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The Syntax of Tonal Music. UMD Syntax Lunch Oct. 3, 2006. Outline of this talk. Background Introduction to musical phenomena Motivating and characterizing the syntactic nature of music Notational evidence Experimental evidence Theories of Heinrich Schenker Theories of Lehrdal & Jackendoff

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The syntax of tonal music

The Syntax of Tonal Music

UMD Syntax LunchOct. 3, 2006


Outline of this talk
Outline of this talk

  • Background

  • Introduction to musical phenomena

  • Motivating and characterizing the syntactic nature of music

    • Notational evidence

    • Experimental evidence

    • Theories of Heinrich Schenker

    • Theories of Lehrdal & Jackendoff

  • Comparison with linguistic theories

Syntax of tonal music


The mental representation of music
The mental representation of music

  • What happens when we listen to music?

Syntax of tonal music


Tonal music
Tonal music

  • European “classical” music, 1600-1900

  • Most modern popular music

  • Highly developed tradition

    • Lots of materials

      • Bach

      • Beethoven

      • Brahms

      • Berlin (Irving)

      • Britney (Spears)

    • Standard notation

  • “Natural” system

  • “Music in a key” (Forte & Gilbert 1982)

Syntax of tonal music


Non tonal music
Non-tonal music

  • Atonalism

    • Developed in Vienna, early 20th century

    • Very short, atmospheric pieces

    • 12-tone composition (Serialism) developed to give “structure” to the pieces

      • Schönberg, Webern, Berg

    • Compositions highly structured

      • Very small number of compositional decisions made, then the piece “writes itself”

      • Little perceptual awareness of the organization

        • “Augenmusik”

Syntax of tonal music


Other music
Other music

  • Music from other cultures

    • Divisions of the octave into larger and smaller numbers of pitch classes

    • The role of harmony generally far less than in (Western) tonal music

    • “Natural” systems

Syntax of tonal music


Musical primitives pitch
Musical primitives: pitch

  • Octaves group pitches into equivalence classes

  • Each octave subdivided into 12 pitch classes

    • A, A# = B, B, C, C# = D, D, D# = E, E, F, F# = G, G, G# = A

  • Exact tuning of intervals may vary

    • Octaves are exact, however

  • Diatonic scales

    • Two variants

    • Mixture of whole steps (-) and half steps (.)

    • Major: - - - . - - - .

    • Minor: - - . - - . - - // - - . - - - - .

Syntax of tonal music


Musical primitives overtones
Musical primitives: overtones

  • Any signal can is equivalent to the sum of sine waves with frequencies related to each other in simple whole-number ratios

  • These simple ratios turn out to be musically significant

    • Observed since Pythagoras

    • Same intervals in many musical cultures

Syntax of tonal music


Musical primitives intervals
Musical primitives: intervals

  • Distance from one pitch to another

  • May be absolute

    • Number of half-steps

  • May be “tonal”, diatonic interval

    • Minor third (m3), major third (M3), perfect fourth (P4), diminished fourth (d4), perfect fifth (P5)

    • Multiple interpretations of one interval (M3 and d4 both have the same number of half steps)

      • Different spellings based on different tonal contexts

      • One is consonant and the other is dissonant

Syntax of tonal music


Musical primitives harmony
Musical primitives: harmony

  • “Synchronic” look at musical context

  • Multiple tones sounding simultaneously

  • Often associated with specific expectation (ie, “functions”)

    • Tonic, dominant, subdominant, submediant (=relative minor)

Syntax of tonal music


Musical primitives harmony1
Musical primitives: harmony

  • Conventionally notated with Roman numerals

    • I = tonic, IV = subdominant, V = dominant

    • Number based on the “root” of the chord

    • Lowercase = minor, Uppercase = major

  • Associated with specific harmonic expectations

  • Tonic example:

Syntax of tonal music


Musical primitives harmony2
Musical primitives: harmony

  • Dominant example:

  • Functional harmony

    • Tonic (I) – goal, stability, complete

    • Dominant (V, V7) – incomplete, expectation for contination

  • Harmony is more abstract than “chords”!

Syntax of tonal music


Musical primitives voice leading
Musical primitives: voice leading

  • “Diachronic” look at musical context

  • Where do the individual pitches “lead” as the music moves from one moment to the next?

  • Complex (perceptual/formal) rules for determining when an interval will be perceived “harmonically” or as voice movement

?

Syntax of tonal music


Musical primitives phrases
Musical primitives: phrases

  • Music groups into phrases, roughly melodic

  • Traditional “classical” melodies have two parts:

    • Antecedent (ending on V)

    • Consequent (ending on I)

  • Example: Mozart, Sonata in A major, K. 331, I

Syntax of tonal music


Understanding music
Understanding music

  • Context

    • Depending on the surrounding music, a particular interval, pitch, or harmony can have vastly different “function”

Syntax of tonal music



What notation tells us about music
What notation tells us about music

  • History of notation is as long and varied as the history of music

  • Constants: pitch (vertical, log scale), duration in time (horizontal, linear)

  • Some indications of hierarchy in notational conventions

    • Ornaments as diacritics

Syntax of tonal music


Notational conventions
Notational conventions

  • Ornaments as diacritics

Aria from the J.S. Bach Goldberg Variations (BWV 988)

Syntax of tonal music


Notional conventions
Notional conventions

  • Hierarchically minor notes notated as grace notes

  • Grace (small) notes should be played with equal length as the notes they are attached to!

  • Their “smallness” indicates their structural value.

Syntax of tonal music


Notational conventions1
Notational conventions

  • Figured bass – structurally unimportant notes were not even written!

  • Usually, a melody given, often used in accompaniment

Syntax of tonal music


Notational conventions2
Notational conventions

  • Guitar tablature

    • Indicates chords, inversions

    • Says nothing about

      • Rhythm

      • Arpeggiation pattern to

  • Extremely common in jazz, pop music

Syntax of tonal music


Schenkerian analysis

Schenkerian Analysis

An impossibly brief introduction


Heinrich schenker
Heinrich Schenker

  • 1868-1935

  • Viennese music theorist

  • Reactionary against post-tonal music (ie, music that violated traditional musical syntax for artistic effect)

  • Sought to explicate the “genius” of great music, especial German music

Syntax of tonal music


Schenkerian analysis1
Schenkerian Analysis

  • First non-prescriptive theory of music with a perceptual angle

    • Previous work on music perception

      • Pythagoras (582-507 BCE)

      • Helmholtz (1821-1894)

    • Other analytic methods in music focused on surface motivic relationships

  • Major innovation: hierarchical organization

Syntax of tonal music


Schenkerian analysis the ursatz
Schenkerian Analysis: the Ursatz

  • Ursatz = Fundamental structure

    • 3 forms

      • All tonal music is really just one of three melodies.

    • Fundamental structure is an elaboration of tonal relationships:

      • Harmonic

      • Voice-leading

    • Tonal relations are not temporal (ie, not rhythmic and not metrical)

Syntax of tonal music


The ursatz
The Ursatz

“Central to Schnker’s work is the notion that the tonic triad, an image of the overtone series generated by the tonic note, functions as a matrix… As Lerdahl & Jackendoff write ‘the tonic is in some sense implicit in every moment of the piece’” - Schachter 1999

Syntax of tonal music


Schenkerian analysis2
Schenkerian analysis

  • Three layers

    • Foreground (surface)

    • Middleground

    • Background (fundamental structure)

  • Series of transformations or “prolongations”

    • Neighbor note

    • Passing tone

    • Arpeggiation

    • Register transfer

    • Composing-out

  • Syntax of tonal music


    Prolongation examples
    Prolongation examples

    • Take a basic melody

    • Certain structure-preserving transformations may be applied:

    Syntax of tonal music


    Schenkerian analysis3
    Schenkerian analysis

    • Goal of Schenkerian analysis: recover underlying structure

      • Explain surface harmonic, voice-leading phenomena (and “problems”) in terms of “deeper” structure

    • Analyses are graphical

      • Several levels of abstraction present in one graph

    Syntax of tonal music


    Schenkerian analysis example
    Schenkerian analysis: example

    • J.S. Bach “Ich bin’s, ich sollte büßen” from the Matthäus-Passion (BWV 244)

    • This middle-ground graph shows the relationship of the surface structure to the fundamental structure

    Syntax of tonal music


    Schenkerian analysis example1
    Schenkerian analysis: example

    • “Hear” foreground

    Syntax of tonal music


    Schenkerian analysis example2
    Schenkerian analysis: example

    • “Hear” middleground

    Syntax of tonal music


    Schenkerian analysis example3
    Schenkerian analysis: example

    • “Hear” background

    Syntax of tonal music


    Lerdahl jackendoff a generative theory for tonal music
    Lerdahl & Jackendoff: A Generative Theory for Tonal Music

    • 1973 – Charles Eliot Norton Lectures at Harvard

      • Leonard Bernstein asks for a “musical grammar” to explain the human capacity for music the same as Chomsky’s approach to linguistic theory had done for language

    • 1983 – A Generative Theory for Tonal Music

    Syntax of tonal music


    L j gttm
    L&J: GTTM

    • Focus on hierarchical dimensions of music

      • Grouping structure

        • Break music in motives, phrases, sections

      • Metrical structure

        • Events in music occur at regular (isochronous) intervals

        • Hierarchy of strong and weak beats at various levels of abstraction

      • Time-span reduction

        • Given metrical and grouping structure assign pitches a hierarchy of structural importance

      • Prolongational reduction

        • Assign pitches a hierarchy based on harmonic and melodic (voice-leading) tension (closest aspect to Schenkerian analysis)

    Syntax of tonal music


    Music theory vs linguistic theory
    Music theory vs. linguistic theory

    • Three rule types in GTTM

      • Well-formedness rules

        • Specify possible SDs

      • Transformational rules

        • “Fudge” the strict hierarchical organization a bit

      • Preference rules

        • Given a set of SDs, which ones will be preferred?

    • The first two establish the SDs for a segment of music

    • What about preference rules?

    Syntax of tonal music


    Preference rules
    Preference rules

    • Structural descriptions not sufficient

    • Ranking various structural descriptions according to coherence is essential

    • Grammaticality far less important for music

    • Almost any passage of music is vastly ambiguous (ie, many possible SDs). Not seemingly the case with language.

    • According to L&J: musical grammar must be able to express preference rules among interpretations (absent from generative theories of language)

    Syntax of tonal music


    Reduction hypothesis
    Reduction hypothesis

    • One musical passage can be hear as an elaboration (or variation) of other passages

    • In some cases, passages may be heard as elaborations of an abstract structure that is never overtly stated

      • Bach Goldberg Variations (BWV 988)

        • Aria + 30 variations

        • Why not 31 separate pieces?

    • Listeners have intuitive understanding of relative structural importance of different pitches

    Syntax of tonal music


    Reduction hypothesis1
    Reduction hypothesis

    • Basic version

      • The listener attempts to organize all pitch-events of a piece into a single coherent structure, such that they are heard in a hierarchy of relative importance.

    • Strong version

      • Pitch-events are heard in a strict hierarchy (partial overlaps are forbidden).

      • Structurally less important events are not heard simply as insertions, but in a specified relationship to surrounding more important events.

    Syntax of tonal music


    Prolongational rules
    Prolongational rules

    • Tension and relaxation as fundamental processes of musical primitives of harmonic/melodic progress

    t r

    t r

    r

    r

    Syntax of tonal music


    Prolongational rules tree notation
    Prolongational rules: tree notation

    Progression Strong prolongation

    Weak prolongation

    x y x y

    x y x y

    t

    r

    t

    r

    x y x y

    t

    r

    Syntax of tonal music


    Prolongational rules1
    Prolongational rules

    • Example

    • Note: strict hierarchy forbids the passing tone from simultaneously prolonging the first and third notes, it must be dominated by one or the other!

    t r

    r

    Syntax of tonal music


    Prolongation and reduction
    Prolongation and reduction

    • All large-scale strong prolongations are right branching.

    • All large-scale weak prolongations are left branching (moving from less consonant to more consonant)

    Syntax of tonal music


    Online processing studies
    Online processing studies

    Mireille Besson, Frédérique Faïta. 1995. “An Event-Related Potential (ERP) Study of Musical Expectancy : Comparison of Musicians With Nonmusicians” J. Exp. Psych: HPP.

    Maess, B., S. Koelsch, T. Gunter, A. Friederici. 2001. “Musical syntax is processed in Broca’s area: an MEG study” Nature Neuroscience.

    Syntax of tonal music


    Maess et al 2001
    Maess, et al. 2001

    • unaltered chord progression

    • Out-of-key chord (Neopolitan 6th) at 3rd position

    • Neopolitan at 5th position

    ?

    *

    Syntax of tonal music


    Maess et al 20011
    Maess, et al. 2001

    Syntax of tonal music


    Maess et al 20012
    Maess, et al. 2001

    • “The ability to perceive distances between chords (and keys, respectively) and to expect certain harmonies (and harmonic functions) to a higher or lower degree can only rely on a representation of the principles of harmonic relatedness described by music theory. These principles, or rules, were reflected in the harmonic expectancies of listeners and may be interpreted as musical syntax.”

    • “The present results indicate that Broca’s area and its right-hemispheric homologue might also be involved in the processing of musical syntax, suggesting that these brain areas process considerably less domain-specific syntactic information than previously believed.”

    Syntax of tonal music


    Interpreting the results
    Interpreting the results

    • Origins of syntactic representations

      • Statistical distributions in input?

        • Possible, but unlikely given rampant experimentation with alternative compositional formalisms

        • Three different common continuations for the leading tone, each with very different expectations satisfied

          • The leading tone (7) is followed conventionally by the tonic (1)

          • In compound melody contexts (extremely common), it may be also followed by the a tone of the dominant chord (2, 4, or 5)

          • It may moved down to the submediant (6)

    Syntax of tonal music


    Innateness of musical syntax
    Innateness of musical syntax

    • Universals in music

      • Isochronous organization extremely common

      • Stresses tend to be heard as strong beats (stresses never are used to suggest weak beats, except to create a marked context)

      • Sensitivity to the overtone series

    • Innateness

      • Tendency to understand music as a hierarchically organized (events are subject to prolongation) is too abstract to be observable

      • Universals

    • Good example of learning without negative evidence: what could it possibly be? (No, Georgie, you didn’t hear that as a consonant passing tone!)

    Syntax of tonal music


    What can linguists take home
    What can linguists take home?

    • Major innovation of L&J: preference rules

      • Similar in structure to OT constraints

      • Find minimal cost

      • Used successfully in subsequent “cognitive” theories of music, e.g. Temperley (2001)

      • Unclear implementation/learnability

        • Computational approaches use dynamic programming

        • Temperley (2001) argues that dynamic programming provides an elegant way of describing “revision” phenomena, but does not go into any detail

    Syntax of tonal music


    What can linguists take home1
    What can linguists take home?

    • Preference rules and language (L&J)

      • Quantifier scope resolution

      • Pragmatics

        • Gricean implicature encoded as preference rules

    • Musical acquisition, unlike language acquisition fails (amusia, arhythmia)

    Syntax of tonal music


    Music vs minimalist syntax
    Music vs. minimalist syntax

    • Conspicuous lack of displacement phenomena

      • Or is there?

        • Musical statement as a kind of derivation unfolded in time

        • Unstable features implicit in initial material checked during the course of the derivation (3 – 2 – 1)

    Syntax of tonal music


    Music vs minimalist syntax1
    Music vs. minimalist syntax

    • What are the interfaces where a derivation can crash?

    • Primitive operations

      • Merge

      • Label/project

      • Differences

        • The process of merging and label seems to be interpreted always as one of dominance/subordination

    Syntax of tonal music


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    Dekelboum Concert Hall, Clarice Smith Center

    Mozart, Symphony No. 35 in D major “Haffner”

    Wagner, Siegfried Idyll

    Beethoven, Piano Concerto No. 5 “Emperor”

    with Santiago Rodriguez, piano

    Syntax of tonal music


    References
    References

    • Lehrdal & Jackendoff. 1983. A Generative Theory of Tonal Music. MIT Press.

    • Forte & Gilbert. 1982. Introduction to Schenkerian Analysis. W. W. Norton & Company.

    • Schachter, C. 1999. Unfoldings: Essays in Schenkerian Analysis

    • Temperley, D. 2001. The Cognition of Basic Musical Structures. MIT Press.

    Syntax of tonal music


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