Musical Instruments 1 Musical Instruments
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Musical Instruments 1
Musical Instruments
Musical Instruments 2
Introductory Question
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
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
Why do strings produce specific notes?
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Oscillations of a Taut String
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
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
Stiffness of a string’s restoring forces are set by
the string’s tension
the string’s curvature (or, equivalently, length)
The inertial characteristics of a string are set by
the string’s mass per length
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Fundamental Vibration
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
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
Why does a vibrating string sound like a string?
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Overtone Vibrations
A string can also vibrate as
two half-strings (one extra antinode)
three third-strings (two extra antinodes)
etc.
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)
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
Second overtone involves three third-strings
Three times the fundamental pitch: 3rd harmonic
An octave and a fifth above the fundamental
Etc.
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A String’s Harmonics (Part 2)
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
Why do stringed instruments need surfaces?
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Projecting Sound
In air, sound consists of density fluctuations
Air has a stable equilibrium: uniform density
Disturbances from uniform density make air vibrate
Vibrating strings barely project sound because
air flows around thin vibrating objects
and is only slightly compressed or rarefied
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
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
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
What is vibrating in a wind instrument?
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Oscillations of Air in a Tube
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
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
Stiffness of the air’s restoring forces are set by
the air’s pressure
the air’s pressure gradient (or, equivalently, length)
The inertial characteristics of the air are set by
the air’s mass per length
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Fundamental Vibration
Open-Open Column
Air column vibrates as a single object
Pressure antinode occurs at column center
Pressure nodes occur at column ends
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
Air column vibrates as a single object
Pressure antinode occurs at closed end
Pressure node occurs at open end
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)
In open-open pipe, the overtones are at
twice fundamental (two pressure antinodes)
three times fundamental (three antinodes)
etc. (all integer multiples or “harmonics”)
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)
Blowing across the column tends to excite a
mixture of fundamental and harmonic
vibrations
Examples
Organ pipes
Recorders
Flutes
Whistles
Reeds and horns also use a vibrating air column
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Surface Instruments
Most 1-dimensional instruments
can vibrate at half, third, quarter length, etc.
harmonic oscillators with harmonic overtones
Most 2- or 3- dimensional instruments
have complicated higher-order vibrations
harmonic oscillators with non-harmonic overtones
Examples: drums, cymbals, bells
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Drumhead Vibrations
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Summary of Musical Instrument
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