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. 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. glottis lips

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 • Diagrammed in terms of velocity: F1 F2

The Big Idea, part 2 • Idea #2: constriction at (or near) a velocity anti-nodedecreases frequency • The constriction slows the velocity down •  constriction at a pressure node decreases frequency • Idea #3: constriction at (or near) a velocity nodeincreases 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 labial glottal 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

Perturbation Theory