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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.
Musical Instruments 3
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
Musical Instruments 4




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?
Musical Instruments 5
Question 1

Why do strings produce specific notes?
Musical Instruments 6
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)!

Musical Instruments 7
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
Musical Instruments 8
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

Musical Instruments 9
Question 2

Why does a vibrating string sound like a string?
Musical Instruments 10
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”

Musical Instruments 11
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.
Musical Instruments 12
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
Musical Instruments 13
Question 3

Why do stringed instruments need surfaces?
Musical Instruments 14
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

Musical Instruments 15
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

Musical Instruments 16
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.
Musical Instruments 17
Question 4

What is vibrating in a wind instrument?
Musical Instruments 18
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)!

Musical Instruments 19
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
Musical Instruments 20
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

Musical Instruments 21
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

Musical Instruments 22
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”)

Musical Instruments 23
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
Musical Instruments 24
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
Musical Instruments 25
Drumhead Vibrations
Musical Instruments 26
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