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!

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)

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• New concept: 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

• For a waveform representation of a standing wave, the x-axis represents distance, not time.

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

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

• Q: What will the relationship between and L be for the next highest resonance?

First ResonanceTime 1: initial impulse is sent down the tubeTime 2: initial impulse bounces at end of tubeTime 3: impulse returns to other end and is reinforced by a new impulse

• Resonant period = Time 3 - Time 1

Time 4: reinforced impulse travels back to far end

Second ResonanceTime 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 reinforcedResonant period = Time 2 - Time 1

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

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

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

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

lipsglottis

“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

Source + Filter = Output

+

=

This is the source/filter

theory of speech production.

Source + Filter(s)

Note:

F0 160 Hz

F1

F2

F3 F4

Schwa at different pitches

100 Hz 120 Hz

150 Hz

More Than Schwa

• Formant frequencies differ between vowels…

• because vowels are produced with different articulatory configurations

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

Vowel Dimensions• For this reason, vowels have traditionally been

described according to four (pseudo-)articulatory parameters:

1. Height (of tongue)

2. Front/Back (of tongue)

3. Rounding (of lips)

4. Tense/Lax

= amount of effort?

= muscle tension?

The Vowel Space

o

The Vowel Space

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

Male Formant Averages

200

300

400

500

600

700

800

900

1000

10001500200025003000

F2

F1

[i][u]

[æ]

(From some old phonetics class data)

Female Formant Averages

200

300

400

500

600

700

800

900

1000

10001500200025003000

F2

F1

[i] [u]

[æ]

(From some old phonetics class data)

Combined Formant Averages

200

300

400

500

600

700

800

900

1000

10001500200025003000

F2

F1

(From some old phonetics class data)

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

Prototypical Voices• Andre the Giant: (very) low F0, low formant frequencies

• Goldie Hawn/Pretty Tiffany: high F0, high formant frequencies

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.

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