The Syntax of Tonal Music

1. The Syntax of Tonal Music UMD Syntax LunchOct. 3, 2006

2. 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

3. The mental representation of music • What happens when we listen to music? Syntax of tonal music

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. Understanding music • Context • Depending on the surrounding music, a particular interval, pitch, or harmony can have vastly different “function” Syntax of tonal music

16. Evidence for hierarchy in music

17. 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

18. Notational conventions • Ornaments as diacritics Aria from the J.S. Bach Goldberg Variations (BWV 988) Syntax of tonal music

19. 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

20. Notational conventions • Figured bass – structurally unimportant notes were not even written! • Usually, a melody given, often used in accompaniment Syntax of tonal music

21. Notational conventions • Guitar tablature • Indicates chords, inversions • Says nothing about • Rhythm • Arpeggiation pattern to • Extremely common in jazz, pop music Syntax of tonal music

22. Schenkerian Analysis An impossibly brief introduction

23. 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

24. 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

25. 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

26. 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

27. 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

28. Prolongation examples • Take a basic melody • Certain structure-preserving transformations may be applied: Syntax of tonal music

29. 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

30. 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

31. Schenkerian analysis: example • “Hear” foreground Syntax of tonal music

32. Schenkerian analysis: example • “Hear” middleground Syntax of tonal music

33. Schenkerian analysis: example • “Hear” background Syntax of tonal music

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. Maess, et al. 2001 • unaltered chord progression • Out-of-key chord (Neopolitan 6th) at 3rd position • Neopolitan at 5th position ? * Syntax of tonal music

46. Maess, et al. 2001 Syntax of tonal music

47. 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

48. 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

49. 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

50. 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