Pitch, tonality, and the missing fundamentals of music cognition

Report
Schenkerien
prolongation and the
emotional connotations
of major-minor tonality
Richard Parncutt
University of Graz, Austria
SysMus Graz
ICME3, Jyväskylä Finland, 2013
Theses
1.
A passage of MmT* is perceived
as a prolonged background triad
2.
This prolonged triad is the origin
of “major-minor = happy-sad”
*major-minor tonality
Questions arising
 Does
this background triad exist?
 Why major=happy and minor=sad?
More questions…
 Why two triads? Why these two?
 Why this specific mapping?
 What about Leonard Meyer’s theory?
Origins of…
…emotion:
evolutionary psychology?
…MmT:
psychohistory?
Evolutionary functional
origins of emotion
Arousal
 High: energy available or necessary
 Low: energy unavailable or unnecessary
Valence
 positive: clear situation  safety, confidence
 negative: unclear situation  fear or anger
Psychohistoric origin of MmT
Five psychohistorical steps
1.
2.
3.
4.
5.
Diatonic scales in ancient music
Leading tones in medieval polyphony
Importance of major and minor triads
Implied pitches in these triads
No consecutive semitones in scales
Major-minor Tonality (MmT)
Most music in MmT can be reduced to chord progressions.
Most chords correspond to one of these diatonic triads.
Pairs of harmonic complex
tones with frequencies or
pitches in common
 Octave and fifth relations
 Circle of fifths
Harmonics in common  perceptual similarity
ethanhein.com
Origin of pentatonic, diatonic,
chromatic scales
Leading tones in early music
Guillaume de Machaut (1300-1377). Rondeau Ma fin est mon commencement
F# tonicizes G - C# tonicizes D
Origin of leading tones?
Consonance / prevalence of individual tones in chant
70000
5
Data
60000
Model
4
50000
3
40000
30000
2
20000
1
10000
0
0
1
A
2
B
3
C
4
D
5
E
6
F
7
G
1
A
2
B
3
C
4
D
5
E
F6
G7
Data: Counted in the Liber Usualis (DDMAL, Fujiniaga et al., McGill)
Model: No. of harmonic pcs (P5, M3, m7, M2) at diatonic scale steps
Comparison: df = 5, r = 0.90, p<.01
Finding: C (F) more prevalent than B (E)  origin of leading tone?
Cf. Parncutt, R. & Prem, D. (2008 ). The relative prevalence of Medieval modes
and the origin of the leading tone (poster). International Conference on Music
Perception and Cognition (ICMPC10), Sapporo, Japan, 25-29 August.
Based on Bryden, & Hughes (1969). An index of Gregorian chant.
Origin of major and minor triads
The most consonant sets of 3 pitch classes
• Include P5  high harmonicity
• No m2 or M2  low roughness
(Parncutt, 1988)
Only two chords satisfy this constraint!
 Major and minor triads dominate MmT
 Our attention is drawn to their difference
Missing fundamentals in triads
(Parncutt, 1988; Terhardt, 1982)
Chord CEG has missing fundamentals at A, F
Chord CEbG has missing fundamentals at F, Ab
Why? e.g. E and G are harmonics of A
Mistuning of a ¼ - ½ tone? No problem in pitch perception
Do these missing fundamentals exist?
 Empirical evidence



E.g. C-major triad goes better with F than F# (Parncutt, 1993)
BUT: can also be explained by musical experience
Logical argument


Brain tries to locate fundamentals in ambiguous sounds
Perception of missing fundamentals is inevitable
Origin of major and minor scales
Compatible with major and minor triads (tonic triads)
 Krumhansl’s key profiles as pitch salience profiles of tonic triads
Parncutt, 2011
∆ Krumhansl’s key profiles
▀ calc. pitch salience in tonic triad
evidence that tonic in MmT is a triad, not a tone
Avoiding consecutive semitones
If there are consecutive semitones in a melody,
 middle tone perceived as passing
 middle tone not a scale step
 no consecutive semitones in (jazz) scales
Pressing (1981)
Common exceptions: #4-5-b6, #7-8-b9
This can explain why C-major & –minor include D & not Db.
Alternative explanations:


Chord V is important. It must have P5 to be consonant.
Use of standard diatonic scales
Psychohistoric origin
of major and minor scales
1. Diatonic scales in ancient music
2. Importance of major and minor triads
 CEG, CEbG
3. Leading tones
 CEGB, CEbGB
4. Implied pitches in these triads
 CEFGAB, CEbFGAbB
5. No consecutive semitones
 CDEFGAB, CDEbFGAbB
Claim: The tonic in MmT is a triad
not a single tone!
Corollary:

Any passage in MmT is perceived
as a prolongation of its tonic triad
Evidence:

Qualitative
• Success of Schenkerian approach

Quantitative
• Correlation between pitch salience in tonic
triad and stability in scale
• Transition probabilities in melodies
Transition probabilities between
scale steps in major-mode melodies
(Huron, 2006, 2012)
Why is the
transition
between
scale
steps 6
and 7
avoided?
“Huron’s stereotype”
5
3
1
7
1
2
3
4
5
Music theory: Embellishment of tonic triad
• Arpeggiation
• Passing notes
• Neighbor notes
Psychological basis: Auditory scene analysis
• Harmonicity
• Pitch proximity
6
Pitch range of major-key melodies
The
lowest
scale
degree
is often
1 or 7.
The
highest
is often
6.
A classical example
Opening
themes of
10 out of 18
Mozart
piano
sonatas
conform to
Huron‘s
stereotype:
KV 279,
280, 281,
283, 284,
310, 331,
332, 333,
545
Schenker’s Ursatz
“Great music” (Bach, Beethoven, Brahms…) is
a prolongation (elaboration, embellishment,
Auskomponierung, Auswicklung) of the Ursatz.



The Ursatz (background) prolongs the tonic triad.
The Ursatz is elaborated  middleground
The middleground is elaborated  foreground
 The whole piece prolongs the tonic triad
Prolongation of prolongation?
 Not only “great” music
Is all music in Mmt is a prolongation of the tonic triad?
 Not only music analysis
Does chord prolongation have a psychological basis?
Major-minor and valence
some basic facts

Positive valence


Negative valence


sadness, anger, fear, tension, solemnity, lament, tragedy, pathos,
mourning, melancholy, frustration, depression, gloom…
Major-minor effect can be overridden by tempo


happiness, contentment, serenity, grace, tenderness, elation, joy,
victory, majesty…
E.g. fast, happy minor or slow, sad major
Applies to tonalities not individual chords


A major triad in a minor key can sound sad
A minor triad in a major key can sound happy
Leonard B. Meyer
Emotion and meaning in music (1956)
 Minor


tonality is more ambiguous
Two versions of scale degrees 6 and 7
Extreme example: Carmen’s Habanera
(“scandalous”: Susan McClary, 2005)
 Uncertainty

 negative emotion
measured by information theory
Sad speech and music
Huron‘s approach. Some salient stuctural features:
Mean pitch is lower than average
2. Smaller pitch intervals between
successive syllables
3. Lower sound level
4. Slower
1.
My hypothesis:
 Only 1 can explain origin of minor-sad
 Other are consequences
Why is major happy?
 Most

(births, weddings, feasts, preparation for war,
entertainment, relaxation) not sad (funerals)
 Most

music is happy
music is major
Major keys/chord more common than minor
• Modulation to relative major; major dominant triad
• Reason: more consonant? (higher harmonicity, less ambiguity)


Minor keys  the tonal Other
Unhappy music  the Other musical emotion
Analogous asymmetries
In everyday life and music,

Positive valence is normal
 Negative valence is abnormal (emotional Other)
In music,

Consonance is normal
 Dissonance is abnormal (needs resolution)
Consonance includes smoothness, harmonicity,
diatonicity, familiarity; major is more consonant
Why is minor sad?
Two theories – competing or complementary?
1. Structural ambiguity (LB Meyer)
Minor: root of tonic and tonic of scale are more ambiguous
Is this dynamic (higher-arousal) negativity? afraid, angry, tense,
frustrated…

2. Expected pitch in speech (Huron)
Minor: 3rd and 6th scale degrees are lower than expected
Is this static (lower-arousal) negativity? sad, solemn, gloomy,
melancholy, depressed, lamenting, mourning, tragic, pathetic…

In both cases,


association between music and situation/speech
role of duration & prolongation: keys not chords
Theses
1.
A passage of MmT is perceived as a
prolonged background triad.
2.
The prolonged triad is the ultimate origin
of “major-minor = happy-sad”.
Read all about it!
Latest issue of Empirical Musicology Review
Special thanks to David Huron and Matthew Davis for their
study on mean interval size, which inspired this study

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