perturbation theory

Perturbation Theory

March 11, 2013

Just So You Know• The Fourier Analysis/Vocal Tract exercise is due on Wednesday.

• Please note: don’t make too much out of rounding off errors!

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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10001500200025003000

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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500

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10001500200025003000

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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300

400

500

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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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10001500200025003000

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