Transcript Pitch, tonality, and the missing fundamentals of music

Pitch, tonality, and the
missing fundamentals
of music cognition
Richard Parncutt
University of Graz, Austria
BRAMS, Université de Montréal
31 May 2012
SysMus Graz
This file has been revised after
discussion and questions
Abstract
What are the psychological foundations of major-minor tonality? Psychologists have
explored how modern listeners perceive its pitch structures, but the psychohistorical origins
of those structures remain unclear. A plausible theory should be able to predict tonal styles
as probability distributions of pitch-time patterns on the basis of a limited number of
psychologically and historically plausible axioms. From a psychological viewpoint, such
axioms should refer to pitches that are perceived (experienced) by audiences and performer
- not pitches notated in scores. Non-notated pitches may include prominent partials, missing
fundamentals, or pitches expected on the basis of short- or long-term experience (e.g.
melodic continuations). Consider a simple example that ignores octave register and tuning.
A C-major triad may have a missing fundamental at A, because E corresponds to the 3rd
harmonic of A and G to the 7th. Other possible missing fundamentals are F and D. The
same chord may have a prominent partial at B, if the 3rd harmonic of E and the 5th
harmonic of G coincide; another prominent partial may be at D. A systematic approach
should consider all such possibilities in a chord’s spectrum, weighting them relative to each
other. Predicted pitches and weights should be consistent with empirical data. But the
psychological reality of non-notated pitches remains unclear because “nature” (predictions
based on psychoacoustics or physiology) and “nurture” (predictions based on musical
experience) are often quantitatively similar. I will present recent data and plans for future
work to separate nature from nurture by systematically manipulating musical expertise,
cultural background, sound type, tone type, onset synchrony, duration, tuning and
background noise. Further strategies include separation of “fundamental listeners” (sensitive
to missing fundamentals) from “spectral listeners” (sensitive to prominent partials), and
modeling musical experience by statistical analysis of symbolic music databases.
Origin of major-minor tonal system
Scientific approach:
Psychologically predetermined?
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Underlying principles?
Why those pc-sets? voicings? progressions?
Can we model frequency of occurrence?
Humanities approach:
Historical accident? If so:
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Why so widespread?
Why so stable?
Assumption
The major-minor system is based on
pitch as subjective experience
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not as physical measurement (frequency)
not as physiological correlate
not as notation in musical scores
.
Thesis
To understand the major-minor system,
we must systematically investigate pitch
as experienced by musicians and
listeners in musical contexts.
Does experience exist?
Visual experience is quite different from

physical world
 info on the retina (upside down, moving)
 neurophysiology of the visual cortex
Visual experience is constructed
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available info is generally incomplete
focus on affordances (survival and reproduction)
Correlates of the color red ≠ red itself
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light wavelengths
physiology of retina
physiology of visual cortex
To study “red”, we must separate experience & physics
Is everything physical?
Modern science is atheist - ok
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Good arguments against existence of gods and spirits
Conscious experience is something else!
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Different from gods/spirits AND brain substrates
Emerges in infancy, disappears when we die
Foundation of arts and aesthetics
The solution: Epiphenomenalism
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Experience is a byproduct of neural substrates
 Both experience and its substrates exist
 Two sides of the same coin, paradoxically inseparable
 Consistent with both neuroscience and philosophy
What is “more real”?
Objective answer: The physical world
It exists without experience - but not vice-versa
Existence of experience depends on physical world
Subjective answer: Experience
Without it we would know nothing (not be human)
Existence of physical world depends on experience
(“Objective”: subject ≠ object, “Subjective”: s=o)
Conclusion
No idea. Can’t compare totally different things
Why scientists reject experience
and why some humanities scholars reject it too
Scientific belief system
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Success of modern physics
 In inherent superiority of objectivity
 Reductionism (belief in simple explanations)
 Grouping of mind-body dualism with theism
Humanities-science conflict
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“Othering” humanities to construct own identity
Refusal to accept own subjectivity (fear?)
Competitive neoliberal research structures
Scientists too arrogant, insecure or busy for philosophy
Three musical representations
and aspects of musical pitch structure you can explain with them
1. Physical: Frequencies and amplitudes
 Room and instrument acoustics, roughness
2. Experiential: Pitches and their salience
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Timbre, fusion, chord roots, harmonic function, harmonic tonality
3. Abstract: Notes in musical scores
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Performance, composition
The “three worlds” of Karl Popper
The broader context of music representations (not “worlds”)
1. Physical
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environment, body, brain
2. Experiential
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sensations, emotions
3. Abstract
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knowledge, info, culture
Assumption:
A clear separation of
3 representations can
clarify discussions of
nature and origin of
• musical structure
• human consciousness
Literature on ecological and evolutionary
psychology versus consciousness &
subjective experience
Gallagher, S. & Zahavi, D. (2010). Phenomenological Approaches
to Self-Consciousness. Stanford Encyclopedia of Philosophy
(online)
Gulick, R. van (2004). Consciousness. Stanford Encyclopedia of
Philosophy (online)
Miller, G. (2007). Reconciling Evolutionary Psychology and
Ecological Psychology: How to Perceive Fitness Affordances.
Acta Psychologica Sinica 39, 546-555.
What I mean by “pitch”
Subjective experience – like the color red
 One-dimensional
 Property of pure/complex tones, noise (+tinnitus)
 May be ambiguous and multiple
 Depends on listener, temporal context
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Here:
In music theory:
pitch = perceived pitch
pitch = notated pitch
What I mean by “chroma”

Octave-generalised perceived pitch
 not D4 or D5 - just D
 Like pitch class, but experienced – not notated
Tone types
 Pure
tone
sinusoidal function of air
pressure against time
 Complex
tone
simultaneity of pure tones in any
frequency relationship
 Harmonic
complex
tone (HCT)
Complex tone whose frequencies
correspond to a harmonic series
The harmonic series
• equally spaced on a linear frequency scale (e.g in Hz)
• unequally on a log frequency scale (e.g. in semitones)
Compared to 12-tone equal temperament:
• 7th harmonic is 1/3 semitone flatter than a m7 above 4th
• 11th harmonic is midway between P4 and TT above 8th
Spectral versus virtual pitch
Pitch perception according to Terhardt
Spectral pitch (SP)
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youtube
church
bells
pitch of a pure tone
pitch of an audible partial of a complex tone
hum tone of a church bell (1s after hammer)
Virtual pitch (VP)
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pitch of a complex tone
most consciously noticed pitches in everyday life
strike tone of a church bell (hammer hitting bell)
pitch at missing fundamental (e.g. voice on telephone)
Spectral versus virtual pitch
This distinction is
 ecological
based on interaction with the environment
 not
physiological
based on peripheral and central processing
The ultimate aim is psychophysical:
understand the relationship between
sound and experience
What about neurophysiology?
We don’t know the functional relationship between
neural states and processes and
conscious experience

Unique nature of this problem!
Never solved (or did I miss the news?)
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Enormous no. of neurons and connections!
Which states/events correspond to experience?
Spectral vs temporal processing
Along auditory pathways, we find both
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temporal representations (phase locking)
spectral representations (tonotopic structures)
Assumptions
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Both are used by neural networks
 Both are inextricable in hidden layers
Conclusion
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Doesn’t help us understand pitch as experience
Bharucha, 1987
Neural processing of pitch
in music and speech
The same neural net can process…
• spectral and temporal patterns
• pitch in speech and music
Virtual objects in vision and hearing
Gestalt principle of closure – filling the gaps in a familiar pattern
Virtual object
(Kanizsa, 1955)
Incomplete triangle
Completed by virtual contours
overtones
Auditory image
(Bregman, 1990; McAdams, 1984)
SPL
missing
fundamental
(f0)
Incomplete harmonic series
Completed by virtual pitch
frequency
Pitch perception: Experimental method
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Listener adjusts frequency of pure tone until the two
sounds have the same pitch
Frequency of pure tone is a measure of pitch of test sound
Results must be consistent within and between listeners
Pitch salience =
probability of matching
Pitch ambiguity
Assumption: The pitch of a pure tone
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is unambiguous
corresponds to frequency (if SPL constant)
Result: The pitch of a complex tone
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is ambiguous
= different pitch in different presentations
and/or multiple
= several pitches perceived simultaneously
 Can explain a lot about musical structure
Pitch salience
 In
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 In
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musical practice:
Pitched versus unpitched percussion
How clear is pitch on a continuous scale?
experimental data:
Probability of noticing a pitch
Subjective clarity of a pitch
 Depends
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high pitch salience
on:
Stimulus (esp. spectrum)
Listener (“spectral” vs “fundamental”)
Temporal context (proximity  expectation)
low pitch salience
Analytic versus holistic perception
You can consciously switch
between two modes
o analytic (strange black shapes)
o holistic (“FLY” in white letters)
Similarly for pitch?
Individual differences in pitch perception
Auditory ambiguity test (Seither-Preisler)
Individual differences  “fundamental listeners” and “spectral listeners”
Auditory Ambiguity Test (AAT)
Seither-Preisler et al. (2007)
You will hear 10 tone pairs
In each pair, does the pitch rise or fall?
Write your answers as arrows:
↑ pitch rises
↓ pitch falls
If you wrote this, you are a “fundamental listener“
If the opposite, you are an “overtone listener”
You may also be a “mixed listener”
Listening strategy
depends on music
experience and
instrument
Research idea:
Study relation to
amusia?
fundamental
listeners
Schneider et al., NY Acad Sci, Vol. 1060, p. 387-395 (2005)
Finding:
overtone
listeners
Pitch dominance regions
Octave register
(piano keyboard)
1
2
3
Salient spectral pitch
(spectral dominance)
Salient virtual pitch
(musical practice)
4
5
f1
6
7
8
f2
middle C
Spectral pitch
Virtual pitch
According to experimental data,
SP salience is highest at F5 (C4-C8).
 speech intelligibility & formants:
f1 ~ 500 Hz ~ C5, f2 ~ 1500 Hz ~ G6
According to model predictions,
VP salience is highest at D4 (C2-C6).
 f0 range of voice and music
Dominance region of spectral pitch
origin: speech perception
after Terhardt et al., 1982
centre at 700 Hz, central band at 300-2000 Hz
Calculated VP salience distribution
After Huron & Parncutt (unpublished)
f0 range of speech and music
Origin of virtual pitch
a bit of history
Before the 1970s many assumed...
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low pitch = combination tone = distortion product
peripheral origin (basilar membrane)
In the 1970s it became clear...
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pitch perception = pattern recognition
mixture of spectral and temporal processing
central origin (brain)
Perception of complex tones
Two separable stages
1. Auditory spectral analysis
 c. 16 audible* or 8 resolvable* harmonics
2. Holistic perception
 (Virtual) pitch, timbre, loudness
*Audible: If you change it, the listener hears something
*Resolvable: Listener can focus attention on it
1
2
Did you hear a bee buzzing in your ear?
trials and tribulations of recorder ensemble performance
?
Combination tones become audible:
• high frequencies, high amplitudes
• little low-frequency masking
Origin: Non-linear distortion in inner ear
Perceptual fusion of HCTs
depends on:
 Tuning
of partials
Mistuning of <1 semitone from harmonic series
 Relative
amplitude of partials
Is spectral envelope like a typical environmental sound?
 Temporal
context
Preceding/following tones can attract attention
 Listener
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Fusion more likely for “holistic” or “fundamental” listeners
Pitch at the missing fundamental
ASA Auditory Demonstrations CD (Houtsma, Rossing, Wagenaars), track 37
2
amplitude
1
Conclusions:
1
1. Pitch does not necessarily
correspond to a partial
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
frequency (kHz)
amplitude
2
2. Pitch is multiple/ambiguous
2
• VP at missing fundamental
• SP at lowest partial
1
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
frequency (kHz)
2
2
2
1
0
0,2
0,4
0,6
0,8
1
1,2
1,4
frequency (kHz)
1,6
1,8
2
5
amplitude
4
amplitude
amplitude
3
1
1
0
0
0,2
0,4
0,6
0,8
1
1,2
1,4
frequency (kHz)
1,6
1,8
2
0,2
0,4
0,6
0,8
1
1,2
1,4
frequency (kHz)
1,6
1,8
2
Sound demo: Masking SP and VP
ASA Auditory Demonstrations CD (Houtsma, Rossing, Wagenaars)
2nd tone in pair
1,2
1,2
1
1
0,8
0,8
amplitude
amplitude
track 40
1st tone in pair
0,6
0,4
0,2
0,6
0,4
0,2
0
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
0,2
0,4
0,6
0,8
frequency (kHz)
1,2
1,4
1,6
1,8
2
1,2
1
1
0,8
0,8
amplitude
41
amplitude
1,2
0,6
0,4
0,2
• Masking is
peripheral
0,6
0,4
0,2
0
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
0,2
0,4
0,6
0,8
frequency (kHz)
1
1,2
1,4
1,6
1,8
2
• Pitch pattern
recognition is
central
frequency (kHz)
1,2
1,2
1
1
0,8
0,8
amplitude
42
amplitude
1
frequency (kHz)
0,6
0,4
0,6
0,4
0,2
0,2
0
0
0,2
0,4
0,6
0,8
1
1,2
frequency (kHz)
1,4
1,6
1,8
2
Conclusion:
Masking and
pitch pattern
recognition
happen in
different
places
0,2
0,4
0,6
0,8
1
1,2
frequency (kHz)
1,4
1,6
1,8
2
Relation between VP and SP pattern
ASA Auditory Demonstrations CD (Houtsma, Rossing, Wagenaars), Track 39
2
amplitude
amplitude
2
1
1
0
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
0,25
2
0,65
0,85
1,05
1,25
1,45
1,65
1,85
2,05
frequency (kHz)
frequency (kHz)
SP3
(Hz)
0,45
Demo SP1
no.
(Hz)
SP2
(Hz)
VP
(Hz)
1
800
1000 1200 200
2
850
1050 1250 210
VP corresponds to:
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best-fit subharmonic of all
partials
NOT frequency difference
small mistuning is no problem
General relation between SP and VP
1. VP lies at fundamental of audible harmonic pattern
2. VP salience depends on SPs at harmonic positions
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how many there are (the more the better)
their salience (the greater the better)
their tuning (mistuning up to a semitone)
their effective harmonic numbers (the lower the better)
Prevalence of individual tones
(scale steps) in chant
70000
60000
no. of
notes
counted
50000
40000
30000
20000
10000
0
1
A
2
B
3
C
4
D
5
E
6
F
7
G
Source: Liber Usualis
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1,900 pages; most versions of ordinary chants for the catholic mass
first edited in 1896 by Solesmes abbot Dom André Mocquereau
Online search by CIRMMT: DDMAL (Ichiro Fujinaga and team)
Prevalence of individual tones
(scale steps) in chant
70000
60000
no. of
notes
counted
50000
40000
30000
20000
10000
0
1
A
2
B
3
C
4
D
5
E
6
F
7
G
Source: Liber Usualis

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1,900 pages; most versions of ordinary chants for the catholic mass
first edited in 1896 by Solesmes abbot Dom André Mocquereau
Online search by CIRMMT: DDMAL (Ichiro Fujinaga and team)
Accidentals are ignored, but less than 1% of Bs are B=-flats
Prevalence of individual tones
(scale steps) in chant
How can we explain the distribution?

Musical structure depends on non-notated chroma
This is just one example
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Listeners have a “feel” for pitches of harmonics
Or at least spectral listeners do
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Tones are preferred if consonant with context
An example of pitches in common (“pitch commonality”)
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Up to ten harmonics are audible (resolvable?)
Almost no masking from other sounds
Prevalence of individual tones
(scale steps) in chant
1. “Octave-generalise” the harmonic series
Harmonic no.
1, 2, 4, 8
3, 6
5, 10
7
9
Interval
P1, P8…
P5, P12…
M3…
m7…
M2, M9…
2. How many “octave-generalised overtones”
correspond to diatonic scale?
Scale step
A
B
C
D
E
F
G
No. of harmonics
3
1
3
3
2
3
4
Prevalence of individual tones
(scale steps) in chant
Data
Model
5
70000
60000
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
F
6
G
7
df = 5, r = 0.90, p<.01
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.
Guillaume de Machaut (1300-1377)
Rondeau Ma fin est mon commencement
What is the origin of (rising) leading tones?
Why do rising semitones “tonicize”?
This is not a popular theory!
Music psychologists:
No “cognitive structures”
 Empirical evidence is unclear
(BUT: consistent with statistical learning)
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Psychoacousticians and
neuroscientists:
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Focuses on subjective experience
Avoids temporal-spectral debate
Music theorists:
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Challenges primacy of musical score
Focuses on tonality (not “modernist”)
Music historians:
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Not based on historic sources
Ignores historic mode classification
Contradicts…
• physical
monism
• established
research
paradigms in
sciences and
humanities
Non-notated chroma in triads
An example of looking carefully at the stimulus (for a change)
1. Spectral synthesis
Build a C major triad from
 first 10 harmonics of C4 (up to E7)
 harmonics of E4 and G4 (up to F#7)
Assume chromatic categorical perception
2. Masking
Assume all partials are equally audible
except inside a chromatic cluster
3. Pitch pattern recognition
At each chromatic scale step:
 Which harmonics are present in chord?
 Synthesize that tone using “SFS Esynth”
C4 E4 G4
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C4E4G4
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C4
C#4 D4
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C7
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G#4 A4
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F#4 G4
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F4
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D#4 E4
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C5
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C4
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C4 E4 G4
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C4E4G4
C3 C#3 D3
D#3 E3 F3 F#3 G3
G#3 A3 A#3 B3
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C3
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Estimating virtual pitch salience
Compromise between


simplicity (parsimony, falsifiability)
accuracy (accounting for all factors)
First approximation

Count the audible harmonics above any pitch (next slide)
Second approximation

Weight each harmonic 1/n, then add weights (slide after that)
Closer approximations



Estimate audibility of partial, normalise salience (Parncutt, 1989)
Consider tuning of partials (Terhardt et al., 1982)
Consider spectral dominance region (Terhardt et al., 1982)
Estimating virtual pitch salience
of pitches within triad C4E4G4
First approximation: number of audible partials
12
10
8
6
4
2
0
1
C3
2
3
D3
4
5
6
E3
F3
7
8
G3
9
10
A3
Predictions
• C3 > E3 and G3
• In both registers, D > C# & D#
• In register 3, A # > B
11
12
13
B3
C4
14
15
D4
16
17
18
E4
F4
19
20
G4
21
22
A4
23
24
B4
Note: Here,
C4 > E4 > G4
is an artefact of
a simple model
Estimating virtual pitch salience
of pitches within triad C4E4G4
2nd approx: Weight each partial 1/n, add weights
350
300
250
200
150
100
50
0
1
C3
2
3
D3
4
5
6
E3
F3
7
8
G3
9
10
A3
11
12 C4
13
B3
14
15
D4
16
17 F4
18
E4
19
Predictions
In both registers, C > E and G, D > C# & D#, F>F#, A>G#
B versus A #: different depending on register
20
G4
21
22
A4
23
24
B4
Estimating virtual pitch salience
of pitches within triad C4Eb4G4 (C minor)
3rd approximation (Parncutt, 1989)
(i) physical representation
(ii) experiential representation
audible partials
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
(masking + pattern rec.)
Gottfried Reichweger
Diplomarbeit Uni Graz 2010
Participants
20 active musicians
Sounds
Test sounds: chords of natural piano tones
Reference tones: octave-complex (Shepard)
Task
How well does the tone go with the chord?
7-point scale
Gottfried Reichweger
Diplomarbeit Uni Graz 2010
Minor triad
Major triad
Root position
1st inversion
2nd inversion
Similarity judgments of
successive tones (Parncutt, 1989)
Effect at octave is greater:
…for complex tones
Evidence for “nature”
…for musicians
Evidence for “nurture”
…for rising complex tones
and falling pure tones
Consistent with prediction that
upper/lower octave more salient for
complex/pure tones
Consistent with implication-realisation
model
Future experiments
to separate “nature” from “nurture”
Listeners


Spectral versus fundamental listeners
Western versus non-Western musicians
Predictions


Psychoacoustic model
Statistical analysis of symbolic music databases
Stimuli



Synchronous versus asynchronous
Pure versus complex tones
With/without background noise
Ideas for future research
PhD students? Postdocs?

Further experiments to separate
nature from nurture

Modeling of empirical data of
Krumhansl and others
Are major and minor triads special?
Especially consonant
A combination of:
1. high harmonicity/fusion (include P5/P4)
2. low roughness (no 2nds)
Part of culture - not “nature”
The result of centuries of experimentation
3. familiarity
 3 psychological components of consonance
Origins of major-minor tonality
Open triangles: chroma stability profile of MmT1
Full squares: chroma salience profile of tonic triad2
From Parncutt (2011, Music Perception)
1Krumhansl,
C. L., & Kessler, E. J. (1982). Tracing the dynamic changes in perceived
tonal organization in a spatial representation of musical keys. Psychological Review
2Parncutt, R. (1988). Revision of Terhardt's psychoacoustical model of the root(s) of a
musical chord. Music Perception
Analysis of C4 E4 G4
Using pitch algorithm of Parncutt (1989)
Each tone is assumed to have many harmonics
Yellow: The notes
1. Spectral pitch saliences
Register
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0: - - - - - - - - - - - 1: - - - - - - - - - - - 2: - - - - - - - - - - - 3: - - - - - - - - - - - 4: 0.08 - - - 0.06 - - 0.07 - - - 5: 0.08 - - - 0.08 - - 0.08 - - - 0.04
6: 0.02 - 0.05 - 0.06 - - 0.05 0.01 - - 0.03
7: - - 0.06 - 0.02 - - - - - - 8: - - - - - - - 0.01 - - - 9: - - - - - - - - - - - -
Analysis of C4 E4 G4
Using pitch algorithm of Parncutt (1989)
Yellow: The notes
2. Virtual pitch saliences
Reg.
Reg.
Reg.
Reg.
Reg.
Reg.
Reg.
Reg.
Reg.
Reg.
0: - - - - - - - - - - - 1: 0.01 - 0.01 - - 0.01 - - 0.02 0.01 - 2: 0.10 - 0.01 0.01 0.02 0.05 - 0.03 0.01 0.06 - 3: 0.29 - 0.01 0.01 0.12 0.03 0.01 0.14 - 0.05 0.02 0.01
4: 0.35 - 0.02 - 0.30 - - 0.28 - 0.02 - 0.02
5: 0.10 - 0.03 - 0.14 - - 0.13 - - - 0.05
6: 0.01 - 0.05 - 0.05 - - 0.02 - - - 0.02
7: - - 0.02 - 0.01 - - - - - - 8: - - - - - - - - - - - 9: - - - - - - - - - - - -
3. Chroma saliences
0.87 0.01 0.19 0.03 0.66 0.09 0.01 0.64 0.05 0.15 0.03 0.12
Analysing different voicings of CEG
Using pitch algorithm of Parncutt (1989)
Which non-notated chromas are implied by CEG?
Procedure: Consider a wide variety of voicings
Root
position
First
inversion
Second
inversion
close
C4 E4 G4
E4 G4 C5
G4 C5 E5
open
C3 G3 E4
E3 C4 G4
G3 E4 C5
skewed
C3 E4 G4
E3 G4 C5
G3 C5 E5
very open
C3 E4 G5
E3 G4 C6
G3 C5 E6
In each voicing, study non-notated chromas
• chroma is not C, E or G
• predicted salience > 0.05 (predicted probability of noticing)
Analysing different voicings of CEG
Pitches whose predicted salience are > 0.05 (Parncutt, 1989)
Root pos. 1st inv.
2nd inv.
Close
position
A2, A3
F2
D6 (D7)
B5
A2 A3
(D7)
F3
Open
position
A2
F1
D5
B5 (B5)
A1
D6
Skewed
position
A2 A3
B5
A1
D6
A3
A1
D6
A2
B7
Very open A2
D6
position
All pitches
are virtual
unless in
brackets
(spectral)
B5
Result: More common voicings have more salient non-notated chromas
Octave generalisation
of the harmonic series template
weight
(Parncutt, 1988)
10
8
6
4
2
0
Five “root-support intervals”
P1
P5
M3
m7
M2
0
1
2
3
4
5
6
7
8
9
10 11
interval class (semitones)
As vector relative to chromatic scale: 10 0 1 0 3 0 0 5 0 0 2 0
Perception of a C-minor triad
Experiential representation for extreme “overtone listeners”
C
C
D
10 0
1
0
E
F
G
A
B
3
0
0
5
0
0
2
0
3
Eb 0
2
0 10 0
1
0
0
0
5
0
0
0
5
0
2
0 10 0
1
0
3
6 10 3
3
0 18 0
1
7
3
G
tot 10 2
0
Implications for music theory
High-register voicings:
• best tone to double: G
• best tones to add: D, Bb ( madd9, m7)
Perception of a C-minor triad
Experiential representation for extreme “fundamental listeners”
C
C
D
10 0
2
0
E
F
G
A
B
0
5
0
0
3
0
1
0
0
Eb 0
1
0 10 0
2
0
5
0
0
3
5
0
0
0
1
0 10 0
2
0
0
2 13 0
8
0 10 8
2
1
3
G
tot 15 1
3
Implications for music theory
Low-register voicing:
• best tone to double: C ( theory of the root)
• best tones to add: F, Ab ( 7, M7)
Are non-notated chromas real?
The evidence

Many people can’t hear notated chromas!



Consider Renaissance vocal polyphony in a church:



But hard to extract notation from signal (MIR transcription problem)
We can experience non-notated chromas directly


Ear has no prior information on which partial belongs to which tone
No easy way to distinguish notated from non-notated
It’s easy to model perception of non-notated chromas


Some music students study “ear training” for years!
Why should non-notated chromas be less “real”?
But not “cognitive structures”
Predictions can explain basic musical structures

modal and major-minor tonality