Physics 1251 The Science and Technology of Musical Sound

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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:

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### Physics 1251The 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.

f1 f2 f3 f4

♩ ♪ ♫

fn

~

~

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.

String Standing Waves

L = 2/₄ λ1

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

Node

Node

Antinode

λ1 /4

λ1 /4

L = 4/₄ λ2

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

Node

Node

Node

Antinode

Antinode

λ2 /4

λ2 /4

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)]

f1 f2 f3 f4

♩ ♪ ♫

fn

~

~

Physics 1251 Unit 2 Session 22 Strings: Guitar, Harp, Piano & Harpsichord

80/20The modes that will be present are those that are excited.

Energy source can excite selected modes

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

Frets

Strings (6)

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 Lo / Ln

1 1.06

2 1.12

3 1.20

4 1.27

5 1.33

Lo

L1

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

f1 f2 f3 f4

♩ ♪ ♫

fn

~

~

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.

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

Neck

Truss rod

Sound hole

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

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

Damper

Plectrum

String

Jack

Key

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

Piano

Strings

Key Board

Sound

Board

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

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.

Back Check

Wood

Felt

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?

f1 f2 f3 f4

♩ ♪ ♫

fn

~

~

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

f Response

f Generation

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.