Pitch, tonality, and the missing fundamentals of music cognition
Pitch, tonality, and the missing fundamentals of music cognition
prolongation and the
of major-minor tonality
University of Graz, Austria
ICME3, Jyväskylä Finland, 2013
A passage of MmT* is perceived
as a prolonged background triad
This prolonged triad is the origin
of “major-minor = happy-sad”
this background triad exist?
Why major=happy and minor=sad?
Why two triads? Why these two?
Why this specific mapping?
What about Leonard Meyer’s theory?
origins of emotion
High: energy available or necessary
Low: energy unavailable or unnecessary
positive: clear situation safety, confidence
negative: unclear situation fear or anger
Psychohistoric origin of MmT
Five psychohistorical steps
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
Origin of pentatonic, diatonic,
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
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
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?
E.g. C-major triad goes better with F than F# (Parncutt, 1993)
BUT: can also be explained by musical experience
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
∆ 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
Common exceptions: #4-5-b6, #7-8-b9
This can explain why C-major & –minor include D & not Db.
Chord V is important. It must have P5 to be consonant.
Use of standard diatonic scales
of major and minor scales
1. Diatonic scales in ancient music
2. Importance of major and minor triads
3. Leading tones
4. Implied pitches in these triads
5. No consecutive semitones
Claim: The tonic in MmT is a triad
not a single tone!
Any passage in MmT is perceived
as a prolongation of its tonic triad
• Success of Schenkerian approach
• 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
Music theory: Embellishment of tonic triad
• Passing notes
• Neighbor notes
Psychological basis: Auditory scene analysis
• Pitch proximity
Pitch range of major-key melodies
1 or 7.
A classical example
10 out of 18
“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
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,
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)
tonality is more ambiguous
Two versions of scale degrees 6 and 7
Extreme example: Carmen’s Habanera
(“scandalous”: Susan McClary, 2005)
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
3. Lower sound level
Only 1 can explain origin of minor-sad
Other are consequences
Why is major happy?
(births, weddings, feasts, preparation for war,
entertainment, relaxation) not sad (funerals)
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
In everyday life and music,
Positive valence is normal
Negative valence is abnormal (emotional Other)
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,
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
A passage of MmT is perceived as a
prolonged background triad.
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