Transcript f 1 - Jacobs University Mathematics

The Perception of Frequency cont'd
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A.Diederich– International University Bremen – USC – MMM – Spring 5
Combination tones
 What happens with our tone sensation when the
frequency of the variable tone f2 increases beyond the
critical band, while f1 is kept constant?
 Combination tones are additional pitch sensations
that appear when two pure tones of frequencies f1 and
f2 are sounded together; they are most easily
perceived if the latter are of high intensity level.
 The additional pitch sensations are not present in the
original sound stimulus.
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 The most easily identifiable combination tone at high
intensity level is one whose frequency is given by the
difference of the component frequencies
fC1= f2 – f1
 This is called difference tone.
 When f2 is an octave above f1, fC1 = 2f1 – f1 = f1 . The
difference tone coincides with the lower component
f1.
 When f2 is halfway between f1 and 2f1, i.e., f2 = 3/2 f1
(a musical interval called the fifth), the difference
tone has a frequency fC1 = 3/2 f1 – f1 = ½ f1 (one
octave below that of f1)
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 Other combination tones that are most easily
identified:
 fC2 = 2 f1 – f2
 fC3 = 3 f1 – 2f2
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A.Diederich– International University Bremen – USC – MMM – Spring 5
Frequencies of combination tones fC1, fC2, fC3,
evoked by two-tone superposition (f1, f2)
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A.Diederich– International University Bremen – USC – MMM – Spring 5
How are these extra tone sensations
generated?
 (They are not present in the original sound vibration)
 There are indeed activated regions on the basilar
membrane at the positions corresponding to the
frequencies of the combination tones.
 They are thought to be caused by a "nonlinear"
distortion of the primary wave form stimulus in the
cochlea (see Prof. Oliver's lecture a couple of weeks
ago)
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 "Fake" combination
tones can be generated
in electronic organs and
low quality hi-fiamplifiers and speakers.
 Apparently this
sensation can be created
by an electronic organ
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A.Diederich– International University Bremen – USC – MMM – Spring 5
Beats of mistuned consonances
 f2 = 2f1 + 
 Beat frequency is equal to 
 Called second order beats of subjective beats
 It is a result of neural processing
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 As long as the octave is
perfectly in tune, the phase
difference remains constant
– our ear does not
distinguish one form from
the another
 When f2 = 2f1 +  the mutual
phase relationship will
change continuously with
time
 The ear is sensitive to a
slow changing phase
difference between two
tones.
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A.Diederich– International University Bremen – USC – MMM – Spring 5
(A) Mistuned unison, (B) Mistuned
octave
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A.Diederich– International University Bremen – USC – MMM – Spring 5
Fundament tracking and missing fundamental
1 period of the lower pitch tone
f2 = 3/2f1
pattern repeats after 0
repetition rate f0 =1/0 =1/2 f1
A.Diederich– International University Bremen – USC – MMM – Spring 5
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Two pure
tones
f2 = 3/2 f1
Repetition
rate
f0=1/2 f1
f2 = 4/3 f1
f0 = 1/3 f1
f2 = 5/4 f1
f0 = 1/4 f1
f2 = 6/5 f1
f0 = 1/5 f1
 Our auditory system is
sensitive to these repetition
rates.
 The perception of the
repetition rate is called
fundamental tracking.
 The tone of frequency f0 is
not present as an original
component.
 This tone is called the
missing fundamental.
 The corresponding pitch
sensation is called
periodicity pitch,
subjective pitch, residue
tone, or virtual pitch
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 The frequency f0 is not present in the
cochlear fluid oscillation (whereas the
combination tones are).
 The missing fundamental is perceived
even if the two component tones are fed
in dichotically, one into each ear.
 This indicates that the missing
fundamental, or periodicity pitch, must be
the result of neural processing at a higher
level.
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 Subjective pitch detection only works in the
lower (but musically most important)
frequency range, below about 1500 Hz.
 The smaller the interval between the
component tones, the more difficult for the
auditory system to identify the missing
fundamental.
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A.Diederich– International University Bremen – USC – MMM – Spring 5
 The individual components of frequency 2f0, 3f0, 4f0,
5f0, …, etc. are called upper harmonics of the
fundamental frequency f0.
 Any two successive tones of the upper harmonic
series form a pair with the same repetition rate of
fundamental frequency f0.
 All upper harmonics , if sounded together, will
produce one single subjective pitch sensation
corresponding to f_0 – even if that latter frequency is
totally absent in the multitone stimulus!
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A.Diederich– International University Bremen – USC – MMM – Spring 5
Implications for music
 Because real musical tones happen to be made
up of a superposition of harmonics,
fundamental tracking is the auditory
mechanism that enables us to assign a unique
pitch sensation to a complex tone of a musical
instrument.
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A.Diederich– International University Bremen – USC – MMM – Spring 5
The perception of consonance and
dissonance
 Consonance and dissonance are subjective
feelings.
 Tonal music of all cultures seems to indicate
that the human auditory system posses a sense
for certain special frequency intervals – the
octave, fifth, fourth, etc.
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A.Diederich– International University Bremen – USC – MMM – Spring 5
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A.Diederich– International University Bremen – USC – MMM – Spring 5
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A.Diederich– International University Bremen – USC – MMM – Spring 5
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A.Diederich– International University Bremen – USC – MMM – Spring 5
Tone chroma
Notes with the same
letters have the same
tone chroma.
1 octave
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A.Diederich– International University Bremen – USC – MMM – Spring 5