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Musical Instruments 1
Musical Instruments
Musical Instruments 2
Introductory Question
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Sound can break glass. Which is most likely to
break:
A.
A glass pane exposed to a loud, short sound
A glass pane exposed to a certain loud tone
A crystal glass exposed to a loud, short sound
A crystal glass exposed to a certain loud tone
B.
C.
D.
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Observations about
Musical Instruments
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They can produce different notes
They must be tuned to produce the right notes
They sound different, even on the same note
They require power to create sound
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4 Questions about
Musical Instruments
Why do strings produce specific notes?
Why does a vibrating string sound like a string?
Why do stringed instruments need surfaces?
What is vibrating in a wind instrument?
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Question 1
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Why do strings produce specific notes?
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Oscillations of a Taut String
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A taut string has
a mass that provides it with inertia
 a tension that provides restoring forces
 a stable equilibrium shape (straight line)
 restoring forces proportional to displacement
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A taut string is a harmonic oscillator
It oscillates about its equilibrium shape
 Its pitch is independent of its amplitude (volume)!
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A Taut String’s Pitch
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Stiffness of a string’s restoring forces are set by
the string’s tension
 the string’s curvature (or, equivalently, length)
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The inertial characteristics of a string are set by
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the string’s mass per length
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Fundamental Vibration
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A string has a fundamental vibrational mode
in which it vibrates as a single arc, up and down,
 with a velocity antinode at its center
 and velocity nodes at its two ends
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Its fundamental pitch (frequency of vibration) is
proportional to its tension,
 inversely proportional to its length,
 and inversely proportional to its mass per length
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Question 2
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Why does a vibrating string sound like a string?
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Overtone Vibrations
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A string can also vibrate as
two half-strings (one extra antinode)
 three third-strings (two extra antinodes)
 etc.
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These higher-order vibrational modes
have higher pitches than the fundamental mode
 and are called “overtones”
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A String’s Harmonics (Part 1)
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A string’s overtones are special: harmonics
First overtone involves two half-strings
Twice the fundamental pitch: 2nd harmonic
 One octave above the fundamental frequency
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Second overtone involves three third-strings
Three times the fundamental pitch: 3rd harmonic
 An octave and a fifth above the fundamental
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Etc.
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A String’s Harmonics (Part 2)
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Integer overtones are called “harmonics”
Bowing or plucking a string excites a mixture of
fundamental and harmonic vibrations, giving the
string its characteristic sound
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Question 3
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Why do stringed instruments need surfaces?
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Projecting Sound
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In air, sound consists of density fluctuations
Air has a stable equilibrium: uniform density
 Disturbances from uniform density make air vibrate
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Vibrating strings barely project sound because
air flows around thin vibrating objects
 and is only slightly compressed or rarefied
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Surfaces project sound much better because
air can’t flow around surfaces easily
 and is substantially compressed or rarefied
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Plucking and Bowing
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Plucking a string transfers energy instantly
Bowing a string transfers energy gradually
Bow does a little work on the string every cycle
 Excess energy builds up gradually in the string
 This gradual buildup is resonant energy transfer
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The string will vibrate sympathetically when
another object vibrates at its resonant frequency
 and it gradually obtains energy from that object
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Introductory Question (revisited)

Sound can break glass. Which is most likely to
break:
A.
A glass pane exposed to a loud, short sound
A glass pane exposed to a certain loud tone
A crystal glass exposed to a loud, short sound
A crystal glass exposed to a certain loud tone
B.
C.
D.
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Question 4
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What is vibrating in a wind instrument?
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Oscillations of Air in a Tube
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Air in a tube has
a mass that provides it with inertia
 a pressure distribution that provides restoring forces
 a stable equilibrium structure (uniform density)
 restoring forces proportional to displacement
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Air in a tube is a harmonic oscillator
It oscillates about its equilibrium density distribution
 Its pitch is independent of its amplitude (volume)!
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Air in a Tube’s Pitch
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Stiffness of the air’s restoring forces are set by
the air’s pressure
 the air’s pressure gradient (or, equivalently, length)
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The inertial characteristics of the air are set by
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the air’s mass per length
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Fundamental Vibration
Open-Open Column
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Air column vibrates as a single object
Pressure antinode occurs at column center
 Pressure nodes occur at column ends
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Pitch (frequency of vibration) is
proportional to air pressure
 inversely proportional to column length
 inversely proportional to air density
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Fundamental Vibration
Open-Closed Column
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Air column vibrates as a single object
Pressure antinode occurs at closed end
 Pressure node occurs at open end
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Air column in open-closed pipe vibrates
as half the column in an open-open pipe
 at half the frequency of an open-open pipe
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Air Harmonics (Part 1)
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In open-open pipe, the overtones are at
twice fundamental (two pressure antinodes)
 three times fundamental (three antinodes)
 etc. (all integer multiples or “harmonics”)
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In open-closed pipe, the overtones are at
three times fundamental (two antinodes)
 five times fundamental (three antinodes)
 etc. (all odd integer multiples or “harmonics”)
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Air Harmonics (Part 2)
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Blowing across the column tends to excite a
mixture of fundamental and harmonic
vibrations
Examples
Organ pipes
 Recorders
 Flutes
 Whistles
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Reeds and horns also use a vibrating air column
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Surface Instruments
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Most 1-dimensional instruments
can vibrate at half, third, quarter length, etc.
 harmonic oscillators with harmonic overtones
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Most 2- or 3- dimensional instruments
have complicated higher-order vibrations
 harmonic oscillators with non-harmonic overtones
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Examples: drums, cymbals, bells
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Drumhead Vibrations
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Summary of Musical Instrument
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use strings, air, etc. as harmonic oscillators
pitches independent of amplitude/volume
tuned by tension/pressure, length, density
often have harmonic overtones
project vibrations into the air as sound