Acoustics of Vowels

24.963 Linguistic Phonetics The acoustics of vowels

1 • 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

200

i 300 u

400 I Ω

500 F1 (Hz)

ε c 600

æ 700 A

800 2500 2000 1500 1000

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.

4 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.

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

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

Image by MIT OCW.

• 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. 6 Low vowels [A, a, œ]

Ab Af

(2n −1)c Fn = 4L lb lf

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

5 € 4

F3 3

nomogram 2 F 2 Frequency (kHz)

Front cavity resonances Back cavity resonances 1

F1 0 0 2 4 6 8 10 12 14 16 Back cavity length (cm)

Image by MIT OCW. Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483. 7 Non-low vowels (e.g. [i, e]) • Short constriction in the mouth

Ab Ac Af b c d b c d

• The back cavity can be approximated by a tube closed at Fn = both ends. 2L • The front cavity is approximated by a tube closed at one (2n −1)c Fn = end. € 4L • 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? 8 Helmholtz resonators

Ab Ac Af b c d b c d

lb lc lf a a Image by MIT OCW. Image by MIT OCW. Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Adapted from Ladefoged, Peter. Elements of Acoustic Phonetics. 2nd ed. Chicago, IL: University of Chicago Press, 1996. Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483. • 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. c A c A • Resonant frequency, f = c = c 2π Vlc 2π Ab lb lc

€ Non-low vowels - nomogram Ab Ac Af

lb lc lf

5 Image by MIT OCW. Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. € Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483. 4

F3 3 (2n −1)c front cavity Fn = 2 F2 4L Frequency (kHz)

Front cavity resonances Back cavity resonances 1 nc back cavity Fn = F1 0 2L 0 2 4 6 8 10 12 14 16 Back cavity length (cm) €

Image by MIT OCW. back cavity + constriction Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden, MA: Blackwell Publishers, 1997. ISBN: 9780631188483. c A f = c € • How would you model a mid vowel? 2π Ab lb lc

€ Perturbation Theory (Chiba and Kajiyama 1941)

' '' • Constriction near a V1 V3 V3 V3 point of maximum F1 F3 V3' velocity (Vn) lowers the V1 V3 associated formant V3'' frequency.

' '' ''' • Constriction near a V2 V2' V4 V4 V4 V4 point of maximum F2 F4

' pressure raises the V4 V2 V4 V4'' ' associated formant V2

''' frequency. V4

Image by MIT OCW. Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden, MA: Blackwell Publishers, 1997. Based on Chiba and Kajiyama 1941.

11 Perturbation Theory (Chiba and Kajiyama 1941)

' '' • What is the effect of a V1 V3 V3 V3 pharyngeal constriction? F1 F3

V3'

• Does this correspond to the V1 V3 tube model above? V3'' • How do you raise F2

' '' ''' maximally? V2 V2' V4 V4 V4 V4

F2 F4

V4'

V2 V4 V4'' V2'

V4'''

Image by MIT OCW. Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden, MA: Blackwell Publishers, 1997. Based on Chiba and Kajiyama 1941.

12 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.

13 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.

14 • 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.

Reproduced from Narayanan, Shrikanth S., Abeer A. Alwan, and Katherine Haker. "Toward articulatory-acoustic models for liquid approximants based on MRI and EPG data. Part I. The laterals." The Journal of the Acoustical Society of America 101, no. 2 (1997): 1064-1077. doi: https://doi.org/10.1121/1.418030, with the permission of the Acoustical Society of America. 15 Perturbation Theory (Chiba and Kajiyama 1941)

A nice story about Am. Eng. [®] ' '' • Three constriction: labial V1 V3 V3 V3 (lip protrusion/rounding), F1 F3 ' palatal (bunching or V3 V1 V3 retroﬂexion), and '' pharyngeal. V3 • All 3 are near velocity ' V ' '' ''' maxima for F3, hence very V2 V2 4 V4 V4 V4 F F low F3. 2 4 V4'

V2 '' • But Espy-Wilson et al V4 V4 ' (2000) argue actual V2 ''' constrictions are in the V4 wrong place Image by MIT OCW. Adapted from Johnson, Keith. Acoustic and Auditory Phonetics. Malden, MA: Blackwell Publishers, 1997. Based on Chiba and Kajiyama 1941.

16 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.

Story et al (1998), MRI

Source: Story, Brad H., Ingo R. Titze, and Eric A. Hoffman. "Vocal tract area functions for an adult female speaker . based on volumetric imaging." The Journal of the Acoustical Society of America 104, no. 1 (1998): 471-487.

Ladefoged & Maddieson (1996) – mean tongue positions

© Walter de Gruyter. All rights reserved. This content is excluded from our Creative Fant (1960), Russian [i] Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ F2 2250 Hz, F3 3200 Hz

18 Hillenbrand et al (1995) – Michigan English vowel formants

3200

3000 i 2800 I eI F3(Hz) 2600 O ø E œ oU a 2400 U u 2200

2000 500 1000 1500 2000 2500 Courtesy of The Acoustical Society of America. Used with permission. F2 (Hz) Source: Hillenbrand, James, Laura A. Getty, Michael J. Clark, and Kimberlee Wheeler. "Acoustic characteristics of American English vowels." The Journal of 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.

20 Lip rounding

• Tube models 2: Fant (1960) suggests the front cavity plus lip constriction can form a helmholtz resonator.

21 Fant’s (1960) nomograms

• A more complex tube model for vowels:

Area A cm2 14 12 2 A=A min*cosh (X-Xmin)/h 10 l /A = 1/4 1/2 1 1 h = 4.75 / arcosh (8/Amin) 8 6 A = 0.25 cm2 min X = 10.5 cm 4 min 2 0 X 16 14 12 10 8 6 4 2 0 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.

22 Nomogram showing variation in constriction location and lip-rounding - 2 narrow constriction (Amin = 0.65 cm )

2 Curve L1 cm A1 cm c/s 2 1 0 8.0 Amin = 0.65cm 2 1 6.0 5000 3 1 2.0 4500 4 1 0.65 5 1 0.16 4000 F5 3500 F4 3000 F3 2500

2000 F2 1500 1000 750 F1 500 0 cm from lip unrounded -3-4 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 opening rounded -2-3 -1 0 1 2 3 4 5 6 7 8 910 11 121314 15 1617 1819 cm from glottis 20 18 16 14 12 10 8 6 4 2 0 -2 Axial 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.

23 Nomogram showing variation in constriction location and lip-rounding - 2 wider constriction (Amin = 2.5 cm )

2 Curve L1 cm A1 cm 1 0 8.0 c/s 2 Amin = 2.6 cm 2 1 6.0 5000 3 1 2.0 4 1 0.65 4500 5 1 0.16 F 4000 5 3500 F4 3000 F3 2500 2000 F2 1500 1000 750 F1 500 0 cm from lip unrounded -3-4 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 opening rounded -2-3 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 cm from glottis 20 18 16 14 12 10 8 6 4 2 0 -2 Axial 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.

24 Nomogram showing variation in constriction location and degree.

2 Amin = 0.32 cm A = 1.3 cm2 min 2 c/s 2 Curve L1 cm A1 cm Amin cm Amin = 5.0 cm 5000 1 0 8.0 0.32 2 0 6.0 1.3 4500 3 0 2.0 5.0 F 4000 5 3500 F4 3000 F3 2500 2000 F2 1500 1000 750 F1 500 0 cm from lip unrounded -3-4 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 opening state cm from glottis 20 18 16 14 12 10 8 6 4 2 0 -2 Axial coordinate of the tongue constriction center.

1 2 3

Image by MIT OCW.

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