Perturbation Theory, part 1
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Transcript of Perturbation Theory, part 1
Perturbation Theory, part 1
March 18, 2014
Just So You Know• The Fourier Analysis/Vocal Tract exercise is due!
• Plus: fun standing wave in water video.
Articulation to Acoustics• Last time, we calculated the formant values for “schwa”, or a neutral vowel.
• Theoretical values (vocal tract length = 17.5 cm)
F1 = 500 Hz
F2 = 1500 Hz
F3 = 2500 Hz
• My values:
F1 = 500 Hz
F2 = 1533 Hz
F3 = 2681 Hz
F4 = 3498 Hz
With a neutral vowel, we’re somewhere in the middle of the acoustic vowel space.
Male Formant Averages
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10001500200025003000
F2
F1
Q: How do we get to the corners of the space?
Perturbation Theory• There are two important theories that answer this question.
• The first of these is Perturbation Theory.
• Remember: formants are resonances of the vocal tract.
• These resonances are the product of standing waves in the resonating tube of the articulatory tract.
lipsglottis
What’s the Big Idea?• Chiba and Kajiyama (1941):
• Formant frequencies can be changed by perturbing the airflow of the standing waves in the vocal tract
• Idea #1: velocity of standing waves is inversely related to pressure
• Sort of like the Bernoulli Effect
Standing Waves in the Vocal Tract
• Remember:
• Vocal tract is a tube with one open end at the lips.
• So:
• Pressure node at the lips
• Pressure anti-node at the glottis
• …for all potential standing waves
• This translates into:
• Velocity anti-node at the lips
• Velocity node at the glottis
Standing Waves in the Vocal Tract
F1 F2
• Diagrammed in terms of velocity:
The Big Idea, part 2• Idea #2: constriction at (or near) a velocity anti-node decreases frequency
• The constriction slows the velocity down
• constriction at a pressure node decreases frequency
• Idea #3: constriction at (or near) a velocity node increases frequency
• The constriction increases the pressure
• This enhances airflow
• constriction at a pressure anti-node increases frequency
Here’s the goal• Let’s figure out how we can perturb the airflow in the articulatory tract to get to the corners of the vowel space.
• We need to:
• Lower F1 and raise F2 --> high, front vowels
• Lower F1 and lower F2 --> high, back vowels
• Raise F1 and raise F2 --> low, front vowels
• Raise F1 and lower F2 --> low, back vowels
• Let’s consider them each in turn…
F1
• Velocity node at glottis
• Velocity anti-node at lips
• To lower F1:
make a constriction closer to the lips than to the glottis
• To raise F1:
make a constriction closer to the glottis than to the lips
F2
• Velocity nodes at:
palate
glottis
• Velocity anti-nodes at:
lips
pharynx
F2
• To raise F2, make a constriction at the:
palate
glottis
• To lower F2, make a constriction at the:
lips
pharynx
1. High, Front Vowels
• Lower F1 and raise F2
• Where should we make a constriction(s)?
Male Formant Averages
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F2
F1
1. High, Front Vowels
• Lower F1 and raise F2
• Where should we make a constriction(s)?
• To lower F1:
1. High, Front Vowels
• Lower F1 and raise F2
• Where should we make a constriction(s)?
• To lower F1:
constrict close to lips
1. High, Front Vowels
• Lower F1 and raise F2
• Where should we make a constriction(s)?
• To lower F1:
constrict close to lips
• To raise F2:
1. High, Front Vowels
• Lower F1 and raise F2
• Where should we make a constriction(s)?
• To lower F1:
constrict close to lips
• To raise F2:
constrict at palate
2. High, Back Vowels
• = Lower F1 and lower F2
• Where should we make a constriction(s)?
Male Formant Averages
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10001500200025003000
F2
F1
2. High, Back Vowels
• = Lower F1 and lower F2
• Where should we make a constriction(s)?
• To lower F1:
2. High, Back Vowels
• = Lower F1 and lower F2
• Where should we make a constriction(s)?
• To lower F1:
constrict at lips
2. High, Back Vowels
• = Lower F1 and lower F2
• Where should we make a constriction(s)?
• To lower F1:
constrict at lips
• To lower F2:
2. High, Back Vowels
• = Lower F1 and lower F2
• Where should we make a constriction(s)?
• To lower F1:
constrict at lips
• To lower F2:
constrict at lips
constrict at “pharynx”
• Note: these vowels are usually rounded
3. Low, Front Vowels
• Raise F1 and raise F2
• Where should we make constriction(s)?
Male Formant Averages
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F2
F1
3. Low, Front Vowels
• Raise F1 and raise F2
• Where should we make constriction(s)?
• To raise F1:
3. Low, Front Vowels
• Raise F1 and raise F2
• Where should we make constriction(s)?
• To raise F1:
constrict close to glottis
3. Low, Front Vowels
• Raise F1 and raise F2
• Where should we make constriction(s)?
• To raise F1:
constrict close to glottis
• To raise F2:
3. Low, Front Vowels
• Raise F1 and raise F2
• Where should we make constriction(s)?
• To raise F1:
constrict close to glottis
• To raise F2:
constrict close to glottis
constrict at palate
4. Low, Back Vowels
• Raise F1 and lower F2
• Where should we make constriction(s)?
Male Formant Averages
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10001500200025003000
F2
F1
4. Low, Back Vowels
• Raise F1 and lower F2
• Where should we make constriction(s)?
• To raise F1:
4. Low, Back Vowels
• Raise F1 and lower F2
• Where should we make constriction(s)?
• To raise F1:
constrict near glottis
4. Low, Back Vowels
• Raise F1 and lower F2
• Where should we make constriction(s)?
• To raise F1:
constrict near glottis
• To lower F2:
4. Low, Back Vowels
• Raise F1 and lower F2
• Where should we make constriction(s)?
• To raise F1:
constrict near glottis
• To lower F2:
constrict at pharynx
SummaryMale Formant Averages
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F2
F1
palatal
glottal
labial
pharyngeal
A Note About F3• What about F3 distinctions?
• They’re unusual.
• For acoustic reasons:
• Intensity of voicing harmonics drops off at the higher end of the frequency scale
• (spectral tilt)
• And also auditory reasons:
• Sensitivity to frequency distinctions drops off in the higher frequency regions
• Note: F2 and F3 often merge for [i]
Decreasing F3
• If we wanted to decrease F3...
• Where we would make constrictions?
Decreasing F3
• If we wanted to decrease F3...
• Where we would make constrictions?
• Constrict at:
lips
“velum”
pharynx
English • English is distinctive because it has a very low F3.
• It has labial, post-alveolar (retroflex), and pharyngeal constrictions.
Synergy• The labial, retroflex and pharyngeal constrictions all work together to lower F3.
• Similarly, both labial and velar constrictions lower F1 and F2 in high, back (round) vowels
• Synergy
• Interestingly, labial-velar vowels are far more common in the languages of the world than either:
• labial vowels
• velar vowels