Applied Psychoacoustics Lecture 4: Pitch

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Applied PsychoacousticsLecture 4 Pitch Timbre
Perception

• Jonas Braasch

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Homework 2 Raw Data
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Homework 2 Mean
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Homework 2 MeanSTD
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Mean and Standard Deviation
mean
Standard deviation
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Homework 2 Mean vs. Median
Mean, blue Median, red
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Homework 2 Median
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Statistics for non-Gaussian distributions

• Median is a number that separates the higher
  half of a sample, a population, or a probability
  distribution from the lower half.
• Quartiles
• first quartile (designated Q1) lower quartile
  cuts off lowest 25 of data 25th percentile
• second quartile (designated Q2) median cuts
  data set in half 50th percentile
• third quartile (designated Q3) upper quartile
  cuts off highest 25 of data, or lowest 75
  75th percentile

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Homework 2 Median
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Homework 2 MedianQuartiles
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Homework 2 Mean vs. Median
Mean, blue Median, red
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Homework 2 MedianQuartiles
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HW 2 MedianQuartilesRange
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HW2 left ear vs. right ear
left ear, blue right ear, red
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Some sounds are higher pitched, being composed of
more frequent and more numerous motions
Euclid (330-275 BC)
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Contents

• Pitch perception
• Pure Tones
• Place and Rate Theory
• Complex Tones
• Timbre

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Equal Temperament

• One semitone equals 12v21.05955.9463
• One cent 1200v21.00060.059
• Perfect fifth 1.5700 cent50
• Perfect Octave 21200 cent100

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Psychometric Function
Describes the relationship between a physical
parameter and its psychological correlate
Example Phon-Sone conversion
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Weber-Fechner Law

• The earliest scientific approach to measuring a
  psychometric function
• Ernst H. Weber (1795-1878) investigated just
  noticeable differences (JNDs) for lifting weights
  with the hand.
• The subjects were blindfolded and the weight was
  gradually increased until they were able to
  detect a difference.
• He noticed that the JNDs were proportional to the
  overall weight. (e.g., if the JND for a 100 g
  weigth was 10 g, the JND for a 1000 g weight was
  100 g). If the mass is doubled, the threshold is
  also doubled.

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Weber-Fechner Law

• Gustav T. Fechner (1801-1887) later developed the
  Weber-Fechner Law from Webers findings
• Sklog(I/I0)
• With I the physical parameter (Intensity), S its
  psychophysical correlate, and k a constant, and
  I0 the detection threshold of I.

The JND is then dSklog(dI/I)
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Fechners indirect scales
0 sensation units (0 JND of sensation) stimulus
intensity at absolute detection threshold 1
sensation unit (1 JND of sensation) stimulus
intensity that is 1 difference threshold above
absolute threshold 2 sensation units (2 JND of
sensation) stimulus intensity that is 1
difference threshold above the 1-unit stimulus
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Fechners Law
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Pitch

• Pitch is often thought to be perceived
  logarithmically

But for other psychophysical correlates, this
logarithmic relationship does not hold true
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Stevens Power Law

• Stevens was able to provide a general formula to
  relate sensation magnitudes to stimulus
  intensity
• S aIm
• Here, the exponent m denotes to what extent the
  sensation is an expansive or compressive function
  of stimulus intensity.
• The purpose of the coefficient a is to adjust for
  the size of the unit of measurement.

log S m log(I-I0) log a
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Examples for Stevens Power Law
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Examples for Stevens Power Law Exponents
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and now in the log-log space
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Definition of Pitch

• Pitch is that attribute of auditory sensation in
  terms of which sounds may be ordered on a scale
  extending from low to high. Pitch depends mainly
  on the frequency content of the sound stimulus,
  but it also depends on the sound pressure and the
  waveform of the stimulus.

ANSI standard 1994
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The mel scale
Stevens, Volkmann Newmann, 1937

• Five listeners were
• asked to judge a the
• frequency of a second
• sinusoidal tone generator
• to be perceived half the
• Magnitude of the first
• oscillator with constant frequency
• (method of
• adjustment)
• Sound was switched between both oscillators
• (2-s interval)
• 60 dB SPL

Stevens, Volkmann Newmann, 1937
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Mel Scale - Raw Data
Geometric means for five observers, and average
error for 2 listeners
Stevens, Volkmann Newmann, 1937
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Def. 1000 mels 1000 Hz at 40 dB
Stevens, Volkmann Newmann, 1937
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Solid line mel scale /2.83 Black squares
integrated difference limens open circles
relative location of the resonant positions on
the basilar membrane
Stevens, Volkmann Newmann, 1937
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Size of Musical interval in terms of Mels
Stevens, Volkmann Newmann, 1937
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Hz/mel conversion

• To convert f hertz into m mel use
• m 1127.01048loge(1 f / 700).
• And the inverse
• f 700(em / 1127.01048 - 1).

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Frequency JNDs
Different symbols show different studies
(Fig.Terhardt 1998)
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Frequency Difference Limens
Wier et al., 1977
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Frequency Difference Limens

• At low sound pressure levels (lt10 dB SPL), the
  JND for pitch Increases.
• The hump at 800 Hz was not confirmed in
  follow-up studies
• At one 1-kHz the difference limens is about 3
  cents (0.2)
• At high and low frequencies, we are less
  sensitive to pitch (e.g., 0.5 at 200 Hz and 1
  at 8 kHz.
• Melody recognition disappears for frequencies
  above 4-5 kHz

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DL compared to a semitone
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Pure Tone Frequency Discrimination
from Cheveigne, 2004
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Effect of signal duration
Large improvement in F0 discrimination with
duration for unresolved harmonics (White and
Plack, 1998)
d relative to 20 ms
Duration (ms)
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SPINC vs. Bark
based on JNDs
(Fig.Terhardt 1998)
SPINCSpectral Pitch Increment
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Theories on Pitch Perception

• Place Theory
• Pitch is determined by the location of the firing
  inner hair cell population on the basilar
  membrane
• Rate Theory
• Pitch is determined by the rate code of the inner
  hair cells (phase locking)

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Place Theory
Excitation on Basilar membrane
from Hartmann, 1996
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Place Theory
Excitation on Basilar membrane
Excitation on Basilar membrane for two sinusoids
of same frequency f but 30 dB level difference
from Hartmann, 1996
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Pitch shift for level variation
according to Terhardt 1982
Pitch variation of a sinusoid as function of SPL
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Autocorrelation
Cross-Correlation Models
t1
S
YY (t) 1/(t1-t0) Y(t)Y(tt)
tt0
Licklider (1951)
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Rate model (Sinusoid analysis)
(from de Cheveigne, 2004)
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Rate Pitch Model
f1/t log(f)log(1/t)
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1-kHz sine tone
FFT
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1-kHz harmonic complex with equally strong
harmonics
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250-Hz harmonic complex with equally strong
harmonics
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250-Hz harmonic complex with missing fundamental
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250-Hz harmonic complex with decreasing harmonic
strength
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Cochlear Implants

• Research on Cochlear Implant users suggest that
  our auditory system makes use of both the rate
  and place when determining the pitch.
• The analysis of the rate code is not possible for
  high frequencies

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Cochlear Implants
Illustration from "Functional Replacement of the
Ear," by Gerald E. Leob, 1985
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Cochlear Implants
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from Plack, Oxenham, 2002
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Which harmonic determines pitch?
from Chris Darwin
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Pitch remains the same without fundamental
(Licklider, 1956)
from Chris Darwin
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Pitch perception of complex sounds
(from de Cheveigne, 2004)
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Figure Explanation

• All spectra (A-E) produce the same magnitude of
  pitch.
• Solution For each harmonic produce subharmonics
  (F/n) and plot these into frequency histogram
  (bottom figure). The perceived pitch typically
  corresponds to the highest value.

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Pitch perception of complex sounds
(from de Cheveigne, 2004)
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Explanation of Figure

• Landmarks do not work well in the time domain
• The pitch does not match the period of the
  envelope if the ratio between carrier frequency
  and modulation frequency is lt 10.

(from de Cheveigne, 2004)
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Pitch perception of formant-like sounds

• Can evoke two pitches
• Diagonal line sinusoids

(from de Cheveigne, 2004)
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Another Pitch Definition
The perceptual correlate of the repetition rate
of a sound.
instead of

• that attribute of auditory sensation in terms
  of which sounds may be ordered on a scale
  extending from low to high (ANSI, 1994).

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F0 discrimination for unresolved harmonics
F0DL ()
(F0 200 Hz)
Lowest Harmonic Number
Low-numbered harmonics (lt10) dominate pitch
perception
Houtsma and Smurzynski, 1990
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Absolute Pitch

• "Passive" absolute pitch
• Persons who are able to identify individual notes
  which they hear,
• They can typically identify the key of a
  composition
• "Active" absolute pitch
• Persons with active absolute pitch will be able
  to sing any given note when asked.
• Usually, people with active absolute pitch will
  not only be able to identify a note, but
  recognize when that note is slightly sharp or
  flat.
• 1 in every 10,000 people in the US posses active
  absolute pitch possessors (1/20 in some other
  locations).

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Motoric Absolute Pitch

• Persons who can reproduce an absolute reference
  tone to determine the pitch of other tones (e.g.
  professional singer knowing their range, persons
  who speak a tone language).

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Timbre
When we hear notes of the same force and same
pitch sounded successively on a piano-forte, a
violin, clarinet, oboe, or trumpet, or by the
human voice, the character of the musical tone of
each of these instruments, notwithstanding the
identify of force and pitch, is so different that
by means of it we recognise with the greatest of
ease which of these instruments was used." (p.
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Helmholtz, 1877
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Timbre
Textbooks customarily believe that loudness,
pitch and timbre correlate directly with sound
intensity, fundamental frequency and overtone
structure..."but these experiments show that a
simple one-to-one relationship does not exist."
(p. 59)
Fletcher, 1934
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Timbre
...harmonics manifest themselves in the specific
quality or timbre of the complex tone. . . .
Timbre is multidimensional. ...we do not have a
unidimensional scale for comparing the timbres of
various sounds.
Plomp, 1976
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Timbre
Timbre is that attribute of auditory sensation in
terms of which a listener can judge that two
sounds similarly presented and having the same
loudness and pitch are dissimilar.
ANSI, 1960
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Comment Bregman (1990)
On the ASA definition "This is, of course, no
definition at all. For example, it implies that
there are some sounds for which we cannot decide
whether they possess the quality of timbre or
not. In order for the definition to apply, two
sounds need to be able to be presented at the
same pitch, but there are some sounds, such as
the scarping of a shovel in a pile of gravel,
that have no pitch at all. We obviously have a
problem Either we must assert that only sounds
with pitch can have timbre, meaning that we
cannot discuss the timbre of a tambourine or of
the musical sounds of many African cultures, or
there is something terribly wrong with the
definition." (p. 92)
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Elusive attributes of timbre

• The range between tonal and noiselike character.
• The spectral envelope.
• The time envelope in terms of rise, duration, and
  decay.
• The changes both of spectral envelope
  (formant-glide) and fundamental frequency
  (micro-intonation).
• The prefix, an onset of a sound quite dissimilar
  to the ensuing lasting vibration.

Schouten, 1968
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Grey (1977)
Multidimensional scaling of orchestral instruments
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Grey (1977)

• Stimuli 16 different orchestral instruments
• Listeners had to judge similarity
• Multidimensional scale analysis
• Three scale axes
• Spectral energy distribution
• Synchronicity of attack transients for different
  harmonics
• Presence of low-amplitude, high frequency energy
  in initial attack segment.

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Harmonic Spectra
Flue pipe (open)
Reed pipe with long cyl. res.
Flue pipe (closed)
Reed pipe with short cyl. res.
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Harmonic Spectra

• Relationship of low harmonics
• Spectral centroid
• Dominance of fundamental sound
• Existence of even partial tones

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Comparison
HM Harmonic centroid
Reed Pipe Stops
Flue pipe Stops
HM
HM
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Comparison
Reed Pipe
Flue pipe
Important synchronicity of attack transients
among partial tones, Initial frequency changes
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Literature

• William M. Hartmann (1996) Pitch, periodicity,
  and auditory organization, J. Acoust. Soc. Am.
  100, 3491-3503.
• Alain de Cheveigne (2004) Pitch Perception
  models, in Pitch (Plack, Oxenham, eds.),
  Springer, New York.