Download Musical Instruments

Musical Instruments 1
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
Musical Instruments
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 5
Musical Instruments 4
„
„
„
„
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 6
Question 1
„
4 Questions about
Musical Instruments
Why do strings produce specific notes?
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)!
1
Musical Instruments 7
Musical Instruments 8
A Taut String’s Pitch
„
Stiffness of a string’s restoring forces are set by
Fundamental Vibration
„
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
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
Musical Instruments 10
Question 2
„
Why does a vibrating string sound like a string?
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
„
„
Musical Instruments 11
Musical Instruments 12
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
„
„
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
Second overtone involves three third-strings
„
„
„
have higher pitches than the fundamental mode
and are called “overtones”
Three times the fundamental pitch: 3rd harmonic
An octave and a fifth above the fundamental
Etc.
2
Musical Instruments 13
Musical Instruments 14
Question 3
„
Why do stringed instruments need surfaces?
Projecting Sound
„
In air, sound consists of density fluctuations
„
„
„
Vibrating strings barely project sound because
„
„
„
„
„
air can’t flow around surfaces easily
and is substantially compressed or rarefied
Musical Instruments 16
Plucking and Bowing
„
air flows around thin vibrating objects
and is only slightly compressed or rarefied
Surfaces project sound much better because
„
Musical Instruments 15
Air has a stable equilibrium: uniform density
Disturbances from uniform density make air vibrate
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
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
„
„
„
The string will vibrate sympathetically when
another object vibrates at its resonant frequency
„ and it gradually obtains energy from that object
„
Musical Instruments 17
B.
C.
D.
Musical Instruments 18
Question 4
„
What is vibrating in a wind instrument?
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)!
3
Musical Instruments 19
Musical Instruments 20
Fundamental Vibration
Open-Open Column
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
„
Air column vibrates as a single object
„
„
„
the air’s mass per length
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
Musical Instruments 22
Air Harmonics (Part 1)
„
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
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
Musical Instruments 24
Air Harmonics (Part 2)
„
„
Blowing across the column tends to excite a
mixture of fundamental and harmonic vibrations
Examples
Organ pipes
„ Recorders
„ Flutes
„ Whistles
„
„
Surface Instruments
„
Most 1-dimensional instruments
„
„
„
Most 2- or 3- dimensional instruments
„
„
„
can vibrate at half, third, quarter length, etc.
harmonic oscillators with harmonic overtones
have complicated higher-order vibrations
harmonic oscillators with non-harmonic overtones
Examples: drums, cymbals, bells
Reeds and horns also use a vibrating air column
4
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
5