Physics 1251
The Science and Technology
of Musical Sound
Unit 2
Session 22 MWF
Strings: Guitar, Harp, Piano
and Harpsichord
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
Foolscap Quiz:
A piano string that has a mass of 6 grams for
every meter and has a length of 1.02 m is
stretched with a tension of 800 N. If the string
is struck by a hammer, what fundamental
frequency will you hear?
f = 1/(2L) √(T/ μ)
f = 1/(2‧1.02) √(800/0.006)
f = 179.0 Hz
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
1′ Lecture:
•
The fundamental frequency of a string is equal
to the reciprocal of twice the length of the string
multiplied by the square root of the ratio of the
tension and the linear mass density:
f1 = [1/(2L)] ‧ √(T/ μ)]
•
Standing waves on the string are responsible for
the harmonic series of stringed instruments.
•
The details of the harmonic recipe depends
upon the excitation of the various harmonics.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
A Standing Wave results from interference of
counter-reflecting waves.
Fundamental Mode
f1 = vstring / λ1 = vstring / 2L
L = 2/₄ λ1
Node
Node
Antinode
λ1 /4
λ1 /4
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
The Vibrating String Demonstration
Physics 1251
Unit 2 Session 21
Scales and Strings
80/20Frequency
of nth harmonic:
fn = [2n/(4L)] ‧ vstring
fn = [n/(2L)] ‧ √(T/ μ)]
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
80/20The
modes that will be present are
those that are excited.
Example: Guitar Overtones (Harmonics)
Plucking, strumming, damping
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
Density of strings sets range of each string
Denser wire: lower pitch
Less dense wire:
higher pitch
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
Frets shorten the string to raise the pitch.
Fret
1
2
3
4
5
Lo / Ln
1.06
1.12
1.20
1.27
1.33
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
The tuning pegs change the string tension.
Tuning Pegs
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
The strings are attached to the body and
sound board by a flat bridge.
Bridge
Strings
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
The response of the body and air cavity
determine the filtering of the harmonics.
Guitar
in cross section
Sound hole
Neck
Truss rod
Bridge
Body
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
Vibration Modes of Guitar Body:
Holographic vibration images
of Guitar body
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
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
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
The principal pitch of each string in a Pedal Harp
is determined by its length.
However, depressing a pedal shortens or lengthens the string
length, raising (or lowering) the pitch incrementally.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
80/20A
Harpsichord is a keyboard stringed
instrument whose strings are plucked.
Each string is used for one pitch only.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
80/20A
Harpsichord is a keyboard stringed
instrument whose strings are plucked.
Each string is used for one pitch only.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
80/20A
Harpsichord is a keyboard stringed
instrument whose strings are plucked.
Each string is used for one pitch only.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
80/20A
Harpsichord is a keyboard stringed instrument
whose strings are plucked.
Each string is used for one pitch only.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
80/20A
Harpsichord is a keyboard stringed instrument
whose strings are plucked.
Each string is used for one pitch only.
Plucked
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
The jack plucks in one direction,
then retracts in the opposite.
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
A Hammer Dulcimer or Cibolo
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
The Piano strings are hammered by a felt
hammer, actuated by the “action.”
String and Damper
Hammer
Jack Key
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
The Piano strings are hammered by a felt
hammer, actuated by the “action.”
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
Piano strings are tensioned by tuning pins in a
strong frame.
Frame
Bass Strings
Tuning Pins
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
Piano hammers are made of felt and wood.
Hammer Head
Felt
Back Check Wood
Pivot hinge
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
The timbre of the piano sound is affected by the
hardness of the hammer felt.
Felt –
hard or soft?
Physics 1251
Unit 2 Session 22
Strings: Guitar, Harp, Piano & Harpsichord
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 22
Strings: Guitar, Harp, Piano & Harpsichord
Summary:
•
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 22
Strings: Guitar, Harp, Piano & Harpsichord
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 that, when depressed, lift a plectrum that
plucks one or more strings.
•
The Piano is a stringed instrument with a
keyboard that causes a felt hammer to strike
the string.