Schenkerian analysis is a method of musical analysis of tonal music based on the theories of Heinrich Schenker. The goal of a Schenkerian analysis is to interpret the underlying structure of a tonal work and to help reading the score according to that structure. The theory's basic tenets can be viewed as a way of defining tonality in music. A Schenkerian analysis of a passage of music shows hierarchical relationships among its pitches, and draws conclusions about the structure of the passage from this hierarchy. The analysis makes use of a specialized symbolic form of musical notation that Schenker devised to demonstrate various techniques of elaboration. The most fundamental concept of Schenker's theory of tonality may be that of "tonal space". The intervals between the notes of the tonic triad form a tonal space that is filled with passing and neighbour notes, producing new triads and new tonal spaces, open for further elaborations until the surface of the work (the score) is reached.
Although Schenker himself usually presents his analyses in the generative direction, starting from the original structure (Ursatz) to reach the score, the practice of Schenkerian analysis more often is reductive, starting from the score and showing how it can be reduced to its original structure. The graph of the Ursatz is arrhythmic, as is a strict-counterpoint cantus firmus exercise. Even at intermediate levels of the reduction, rhythmic notation (open and closed noteheads, beams and flags) is used to show hierarchical relationships between the pitch–events being analyzed.
Schenkerian analysis is subjective. There is no mechanical procedure involved and the analysis reflects the musical intuitions of the analyst. The analysis represents a way of hearing (and reading) a piece of music.
Schenker was convinced of the superiority of music of the common practice period (especially the music of Johann Sebastian Bach, Carl Philipp Emanuel Bach, Joseph Haydn, Wolfgang Amadeus Mozart, Ludwig van Beethoven, Franz Schubert, Fryderyk Chopin, and Johannes Brahms). This led him to seek the key to an understanding of music in the traditional discipline of counterpoint, the type of theory the Masters themselves had studied.
Schenker's project was to show that free composition (freier Satz) was an elaboration, a "prolongation", of strict composition (strenger Satz), by which he meant species counterpoint, particularly two-voice counterpoint. He did this by developing a theory of hierarchically organized levels of elaboration (Auskomponierung), called prolongational levels, voice–leading levels (Stimmführungsschichten), or transformations (Verwandlungen), the idea being that each of the successive levels represents a new freedom taken with respect to the rules of strict composition.
Because the first principle of the elaboration is the filling in of the tonal space by passing notes, an essential goal of the analysis is to show linear connections between notes which, filling a single triad at a given level, remain closely related to each other but which, at subsequent levels, may become separated by many measures or many pages as new triads are embedded in the first one. The analyst is expected to develop a "distance hearing" (Fernhören), a "structural hearing".
The original triad, that from which the work as a whole arises, takes its model in the harmonic series. However, "the mere duplication of nature cannot be the object of human endeavour. Therefore [...] the overtone series [...] is transformed into a succession, a horizontal arpeggiation, which has the added advantage of lying within the range of the human voice. Thus the harmonic series is condensed, abbreviated for the purposes of art". Linking the (major) triad to the harmonic series, Schenker merely pays lip service to an idea common in the early 20th century. He confirms that the same derivation cannot be made for the minor triad: "Any attempt to derive even as much as the first foundation of this [minor] system, i.e., the minor triad itself, from Nature, i.e., from the overtone series, would be more than futile. [...] The explanation becomes much easier if artistic intention rather than Nature herself is credited with the origin of the minor mode.".
The basic component of Schenkerian harmony is the Stufe (scale degree, scale-step), i.e. a chord having gained structural significance. Chords arise from within chords, as the result of the combination of passing notes and arpeggiations: they are at first mere embellishments, mere voice-leading constructions, but they become tonal spaces open for further elaboration and, once elaborated, can be considered structurally significant: they become scale-steps properly speaking. Schenker recognizes that "there are no rules which could be laid down once and for all" for recognizing scale steps, but from his examples one may deduce that a triad cannot be recognized as a scale-step as long as it can be explained by passing or neighboring voice-leading.
Schenkerian analyses label scale-steps with roman numerals, a practice common in 19th- and 20th-century Vienna, developed by the theoretic work of Georg Joseph Vogler and his student Gottfried Weber, transmitted by Simon Sechter and his disciple Anton Bruckner, the classes of which Schenker had followed in the Konservatorium in Vienna.
Schenker’s theory is monotonal: the Ursatz, as the unfolding of the tonic triad, by definition cannot include modulation. Local "tonicisation" may arise when a scale-step is elaborated to the point of becoming a local tonic, but the work as a whole projects a single key and ultimately a single Stufe (the tonic).
Counterpoint, voice-leading 
Two-voice counterpoint remains for Schenker the model of strict writing. Free composition is a freer usage of the laws of strict counterpoint. One of the aims of the analysis is to trace how the work remains subject to these laws at the deepest level, despite the freedom taken at subsequent levels.
One aspect of strict, two-voice writing that appears to span Schenker’s theory throughout the years of its elaboration is the rule of "fluent melody" (fliessender Gesang), or "melodic fluency″. Schenker attributes the rule to Cherubini, who would have written that "fluent melody is always preferable in strict counterpoint" Melodic fluency, the preference for conjunct (stepwise) motion, is one of the main rules of voice leading, even in free composition. It avoids successive leaps and produces "a kind of wave-like melodic line which as a whole represents an animated entity, and which, with its ascending and descending curves, appears balanced in all its individual component parts". This idea is at the origin of that of linear progression (Zug) and, more specifically, of that of the Fundamental Line (Urlinie).
Ursatz (usually translated as "fundamental structure") is the name given by Schenker to the underlying structure in its simplest form, that from which the work as a whole originates. In the canonical form of the theory, it consists of the Urlinie, the "fundamental line", supported by the Bassbrechung, the "arpeggiation of the bass". The fundamental structure is a two-voice counterpoint and as such belongs to strict composition. In conformity with the theory of the tonal space, the fundamental line is a line starting from any note of the triad and descending to the tonic itself. The arpeggiation is an arpeggiation through the fifth, ascending from I to V and descending back to I. The Urlinie unfolds the tonal space in a melodic dimension, while the Bassbrechung expresses its harmonic dimension.
The theory of the fundamental structure is the most criticized aspect of Schenkerian theory: it has seemed unacceptable to reduce all tonal works to one of a few almost identical background structures. This is a misunderstanding: Schenkerian analysis is not about demonstrating that all compositions can be reduced to the same background, but about showing how each work elaborates the background in a unique, individual manner, determining both its identity and its "meaning". Schenker has made this his motto: Semper idem, sed non eodem modo, "always the same, but never in the same manner".
The fundamental line 
The idea of the fundamental line comes quite early in the development of Schenker’s theory. Its first printed mention dates from 1920, in the edition of Beethoven’s Sonata op. 101, but the idea obviously links with that of "fluent melody", ten years earlier. Schenker first conceived the Urlinie, the "original line", as a kind of motivic line characterized by its fluency, repeated under different guises throughout the work and ensuring its homogeneity. He later imagined that a musical work should have only one original line, unifying it from beginning to end. The realization that such original lines usually were descending led him to formulate the canonical definition of the fundamental line as necessarily descending. It is not that he rejected ascending lines, but that he came to consider them hierarchically less important.
"The fundamental line begins with , or , and moves to via the descending leading tone ". The initial note of the fundamental line is called its "head tone" (Kopfton) or "primary tone". The head note may be elaborated by an upper neighbour note, but not a lower one. In many cases, the head note is reached through an ascending line (Anstieg, "initial ascent") or an ascending arpeggiation, which do not belong to the fundamental structure properly speaking.
The arpeggiation of the bass and the divider at the fifth 
The arpeggiation through the fifth is an imitation of the overtone series, adapted to man "who within his own capacities can experience sound only in a succession". The fifth of the arpeggiation coincides with the last passing note of the fundamental line. This at first produces a mere "divider at the fifth", a complex filling in of the tonal space. However, as a consonant combination, it defines at a further level a new tonal space, that of the dominant chord, and so doing opens the path for further developments of the work. It would appear that the difference between the divider at the fifth and the dominant chord properly speaking really depends on the level at which the matter is considered: the notion of the divider at the fifth views it as an elaboration of the initial tonal space, while the notion of dominant chord conceives it as a new tonal space created within the first. But the opinions of modern Schenkerians diverge on this point.
Schenkerian notation 
Graphic representations form an important part of Schenkerian analyses: "the use of music notation to represent musical relationships is a unique feature of Schenker’s work". Schenkerian graphs are based on a "hierarchic" notation, where the size of the notes, their rhythmic values and/or other devices indicate their structural importance. Schenker himself, in the Foreword to his Five Graphic Analyses, claimed that "the presentation in graphic form has now been developed to a point that makes an explanatory text unnecessary". Even so, Schenkerian graphs represent a change of semiotic system, a shift from music itself to its graphical representation, akin to the more usual change from music to verbal (analytic) commentary; but this shift already exists in the score itself, and Schenker rightly noted the analogy between music notation and analysis. One aspect of graphic analyses that may not have been enough stressed is the desire to abolish time, to represent the musical work as something that could be apprehended at a glance or, at least, in a way that would replace a "linear" reading by a "tabular" one.
The first step of the analytic rewriting often takes the form of a "rhythmic" reduction, that is one that preserves the score, but "normalizes" its rhythm and its voice-leading content. This type of reduction has a long tradition, not only in counterpoint treatises or theory books, but also in the simplified notation of some Baroque works, e.g. the Prelude to Händel’s Suite in A major, HWV 426, or early versions of Bach’s C major Prelude of Book I of the Well Tempered Keyboard. One indirect advantage of rhythmic reduction is that it helps reading the voice leading: Czerny’s example hereby transforms Chopin’s arpeggios into a composition in four (or five) voices. Edward Aldwell and Carl Schachter write that the first rewriting should "produce a setting that is reasonably close to note-against-note." Allen Cadwallader and David Gagné suggest a special type of rhythmic reduction that they call "imaginary continuo", stressing the link between the rhythmic reduction and a notation as a melody with figured bass. Basically, it consists in imagining a figured bass line for the work analyzed, and writing a chordal realization of it.
Schenker himself usually began his analyses with a rhythmic reduction that he termed Urlinietafel. From 1925 onwards, he complemented these with other levels of representation, corresponding to the successive steps leading to the fundamental structure. At first, he mainly relyed on the size of the note shapes to denote their hierarchic level, but later abandoned this system as it proved too complex for contemporary techniques of musical engraving. Allen Cadwallader and David Gagné propose a description of Schenker’s system of graphic notation which, they say, "is flexible, enabling musicians to express in subtle (and sometimes different) ways what they hear and how they interpret a composition". They discuss open noteheads, usually indicating the highest structural level, and filled-in noteheads for tones of lower levels; slurs, grouping tones in an arpeggio or in linear motions with passing or neighbor tones; beams, for linear motions of higher structural level or for the arpeggiation of the bass; broken ties, for repeated or sustained tones; diagonal lines to realign displaced notes; diagonal beams, connecting successive notes that belong to the same chord ("unfolding"); etc.
Techniques of prolongation 
The meat of a Schenkerian analysis is in showing how a background structure expands until it results in the succession of musical events on the surface of the composition itself. Schenker refers to this process under the term Auskomponierung, literally "composing out", but more often translated as "elaboration". Modern Schenkerians usually prefer the term "prolongation", stressing that elaborations develop the events along the time axis.
- In practical art the main problem is how to realize the concept of harmony in a live content. In Chopin’s Prelude, op. 28, No 6, thus, it is the motif
- that gives life to the abstract concept of the triad, B, D, F-sharp.
The elaboration of the triad, here mainly in the form of an arpeggio, loads it with "live content", with meaning. Elaborations take the form of diminutions, replacing the total duration of the elaborated event by shorter events in larger number. By this, notes are displaced both in pitch and in rhythmic position. The analysis to some extent aims at restoring displaced notes in their "normal" position and explaining how and why they were displaced.
One fascinating aspect of Schenkerian analysis is that it does not view the work as built from a succession of events, but as the growth of new events from within events of higher level, much as a tree develops twigs from its branches and branches from its trunk: it is in this sense that Schenkerian theory must be considered organicist. The example hereby may at first be considered a mere elaboration of an F major chord, an arpeggiation in three voices, with passing notes (shown here in black notes without stem) in the two higher voices: it is an exemplification of the tonal space of F major. The chord labelled (V) at first merely is a "divider at the fifth". However, the meeting of the fifth (C) in the bass arpeggiation with the passing notes may also be understood as producing a dominant chord, V, arising from within the tonic chord I.
This is the situation found at the beginning of Haydn’s Sonata in F major, Hob. XVI:29, where the (incomplete) dominant chord appears at the very end of bar 3, while the rest of the fragment consist of arpeggios (with neighbor notes) of the F chord:
Arpeggiation, neighbour note, passing note 
Arpeggiation is the simplest form of elaboration. It delimitates a tonal space for elaboration, but lacks the melodic dimension that would allow further developments: it "remains a harmonic phenomenon". From the very structure of triads (chords), it follows that arpeggiations remain disjunct and that any filling of their space involves conjunct motion. Schenker distinguishes two types of filling of the tonal space: – neighbor notes (Nebennoten), ornamenting one single note of the triad by being adjacent to it. – passing notes, which pass by stepwise motion from one note to another and fill the space in between. Both neighbor notes and passing notes are dissonances. They may be made consonant by their meeting other notes (as in the Haydn example above) and, once consonant, may delimitate further tonal spaces open to further elaborations. Insofar as chords consist of several voices, arpeggiations and passing notes always involve passing from one voice to another.
Linear progression (‘’Zug’’) 
A linear progression (Zug) is the stepwise filling of some consonant interval. It usually is underlined in graphic analyses with a slur from the first note of the progression to the last. The most elementary linear progressions are determined by the tonal space that they elaborate: they span from the prime to the third, from the third to the fifth or from the fifth to the octave of the triad, in ascending or descending direction. Schenker writes: "there are no other tonal spaces than those of 1–3, 3–5, and 5–8. There is no origin for passing-tone- progressions, or for melody" Linear progressions, in other words, may be either third progressions (Terzzüge) or fourth progressions (Quartzüge); larger progressions result from a combination of these. Linear progressions may be incomplete (deceptive) when one of their tones is replaced by another, but nevertheless suggested by the harmony. In the example below, the first bars of Beethoven’s Sonata op. 109, the bass line descends from E3 to E2. F#2 is replaced by B1 in order to mark the cadence, but it remains implicit in the B chord. In addition, the top voice answers the bass line by a voice exchange, E4-F#4-G#4 above G#2-(F#2)-E2, in bar 3, after a descending arpeggio of the E chord. The bass line is doubled in parallel tenths by the alto voice, descending from G#4 to G#3, and the tenor voice alternatively doubles the soprano and the bass, as indicated by the dotted slurs. It is the bass line that governs the passage as a whole: it is the "leading progression", on which all the other voices depend and which best expresses the elaboration of the E major chord.
Schenker describes lines covering a seventh or a ninth as "illusory", considering that they stand for a second (with a register transfer): they do not fill a tonal space, they pass from one chord to another.
Lines between voices, reaching over 
Passing tones filling the intervals of a chord may be considered forming lines between the voices of this chord. At the same time, if the chord tones themselves are involved in lines from one chord to another (as usually is the case), lines of lower level unfurl between lines of higher level. The most interesting case is when the lines link an inner voice to the upper voice. This may happen not only in ascending (a case usually described as a "line from an inner voice"), but also in descending, if the inner voice has been displaced above the upper line by a register transfer, a case known as "reaching over" (Übergreifen). In the example from Schubert’s Wanderers Nachtlied below, the descending line Gb–F–Eb–Db at the end of the first bar may be read as a reaching over.
Unfolding (Ausfaltung) is an elaboration by which several voices of a chord or of a succession of chords are combined in one single line "in such a manner that a tone of the upper voice is connected to a tone of the inner voice and then moves back, or the reverse". At the end of Schubert’s Wanderers Nachtlied op. 4 n. 3, the vocal melody unfolds two voices of the succession I–V–I; the lower voice, Bb–Ab–Gb, is the main one, expressing the tonality of Gb major; the upper voice, Db–Cb–Bb, is doubled one octave lower in the right hand of the accompaniment:
In his later writings (from 1930 onwards), Schenker sometimes used a special sign do denote the unfolding, an oblique beam connecting notes of the different voices that are conceptually simultaneous, even if they are presented in succession in the single line performing the unfolding.
Register transfer, coupling 
Register transfer is the motion of one or several voices into a different octave (i.e. into a different register). Schenker considers that music normally unfolds in one register, the "obligatory register", but at times is displaced to higher or lower registers. Register transfers are particularly striking in piano music (and that for other keyboard instruments), where contrasts of register (and the distance between the two hands) may have a striking, quasi orchestral effect. Coupling is when the transferred parts retain a link with their original register. The work, in this case, appears to unfold in two registers in parallel.
Voice exchange 
Voice exchange is a common device in counterpoint theory. Schenkers view it as a means of elaborating a chord by modifying its position. Two voices exchange their notes, often with passing notes in between. At the end of the example of Beethoven’s Op. 109 above, the bass and soprano exchange their notes: G# is transferred from bass to soprano, while E is transferred from soprano to bass. The exchange is marked by crossed lines between these notes.
Elaboration of the Fundamental Structure 
The elaborations of the fundamental structure deserve a specific discussion because they may determine the form of the work in which they occur.
Initial ascent, initial arpeggiation 
The starting point of the fundamental line, its "head note" (Kopfton), may be reached only after an ascending motion, either an initial ascending line (Anstieg) or an initial arpeggiation, which may take more extension than the descending fundamental line itself. This results in melodies in arch form. Schenker decided only in 1930 that the fundamental line should be descending: in his earlier analyses, initial ascending lines often are described as being part of the Urlinie itself.
First order neighbor note 
Schenker stresses that the head note of the fundamental line often is decorated by a neighbor note "of the first order", which must be an upper neighbor because "the lower neighboring note would give the impression of the interruption". The neighbor note of the first order is –– or ––: the harmony supporting it often is the IVth or VIth degree, which may give rise to a section of the work at the subdominant.
Articulation of the span from I to V in the bass arpeggiation 
The canonic form of the bass arpeggiation is I–V–I. The second interval, V–I, forms under – the perfect authentic cadence and is not susceptible of elaboration at the background level. The first span, I–V, on the other hand, usually is elaborated. The main cases include:
This is the complete arpeggiation of the triad. Once elaborated, it may consist in a succession of three tonalities, especially in pieces in minor. In these cases, III stands for a tonicisation of the major relative. This often occurs in Sonata forms in minor, where the first thematic group elaborates degree I, the second thematic group is in the major relative, degree III, and the development leads to V before the recapitulation in the tonic key.
I–IV–V or I–II–V 
Even though he never discussed them at length, these elaborations occupy a very special place in Schenker’s theory. One might even argue that no description of an Ursatz properly speaking is complete if it does not include IV or II at the background level. Schenker uses a special sign to denote this situation, the double curve shown in the example hereby, crossing the slur that links IV (or II) to V. That IV (here, F) is written as a quarter note indicates that it is of lower rank than I and V, notated as half notes. There exists here an unexpected link between Schenkerian theory and Riemann’s theory of tonal functions, a fact that might explain Schenker’s reluctance to be more explicit about it. In modern Schenkerian analysis, the chord of IV or II is often dubbed the "predominant" chord, as the chord that prepares the dominant one, and the progression may be labelled "T–P–D–T", for Tonic–Predominant–Dominant–Tonic.
The dominant chord may be linked to the tonic by a stepwise linear progression. In such case, one of the chords in the progression, II, III or IV, usually takes preeminence, reducing the case to one or the other described above.
The interruption (Unterbrechung) is an elaboration of the fundamental line, which is interrupted at its last passing note, , before it reaches its goal. As a result, the bass arpeggiation itself is also interrupted at the divider at the fifth (V). Both the fundamental line and the bass arpeggiation are bound to return to their starting point and the fundamental structure repeats itself, eventually reaching its goal. The interruption is the main form-generating elaboration: it often is used in binary forms (when the first part ends on the dominant) or, if the elaboration of the "dividing dominant", above V, takes some importance, it may produce ternary form, typically sonata form.
Schenker calls "mixture" (Mischung) the change of mode of the tonic, i.e. the replacement of its major third by the minor one, or of its minor third by the major one. The elaboration of the resulting chord may give rise to a section in minor within a work in major, or the reverse.
Transference of the fundamental structure 
The forms of the fundamental structure may be repeated at any level of the work. "Every transferred form [of the fundamental structure] has the effect of a self-contained structure within which the upper and lower voices delimit a single tonal space". That is to say that any phrase in a work could take the form of a complete fundamental structure. Many classical themes (e.g. the theme to the set of variations in Mozart’s K331 piano sonata) form self-contained structure of this type. This resemblance of local middleground structures to background structures is part of the beauty and appeal of Schenkerian analysis, giving it the appearance of a recursive construction.
Legacy and responses 
|This section requires expansion. (June 2008)|
In the academic generations after Schenker, other music theorists have both added to and disseminated Schenker's ideas. In the second generation (Schenker himself being the first), the fierce philosophical opposition between Oswald Jonas and Felix Salzer set the stage for a conservative–liberal split among Schenkerians that persists to this day. Jonas, a traditional disciple who was more strict about the theory than Schenker himself, promoted the viewpoint that the analysis belonged only in the realm of triadic tonal music. This camp is generally responsible for the codification and clarification of the theory's principles, epitomized as the "New York" model by theorists such as Carl Schachter and recently Allen Cadwallader and David Gagne. Salzer, on the other hand, was the first Schenkerian to attempt to use elements of the theory to explain music that is not strictly tonal, an approach that has since engendered structural and linear analysis of early music as well as post-tonal music of the twentieth and twenty-first centuries.
Milton Babbitt admired Schenker's work and his own work may be seen as part response, revision, and alternative to Schenker's. For example, he suggests that the properties described as natural phenomena by Schenker be considered axioms and he also formulated a system to compose twelve-tone music that was "equally intricate and fruitful." Allen Forte also responded to Schenker by providing an alternative system applicable to the analysis of nontonal nontwelve tone music. (ibid, p. 162–163)
Hampered by limited availability in and after the war years, by the 1960s Schenkerian analysis had begun to attract renewed interest, and by the 1980s it had become one of the main analytical methods used by many North American music theorists. While Schenker's theories have been increasingly challenged since the mid-century for their rigidity and organicist ideology, the wider analytical tradition that they inspired has remained central to the study of tonal music in North America.
Fred Maus (2004, p. 162) compares Schenker's "creation of an elaborate tonal theory in response to post-tonal music" with "sexologists' back-formation of the concept of heterosexuality as a complement to their new concept of homosexuality." Finding similarities, "to some extent" including the "conceptualization of the normative or unmarked category" following "awareness of an alternative." Schenker considered nontonal or atonal music unnatural.
For more on issues pertaining to the dissemination of Schenkerian thought in the U.S., see several essays by David Carson Berry, including:
- "The Role of Adele T. Katz in the Early Expansion of the New York 'Schenker School'". Current Musicology 74: 103–51. 2002.
- Berry, David Carson (2003). "Hans Weisse and the Dawn of American Schenkerism". Journal of Musicology 20 (1): 104–56. doi:10.1525/jm.2003.20.1.104.
- "Victor Vaughn Lytle and the Early Proselytism of Schenkerian Ideas in the U.S". Journal of Schenkerian Studies 1: 92–117. 2005.
- "Schenkerian Theory in the United States: A Review of Its Establishment and a Survey of Current Research Topics". Zeitschrift der Gesellschaft für Musiktheorie 2 (2–3): 101–37. 2005.
- Eybl, Martin; Fink-Mennel, Evelyn, eds. (2006). "Hans Weisse (1892–1940)". Schenker-Traditionen: Eine Wiener Schule der Musiktheorie und ihre internationale Verbreitung [Schenker Traditions: A Viennese School of Music Theory and Its International Dissemination]. Vienna: Böhlau Verlag. pp. 91–103. ISBN 978-3-205-77494-5.
Further reading 
A thorough documentation of Schenker-related research and analysis is provided in David Carson Berry, A Topical Guide to Schenkerian Literature: An Annotated Bibliography with Indices (Hillsdale, NY: Pendragon Press, 2004). The largest Schenkerian reference work ever published, it has 3600 entries (2200 principal, 1400 secondary) representing the work of 1475 authors. It is organized topically: fifteen broad groupings encompass seventy topical headings, many of which are divided and subdivided again, resulting in a total of 271 headings under which entries are collected.
See also 
- Schenker described the concept in a paper titled Erläuterungen (“Elucidations”), which he published four times between 1924 and 1926: Der Tonwille vol. 8-9, pp. 49-51, vol. 10, pp. 40-2; Das Meisterwerk in der Musik, vol. 1, pp. 201-5; 2, p. 193-7. English translation, Der Tonwille, vol. 2, p. 117-8 (the translation, although made from vols. 8-9 of the German original, gives as original pagination that of Das Meisterwerk 1; the text is the same). The concept of tonal space is still present in Free Composition, especially § 13, but less clearly than in the earlier presentation.
- Free Composition, § 21.
- Robert Snarrenberg, Schenker’s Interpretive Practice, Cambridge Studies in Music Theory and Analysis 11, 1997.
- Wilhelm Furtwängler, Wort und Ton, Wiesbaden, Brockhaus, 1954, pp. 201-202.
- Felix Salzer, Structural Hearing, New York, Boni, 1952.
- Free Composition, § 1. See also Harmony, § 13.
- The same link is made, for instance, in Schoenberg’s Harmonielehre, Wien, Universal, 1911, 7/1966, p. 16.
- Harmony, § 23
- Harmony, § 79.
- Robert E. Wason, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg, Ann Arbor, London, UMI Research Press, 1982/1985.
- Counterpoint, vol. I, p. 74. J. Rothgeb and J. Thym, the translators, quote Cherubini from the original French, which merely says that "conjunct motion better suits strict counterpoint than disjunct motion", but Schenker had written: der fliessende Gesang ist im strengen Stile immer besser as der sprungweise (Kontrapunkt, vol. I, p. 104) ("the fluent melody is always better in strict style than the disjunct one"). Fliessender Gesang not only appears in several 19th-century German translations of Cherubini, but may be common in German counterpoint theory from the 18th century and might go back to Fuchs’ description of the flexibili motuum facilitate, the "ease of flexible motion" (Gradus, Liber secundus, Exercitii I, Lectio quinta) or even earlier.
- Counterpoint, vol. I, p. 94.
- The canonical Ursatz is discussed in Free Composition, §§ 1-44, but it was first described in Das Meisterwerk in der Musik, vol. III (1930), p. 20-21 (English translation, p. 7-8). The word Ursatz already appeared in Schenker’s writings in 1923 (Der Tonwille 5, p. 45; English translation, vol. I, p. 212).
- Free Composition, § 10.
- Free Composition, § 106.
- Free Composition, § 16.
- David Beach, "Schenker’s Theories: A Pedagogical View", Aspects of Schenkerian Theory, D. Beach ed., New Haven and London, Yale University Press, 1983, p. 27.
- H. Schenker, Fünf Urlinie-Tafeln (Five Analyses in Sketchform), New York, Mannes Music School, 1933; Five Graphic Analyses, New York, Dover, 1969. The Foreword is dated 30 August 1932.
- On this most interesting topic, see Kofi Agawu, "Schenkerian Notation in Theory and Practice", Music Analysis 8/3 (1989), pp. 275-301.
- William Rothstein, "Rhythmic Displacement and Rhythmic Normalization", Trends in Schenkerian Research, A. Cadwallader ed., New York, Schirmer, 1990, pp. 87-113. Rothstein’s idea is that ornamentations such as retardations or syncopations result from displacements with respect to a "normal" rhythm; other diminutions (e.g. neighbor notes) also displace the tones that they ornate and usually shorten them. Removing these displacements and restoring the shortened note values operates a "rhythmic normalization" that "reflects an unconscious process used by every experienced listener" (p. 109).
- Kofi Agawu, "Schenkerian Notation in Theory and Practice", op. cit., p. 287, quotes Czerny’s representation of the "ground-harmony" of Chopin’s Study op. 10 n. 1 (in his School of Practical Composition, 1848), reproduced here in a somewhat simplified version.
- Edward Aldwell, Carl Schachter and Allen Cadwallader, Harmony and Voice Leading, 4th edition, Schirmer, Cengage Learning, 2011, p. 692.
- Allen Cadwallader and David Gagné, Analysis of Tonal Music: A Schenkerian Approach, New York, OUP, 3/2011, pp. 66-8.
- Op. cit., p. 384.
- Harmonielehre, p. 281; English translation, p. 211.
- William Rothstein, "Rhythmic Displacement and Rhythmic Normalization", Trends in Schenkerian Research, op. cit.
- Heinrich Schenker, “Elucidations”, Der Tonwille 8-9, English translation, vol. II, p. 117 (translation by Ian Bent)
- "Erläuterungen", Der Tonwille 8-9, English translation, vol. I, p. 117 (translation by Ian Bent).
- Free composition, p.78, §221.
- Free Composition, p.74-5, §205-7. Schenker’s German term is scheinbare Züge, litterally "apparent linear progressions"; Oster’s translation as "illusory" may overstate the point.
- The matter of the elaboration of seventh chords remains ambiguous in Schenkerian theory. See Yosef Goldenberg, Prolongation of Seventh Chords in Tonal Music, Lewinston, The Edwin Mellen Press, 2008.
- Free composition, p. 50, §140.
- For a detailed study of "unfolding", see Rodney Garrison, Schenker’s Ausfaltung Unfolded: Notation, Terminology, and Practice, PhD Thesis, State University of New York at Buffalo, 2012.
- See David Gagné, "The Compositional Use of Register in Three Piano Sonatas by Mozart", Trends in Schenkerian Research, A. Cadwallader ed., New York, Schirmer, 1990, pp.23-39.
- Free Composition, p.42, §106.
- Free Composition, p.87, §242.
- Schenker, Heinrich; ed. and annot. Oswald Jonas (1954) . Harmony. trans. Elisabeth Mann-Borgese. Chicago: University of Chicago Press. ISBN 0-226-73734-9. OCLC 280916.
- Schenker, Heinrich (1989) [1910, 1922]. Counterpoint. trans. John Rothgeb, Jürgen Thym. New York, London: Schirmer Books. ISBN 0-02-873220-0.
- Schenker, Heinrich; ed. Oswald Jonas (1979) . Free Composition. trans. Ernst Oster. New York, London: Longman.
Further reading 
|This article lacks ISBNs for the books listed in it. (March 2012)|
- Beach, David, ed. (1983). Aspects of Schenkerian Theory. New Haven: Yale University Press.
- Berry, David Carson (2004). A Topical Guide to Schenkerian Literature: An Annotated Bibliography with Indices. Hillsdale, NY: Pendragon Press.
- Eybl, Martin and Fink-Mennel, Evelyn, eds. (2006). Schenkerian traditions. A Viennese school of music theory and its international dissemination. Vienna, Cologne, Weimar: Böhlau. ISBN 3-205-77494-5.
- Maus, Fred (2004). "Sexual and Musical Categories", The Pleasure of Modernist Music. ISBN 1-58046-143-3.
- Middleton, Richard (1990/2002). Studying Popular Music. Philadelphia: Open University Press. ISBN 0-335-15275-9.
- Jonas, Oswald (1982). Introduction to the theory of Heinrich Schenker : the nature of the musical work of art. Translated by John Rothgeb. New York and London: Longman. "Most complete discussion of Schenker's theories." (Beach 1983)
- Brown, Matthew (2005). Explaining Tonality: Schenkerian Theory and Beyond. University of Rochester Press. ISBN 1-58046-160-3.
- Cook, Nicholas (2007). The Schenker Project: Culture, Race, and Music Theory in Fin-de-siècle Vienna. Oxford University Press. ISBN 0-19-974429-7.
- Forte, A. (1959). "Schenker's Conception of Musical Structure". Journal of Music Theory 3 (1): 1–30. doi:10.2307/842996. JSTOR 842996.
- Katz, A. T. (1935). "Heinrich Schenker's Method of Analysis". The Musical Quarterly 21 (3): 311–329. doi:10.1093/mq/XXI.3.311. JSTOR 739052.
Pedagogical works 
- Forte, Allen and Gilbert, Steven E. (1982). Introduction to Schenkerian Analysis. W. W. Norton & Company. ISBN 0-393-95192-8. Schenker never presented a pedagogical presentation of his theories, this being the first according to its authors.
- Snarrenberg, Robert (1997). "Schenker's Interpretive Practice." Cambridge: Cambridge University Press. ISBN 0-521-49726-4.
- Cadwallader, Allen and Gagné, David (1998). Analysis of Tonal Music: A Schenkerian Approach, Oxford: Oxford University Press, ISBN 0-19-510232-0. The second major English-language textbook on Schenkerian analysis"
- Kalib, Sylvan (1973). Thirteen Essays from The Three Yearbooks “Das Meisterwerk in Der Musik,” by Heinrich Schenker: An Annotated Translation. (Vol.’s I–III). Ph.D. diss., Northwestern University.
- Westergaard, Peter (1975). An Introduction to Tonal Theory. New York: W.W. Norton.
- Pankhurst, Tom (2008), SchenkerGUIDE: A Brief Handbook and Web Site for Schenkerian Analysis, New York: Routledge. ISBN 0-415-97398-8 – an introduction for those completely new to the subject.
- Salzer, Felix (1952). Structural Hearing: Tonal Coherence in Music. New York: Charles Boni. "The first book to present a reorganization (as well as modification and expansion) of Schenker's writings from a pedagogical standpoint." Beach (1983)
- Westergaard, Peter (1975). An Introduction to Tonal Theory. New York: W.W. Norton.
- Yeston, Maury, ed. (1977). Readings in Schenker Analysis and Other Approaches. New Haven: Yale University Press.
- Beach, David, ed. (1983). Aspects of Schenkerian Theory. New Haven: Yale University Press.
- Epstein, David (1979). Beyond Orpheus – Studies in Musical Structure. Cambridge, MA: The MIT Press.
Post-tonal expansions 
- Travis, R. (1959). "Towards a New Concept of Tonality?". Journal of Music Theory 3 (2): 257–284. doi:10.2307/842853. JSTOR 842853.
- Travis, R. (1966). "Directed Motion in Schoenberg and Webern". Perspectives of New Music 4 (2): 85–89. doi:10.2307/832217. JSTOR 832217.
Rhythmic expansions 
- Komar, Arthur (1971/1980). Theory of Suspensions: A Study of Metrical Pitch Relations in Tonal Music. Princeton: Princeton University Press/Austin, Texas: Peer Publications. (Beach 1983)
- Yeston, Maury (1976). The Stratification of Musical Rhythm. New Haven: Yale University Press. (Beach 1983)
- Narmour, Eugene (1977). Beyond Schenkerism: The Need for Alternatives in Music Analysis. Chicago: The University of Chicago Press.
|Wikiquote has a collection of quotations related to: Schenkerian analysis|
- Schenker Guide by Tom Pankhurst
- List of Schenker's writings concerning analysis on the Schenkerian site of Luciane Beduschi and Nicolas Meeùs (Paris)
- Tim Smith's introduction to Schenkerian concepts in a Bach fugue: Shockwave movie or pdf
- Yale University's Gilmore Music Library provides a good introduction to important primary and secondary sources