Pitch, tonality, and the missing
fundamentals of music cognition
Richard ParncuttUniversity of Graz, Austria
BRAMS, Université de Montréal31 May 2012
This file has been revised after discussion and questions
SysMus Graz
AbstractWhat 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? Underlying principles? Why those pc-sets? voicings? progressions? Can we model frequency of occurrence?
Humanities approach:Historical accident? If so: Why so widespread? Why so stable?
AssumptionThe major-minor system is based on pitch as subjective experience 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 available info is generally incomplete focus on affordances (survival and reproduction)
Correlates of the color red ≠ red itself light wavelengths physiology of retina physiology of visual cortexTo study “red”, we must separate experience & physics
Is everything physical?Modern science is atheist - ok Good arguments against existence of gods and spirits
Conscious experience is something else! Different from gods/spirits AND brain substrates Emerges in infancy, disappears when we die Foundation of arts and aesthetics
The solution: Epiphenomenalism 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 worldIt exists without experience - but not vice-versaExistence of experience depends on physical world
Subjective answer: ExperienceWithout it we would know nothing (not be human)Existence of physical world depends on experience
(“Objective”: subject ≠ object, “Subjective”: s=o)
ConclusionNo idea. Can’t compare totally different things
Why scientists reject experienceand why some humanities scholars reject it too
Scientific belief system Success of modern physics In inherent superiority of objectivity Reductionism (belief in simple explanations) Grouping of mind-body dualism with theism
Humanities-science conflict “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 representationsand 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 Timbre, fusion, chord roots, harmonic function, harmonic tonality
3. Abstract: Notes in musical scores Performance, composition
The “three worlds” of Karl PopperThe broader context of music representations (not “worlds”)
1. Physical environment, body, brain
2. Experiential sensations, emotions
3. Abstract 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 experienceGallagher, 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
Here: pitch = perceived pitchIn music theory: 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 pitchPitch perception according to Terhardt
Spectral pitch (SP) 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) 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)
youtubechurch
bells
Spectral versus virtual pitch
This distinction is
ecologicalbased on interaction with the environment
not physiologicalbased 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?)
Enormous no. of neurons and connections!Which states/events correspond to experience?
Spectral vs temporal processingAlong auditory pathways, we find both temporal representations (phase locking) spectral representations (tonotopic structures)
Assumptions Both are used by neural networks Both are inextricable in hidden layers
Conclusion Doesn’t help us understand pitch as experience
Neural processing of pitch in music and speech
The same neural net can process…• spectral and temporal patterns• pitch in speech and music
Bha
ruch
a, 1
987
Virtual object (Kanizsa, 1955)
Incomplete triangleCompleted by virtual contours
Auditory image (Bregman, 1990; McAdams, 1984)
Incomplete harmonic seriesCompleted by virtual pitch
missingfundamental
(f0) overtones
frequency
Virtual objects in vision and hearingGestalt principle of closure – filling the gaps in a familiar pattern
SP
L
Pitch perception: Experimental method
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 is unambiguous corresponds to frequency (if SPL constant)
Result: The pitch of a complex tone is ambiguous
= different pitch in different presentations and/or multiple
= several pitches perceived simultaneously
Can explain a lot about musical structure
Pitch salience In musical practice:
Pitched versus unpitched percussion How clear is pitch on a continuous scale?
In experimental data: Probability of noticing a pitch Subjective clarity of a pitch
Depends on: Stimulus (esp. spectrum) Listener (“spectral” vs “fundamental”) Temporal context (proximity expectation)
high pitch salience
low pitch salience
Analytic versus holistic perception
You can consciously switch between two modeso analytic (strange black shapes)o holistic (“FLY” in white letters)
Similarly for pitch?
Individual differences in pitch perceptionAuditory 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 pairsIn 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”
Schn
eide
r et a
l., N
Y Ac
ad S
ci, V
ol. 1
060,
p. 3
87-3
95 (2
005)
overtone listeners
fundamental listeners
Finding: Listening strategy depends on music experience and instrument
Research idea: Study relation to amusia?
Pitch dominance regionsOctave register(piano keyboard)
1 2 3 4 5 6 7 8
Salient spectral pitch (spectral dominance)
Salient virtual pitch (musical practice)
Spectral pitchAccording to experimental data,SP salience is highest at F5 (C4-C8). speech intelligibility & formants: f1 ~ 500 Hz ~ C5, f2 ~ 1500 Hz ~ G6
Virtual pitchAccording to model predictions,VP salience is highest at D4 (C2-C6). f0 range of voice and music
f1 f2
middle C
Dominance region of spectral pitchorigin: speech perception
centre at 700 Hz, central band at 300-2000 Hz
afte
r Ter
hard
t et a
l., 1
982
Calculated VP salience distribution
f0 range of speech and music
Afte
r Hur
on &
Par
ncut
t (un
publ
ishe
d)
Origin of virtual pitcha bit of history
Before the 1970s many assumed... low pitch = combination tone = distortion product peripheral origin (basilar membrane)
In the 1970s it became clear... pitch perception = pattern recognition mixture of spectral and temporal processing central origin (brain)
Perception of complex tonesTwo 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 HCTsdepends on:
Tuning of partialsMistuning 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 Fusion more likely for “holistic” or “fundamental” listeners
Pitch at the missing fundamentalASA Auditory Demonstrations CD (Houtsma, Rossing, Wagenaars), track 37
0
1
2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
1
2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
Conclusions:
1. Pitch does not necessarily correspond to a partial
2. Pitch is multiple/ambiguous• VP at missing fundamental• SP at lowest partial
0
1
2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
1
2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
1
2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
1
2
3 4 5
Sound demo: Masking SP and VPASA Auditory Demonstrations CD (Houtsma, Rossing, Wagenaars)
0
0,2
0,4
0,6
0,8
1
1,2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
0,2
0,4
0,6
0,8
1
1,2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
0,2
0,4
0,6
0,8
1
1,2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
0,2
0,4
0,6
0,8
1
1,2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
0,2
0,4
0,6
0,8
1
1,2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
track 40 41 42
0
0,2
0,4
0,6
0,8
1
1,2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
Conclusion:Masking and pitch pattern recognition happen in different places
• Masking is peripheral
• Pitch pattern recognition is central
1st tone in pair 2nd tone in pair
Relation between VP and SP pattern ASA Auditory Demonstrations CD (Houtsma, Rossing, Wagenaars), Track 39
VP corresponds to: best-fit subharmonic of all
partials NOT frequency difference small mistuning is no problem
Demo no.
SP1 (Hz)
SP2 (Hz)
SP3 (Hz)
VP (Hz)
1 800 1000 1200 200
2 850 1050 1250 210
0
1
2
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
frequency (kHz)
ampl
itude
0
1
2
0,25 0,45 0,65 0,85 1,05 1,25 1,45 1,65 1,85 2,05
frequency (kHz)
ampl
itude
General relation between SP and VP1. VP lies at fundamental of audible harmonic pattern2. VP salience depends on SPs at harmonic positions
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
Source: Liber Usualis 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)
1 2 3 4 5 6 70
10000
20000
30000
40000
50000
60000
70000
A B C D E F G
no. of notescounted
Prevalence of individual tones (scale steps) in chant
Source: Liber Usualis 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
1 2 3 4 5 6 70
10000
20000
30000
40000
50000
60000
70000
A B C D E F G
no. of notescounted
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
Listeners have a “feel” for pitches of harmonics Or at least spectral listeners do
Tones are preferred if consonant with contextAn example of pitches in common (“pitch commonality”)
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
2. How many “octave-generalised overtones” correspond to diatonic scale?
Harmonic no. 1, 2, 4, 8 3, 6 5, 10 7 9Interval P1, P8… P5, P12… M3… m7… M2, M9…
Scale step A B C D E F GNo. of harmonics 3 1 3 3 2 3 4
Prevalence of individual tones (scale steps) in chant
1 2 3 4 5 6 70
10000
20000
30000
40000
50000
60000
70000
A B C D E F G
1 2 3 4 5 6 70
1
2
3
4
5Data
A B C D E F G
Model
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)
Psychoacousticians and neuroscientists: Focuses on subjective experience Avoids temporal-spectral debate
Music theorists: Challenges primacy of musical score Focuses on tonality (not “modernist”)
Music historians: Not based on historic sources Ignores historic mode classification
Contradicts…
• physical monism
• established research paradigms in sciences and humanities
Non-notated chroma in triadsAn example of looking carefully at the stimulus (for a change)
1. Spectral synthesisBuild 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. MaskingAssume all partials are equally audibleexcept inside a chromatic cluster
3. Pitch pattern recognitionAt each chromatic scale step: Which harmonics are present in chord? Synthesize that tone using “SFS Esynth”
C4 E4 G4 C4E4G4 C4 C#4 D4 D#4 E4 F4 F#4 G4 G#4 A4 A#4 B4
C7
C6
C5
C4
C4 E4 G4 C4E4G4 C3 C#3 D3 D#3 E3 F3 F#3 G3 G#3 A3 A#3 B3
C7
C6
C5
C4
C3
Estimating virtual pitch salienceCompromise 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 salienceof pitches within triad C4E4G4
First approximation: number of audible partials
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240
2
4
6
8
10
12
Predictions• C3 > E3 and G3• In both registers, D > C# & D#• In register 3, A # > B
C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4
Note: Here, C4 > E4 > G4 is an artefact of a simple model
Estimating virtual pitch salienceof pitches within triad C4E4G4
2nd approx: Weight each partial 1/n, add weights
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240
50
100
150
200
250
300
350
C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4
PredictionsIn both registers, C > E and G, D > C# & D#, F>F#, A>G#B versus A #: different depending on register
Estimating virtual pitch salienceof pitches within triad C4Eb4G4 (C minor)
3rd approximation (Parncutt, 1989)(i) physical representation
(ii) experiential representation
audible partials
Experimental dataParncutt, 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 ReichwegerDiplomarbeit Uni Graz 2010
Participants20 active musicians
SoundsTest sounds: chords of natural piano tonesReference tones: octave-complex (Shepard)
Task How well does the tone go with the chord?7-point scale
Gottfried ReichwegerDiplomarbeit Uni Graz 2010
Maj
or tr
iad
Min
or tr
iad
1st inversionRoot position 2nd inversion
Similarity judgments of successive tones (Parncutt, 1989)
Effect at octave is greater:
…for complex tonesEvidence for “nature”
…for musiciansEvidence 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 experimentsto 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 researchPhD students? Postdocs?
Further experiments to separate nature from nurture
Modeling of empirical data of Krumhansl and others
Are major and minor triads special?
Especially consonantA 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 tonalityOpen triangles: chroma stability profile of MmT1
Full squares: chroma salience profile of tonic triad2
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
From P
arncutt (2011, Music P
erception)
Analysis of C4 E4 G4 Using pitch algorithm of Parncutt (1989)
Each tone is assumed to have many harmonicsYellow: The notes
1. Spectral pitch saliencesRegister 0: - - - - - - - - - - - -Register 1: - - - - - - - - - - - -Register 2: - - - - - - - - - - - -Register 3: - - - - - - - - - - - -Register 4: 0.08 - - - 0.06 - - 0.07 - - - -Register 5: 0.08 - - - 0.08 - - 0.08 - - - 0.04Register 6: 0.02 - 0.05 - 0.06 - - 0.05 0.01 - - 0.03Register 7: - - 0.06 - 0.02 - - - - - - -Register 8: - - - - - - - 0.01 - - - -Register 9: - - - - - - - - - - - -
Analysis of C4 E4 G4 Using pitch algorithm of Parncutt (1989)
Yellow: The notes
2. Virtual pitch saliencesReg. 0: - - - - - - - - - - - -Reg. 1: 0.01 - 0.01 - - 0.01 - - 0.02 0.01 - -Reg. 2: 0.10 - 0.01 0.01 0.02 0.05 - 0.03 0.01 0.06 - -Reg. 3: 0.29 - 0.01 0.01 0.12 0.03 0.01 0.14 - 0.05 0.02 0.01Reg. 4: 0.35 - 0.02 - 0.30 - - 0.28 - 0.02 - 0.02Reg. 5: 0.10 - 0.03 - 0.14 - - 0.13 - - - 0.05Reg. 6: 0.01 - 0.05 - 0.05 - - 0.02 - - - 0.02Reg. 7: - - 0.02 - 0.01 - - - - - - -Reg. 8: - - - - - - - - - - - -Reg. 9: - - - - - - - - - - - -
3. Chroma saliences0.87 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
In each voicing, study non-notated chromas • chroma is not C, E or G• predicted salience > 0.05 (predicted probability of noticing)
Root position
First inversion
Second inversion
close C4 E4 G4 E4 G4 C5 G4 C5 E5open C3 G3 E4 E3 C4 G4 G3 E4 C5skewed C3 E4 G4 E3 G4 C5 G3 C5 E5very open C3 E4 G5 E3 G4 C6 G3 C5 E6
Analysing different voicings of CEG Pitches whose predicted salience are > 0.05 (Parncutt, 1989)
Root pos. 1st inv. 2nd inv. Close position
A2, A3F2D6 (D7)B5
A2 A3(D7)
F3
Open position
A2F1D5B5 (B5)
A1D6
Skewed position
A2 A3B5
A1D6
A3
Very open position
A2D6B5
A1D6
A2B7
All pitches are virtual unless in brackets (spectral)
Result: More common voicings have more salient non-notated chromas
Octave generalisation of the harmonic series template
(Parncutt, 1988)
02468
10
0 1 2 3 4 5 6 7 8 9 10 11
interval class (semitones)
wei
ght
m7
Five “root-support intervals”P1
M2M3
P5
As vector relative to chromatic scale: 10 0 1 0 3 0 0 5 0 0 2 0
Perception of a C-minor triadExperiential representation for extreme “overtone listeners”
C D E F G A B
C 10 0 1 0 3 0 0 5 0 0 2 0
Eb 0 2 0 10 0 1 0 3 0 0 5 0
G 0 0 5 0 0 2 0 10 0 1 0 3
tot 10 2 6 10 3 3 0 18 0 1 7 3
Implications for music theoryHigh-register voicings:• best tone to double: G• best tones to add: D, Bb ( madd9, m7)
Perception of a C-minor triadExperiential representation for extreme “fundamental listeners”
C D E F G A B
C 10 0 2 0 0 5 0 0 3 0 1 0
Eb 0 1 0 10 0 2 0 0 5 0 0 3
G 5 0 0 3 0 1 0 10 0 2 0 0
tot 15 1 2 13 0 8 0 10 8 2 1 3
Implications for music theoryLow-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! Some music students study “ear training” for years! Why should non-notated chromas be less “real”?
Consider Renaissance vocal polyphony in a church: 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 But hard to extract notation from signal (MIR transcription problem)
We can experience non-notated chromas directly But not “cognitive structures”
Predictions can explain basic musical structures modal and major-minor tonality