24.963

Linguistic PhoneticsThe acoustics of vowels

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•  No class on Tuesday 10/13 (Tuesday is a Monday) Readings: •  Johnson chapter 6 (for this week) •  Liljencrants & Lindblom (1972) (for next week) Assignment: •  Modeling lip-rounding, due 10/15

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F2 (Hz)

F1 (Hz)

200

300

400

500

600

700

8002500 2000 1500 1000

iu

I

ε

Ω

c

Aæ

Image by MIT OCW.Adapted from Peter Ladefoged. A Course in Phonetics. 5th ed. Berlin, Germany: Heinle, 2005.ISBN: 9781413006889. Available at: https://www.phonetics.ucla.edu/course/contents.html.

The Acoustics of Vowels

Source-Filter models: •  Source: voicing (usually) •  Filter characteristics can be given a basic but

useful analysis using simple tube models. •  Tube models can be supplemented by perturbation

theory for approximate analysis of the effects of wide constrictions.

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Low vowels [A, a, œ] •  Pharyngeal constriction

•  Since the back tube is much narrower than the front tube, each can reasonably be approximated by a tube closed at one end and open at the other.

•  The resonances of the combined tubes deviate from the values we would calculate for these configurations in isolation because the resonators are acoustically coupled.

•  The degree of coupling depends on the difference in cross-sectional areas.

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ɑThe shape of the vocal tract in the vowel [ ] as in father schematized as two tubes.

Image by MIT OCW.

Low vowels [A, a, œ]

€

Fn =(2n −1)c4L

nomogram

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lb

b f

lf

A A

Front cavity resonances Back cavity resonances

Freq

uenc

y (k

Hz)

Back cavity length (cm)

5

4

3

2

1

00 2 4 6 8 10 12 14 16

F3

F2

F1

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics.Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483.

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics.Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483.

Non-low vowels (e.g. [i, e]) • Short constriction in the mouth

•  The back cavity can be approximated by a tube closed at both ends.

•  The front cavity is approximated by a tube closed at one end.

•  Neglects coupling. The degree of coupling depends on the cross-sectional area of the constriction.

•  How do we account for the F1 of high vowels?

€

Fn =nc2L

€

Fn =(2n −1)c4L

8

lb lc lf

Ab Ac Af

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics.Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483.

aa

b

b

c

cd

d

Image by MIT OCW.Adapted from Ladefoged, Peter. Elements of Acoustic Phonetics.2nd ed. Chicago, IL: University of Chicago Press, 1996.

Helmholtz resonators

•  The back cavity and the constriction together form a resonant system called a Helmholtz resonator.

•  If the length of the constriction is short, the air in it vibrates as a mass on the ‘spring’ formed by the air in the back cavity.

•  Resonant frequency, f = c

2π Vl= c

c 2π Ablblc

€

c A c A

9

lb lc lf

Ab Ac Af

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics.Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483.

b cd

b c

d

aa

Image by MIT OCW.Adapted from Ladefoged, Peter. Elements of Acoustic Phonetics.2nd ed. Chicago, IL: University of Chicago Press, 1996.

Non-low vowels - nomogram

€

f =c2π

Ac

Ablblc€

Fn =nc2L

€

Fn =(2n −1)c4L

front cavity

back cavity

back cavity + constriction

•  How would you model a mid vowel? 10

lb lc

Ab Ac Af

lf

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics.

Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483.€

Front cavity resonances Back cavity resonances

Freq

uenc

y (k

Hz)

Back cavity length (cm)

5

4

3

2

1

00 2 4 6 8 10 12 14 16

F3

F2

F1

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics.Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483.

Perturbation Theory (Chiba and Kajiyama 1941) •  Constriction near a

point of maximum velocity (Vn) lowers the associated formant frequency.

•  Constriction near a point of maximum pressure raises the associated formant frequency.

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V1

V2

V2

F2 F4

V1 V3

V4

V4

F1 F3

V3 V3'

V2'

V2'

V3'

V4'

V4'

V4''

V4''

V4'''

V4'''

V3''

V3''

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden,

MA: Blackwell Publishers, 1997. Based on Chiba and Kajiyama 1941.

Perturbation Theory (Chiba and Kajiyama 1941) •  What is the effect of a

pharyngeal constriction? •  Does this correspond to the

tube model above? •  How do you raise F2

maximally?

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V1

V2

V2

F2 F4

V1 V3

V4

V4

F1 F3

V3 V3'

V2'

V2'

V3'

V4'

V4'

V4''

V4''

V4'''

V4'''

V3''

V3''

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden,

MA: Blackwell Publishers, 1997. Based on Chiba and Kajiyama 1941.

Perturbation Theory vs. two-tube models •  Our simple tube models ignore acoustic coupling and are

therefore most valid where constrictions are narrow. Perturbation theory accounts for the effects of small perturbations of a uniform tube, and thus is most accurate for open constrictions. Mrayati et al (1988): perturbation theory is generally valid for constrictions greater than 0.8 cm2, and two-tube models are valid for a constriction of 0.05 cm2 or less, with a transitional region in between.

Mrayati, Carré & Guérin (1988). Distinctive regions and modes. Speech Communication 7, 257-286.

• 

• 

• 

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American English [ɹ] •  American English [ɹ] is characterized by an exceptionally

low F3 (<2000 Hz).

Reproduced from Espy-Wilson, Carol Y., Suzanne E. Boyce, Michel Jackson, Shrikanth Narayanan, and Abeer Alwan."Acoustic modeling of American English/r." The Journal of the Acoustical Society of America 108, no. 1 (2000):343-356. doi: https://doi.org/10.1121/1.429469, with the permission of the Acoustical Society of America.

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•  American English [ɹ] is produced in a variety of ways across speakers and contexts (Alwan et al 1997 JASA, Westbury et al 1998, Speech Comm.).

•  A basic distinction that is often made: ‘bunched’ vs. ‘retroflex’. –  But there appears to be a

continuum of variants.

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Reproduced from Narayanan, Shrikanth S., Abeer A. Alwan, and Katherine Haker."Toward articulatory-acoustic models for liquid approximants based on MRI andEPG data. Part I. The laterals." The Journal of the Acoustical Society of America101, no. 2 (1997): 1064-1077. doi: https://doi.org/10.1121/1.418030, withthe permission of the Acoustical Society of America.

Perturbation Theory (Chiba and Kajiyama 1941) A nice story about Am. Eng.

[®] •  Three constriction: labial

(lip protrusion/rounding), palatal (bunching or retroflexion), and pharyngeal.

•  All 3 are near velocity maxima for F3, hence very low F3.

•  But Espy-Wilson et al (2000) argue actual constrictions are in the wrong place

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V1

V2

V2

F2 F4

V1 V3

V4

V4

F1 F3

V3 V3'

V2'

V2'

V3'

V4'

V4'

V4''

V4''

V4'''

V4'''

V3''

V3''

Image by MIT OCW.Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden,

MA: Blackwell Publishers, 1997. Based on Chiba and Kajiyama 1941.

Espy-Wilson et al (2000) argue from MRI data that:

•  Actual constrictions are in the wrong places, e.g. pharyngeal constriction is too high.

•  Constrictions are too narrow to apply perturbation theory.

•  Argue that F3 is a front cavity resonance.

•  Low due to length (bunched) or sub-lingual cavity (retro) + lip constriction. (How long?)

•  Or: lip constriction is narrow enough for the front cavity to form a Helmholtz resonator.

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Reproduced from Narayanan, Shrikanth S., Abeer A. Alwan, and Katherine Haker."Toward articulatory-acoustic models for liquid approximants based on MRI andEPG data. Part I. The laterals." The Journal of the Acoustical Society of America101, no. 2 (1997): 1064-1077. doi: https://doi.org/10.1121/1.418030, withthe permission of the Acoustical Society of America.

Constriction locations and area functions for [i] vowels

Story et al (1998), MRI

Ladefoged & Maddieson (1996) – mean tongue positions

Fant (1960), Russian [i]F2 2250 Hz, F3 3200 Hz

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.

© Journal of the Acoustical Society of America. All rights reserved. This content is excluded fromour Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/.Source: Story, Brad H., Ingo R. Titze, and Eric A. Hoffman. "Vocal tract area functions for an adult female speakerbased on volumetric imaging." The Journal of the Acoustical Society of America 104, no. 1 (1998): 471-487.

© Walter de Gruyter. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/

© MIT Press. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/

Hillenbrand et al (1995) – Michigan English vowel formants

2000

2200

2400

2600

2800

3000

3200

500 1000 1500 2000 2500

F3(Hz)

F2 (Hz)

a

ø

U

O

u

oU

i

eI I

œ œ E

Courtesy of The Acoustical Society of America. Used with permission.Source: Hillenbrand, James, Laura A. Getty, Michael J. Clark, and KimberleeWheeler. "Acoustic characteristics of American English vowels." The Journalof the Acoustical society of America 97, no. 5(1995): 3099-3111.

Lip rounding •  Lip-rounding also involves lip protrusion so it both

lengthens the vocal tract and introduces a constriction at the lips.

•  Perturbation theory: All formants have a velocity maximum at the lips, so a constriction at the lips should lower all formants.

•  Lengthening the vocal tract also lowers formants. •  Tube models: The effect of a constriction at the lips is

equivalent to lengthening the front cavity. Protrusion actually lengthens the front cavity.

•  This lowers the resonances of the front cavity - in front vowels the lowest front cavity resonance is usually F3, in back vowels it is F2.

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Lip rounding •  Tube models 2: Fant (1960) suggests the front cavity plus

lip constriction can form a helmholtz resonator.

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Fant’s (1960) nomograms

• A more complex tube model for vowels:

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14 12

12

10

8

8

6

6

4

4

2

2 0

0

10

1016 14X

Area A

cm2

Amin = 0.25 cm2

Xmin = 10.5 cm

l1/A1 = 1/4 A = Amin*cosh2 (X-Xmin)/h

h = 4.75 / arcosh (8/Amin)1/2

X = Constriction coordinate in cm from glottis Image by MIT OCW.

Based on Fant, Gunnar. Acoustic Theory of Speech Production. The Netherlands: Mouton De Gruyter, 1960.

Nomogram showing variation in constriction location and lip-rounding - narrow constriction (Amin = 0.65 cm2)

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c/s

5000

4500

4000

3500

3000

2500

2000

1500

1000 750 500

0 cm from lip unrounded

opening rounded -3-4 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 -2-3 -1 0

F5

F4

F3

F2

F1

Amin = 0.65cm2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Curve L1 cm A1 cm2

1 2 3 4 5

0 1 1 1 1

8.0 6.0 2.0 0.65 0.16

cm from glottis 20 18 16 14 12 10 8 6 4 2 0 -2Axial coordinate of the tongue constriction center.

1 2 3 4 5<

Image by MIT OCW.Based on Fant, Gunnar. Acoustic Theory of Speech Production. The Netherlands: Mouton De Gruyter, 1960.

Nomogram showing variation in constriction location and lip-rounding - wider constriction (Amin = 2.5 cm2)

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20

-3-4cm from opening

unroundedrounded

cm from glottisAxial coordinate of the tongue constriction center.

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18-2-3 -1 0

0

500750

1000

1500

2000

2500

3000

3500

4000

4500F5

F4

F3

F2

F1

5000c/s Amin = 2.6 cm2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1918 16 14 12 10 8 6 4 2 0 -2

Curve L1 cm A1 cm2

12345

01111

8.06.02.00.650.16

1 2 3 4 5

lip

Image by MIT OCW.Based on Fant, Gunnar. Acoustic Theory of Speech Production. The Netherlands: Mouton De Gruyter, 1960.

Nomogram showing variation in constriction location and degree.

25

20

-3-4cm from opening

unroundedstate

cm from glottisAxial coordinate of the tongue constriction center.

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180

500750

1000

1500

2000

2500

3000

3500

4000

4500F5

F4

F3

F2

F1

5000c/s

Amin = 0.32 cm2

Amin = 1.3 cm2

Amin = 5.0 cm2

18 16 14 12 10 8 6 4 2 0 -2

Curve123

L1 cm000

A1 cm8.06.02.0

Amin cm2

0.321.35.0

1 2 3

lip

Image by MIT OCW.

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