As mentioned above, Schenker’s thinking about functional tonality is dominated by the basic idea that complex tonal progressions can be explained as transformations of simple tonal prototypes. For him,
the processes of transformation have several implications. First, they imply that whenever a prototype is transformed, it will be elaborated with new harmonic and melodic material. Schenker drew atten-tion to this idea by invoking a number of colorful terms, such as Auskomponierung (composing out), Mehrung (increase of content), and Auswicklung (unwinding).2Second, these processes imply that when a prototype is elaborated, the new material may behave some-what differently from the original prototype. For example, although Schenker’s prototypes are fundamentally consonant and diatonic, they can be transformed to create progressions that are highly disso-nant and chromatic. Such changes are what Schenker had in mind when he used terms like Umwandlung (reshaping) and Umbildung (recasting).3
Schenker was not, of course, the first person to think about music in terms of prototypes, transformations, and levels; on the contrary, the same ideas lie at the heart of many theoretical proj-ects, including Fuxian species counterpoint.4Fux’s interest in the process of elaboration is obvious enough; in codifying the Five Species he set out to show how cantus firmi can be elaborated with simple or florid counterpoints. But as we saw in chapter 1, Fux also realized that the behavior of the contrapuntal lines alters from one species to the next; whereas counterpoints are always strictly con-sonant with the cantus firmus in First Species, they can be disso-nant in Fifth Species. Furthermore, as counterpoints become more elaborate, so they can include progressively more repeated tones and even chromaticisms.
But while Fuxian species counterpoint has some obvious connections with Schenker’s system of prototypes, transformations, and levels, it nonetheless differs in crucial respects. For one thing, the theoretical status of a cantus firmus is very different from that of a Schenkerian prototype. Fux’s cantus firmi are essentially examples of effective modal melodies. Each one begins and ends on the final of a given mode and each one moves onto the reciting tone at points of structural significance. Meanwhile, Schenker’s prototypes are far more general in scope. Instead of representing a specific melody per se, they encapsulate certain underlying principles of melodic construction. These principles guide both the local behavior of indi-vidual melodies and the global behavior of entire pieces. Schenker
conveyed such levels of abstraction very beautifully at the start of Der freie Satz, in a chart given here as figure 2.1 (Schenker’s concept of prototypes, transformation, and levels). This chart shows that the tonality of a given foreground (Vordergrund) can be generated from the diatony of the given background (Hintergrund) through various levels of the middleground (Mittelgrund). Each prototype (Ursatz) contains an upper line (Urlinie) in the upper register and a bass arpeggiation (or Bassbrechung) that articulates the upper fifth, i.e., it moves from I to V and back to I. To reinforce the global nature of his prototypes, Schenker even included a paragraph in Der freie Satz that differentiates prototypes from cadences.5For convenience, we will refer to the notion that functional monotonal pieces derive from a single prototype as ‘The Global Paradigm.’
As it happens, ‘The Global Paradigm’ did not come to Schenker overnight; it actually developed in his mind over a period of at least twenty years. In fact, he started to toy with the concept of a prototype as early as the Harmonielehre and Kontrapunkt I and even criticized C. P. E. Bach for underestimating their significance.
But at this point in time his prototypes were purely local phenom-ena and certainly did not control an entire piece.6 Over the next decade, however, they became wider in scope.7 In his edition of Beethoven’s Piano Sonata, Op. 110 (1914), for example, Schenker explained how the development section of the second movement
Figure 2.1. Schenker’s concept of prototypes, transformation, and levels. From Schenker, Der freie Satz, Fig. 1.
could be derived from a single 32-bar progression. Six years later, in his comments to Beethoven’s Piano Sonata, Op. 101 (1920), he even coined the term Urlinie. Nevertheless, it was not until the early issues of Der Tonwille (1921) that he proposed a single Urlinie for an entire piece and the fifth issue (1923) that he derived a whole work from a single Ursatz.8
Another important difference between Fux’s conception of species counterpoint and Schenker’s system of prototypes, transfor-mations, and levels is that the latter is recursive and rule preserving, whereas the former is not. Recursion is a term borrowed from mathe-matics. In very general terms, a musical system is recursive if it posits certain starting states, such as a prototypical harmonic progression, and derives more complex states, or progressions, by repeatedly applying a given set of transformations. This system is also rule preserving if every derived state or progression conforms to the same underlying laws of voice leading and harmony as the prototype. If the musical system is indeed recursive and rule preserving, then the processes of generation and reduction will be the reverse of each other. We will refer to these ideas as ‘The Recursive Model.’
There are good reasons to suppose that Fuxian species counter-point is neither recursive nor rule preserving. To begin with, Fux never claimed that Fifth Species counterpoints can be generated by repeatedly elaborating First Species prototypes, nor did he suggest that Second, Third, and Fourth Species are intermediate stages of generation. Furthermore, if Fuxian species counterpoint were indeed recursive and rule preserving, then we would be able to reduce all of the species to First Species prototypes. But this is not necessarily the case. Figure 2.2 (The non-recursive nature of Fuxian species counterpoint) gives settings of a single cantus firmus in First, Second, Third, and Fourth Species. Although these settings all satisfy the laws of strict counterpoint, they cannot be reduced to a plausible First Species prototype, like the one in figure 2.2a. Take, for example, the counterpoint given in figure 2.2b. When we try to reduce the counterpoint in mm. 3–5 from two notes per measure to one, we soon run into difficulties. If we regard the G in m. 4 as ornamental, then parallel octaves occur between the downbeats of mm. 4 and 5. Conversely, if we regard the C in m. 4 as ornamental, then we create parallel fifths with the main note A in m. 3. Many
of the same issues arise in mm. 4–5 of figure 2.2c. But the problems are even more acute in figure 2.2d. Here, the string of suspensions is created by displacing the counterpoint over the cantus firmus. If we try to normalize this displacement, then the resulting First Species prototype consistently violates the law prohibiting parallel perfect fifths (c.f., mm. 3–7).
Whatever their similarities, it seems that Fux’s concept of species counterpoint and Schenker’s system of prototypes, transfor-mations, and levels have quite different goals. Whereas Fux used his cantus firmi to illustrate a well-composed melody in each mode, Schenker used his prototypes to explain the general principles of
Figure 2.2. The non-recursive nature of Fuxian species counterpoint. From Fux, The Study of Counterpoint, Figs. 11, 36, 57, 76.
melodic construction as they operate locally within a phrase and globally across an entire piece. And, whereas Fux’s species teach students certain general techniques of elaboration, Schenker’s system of prototypes, transformations, and levels explains how the extraor-dinary diversity of functional tonality stems from the working out of certain underlying principles of counterpoint and harmony. To show how this system is, in principle at least, both recursive and rule preserving, let us now consider its main components in detail, starting with Schenker’s prototypes.
Prototypes
For anyone concerned with the explanatory scope of Schenkerian theory, it is important to reconsider the various covering laws mentioned in chapter 1. In very general terms, these laws cover six areas: 1) how individual lines move and reach closure; 2) how polyphonic lines move in relation to one another; 3) how unstable tones behave in relation to stable tones; 4) how stable harmonies are distinguished from unstable harmonies; 5) how successive har-monies are arranged to create typical functional progressions; and 6) how chromatic harmonies arise in functional tonal contexts. As such, these laws seem to be necessary and sufficient for explaining tonal relations. Within each domain, we classified these laws in several ways. On the one hand, we distinguished main laws from subordinate laws: the former explain how melodies normatively behave, whereas the latter explain significant exceptions to that norm. On the other hand, we distinguished local laws from global laws; the former explain how one note moves to the next, whereas the latter explain how the melody moves as a whole.
The great advantage of classifying the covering laws in this way is that it allows us to structure our knowledge about tonal music; this structure becomes very important when we reformulate our laws as prototypes, transformations, and levels. As we saw in chapter 1, Schenker believed that, locally, melodies mainly move by step and, globally, they are maximally closed if they begin on
8, 5, or 3 andend
2–1. Similarly, he acknowledged that contrapuntal lines tend to move in contrary motion or in parallel thirds, sixths, and tenths andnever produce parallel perfect octaves or fifths between successive Stufen. Schenker also accepted that the only stable harmonies are functional triads in root or first inversion. According to him, chord successions are controlled contrapuntally, with closure most strongly articulated by the harmonic progression V–I. And, he recognized that functional progressions are primarily diatonic in nature.
If we now look at Schenker’s prototypes, we soon see that they summarize these main laws in an optimally compact way. For con-venience, the three parts of figure 2.3 (Schenkerian Ursätze in C Major) show the three basic prototypes for C major. Each one consists of a stepwise descent from a headtone (Kopfton), either
3, 5, or 8, to 1. This simple passing motion is supported by a bass arpeggiation I–V–I. Whereas Schenker notated the upper line and bass arpeggiation in whole notes, he included a prototypical inner voice in black note heads. Elsewhere, he referred to the precise location of the prototype as the obligatory register (obligate Lage).9Figure 2.3. Schenkerian Ursätze in C Major. Adapted from Schenker, Der freie Satz, Figs. 9, 10, 11.
With respect to the laws of melodic motion and closure, it is clear that the upper line follows the local law of moving by step and the global law of beginning on
8, 5, or 3 and ending 2–1. The upper line and bass arpeggiation likewise obey the main laws of relative motion: the three essential lines close 2–1, 7–1, and 5–1, whereasthe outer voices essentially move in contrary or oblique motion with the upper line descending from the headtone to
1 and the bass arpeggiation ascending from I to V. Similarly, each prototype follows the main laws of vertical alignment by beginning and ending on members of the tonic Stufe. Schenker’s prototypes also conform to the main laws of functional harmony: each one contains three Stufen arranged to form the quintessential functional progression I–V–I. This progression is not only diatonic, but it also defines the tonic C in the most unambiguous manner possible.Having shown how Schenker’s prototypes summarize the main laws of functional tonality in an optimally compact way, we can now respond to several alternative accounts of their structure. For starters, although we have seen certain connections between Schenkerian theory and Fuxian species counterpoint, one should not read too much into Carl Schachter’s claim that Schenker’s pro-totypes are basically Second-Species constructs.10While Schachter is certainly right to suggest that their upper lines derive some of their logic from the principle of passing motion, it is important to remember that these lines do not belong to the purely intervallic world of strict counterpoint; on the contrary, they clearly belong to the world of Stufen, something that is underscored by the Roman numerals marked under the bass arpeggiation in figures 2.3a–c.11
We can use similar arguments to rebuff Peter Westergaard’s critique of
5 and 8 lines.12 Westergaard has suggested that, since these upper lines contain unsupported stretches (Leerlaufen), they are conceptually inferior to 3 lines. This notion seems to be con-firmed by the fact that the Fünf Urlinie-Tafeln and Der freie Satz contain a deluge of 3-line graphs and a dearth of 8-line readings.The chief problem with Westergaard’s position is that Schenker himself did not insist the upper line should be completely sup-ported; on the contrary, in par. 69 of Der freie Satz, he specifically acknowledged that they can contain passing tones.13Since
5 and 8lines do indeed conform to the general laws of tonal voice leading
outlined in chapter 1, there are no grounds for discriminating against them. This does not mean that
5 and 8 lines will necessarily be as common as 3 lines; on the contrary, there are good reasons why they may be rare. For example, 8-lines present the composer with a fairly narrow set of options; it is perhaps small wonder that they tend to be found in specific types of music, such as Baroque preludes.14By the same token, we can respond to David Neumeyer’s revised list of prototypes. In a series of thought-provoking papers, Neumeyer has proposed that: 1) there are other forms of upper line, including some that rise, for example,
5–6–7–8; 2) 8 lines properly belong to the middleground not the background; and 3) some pieces are controlled by three-voice prototypes consisting of “a structural soprano and a structural alto—above a bass.”15 These alternative prototypes stem in part from the work of Schenker’s pupil, Felix-Eberhard von Cube.16 The chief drawback with Neumeyer’s additions is that they no longer conform to the laws of tonal motion mentioned above. Most obviously, his rising lines contradict the law that melodies reach maximum closure when they descend 3–2–1. Since 8 lines satisfy this and our other laws, it is hard to see how they can be rejected as Neumeyer suggests.17 Having said that, Neumeyer’s case for three-part prototypes is utterly convincing. For one thing, it is certainly borne out by the examples shown in figures 2.3a and b. For another, Joseph Lubben has noted that in Der Tonwille, Schenker often used the term Aussensatz to denote “a structure of two upper voices above the Stufen.”18Furthermore, since Schenker clearly used so-called poly-phonic transformations (for example, unfolding and motion from an inner voice) at the deep middleground, it suggests certain inner voices must be present at some prior level or derivation. As we will see in chapter 3, these inner voices also help us resolve certain inconsistencies in Schenker’s generation of sequences.Finally, we can address Eugene Narmour’s charge that Schenkerian theory commits the fallacy of affirming the conse-quent.19 Narmour claims that Schenkerian theory is circular because it sets out to show that all tonal compositions can be gen-erated from various prototypes. According to Narmour, however, the analyst must know the nature of these prototypes in advance in
order to make a reduction. Since Schenkerian analyses seem to bend the notes to fit the theory, they are, in Narmour’s opinion, specious. Given the ways in which Schenkerian analyses are normally presented, it is hard to disagree with Narmour; at times the arguments do indeed seem viciously circular. But the foregoing account minimizes these problems by suggesting that the process of confirming Schenkerian theory is a lot more complex than Narmour suggests. We have seen that the explanatory laws underpinning Schenkerian theory were actually discovered empirically in the Harmonielehre and Kontrapunkt I, long before Schenker formulated his concept of a single tonal prototype. These laws are, in fact, extensions or transformations of well-established laws of counter-point. By classifying them along the lines suggested in chapter 1, we have good reason to suppose that they are both necessary and sufficient for all functional tonal music. After spending the next decade studying a broad range of functional monotonal composi-tions, Schenker discovered empirically that he could reformulate this set of explanatory laws in terms of prototypes, transformations, and levels. There is a sense, then, in which the principles govern-ing Schenker’s prototypes can be confirmed independently without the need for graphing; whatever circularity remains stems from the kind of bootstrapping process by which Schenker came up with his results.
Transformations
So far, we have seen that Schenker’s prototypes summarize the main laws of tonal voice leading and harmony in an optimally compact way. They are the simplest possible expressions of a given key. But we also know from chapter 1 that these particular laws do not explain every aspect of functional tonality; on the contrary, we also introduced a number of subordinate laws to cover deviations from these norms. Among other things, these exceptions allow us to explain why leaps can occur in melodic lines, why melodic lines can contain a whole host of dissonances and not just the simple passing tone, why functional progressions can include harmonies other than I and V, and why these progressions can contain a variety of
chromaticisms. To explain these diverse phenomena and satisfy the various subordinate laws, Schenker invoked a set of transformations or prolongations.
Although Schenker did not define his arsenal of transformations as precisely as we might wish, we can divide them into four main groups. The first group horizontalizes a given Stufe by presenting the constituent harmonic tones successively rather than simultaneously (see figure 2.4a), the first of Horizontalizing transformations.20 Of these, the simplest is repetition (Wiederholung). In a nutshell, repeti-tion expands a particular Stufe by taking a note in the soprano or bass voice and duplicating it exactly. In figure 2.4b, the tonic chord in C major is composed out by repeating the soprano pitch G. Sig-nificantly, repetition can generate new material before or after the original Stufe: this new material is said to be “front-related” if it appears before and “back-related” if it appears after.
Whereas repetition duplicates a particular tone in a given regis-ter, the other members of this group create leaps. To begin with, reg-ister transfer creates octave leaps by projecting a tone from one register to another. Schenker referred to this idea in general by the term Lagenwechsel and introduced specific transformations to denote ascending register transfers (Höherlegung), descending transfers (Tieferlegung), and alternations between a given pair of registers (Koppelung). An ascending register transfer is given in figure 2.4c.
Like repetition, register transfer can occur in the soprano or the bass.
Meanwhile, arpeggiation (Brechung) creates leaps by taking the soprano or bass note and moving to another member of the same Stufe. For example, in figure 2.4d the soprano voice arpeggiates the tonic Stufe by leaping from the third to the root. Significantly, arpeg-giations can appear successively and even in the same direction. Just like repetition, they can be applied before or after the original Stufe.
Just as register transfer and arpeggiation create leaps by moving from one harmonic tone to another, so unfolding, voice exchange,
Just as register transfer and arpeggiation create leaps by moving from one harmonic tone to another, so unfolding, voice exchange,