Acoustic and Auditory Phonetics
Chapter 2 part 2
Jeffrey Heinz
[email protected]
University of Delaware
February 19, 2013
1 / 24
Source-Filter Theory of Speech
2 / 24
Modeling the schwa [@]
Last week we modeled the vocal tract as a uniform tube closed
at one end (the glottis) and open at the other (the lips).
The first three formants of
[@] for a tube 17.5cm long
F1 = 500Hz
F2 = 1500Hz
F3 = 2500Hz
For shorter vocal tracts, (e.g. 15cm) the formant frequencies
increase.
3 / 24
Resonance
Objects resonate at their own particular frequencies.
1. This is how sopranos are said to be able to break a glass by
singing at the right pitch.
2. It’s also the idea behind how you can swing yourself higher
on a swing by “pumping” at just the right time.
3. It’s like positive feedback.
4 / 24
Resonances in Tubes
1. Columns of air within a tube also have resonances (called
formants when speaking of the vocal tract).
2. As sound is passed through a tube, frequencies which
match the resonant frequencies are enhanced whereas those
which do not match are damped.
5 / 24
Source-Filter Theory
Source of speech sounds
The speech source is a complex “sawtooth-shaped” wave, which is the sum of
many simple waves with different frequencies.
6 / 24
Source-Filter Theory
Spectrum of the source
The spectrum of this sawtooth-waved reveals it is the sum of
multiple simple waves. The peaks in the spectrum are called
harmonics.
7 / 24
Source-Filter Theory
Spectrum of the filterered source
Here we see the spectrum of this sound after it has been filtered.
8 / 24
Calculating resonant frequencies
1. Tube closed at both ends
fk =
kc
2L
(1)
2. Tube closed at one end and open at the other (resembles
the vocal tract)
(2k − 1)c
fk =
(2)
4L
• L = length of tube, c = speed of sound, k is a positive
integer
• These resonant frequencies fk are called the kth formant
frequencies.
9 / 24
Uniform tube open at one end is like [@]
Formants are resonant frequencies of the vocal tract
1. Approximate length of average male’s vocal tract
L=17.5cm
1.1 The first formant is 35,000/70 = 500Hz
1.2 The second formant is 3× 35000/70 = 3×500 = 1500 Hz
1.3 The third formant is 5 × 35000/70 = 5×500 = 2500 Hz
2. Approximate length of average female’s vocal tract
L = 15cm
2.1 The first formant is
2.2 The second formant is
2.3 The third formant is
10 / 24
Uniform tube open at one end is like [@]
Formants are resonant frequencies of the vocal tract
1. Approximate length of average male’s vocal tract
L=17.5cm
1.1 The first formant is 35,000/70 = 500Hz
1.2 The second formant is 3× 35000/70 = 3×500 = 1500 Hz
1.3 The third formant is 5 × 35000/70 = 5×500 = 2500 Hz
2. Approximate length of average female’s vocal tract
L = 15cm
2.1 The first formant is 35,000/60 = 583.3Hz
2.2 The second formant is 3 × 35000/60 = 3×583.3 = 1750 Hz
2.3 The third formant is 5 × 35000/60 = 5×583.3 = 2917 Hz
10 / 24
A little more physics
These vibrating columns of air which resonate are a property of
standing longitudinal waves.
11 / 24
Resonance in tubes closed at both ends
Recall what happens when the compression is sent down the
length of the tube closed at both ends:
It bounces back, canceling out the area of rarefaction that is
following it.
12 / 24
Standing Waves in Tubes Closed at both ends
The standing
wave pattern of
the 1st resonance
in a tube closed
from both ends
(Figure 5.8 from
Johnson (2004)).
• The nodes are places where the waves constantly out of
sync, leaving the pressure exactly neutral.
• The antinodes are places where the waves are constantly in
sync, creating maximal areas of compression and
rarefaction.
13 / 24
Standing Waves in Tubes Closed at both ends
The standing
wave pattern of
the 2nd
resonance in a
tube closed from
both ends
(Figure 5.8 from
Johnson (2004)).
• The nodes are places where the waves constantly out of
sync, leaving the pressure exactly neutral.
• The antinodes are places where the waves are constantly in
sync, creating maximal areas of compression and
rarefaction.
13 / 24
Demo: Nodes and Antinodes
http://www.physicsclassroom.com/Class/waves/u10l4c.cfm
http://www.youtube.com/watch?v=HpovwbPGEoo
14 / 24
Standing Waves in Tubes Closed at One End and Open
at the Other
The standing wave pattern of the 1st resonance in a tube closed
at one end, open at the other (Figure 5.9 from Johnson (2004)).
15 / 24
Standing Waves in Tubes Closed at One End and Open
at the Other
The standing wave pattern of the 2nd resonance in a tube closed
at one end, open at the other (Figure 5.9 from Johnson (2004)).
15 / 24
One more technical consideration
Pressure vs. velocity
1. Standing waves can also be described in terms of particle
displacement, i.e. velocity.
2. The nodes of one kind are the antinodes of the other and vice versa.
16 / 24
Velocity Standing Waved Demo
http://www.physics.smu.edu/~ olness/www/05fall1320/applet/pipe-waves.html
17 / 24
Why we care
• The vocal tract is clearly nothing like a uniform tube most of the time.
• Recall that in source-filter theory the vocal tract acts as a filter and
resonator on the glottal source.
Figure: Vocal tract configurations and filters (from Ladefoged 1996)
How can we predict which vocal tract shapes will have which resonances?
18 / 24
Preview of what’s coming after Spring Break
1. In the second part of this course, we will look at two ways
to predict the resonances of nonuniform tubes:
1.1 Multitube models, which treat the nonuniform vocal
tract as if it were a concatenation of small tube-like sections.
1.2 Pertubation theory, which lets us determine how
formants deviate from the norm if there are constrictions
near the nodes or antinodes of the standing wave patterns
(which recall are a result of the resonant frequencies)
19 / 24
Illustration of Main Idea in Quantal Theory (From
Stevens 1998)
Articulatory space is necessarily continuous, but slightly
different articulations can produce qualitatively different
acoustic effects, especially in region II.
20 / 24
Illustration of Main Idea in Quantal Theory (From
Stevens 1998)
There is a large acoustic (and auditory) difference between
regions I and III.
20 / 24
Illustration of Main Idea in Quantal Theory (From
Stevens 1998)
Within regions I and III, however, the acoustic parameter is
relatively insensitive to change in the articulatory parameter. In
other words, changes in articulation don’t have much effect on
the speech output.
20 / 24
Illustration of Main Idea in Quantal Theory (From
Stevens 1998)
Stevens’ claim is that linguistic contrasts involve differences
between ‘quantal’ regions; i.e. regions of acoustic stability (I
and III).
20 / 24
Quantal Theory (Stevens 1972, 1989)
1. Articulations don’t have to be precise to produce a certain
output.
2. Continuous movement through a quantal region will yield
an acoustic steady state.
21 / 24
Quantal Theory (Stevens 1972, 1989)
1. Strong Hypothesis of Quantal Theory: All quantal regions
(I and III) define contrastive sounds and all contrastive
sounds differ quantally.
21 / 24
Predictions of Quantal Theory
i
u
a
These three vowels are preferred cross-linguistically, so they
should be ‘quantal vowels’. Later we will see that in fact they
are.
22 / 24
Why do quantal regions influence sound systems?
1. Speakers can get away with articulatory sloppiness if
languages make use of quantal phonetic categories.
2. Near-continuous articulations yield discrete acoustic
(perceptual) categories.
23 / 24
Summary
1. Vibrating air within tubes resonate at certain frequencies.
2. We can calculate the resonant frequencies if we know the
the length of the tube, the speed of sound, and which
resonant frequency we want (the first, second or third etc.)
3. These resonances also create standing wave patterns with
nodes and antinodes.
4. The physics we have studied here provides the basis for the
acoustic analysis of speech sounds, which we will return to
after spring break.
• Quantal theory explores the hypothesis that speech sounds
occur in regions of acoutic stability.
24 / 24