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Schenkerian analysis

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 reveal the underlying structure of a tonal masterpiece; in fact its basic tenets can be viewed as a way of defining tonality in music. The primary means of describing the structure of a musical passage for the Schenkerian analyst is to show hierarchical relationships among the pitches of the passage. This can be done through making reductions of the music and through a specialized symbolic form of musical notation that Schenker devised to demonstrate various prolongational techniques.

The musical reductions of Schenkerian analysis are usually arrhythmic. This reflects Schenker's belief that the deep, long-range structure of a piece of music has no particular rhythm. This long-range structure is called the Fundamental Structure (Ursazt) in Schenkerian analysis, while the more surface aspects of the music are called the foreground or surface layer. So one could rephrase the previous statement as "the background of a musical composition is arhythmic," or, better yet, "rhythm is a characteristic of the musical foreground" (See Der Freie Satz section 21 and chapter 4). Open and closed noteheads, beams, and flags, which show rhythm in ordinary musical notation, are used in Schenkerian analysis to show hierarchical relationships between the pitch-events being analyzed.

Schenkerian analysis is a subjective, not an objective, method. This means that there is no mechanical procedure for arriving at an analysis for a given piece of music; rather, the analysis reflects the musical intuitions of the analyst. Therefore, this form of analysis is more art than science. The analysis represents a way of hearing a piece of music. Schenker himself was certain that a tonal masterpiece contains an inner truth-content, although few are sufficiently gifted to appreciate it. Although it is a subject of debate among music theorists whether there is a single correct hearing and analysis of a piece of tonal music, even those who hold that there is a unique correct analysis agree that the analysis can only be arrived at and evaluated subjectively by an expert listener. Therefore learning how to do Schenkerian analysis is above all else learning a way of hearing and understanding tonal music, and it requires study and practice just as learning to play an instrument does.

Schenker's goals

Schenker's primary theoretic aims were to prove the superiority of German music of the common practice period (especially the music of Johann Sebastian Bach, Carl Philipp Emanuel Bach, Franz Josef Haydn, Wolfgang Amadeus Mozart, Ludwig van Beethoven, and Johannes Brahms) over more modern music such as that of Richard Wagner, Igor Stravinsky, and Arnold Schoenberg, and to show that most of the established music theory teaching of the time, with an emphasis on the theories of his contemporary Hugo Riemann, was misleading and useless for an understanding of the "masterworks." These premises led Schenker to seek the key to an understanding of music in the traditional discipline of counterpoint, since this is the type of theory the "German Masters" themselves had studied. While Schenker's theory has been tremendously influential, particularly in North America thanks in part to his emigre students Oswald Jonas and Felix Salzer, most "Schenkerians" do not share his exceedingly narrow and nationalistic view of musical excellence, and his ideas and methods have been applied to a wide range of composers.

Schenker's project, thus, was to show that free composition (freier Satz) was an elaboration of strict composition (strenger Satz), by which Schenker meant species counterpoint. He did this by developing a theory of hierarchically organized reductional levels, called prolongational levels, voice-leading levels (Stimmführungsschichten), or transformations (Verwandlungen), the idea being that at higher levels in the structure the musical materials conform more closely to those of strict composition. A primary goal in constructing these levels therefore is to show linear connections between notes that may be separated by many measures on the musical surface (since linearity or step-wise motion is the most important characteristic of good voice leading).

The basic components of Schenkerian theory and analysis therefore are the nature of the background—that is, the highest voice leading level—and the ways in which the background may be prolonged (elaborated, transformed) to arrive at the foreground—i. e. the musical composition.

Schenkerian harmony

Schenker's magnum opus, Neue Musikalische Theorien und Phantasien ("New Musical Theories and Fantasies"), spans his entire publication career from the early work, Harmonielehre ("Harmony") (volume I) through the formative Kontrapunkt ("Counterpoint") (volumes II.1 and II.2) to the posthumously published Der Freie Satz ("Free Composition") (volume II.3). The organization of this work reflects the organization of Schenkerian analysis itself. The tenets of harmony and counterpoint, given by nature, combine through art to produce the musical work in free composition.

The first tenet of Schenkerian harmony is that nature, through the harmonic series, gives us the triad as the ultimate (and only possible) basis for musical composition. In fact, Schenker's explanation only secures "naturalness" for the major triad, whereas Schenker describes the minor triad as an artificial construction of musicians. Despite this difference, in practice the major and minor triads are treated equally in Schenkerian analysis.

The basic component of Schenkerian harmony is the Stufe (scale degree, scale-step). The Stufe is an abstraction of the idea of a chord and a revision of Jean-Philippe Rameau's idea of basse fondementale (fundamental bass). A chord in a piece of music may represent the stufe corresponding to its root. However, many surface phenomena in music that appear to be chords are not actually representative of Stufen themselves but are voice-leading constructions of a passing nature whose real function is the prolongation of some other Stufe. In short, not all chords represent Stufen. Furthermore, a literal chord is not necessary for the representation of a Stufe. The chord may be arpeggiated, so that all its tones are not present simultaneously. This arpeggiation may occur at a very background level so that it is not apparent on the musical surface. (In other words, the arpeggiation or chord may be prolonged, e.g. by passing motions). Sometimes a Stufe may be represented by only a single note.

Schenkerian analyses show Stufen with roman numerals. E. g., "I" indicates the tonic Stufe, "V" indicates the dominant Stufe, and so on. The practice of roman numeral analysis originated in the theoretic work of Gottfried Weber and was common in Schenker's time. However it should be emphasized that Schenker and Schenkerians after him are generally at odds with the practice of roman numeral analysis, mainly because they believe that it fails to recognise the sensitivity of the meaning of a chord to its musical context (particularly its rhythmic and voice-leading context) and that it tends to project an insufficiently sophisticated theory of modulation and tonicization.

The Stufe is an elusive but important concept in Schenkerian theory. Its formulation in Schenker's earliest significant work, Harmony, is associated with the idea of "contrapuntal" or "passing" chords. That is, some chords in music are not harmonic in nature (do not represent real Stufen) but arise by contrapuntal-melodic processes of a passing nature (Scheinharmonie). In other words, they are made up of notes that are treated like dissonant notes, even though they may appear consonant. Thus, the most important aspect of the Stufe concept is the negative one: not all chords represent Stufen. Schenker gives a more detailed explanation of such passing chords in the second volume of Kontrapunkt, a more mature work. Here, the Stufen is seen as an imaginary cantus firmus tone against which the passing chords are constructed in multiple parts, dissonant with the cantus firmus but consonant with one another. This is the most accurate way to think of the roman numerals that sometimes are placed below a Schenkerian analysis, rather than thinking of them as chord roots.

Schenkerian harmonic theory holds that modulation is an illusory phenomenon in music (or at least in musical "masterworks"). Every complete musical piece projects a single key and ultimately a single stufe (the tonic). (See Ursatz). What appear to be modulations in a musical work are actually the result of prolongations. Whenever harmonic progressions suggest new tonics without disrupting the unity of a tonal background in the home key, Schenkerian analysts prefer the weaker term "tonicization" to "modulation."

Ursatz

The Ursatz (usually translated as "fundamental structure"; see also satz) is the basic form of the background in Schenkerian analysis. That is, it represents the most reduced contrapuntal version of a piece of music and shows its most skeletal form, the essential pillars of the tonal structure.

One of the tenets of Schenkerian theory is that this basic background counterpoint can only come in a few different versions. This is also the most common point of contention with Schenkerian theory, that there is a basic structure to which all tonal compositions conform. Supporters of Schenkerian theory have defended against these attacks on the idea of the Ursatz, among them Allen Forte's statement that "Schenker's major concept is not that of the Ursatz, as it is sometimes maintained, but that of structural levels, a far more inclusive idea." It is important to realize that the Ursatz itself is not a description of the piece of music; rather the art of tonal composition is (according to Schenkerian theory) in the way in which the Ursatz is expressed and elaborated. Correspondingly, the art of understanding music, of analysing music, is in penetrating the musical surface and hearing through the foreground to the background—in other words, hearing how the foreground and the background are connected by a series of prolongations.

Forms

For Schenker, the simplest type of "musical composition" is represented by a structure of the following form:

This prototypical counterpoint consists of a melodic prototype (the "Urlinie") and a harmonic prototype (the "Bassbrechung"). The Urlinie is "a stepwise descent from one of the notes of the tonic triad to the tonic (hence, 3-2-1, 5-4-3-2-1, 8-7-6-5-4-3-2-1)" (Middleton 1990, p.193). The bass arpeggiation (Bassbrechung) is a two-stage progression: first moving from I to V and then from V back to I. Schenker came to understand every tonal work to be an embellishment of such an Ursatz, making the claim that a tonal work unfolds in a particular triad or key more specific.

The initial note of the Ursatz is called the head-tone (Kopfton). The only possible head-tones for a piece of music are scale-degrees , , and the tonic (in which case the urlinie is an octave descent). The three corresponding forms of the Urlinie are often called the 3-line (--), the 5-line (----) and the octave-line (-------). In all three cases, the V (dominant key area) of the bass arpeggiation corresponds to the penultimate scale-degree in the Urlinie. (This is the most basic form of the background. At the next level of elaboration, it is possible to fill out this bass part.) While Schenker often used the octave-line in his analyses, it has generally fallen out of favor with later analysts and is rarely used nowadays.

Initial ascent

Frequently the head-tone of the Urlinie does not coincide with the beginning of the piece or the initial statement of the tonic harmony in the piece. In such cases the background of the analysis includes approach to the head-tone of the Urlinie and this is known as the initial ascent (Anstieg). The initial ascent takes the form of either an arpeggiation or a linear progression from one tone of the tonic triad to another. For instance, if the head-tone of the piece is scale-degree 3, the initial ascent could take the form of an arpeggiation from scale-degree 5 (5-1-3), or a linear progression from the tonic (1-2-3), among others.

Schenkerian notation

Schenker created a symbolic language from modified musical notation creating graphic analyses, or graphs. Forte groups Schenker's graphs in "Free Composition" into "rhythmic" and "structural" types. In rhythmic reduction, often called metric reduction, the original note durations and their meanings are kept, while in structural analysis longer rhythmic values indicate greater structural importance or level. (Beach 1983) In modern Schenkerian analysis structural graphs are the norm. In the language of Schenkerian symbols open noteheads are used for notes of the fundamental line and their supporting bass notes. These are usually stemmed and beamed together and accompanied by careted scale-degree numbers. Flagged open notes indicate the deeper middleground prolongations of the fundamental structure, frequently neighbor-note embellishments of the Urlinie. Other stemmed closed notes show a subsequent stage of middleground prolongation. Occasionally these may be beamed together beneath the beamed open notes to show a middleground image of the fundamental structure. In this case the careted scale-degree numbers should be in parentheses to show that this is not the true fundamental line. Slurs indicate a variety of middleground connections, especially linear progressions, and dotted slurs show the retention of a single note over a long span or the registral displacement (or octave coupling) of a particular note. Other symbols include those for interruptions, unfolding, and voice exchange.

Techniques of prolongation

The meat of a Schenkerian analysis is in showing how a background structure may be expanded step-by-step until it results in the succession of musical events on the surface of the composition itself. We refer to this as the process of prolongation. There are many techniques for prolonging musical structures, and it is impossible to make a complete list of them given the unlimited nature of compositional ingenuity. This section includes most of the more common techniques one might encounter in a Schenkerian analysis. The first few focus on those techniques, arpeggiation, interruption, and neighbor note, which are important techniques at the earliest levels of prolongation—that is, those that may prolong the Ursatz itself. These techniques are common at later levels of elaboration also.

Arpeggiation

One important means of prolongation is the composing-out (Auskomponierung) of a triad. This means that two or more notes of the triad are presented in succession rather than simultaneously. (In other words the vertical is made horizontal). This is called arpeggiation.

The Ursatz includes an example of arpeggiation in the lower voice, called the Bass Arpeggiation. That is, Schenkerian theory views the I - V - I motion in the lower voice of the Ursatz as an arpeggiation of the root and fifth of the tonic triad.

This basic bass arpeggiation may be further prolonged by the addition of the note III in the bass, transforming the Ursatz to

       ^              ^      ^
       3              2      1
       I      III     V      I

or,

       ^      ^      ^      ^      ^
       5      4      3      2      1
       I            III     V      I

In music of the romantic period this frequently takes the form of a section in the key of the chromatic mediant III# (when the tonic key is major). Such an arpeggiation in the bass is especially common in minor key pieces, which move easily to the relative major, III.

Arpeggiation can occur as a prolongational technique at the foreground also, and at various levels in between. An example of a common middleground use of arpeggiation is in the initial ascent to the head-tone of the Urlinie. Another common use of arpeggiation at later middleground levels is in the service of a register transfer.

Interruption

Another fundamental prolongational technique, one that applies directly to Ursatz itself is called Interruption (Unterbrechung). Interruption is an important form-generating prolongation.

It works as follows: beginning with the Ursatz:

       ^        ^        ^
       3        2        1
       I        V        I

one prolongs the resolution of scale-degree 2 to 1 by going back to degree 3 and replaying the whole sequence:

                    ||
       ^        ^        ^       ^        ^
       3        2        3       2        1
       I        V        I       V        I

The symbol || indicates the point of interruption. Similarly, the 5-line Ursatz:

       ^        ^        ^       ^        ^
       5        4        3       2        1
       I                         V        I

can become,

                                      ||
       ^        ^        ^       ^        ^        ^        ^       ^        ^
       5        4        3       2        5        4        3       2        1
       I                         V        I                         V        I

When this is used as a form-generating prolongation, the interruption (||) marks the major formal division that it creates. A typical example is sonata form: the exposition of a major key sonata conventionally ends on the dominant (V). The beginning of the recapitulation brings the return to the head tone (scale-degree 3 or 5). On a smaller scale, form-generating interruption is typical in movements of a dance suite. The octave-line (the third form of the Ursatz) doesn't generally yield to prolongation by interruption (Instead, the arrival at scale-degree 5 in the principle voice may provide a point of formal division).

Neighbour note

The neighbour note (Nebennote) is also an important way of prolonging the fundamental structure (Ursatz). In the principal voice it is always an upper neighbor to scale-degree three or five, as in,

       ^        ^        ^       ^        ^
       3        4        3       2        1
       I                         V        I

       ^        ^        ^        ^        ^       ^        ^
       5        4        3        4        3       2        1
       I                                           V        I

       ^        ^        ^        ^        ^       ^        ^
       5        6        5        4        3       2        1
       I                                           V        I

Of course, other prolongations of the bass-voice will usually accompany this, such as,

       ^        ^        ^       ^        ^
       3        4        3       2        1
       I        IV       vi      V        I

And the resumption of the Urlinie tone may be eliminated (yielding an incomplete neighbour figure):

       ^        ^       ^        ^
       3        4       2        1
       I        II      V        I

Note that the II itself is an incomplete neighbour in the bass. (see Schenkerian analysis#Prolongations of the bass arpeggiation). Neighbour notes can happen at all prolongational levels, not just the primary prolongations of the Urlinie as illustrated here. (The limitation to upper neighbors of notes of the tonic triad applies only to the technique as a prolongation of the Urlinie)

Prolongations of the bass arpeggiation

Schenkerian theory regards the I-V-I progression of the Ursatz as the basic harmonic progression for all tonal music, derived from the most fundamental arpeggiation of the tonic triad in the bass. Thus, it recognises the harmonic nature of the dominant, but not the subdominant chord. This contrasts with the theory of functional harmony of Hugo Riemann, which regards the tonic-subdominant relationship as equally fundamental as the tonic-dominant relationship, and takes the progression I-IV-V-I to be basic.

In Schenkerian theory, the I-IV-V-I progression is suitable for a background structure, but the IV is derived as prolongation of the bass arpeggiation I-V. The IV is an incomplete neighbor to the V in the bass. Thus, I-IV-V-I is no more basic than I-II-V-I, where the II in the bass is a neighbor to the initial I.

Other prolongations of the bass arpeggiation include the arpeggiation I-III-V-I, and versions of this arpeggiation with passing tones included, as in I-III-IV-V-I or I-II-III-IV-V-I (note that the roman numerals in this case refer to bass-notes, not necessarily harmonies. For instance, "III" could refer to the bass of a first inversion tonic chord).

Linear progressions

A Linear progression (Zug) is the step-wise filling in of some interval. It is shown symbolically with a slur from the first note of the progression to the last.

The most common linear progression is the third progression. This is essentially a passing tone figure, but calling it a "third progression" implies the composing-out of a third–that is, the third interval has some harmonic significance.

The fourth progression fills in the space between the fifth and root of a triad with stepwise melodic motion.

The fifth progression functions like a fourth progression. That is, it outlines the same interval in inversion. A fifth progression can be built from two third progressions.

Likewise, the sixth progression functions like the third progression.

"Seventh progressions" and "octave progressions" are not true linear progressions (according to Schenker), but rather examples of register transfer. Regardless of this, Schenker and others do speak of seventh progressions and show them in their analyses, and sometimes even regard the seventh as a harmonic interval (when it is made up of the root and seventh of a dominant chord).

The Urlinie is an example of a linear progression. It outlines the interval of a third from scale-degree three to one, or a fifth from scale-degree five to one.

Schenker also sometimes uses the more elaborate term Auskomponierungzug for linear progressions. This indicates that one is to think of the linear progression as the composing-out of an interval, much as an arpeggiation is the composing-out, or "horizontalization," of a chord.

Images of the fundamental structure

Frequently a piece of music will include a self-contained section that has its own small-scale version of the Ursatz (fundamental structure). A good example is the theme to the set of variations in Mozart's K331 piano sonata. (See Forte/Gilbert 133-138). This theme divides into two 8 measure sections. The first of these includes a complete 5-line in A major with an interruption in measure 4. This is followed by 4 contrasting measures exploring a higher range E2-A2, then a repeat of measures 5-8 bringing the theme to a close. The "real" Urlinie descent for the entire theme is the one in measures 13-16 even though it's identical to measures 5-8. The Urlinie descent of measures 5-8 provides a local musical goal that ultimately (in the context of the whole theme) subserves the prolongation of the head-tone E2. The "proof" of this larger goal for the first eight measures is the fact that the note E2 is regained in the melody in measure 9.

This is a typical example, where the middleground image of the Ursatz is identical to the background Ursatz, but because of its placement at the beginning of the piece we regard it as a prolongation of the head-tone, a way of providing goal-directed motion locally without advancing the background progression. Images of the fundamental structure may also occur in different keys, prolonging other elements of the background structure.

All of the prolongational techniques that apply directly to the Ursatz, such as interruption and neighbor note and the prolongations of the bass arpeggiation, apply in the same way to these images of the fundamental structure.

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 or fractal-like construction.

Register transfer

Register transfer refers to the motion of a voice into a different octave (i.e. into different register). Register transfer is an important technique at later middleground prolongational levels. The motion up or down by an octave may occur directly, but often the octave is filled in by step-wise motion or arpeggiation. Register transfer can also take the form of a motion through the interval of a seventh or ninth in a voice. This is thought of as the elision of a step-wise motion in that voice and a change of register.

Unfolding

The unfolding (Ausfaltung) of an interval is a form of composing-out in which two notes that are presented successively in a single voice at the foreground are thought of as a simultaneity in different voices at a more background level. The difference between unfolding and arpeggiation is that each note of the unfolding usually has an independent melodic importance at some deeper level of the analysis. An unfolding is indicated in Schenkerian notation by stemming each note in opposite directions and then connecting these stems with a diagonal beam.

Voice exchange

Voice exchange occurs when two voices (one of which is usually the bass) trade pitch-classes that are a third apart, in the service of prolonging some harmony. For instance, if a tonic chord with scale-degree 3 in the upper voice moves to a first-inversion tonic with scale-degree 1 in the upper voice, this may be regarded as a voice exchange prolonging the tonic harmony. Typically the voice exchange will occur through a third-progression or sixth-progression in one or both voices. Voice exchange is indicated notationally by drawing two crossing lines connecting the identified pitch-classes.

Diminution

In Schenkerian theory, the process of elaborating basic structures by generating subordinate tones is often called diminution. In the pedagogical method of Forte/Gilbert diminutions are classified as passing notes (P), neighboring notes (N, upper or lower, complete or incomplete, direct or indirect), consonant skips (CS), and arpeggiations (Arp). Forte and Gilbert developed this list of "diminutions" in the interest of creating a complete system for analytical reduction of the musical surface (foreground). The term "consonant skip" is their own invention. (Forte and Gilbert, 1982).

"The function of a note is determined by its harmonic and contrapuntal setting." Thus, whether a note is part of a diminution is determined by its context. For example, if two adjacent notes alternate, the one which is 'unsupported' by the harmony is [often] a neighboring note. (Forte and Gilbert 1982)

Legacy and responses

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." Though Schenker considered nontonal or atonal music unnatural, unlike the sexologists who focused more on minoritized categories, he focused on elaborating his theory of tonal music.

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)

Carl Schachter, who is on faculty at the Mannes College of Music, is considered one of the current masters and advocates of the practice.

Notes

References

Further reading

  • 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.
  • 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)

Summaries

  • Forte, Allen (1959). "Schenker's Conception of Musical Structure", Journal of Music Theory 3. (Beach 1983)
  • Katz, Adele (1935). "Heinrich Schenker's Method of Analysis," The Musical Quarterly 31. (Beach 1983)

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. ISBN 0-19-510232-0. The second major English-language textbook on Schenkerian analysis.
  • 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 0415973988 - an introduction for those completely new to the subject.

Expansions

  • 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.

Post-tonal expansions

  • Travis, Roy (1959). "Toward a New Concept of Tonality", Journal of Music Theory 3. (Beach 1983)
  • Travis, Roy (1966). "Directed Motion in Schoenberg and Webern", Perspectives of New Music 4. (Beach 1983)

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)

Criticisms

  • Narmour, Eugene (1977). Beyond Schenkerism: The Need for Alternatives in Music Analysis. Chicago: The University of Chicago Press.

External links

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