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Resonance, Revisited (Again)

Resonance, Revisited (Again). March 13, 2014. Practicalities. I’m still working through the pile of grading… Although I can report that most of the third course project reports were really good. For today: let’s figure out how vocal tract length determines formant frequencies!.

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Resonance, Revisited (Again)

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  1. Resonance, Revisited (Again) March 13, 2014

  2. Practicalities • I’m still working through the pile of grading… • Although I can report that most of the third course project reports were really good. • For today: let’s figure out how vocal tract length determines formant frequencies!

  3. Resonant Frequencies • Remember: a standing wave can only be set up in the tube if pressure pulses are emitted from the loudspeaker at the appropriate frequency • Q: What frequency might that be? • It depends on: • how fast the sound wave travels through the tube • how long the tube is • How fast does sound travel? • ≈ 350 meters / second = 35,000 cm/sec • ≈ 1260 kilometers per hour (780 mph)

  4. Calculating Resonance • A new pressure pulse should be emitted right when: • the first pressure peak has traveled all the way down the length of the tube • and come back to the loudspeaker.

  5. Calculating Resonance • Let’s say our tube is 175 meters long. • Going twice the length of the tube is 350 meters. • It will take a sound wave 1 second to do this • Resonant Frequency: 1 Hz 175 meters

  6. Wavelength • New concept: a standing wave has a wavelength • The wavelength is the distance (in space) it takes a standing wave to go: • from a pressure peak • down to a pressure minimum • back up to a pressure peak • For a waveform representation of a standing wave, the x-axis represents distance, not time.

  7. First Resonance • The resonant frequencies of a tube are determined by how the length of the tube relates to wavelength (). • First resonance (of a closed tube): • sound must travel down and back again in the tube • wavelength = 2 * length of the tube (L) •  = 2 * L L

  8. Calculating Resonance • distance = rate * time • wavelength = (speed of sound) * (period of wave) • wavelength = (speed of sound) / (resonant frequency) •  = c / f • f  = c • f = c /  • for the first resonance, • f = c / 2L • f = 350 / (2 * 175) = 350 / 350 = 1 Hz

  9. Higher Resonances • It is possible to set up resonances with higher frequencies, and shorter wavelengths, in a tube. •  = L

  10. Higher Resonances • It is possible to set up resonances with higher frequencies, and shorter wavelengths, in a tube. •  = L •  = 2L / 3 • Q: What will the relationship between  and L be for the next highest resonance?

  11. First Resonance Time 1: initial impulse is sent down the tube Time 2: initial impulse bounces at end of tube Time 3: impulse returns to other end and is reinforced by a new impulse Time 4: reinforced impulse travels back to far end • Resonant period = Time 3 - Time 1

  12. Second Resonance Time 1: initial impulse is sent down the tube Time 2: initial impulse bounces at end of tube + second impulse is sent down tube Time 3: initial impulse returns and is reinforced; second impulse bounces Time 4: initial impulse re-bounces; second impulse returns and is reinforced Resonant period = Time 2 - Time 1

  13. Doing the Math • It is possible to set up resonances with higher frequencies, and shorter wavelengths, in a tube. •  = L f = c /  f = c / L f = 350 / 175 = 2 Hz

  14. Doing the Math • It is possible to set up resonances with higher frequencies, and shorter wavelengths, in a tube. •  = 2L / 3 f = c /  f = c / (2L/3) f = 3c / 2L f = 3*350 / 2*175 = 3 Hz

  15. Patterns • Note the pattern with resonant frequencies in a closed tube: • First resonance: c / 2L (1 Hz) • Second resonance: c / L (2 Hz) • Third resonance: 3c / 2L (3 Hz) • ............ • General Formula: • Resonance n: nc / 2L

  16. Different Patterns • This is all fine and dandy, but speech doesn’t really involve closed tubes • Think of the articulatory tract as a tube with: • one open end • a sound pulse source at the closed end • (the vibrating glottis) • At what frequencies will this tube resonate?

  17. Anti-reflections • A weird fact about nature: • When a sound pressure peak hits the open end of a tube, it doesn’t get reflected back • Instead, there is an “anti-reflection” • The pressure disperses into the open air, and... • A sound rarefaction gets sucked back into the tube.

  18. Open Tubes, part 1

  19. Open Tubes, part 2

  20. The Upshot • In open tubes, there’s always a pressure node at the open end of the tube • Standing waves in open tubes will always have a pressure anti-node at the glottis • First resonance in the articulatory tract glottis lips (open)

  21. Open Tube Resonances • Standing waves in an open tube will look like this: •  = 4L •  = 4L / 3 •  = 4L / 5 L

  22. Open Tube Resonances • General pattern: • wavelength of resonance n = 4L / (2n - 1) • Remember: f = c /  • fn = c • 4L / (2n - 1) • fn = (2n - 1) * c • 4L

  23. Deriving Schwa • Let’s say that the articulatory tract is an open tube of length 17.5 cm (about 7 inches) • What is the first resonant frequency? • fn = (2n - 1) * c • 4L • f1 = (2*1 - 1) * 350 = 1 * 350 = 500 • (4 * .175) .70 • The first resonant frequency will be 500 Hz

  24. Deriving Schwa, part 2 • What about the second resonant frequency? • fn = (2n - 1) * c • 4L • f2 = (2*2 - 1) * 350 = 3 * 350 = 1500 • (4 * .175) .70 • The second resonant frequency will be 1500 Hz • The remaining resonances will be odd-numbered multiples of the lowest resonance: • 2500 Hz, 3500 Hz, 4500 Hz, etc. • Want proof?

  25. The Big Picture • The fundamental frequency of a speech sound is a complex periodic wave. • In speech, a series of harmonics, with frequencies at integer multiples of the fundamental frequency, pour into the vocal tract from the glottis. • Those harmonics which match the resonant frequencies of the vocal tract will be amplified. • Those harmonics which do not will be damped. • The resonant frequencies of a particular articulatory configuration are called formants. • Different patterns of formant frequencies = • different vowels

  26. Vowel Resonances • The series of harmonics flows into the vocal tract. • Those harmonics at the “right” frequencies will resonate in the vocal tract. • fn = (2n - 1) * c • 4L • The vocal tract filters the source sound glottis lips

  27. “Filters” • In speech, the filter = the vocal tract • This graph represents how much the vocal tract would resonate for sinewaves at every possible frequency: • The resonant frequencies are called formants

  28. Source + Filter = Output + This is the source/filter theory of speech production. =

  29. Source + Filter(s) F1 F2 F4 F3 Note: F0  160 Hz

  30. Schwa at different pitches 100 Hz 120 Hz 150 Hz

  31. More Than Schwa • Formant frequencies differ between vowels… • because vowels are produced with different articulatory configurations

  32. Remember… • Vowels are articulated with characteristic tongue and lip shapes.

  33. Vowel Dimensions • For this reason, vowels have traditionally been described according to four (pseudo-)articulatory parameters: • Height (of tongue) • Front/Back (of tongue) • Rounding (of lips) • Tense/Lax = amount of effort? = muscle tension?

  34. The Vowel Space The Vowel Space o

  35. Formants and the Vowel Space • It turns out that we can get to the same diagram in a different way… • Acoustically, vowels are primarily distinguished by their first two formant frequencies: F1 and F2 • F1 corresponds to vowel height: • lower F1 = higher vowel • higher F1 = lower vowel • F2 corresponds to front/backness: • higher F2 = fronter vowel • lower F2 = backer vowel

  36. [i] [u] [æ] (From some old phonetics class data)

  37. [i] [u] [æ] (From some old phonetics class data)

  38. (From some old phonetics class data)

  39. Women and Men • Both source and filter characteristics differ reliably between men and women • F0: depends on length of vocal folds • shorter in women  higher average F0 • longer in men  lower average F0 • Formants: depend on length of vocal tract • shorter in women  higher formant frequencies • longer in men  lower formant frequencies

  40. Prototypical Voices • Andre the Giant: (very) low F0, low formant frequencies • Goldie Hawn/Pretty Tiffany: high F0, high formant frequencies

  41. F0/Formant mismatches • The fact that source and filter characteristics are independent of each other… • means that there can sometimes be source and filter “mismatches” in men and women. • What would high F0 combined with low formant frequencies sound like? • Answer: Julia Child.

  42. F0/Formant mismatches • Another high F0, low formants example: • Roy Forbes, of Roy’s Record Room (on CKUA 93.7 FM) • The opposite mis-match = • Popeye: low F0, high formant frequencies

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