Formants, Resonance, and Deriving Schwa

Formants, Resonance,and Deriving Schwa

March 10, 2009

Loose Ends• Course Project reports!

• Hand back mid-terms.

• New guidelines to hand out…

• As well as an extra credit assignment.

• Any questions?

Mid-Term

Rehash

Mid-Term

Rehash

Mid-Term

Rehash

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

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

• Check out Peter Frampton.

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.

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.

“Band”?• Way back when, we discussed low-pass filters:

• This filter passes frequencies below 250 Hz.

• High-pass filters are also possible.

Band-Pass Filters• A band-pass filter combines both high- and low-pass filters.

• It passes a “band” of frequencies around a center frequency.

Band-Pass Filtering

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

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.

Different Bandwidths

narrow band wide band

Your Grandma’s Spectrograph

• Originally, spectrographic analyzing filters were constructed to have either wide or narrow bandwidths.

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

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

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

modal voice tense voice

A Real Vowel Spectrum

Why does the “spectral tilt” go up and down

in this example?

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.

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.

Wide Band Spectrogram• Formants appear as dark horizontal bars in a wide band spectrogram.

• Each formant has both a center frequency and a bandwidth.

formants

F1

F2

F3

Narrow-Band Spectrogram• A “narrow-band spectrogram” clearly shows the harmonics of speech sounds.

• …but the formants are less distinct.

harmonics

A Static Spectrum

Note:

F0 160 Hz

F1

F2

F3 F4

Questions1. How does resonance occur?

• And how does it occur in our vocal tracts?

2. Why do sounds resonate at particular frequencies?

3. How can we change the resonant frequencies of the vocal tract? (spectral changes)

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.

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.

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

Sound in a Closed Tube

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?

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.

Tacoma Narrows Movie

Standing Wave Terminology

node: position of zero pressure change in a standing wave

node

Standing Wave Terminology

anti-node: position of maximum pressure change in a standing wave

anti-nodes

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)

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.

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

Wavelength• A standing wave has a wavelength

• The wavelength is the distance (in space) it takes a standing wave to go:

1. from a pressure peak

2. down to a pressure minimum

3. back up to a pressure peak

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

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

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

= L

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

= L

= 2L / 3

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

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

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

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?

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.

Open Tubes, part 1

Open Tubes, part 2

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

glottislips (open)

Open Tube Resonances• Standing waves in an open tube will look like this:

= 4L

L

= 4L / 3

= 4L / 5

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

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

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?

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