Multiple levels Multiple levels of implication-realisation of implication-realisation at the authentic cadenceat the authentic cadence

Richard Parncutt and Annemarie Seither-PreislerRichard Parncutt and Annemarie Seither-Preisler

University of Graz, AustriaUniversity of Graz, Austria

Presented at: Music & Emotion, Durham, England, 31 August - 4 September 2009Presented at: Music & Emotion, Durham, England, 31 August - 4 September 2009

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Why is musical syntax like it is?Why is musical syntax like it is?

Can we predict probability distributions of Can we predict probability distributions of pitch-time patterns in a well-defined style? pitch-time patterns in a well-defined style?

...starting from a few “first principles”?...starting from a few “first principles”?• perceptual, cognitive, social, historical...perceptual, cognitive, social, historical...

Whence the authentic cadence?Whence the authentic cadence? Clarification of tonic Clarification of tonic cognitive efficiency cognitive efficiency But why But why thisthis harmony, harmony, thisthis voice leading? voice leading?

Historical accountHistorical account1. Medieval 2-part cadence M6-P8 (e.g. DB-CC)1. Medieval 2-part cadence M6-P8 (e.g. DB-CC)

2. add a third voice2. add a third voice double leading-tone cadence double leading-tone cadence

1414thth Century Ars Nova: Vitry, Machaut Century Ars Nova: Vitry, Machaut

authentic cadence authentic cadence

1515th th Century: Dunstable, Dufay, OckeghemCentury: Dunstable, Dufay, Ockeghem

Why this change? Why is authentic cadence so stable?Why this change? Why is authentic cadence so stable?

Avoid circular argumentsAvoid circular arguments

Explain by non-musical phenomenaExplain by non-musical phenomena mathematics of frequency ratios (Pythagoras)mathematics of frequency ratios (Pythagoras)

psychophysics of pitch perception (Aristoxenus)psychophysics of pitch perception (Aristoxenus)

Implication-realisation theory

fulfilment of expection

= realisation of implication

emotion

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Realised implications in tonal musicRealised implications in tonal music

melodic gap-fill, rising leapmelodic gap-fill, rising leapfalling stepfalling step authentic cadence, chains of falling fifthsauthentic cadence, chains of falling fifths rising leading tones, falling appogiaturasrising leading tones, falling appogiaturas thematic repetitionthematic repetition

...and people like it! ...and people like it! (Sloboda, 1991) (Sloboda, 1991)

Thwarted expectationsThwarted expectationsexceptional, but essentialexceptional, but essential

manipulate attention and emotion manipulate attention and emotion create conflict (social metaphor)create conflict (social metaphor)

new expectation of “happy end” (the norm)new expectation of “happy end” (the norm)

ExamplesExamples delayed melodic gap fill delayed melodic gap fill ((baa baa black sheepbaa baa black sheep)) interrupted cadences interrupted cadences (Mozart arias)(Mozart arias) tonic avoidance tonic avoidance (Wagner: (Wagner: TristanTristan))

Authentic cadence V-IAuthentic cadence V-Iimplication-realisation implication-realisation “ultimate” satisfaction? “ultimate” satisfaction?

1.1. Rising semitone (leading tone to tonic)Rising semitone (leading tone to tonic)

2.2. If seventhIf seventhtriad: tension-relaxationtriad: tension-relaxation

3.3. Entire passage Entire passage final triad (Schenker) final triad (Schenker)

4.4. Falling fifth between rootsFalling fifth between roots

1.1. Why do leading tones tend to Why do leading tones tend to riserise by m2? by m2?

2.2. Why Mm7? Why major or minor triad?Why Mm7? Why major or minor triad?

3.3. What aspect of passage? Of final triad? What aspect of passage? Of final triad?

4.4. Why P5? Why fall rather than rise? Why P5? Why fall rather than rise?

1. Origin of the leading tone1. Origin of the leading tonePrevalence of scale steps in Gregorian chant Prevalence of scale steps in Gregorian chant

(Parncutt & Prem, ICMPC 2008)(Parncutt & Prem, ICMPC 2008)

Most prevalent: G and D. C>B, F>E (exception: E as final)Theory: tones are preferred if their harmonics are in diatonic scale

2. Why major and minor triads?2. Why major and minor triads?

19 Tn-types of cardinality 319 Tn-types of cardinality 3after Rahn (1980)after Rahn (1980)

prime form 012 013 014 015 016 024 025 026 027 036 037 048

inversion 023 034 045 056 035 046 047

Most consonant Tn-types of cardinality 3• fourth/fifth (fusion)

• no major/minor second (roughness).

3. Chroma prevalence anticipates 3. Chroma prevalence anticipates chroma saliencechroma salience

Aarden, B. (2003). Dynamic melodic expectancy. PhD dissertation, Ohio State University.

major key minor key

4. Why falling fifth between roots? 4. Why falling fifth between roots?

competing theoriescompeting theories

Common notes or pitchesCommon notes or pitcheschords 1 and 2 have something in commonchords 1 and 2 have something in common

Root newnessRoot newnessroot of chord 2 is not a note in chord 1root of chord 2 is not a note in chord 1

Implication-realisationImplication-realisationimplied pitches* in 1 implied pitches* in 1 real pitches in 2 real pitches in 2

*missing fundamentals*missing fundamentals

Prevalence of diatonic progressions

rising P4

falling P4

rising 3rd

falling 3rd

rising M2

falling M2

total

maj-maj 64 19 0 0 6 2 91maj-min 60 1 2 9 5 0 77min-maj 5 20 1 15 5 3 49

min-min 21 5 0 0 1 0 27

total 150 45 3 24 17 5 244

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J. S. Bach 7 chorales; kleine harmonische Labyrinth

Händel Trio sonata Op. 5 No. 5

Mozart Missa brevis KV 65 (Kyrie, Gloria, Agnus Dei)

Beethoven Mass in C (Kyrie, Gloria)

Mendelssohn Motets Op. 78, Nos. 1 & 2

Assumption: Asymmetry began in 15th Century and grew

Why fourth/fifth intervals?Common notes?

No. of common notes

Non-directional interval between roots

prevalence

0 second low

1 fourth high

2 third medium

3 unison high*

*= sustained chord

Practical constraints on common notesPractical constraints on common notes

Does a “progression” imply same rhythm in Does a “progression” imply same rhythm in each voice? each voice? zero common notes? zero common notes?

Just one common note is better:Just one common note is better: helps perceptual coherencehelps perceptual coherence helps tuning in performancehelps tuning in performance avoids parallel fifthsavoids parallel fifths

Is that why one common note is preferred?Is that why one common note is preferred?Does that in turn explain why fourth/fifths preferred?Does that in turn explain why fourth/fifths preferred?

But what about the cycle of fifths?But what about the cycle of fifths?

Neural net model Neural net model (Bharucha)(Bharucha)

Spontaneous emergence of cycle of fifths from exposure to triads or tonal music?Psychological reality of cycle of fifths?

Interval asymmetry: Root newnessInterval asymmetry: Root newnesse.g. dominant preparation: imply tonic without playing it e.g. dominant preparation: imply tonic without playing it tension tension

Diatonic interval between roots

Preferred direction

Predicted Actual

second -* rising

third falling rising

fourth rising rising

* BUT: 2 rising fourths + rising second = octave

Virtual triangle (Kanizsa, 1955)Reconstruction of foreground object

from elements

Virtual pitch (Terhardt, 1976)Reconstruction of a missing fundamental frequency (F0)

from harmonics

fundamental (F0) overtones

frequency

Virtual objects – Virtual pitchS

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Basics of pitch perceptionBasics of pitch perceptionThings that everyone agrees aboutThings that everyone agrees about

Pitches correspond either toPitches correspond either to individual spectral components (spectral)individual spectral components (spectral) harmonic patterns of components (virtual)harmonic patterns of components (virtual)

Pitches vary in saliencePitches vary in salience Predictions of spectral and temporal models Predictions of spectral and temporal models

are about the sameare about the same

Missing fundamentals in major triadsMissing fundamentals in major triads

harmonics above pitch that are present in the chord

P5 M3 m7 M2

pitch relative to root

M2 - - P1 M3

P4 P1 - - P5

M6 M3 - P5 -

m7 - - - P1

Rank order of salience: M6, P4, M2, m7

Missing fundamentals in minor triadsMissing fundamentals in minor triads

harmonics above pitch that are present in the chord

P5 M3 m7 M2

pitch relative to root

M2 - - P1 -

P4 P1 - m3 P5

m6 m3 P1 - -

m7 - - - P1

Rank order of salience: P4/m6, M2, m7

Experimental dataExperimental dataParncutt, 1993Parncutt, 1993

Stimuli in one trial:A chord of OCTs,then a single OCT

Listeners rate how well tone follows chord

Diamonds: Mean ratings

Squares : Theoretical predictions

Physical Physical spectra spectra

and and

calculated calculated experiential experiential

spectraspectra

“Pitch category”: “Pitch category”: 48 = C4, 60 = C5 etc.48 = C4, 60 = C5 etc.

(Parncutt, 1989)(Parncutt, 1989)

Pure tone

Harmonic complex tone

Octave complex tone

Physical Physical spectra spectra

and and

calculated calculated experiential experiential

spectraspectra

“Pitch category”: “Pitch category”: 48 = C4, 60 = C5 etc.48 = C4, 60 = C5 etc.

(Parncutt, 1989)(Parncutt, 1989)

Minor triad

Tristan chord

Implication-realisation model of Implication-realisation model of falling fifth progressionsfalling fifth progressions

CEG implies F and ACEG implies F and A in CEG-CFA, implications are realisedin CEG-CFA, implications are realised

CEbG implies F and AbCEbG implies F and Ab in CEbG-CFAb, implications are realisedin CEbG-CFAb, implications are realised

Also explains falling third progressionsAlso explains falling third progressions prevalent because of IRprevalent because of IR less prevalent than fifths because less IRless prevalent than fifths because less IR

Pitch salience and common notesPitch salience and common notes

Consider two chords: C and Am/C Consider two chords: C and Am/C

Prediction: Most salient pitch in both is CPrediction: Most salient pitch in both is C

Chords with 2 common notes are not differentChords with 2 common notes are not differentrelative major-minor (Riemann: relative major-minor (Riemann: parallelparallel))

Fourth progressions > third progressionsFourth progressions > third progressions

Overtone spectrum: Elementary physical dimension

Virtual pitch:

Musical gestalt dimension

2.- 4.

5.- 10.

1.

1.

Individual differences in pitch perceptionAuditory ambiguity test (Seither-Preisler)

Pitch salience and music historyPitch salience and music history

Revised thesis: Revised thesis: Missing fundamentals influence historical development of Missing fundamentals influence historical development of

syntax because syntax because some (not all)some (not all) listeners, performers, listeners, performers, composers perceive themcomposers perceive them

Advantages of virtual pitch approach Advantages of virtual pitch approach pitch commonality - implication-realisationpitch commonality - implication-realisation

Bottom-up: underlying scale not assumedBottom-up: underlying scale not assumed not circularnot circular prevalence of any chromatic progression?prevalence of any chromatic progression?

Same model explains similarity of successive Same model explains similarity of successive tones, chords, keystones, chords, keys

Explain perceptual coherence of progressionExplain perceptual coherence of progression

IR explains why progressions are emotionalIR explains why progressions are emotional

Why is authentic cadence based on a falling fifth between roots?

Why fifth interval? pitch commonality perceptual coherence

one common note feeling of progression

two implied pitches are realised

Why falling? Harmonic aspect: root newness or IR

Melodic aspect: leading tone rises