Physics 1251
The Science and Technology
of Musical Sound
Unit 2
Session 24 MWF
Review of Perception
and Strings
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Why do the characters in the film clip
concern themselves so much with the
tones produced by tapping on the plate?
The vibration modes of the plates determine
the timbre of the finished instrument.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Wolf Tones
String
Harmonics
Body
Harmonics
Beats !#@%!!!
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Fighting the Wolf in a Famous Violin
Stradivarius Violin
The solution: a reflector
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
1′ Lecture (Unit 2 Review):
•
The function of the Human Ear determines the
essential features of acoustics.
•
Sound Intensity Level (in dB) is a logarithmic measure
of intensity.
•
Loudness is a subjective measure of the dynamic of
sound.
•
Pitch, which is determined by frequency, is a measure
of the highness or lowness of a musical sound.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
1′ Lecture (continued):
•
•


Waves are characterized by
 Frequency
 Wavelength
 Velocity
Reflection
 Specular
 Diffuse
Refraction
Difraction
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
1′ Lecture (continued):
•
•
•
Doppler effect
Beats
Interference
•
Room acoustics are determined by the volume,
surface area, shape and absorption properties of its
contents.
•
The reverberation time of a room can be calculated
using the Sabine Equation.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
1′ Lecture (continued):
•
Scales are based on tones whose frequencies are the
ratio of whole numbers.
•
Strings produce a harmonic series of vibrations
described by the Marsenne equation.
•
The vibration modes of the body and air cavity of a
stringed instrument strongly modify the harmonic
recipe produced by the string vibrations.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
How to organize so much information?
By Topic:
• Physiology
- the ear is a traducer: from pressure fluctuations
to neural stimulation
• Perception - non-linear in intensity and frequency sensitivity
• Room Acoustics - properties determine quality of sound
• Wave Properties - reflection, refraction, diffraction, Doppler
effect, beats and inference
• Strings - f determined by standing waves
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
By Equation:
•
•
•
•
•
•
•
•
SIL
Pitch
Inverse Square Law
Reverberation Time
Harmonics
Law of Reflection
Beats
Marsenne Equation
SIL=10‧Log(I / Ithreshold )
Pitch Interval=3986¢‧Log(f1 /f2 )
I /I0 = (r0 /r ) 2
TR = 0.16 V/Se
fn = nf1
Angle in = Angle out
fmean= (f1 +f 2)/2; fbeat =⃒f1 –f2⃒
fn = n/(2L) ‧ √(T/μ)
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
By Chronological facts:
Anatomy of the Ear
• 80/20Middle
Ear: The Ossicles (little bones)
1. Malleus -- (the hammer) moved by
Tympanium.
2. Incus -- (the anvil) supported by ligaments
that protect against loud percussion.
3. Stapes – (the stirrup) force multiplied by
1.3 because of lever action.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Anatomy of the Ear
• 80/20Inner
Ear:
Cochlea – (the Snail) converts
displacement into neural impulses.
Auditory Nerve – neural impulses to
brain
Semicircular canals – detect motion
and orientation
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Frequency Discrimination in Cochlea
•
20 Hz to 20 kHz (typical in Humans)
•
Resonances in Basilar membrane and in HC
cause spatial separation by frequency.
•
Differential movement of membranes
stimulate HC.
•
Minimum stimulation required for response.
Inhibition of neighbors causes non-linear
response.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
frequency range of detectability
for humans is approximately 20 Hz to
20 kHz.
80/20Humans
are most sensitive in the
frequency range 2 kHz to 5 kHz.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The objective relative intensity level of
sound is quantified as the Sound
Intensity Level (SIL) and is measured in
deciBel (dB), where
SIL = 10 Log( I / Ithreshold )
80/20The
Intensity of a sound wave is the energy
radiated per unit time per unit area. [W/m2]
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
lowest detectable intensity (the
threshold of hearing Ithreshold ) is about
1 pW/m 2 or 1x10 -12 W/m 2 [SIL = 0dB].
80/20The
intensity at which one experiences
pain (the threshold of pain) is about
1 W/m 2 [120 dB].
Physics 1251
80/20Sound
Unit 2 Session 24
Review of Perception and Strings
Intensity Level:
SIL = 10 Log ( I / I threshold ).
The Sound Intensity Level is 10 times the
logarithm of the ratio of the intensity of a
sound and the threshold of hearing.
The units of SIL are deciBel or dB.
I = I threshold 10 SIL/10
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Application:
What is the Sound Intensity Level of a
tone that has an intensity of
I = 1.26 mW/m2 ?
SIL = 10 Log ( I / Ithreshold)
( I / Ithreshold) = (1.26 x10 -3 W/m2/1 x10 –12 W/m2)
= 1.26 x 10 9
Log ( I / Ithreshold) = Log(1.26 x 10 9)= 9.1
SIL = 10 Log ( I / Ithreshold ) = 10 (9.1) = 91 dB
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Application:
What is the Intensity Level of a tone that
has an SIL of 35 dB?
I = Ithreshold 10SIL/10
SIL/10 = 35/10 = 3.5
10SIL/10=10 3.5= 3.2 x 10 3
I = Ithreshold 10SIL/10
= (1 x 10-12 W/m 2 )( 3.2 x 10 3 ) = 3.2 x 10 -9 W/m 2
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20Just
Noticeable Difference (JND) is the
limen of difference that elicts 75% in a
Two Alternative Forced-Choise test
(2AFC test).
The limen of intensity is a ratio of about 1.26
which corresponds to a SIL difference of 1 dB.
10 Log( 1.26 ) = 1.0
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Musical Dynamics
Pianissimo:
pp
very soft:
50 dB
Piano:
Mezzopiano:
Mezzoforte:
Forte:
Fortissimo:
Fortississimo:
p
mp
mf
f
ff
fff
soft:
medium soft:
medium loud:
loud:
very loud:
Very, very loud:
60 dB
66 dB
76 dB
80 dB
90 dB
100 dB
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
Fletcher-Munson Diagram is a plot of
the SIL (in dB) versus frequency for the SIL
required to produce an equal sensation as
that produced at 1000 Hz.
80/20The
contours are of equal loudness level.
80/20The
unit of loudness level is the phon.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Fletcher- Munson Diagram
SIL
(dB)
130
120
110
100
90
80
70
60
50
40
30
20
10
Loudness (phon)
Frequency (Hz)
Fletcher and Munson (1933) J. Acoust. Soc. Am. 5, 82-108
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Loudness
A subjective measure of the magnitude of
auditory sensation is called Loudness and is
measured in sone. In this system, one listens
to two sounds and judges how much louder or
softer a test sound is compared to the
reference.
80/20
For example, a sound of 2 sone sounds twice as loud as
a tone of 1 sone.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Loudness Scaling
Loudness ∝∛ I
80/20Thus,
an eight (8) singer ensemble sounds
about twice (2x) as loud as a soloist. Because
∛8 = 2.
Likewise, a choir of sixty-four (64) sounds about
four (4x) times louder than a soloist. ∛64 = 4.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
Just Noticeable Difference (JND) or
difference limen is the difference in pitch (or
loudness) that will elicit 75% correct responses
in a Two-Alternative Forced-Choice test
(2AFC) test.
80/20The
Difference limen for pitch is about 1/30
the critical band width and varies from ~40¢ at
low frequencies (<62 Hz, C2) to ~6¢ at high
frequencies (>8kHz, C9 ).
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (SIL and Loudness):
•
•
•
Loudness is the magnitude of the sensation
produced by a sound; it is measured in sone.
Loudness Level (in phon) is equal to the SIL
at 1000 Hz that produces the same magnitude
of sensation.
Loudness increases approximately with the
cube root of intensity.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary:
•
The loudness depends on frequency.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Pitch Interval
80/20An
equal ratio of frequencies sounds
like an equal difference or interval of
Pitch.
80/20An
octave is the pitch interval
corresponding to a frequency ratio of 2:1.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Pitch Interval
80/20A
semitone is 1/12 of an octave.
80/20A
cent (¢) is 1/100 of a semitone or 1/1200 of
an octave. 1 octave = 1200 ¢.
80/20Pitch
Interval corresponding to f2 and f1 :
₧ = [1200¢/Log 2] ‧ Log (f2 /f1 )
₧ = 3986 ¢ ‧ Log (f2 /f1 )
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Pitch Interval – Application
What is the pitch interval (in ¢) that corresponds
to a perfect 5th?
An interval of a perfect 5th corresponds to a ratio
of f2 /f1 =3/2 = 1.5.
₧ = 3986 ¢ ‧ Log (f2 /f1 )= 3986 ¢ ‧ Log(1.5)
= 3986 ¢ ‧ (0.176) = 702 ¢
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
sensation of pitch is a property of
human auditory perception that infers pitch
from the repeat period.
80/20Harmonics
(including the fundamental) may
be missing but we hear the difference
frequency as well as the harmonics.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Critical Band Width wcritical
80/20If
Δf < wcritical , sounds like one sound.
If Δf > wcritical , sounds like separate sounds.
80/20If
Δf < wcritical , sounds softer than
if Δf > wcritical .
For example, a choir singing in “prime unison” (exactly
the same pitch) sounds softer than when singing parts.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (Pitch):
•
Tones are indistinguishable if they are closer than a
critical band width.
•
Pitch interval is proportional to the logarithm of the
ratio of the frequencies.
•
An octave corresponds to a ratio of 2/1.
•
A semitone is 1/12 of an octave.
•
A cent ¢ is 1/100 of a semitone.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary:
•
The sensation of pitch is a property of human
auditory perception that infers pitch from the
repeat period.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The intensity of a spherical sound wave decreases as
the distance from the source increases. [“Inverse”]
d [m]
1.0
I [W/m2]
1.0
=12 /12
2.0
0.25 =12/2 2
3.0
0.11
4.0
0.068 = 12 /4 2
5.0
0.040 = 12 /5 2
=12 /3 2
As 1/r 2
Physics 1251
80/20Inverse
Unit 2 Session 24
Review of Perception and Strings
Square Law:
The intensity of sound (originating from a
point source in an open environment)
diminishes as the square of the inverse
ratio of the distances from a source.
I / I0 = (r0 /r) 2
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20Inverse
Square Law:
The Sound Intensity Level (SIL) decreases
by 20 dB for every 10x increase in
distance.
SIL = SIL0 – 20 Log (r0 / r)
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Application—Inverse Square Law:
What is the intensity of the sound of an explosion
heard 100 m away when the intensity is 1
W/m2 at a distance of 10 m?
I = I0 (r0 /r) 2
I = (1 W/m2)(10m/100m)2 = (1 W/m2)(.01)
= 0.01 W/m2 =10. mW/m2
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Reflection:
80/20Reflection,
a “bouncing back,” occurs
whenever there is an abrupt change in
the medium.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
What happens when a wave “hits” a
change in the medium?
Reflection
Θin = Θout
Θout
Θin
Transmission
Medium 1
Medium 2
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
When the surface is smooth we have
“specular” (mirror-like) reflection.
Specular
Reflection
Smooth Surface
Roughness ≲ λ
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
What if the surface is rough?
Diffuse
Reflection
Rough Surface
Roughness > λ
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Refraction:
80/20Refraction,
a “bending aside,” occurs
whenever there is a change in velocity
along the wavefront.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Refraction occurs when a wave “enters” a
medium that has a different velocity?
Refraction
V1 <
V2
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
In the second medium the wavefront
“races ahead,” changing the direction.
Refraction
V1
V2
Physics 1251
Wind Speed
Unit 2 Session 24
Review of Perception and Strings
Refraction
velocity varies
along
wavefront
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Cold Aloft
v - slower
Refraction:
velocity varies
with
temperature
v - faster
Warm below
silence
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Diffraction:
80/20Diffraction,
a “bending around” obstacles
because every point on a wave is a source;
waves cannot terminate abruptly.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20Huygens
Principle: every point on
the wave is the source of a new
(spherical) “wavelet.”
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
What happens when a wave “is partially
obstructed?
Diffraction
Physics 1251
80/20The
Unit 2 Session 24
Review of Perception and Strings
Doppler Shift:
Doppler shift is the change in the
frequency of a source fs to that observed fo
due to the relative velocity of the source vs , or
observer vo ,where the velocity of sound in the
medium is v.
f o= f s ‧ [v - vo ] / [v - vs],
[sign is + in direction of v, e.g. for observer moving toward source
vo <0, source moving toward observer vs >0.]
Physics 1251
Unit 2 Session 18
Room Acoustics
Doppler Shift:
Lower f
Moving source
Higher f
fobserver = fsource [v - vobserver] / [v – vsource]
Physics 1251
80/20Coherent
Unit 2 Session 24
Review of Perception and Strings
Addition of Waves:
addition of waves is the addition of
the amplitudes of waves of the same or nearly
the same frequency; the intensity then is
proportional to the square of the the combined
amplitude.
80/20Incoherent
addition of waves is the addition of
their independent intensities if they are not
within the critical bandwidth.
Physics 1251
80/20Beats
Unit 2 Session 24
Review of Perception and Strings
Beats
are a modulation of intensity due to the
coherent addition of two sound waves whose
frequency lie with a critical band width of each
other but differ slightly in frequency.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Beat Frequency
80/20Two
tones of frequency f1 and f2 sound like
one tone of mean frequency
f mean= (f1 +f 2)/2
that beats at a beat frequency of
f beat= f1 – f2 .
Physics 1251
Unit 2 Session 18
Room Acoustics
Beats
In phase
Out of phase
f1
f2
fmean
fbeat
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Interference:
80/20Interference
is the coherent addition of
waves at various points in space leading
to spatial modulation of intensity.
Physics 1251
Unit 2 Session 18
Room Acoustics
Interference
Constructive
Destructive
Softer
Louder
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (Propagation)
• Reflection is the return of a wave from a
change in a medium.
• Refraction is bending of the wave due to
velocity variation along the wavefront.
• Diffraction is the bending of the wave around
obstacles because of Huygen’s Principle.
• Doppler Shift the change in frequency due to
the relative motion
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary
• Beats are intensity modulations in time
due to coherent addition of waves of
near equal frequency.
• Interference is spatial modulation of
intensity due to the coherent addition of
nearly equal frequency waves at different
points in space.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20Standards
•
•
•
•
•
•
•
for “Good” Acoustics:
Clarity …little overlap of sounds
Uniformity …everywhere the same
Envelopment …sound from all directions
Smoothness …no echoes
Reverberation …appropriate length of time
Performer satisfaction …reflected to stage
Freedom from noise …no competition
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20 Haas
or Precedence Effect
The earliest sound that arrives determines the
sense of the origin of a sound, even if the later
(<100 ms) reflections are louder.
The direct sound should arrive first.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Room Acoustics: Reverberation
Direct Sound
Speaker
Hearer
Reverberant Sound
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Energy Lost in Reflections:
• The sound reflects many times, each time
losing energy to the reflecting surfaces.
• The quantity α is the absorptivity of the
surface.
• The intensity of the sound that is lost in a
reflection is ΔIlost = α Iin.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
• The intensity of the reflected wave is
Ireflected= (1-α) Iin.
• Values for α, the absorptivity, for many
types of surfaces have been measured
and appear in extensive tables.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Absorptivity
Material
α (at 500 Hz)
Acoustic tile
Plaster wall
Concrete
Person
0.6
0.1
0.02
0.8 (x1 m2)
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Wallace Sabine
(Harvard professor 1868-1919)
Asked: “How long will it take for the
sound to die down to 1 millionth (-60 dB)
of the initial value?”
80/20The
Wallace
Sabine
reverberation time is the time for the
intensity to decay by a factor of 10 –6 (- 60dB)
of its initial value.
Physics 1251
80/20The
Unit 2 Session 24
Review of Perception and Strings
Sabine Equation:
I = Io ‧ 10 – 6 (t/TR)
TR = 0.16 V/Se
•
•
•
V is the volume of the room.
Se is the “effective surface area” of the walls
S1 , floor S2 and ceiling S3 (in sabin) etc.
α is the absorptivity of the surface (in table)
Se = α1 S1 + α2 S2 + α3 S3 + α4 S4 +…
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (Reverberation):
Direct sound should come first
•
•
•
•
•
•
•
•
•
Haas or precedence effect
Reverberant sound
TR = 0.16 V/Se
0.8 sec for clear speech, 1-2 for music
Freedom from echo and interference
Use diffuse and random reflectors
Background noise level
Assure good acoustic isolation
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Sabine Equation—Application
What will be the relative intensity of a sound after
1.0 second in room with a reverberation time
of 3.0 seconds?
Sound decays exponentially in a room:
I = Io ‧ 10 – 6 (t/TR)
I /Io = 10 – 6 (t/TR) = 10 – 6 (1.0 sec/3.0 sec)=10 – 2
I /Io = 0.01 = 1 %
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Sabine Equation—Application
What is the reverberation time of a room that is
1000 m3 in volume with a surface area of 125
m2 of highly absorbent acoustical material with
an α of 0.95?
Sabine Equation
TR = 0.16 V/Se
Se = α S = 0.95 (125 m 2) = 119 sabine
TR = 0.16 (1000/119)= 1.3 seconds
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Build a Scale from Whole Number
Ratios of Frequencies!
80/20A
scale is a series of tones arranged in
ascending pitch.
80/20In
Pythagorean (or “Just”) intonation the
frequencies of the tones of the scale are
integer ratios of each other.
80/20The
beginning pitch of the scale is called the
“tonic.”
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Musical Notation
♩
♩
♩
♩♩
♩
♩
♩
♩ F2
C3
♩
♩
♩
♩♩
♩
♩
♩
♩
440 Hz
♩
♩
♩
♩
♩
♩
♩
♩
♩ ♩ ♩G
D
A C
D4E4
F3G3 B3C4
E
A
B
D2 E2 G A2 2 D
3
C2
2
3
4
F4
4
C5
A4B4
5
F G5
E5
5
5
B5
6
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Musical Notation
♩
♩
♩
♩
♩
♩
♩
♩
♯
♩
♩
♯ ♩♯
♩
♯
♩♯ ♩ G4 ♯ B
♩♯ E F4♯ A ♯
♩
♩
♩
♩
♩
♭ ♩♭
♩
♭
♩
♩
♭
♩
♭
♩
♩
♩
♩
4
D4♯ 4
A3♭ C ♯
E3♭
4
D3♭ F3 G ♭
3
B3♭
C3
4
C5♯
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20With
Just Temperament it is impossible to
tune all notes or tones of the chromatic scale
so that they are all in tune, that is, the correct
ratio for all keys.
80/20Equal
Temperament intonation is a
compromise in which the semitones are
precisely 100 ¢ or a ratio of 1.05946… of its
neighbor.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (scales) :
A scale is a progression of tones arranged in an
ascending order of pitch.
•
The tonic of a scale is the pitch that is the
basis of the scale.
•
The chromatic scale is a series of tones,
each separated from the next by a pitch
interval of a semitone.
•
The diatonic scale contains a series of
(whole) tone and semitone intervals.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary:
•
•
Just Temperament uses integer ratios
of adjacent notes in the chromatic
scale.
Equal Temperament uses a constant
ratio of adjacent notes in the chromatic
scale.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
What is a scale?
•
“Gamut” {Note “G-Clef”}
♩
♩
♩
♩
♩
♩
♩♩
Do Re Mi Fa So La Ti Do
C-major
♩
♩
♯
♩
♩
♩
♩
♩♩
Do Re Mi Fa So La Ti Do
G-major
Guido d’Arezzo: “gamma ut→gamut” • Solfeggio
G is “Do” in the G-scale
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
What is a scale?
•
Solfege
Ti
Do
La
Fa
Do
Re
So
Mi
Zoltan Kodaly
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Circle of Fifths:
633.6
950.4
440
A
[tonic]
352
528
792
2:3
[major 3 rd]
[perfect 5th]
528
792
1188
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20It
is impossible to tune perfectly scales
in all keys using the same frequencies
and just temperament.
80/20The
Equal Tempered Scale sets each
semitone exactly 100 ¢ apart or at a
ratio of 1.05946….
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Musical Notation
♯
♭♩♭♩
♭
♩
♩♭♩
♩♭♩
♯♩
♩
♯♩ ♩
♯
♩
♩ ♯♩
G4♯ B4 C5♯
F
♯
E 4
A♯
D4♯ 4
A3♭ C ♯
E3♭
4
D3♭ F3 G ♭
3
B3♭
C3
♯♩
4
Physics 1251
80/20Velocity
Unit 2 Session 24
Review of Perception and Strings
of a wave on a string:
vstring = √T/ μ
T is the tension in the string [N].
μ is the mass per unit length [kg/m]
Physics 1251
80/20f
1
Unit 2 Session 24
Review of Perception and Strings
= [1/(2L)] ‧ vstring
f1 = [1/(2L)] ‧ √(T/ μ)]
Tuning pegs
More Tension:
Raises pitch
Physics 1251
80/20f
1
Unit 2 Session 24
Review of Perception and Strings
= [1/(2L)] ‧ √(T/ μ)]
Denser wire: lower pitch
Less dense wire:
higher pitch
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (scales and strings):
•
A Just Tempered Scale sets the frequencies
of the notes in the scale at precise ratios of
whole numbers.
•
The Equal Tempered Scale is the
compromise that sets all notes an equal
interval apart (100¢, frequency
ratio:1.05946).
•
f1 = [1/(2L)] ‧ √(T/ μ)]
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Application—Marsenne Equation:
What is the fundamental frequency of a 1.00 m
long string stretched with a tension of 700 N
that has a mass density of 0.005 kg/m.
f = 1/(2L) √ (T/ μ)
= 1/(2(1.00 m)) √ (700/ 0.005)
= 1/ ( 2.00) • (374.) = 187. Hz
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20A
musical instrument comprises four
important components:
(1) a mechanical energy source;
(2) a frequency generator;
(3) a frequency filter;
(4) an antenna.
♩♪♫
~
~
f1 f2 f3 f4
fn
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20A
stringed instrument uses standing
waves on a string to provide the
frequency generation.
♩♪♫
~
f1 f2 f3 f4
~
String Standing Waves
fn
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
distance between neighboring nodes &
antinodes is ¼ λ. [ “N-A d = ¼ λ” ]
Second Harmonic f2 = vstring / λ2 = vstring / L
L = 4/₄ λ2
Node
Node
Antinode
λ2 /4
λ2 /4
Node
Antinode
λ2 /4
λ2 /4
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20Frequency
of nth harmonic:
Marsenne Equation
fn = [n/(2L)] ‧ √(T/ μ)]
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Application—Marsenne Equation:
What are the harmonics of the string discussed
earlier? (L =1.00 m, T= 700 N, μ = 0.005 kg/m)
f1 = 1/(2L) √ (T/ μ)
= 1/(2(1.00 m)) √ (700/ 0.005)
= 1/ ( 2.00) • (374.) = 187. Hz
f2 = 2 f1 = 374 Hz
f3 = 3 f1 = 561 Hz
Etc…
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
modes that will be present are
those that are excited.
♩♪♫
~
~
f1 f2 f3 f4
Energy source can excite
selected modes
fn
Physics 1251
80/20The
Unit 2 Session 24
Review of Perception and Strings
modes that will be present are
those that are excited.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Guitar is a fretted stringed instrument
that is plucked or strummed.
Tuning pegs
Neck
Strings (6)
Frets
Body
Bridge
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
coupling of the string to the sound
board will actuate or attenuate various
frequencies.
♩♪♫
~
~
f1 f2 f3 f4
fn
Sound Board and Acoustics filter
harmonics
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20A
Harp is a stringed instrument that is
plucked. Its strings are of various
lengths.
Strings
Traditional
Irish Harp
Sound Board
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The principal pitch of each string in a Pedal Harp
is determined by its length.
Pedal Harp
Strings
Pedals
However, depressing a pedal shortens or lengthens the string
length, raising (or lowering) the pitch incrementally.
Physics 1251
80/20A
Unit 2 Session 24
Review of Perception and Strings
Harpsichord is a keyboard stringed
instrument whose strings are plucked.
Each string is used for one pitch only.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20A
Harpsichord is a keyboard stringed instrument
whose strings are plucked.
Each string is used for one pitch only.
Harpsichord
Action
String
Key
Damper
Plectrum
Jack
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20A
Piano is a keyboard stringed instrument
whose strings are hammered. Each string is
used for one pitch only.
Fortepiano
Strings
Key Board
Sound
Board
Piano
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20A
Piano has multiple strings (in the treble) to
provide a more powerful sound.
Multiple Strings
Sound Board
Bridge
Frame
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Piano strings are tensioned by tuning pins in a
strong frame.
Frame
Bass Strings
Tuning Pins
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Piano hammers are made of felt and wood.
Hammer Head
Felt
Back Check Wood
Pivot hinge
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The timbre of the piano sound is affected by the
hardness of the hammer felt.
Felt –
hard or soft?
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
timbre of a musical instrument is
determined by the excitation, frequency
generation, filtering and broadcast of the
harmonics.
♩♪♫
Excitation
~
~
f1 f2 f3 f4
fn
f Response
f Generation
Broadcast
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (strings):
•
fn = [n/(2L)] ‧ √(T/ μ)
•
The node-antinode distance is ¼ λ.
•
The timbre of a stringed instrument is
determined by its harmonic recipe.
•
The harmonic recipe of a stringed instrument
is determined by its mode of excitation, the
harmonics of a string, and the natural
frequency response of the instrument.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary:
•
A Guitar is a stringed instruments with frets
played by plucking.
•
A Harp is a fretless stringed instrument
played by plucking.
•
The Harpsichord is a stringed instrument with
keys which when depressed lift a plectrum
that plucks one or more strings.
•
The Piano is a stringed instrument with a
keyboard which causes a felt hammer to
strike the string.
Physics 1251
80/20The
Unit 2 Session 24
Review of Perception and Strings
timbre of bowed strings is
affected by the properties of strings,
by bowing and by the frequency
resonances of the instrument.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Bowed String Instruments
Violin
•
•
•
•
G3 , D4 , A4 , E5 (5ths)
Viola
C3 , G3 , D4 , A4
(5th below violin)
Cello
•
•
•
Bow
C2 , G2 , D3 , A3
(8vo below viola)
Bass (Viol)
•
Bridge
E2 , A2 , D3 , G3 (4ths)
Strings
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Pitch is changed by shortening the speaking
length of the strings by “fingering”
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The strings are tensioned
by means of tuning pegs
in a “scroll.”
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Density of strings sets range of each string
Viola Strings
Denser wire: lower pitch
C3
Metal over “catgut”
G3
D4
Less dense wire:
higher pitch
A4
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Different pitches can
be played on different
strings or on the same
string.
Example:
“Air on the G-String”
J.S. Bach
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Anatomy of a “String”
Stradivarius Violin
Scroll
Fingerboard
Body
Bridge
“f” Holes
Tail piece
Guarneri Viola
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
sound post transfers vibration from front
plate to back plate and supports bridge.
Front plate
Sound post
Back plate
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
A rosined horse hair bow rubs the string.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Action of the Bow
The rubbing of the bow
alternately pulls the string
forward, then releases
it when the string wave
reaches the bow.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The Stick-Slip Mechanism causes the
string to vibrate when rubbed by the
bow.
Horse Hair
of Bow
String
Slip point
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
A complicated wave shoots down the string
when the string slips from the bow.
Scroll
Moving Bow
Waveform envelope
Bridge
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
80/20The
coupling of the string to the sound
board will accentuate or attenuate
various frequencies.
♩♪♫
~
~
f1 f2 f3 f4
fn
Sound Board and Acoustics filter
harmonics
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Cross Section of Violin
Scroll with
Tuning pegs
Finger board
Body
f-hole
Sound Post
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Plates are of varying thickness and are
“hinged” at edge (“Purfling”)
Purfling
Thicker
Thinner
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
The bridge is essential to transforming the lateral
motion of the string into vibrations of the top
plate.
Bridge
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Bridge “Action”
Bass
bar
Sound
post
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Vibration is transmitted to the body and air
cavity by the action of the Bridge.
Rocking
motion
Helmholtz
Resonance
Sound post
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Chladni plates simulate the vibrational
modes of Violin Plates.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary (Bowed Strings):
•
fn = [n/(2L)] ‧ √(T/ μ)
•
The bow works by the Slip-slick mechanism.
•
The timbre of a bowed stringed instrument is
determined by its harmonic recipe.
•
The harmonic recipe of a bowed stringed
instrument is determined by its modes of
excitation, the harmonics of a string, and the
natural frequency response of the instrument.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary:
•
A violin, viola, cello and bass are stringed
instruments with a finger board played by
bowing.
•
The string vibrates the bridge which vibrates
the top plate.
•
The sound post transmits the vibration to the
back plate.
•
The timbre of the violin family is strongly
affected by the modes of vibration of the
instrument.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary of Unit:
Human sound perception sets the priorities in
musical acoustics
•
•
•
•
•
SIL in dBA is more like human hearing intensity.
Pitch is inferred from the place that resonance
occurs in the Basilar membrane of the Cochlea.
The range of human hearing is 20 Hz to 20 kHz.
The softest sound that is audible has an SIL of
0 dB and an Intensity of 1 x 10 -12 W/m2.
Waves have the following properties
•
•
Reflection– specular and diffuse, a bouncing back.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary of Unit:
•
•
•
•
•
Refraction– bending due to velocity changes.
Diffraction– bending around obstructions.
Doppler shift– frequency change due to motion of
the source or observer.
Beats– amplitude modulation due to the coherent
addition of two tones of nearly equal frequency.
Interference– spatial modulation of intensity due to
coherent addition of two or more waves of same f.
The acoustics of a room depend on its shape
and size.
•
•
The reverberation time is proportional to the ratio
of the volume and its effective surface area.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary of Unit:
A Scale is the arrangement of tones in a
progression: lower to higher pitch.
•
•
•
•
•
The frequency of notes in the scale is determined
by the ratio of whole numbers in Just Intonation or
temperament.
The ratios of frequencies of notes in a scale using
Equal Intonation or Temperament are constant;
Semitone ratio = 1.05946.
The pitch interval is proportional to the logarithm of
the ratio of the fundamental frequencies.
A semitone corresponds to 100¢.
Physics 1251
Summary of Unit:
•
•
•
•
Unit 2 Session 24
Review of Perception and Strings
Strings obey the Marsenne Equation.
In Standing Waves the distance from a node to an
adjacent antinode is ¼ wavelength.
The frequency of a bowed instrument is determined by
the feedback from the string.
The harmonic recipe of a stringed instrument is
determined by:
• The means of excitation of the instrument.
• The normal modes of vibration of a string.
• The filtering by the resonances of the instrument.
• The coupling efficiencies of the instrument to the air.
Physics 1251
Unit 2 Session 24
Review of Perception and Strings
Summary of Unit:
One can distinguish the instruments:
•
•
Guitar, Harp, Harpsichord and Piano by their
respective acoustic construction and excitation and
by their resulting timbre.