuxian species counterpoint, and how do the derivations in figures 3.3–3.11 fit with Schenkerian theory? In order to answer these questions, we must take a closer a look at the principles of strict counterpoint as formulated by Fux and assess their impact on Schenker’s thinking.