Multiple levels of implication-realisation at the authentic cadence

Richard Parncutt and Annemarie Seither-Preisler University of Graz, Austria

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

Why is musical syntax like it is?

Can we predict probability distributions of pitch-time patterns in a well-defined style? ...starting from a few “first principles”?

• perceptual, cognitive, social, historical...

Whence the authentic cadence?

 Clarification of tonic  cognitive efficiency  But why

this

harmony,

this

voice leading?

Historical account 1. Medieval 2-part cadence M6-P8 (e.g. DB-CC) 2. add a third voice  double leading-tone cadence 14 th Century Ars Nova: Vitry, Machaut  authentic cadence 15 th Century: Dunstable, Dufay, Ockeghem Why this change? Why is authentic cadence so stable?

Avoid circular arguments Explain by non-musical phenomena  mathematics of frequency ratios (Pythagoras)  psychophysics of pitch perception (Aristoxenus)

Implication-realisation theory

fulfilment of expection = realisation of implication  emotion  Example: melodic gap-fill  implication: rising leap  realisation: stepwise descent

Realised implications in tonal music  melodic gap-fill, rising leap  falling step  authentic cadence, chains of falling fifths  rising leading tones, falling appogiaturas  thematic repetition ...and people like it! (Sloboda, 1991)

Thwarted expectations

exceptional, but essential

 manipulate attention and emotion  create conflict (social metaphor)  n ew expectation of “happy end” (the norm) Examples  delayed melodic gap fill (

baa baa black sheep

)  interrupted cadences (Mozart arias)  tonic avoidance (Wagner:

Tristan

)

Authentic cadence V-I

implication-realisation



“ultimate” satisfaction?

1.

2.

3.

4.

Rising semitone (leading tone to tonic) If seventh  triad: tension-relaxation Entire passage  final triad (Schenker) Falling fifth between roots 1.

2.

3.

4.

Why do leading tones tend to

rise

by m2?

Why Mm7? Why major or minor triad?

What aspect of passage? Of final triad? Why P5? Why fall rather than rise?

1. Origin of the leading tone

Prevalence of scale steps in Gregorian chant (Parncutt & Prem, ICMPC 2008) 350 300 250 initial tone final tone any tone /10 200 150 100 50 0 0 1 2 3 4 5 6 7 8

chroma (semitones above C)

9 10 11 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?

19 Tn-types of cardinality 3

after Rahn (1980)

prime form inversion 012 013 014 015 016 024 025 026 027 036 037 048 023 034 045 056 035 046 047 Most consonant Tn-types of cardinality 3 • fourth/fifth (

fusion

) • no major/minor second (

roughness

).

3. Chroma prevalence anticipates chroma salience

major key minor key Aarden, B. (2003).

Dynamic melodic expectancy.

PhD dissertation, Ohio State University.

4. Why falling fifth between roots?

competing theories

Common notes or pitches chords 1 and 2 have something in common Root newness root of chord 2 is not a note in chord 1 Implication-realisation implied pitches* in 1  real pitches in 2

*missing fundamentals

Prevalence of diatonic progressions

J. S. Bach Händel Mozart Beethoven Mendelssohn 7 chorales;

kleine harmonische Labyrinth

Trio sonata Op. 5 No. 5

Missa brevis

KV 65 (Kyrie, Gloria, Agnus Dei) Mass in C (Kyrie, Gloria) Motets Op. 78, Nos. 1 & 2

maj-maj maj-min min-maj rising P4 falling rising falling rising falling total P4 3rd 3rd M2 M2

64 19 0 0 6 2

91

60 5 1 20 2 1 9 15 5 5 0 3

77 49 min-min total

21

150

5

45

0

3

0

24

1

17

0

5 27 244

Assumption: Asymmetry began in 15 th Century and grew

Why fourth/fifth intervals?

Common notes?

No. of common notes

0 1 2 3

Non-directional interval between roots

second fourth third unison

prevalence

low high medium high* *= sustained chord

Practical constraints on common notes Does a “progression” imply same rhythm in each voice?  zero common notes?

 Just one common note is better:    helps perceptual coherence helps tuning in performance avoids parallel fifths  Is that why one common note is preferred?

Does that in turn explain why fourth/fifths preferred?

But what about the cycle of fifths?

Neural net model

(Bharucha)

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

Psychological reality of cycle of fifths?

Interval asymmetry: Root newness e.g. dominant preparation: imply tonic without playing it  tension

Diatonic interval between roots Preferred direction

Predicted Actual second -* rising third fourth falling rising rising rising * BUT: 2 rising fourths + rising second = octave

Virtual objects – Virtual pitch

Virtual triangle (Kanizsa, 1955)

Reconstruction of foreground object from elements fundamental (F0) overtones

Virtual pitch (Terhardt, 1976)

Reconstruction of a missing fundamental frequency (F0) from harmonics frequency

Basics of pitch perception

Things that everyone agrees about

 Pitches correspond either to   individual spectral components (spectral) harmonic patterns of components (virtual)  Pitches vary in salience  Predictions of spectral and temporal models are about the same

Missing fundamentals in major triads pitch relative to root

M2 P4 M6 m7

harmonics above pitch that are present in the chord

P5 M3 m7 M2

P1 M3 P1 P5 M3 P5 P1 Rank order of salience: M6, P4, M2, m7

Missing fundamentals in minor triads pitch relative to root

M2 P4 m6 m7

harmonics above pitch that are present in the chord

P5 M3 m7 M2

P1 m3 P1 P1 m3 P5 P1 Rank order of salience: P4/m6, M2, m7

Experimental data

Parncutt, 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

Pure tone

Physical spectra

and

calculated experiential spectra

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

(Parncutt, 1989) Harmonic complex tone Octave complex tone

Minor triad

Physical spectra

and

calculated experiential spectra

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

(Parncutt, 1989) Tristan chord

Implication-realisation model of falling fifth progressions

 CEG implies F and A  in CEG-CFA, implications are realised  CEbG implies F and Ab  in CEbG-CFAb, implications are realised  Also explains falling third progressions   prevalent because of IR less prevalent than fifths because less IR

Pitch salience and common notes Consider two chords: C and Am/C Prediction: Most salient pitch in both is C  Chords with 2 common notes are not different relative major-minor (Riemann:

parallel

)  Fourth progressions > third progressions

Individual differences in pitch perception Auditory ambiguity test (Seither-Preisler)

5.- 10.

2.- 4.

1.

1.

Overtone spectrum: Elementary physical dimension Virtual pitch: Musical gestalt dimension

Pitch salience and music history

Revised thesis: Missing fundamentals influence historical development of syntax because some (not all) listeners, performers, composers perceive them

Advantages of virtual pitch approach

pitch commonality - implication-realisation

 Bottom-up: underlying scale not assumed   not circular prevalence of any chromatic progression?

 Same model explains similarity of successive tones, chords, keys  Explain perceptual coherence of progression  IR 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