Transcript Document

Multiple levels
of implication-realisation
at the authentic cadence
Richard Parncutt and Annemarie Seither-Preisler
University of Graz, Austria
SysMus Graz
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
14th Century Ars Nova: Vitry, Machaut

authentic cadence
15th 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)
Meyer, L. B. (1956).
Emotion and meaning in music.
Chicago: U Chicago Press.
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)
 new 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.
1.
2.
3.
4.
Rising semitone (leading tone to tonic)
If seventhtriad: tension-relaxation
Entire passage  final triad (Schenker)
Falling fifth between roots
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
initial tone
prevalence
300
final tone
250
any tone /10
200
150
100
50
0
0
1
2
3
4
5
6
7
8
9
10 11
chroma (semitones above C)
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 012 013 014 015 016 024 025 026 027 036 037 048
023 034 045 056
035 046
047
inversion
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
Eberlein, R. (1994). Die Entstehung
der tonalen Klangsyntax (pp. 422423). Frankfurt: Peter Lang.
Prevalence of diatonic progressions
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
rising falling rising falling rising falling total
P4
P4
3rd
3rd
M2
M2
maj-maj
maj-min
min-maj
min-min
total
64
60
5
19
1
20
0
2
1
0
9
15
6
5
5
2
0
3
91
77
49
21
150
5
45
0
3
0
24
1
17
0
5
27
244
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
1
2
second
fourth
third
low
high
medium
3
unison
high*
*= sustained chord
Practical constraints on common notes
Does a “progression” imply same rhythm in
each voice?  zero common notes?
 Just



 Is
one common note is better:
helps perceptual coherence
helps tuning in performance
avoids parallel fifths
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
Preferred direction
interval
Predicted
Actual
between roots
second
-*
rising
third
falling
rising
fourth
rising
rising
* BUT: 2 rising fourths + rising second = octave
Virtual objects – Virtual pitch
Virtual triangle (Kanizsa, 1955)
Reconstruction of foreground object
from elements
overtones
SPL
fundamental
(F0)
frequency
Virtual pitch (Terhardt, 1976)
Reconstruction of a missing
fundamental frequency (F0)
from harmonics
Basics of pitch perception
Things that everyone agrees about
 Pitches


individual spectral components (spectral)
harmonic patterns of components (virtual)
 Pitches

correspond either to
vary in salience
Predictions of spectral and temporal models
are about the same
Missing fundamentals in major triads
pitch
M2
relative P4
to root
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
M2
relative P4
to root
m6
m7
harmonics above pitch
that are present in the chord
P5
M3
m7
M2
P1
P1
m3
P5
m3
P1
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

in CEbG-CFAb, implications are realised
 Also


implies F and Ab
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