Formants, Resonance, and Deriving Schwa

1. Formants, Resonance,and Deriving Schwa March 10, 2009

2. Loose Ends • Course Project reports! • Hand back mid-terms. • New guidelines to hand out… • As well as an extra credit assignment. • Any questions?

3. Mid-Term Rehash

4. Mid-Term Rehash

5. Mid-Term Rehash

6. For the Skeptics • Sounds that exhibit spectral change over time sound like speech, even if they’re not speech • Example 2: wah pedal • shapes the spectral output of electrical musical instruments

7. Phonetics Comes Alive! • It is possible to take spectral change rock one step further with the talk box. • Check out Peter Frampton.

8. All Sorts of Trade-Offs • The problem with Fourier Analysis: • We can only check sinewave frequencies which fit an integer number of cycles into the window • Window length = .005 seconds  • 200 Hz, 400 Hz, 600 Hz, … 10,000 Hz • We can increase frequency resolution by adding zeroes to the end of a window. • At the same time, zero-padding smooths the spectrum. • We can increase frequency accuracy by lengthening the window. • We can increase the frequency range by increasing the sampling rate.

9. Morals of the Fourier Story • Shorter windows give us: • Better temporal resolution • Worse frequency resolution • = wide-band spectrograms • Longer windows give us: • Better frequency resolution • Worse temporal resolution • = narrow-band spectrograms • Higher sampling rates give us... • A higher limit on frequencies to consider.

10. “Band”? • Way back when, we discussed low-pass filters: • This filter passes frequencies below 250 Hz. • High-pass filters are also possible.

11. Band-Pass Filters • A band-pass filter combines both high- and low-pass filters. • It passes a “band” of frequencies around a center frequency.

12. Band-Pass Filtering • Basic idea: components of the input spectrum have to conform to the shape of the band-pass filter.

13. Bandwidth • Bandwidth is the range of frequencies over which a filter will respond at .707 of its maximum output. bandwidth • Half of the acoustic energy passed through the filter fits within the bandwidth. • Bandwidth is measured in Hertz.

14. Different Bandwidths wide band narrow band

15. Your Grandma’s Spectrograph • Originally, spectrographic analyzing filters were constructed to have either wide or narrow bandwidths.

16. Narrow-Band Advantages • Narrow-band spectrograms give us a good view of the harmonics in a complex wave… • because of their better frequency resolution. modal voicing EGG waveform

17. Narrow-Band Advantages • Narrow-band spectrograms give us a good view of the harmonics in a complex wave… • because of their better frequency resolution. tense voicing EGG waveform

18. Comparison • Remember that modal and tense voice can be distinguished from each other by their respective amount of spectral tilt. modal voice tense voice

19. A Real Vowel Spectrum Why does the “spectral tilt” go up and down in this example?

20. The Other Half • Answer: we filter the harmonics by taking advantage of the phenomenon of resonance. • Resonance effectively creates a series of band-pass filters in our mouths. + = • Wide-band spectrograms help us see properties of the vocal tract filter.

21. Formants • Rather than filters, though, we may consider the vocal tract to consist of a series of “resonators”… • with center frequencies, • and particular bandwidths. • The characteristic resonant frequencies of a particular articulatory configuration are called formants.

22. Wide Band Spectrogram • Formants appear as dark horizontal bars in a wide band spectrogram. • Each formant has both a center frequency and a bandwidth. F3 formants F2 F1

23. Narrow-Band Spectrogram • A “narrow-band spectrogram” clearly shows the harmonics of speech sounds. • …but the formants are less distinct. harmonics

24. A Static Spectrum F1 F2 F4 F3 Note: F0  160 Hz

25. Questions • How does resonance occur? • And how does it occur in our vocal tracts? • Why do sounds resonate at particular frequencies? • How can we change the resonant frequencies of the vocal tract? (spectral changes)

26. Some Answers • Resonance: • when one physical object is set in motion by the vibrations of another object. • Generally: a resonating object reinforces (sound) waves at particular frequencies • …by vibrating at those frequencies itself • …in response to the pressures exerted on it by the (sound) waves. • In the case of speech: • The mouth (and sometimes, the nose) resonates in response to the complex waves created by voicing.

27. Traveling Waves • Resonance occurs because of the reflection of sound waves. • Normally, a wave will travel through a medium indefinitely • Such waves are known as traveling waves.

28. Reflected Waves • If a wave encounters resistance, however, it will be reflected. • What happens to the wave then depends on what kind of resistance it encounters… • If the wave meets a hard surface, it will get a true “bounce” • Compressions (areas of high pressure) come back as compressions • Rarefactions (areas of low pressure) come back as rarefactions

29. Sound in a Closed Tube

30. Wave in a closed tube • With only one pressure pulse from the loudspeaker, the wave will eventually dampen and die out • What happens when: • another pressure pulse is sent through the tube right when the initial pressure pulse gets back to the loudspeaker?

31. Standing Waves • The initial pressure peak will be reinforced • The whole pattern will repeat itself • Alternation between high and low pressure will continue • ...as long as we keep sending in pulses at the right time • This creates what is known as a standing wave.

32. Tacoma Narrows Movie

33. Standing Wave Terminology node node: position of zero pressure change in a standing wave

34. Standing Wave Terminology anti-nodes anti-node: position of maximum pressure change in a standing wave

35. 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 • ≈ 780 miles per hour (1260 kph)

36. 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.

37. 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

38. Wavelength • 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

39. 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 •  = 2 * L L

40. 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

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

42. Higher Resonances • It is possible to set up resonances with higher frequencies, and shorter wavelengths, in a tube. •  = L •  = 2L / 3

43. Higher Resonances • 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

44. Higher Resonances • 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

45. 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

46. 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?

47. 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.

48. Open Tubes, part 1

49. Open Tubes, part 2

50. 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)