MUSIC 672: Schenkerian Analysis

These notes draw on Adrian Cadwallader and David Gagné’s Analysis of Tonal Music: A Schenkerian Approach (4th ed., 2020), Tom Pankhurst’s SchenkerGUIDE: A Brief Handbook and Website for Schenkerian Analysis (2008), Heinrich Schenker’s Free Composition (Der freie Satz, trans. Ernst Oster, 1979), Allen Forte and Steven Gilbert’s Introduction to Schenkerian Analysis (1982), Carl Schachter’s Unfoldings: Essays in Schenkerian Theory and Analysis (1999), and supplementary materials from Indiana University Music Theory doctoral curriculum and Yale University graduate music theory seminars.

Chapter 1: Schenker’s Intellectual Project and Historical Context

1.1 Biography and Musical Environment

Heinrich Schenker was born on June 19, 1868, in Wisniowczyki, a small town in Galicia — then part of the Austro-Hungarian Empire, now western Ukraine. His early musical training was thoroughly Viennese in orientation: he studied at the Vienna Conservatory under Anton Bruckner, taking piano lessons and music theory, and he subsequently pursued law at the University of Vienna while continuing to compose and perform. By the mid-1890s he had abandoned any serious ambition for a performing career and settled into the dual role for which he would become famous: private music theory instructor and polemical critic-theorist. He taught an intimate circle of advanced pupils in Vienna for four decades, earning a reputation simultaneously for extraordinary pedagogical rigor and for an often savage critical temperament. He died in Vienna on January 13, 1935, having witnessed — and bitterly opposed — virtually every important development in twentieth-century musical modernism, from Debussy’s impressionism and Schoenberg’s atonalism to the neoclassicism of Stravinsky and Hindemith.

Schenker’s intellectual world was shaped by the cultural ferment of fin-de-siècle Vienna. He was a close friend and collaborator of the philosopher and music critic Eduard Hanslick in his later years, was acquainted with the musicologists Guido Adler and Hugo Riemann, and maintained complex relationships with the composers of his day — admiring Brahms unreservedly while dismissing Reger, Pfitzner, and virtually every living composer as incapable of understanding tonal logic. His correspondence and published essays reveal a thinker of fierce, even pathological, confidence in his own interpretive authority, combined with a genuine and profound engagement with the musical scores he analyzed. He was also a distinguished editorial scholar: his editions and performance suggestions for the Beethoven piano sonatas and his edition of C.P.E. Bach’s keyboard works are substantive contributions to Urtext editing that retain value independent of his theoretical system.

Among his students in Vienna were Oswald Jonas, Hans Weisse, Felix Salzer, and — through the diaspora following Schenker’s death and the rise of National Socialism — a generation of American theorists who would carry the Schenkerian tradition to the United States. Weisse brought Schenkerian pedagogy to the Mannes School of Music in New York; Salzer continued it there and at Queens College; and through their students, Schenker’s analytical methods entered the American music theory curriculum, reaching their most influential institutional expression in the work of Allen Forte, Milton Babbitt, and their students at Yale and Princeton in the 1960s through the 1980s. The story of Schenkerian analysis in America is thus a story of transatlantic diaspora as much as of intellectual transmission.

1.2 The Three Major Theoretical Works

Schenker’s theoretical output falls into three great phases, each associated with a major treatise, and the three together constitute a single evolving argument about the nature of tonal music.

Harmonielehre (1906), translated as Harmony, is the first phase. In it Schenker presents what initially looks like a conventional theory of tonal harmony — scale degrees, chord construction, modulation — but presses it in an unconventional direction by emphasizing the organic generation of harmonic vocabulary from overtone structure and by insisting that harmonic function cannot be understood apart from voice-leading context. The concept of Stufen (scale steps), borrowed from Rameau and Riemann but significantly transformed, already hints at the hierarchical thinking that would characterize his mature work: a Stufe is not merely a chord but a span of harmonic space that may contain many surface chords within it.

Kontrapunkt, published in two volumes (1910 and 1922), translated as Counterpoint, is the second phase and in some respects the technical foundation of everything that follows. Schenker here presents a rigorous species-counterpoint pedagogy clearly indebted to Fux’s Gradus ad Parnassum, but he develops it with far greater specificity about the connection between strict contrapuntal discipline and free composition. The concept of Auskomponierung — composing-out, or prolongation — begins to emerge here explicitly: the claim that a composition at any structural level can be understood as a contrapuntally strict elaboration of a simpler underlying structure. The two-voice strict counterpoint of Fux’s species exercises becomes, for Schenker, not merely a pedagogical tool but a theoretical prototype: every great composition is, at the background level, a species-counterpoint exercise, and the art of tonal composition consists in the inventive elaboration of that strict framework into a rich and complex foreground.

Der freie Satz (1935), translated by Ernst Oster as Free Composition, is the culminating synthesis, published in the year of Schenker’s death. This is the work in which the theory of hierarchical structural levels, the Ursatz, the Urlinie, and the Bassbrechung are presented in their definitive form. The treatise is accompanied by a large atlas of musical graphs — the Erläuterungsausgabe (elucidatory edition) — that demonstrates the theory’s application to dozens of works from the tonal canon. Free Composition is a demanding, sometimes elliptical text; Oster’s English translation (1979), with its extensive editorial footnotes, remains the standard scholarly edition, and it is the version cited throughout these notes. The title itself encapsulates Schenker’s argument: “free” composition, as practiced by the masters, is not truly free in the sense of unconstrained — it is the freedom of a deep organic logic expressing itself through surface variety, just as the growth of a living organism is both determinate (ruled by its genetic structure) and endlessly varied in its outward form.

1.3 The Organic Metaphor

The most persistent figure in Schenker’s theoretical writing is that of the musical work as a living organism. A great tonal composition, he argued, is not assembled from pre-existing blocks — from phrases, periods, themes, and sections as though from bricks — but grows from a generative seed that contains within itself, in potential, the entire structure of the work. This seed is the Ursatz (fundamental structure), which Schenker conceived as the germinal content from which every element of the foreground is “grown” through successive acts of elaboration. The analogy is explicitly botanical: just as a tree is the unfolding of the genetic information contained in a seed — with its particular pattern of root, trunk, branch, and leaf all determined by an inner necessity — so a Bach fugue or a Beethoven symphony is the unfolding of the harmonic-contrapuntal potential of its fundamental structure.

This organic metaphor served Schenker polemically as well as analytically. It allowed him to argue that the great works of the tonal canon are not merely stylistically accomplished or historically important, but organically unified in a deeper sense than any music theory before his had been able to demonstrate. By contrast, music that could not be shown to participate in this organic generative process — which for Schenker meant virtually all music outside the German-Austrian tonal tradition from Bach through Brahms — was by definition not truly organic, and therefore not truly great. The metaphor thus encodes a value judgment that is simultaneously a theoretical claim and a piece of cultural nationalism, a point to which we return in Chapter 8.

It is worth noting that the organic metaphor was not original to Schenker: the German Romantic tradition, from Goethe’s Metamorphosis of Plants through A.B. Marx’s Musical Form in Composition and in Life (1837), had applied organic imagery to music extensively. What Schenker added was a technical specificity absent from most Romantic organicism: he gave precise formal content to the claim that music “grows” from a seed, by identifying the Ursatz as the seed and specifying the prolongational techniques as the growth mechanisms. The organic metaphor, in Schenker’s hands, is not merely an evocative image but a claim with analytical cash value — one that can be tested against specific pieces and can generate specific predictions about their structural organization.

1.4 Auskomponierung and Structural Levels

The concept of Auskomponierung (literally “composing-out,” usually translated as “prolongation” in English-language scholarship) is the technical lynchpin of Schenker’s theory. To compose out a harmony means to extend it through time by deploying contrapuntal and melodic elaborations — passing tones, neighbor notes, arpeggiations, linear progressions — so that it governs a span of musical time far longer than a single chord. A C-major triad can be composed out across an entire movement of a symphony; the movement’s many surface harmonic events are, at the relevant structural level, understood as elaborations of that prolonged tonic rather than as independent harmonic events in their own right.

Prolongation operates at every level of structure simultaneously. At the broadest level, the entire piece is a composing-out of the Ursatz, the fundamental structure. At intermediate levels, each prolonged harmony in the middleground is itself composed out by a set of more detailed prolongations. And at the finest level, the individual surface melody is a composing-out of the middleground harmonies beneath it. This nesting of prolongational processes at multiple levels is what Schenker calls the system of Schichten (layers or levels).

The relationship between levels is one of elaboration: each foreground detail is an elaboration of a simpler middleground structure, which in turn elaborates the background. Crucially, this relationship is asymmetrical in time. The background does not precede the foreground as cause precedes effect in a temporal sequence of events; rather, the levels are simultaneously present in a complete performance, with the background providing the deepest structural coherence that the foreground articulates moment by moment. Schenker’s concept of musical time is thus hierarchical rather than merely linear: a phrase in the first bar of a sonata can be “structurally connected” to a phrase in the last bar not because they are adjacent in time but because they share a background prolongational function. This hierarchical conception of musical time has been enormously influential in subsequent music theory, particularly in the work of Fred Lerdahl and Ray Jackendoff, whose A Generative Theory of Tonal Music (1983) draws explicitly on Schenkerian hierarchical concepts while grounding them in cognitive science.

Chapter 2: The Ursatz — Fundamental Structure

2.1 The Concept of the Ursatz

The Ursatz (fundamental structure) is the most radical and controversial element of Schenker’s theory. It is his claim that every great tonal composition, regardless of its surface style, length, genre, or historical period, rests on a single, simple, two-voice contrapuntal framework: a stepwise descending melodic line in the upper voice, coordinated with a bass arpeggiation from tonic to dominant and back to tonic. This framework is not a theme or a motive that appears on the musical surface; it is a structural abstraction, audible only through the analytical act of reading through the layers of prolongational elaboration to discover what they are elaborating.

The claim is analogous, Schenker suggests, to the claim that all organisms share a common cell structure, or that all sentences in a natural language share a common deep syntactic structure. The Ursatz represents the tonal background that gives every piece its coherence, its sense of departure and return, its harmonic purposefulness, and its ability to sustain a long-range melodic trajectory. Without this framework, Schenker argues, the music would be, in his charged phrase, “a random succession of chords” — a surface without a substratum, mere succession without structure.

2.2 The Urlinie

The Urlinie (fundamental line) is the melodic component of the Ursatz. It is always a stepwise, diatonic, descending line that concludes on the first scale degree, \(\hat{1}\), over the final tonic chord. Schenker recognized only three possible primary tones (Kopftöne, literally “head tones”) from which the Urlinie may begin: \(\hat{3}\), \(\hat{5}\), and \(\hat{8}\).

The third-line Urlinie descends from \(\hat{3}\): the fundamental line traverses the three scale degrees \(\hat{3}–\hat{2}–\hat{1}\). This is the most compact Urlinie and is associated with shorter, simpler works or movements. A great many minuets, short character pieces, and song forms exhibit a third-line Urlinie.

The fifth-line Urlinie descends from \(\hat{5}\): the fundamental line traverses five scale degrees, \(\hat{5}–\hat{4}–\hat{3}–\hat{2}–\hat{1}\). This is the most common Urlinie in Schenker’s analyses of the standard tonal repertoire, appearing in works of moderate length and complexity. The fifth-line is in a sense the most natural Urlinie: the fifth scale degree is a member of the tonic triad and is supported by the dominant, so the descent from \(\hat{5}\) over I through \(\hat{4}\) (harmonically active), \(\hat{3}\) (again a tonic chord tone), \(\hat{2}\) (over the structural dominant V), and finally \(\hat{1}\) (over the closing I) traces out the fundamental harmonic drama of tonality in the most complete way.

The octave-line Urlinie descends from \(\hat{8}\): the fundamental line traverses the entire octave, \(\hat{8}–\hat{7}–\hat{6}–\hat{5}–\hat{4}–\hat{3}–\hat{2}–\hat{1}\). This is relatively rare and is associated with works of particular breadth and scale. Schenker was somewhat reluctant to posit octave-line structures and did so only in a handful of analyses; later Schenkerian scholars have been more liberal. The octave-line presents analytical challenges: the long descent, traversing every diatonic step, requires that each intermediate scale degree be accounted for structurally, and the risk of confusing the structural Urlinie tones with foreground passing tones is correspondingly greater.

The requirement that the Urlinie descend — and always descend, never leap upward — is fundamental to Schenker’s theory. It reflects his conviction that tonal music has a fundamental directional bias toward resolution and closure: the primary tone is a point of tension achieved at the outset (or after an initial ascent), and the entire compositional duration is spent in a controlled, purposeful descent toward the tonic resolution at the close. Upward leaps in the Urlinie would introduce new melodic initiative — new structural “beginnings” — that would undermine the sense of a single, unified, goal-directed trajectory.

2.3 The Bassbrechung

The Bassbrechung (bass arpeggiation) is the harmonic component of the Ursatz. It consists of a bass motion that arpeggiates the tonic triad from the root up to the fifth (I to V) and back down to the root (I). This motion, \(\text{I} \to \text{V} \to \text{I}\), is the simplest possible harmonic framework that embodies the fundamental tonal drama of tension and resolution. The dominant is the pole of tension — the point of maximum harmonic distance from the tonic within the tonal system — and the bass arpeggiation to V and back creates the large-scale harmonic trajectory of the work.

The coordination between the Urlinie and the Bassbrechung is precisely specified. The scale degree \(\hat{2}\) of the Urlinie is harmonically supported by the dominant (V) that the Bassbrechung has reached; the concluding \(\hat{1}\) of the Urlinie coincides with the return to I in the bass. This simultaneous arrival of \(\hat{2}\) over V and then \(\hat{1}\) over I in the bass creates the deep structural authentic cadence that closes the Ursatz and, through prolongational elaboration, governs the harmonic closure of the work.

2.4 Two-Voice Counterpoint at the Background

The relationship between the Urlinie and the Bassbrechung is governed by the rules of strict two-voice counterpoint — specifically, the rules of first species (note against note) as codified by Fux. At the background level, the Ursatz is a two-voice first-species counterpoint: the soprano voice (Urlinie) moves in intervals of the sixth and third against the bass voice (Bassbrechung) at the structurally significant points. The opening of the Ursatz presents the primary tone of the Urlinie over the root of the tonic triad, forming an interval of the third (if the primary tone is \(\hat{3}\)), a fifth (if \(\hat{5}\)), or an octave (if \(\hat{8}\)) with the bass. The concluding point presents \(\hat{1}\) over I in both voices, a unison at the octave. The medial point — \(\hat{2}\) over V — presents a seventh between the Urlinie and bass if the primary tone was \(\hat{5}\) (since \(\hat{2}\) above V creates the seventh of a dominant-seventh chord), or a sixth (from \(\hat{2}\) above the leading tone \(\hat{7}\) in the bass, if the bass carries the third of V).

This contrapuntal precision is not accidental. Schenker’s insight — or claim — is that the deepest structural level of tonal music is governed by the same contrapuntal laws as the pedagogical exercises in which students learn the foundations of voice leading. Free composition is not truly free; it is, at the background, strict species counterpoint.

2.5 Graph Notation for the Ursatz

The Ursatz is typically represented in a two-staff graph, with the treble staff showing the Urlinie and the bass staff showing the Bassbrechung. The conventions for this notation are described in detail in Chapter 3, but a few fundamental points belong here. Structural tones — those belonging to the Ursatz itself — are represented by open noteheads (resembling whole notes without flags or beams), labeled with caret notation (\(\hat{3}\), \(\hat{2}\), \(\hat{1}\)) above the staff. A beam drawn above the noteheads of the Urlinie tones connects them visually, indicating their sequential participation in the descending linear progression. Roman numerals below the bass staff indicate the harmonic scale steps — I, V, I — that the Bassbrechung traverses. A slur in the bass connects the three bass notes to indicate their membership in a single arpeggiational gesture.

Chapter 3: Graphic Notation in Detail

3.1 The Schenkerian Graph as a Notational System

The Schenkerian graph is a specialized musical notation unlike conventional scores or lead sheets. It is a reductive notation whose purpose is to display prolongational relationships at different structural levels simultaneously, within a single written representation. Learning to read Schenkerian graphs is an essential analytical skill and requires considerable practice; it is not self-evident from a background in conventional music notation alone.

A fully realized Schenkerian analysis typically presents multiple layers of notation. The standard presentation in Forte and Gilbert’s Introduction to Schenkerian Analysis (1982), and more recently in Cadwallader and Gagné’s textbook, shows a “foreground graph” (a rhythmically simplified reduction of the score, preserving most surface events but abstracting away performance nuances) alongside one or two “middleground graphs” (showing progressively more reduced versions) and sometimes a separate “background” diagram showing the Ursatz alone. Reading downward through these successive reductions is reading toward the background; reading upward from the background toward the foreground is reading toward greater compositional specificity.

3.2 Noteheads and Their Structural Meaning

The most important notational distinction in a Schenkerian graph is between open noteheads and filled noteheads.

Open noteheads (stemless, whole-note-like symbols) represent structurally primary tones — the tones that belong to the Ursatz or to the first and second middleground. An open notehead says: this pitch occupies a structural position at a high level; it is not merely an elaboration of something more primary, but is itself among the primary structural events of the piece or section.

Filled noteheads represent tones at progressively lower structural levels — passing tones, neighbor notes, arpeggiations, and other elaborating figures. The filled notehead says: this pitch is present at the foreground but is an elaboration of an open-notehead structural tone above it. As analysis moves from foreground toward background, filled noteheads are progressively eliminated, until only the open noteheads of the Ursatz remain.

Within filled noteheads, stem length and flag conventions indicate relative structural importance. A filled notehead with a single stem and no flag is closer to the middleground; a filled notehead with a stem and flag, or a beam group, is closer to the foreground. Some analysts use filled noteheads of different sizes to indicate different levels, though this is not standardized across the literature. Pankhurst’s SchenkerGUIDE provides a useful visual glossary of the principal graphing conventions as used by different analysts.

3.3 Beams, Slurs, and Linear Progressions

Beams drawn above or below noteheads connect a sequence of tones into a linear progression (German: Zug, meaning “pull” or “draw”). A beam connecting two or more noteheads — typically drawn in the treble staff above the note stems — indicates that those tones are members of a single stepwise melodic motion from one structural point to another. The Urlinie itself is shown as a beamed sequence of open noteheads. At lower structural levels, beams connect filled noteheads into smaller-scale linear progressions that elaborate the structural tones.

Slurs serve a different function. A curved slur connecting two noteheads indicates either (a) a neighbor-note relationship — a structural tone departing to a step-neighbor and returning — or (b) a transfer of a structural tone from one register to another (register transfer). In either case, the slur signals that the two connected notes belong to a single structural event, with the first note being prolonged by or connected to the second.

3.4 Roman Numerals and Harmonic Labels

Roman numerals appear below the bass staff and indicate the harmonic scale steps (Stufen) that govern a given span of music at a particular structural level. At the background level, only I and V appear (the Bassbrechung’s two structural bass points). At the first middleground, additional harmonic scale steps appear — II, IV, VI — as the prolonged tonic and dominant are articulated by intermediate harmonies. At the foreground, the full Roman-numeral analysis of the score becomes visible.

The crucial point is that Roman numerals in a Schenkerian graph do not necessarily correspond to single chords. A Roman numeral at the middleground level may govern several bars, an entire section, or even an entire movement. The Roman numeral indicates not “there is a chord here” but “this span of music is governed by this harmonic scale step.” This is what Schenker means by Stufe in Harmonielehre: a scale step is a prolonged harmonic region, not a momentary chord.

3.5 Caret Notation and Scale-Degree Labels

Caret notation (the circumflex accent above a numeral, written \(\hat{n}\)) is used in Schenkerian graphs to label the scale degrees of the Urlinie. The primary tone at the beginning of the Urlinie is labeled \(\hat{3}\), \(\hat{5}\), or \(\hat{8}\) depending on which Urlinie form is operative; each subsequent Urlinie tone is labeled with its scale degree number in sequence. The notation \(\hat{2}|\) (with a vertical stroke) indicates the interrupted \(\hat{2}\) at the end of the first part of an interrupted Ursatz.

3.6 Reading a Multi-Level Graph: A Practical Guide

The discipline of reading multi-level Schenkerian graphs requires simultaneous attention to several distinct layers of information. A complete analysis of a piece typically presents three or four stages of reduction side by side, and the student must learn to move fluidly among them. The recommended procedure, based on the pedagogical approach in Cadwallader and Gagné, is as follows.

Begin with the score itself. Before consulting any graph, play through the piece at the piano several times, attending first to the surface melody and bass, then to the harmonic rhythm, then to the long-range melodic trajectory of the soprano voice. Ask which melodic notes seem structurally prominent — which notes the melody consistently returns to, which notes the phrase structure emphasizes as points of arrival or departure. These observations will be the raw material from which the Schenkerian analysis is constructed.

Next, consult the foreground graph. This is the closest to the score and should be largely self-explanatory given a knowledge of the graphing conventions. Check that every note you hear as significant appears in the foreground graph, and that the notehead types assigned to different notes (open versus filled) accord with your sense of their structural weight. Where you disagree with the graph’s assignment of structural weight, try to understand the analyst’s reasoning before concluding that it is wrong.

Finally, move to the middleground graph and then to the background, tracing upward from the detail of the foreground to the simplicity of the background. At each stage, identify which events have been eliminated (reduced away) and which have been retained; ask why the retained events are treated as more structurally primary than the eliminated ones.

Chapter 4: Prolongational Techniques

4.1 Neighbor Notes

The neighbor note (German: Nebennote) is the simplest prolongational technique. A structural tone is prolonged by departing a step above or below it and returning. The departing tone is the neighbor note; the returning tone restores the structural pitch. An upper neighbor note is a step above the structural tone; a lower neighbor note is a step below. In either case, the structural tone sounds at the beginning of the neighbor-note figure, is “left” by the step motion, and then returns to reassert itself.

The neighbor note prolongs its structural tone because it does not introduce a new structural initiative — it merely elaborates the existing one. At the background or first middleground level, a neighbor note to the primary tone (\(\hat{3}\), \(\hat{5}\), or \(\hat{8}\)) can account for a vast span of musical time: the structural tone is established, moves to the neighbor (which may itself be elaborated by many foreground events), and then returns. The entire span governed by the neighbor-note motion is a single prolongation of the structural tone.

Neighbor notes may be complete (the structural tone — neighbor — structural tone sequence is fully presented) or incomplete (the approach from the structural tone is absent, so the figure moves to the neighbor and then to the structural tone, without first presenting the structural tone itself). Incomplete neighbor notes are common at phrase beginnings, where the structural tone is not yet established; complete neighbor notes are more common in the interior of prolongational spans.

4.2 Linear Progressions and Passing Tones

The linear progression (Zug) is the second fundamental prolongational technique. Where the neighbor note prolongs a single structural tone by circular departure and return, the linear progression prolongs the harmonic space between two structural tones by filling it with stepwise motion. A third-progression from C to E, for instance, passes through D; a fifth-progression from C to G passes through D, E, and F. In each case, the intermediate tones are passing tones (Durchgangstöne) — they fill the interval, connecting the endpoints, without constituting new structural events.

The names of linear progressions reflect their interval sizes. The Terzug (third-progression) traverses a third; the Quartzug (fourth-progression) traverses a fourth; the Quintzug (fifth-progression) traverses a fifth; the Sextzug (sixth-progression) traverses a sixth; and the Oktavzug (octave-progression) traverses an octave. The Quintzug is of special structural importance because the fifth-line Urlinie is itself a descending Quintzug: \(\hat{5}–\hat{4}–\hat{3} –\hat{2}–\hat{1}\). The recognition of linear progressions at multiple structural levels — a surface third-progression, a middleground fifth-progression, and the background Urlinie as Quintzug — is one of the most characteristic moves in Schenkerian analysis.

4.3 Arpeggiation

Arpeggiation (Brechung) is the technique by which a structural tone elaborates the other members of its supporting triad. Rather than moving stepwise to a neighbor or filling an interval with passing tones, arpeggiation moves by skip to another chord member. The motion C–E–G in the soprano over a sustained C-major harmony is an arpeggiation: the soprano “composes out” the C-major triad by traversing its members in melodic succession. At the background level, the Bassbrechung itself is an arpeggiation of the tonic triad in the bass: the bass moves from the root (I) to the fifth (V) and returns.

Arpeggiation differs from linear progression in that it involves a leap rather than stepwise motion. This means that arpeggiation does not connect its endpoint tones in the same way that a Zug does — it does not “fill” an interval but rather “expands” a chord into melodic time. The difference is musically important: a linear progression suggests directed, purposeful stepwise motion; an arpeggiation suggests a more static, harmonically self-contained expansion. The two techniques work together in tonal music: arpeggiation establishes the harmonic environment within which linear progressions move directionally.

4.4 Register Transfer and Coupling

Register transfer (German: Übergreifen for upward transfer, Untergreifen for downward transfer) is the technique of moving a structural melodic tone from one octave to another while preserving its harmonic identity. If the primary tone \(\hat{5}\) of a C-major piece is established as E5 (E in the fifth octave), but the piece’s melodic range requires the Urlinie to operate partly from E4, then a register transfer connects these two octave positions. The tone is “the same” structural tone — E, the third scale degree — regardless of its octave placement.

Register transfer allows the Urlinie to operate across a large melodic range while maintaining the integrity of its descending trajectory. A piece whose primary tone is \(\hat{5}\) at a high register can “borrow” a lower octave for much of its melodic content; the structural \(\hat{5}\) remains active at the higher register even when the foreground melody is operating in the lower one. Register transfers are shown in the graph by a slur connecting the original-register notehead to the transferred-register notehead.

Coupling (Koppelung) is a related technique in which two octave registers are “coupled” so that motion between them is heard as structurally continuous. Rather than a single register transfer, coupling involves a back-and-forth between two registers that are heard as two manifestations of the same structural voice. This technique is particularly important in Schenker’s analysis of keyboard music, where the instrument’s large range encourages composers to exploit octave doubling and register exchange extensively.

4.5 Initial Ascent and Cover Tones

The initial ascent (Anstieg) is a foreground or middleground figure that precedes the establishment of the primary tone. The Urlinie, by definition, must begin on the primary tone; but many pieces open with a melodic ascent from the tonic scale degree toward the primary tone before that tone is firmly established. This ascending gesture is not part of the Urlinie — it precedes it — but it is an important formal and expressive element of many works, and it must be accounted for analytically. The initial ascent “earns” the primary tone by approaching it from below, giving the structural beginning a sense of arrival rather than assumption.

The initial ascent is shown in the graph with filled noteheads (since it is not itself part of the Urlinie), connected by a beam (since it is a linear progression — an ascending Anstieg) that leads to the open notehead of the primary tone. The contrast between the filled noteheads of the ascent and the open notehead of the primary tone visually signals the distinction between the preparatory motion and the structural beginning.

A cover tone (Deckton) is a prominent foreground note that lies above the Urlinie structural tone, “covering” it in the sense of carrying the melodic interest at the surface while the Urlinie proceeds in an inner voice beneath. The structural Urlinie tone is present — it is the tone that matters structurally — but it is not the most prominent melody note at that point; rather, a higher note in an upper register draws the listener’s attention away from the structural tone while the structural progression continues quietly in a lower register. The cover tone is a source of melodic richness in tonal music: it allows composers to write a melodically varied and interesting surface while the underlying structural voice maintains its precise, goal-directed trajectory.

4.6 Unfolding

Unfolding (Ausfaltung) is one of the more conceptually distinctive prolongational techniques in Schenker’s system. In an unfolding, two voices of a chord are presented in alternation as a single melodic line: the “melody” moves back and forth between two structural voices, alternately presenting a tone from the upper voice and a tone from the lower voice in rapid succession. The single melodic line thus “unfolds” a two-voice harmonic interval — a third, a sixth, a tenth — by presenting the two voices one after the other rather than simultaneously.

Chapter 5: Interruption and the Two-Part Structure

5.1 The Concept of Interruption

Interruption (Unterbrechung) is the most structurally consequential prolongational technique in Schenker’s theory — the one most directly responsible for mapping the Ursatz onto the large-scale formal structures of tonal music. Without interruption, every tonal piece would have to complete its Urlinie descent in a single, continuous motion from primary tone to \(\hat{1}\). With interruption, the Urlinie can be “suspended” at \(\hat{2}\) — left incomplete, suspended over the dominant — and then restarted from the primary tone to complete its descent in a second motion. This division of the Ursatz into two halves creates a large-scale two-part structure that maps with remarkable consistency onto the formal architectures of Western tonal music.

The mechanics of interruption are precise. The Urlinie begins on the primary tone and descends toward \(\hat{1}\). At some point — which in the score corresponds to a structural half cadence or dominant arrival — the descent reaches \(\hat{2}\) over V and halts. The notation for interruption in the graph is a short diagonal stroke, resembling a division sign, placed between the \(\hat{2}\) and the resumption of the Urlinie. After the interruption, the primary tone is re-established (the beginning of the “second part”), and the Urlinie descends again from the primary tone all the way to \(\hat{1}\) over the final I. The second descent is not a mere repetition of the first; it is the structural completion of what the first descent began.

5.2 The Two-Part Structure and Formal Archetypes

The two-part structure (zweiteilige Gliederung) generated by interruption has a formal logic that corresponds with extraordinary precision to the standard formal archetypes of tonal composition. The first part runs from the beginning of the piece (or movement) through the interrupted \(\hat{2}\) over V; the second part begins with the re-establishment of the primary tone and tonic, and runs through the final \(\hat{1}\) over I.

This two-part division maps directly onto binary form: the first reprise of a binary movement leads to a structural half cadence (the interruption point), and the second reprise begins with a fresh tonic and concludes the harmonic and melodic trajectory. In rounded binary form (the minuet, the scherzo, and the prototype of sonata form), the return of the opening material at the beginning of the second reprise corresponds exactly to the re-establishment of the primary tone at the beginning of the second part of the interrupted Ursatz. In sonata form, the exposition’s half cadence or dominant arrival, or the half cadence that ends the exposition, corresponds to the interruption; the recapitulation’s reassertion of the tonic and the return of the primary theme corresponds to the re-establishment of the primary tone.

5.3 Continuous Structures

Not every tonal composition is governed by an interrupted Ursatz. A continuous structure is one in which the Urlinie descends from the primary tone to \(\hat{1}\) in a single, uninterrupted motion, without halting at \(\hat{2}\) over V. Continuous structures are most common in short, single-phrase forms — a period, a phrase, or a very brief movement — and are associated with formal designs that do not require a large-scale articulation at the dominant. They are considerably rarer in Schenkerian analysis than interrupted structures; the interruption technique is so ubiquitous because the formal conventions of Western tonal composition so consistently require a dominant arrival before the final tonic.

5.4 Inner-Voice Motions and the Interruption Boundary

A consequence of the interruption that is often overlooked in introductory presentations is its effect on the inner voices — the alto and tenor voices between the Urlinie soprano and the Bassbrechung bass. When the Urlinie reaches \(\hat{2}\) over V at the interruption point, the inner voices typically present the leading tone (\(\hat{7}\)) of the key, which strongly implies a resolution to \(\hat{8}\) = \(\hat{1}\). But the interruption prevents this resolution: instead of continuing to \(\hat{1}\), the structure restarts. The frustrated expectation created by the inner voice leading — the leading tone that does not resolve — is a major source of the “suspense” that interruption produces, and it contributes to the formal energy that drives the second part of the structure. Understanding the inner-voice implications of the interruption is thus essential to a full appreciation of what the interruption accomplishes formally and expressively.

Chapter 6: Middleground Analysis

6.1 From Background to First Middleground

Moving from the background Ursatz to the first middleground means adding the first layer of prolongational elaboration to the skeletal I–V–I Bassbrechung and its associated Urlinie. This is the analytical step at which the specific character of a piece begins to emerge: different pieces will elaborate the same background Ursatz in different ways at the first middleground level, and it is at this level that the analyst begins to see how the piece’s formal structure, its harmonic language, and its melodic character arise from the background seed.

The tonic opening — the initial span of tonic prolongation before the first structural dominant is reached — is the first region to be analyzed at the middleground. The tonic may be prolonged in several characteristic ways. A neighbor-note motion to the dominant chord (I to V and back within the tonic prolongation, before the structural V of the Bassbrechung is reached) is a common first-middleground elaboration; the prominent V chord in the middle of an opening phrase may be a neighbor-note chord that prolongs the tonic rather than a structural dominant. A linear progression filling the interval from the tonic to the structural dominant in the bass — for example, a rising fourth-progression C–D–E–F in the bass, from I to IV, passing through II as a passing chord — is another common first-middleground elaboration. The subdominant chord (IV) heard at the first middleground as a passing event within the tonic prolongation is one of the most characteristic features of the classical style.

6.2 Elaborating the Dominant: The Pre-Dominant Region

The dominant prolongation — the span of music governing the structural V that supports \(\hat{2}\) over the interruption — is a crucial analytical region. In sonata form, this region corresponds roughly to the development section, or to the transition and secondary theme group of the exposition. In binary form, it corresponds to the dominant arrival that closes the first reprise. The dominant may be prolonged by a variety of means: by a linear progression in the bass leading to it (a fifth-progression from I down to V, for instance), by inner-voice passing motions and neighbor notes within the V harmony, and especially by the leading-tone motion (\(\hat{7}–\hat{8}\) in the context of the dominant chord) that characterizes the approach to the structural tonic return.

A particularly important feature of the dominant prolongation is the pre-dominant harmony that precedes the structural V. The subdominant (IV) or supertonic (II or ii\(^{\varnothing 7}\)) frequently appear as pre-dominant harmonies at the middleground, providing the stepwise bass motion from scale degree \(\hat{4}\) or \(\hat{2}\) to \(\hat{5}\) (the bass of the dominant) that connects the prolonged tonic to the structural dominant in a smooth linear progression.

6.3 Prolonging the Tonic Return: Structural Closure

The re-establishment of the tonic at the beginning of the second part of an interrupted structure is one of the most structurally charged moments in tonal composition. In sonata form, this is the beginning of the recapitulation; in rounded binary form, it is the return of the opening material in the second reprise. Schenker’s analysis reveals why these moments are so powerful: they simultaneously re-establish the primary tone of the Urlinie (the top voice returns to \(\hat{3}\), \(\hat{5}\), or \(\hat{8}\) over I) and begin the final, goal-directed descent toward \(\hat{1}\) and the closing I. The listener has been waiting — structurally, not merely formally — for this moment; and when it arrives, the entire prolongational logic of the piece is fulfilled.

The coda of a large tonal work, in the Schenkerian reading, is typically a foreground elaboration of the final tonic — a prolongation of the \(\hat{1}\) and I that the Urlinie has already reached at the structural close. The coda does not add new structural content; it deepens and reinforces the closure already achieved. This reading may seem to diminish the coda’s importance, but Schenker argues otherwise: the coda’s function is to ensure that the structural weight of the tonic arrival is felt fully by the listener, who needs time and emphasis to register the completion of a structural motion that may have been in progress for many minutes.

6.4 Motivic Parallelism Across Levels

One of the most analytically productive — and musically compelling — aspects of Schenkerian analysis is its ability to reveal motivic parallelism across structural levels: the same interval or melodic shape appears at multiple structural levels simultaneously, creating a kind of structural resonance or self-similarity that Schenker viewed as the deepest mark of organic unity. A descending third that appears as a surface melodic figure in the first phrase may also be discoverable as a first-middleground linear progression governing the first section of the movement, and perhaps — in the most profound cases — as a reflection of the background Urlinie descent itself.

Schenker did not use the word “motive” in quite the sense it has in conventional analysis; he preferred the term Motiv der Urlinie (motive of the fundamental line) for these cross-level parallelisms, and he sometimes spoke of verborgene Wiederholung (hidden repetition) when the same shape recurred at a deeper level. The implication is that what we call “motivic development” in conventional analysis is, in Schenker’s view, partly an artifact of the deeper structural parallelism: the foreground motive recurs because it mirrors a background structural necessity.

Chapter 7: Complete-Movement Analyses

7.1 Bach, Prelude in C Major, WTC I, BWV 846

The Prelude in C major that opens the first volume of Bach’s Well-Tempered Clavier is one of the most analyzed pieces in the Schenkerian literature, and for good reason: its texture is almost perfectly transparent. Every bar consists of a single broken-chord figure — an arpeggiated harmonic pattern in which the notes of a chord are spread across a bar in a fixed rhythmic pattern — and this texture makes it possible to read the harmonic substratum almost directly from the score. The challenge for Schenkerian analysis is not to discover the harmonic progressions (they are nearly audible as block chords if one sustains the pattern mentally) but to determine which of those harmonies are structural and which are passing or neighbor events at the middleground.

The Urlinie of the Prelude is generally read as a third-line descending from \(\hat{3}\) (E5) to \(\hat{1}\) (C5). The primary tone E is established in the first bar over the opening C-major tonic. The descent moves to \(\hat{2}\) (D5) over the structural dominant (V, in the form of G major or G-dominant-seventh harmony) and concludes on \(\hat{1}\) (C5) over the final tonic. The Bassbrechung traverses I–V–I in the bass: the opening C-major arpeggiation establishes the tonic; the dominant is reached and prolonged across a substantial section of the piece; and the final bars restore the tonic.

At the first middleground, the most significant analytic decision concerns the long series of seventh chords in bars 11–19. Schenker’s own graph (published in Der freie Satz) reads these as a prolonged dominant — specifically, as a dominant prolongation supported by a descending bass fifth-progression from G (the dominant root) through F, E, D, and C, with the harmonic surface in those bars consisting of the seventh chords that elaborating voices create above this bass. The descending bass fifth-progression is a Quintzug in the bass that composes out the structural dominant across many bars, giving the prelude its sense of sustained harmonic tension before the final tonic return.

7.2 Chopin, Prelude in E Minor, Op. 28, No. 4

Chopin’s Prelude in E minor is one of the most analyzed examples in the Schenkerian literature on Romantic music, and its brevity (25 bars) makes it an ideal object for demonstrating how a large amount of harmonic surface complexity can rest on a very simple structural foundation. The piece has a peculiar, almost static quality at the surface: the right hand sustains a slow-moving soprano line while the left hand plays a pattern of repeated chords that descend chromatically in a bass that proceeds with relentless, funeral pacing. The chromatic inner voices create a sense of continuous harmonic motion, yet the piece never seems to “go anywhere” until the final bars — and this quality is precisely what the Schenkerian analysis illuminates.

The Urlinie is most naturally read as a fifth-line descending from \(\hat{5}\) (B4) in E minor. The primary tone B is established in bar 1 over the opening E-minor tonic. The descent of the Urlinie is notably compressed: \(\hat{5}\) persists for an extended span while the bass slowly descends, and the Urlinie tones \(\hat{4}\), \(\hat{3}\), \(\hat{2}\) arrive in relatively quick succession near the end, with the final \(\hat{1}\) (E4) coinciding with the closing cadence.

The slowly descending bass — moving from E (I) downward through a chromatic series of harmonies — is best analyzed at the first middleground as a linear progression in the bass from E down to B (a fourth-progression), elaborated at the foreground by the many chromatic passing chords that fill the descent with harmonic tension. These chromatic passing chords are not themselves structural; they are foreground elaborations of the stepwise bass descent from I toward V.

7.3 Brahms, Intermezzo in A Major, Op. 118, No. 2

The Intermezzo in A major, Op. 118, No. 2, is perhaps the most beloved of Brahms’s late piano character pieces, and Carl Schachter’s analysis of it in Unfoldings stands as one of the finest examples of Schenkerian analysis in the literature. The piece is formally an ABA’ ternary form with a brief coda. The A section (bars 1–48) is in A major; the B section (bars 49–76) is in F-sharp minor; and the A’ section (bars 77–116) returns to A major with significant variations.

The primary tone of the A section is \(\hat{3}\) (C-sharp5), and the Urlinie is a third-line descending \(\hat{3}–\hat{2}–\hat{1}\). The interruption occurs at the half cadence near the center of the A section, where the Urlinie arrives at \(\hat{2}\) (B4) over V (E major) and is suspended. The second half of the A section restores the primary tone \(\hat{3}\) and descends to \(\hat{1}\) in the cadential close. The A’ section repeats this structure with elaborations, and its final descent to \(\hat{1}\) constitutes the structural close of the entire piece.

The B section introduces what at first seems a digression to a distant harmonic region (F-sharp minor). At the background, however, F-sharp minor is the submediant of A major (VI in A major), and Schenker’s analysis reads the B section as a prolongation of the submediant harmony within the overall tonic prolongation. The A’ section’s return to A major thus restores not merely a formal reprise but the structural tonic that has been temporarily composed-out through the B section’s detour to VI.

Schachter’s analysis draws attention to a remarkable motivic parallelism. The opening motive of the Intermezzo — a descending sixth (C-sharp to E, with intermediate steps) heard as the first melodic gesture — recurs at multiple structural levels: as a foreground surface motive, as a middleground linear progression, and as a shadow of the background Urlinie descent itself. This cross-level parallelism is what Schachter identifies as the piece’s deepest structural feature, the evidence that Brahms’s surface motive is not merely a theme but an organic outgrowth of the fundamental structure. The interval of the sixth is present at the surface (the opening melodic figure), at the middleground (the span between the primary tone \(\hat{3}\) and the \(\hat{5}\) that the initial ascent approaches from below), and as a shadow of the Urlinie in the relationship between the primary tone and the tonic.

7.4 Using Cadwallader and Gagné’s Textbook Graphs

The primary pedagogical resource for this course is Cadwallader and Gagné’s Analysis of Tonal Music: A Schenkerian Approach (4th ed., 2020). Their textbook takes a systematic approach: each chapter introduces a new prolongational technique, illustrates it with one or more complete analyses, and provides exercises at the end. The graphs in the textbook are carefully constructed to be pedagogically accessible, presenting middleground reductions alongside foreground graphs in a format designed to make the analytical decisions visible.

When using the textbook graphs, the student should adopt the following practice. First, play through the original score at the piano (or listen to a recording) several times, until the surface melody and harmony are thoroughly internalized. Second, read the foreground graph while listening to or playing the piece, checking the graph’s notational decisions against the score. Third, read the middleground graph and identify which surface events the graph has eliminated — which chords and melodic tones have been reduced away — and ask why. Fourth, read the background diagram and trace how it is elaborated by the middleground. Only when one can move fluently back and forth among all three levels, hearing the structural relationships the graphs represent, has one understood the analysis.

Chapter 8: Critical Perspectives and Post-Schenkerian Developments

8.1 The Cultural Specificity of Schenker’s Canon

Schenkerian analysis was developed by a late-nineteenth-century Viennese theorist to explain the music of German and Austrian tonal masters — Bach, Handel, Haydn, Mozart, Beethoven, Schubert, and Brahms. This origin is not merely biographical but is inscribed in the theory’s deepest structure. The Ursatz is posited as a universal structure, common to all great tonal music; but “tonal music” in Schenker’s usage is effectively coextensive with the diatonic, functionally harmonic music of the German-Austrian tradition from approximately 1700 to 1900. Music outside this tradition — Renaissance polyphony, Gregorian chant, modal music, non-Western music, blues, jazz, folk music — falls outside the theory’s explanatory scope, not because Schenker forgot to include it, but because his theory is constitutively built on assumptions (diatonic tonality, harmonic function, the structural primacy of I and V) that do not apply to these repertoires.

This cultural specificity has attracted sustained criticism from music theorists and musicologists who argue that Schenkerian analysis, by treating the German tonal canon as the universal norm, implicitly marginalizes all other musical traditions. The theory presents itself as a discovery of universal structural truth — the claim is that the Ursatz is not a stylistic convention but a law of tonal coherence, as necessary as the laws of acoustics — but critics argue that this universalist claim is ideological, a projection of one culture’s musical norms onto the concept of musicality as such. These criticisms have intensified in recent decades as music theory has become more attentive to issues of race, gender, and colonialism in its own history and methodologies.

The debate has practical consequences for music theory pedagogy. If Schenkerian analysis is taught as the central methodology of graduate music theory, as it was throughout most of American music theory programs from the 1960s through the 1990s, then it inevitably shapes what counts as musically “deep” or “sophisticated” — and by implication, what repertoires are worth serious theoretical attention. The musicologist Philip Bohlman has argued that the dominance of Schenkerian analysis in American music theory programs reflects and reinforces the canonical privilege of the European tonal art-music tradition in ways that have had real effects on what music scholars study and what musicians are trained to hear.

8.2 The Political Dimensions of Schenker’s Work

Schenker’s theoretical writings are saturated with German nationalist ideology of a particularly virulent kind. He regularly described great tonal music as a specifically German achievement, a product of German genius that reflected the superiority of German culture over all other national traditions. His published essays from the 1920s and early 1930s contain passages of explicit antisemitism, anti-democratic polemic, and contemptuous dismissal of non-German music. He was a vocal opponent of the Weimar Republic and an admirer of German conservatism; his politics combined cultural elitism with ethnic nationalism in a manner that is deeply troubling to contemporary readers.

The question of how Schenker’s ideology affects his theory has been much debated since the 1980s. Some scholars — including, at times, his closest students and advocates — have argued that the technical content of the theory can be separated cleanly from its ideological envelope: the analytical method works, or fails to work, regardless of what Schenker believed about German cultural supremacy. This position was the dominant view in American Schenkerian scholarship for several decades, particularly among theorists trained in the apolitical, formalist tradition associated with the Journal of Music Theory and the Society for Music Theory.

A contrary view argues that the theory and the ideology are not cleanly separable. The very categories of Schenkerian analysis — the privileged status of the Ursatz as a universal law, the organic unity of the masterwork, the relegation of non-canonical music to the status of “incoherent” or “structurally deficient” — are ideological constructs that reproduce and reinforce the hierarchies of cultural value that Schenker’s politics express. On this view, there is no purely technical Schenkerian analysis unsullied by the politics; the method already encodes the ideology in its selection of repertoire, its definition of structural “depth,” and its implicit claim that pieces not amenable to Ursatz reduction are structurally inferior.

8.3 Felix Salzer and Structural Hearing

Felix Salzer (1904–1986) was one of Schenker’s most distinguished students, and his Structural Hearing (1952) represents the most ambitious attempt to extend Schenkerian prolongational thinking beyond the boundaries that Schenker himself had set. Where Schenker restricted his analyses almost entirely to the German tonal canon, Salzer applied prolongational analysis to medieval monophony, Renaissance polyphony, and twentieth-century music — including Bartók, Hindemith, and Prokofiev.

The theoretical key to Salzer’s extension is the concept of directed motion: the idea that any music can be said to have a structural direction — a sense of departure, motion, and arrival — even if it does not participate in the specific harmonic language of diatonic tonality. Salzer argues that prolongation, understood as the extension of a structural point through motion away from and back to it, is a general feature of coherent musical construction, not specific to tonal music. A Gregorian chant can prolong a pitch through neighbor-note and passing motions; a Bartók fugue subject can prolong a structural pitch-class through similar means; and these prolongations can be represented in Schenkerian-style graphs.

The reception of Structural Hearing was mixed. Many Schenkerian theorists objected that Salzer’s extension violated the logic of the theory: the Ursatz, with its specific claim about the I–V–I Bassbrechung, is not separable from the diatonic tonal system, and a “prolongation” in Bartók that does not involve that harmonic framework is not a prolongation in Schenker’s sense at all. If one dilutes the concept of prolongation to mean merely “any directed motion,” one loses the explanatory power of the concept as Schenker defined it. The critic William Benjamin put the objection with characteristic sharpness: Salzer’s “prolongations” in modal and atonal music are suggestive analogies, not structural equivalents.

8.4 Straus and the Problem of Post-Tonal Prolongation

Joseph Straus’s 1987 article “The Problem of Prolongation in Post-Tonal Music” (published in the Journal of Music Theory) is the most systematic critique of Salzerian extensions, and it has been enormously influential in post-tonal music theory. Straus’s argument is essentially structural: he identifies four conditions that must be met for prolongation to be a coherent analytical concept, and argues that post-tonal music fails to satisfy them.

The first condition is consonance and dissonance: in tonal music, the distinction between structural tones (consonant with the underlying harmony) and elaborating tones (dissonant) provides the criterion for determining which tones are structural and which are passing or neighboring. In post-tonal music, where the distinction between consonance and dissonance is often systematically undermined, there is no such criterion. The second condition is scale-degree identity: in tonal music, each pitch has an identity as a scale degree (\(\hat{1}\), \(\hat{2}\), etc.) that determines its structural function. In a post-tonal context where no key is operative, scale degrees cannot be assigned, and the concept of a “primary tone” or an Urlinie becomes incoherent.

The third condition is harmonic function: the I–V–I progression of the Bassbrechung relies on a hierarchical system of harmonic function in which I and V occupy special roles. In post-tonal music, this functional hierarchy is absent. The fourth condition is the distinction between chord and non-chord tones: in tonal music, we can determine whether a given pitch is a member of the underlying chord or an elaboration of it. In atonal music, no such distinction is available without appeal to a conventional theory that does not apply.

8.5 Schachter and the Richness of Schenkerian Application

Carl Schachter’s essays, collected in Unfoldings: Essays in Schenkerian Theory and Analysis (1999), represent a different tradition of post-Schenkerian scholarship: not the extension of the theory to new repertoires, and not the critical deconstruction of its ideology, but the deepening and refinement of the method as applied to its home territory. Schachter is widely regarded as the most gifted Schenkerian analyst of the late twentieth century, and his essays on Mozart, Schubert, and Brahms are models of analytical rigor combined with musical sensitivity.

Schachter’s most important theoretical contribution beyond Schenker’s own work is his sustained attention to rhythm and meter as structural dimensions that interact with Schenkerian prolongation in complex ways. Schenker’s own theory was notably weak on rhythm: he tended to treat structural tones as existing outside of metric time, their positions in the graph determined by their structural role rather than their metrical placement. Schachter showed, through a series of careful analyses, that the metrical position of a structural tone significantly affects the listener’s experience of its structural weight, and that structural tones and elaborating tones interact with the metric grid in ways that produce subtle and profound musical effects.

His essay “Rhythm and Linear Analysis” (reprinted in Unfoldings) introduces the concept of tonal rhythm — the rhythm created by the succession of structural events at a given prolongational level — as distinct from surface rhythmic pattern. A slow harmonic rhythm at the middleground creates a sense of ponderous, sustained motion even if the surface rhythm is rapid and ornate; a fast harmonic rhythm at the middleground creates urgency even if the surface is static. This insight enriches Schenkerian analysis considerably, giving it a purchase on the rhythmic expressivity of tonal music that the theory in its original form lacks.

Another important contribution of Schachter’s work is his treatment of form and prolongation as mutually illuminating. Rather than treating formal analysis as prior to or separate from prolongational analysis — the traditional view in which one first identifies the form (binary, ternary, sonata) and then applies Schenkerian analysis — Schachter argues that the two kinds of analysis are genuinely interdependent. The identification of the interruption point, for example, is partly a formal observation (there is a structural cadence here) and partly a prolongational one (the Urlinie has descended to \(\hat{2}\) over V), and the two observations reinforce each other. This dialogical conception of the relationship between form and structure is one of the most nuanced aspects of Schachter’s analytical practice.

8.6 Analytical Validity and Multiple Readings

One of the most persistent methodological concerns raised about Schenkerian analysis is the question of analytical validity: since Schenkerian graphs require many analytical decisions that are not fully determined by the theory — the choice of primary tone, the placement of the interruption, the level at which a particular harmony is structural — how do we adjudicate between competing readings of the same piece?

Two analysts can produce different, incompatible graphs of the same piece, each of which is internally consistent with Schenkerian principles. One analyst may read the primary tone as \(\hat{3}\); another may read it as \(\hat{5}\). One may place the interruption at bar 24; another at bar 32. These are not merely notational differences; they represent substantively different claims about the piece’s structure. But the theory does not, in its published form, provide an explicit decision procedure for resolving such disagreements.

Schenker himself acknowledged the multiplicity of valid readings in some contexts, arguing that the richness of great music allows for multiple structural interpretations at lower levels while the background Ursatz remains fixed. But in practice, he tended to present his own readings as authoritative, and he did not provide formal criteria for preferring one reading over another. Later Schenkerian theorists have proposed various criteria: prefer the reading that accounts for more surface events without exceptions; prefer the reading that reveals the most coherent formal structure; prefer the reading that discloses the most interesting motivic parallelisms across levels. None of these criteria is uncontroversial, and none is sufficient on its own to determine a unique correct reading in all cases.

8.7 Computational Approaches to Schenkerian Analysis

The last two decades have seen increasing interest in computational and algorithmic approaches to Schenkerian analysis, motivated in part by the availability of large digital corpora of tonal music and in part by developments in machine learning that make pattern recognition across large datasets feasible. These approaches take two broad forms.

The first is rule-based automated analysis: systems that implement Schenkerian analytical rules as formal algorithms and apply them to encoded musical scores to produce graphs automatically. The Voice-leading Suite of Algorithms (VoSA), developed by a team of music theorists and computer scientists in the 2010s, is perhaps the most sophisticated such system. VoSA identifies structural tones by applying formalized versions of Schenker’s contrapuntal and harmonic criteria, constructs prolongational hierarchies, and outputs graphic representations of the resulting analysis. The results are mixed: VoSA performs reasonably well on simple, short pieces whose structure is relatively unambiguous, but struggles with the analytical decisions that human analysts find difficult — precisely because those decisions require interpretive judgment that formal rules cannot fully capture.

The second approach is machine-learning-based analysis: systems trained on corpora of expert-annotated Schenkerian analyses to learn the patterns of analytical decision-making from examples, rather than from explicit rules. These systems can, in principle, learn the stylistic preferences of particular analysts (Schenker himself, Schachter, Cadwallader) and reproduce analyses in that style. Whether such systems are “doing Schenkerian analysis” in any meaningful sense, or merely pattern-matching on the surface features of analytical graphs without understanding the musical-structural concepts those graphs represent, is a philosophical question that remains open.

Both approaches share a fundamental limitation that is also a theoretical insight: the difficulty of automating Schenkerian analysis reveals precisely where the theory’s explanatory power lies. The decisions that machines find hardest — choosing between competing readings, determining which structural level a particular event belongs to, adjudicating the boundary between a meaningful motivic parallelism and a coincidence — are exactly the decisions that require deep musical understanding. That these decisions resist formalization suggests that Schenkerian analysis, at its best, is not a mechanical procedure but a practice of musical interpretation that draws on knowledge and hearing of a kind that current computational systems do not possess.

8.8 The Ongoing Relevance of Schenkerian Analysis

Despite all the criticisms surveyed in this chapter — its cultural specificity, its ideological loading, its methodological underdetermination, its resistance to computational formalization — Schenkerian analysis remains the most powerful and influential analytical methodology available for the study of tonal music. No other method provides the same combination of structural depth, formal explanatory power, and sensitivity to the long-range melodic and harmonic trajectories of tonal composition.

The reason is not that Schenker’s theory is perfect or that its ideology is harmless. Rather, it is that the phenomena the theory was designed to explain are real: tonal music does have hierarchical prolongational structure; the long-range harmonic trajectories of Beethoven and Brahms do operate across large time-spans in ways that conventional Roman-numeral analysis cannot capture; the sense of structural closure at the end of a sonata movement is not merely a foreground cadential event but the culmination of a background trajectory that has been in motion for many minutes. These phenomena call for an analytical apparatus capable of tracking multiple structural levels simultaneously and tracing the relationships among them. Schenkerian analysis provides that apparatus, and no alternative methodology has yet replaced it for this purpose.

Graduate study in music theory, which necessarily involves deep engagement with the tonal canon, therefore requires serious engagement with Schenker — not uncritical acceptance, but informed, rigorous, critical use of the method. The student who completes this course should be able to produce competent Schenkerian analyses of tonal works from the common practice period, to read and evaluate the analyses of others, to understand the theoretical foundations on which the method rests, and to engage with the critical literature about the method’s scope and limitations. With these skills in hand, Schenkerian analysis becomes not a dogma but a resource — one of the most powerful analytical lenses available to the student of tonal music.

Appendix A: Schenkerian Notation Quick Reference

The following table summarizes the principal graphic symbols used in Schenkerian analysis, as standardized in Cadwallader and Gagné’s textbook and consistent with Forte and Gilbert’s Introduction to Schenkerian Analysis.

An open notehead (stemless whole-note-like symbol) indicates a structural tone at the background or first middleground level. These are the tones of the Ursatz and its first layer of prolongation.

A filled notehead with a stem indicates a structural tone at the second or third middleground level — still primary within its section or phrase, but subordinate to the open-notehead tones above.

A filled notehead with a stem and beam (part of a beam group) indicates a foreground or deeper middleground tone, typically a passing tone or an arpeggiation within a prolongational figure.

A beam drawn above noteheads connects the tones of a linear progression (Zug), indicating that they participate in a single stepwise directed motion. The direction of the beam (left to right, descending or ascending) indicates the direction of the progression.

A curved slur between two noteheads indicates a neighbor-note relationship (the note under the right end of the slur is the neighbor, returning from its departure to the left) or a register transfer (the note under the right end of the slur is the transferred-register version of the note under the left end).

A vertical stroke (\(|\)) placed in the upper voice of the graph between two noteheads indicates the interruption — the suspension of the Urlinie descent at \(\hat{2}\) before the restart.

Roman numerals below the bass staff, without additional figured-bass notation, indicate structural scale steps (Stufen) at the middleground or background level. Roman numerals with figured-bass superscripts (e.g., I\(^6\), V\(^7\)) indicate foreground harmonic details.

Caret notation (\(\hat{n}\)) above Urlinie noteheads labels each Urlinie scale degree in sequence.

Appendix B: Glossary of Key Schenkerian Terms

The following terms are fundamental to the vocabulary of Schenkerian analysis. Students should be able to define each term precisely and illustrate it with a musical example drawn from the course repertoire.

Anstieg (initial ascent): a melodic ascent from the tonic toward the primary tone, preceding the establishment of the Urlinie proper. The Anstieg is shown in the graph with filled noteheads leading to the open notehead of the primary tone.

Ausfaltung (unfolding): the technique in which two voices of a chord are presented alternately in a single melodic line, the line weaving back and forth between the registers of the two voices and thus traversing the harmonic interval between them.

Auskomponierung (composing-out, prolongation): the technique of extending a harmonic event through time by deploying melodic and contrapuntal elaboration. Prolongation is the foundation of all Schenkerian analysis.

Bassbrechung (bass arpeggiation): the bass component of the Ursatz, consisting of the arpeggiated motion I–V–I. The Bassbrechung supports the Urlinie descent and provides the harmonic framework of the fundamental structure.

Brechung (arpeggiation): the elaboration of a structural tone or harmony by motion through the other members of the governing chord. Brechung differs from the Zug in that it involves leaps rather than stepwise motion.

Deckton (cover tone): a prominent foreground tone above the Urlinie structural tone, carrying the melodic interest at the surface while the Urlinie proceeds in a lower, less prominent voice.

Durchgangston (passing tone): in Schenkerian usage, any tone that fills the interval between two structural tones in a linear progression, regardless of metric position or harmonic support.

Hintergrund (background): the deepest structural level, containing only the Ursatz.

Kopfton (head tone, primary tone): the initial tone of the Urlinie — \(\hat{3}\), \(\hat{5}\), or \(\hat{8}\) — which must be a member of the tonic triad.

Koppelung (coupling): the linking of two octave registers so that structural voice-leading moves coherently between them as a sustained structural event.

Mittelgrund (middleground): the intermediate structural level(s) between background and foreground, typically divided into first middleground, second middleground, and sometimes a third middleground for longer works.

Motiv der Urlinie (motive of the fundamental line): Schenker’s term for a motivic figure at the foreground that mirrors the shape of the Urlinie or a middleground linear progression — an instance of what Schenker also calls verborgene Wiederholung (hidden repetition).

Nebennote (neighbor note): a step above or below a structural tone that elaborates it by departure and return, without introducing a new structural initiative.

Quintzug (fifth-progression): a linear progression traversing a fifth, with the three intermediate tones filling the interval. The fifth-line Urlinie is a descending Quintzug.

Schicht (layer, level): one of the structural strata of a Schenkerian analysis. The three broad Schichten are Hintergrund, Mittelgrund, and Vordergrund.

Stufe (scale step): in Schenker’s usage, a prolonged harmonic region — not a single chord but a span of harmonic space governed by a single harmonic scale degree.

Terzug (third-progression): a linear progression traversing a third, with the single intermediate tone filling the interval. The third-line Urlinie is a descending Terzug.

Übergreifen / Untergreifen (register transfer, upward / downward): the technique of moving a structural tone from one octave to another while preserving its harmonic identity.

Unterbrechung (interruption): the suspension of the Urlinie descent at \(\hat{2}\) over V, followed by a restart from the primary tone and a completion of the descent to \(\hat{1}\). The interruption divides the Ursatz into two structurally parallel halves and is the analytical foundation of two-part and binary formal structures.

Urlinie (fundamental line): the melodic component of the Ursatz — a stepwise descending diatonic line from the primary tone (\(\hat{3}\), \(\hat{5}\), or \(\hat{8}\)) to \(\hat{1}\).

Ursatz (fundamental structure): the background structure common to all tonal masterworks, consisting of the Urlinie in the upper voice and the Bassbrechung in the bass, in strict two-voice counterpoint at the background level.

Verborgene Wiederholung (hidden repetition): Schenker’s term for the recurrence of a melodic or intervallic shape at a deeper structural level than its first appearance, revealing the structural parallelism between surface and background.

Vordergrund (foreground): the structural level closest to the musical surface, as notated in the score.

Zug (linear progression): a stepwise motion through a diatonic interval that prolongs the structural harmony at its endpoints. The types of Zug are named after their intervals: Terzug (third), Quartzug (fourth), Quintzug (fifth), Sextzug (sixth), Oktavzug (octave).

Appendix C: Suggested Repertoire for Analysis Practice

The following works are recommended for in-depth analysis practice throughout the course. They are listed in approximate order of analytical difficulty, from simpler to more complex, and are all discussed in Cadwallader and Gagné’s textbook or in Schachter’s Unfoldings.

Bach, Well-Tempered Clavier, Volume I: Preludes in C major (BWV 846), C minor (BWV 847), and D major (BWV 850) offer excellent introductory material because their textures are transparent and their harmonic structures are clearly articulated.

Mozart, Piano Sonata in A major, K. 331, first movement (theme and first two variations) provides a clear example of a third-line Urlinie and an interrupted two-part structure within a simple theme form.

Beethoven, Bagatelle in A minor, Op. 119, No. 9, is a short, single-phrase piece that illustrates a continuous (non-interrupted) Ursatz structure.

Schubert, “Der Leiermann” from Winterreise, D. 911, No. 24, demonstrates the Schenkerian treatment of an obstinate, repetitive foreground texture and raises interesting questions about structural closure in a song that deliberately refuses conventional tonal resolution.

Brahms, Intermezzo in A major, Op. 118, No. 2, as analyzed in detail in Chapter 7, is the most challenging work on this list and should be approached after thorough familiarity with the simpler examples. Schachter’s analysis in Unfoldings should be consulted after completing an independent analysis.

End of MUSIC 672 notes. The analytical skills developed in this course — the ability to read Schenkerian graphic notation, to produce independent background, middleground, and foreground reductions, and to engage critically with the theory’s assumptions and limitations — constitute one of the most rigorous and rewarding disciplines available to the music theorist. Students wishing to extend their study should work through Cadwallader and Gagné’s complete set of analytical exercises, read Schachter’s essays in Unfoldings for the most penetrating post-Schenkerian analytical writing in English, and consult Schenker’s own Free Composition in the Oster translation for the primary theoretical source.