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At first sight, it seems utterly perverse to search for the origins of sequences in the pages of Gradus ad Parnassum. After all, Fux called for variety in melodic writing and discouraged the use of recurring melodic patterns (or monotonia) in strict counterpoint.⁹ And yet, his examples often contain an important component of the sequence, namely, strings of notes that move in a single direc-tion by some fixed interval. This point is clear from figure 3.12 (Fux’s prototypical cantus firmi): although these melodies avoid exact repetitions of given melodic shapes, many of them end with a string of descending steps. In fact, Melody 6 ends with a string of three, Melody 1 with a string of four, and Melody 5 with a string of five!

The significance of these stepwise descents becomes more apparent when we add a simple counterpoint to the texture (see figure 3.13, Typical two-voice counterpoint in First Species).

Indeed, although both voices primarily move in contrary motion, they often move by parallel thirds or sixths when the cantus firmus descends by step at the cadence. These parallel strings actually cre-ate a strong feeling of motion that makes the closure at the cadence all the more compelling. In Fourth Species, this sense of propulsion is heightened when the parallel intervals are displaced to create chains of suspensions (see figure 3.14 , Typical two-voice counter-point in Fourth Species).

Textures with three or more voices provide even more opportu-nities for parallel motion by step. The reasons are clear: if the new counterpoint moves in contrary motion with the cantus firmus, then it will inevitably move in similar motion to the existing coun-terpoint; and if it moves in contrary motion with the counterpoint, then it will inevitably move in similar motion with the cantus

Figure 3.12. Fux’s prototypical cantus firmi.

Figure 3.13. Typical two-voice counterpoint in First Species. From Fux, The Study of Counterpoint, Fig. 22.

firmus. Fux himself offered some intriguing examples in his discus-sion of First Species in three voices: instead of recycling one of the cantus firmi from figure 3.12, he introduced some new prototypes in which the lowest voice simply ascends by step from C through D, E, and F to G, before ending back on C (see figure 3.15, Fux’s three-voice prototypes). The first prototype (figure 3.15a) has three des-cending thirds between the upper parts, whereas the second (figure 3.15b) has three ascending thirds between the lowest voices. In both of these cases, the soprano essentially moves in contrary motion with the bass. It seems very likely that Schenker had these proto-types in mind when he showed how Ursätze are usually transformed at the deep middleground.¹⁰ Similar examples are shown in figure 3.16 (Typical three-voice counterpoints in Fourth Species). Figure 3.16a is interesting for a couple reasons. In the first place, Fux har-monized the chain or 7–6 suspensions in the upper-voice parallels at the end of the passage with a sequential bass line F–C–D–A–D.

For another, he included a pair of 7–6 suspensions against the repeated tone G in the bass in mm. 3–4. In his text, Fux singled out this passage, noting that the bass can be extended to create a pedal

Figure 3.14. Typical two-voice counterpoints in Fourth Species. From Fux, The Study of Counterpoint, Figs. 73, 74.

Figure 3.15. Fux’s three-voice prototypes. From Fux, The Study of Counterpoint, Figs. 91, 92.

Figure 3.16. Typical three-voice counterpoints in Fourth Species. From Fux, The Study of Counterpoint, Figs. 141, 142.

point as shown in figure 3.16b. According to him, the results are

“not only correct but even very beautiful.”¹¹

As it happens, figures 3.16a–b open the door for mixed species.

Although Fux did not address this topic systematically in Gradus ad Parnassum, he did include a few tantalizing examples, given here in figure 3.17 (Parallel motion in mixed species with three and four voices). In figure 3.17a, he put the cantus firmus in the bass, the soprano in Second Species, the alto in Third, and the tenor in Fourth. Notice how the stepwise descent in the bass near the end of the example supports quasi-sequential motion in the upper voices.

Similarly, in figure 3.17b, he showed a three-voice texture in which the melody moves in First Species, the alto in Fourth, and the bass in Second. In this case, the parallel motion of the upper voices starting in m. 4 is supported by a sequential bass part, F–G, E–F, D–E, and C–D. These examples suggest that pedals and sequences often spring from the same source: pedals can sometimes be created when chains of stepwise parallel lines move against a static voice, whereas sequences can arise when such strings are supported by a recurring bass pattern. This ties in nicely with our discussion of fig-ures 3.3–3.11.

Whereas Fux avoided sequences, Schenker was openly hostile to them. His response was simply to reject them altogether. For example, when discussing the passage in figure 3.1a, he scoffed at the idea that the progression I–IV–VII–III–VI–II–I in mm. 1–8 is an “idle sequence” and insisted that it simply defines the tonic D

Figure 3.16 (continued).

minor.¹² Later, in the essay “Das Organische der Fuge” (1926), Schenker claimed that the word sequence “has no validity” and that “the mere fact of its existence as a theoretical term does not lend it any credibility as a concept.”¹³And, in Der freie Satz (1935), he rejected the term on the grounds that it described local details without explaining global processes.¹⁴ Yet, like Fux, Schenker cer-tainly recognized the significance of parallel stepwise counterpoint, especially over a pedal. This is already apparent in his discussion of mixed species in the latter portions of Kontrapunkt II (1922). Here, Schenker specifically referred to examples from Fux like those shown in figures 3.16–3.17. According to him, it is irrelevant whether the florid counterpoints are consonant or dissonant with the bass; all that matters is that they make contrapuntal sense with

Figure 3.17. Parallel motion in mixed species with three and four voices.

a. From Fux, The Study of Counterpoint, Fig. 204.

b. From Fux, The Study of Fugue, Ex. 61.

each other.¹⁵ This crucial idea fits with our observation that the soprano and alto parts in figure 3.3 make contrapuntal sense in their own right.

Matters become more complex when Schenker shifted his attention from the purely intervallic world of strict counterpoint to the triadic world of functional monotonality. Now the stepwise par-allel strings are constrained by the underlying harmonic motion of the music. Schenker took up this particular issue with a vengeance in his discussion of parallel linear progression in par. 224–26 of Der freie Satz.¹⁶In fact, these paragraphs appear in a general discussion of the various ways in which two or more linear progressions can be combined (par. 221–29); they mention sequences only in passing.¹⁷ The main thrust of Schenker’s argument is clear: when two or more linear progressions are combined, one is primary (or leading) and the others are secondary. To prioritize one linear progression over another, Schenker insisted that each progression must be evaluated according to the order in which it is generated from the back-ground.¹⁸ This is readily apparent from figure 3.18 (Parallel linear progressions within a single Stufe). Here Schenker sketched a short passage from the third movement of Mozart’s Piano Sonata in A Minor, K. 310. He suggested that the soprano voice leads and the alto follows, and that both voices move over a pedal A. Given the essential role that Urlinien play in generating the melodic profile of a piece, leading progressions are frequently found in the soprano voice. But, as Schenker was quick to point out, they need not be confined to this register: “once one has decided whether the leading linear progression is in the lower or in the upper voice, one must understand the counterpointing progressions as upper or

Figure 3.18. Parallel linear progressions within a single Stufe. From Schenker, Der freie Satz, Fig. 97.2.

lower thirds, tenths, or sixths.”¹⁹These points are implied in figures 3.3–3.11. Since these derivations proceed from a single stepwise descent that we subsequently harmonized in thirds or sixths, they necessarily treated one linear progression as the leader. Further-more, we have already noted that the various parallel voices can be redistributed in the texture and even inverted contrapuntally. We used these strategies to derive the sequences in figure 3.4 from those in figure 3.3.

Besides evaluating the status of linear progressions, Schenker also classified parallel linear progressions in two ways.²⁰The first way is according to the length of the leading progression. Since he believed that genuine linear progressions can span only the intervals contained within triads, this means that the leader will normally span a third, fourth, fifth, sixth, or octave; in the case of figure 3.18, for example, it articulates an octave span. That being said, it is important to note that the follower need not be the same length as the leader. Thus, whereas the leader in figure 3.18 projects an octave line, the follower is far more ad hoc in nature and barely a linear progression at all. Indeed, as William Rothstein has recently pointed out: “[T]he one instance in which Schenker admits non-harmonic Züge is in the case of two Züge moving in parallel thirds, sixths, or tenths.”²¹In this respect, the “leading” Zug arpeggiates the harmony;

the “follower” goes along passively for the contrapuntal ride.

The second way in which Schenker classified parallel linear progressions is according to whether they horizontalize a single Stufe or whether they fill in the space between two Stufen. We have already seen the first possibility in figure 3.18; in this case the par-allel linear progressions compose out a single A-major chord. The second possibility is shown in figure 3.19 (Parallel linear progres-sions between different Stufen). This figure derives an ascending 5–6 sequence along the lines shown in figure 3.9. Now, however, the phrase as a whole is controlled by a stepwise descent







the soprano and a cadential progression I–II5^–V⁷–I (see figure 3.19a). Before reaching the cadence, the alto line ascends G–A–

B–C–D–E (see figure 3.19b). In figure 3.19c, every note of the alto voice is repeated, and in figure 3.19d this string is harmonized with alternating thirds and fourths. Finally, in figure 3.19e the pedal is replaced by a string of alternating root and first-inversion chords.

Figure 3.19. Parallel linear progressions between different Stufen.

As a result, the alto voice appears to span a minor seventh from G to F. But this span is not, in Schenker’s terms, a true linear progres-sion; instead of composing out a single line of counterpoint, the span actually connects the alto voice of the opening tonic Stufe with the soprano voice of the II5^. The span is therefore an example of what Schenker termed “motion from an inner voice.” Although he devoted a paragraph of Der freie Satz to so-called seventh pro-gressions (Die Septzüge), Schenker conceded that they usually arise from changes of register between voices, rather than from genuine linear progressions within a single voice.²²