resonance, revisited (again)
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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!. - PowerPoint PPT PresentationTRANSCRIPT
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Resonance, Revisited (Again)
March 13, 2014
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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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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)
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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.
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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
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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.
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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
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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
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Higher Resonances• It is possible to set up resonances with higher frequencies, and shorter wavelengths, in a tube.
= L
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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?
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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
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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
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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
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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
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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
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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?
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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.
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Open Tubes, part 1
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Open Tubes, part 2
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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)
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Open Tube Resonances• Standing waves in an open tube will look like this:
= 4L
L
= 4L / 3
= 4L / 5
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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
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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
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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?
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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
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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
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“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
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Source + Filter = Output
+
=
This is the source/filter
theory of speech production.
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Source + Filter(s)
Note:
F0 160 Hz
F1
F2
F3 F4
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Schwa at different pitches
100 Hz 120 Hz
150 Hz
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More Than Schwa
• Formant frequencies differ between vowels…
• because vowels are produced with different articulatory configurations
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Remember…• Vowels are articulated with characteristic tongue and lip shapes.
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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?
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The Vowel Space
o
The Vowel Space
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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
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Male Formant Averages
200
300
400
500
600
700
800
900
1000
10001500200025003000
F2
F1
[i][u]
[æ]
(From some old phonetics class data)
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Female Formant Averages
200
300
400
500
600
700
800
900
1000
10001500200025003000
F2
F1
[i] [u]
[æ]
(From some old phonetics class data)
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Combined Formant Averages
200
300
400
500
600
700
800
900
1000
10001500200025003000
F2
F1
(From some old phonetics class data)
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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
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Prototypical Voices• Andre the Giant: (very) low F0, low formant frequencies
• Goldie Hawn/Pretty Tiffany: high F0, high formant frequencies
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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.
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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