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”

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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

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Calculated VP salience distribution

f0 range of speech and music

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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

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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

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0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

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itude

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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

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ampl

itude

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itude

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itude

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0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

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itude

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itude

track 40 41 42

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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

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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