From the preceding discussion, it should be clear that ‘The Hein-rich Maneuver’ ties the principles of tonal voice leading to the behavior of tonal harmonies. This observation reflects Schenker’s general belief that it is impossible to understand functional tonality adequately from a purely contrapuntal or a purely harmonic per-spective. Indeed, as he put it in Harmonielehre: “[O]ne note, or even more, may be heard merely horizontally, while the vertical is to be totally disregarded; for other notes, on the contrary, the vertical concept is far more important.”45For convenience, we will refer to this interrelation of line and chord as ‘The Complementarity Principle.’
Figure 1.23. Beethoven, Piano Sonata, Op. 81a, 1st movement, mm. 230–42.
From Schenker, Harmonielehre, Ex. 132.
This principle can, in fact, be stated even more strongly: although every tonal event can be interpreted contrapuntally or harmoni-cally, any account that is purely contrapuntal or purely harmonic will necessarily be incomplete.46 ‘The Stufe Constraint’ does not simply list what essential harmonies are possible, it also explains how they conform to “their own secret law of progression.”47 To understand what ‘The Stufe Constraint’ involves, let us briefly com-pare traditional explanations of functional harmony with those offered by Schenker.
In general terms, tonal theorists usually address three main issues. First, they are concerned with showing how stable harmonies are distinguished from unstable harmonies. Most theorists believe that functional tonal music is basically built from triads and seventh chords, which they classify in terms of their quality (major, minor, diminished, and augmented) and their inversion (root, first, second, and third). We can use the term ‘inversional equivalence’ to refer to the notion that the identity of a triad is independent of its vertical ordering. Most theorists also classify triads into seven types identified by seven Roman numerals. Second, tonal theorists are interested in determining how successive harmonies are arranged to create typical functional progressions. To do this, many theorists invoke the idea of
‘functional equivalence’: they propose that the seven essential harmonies fulfill three basic functions—tonic (I, VI, and III), domi-nant (V and VII), and subdomidomi-nant (IV, II, and VI). They then sug-gest that prototypical tonal progressions follow the scheme tonic (T)–subdominant (S)–dominant (D)–tonic (T). Third, tonal theo-rists are concerned with understanding how chromatic harmonies arise in functional tonal contexts. Since most theorists assume that functional tonal music is fundamentally diatonic, they usually regard chromaticisms as surface deviations from this basic system.
Schenker’s outlook on these three topics was, however, anything but conventional; by including voice-leading elements in his theory of harmony, he was able to simplify the principles of harmonic classification and harmonic progression considerably, thereby increasing their flexibility and accuracy. In the first case, he used the laws of tonal voice leading to limit the number of essential harmonies to major, minor, and diminished triads and he was very suspicious of inversional equivalence.48 To be specific, since the
interval of the perfect fourth is dissonant when it occurs above the bass, Schenker seldom treated 6/4 sonorities as functional har-monies, and since augmented triads do not appear within the major or minor systems, he did not count them as essential either. Finally, and perhaps most importantly, Schenker rejected the idea that seventh chords normally behave as functional harmonies. To quote from the start of Kontrapunkt I:
The Stufe exists in our perception only as [a] triad; that is, as soon as we expect a Stufe, we expect it first of all only as [a] triad, not as a seventh chord. In this sense, the seventh is absolutely not an a priori element of our perception comparable to the fifth and the third; it is rather an event a posteriori, which we understand best of all with reference to the function associated with it; that is, we understand it in retrospect as a passing tone, or as a means of chromaticization, or the like.49
Obviously, if seventh chords are ultimately created by contrapuntal motion, then so must more abstruse sonorities, such as ninths, thir-teenths, and augmented-sixths. These are shown in figure 1.24 (Laws of harmonic classification).
Schenker also used the laws of tonal voice leading to shed light on the behavior of harmonic progressions (or Stufengang). He was quite clear that cadential closure was not simply a contrapuntal phenomenon, it also depends on the distinctive motion from dom-inant to tonic:
In order to gain insight into cadences in [tonal] composition it is important to recognize that there the closure is no longer based on the horizontal line alone but rather (and to a larger degree) on the harmony of the vertical [dimension], or, more precisely, on the succession from the V Stufe to I.50
But while he certainly accepted the functional priority of the tonic and dominant Stufen, Schenker rejected the notion of functional equivalence: he denied that the seven Stufen necessarily fulfill just three basic functions—tonic, subdominant, and dominant. To quote from Kontrapunkt I, “How can one claim to have understood the [tonal] ‘system’ if its individual Stufen, except I, IV, and V, are deprived of their independence and thus of their attractive capabil-ity of assuming various functions?”51 Schenker added, “[I]t is the functional versatility of the Stufe that is the basis of [tonal] practice,
and this, of course, at least presupposes its independence!”52 Although there will certainly be times when II chords behave the same way as IV chords, there will be other times when they do not;
this means that the notion of functional equivalence cannot be generalized across the entire range of tonal progressions.53
Taking Schenker’s argument further, even if two Stufen do behave in the same way, they need not arise for the same reasons (see figure 1.25, Chord function vs. chord derivation). Figure 1.25a gives a simple progression I–V–I with a descent
3–2–1 in thesoprano. Next, figure 1.25b shows how we can compose out this progression by a leaping passing tone in the bass, thereby generat-ing an incomplete upper neighbor motion in the soprano. We can assume that the II6in figure 1.25c is generated in much the same
a. Traditional Laws of Harmony b. Revised Laws of Tonal Harmony If a melody is harmonized, then it is L M If a melody is harmonized, then it mainly supported by major, minor, is mainly supported by major, minor, diminished, or augmented triads, and diminished triads on seven degrees.
seventh chords on seven degrees.
If these triads appear in succession, L M then these seven degrees serve one
of three functions—tonic (T), subdominant (S), or dominant (D) (functional equivalence).
If a triad appears, then it always has the L S If a triad appears, then it has the root and the third, with any member in root and the third, with only these the bass (inversional equivalence). members in the bass.
If the triad doubles notes, then it S If the triad doubles notes, then it normally doubles the root, then the normally doubles the root, then fifth, then the third, but not 7. the fifth, then the third, but not 7.
If non-harmonic tones appear, then L S If non-harmonic tones appear, they arise from seventh chords or then they arise from motion
motion between triads. between harmonic tones or
contrapuntal voices.
(G⫽global, L⫽local, M⫽main, S⫽subordinate) Figure 1.24. Laws of harmonic classification.
Figure 1.25. Chord function vs. chord derivation.
way. But what about the II Stufe in figure 1.25d? This one seems to be derived from the upcoming dominant rather than from the pre-ceding tonic.54The same can also be said of the D-major Stufe given in figure 1.25e. In other words, figure 1.25 indicates that different predominant chords may serve the same function, even though they may be generated in quite different ways. Schenkerian deriva-tions are simply more accurate than functional explanaderiva-tions.55
Assuming that Schenker’s seven Stufen cannot be reduced to three functional categories, how can we explain the behavior of har-monic progressions? The answer is, in fact, surprisingly easy; accord-ing to Schenker, they arise from the process of composaccord-ing out:
As a consequence of voice-leading constraint[s], all those individual har-monies that arise from the progression of the various voices are forced to move forward. All the transient harmonies which appear in the course of a work have their source in the necessities of voice-leading [par. 178, 180].56
Since “Stufen are inextricably bound up with counterpoint,” we can reformulate some new laws, as shown in figure 1.26 (Laws of har-monic progression).57
Lastly, Schenker recognized that tonal voice leading has an important influence on the behavior of tonal chromaticisms. Obvi-ously, one of the biggest differences between strict counterpoint and functional tonal composition lies in the area of chromaticism.
As Schenker himself pointed out, whereas strict counterpoint is primarily diatonic and avoids direct chromatic successions, functional tonal composition uses mixture and tonicization to create the entire spectrum of chromaticisms.58In some cases, these chromatic
a. Traditional Laws of Harmony b. Revised Laws of Tonal Harmony If triads appear in succession, then GM If a tonal progression is maximally they are normally arranged as T-S-D-T. closed, then it ends by moving
from V to I.
LS If another essential harmony occurs, then it does so from motion between I and V.
Figure 1.26. Laws of harmonic progression.
successions will be direct: “In contrast to strict counterpoint (see Kpt. I, II/2, par. 28, and Kpt. II, III/1, par. 25), [tonal] composition permits a succession of chromatic tones.”59 But, since tonal com-posers still try to avoid juxtapositions of this latter sort, Schenker believed that “the prohibition is in a certain sense reestablished.”60 In particular, he suggested that they can be avoided by techniques such as motion from an inner voice, neighbor motions, and linear progressions.61 This point is summarized in figure 1.27 (Laws of chromatic generation).
To illustrate what Schenker had in mind, we need only consider his discussion of the Phrygian II. As we all know, the supertonic Stufen are often used to reinforce the dominant Stufe in perfect authentic cadences (see figure 1.28a, Rectification of Phrygian II).
In such contexts, the soprano voice will normally descend from
2 to 1 as the bass moves from V to I. But when the Phrygian II is added before the dominant, the voice leading becomes more complicated (see figure 1.28b). Since the Phrygian II typically has 2 in thesoprano and since changing this scale degree to
2 for the dominant Stufe would create the direct chromatic succession 2–2, Schenker insisted that the melody should descend from 2 to 7 in the alto voice. He referred to the modification of 2 as rectification (or Die Richtigstellung).62Occasionally, Schenker even used this same princi-ple in reverse. For examprinci-ple, to explain the special ‘Phrygian’ effects at the end of Chopin’s Mazurka, Op. 41, no. 2, in E Minor, Schenkera. Traditional Laws of Harmony b. Revised Laws of Tonal Harmony If a melody is harmonized by L M If a melody is harmonized by triads, triads, then these triads are then these triads are mainly
mainly diatonic. diatonic.
If chromaticisms occur, then they L S If chromaticisms occur, then they substitute for or elaborate diatonic arise from mixture or tonicization.
triads.
L S If harmonies appear on IV/V, then they are always indirectly related to I.
Figure 1.27. Laws of chromatic generation.
suggested that
2 or F is rectified in reverse to create 2 or F, though once again the music avoids a direct chromatic succession.63In a paper cowritten with Douglas Dempster and Dave Head-lam, I have described another more remarkable situation in which contrapuntal devices are used to eliminate one specific type of direct chromatic succession.64 We began by showing that, according to Schenkerian theory, direct connections between I and IV/V can-not occur in tonal contexts. We then considered examples in which Stufen on IV/V arise from contrapuntal motion. Among the most remarkable examples of this appears in the Scherzo to Beethoven’s String Quartet, Op. 59, no. 1. Although this movement is in B
major, the coda contains a brief excursion to E minor. We argue that this outburst actually arises from a contrapuntal expansion of the dominant F, which is reached in m. 445. Instead of resolving onto a tonic B, it shifts to a diminished seventh on F (m. 446). This latter sonority tonicizes E (m. 450) before returning to F (m. 458) to set up the final arrival on B (m. 460). The F and E therefore serve as dou-ble neighbors to the dominant F, and, as if to draw attention to the significance of this pattern F–F–(G)–E–F, Beethoven presents it locally within the final cadence (mm. 469–76). In other words, the
Figure 1.28. Rectification of Phrygian II. Adapted from Schenker, Five graphic Analyses, No. 5, Chopin, Étude in C Minor, Op. 10, no. 12.
same procedures govern the behavior of surface melodies and larger harmonic progressions.
All in all, ‘The Heinrich Maneuver’ and ‘The Complementarity Principle’ demonstrate the intimate connections between voice leading and harmony in functional tonality. These two basic princi-ples allowed Schenker to show not only how the traditional laws of strict counterpoint are transformed in functional contexts by the influence of Stufen, but also how the traditional laws of functional harmony can be modified by grounding them in the laws of tonal voice leading. By classifying these laws along the lines suggested above, he had good reason for supposing that these new laws are both necessary and sufficient for explaining functional tonality.
Schenker was able to make these connections between line and chord because he recognized that Stufen play a crucial role in both domains: in his words, they are “the essential generator of all [musical]
content.”65 He was prompted to take these steps precisely because traditional theories of counterpoint and harmony proved to be inaccurate.
Such observations are significant for several reasons. From a pedagogical perspective, we have good reason to update current textbooks on tonal theory. On the one hand, we can provide stu-dents with a more persuasive account of why Fuxian species coun-terpoint helps us understand functional tonality. We can tell them that Fux teaches us about the behavior of contrapuntal lines as they exist in the simplified world of intervals, whereas functional tonal-ity exists in the messy world of functional triads, or Stufen. On the other hand, we can offer students an explicit list of laws that cover the behavior of functional voice leading. These new laws allow us to abandon the rather dubious notion that in free composition, great composers sometimes break the rules of strict counterpoint simply because they are great composers.
From a methodological perspective, our observations also under-score the importance of accuracy to the development of music theo-ries. Schenker was obviously very concerned about whether the laws of strict counterpoint were adequate for explaining the behavior of tonal voice leading. Instead of dismissing abnormalities by appealing to the liberties of genius or to the extra-musical allusions of specific
pieces, he tried to build a theory that provided a more accurate fit with the music he was studying. In this sense Schenker’s work falls within a quite normal pattern of theoretical inquiry. Indeed, as Quine puts it:
The tension between law and anomaly is vital to the progress of science.
The scientist goes out of his way to induce it. Sir Karl Popper well depicts him as inventing hypotheses and then making every effort to falsify them by cunningly devised experiments.66
Quine adds that “it is the tension between the scientist’s laws and his own breaches of them that powers the engines of science and makes it forge ahead.”67 Given that Schenker left us with an empirically testable theory of functional monotonality, our next job is to find the anomalies that it surely contains; if we are able to fix them up, then we can keep the engines of music theory firing on all cylinders.