# 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.

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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. 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 F1F2 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)?

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)?

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)?

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)?

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

Summary 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

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