Physics 1251 The Science and Technology of Musical Sound

1
Physics 1251The Science and Technology of
Musical Sound

• Unit 3
• Session 35
• Pipes, Voice and Percussion
• Review

2
Physics 1251 Unit 3 Session 35 Pipes, Voice
and Percussion

• Foolscap Quiz
• What are the first three harmonics that speak
  in the harmonic series of a Tympani tuned to A3
  (110 Hz)?

Answer approximately 220 Hz, 330 Hz, 440 Hz. The
fundamental 110 Hz is missing.
3
Physics 1251 Unit 3 Session 35 Pipes, Voice
and Percussion

• 1' Lecture
• The pitch of pipes and the voice is determined by
  a harmonic series.
• Percussion most often have no pitch because they
  lack a harmonic series.
• Vibration recipes of pipes, the voice and
  percussion arise from the modes of vibration the
  air column, of the vocal folds and of the
  membrane, plate, or block, respectively.

4
Physics 1251 Unit 3 Session 35 Pipes, Voice
and Percussion

• The key to understanding the
• Sound of Music!

6
2
4
1
3
7
5
Harmonic Series produce a sense of musical
intonation.
5
Physics 1251 Unit 3 Session 35 Pipes, Voice
and Percussion

• 80/20The timbre of an instruments sounds depends
  on its vibration recipe.

fn n f1
Pitched
Amplitude
f1
2f1
3f1
4f1
fn m xn m f1
Unpitched
Amplitude
f01
Frequency
6
Physics 1251 Unit 3 Session 26 Sound in Pipes

• 1' Lecture
• Sound in pipes can produce standing waves in the
  air column.
• Standing waves in air columns produce pressure
  nodes and displacement nodes (and antinodes) at
  different places.
• A change in the acoustic impedance of the air
  column produces a reflection.
• Organ pipes and the flute are examples of open or
  unstopped pipes.

7
Physics 1251 Unit 3 Session 26 Sound in Pipes

• Standing Waves in a Cylindrical Pipe
• A Closed or Stopped Pipe the pressure wave
  reflects without inversion, but the displacement
  wave inverts upon reflection.
• Thus, a pressure anti-node will occur at the
  wall but, on the other hand, a displacement node
  will occur at the same place.

8
Physics 1251 Unit 3 Session 26 Sound in Pipes

• Comparison of Pressure and Displacement Standing
  Wave in a Double Stopped Pipe

?/4
?/4
Pressure Wave
Displacement Wave
9
Physics 1251 Unit 3 Session 26 Sound in Pipes

• 80/20Acoustic Impedance
• Z p/U
• Acoustic Impedance is the ratio of the pressure p
  of a sound wave to the flow U ( u S) that
  results.
• For a plane wave in a tube of cross section S
  (m2) in air the acoustic impedance is
• Z ?v/S 415/ S rayl

10
Physics 1251 Unit 3 Session 26 Sound in Pipes

• 80/20For Stopped Pipe
• Nna odd number 2n-1, n1,2,3,4
• ?n 4L/ Nna 4L / (2n-1)
• fstopped f2n-1 v/ ?2n-1 (2n-1) v/ 4L
• 80/20Only odd harmonics of fstopped 1 v/4L.

11
Physics 1251 Unit 3 Session 26 Sound in Pipes

• 80/20For Open Pipe
• Nna even number 2n, n1,2,3,4
• ?n 4L / Nna 4L/(2n) 2L/ n
• fopen fn v/ ?n n ? v/2L
• 80/20All harmonics of fopen 1 v/2L 2
  fstopped 1

12
Physics 1251 Unit 3 Session 26 Sound in Pipes

• End Correction for Open Pipe without Flange
• d 0.6 a for a ? d 0 a for a gt ? / 4

a Radius
L d
d
13
Physics 1251 Unit 3 Session 26 Sound in Pipes

• Transverse Flute
• 80/20The transverse flute is a cylindrical open
  pipe.

Mouthpiece is open
14
Physics 1251 Unit 3 Session 26 Sound in Pipes

• Summary
• fopen fn n ? v/2L
• fstopped f2n-1 (2n-1) v/ 4L
• Stopped and open cylindrical pipes have different
  timbres.
• Impedance Z p/U
• An abrupt change in Z is responsible for the
  reflections that lead to standing waves in pipes.

15
Physics 1251 Unit 3 Session 27 Flutes et cetera

• 1' Lecture
• Flutes and flue pipes are driven by fluid flow
  instabilities at their mouth.
• Standing waves in open air columns of flutes
  determine the pitch.
• Open holes in the flute tube change the effective
  length of the air column.

16
Physics 1251 Unit 3 Session 27 Flutes et cetera

• The Flute
• The transverse flute is acoustically driven by
  the fluid flow instabilities whose frequency is
  controlled by the feedback of the resonances of
  the pipe.

Standing wave frequencies
Flow Instability
Feedback
17
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Transverse Flute
• 80/20The flute is driven by air flow against the
  edge of the embrochure.

Air flow
Embrochure
18
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Edge Tone
• 80/20An air stream striking against an edge
  produces a fluctuating instability in flow.

Air Stream
Edge
The flow alternates sides.
19
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Why does the stream oscillate?

Short answer positive feedback.

• When the stream bends to the left, the stream
  moves faster on the right side.
• Bernoullis Principle tells us that the faster
  the flow, the lower the pressure.
• Therefore, the left-flowing stream will bend
  back to the right

20
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Bernoulli Effect
• 80/20The pressure in a fluid decreases as the
  velocity increases.

Hold the foolscap by the edge and blow across the
top. What do you observe?
21
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Edge Tone

fedge 0.4 vjet / 2 b 0.2 vjet /b
u 0.4 vjet
b
b
vjet
u
22
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Feedback from the acoustic standing wave locks
  the frequency of the oscillation if the edge tone
  is near the fundamental frequency.

fedge 0.2 vjet /b fn n v/ 2L
fedge fn
Displacement wave
23
Physics 1251 Unit 3 Session 27 Flutes et cetera

• The Problem with Flutes
• Only about 1 of the energy of the air stream
  produces sound.
• Playing louder means more air flow.
• More air flow means higher jet velocity
• Edge tone goes sharp
• Worse in Recorder than in Transverse Flute
• Player must lip tone into tune

24
Physics 1251 Unit 3 Session 27 Flutes et cetera

• How does one play the notes?

• By effectively changing the length of the air
  column.

• Opening holes introduces reflections that change
  the standing wave length.

Displacement wave
f n' n ? v/2Leff
25
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Cross Fingering
• 80/20The position and size of the open holes
  modify the effective length of the air column and
  consequently the pitch.

26
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Why does the size of the hole matter?
• Z p/U
• Impedance pressure/flow

Displacement ?Flow U
Z '
Z
27
Physics 1251 Unit 3 Session 27 Flutes et cetera

• Summary
• Flutes and flue pipes are open columns of air,
  with fn n v/2L, n 1,2,3,4.
• Flue pipes are excited by flow instabilities of
  the air steam in the embrochure or fipple.
• The frequency range is selected by the edge tone.
• The pitch is determined by the effective length
  of the pipe.
• Open holes determine the effective length of the
  pipe.

28
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• 1' Lecture
• Reed instruments are stopped pipes.
• The clarinet has a cylindrical bore and is a
  stopped pipe consequently, only odd harmonics
  are significant.
• Conical pipes exhibit all harmonics, even in
  stopped pipes.
• The saxophone, oboe and bassoon?all have conical
  bores.

29
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• Comparison
  of Flute and Clarinet Registers
• Overblown flutes jump from a fundamental f1 v/2L
  to an octave f2 2f1 in the second register an
  octave (2x) and a perfect fifth (3/2) f3 3 f1
  3 (v/2L) in the third register.
• Overblown clarinets jump from a fundamental
  f1 v/4L to an octave (2x) and a fifth (3/2
  )?the twelfth? in the second register, because
  only odd harmonics produce standing waves in a
  stopped cylindrical pipe.

30
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• Reed Instruments
• The reed produces a pulsation in the pressure
  admitted to the pipe the pressure standing wave
  feeds back to control the oscillations of the
  reed.

Standing wave frequencies
Reed pulsations
Feedback
31
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• The Single Reed
• 80/20The reed opens and closes like a valve,
  pressurizing the pipe when open, closing due to
  the Bernoulli effect when the air flows.

Reed
32
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• Hard and Soft Reeds
• 80/20A hard reed is one for which the frequency
  is determined by its stiffness and dimensions.
• A soft reed flexes easily and vibrates at the
  frequency of external pressure fluctuations.

Soft Reeds
Hard Reed Harmonica
Clarinet
Oboe
33
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• The Double Reed
• 80/20The reed opens and closes like a valve,
  pressurizing the pipe when open, closing due to
  the Bernoulli effect when the air flows.

Pressure Pulses
Reed Tip
34
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• Bernoulli Effect
• 80/20The pressure in a fluid decreases as the
  velocity increases.

Thus, as the air flows past the reed, it is
forced closed.
Bernoulli Effect
35
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• 80/20Feedback from the pressure standing wave
  locks the frequency of the oscillation of the
  reed.

f2n-1 (2n-1) v/ 4L'
Pressure wave
L' L 0.3 d
0.3 d
36
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

80/20For a stopped conical pipe fn n v / 2(L'
c) if c ltlt ? L' L 0.3 d
L'
d
c
0.3 d
37
Physics 1251 Unit 3 Session 28 Clarinets et
cetera

• Summary
• Reed Instruments are stopped pipes.
• L' L 0.3 d
• f2n-1 (2n-1) v/4L' for stopped cylindrical
  pipes such as the clarinet.
• fn n v/ 2(L'c) for stopped conical pipes such
  as the saxophone, oboe, bassoon, etc.
• Soft reeds act as pressure valves that respond to
  the frequency fed back from the standing waves of
  the pipe.

38
Physics 1251 Unit 3 Session 29 Brass
Instruments

• Brass Instruments
• The lips produce a pulsation in the pressure
  admitted to the pipe the pressure standing wave
  feeds back to control the oscillations of the
  plays lips.

Lip-valve pulsations
Standing wave frequencies
Feedback
39
Physics 1251 Unit 3 Session 29 Brass
Instruments

• The Lip Valve
• 80/20Brass instruments are played by the players
  lips.
• Breath pressure, muscle tension and pressure
  feedback from the pipe determine the frequency of
  the opening and closing of the lips.

Louis Armstrong trumpet (1901-1971)
40
Physics 1251 Unit 3 Session 29 Brass
Instruments

• Lip Valve
• The lips of the player act as a valve that admits
  pressure pulses into the pipe.
• The frequency is determined by the breath air
  pressure, the lip tension and the resonances of
  the pipe.

41
Physics 1251 Unit 3 Session 29 Brass
Instruments

• 80/20Brass Instruments
• are stopped pipes.
• The players lips produce a
  displacement node
  (pressure antinode)
  at the mouthpiece.
• A displacement
  anti-node (pressure node)
  exists at the bell.

Winton Marsalis Trumpet
42
Physics 1251 Unit 3 Session 29 Brass
Instruments

• The Mouthpiece

Cup Volume
80/20The Cup Volume and the diameter of the
constriction leading to the back bore are more
important than the shape of the cavity.
Diameter
43
Physics 1251 Unit 3 Session 29 Brass
Instruments

• Resonance for Combination Pipes

80/20The Brass mouthpiece lowers the high
frequency resonances.
f
Cone with mouthpiece
Cone
44
Physics 1251 Unit 3 Session 29 Brass
Instruments

• The pitch is changed by pipe length and
  excitation of resonances.

By means of slides and valves the length is
changed.
45
Physics 1251 Unit 3 Session 29 Brass
Instruments

• Resonance for Combination Pipes

f
Cone/ Cylinder
0/100
25/75
50/50
40/60
20/80
100/0
46
Physics 1251 Unit 3 Session 29 Brass
Instruments

• Resonances for Combination Bores
  in Brass Instruments
• 80/20A 50 cylindrical ? 50 conical bore has a
  nearly harmonic series.

47
Physics 1251 Unit 3 Session 29 Brass
Instruments

• The Bell

Exponential Horn
a ao exp(m x) b
80/20m is called the
flare constant. Larger m means more rapid flare.
48
Physics 1251 Unit 3 Session 29 Brass
Instruments

• The Bell

Bessel Horns
a ao e-(ex) b
80/20Called Bessel Horns because the standing
wave follows a Bessel Function.
49
Physics 1251 Unit 3 Session 29 Brass
Instruments

• Summary
• Brass Instruments are stopped pipes.
• The pipe bore is designed to give resonances that
  are harmonic.
• The pedal tone (the lowest note) is not harmonic.
• The players lips are a soft reed.
• The pitch is changed by changing the length and
  exciting resonances.

50
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• 1' Lecture
• The pitch of a wind instrument is determined by
  the length and shape of its air column.
• The effective length of the air column is
  controlled with holes, valves and slides.
• Feedback from the resonances of the pipe select
  the frequency of oscillation of the jet, reed or
  lip-valve.
• The excitation, transmission and emittance of the
  sound in the horn determine the timbre of the
  instrument.

51
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Transverse Flute
• 80/20The flute is driven by air flow against the
  edge of the embrochure hole.
• 80/20A pressure node exists at the open hole.

Air flow
Embrochure
52
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• The Single Reed
• 80/20The reed opens and closes like a valve,
  pressurizing the pipe when open, closing due to
  the Bernoulli effect when the air flows.
• 80/20A pressure anti-node exists at the reed.

Reed
53
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• The Double Reed
• 80/20The reed opens and closes like a valve,
  pressurizing the pipe when open, closing due to
  the Bernoulli effect when the air flows.
• 80/20A pressure anti-node exists at the reed.

Pressure Pulses
Reed Tip
54
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• The Lip Valve
• 80/20Brass instruments are played by the players
  lips that form a lip valve.
• 80/20A pressure anti-node exists at the players
  lips.

Louis Armstrong trumpet (1901-1971)
55
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Comparison of Wind Instruments

f
Pedal Tone
fo (1?)v/4(Lc)
L
f1 v/2L
f1 v/4L
f1 v/2(Lc)
Other Woodwinds
Clarinet
Flute
Brass
c
56
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Comparison of Wind Instruments (contd.)

Open Cylinder Np Np fn nf1 f1 v/2L
Stopped Cylinder Ap Np f2n-1
(2n-1)f1 f1 v/4L
Stopped Cone Ap Np fn nf1
f1 v/2(Lc)
Stopped Combination Ap Np
fn nf0 f0 (1?)v/4(Lc)
fo (1?)v/4(Lc)
L
f1 v/2L
f1 v/4L
f1 v/2(Lc)
Other Woodwinds
Clarinet
Flute
Brass
c
57
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• 80/20In the flute, feedback from the acoustic
  standing wave locks the frequency of the
  oscillation if the edge tone is near the
  fundamental frequency.

Displacement wave
fedge 0.2 vjet /b fn n v/ 2L
fedge fn
58
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• 80/20IIn reed instruments, feedback from the
  pressure standing wave locks the frequency of the
  oscillation of the reed.

f2n-1 (2n-1) v/ 4L'
Pressure wave
L' L 0.3 d
0.3 d
59
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Feedback from Resonaces
• 80/20The pitch of a wind instrument is determined
  by the influence on the jet/reed/lip-valve of
  feedback from the pressure/displacement standing
  waves in the pipe.

60
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Wind Instruments
• A jet produces a fluctuating air flow, while a
  reed or the lips produce pressure pulsations, the
  frequencies of which are controlled by feedback
  from standing waves in the horn.

Standing waves in horn
Flow fluctuations or Pressure pulsations
Feedback
61
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Effect of Excitation
• The mode of excitation of the horn significantly
  influences the harmonic recipe of the air column.
• The harmonics will only be as strong as the
  excitation of the jet/reed/lip-valve.

62
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• The Mouthpiece

Cup Volume
80/20The Cup Volume and the diameter of the
constriction leading to the back bore are the
most important factors in determining the
frequency spectrum of the mouthpiece.
Diameter
63
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Driven Pipe Vibration Recipe

Pipe Spectrum
A
Mouthpiece Spectrum
A
Driven Pipe Spectrum
A
Frequency
64
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Effect of the Pipe
• A pipe is three dimensional therefore, 3-D modes
  of oscillation are possible in the pipe.
• 80/20Only those modes with frequency above a
  Cut-off Frequency fc will exist in the pipe.
• f gt fc for propagation.

65
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Modes of Vibration of a Column of Air

(0,0)
D
(1,0)
(2,0)
Cut Off Frequency fc qn m v/D
for f lt fc no propagation q00 0 q10 0.59
q20 0.97
66
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Effect of Modes on Spectrum
• More modes implies more intensity.
• Most influential in high f harmonics.
• Shape and relative diameter of pipe influence
  modes.
• Thus, a square organ pipe has a different timbre
  than does a round organ pipe because of the modes.

67
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Reflections from the array of holes in a woodwind
  affect the relative strength of the high
  frequency harmonics in the pipe.

Displacement wave
Reflections from holes (closed and open)
68
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Effect of Holes on Transmission
• Larger holes have greater effect.
• A high pass filter Low frequencies tend to be
  reflected more and high frequencies transmitted
  more.
• The holes make a brighter sounding instrument.

69
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Reflections from joints and imperfections affect
  the relative strength of the high frequency
  harmonics in the pipe.

Reflections
70
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Filtering of Wind Instrument Sound
• The vagaries of transmission of the various
  frequency components in the pipe produce a
  filtering effect on the frequency spectrum of the
  sound.

Transmission through horn
71
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Radiation of Sound from Wind Instruments
• The radiation characteristics of the bell shape
  the harmonic recipe and strongly influence the
  timbre of the instrument.

Radiation Characteristics
72
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• 80/20The diameter of the mouth and the flare rate
  of the bell determine the radiation
  characteristics of brass instruments.

• The larger the bore diameter, the more intense
  the low frequency harmonics.
• The more rapid the flare, the more the low
  frequencies are reflected, and thus, the more
  high frequency harmonics are radiated.

Trumpet
Cornet
Flugel Horn
73
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• The Bell

Exponential Horn
a ao exp(m x) b
80/20m is called the
flare constant. Larger m means more rapid flare.
74
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• The Bell

Bessel Horns
a ao e-(ex) b
80/20Called Bessel Horns because the standing
wave follows a Bessel Function.
75
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Mutes
• The French Horn players hand modifies the
  radiation characteristics of the horn, as well as
  the effective flare.
• Mutes reduce the effective area of the horn and,
  therefore, reduce the intensity.
• Mutes tend to reduce more the first and second
  harmonic of the pipe than higher frequency
  harmonics due to their internal modes of
  oscillation.
• Mutes make brass sound thin and reedy.

76
Physics 1251 Unit 3 Session 30 The Timbre
of Wind Instruments

• Summary
• The pitch of a wind instrument is determined by
  the length and shape of its air column.
• Feedback from the resonances of the pipe select
  the frequency of oscillation of the jet, reed or
  lip-valve.
• The excitation, transmission and emittance of the
  sound in the horn determine the timbre of the
  instrument.

77
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• 1' Lecture
• The vocal folds, located in the larynx, produce
  vibrations in the vocal tract.
• The vocal tract is a stopped air column
  approximately 17 cm long. It resonates at a
  fundamental frequency of about 500 Hz.
• The shape of the vocal tract provides an acoustic
  filter of the harmonics produced by the vocal
  folds.

78
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Anatomy of the Human Voice
• 80/20The vocal tract is the instrument of
  the human voice.

Vocal Tract

• Lungssource of air

Pharynx

• Tracheawind pipe

• Larynxvoice box

Larynx
Trachea

• Pharynxmouth and nose

Lungs
79
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Anatomy of the Human Voice
• 80/20The sound of the human voice originates in
  the larynx.

Larynx
Larynx
80
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Anatomy of the Human Voice
• 80/20The larynx (or voice box) contains the vocal
  folds.

Vocal Folds
Larynx
81
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Anatomy of the Human Voice
• 80/20The vocal folds rapidly open and close,
  introducing pulsations of air into the vocal
  tract.

Vocal Folds
Lower Vocal Tract
82
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Anatomy of the Human Voice
• 80/20The vocal folds rapidly open and close,
  introducing pulsations of air into the vocal
  tract.

Vocal Folds
Lower Vocal Tract
83
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• The Vocal Folds--Function
• 80/20The vocal folds are controlled by muscle and
  actuated by air moving between them, closing due
  to the Bernoulli Effect, opening by tension.

When flow is interrupted folds open.
Vocal Folds
Air Flow
84
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• The Vocal Folds
• 80/20The pressure waveform produced by the action
  of the vocal folds is an asymmetrical sawtooth,
  rich in harmonics.
• 80/20The fundamental frequency of the voice is
  determined by the properties of the vocal folds,
  not the vocal tract.

Vocal Folds snap open
are pulled shut by air flow
are pulled shut by air flow
are pulled shut by air flow
are pulled shut by air flow
are pulled shut by air flow
85
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Resonances of Vocal Tract

L 17 cm
f1 v/4L 354/(4 ? 0.17) 521 Hz f3 3f1 ,
f5 5f1
80/20The Vocal Tract is a lossy stopped pipe
17 cm long with a fundamental frequency of 500
Hz.
86
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Formants of Vocal Tract

Amplitude
Formant
L 17 cm
Frequency
f1 v/4L 354/(4 ? 0.17) 521 Hz f3 3f1 ,
f5 5f1
80/20The Vocal Tract filters the spectrum
generated by the vocal folds the frequency
filter is called the Formant.
87
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Speech
• 80/20The individual units of speech are called
  phonemes.
• The classes of (English) phonemes are
• Unvoiced Plosives? p, t, k (c, q, x)
• Voiced Plosives? b, d, g
• Fricatives? unvoiced/voiced f/v, th/th,
• Sibilants? unvoiced/voiced s(c)/z, sh/zh (j),
  h/kh
• Liquids? l, r
• Nasals? m, n, ng
• Semi-vowels? w, y
• Vowels? a, e, i, o, u

88
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• 80/20The shape of the Vocal Tract determines the
  frequency of the Formants.

ah
eh
oh
oo
89
Physics 1251 Unit 3 Session 31 The
Fundamentals of the Human Voice

• Summary
• The vocal folds, located in the larynx, produce
  vibrations in the vocal tract.
• The vocal tract is a stopped air column
  approximately 17 cm long, that resonates at
  500, 1500 and 2500 Hz.
• The shape of the vocal tract provides an acoustic
  filter, called the formant, that modifies the
  amplitude of the harmonics produced by the vocal
  folds.

90
Physics 1251 Unit 3 Session 32 The Singing
Voice

• 1' Lecture
• The pitch range of the singing voice is
  determined by the properties of the vocal folds.
• The intelligibility of words is due to the
  relationship of the first two formants.
• Modification of the shape of the vocal tract
  significantly affects the timbre of the singing
  voice.

91
Physics 1251 Unit 3 Session 32 The Singing
Voice

• The Mechanics of the Vocal Folds
• 80/20The properties of the vocal folds determine
  their vibration frequency.

Larynx
Larynx
92
Physics 1251 Unit 3 Session 32 The Singing
Voice

• The Mechanics of the Vocal Folds
• 80/20The properties of the vocal folds determine
  their vibration frequency.

Vocal Folds
Larynx
fvocal 1/2p vk/ m
93
Physics 1251 Unit 3 Session 32 The Singing
Voice

• The Mechanics of the Vocal Folds
• 80/20The properties of the vocal folds determine
  their vibration frequency.

fvocal 1/2p vk/ m
Vocal Folds
Density ?
k fold stiffness m effective mass
For a cord f 1/2LvT/ µ T s (t?d)
µ ?(t?d) f
1/2Lv s / ?
Stress s
Length L
fvocal 1/2p vk/ m
94
Physics 1251 Unit 3 Session 32 The Singing
Voice

• The Mechanics of the Vocal Folds
• 80/20The properties of the vocal folds determine
  their vibration frequency.

fvocal 1/2p vk/ m
f v s / (4L2 ?)
k fold stiffness m effective mass
k p2 s m/ L2 ? p2 T/ L m ? L(t?d)
For a cord f (1/2L)vT/ µ T s
(t?d) µ ?(t?d)
f (1/2L)v s / ?
L 0.017 m ? 1040 kg/m3
s 12 kPa
f 100 Hz m 200 mg T
0.14 N
95
Physics 1251 Unit 3 Session 32 The Singing
Voice

• The Mechanics of the Vocal Folds
• 80/20The properties of the vocal folds determine
  their vibration frequency.

f1 (1/2L)v s / ?

• 80/20Conclusions
• Resting length, stress and density set voice
  range
• Stress (tension) can be increased external to
  the vocal fold or internal to it.
• Overall, increased tension increases stiffness,
  pitch

96
Physics 1251 Unit 3 Session 32 The Singing
Voice

• Anatomy of the Human Voice
• 80/20During adolescent the vocal folds grow
  longer and the voice lowers in pitch.

Vocal Folds lengthen at puberty
f1 v s / (4L2 ?) f 1 1700/L (mm) Pitch
lowers at puberty.
97
Physics 1251 Unit 3 Session 32 The Singing
Voice

• Anatomy of the Human Voice
• 80/20The vocal folds comprise muscle, lamina
  propria and epithelium.

Cover
Body
Epithelium
Lamina Propria (3 layers)
Thyroarytenoid Muscle
98
Physics 1251 Unit 3 Session 32 The Singing
Voice

• 80/20Pitch is raised by increasing tension on
  vocal folds, both external to the vocal fold
  (Cricothyroid muscle) and internal to it
  (Thyroarytenoid muscle).
• f1 (1/2L)v s / ?

The nature of the stress in the vocal fold
(internal or external tension) permits phonation
in different registers.
99
Physics 1251 Unit 3 Session 32 The Singing
Voice

• 80/20Vocal Registers
• f1 (1/2L)v s / ?
• s sexternal sinternal

Terminology Speaking Pulse Modal Falsetto Singi
ng Chest Head Falsetto (alternative) Fry Middl
e Whistle Stohbass flageolet
100
Physics 1251 Unit 3 Session 32 The Singing
Voice

• Vowels and Formants
• 80/20The relative frequency of the 1 st and 2 nd
  vowels formants are unique to various vowels.

i
I
e
æ
e
Second formant frequency
?
D
U
u
c
First formant frequency
101
Physics 1251 Unit 3 Session 32 The Singing
Voice

• Control of Formants
• 80/20Tongue and lip placement and the shape of
  the pharanx are most important in vowel formation.

Corner Vowels
D
i
u
A
A
A
f
f
f
102
Physics 1251 Unit 3 Session 32 The Singing
Voice

• Formants and Singing

Harmonics align with Formants
Singers Formant

• Vowel modification shifts formats.
  Alignment of formants with harmonics
  intensifies pitch.
  Dilation of vocal tract causes
  Singers Formant.

103
Physics 1251 Unit 3 Session 32 The Singing
Voice

• Summary
• The pitch range of the singing voice is
  determined by the size, tension, and density of
  the vocal folds.
• Vocal registers and breaks in the voice result
  from changing modes of oscillation of the vocal
  folds.
• Vowels are distinguished by the frequency
  relationship of the first two formants.
• Modification of the vocal tract shape sets the
  timbre of the singing voice.

104
Physics 1251 Unit 3 Session 33 Percussion

• 1' Lecture
• Percussion instruments are instruments that are
  struck.
• The timbre of their sound is determined by their
  vibration recipe.
• Their vibration recipe is determined by the modes
  of oscillation that are excited.
• Often percussion instruments do not have pitch.

105
Physics 1251 Unit 3 Session 33 Percussion

• 80/20The timbre of an instruments sounds depends
  on its vibration recipe.

fn n f1
Pitched
Amplitude
f1
2f1
3f1
4f1
fn m xn m f1
Unpitched
Amplitude
f01
Frequency
106
Physics 1251 Unit 3 Session 33 Percussion

• The Oscillation of a Clamped Membrane

Mode (0,1)
d
f0 1 v/? v v(S/ s) f0 1 x0 1 /(p d) ?
v(S/ s) x0 1 2.405
Surface density s mass/area s density ?
thickness
Surface Tension S force/length
107
Physics 1251 Unit 3 Session 33 Percussion

• The Modes of Oscillation
  of an (Ideal) Clamped Membrane

Mode (0,1)
f0 1 x0 1 /(p d) ? v(S/ s) x0 1 2.405
Mode (1,1)
Mode (2,1)
f1 1 (x1 1 / x0 1) f0 1 x1 1 / x0 1 1.594
f2 1 (x2 1 / x0 1) f0 1 x2 1 / x0 1 2.136
108
Physics 1251 Unit 3 Session 33 Percussion

• The Modes of Oscillation
  of a Clamped Membrane

Mode (0,1) xn m / x0 1 1
(1,1)1.594
(2,1)2.136
(0,2)2.296
(3,1)2.653
(1,2)2.918
(4,1)3.156
(2,2)3.501
(0,3)3.600
(5,1)3.652
109
Physics 1251 Unit 3 Session 33 Percussion

• 80/20Membrane Acoustics
• The overtones of a circular membrane clamped at
  the edge are not harmonic and, therefore, they
  have no pitch.
• fnm (xn m /x01)f01
• The frequencies fnm of a membrane are (1)
  proportional to the square root of the ratio of
  surface tension of the head to the surface
  density ?v(S / s) and (2) inversely proportional
  to its diameter ?1/d.

110
Physics 1251 Unit 3 Session 33 Percussion

• Ideal vs Real Membranes
• 80/20Real membranes have a lower frequencies than
  predicted for ideal membranes because of air
  loading the lowest frequencies are lowered the
  most.

111
Physics 1251 Unit 3 Session 33 Percussion

• Mode Excitation
• 80/20Only those frequencies for which the modes
  were excited will appear in the vibration recipe.
• 80/20The highest frequency that can be excited by
  a mallet that is in contact with the surface for
  a period of Tcontact is
• f max 2/Tcontact

112
Physics 1251 Unit 3 Session 33 Percussion

• Mode Excitation
• 80/20The highest frequency that can be excited by
  a mallet that is in contact with the surface for
  a period of Tcontact is
• f max 2/Tcontact

Tcontact ½ Tperiod 1/(2fmax )
113
Physics 1251 Unit 3 Session 33 Percussion

• Bending Wave in a Plate

vbend
h thickness
? density
E Youngs Modulus
Density ? mass/volume
vL vE/(.91 ?) vbend v1.8 f h vL

Youngs Modulus E stress/elongation
stiffness
fnm 0.0459 h vL( ynm /d)2
114
Physics 1251 Unit 3 Session 33 Percussion

• The Modes of Oscillation
  of a Flat Cymbal

Mode (2,0) fn m / f0 1 1
(0,1)1.730
(3,0)2.328
(1,1)3.910
(4,0)4.110
(5,0)6.30
(2,1)6.71
(0,2)3.600
115
Physics 1251 Unit 3 Session 33 Percussion

• 80/20 Plate Acoustics
• The overtones of a circular plate clamped in the
  center are not harmonic and, therefore, have no
  pitch.
• fn m (yn m /y20)2 f20
• The frequencies fnm of a circular plate are (1)
  proportional to the thickness ?h and (2) to the
  square root of the ratio of the stiffness and the
  density ?vE/? and (3) inversely proportional to
  the square of the diameter ?1/d2 .

116
Physics 1251 Unit 3 Session 33 Percussion

• Summary
• Percussion instruments are instruments that are
  struck.
• Their vibration recipe is often not harmonic and,
  therefore, they do not have a definite pitch.
• For ideal circular edge-clamped membranes
  fnm ?(xnm /d)v(S/s).
• For circular plates free at the edge
  fnm ?h ? (ynm /d) 2 v(E/?).
• The maximum frequency excited by a mallet is f
  max 2/Tcontact.

117
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 1' Lecture
• Piano strings exhibit inharmonicity because of
  the stiffness of the wire.
• Some percussion instruments have pitch.
• Pitch results from a harmonic series of
  overtones.
• Tympani and Tabla are pitched drums.
• Orchestra Chimes, Glockenspiel, Xylophone,
  Marimba and Vibraphone have intonation.

118
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 80/20The task of producing pitch in a percussion
  instrument is an exercise in manipulating the
  overtones into a harmonic series.

Frequency
119
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• The Modes of vibration of an ideal string are
  harmonic.

Linear density µ mass/length
The stiffness of the wire increases the frequency
of the higher frequency harmonics.
Tension T force
fn n /(2 L) ? v(T/ µ) n 1, 2, 3, 4,
5, 6, 7.
P 3986 Log(nf1 /440) I(P) I(P)
Inharmonicity
120
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Inharmonicity of Piano

40
20
Inharmonicity
-20
Pitch ()
Because of the inharmonicity of strings the
octaves are stretched in a piano.
121
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Tympani are tuned by adjusting the tension
  on the head.

Tension device
Tension pedal
122
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Air Loading of a Clamped Membrane

The mass of air moved by the membrane adds to the
effective surface density, lowering the frequency.
Air mass
123
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 80/20The kettle of Tympani modifies the membrane
  frequencies by the interaction of the air
  resonances with the surface modes.

Modes of air vibration
124
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• The Modes of Oscillation
  of Tympani

Mode (0,1) fn m/f01 1
(1,1)1.594
(2,1)2.136
(0,2)2.296
(3,1)2.653
(1,2)2.918
(4,1)3.156
(2,2)3.501
(0,3)3.600
(5,1)3.652
125
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 80/20Tympani achieve pitch by (1) suppression of
  radial modes (2) modification of other mode
  frequencies by air loading and the effect of the
  kettle (3) attenuation of the lowest mode.

Amplitude
Frequency
126
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Metalophones
• Glockenspiels, Xylophones and Marimbas

Bar
h thickness
w width
L Length
Density ? mass/volume Youngs Modulus E
Force/elongation
127
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Metalophones
• Glockenspiels, Xylophones and Marimbas

Longitudinal Waves in a Bar
vL vE/ ? Longitudinal Wave Velocity
node
Anti-node
Anti-node
fn n/2LvE/ ? like an open pipe
Density ? mass/volumeYoungs Modulus E
Stress/Elongation
128
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Bending Modes in Bars

129
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Bending Modes in Bars

Free Ends

f1 1.133 fo f2 3.125 fo f36.125 fo fo ? h/L2
.224 L
130
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Orchestral Chimes

Free Ends
End Plug
f1 1.133 fo f2 3.125 fo f36.125 fo
131
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Mode Frequencies in Undercut Bar

Undercut Bar in Xylophone, Marimba and Vibraphone
Xylophone f1/f1 1.00f2 /f1 3.00f3 /f1 6.1
Marimba/Vibes f1 /f1 1.00f2 /f1 4.00f3 /f1
6.5
132
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 80/20What is the different between a Xylophone, a
  Marimba and a Vibraphone?
• The depth of the undercut a marimba is undercut
  more than a xylophone.
• The first harmonic of a xylophone is 3x the
  fundamental, for a marimba and vibe it is 4x.
• The xylophone sounds brighter and the marimba
  more mellow.
• Vibes have a tremolo mechanism.

133
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Summary
• Piano strings exhibit inharmonicity because of
  the stiffness of the wire.
• Some percussion instruments have pitch.
• Pitch results from a harmonic series of
  overtones.
• Tympani and Tabla are pitched drums.
• Orchestra Chimes, Glockenspiel, Xylophone,
  Marimba and Vibraphone have intonation.
• Marimba are undercut more than xylophones.