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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 1251 the science and technology of musical sound

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
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 harpsichord3
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 harpsichord4

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.

physics 1251 unit 2 session 22 strings guitar harp piano harpsichord5

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

physics 1251 unit 2 session 22 strings guitar harp piano harpsichord6

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

physics 1251 unit 2 session 22 strings guitar harp piano harpsichord7

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 harpsichord8
Physics 1251 Unit 2 Session 22 Strings: Guitar, Harp, Piano & Harpsichord

The Vibrating String Demonstration

physics 1251 unit 2 session 21 scales and strings
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 harpsichord10

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 harpsichord11
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 harpsichord12
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 harpsichord13
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 harpsichord14
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 harpsichord15
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 harpsichord16
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 harpsichord17

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 harpsichord18
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 harpsichord19
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 harpsichord20
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 harpsichord21
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 harpsichord22
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 harpsichord23
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 harpsichord24
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 harpsichord25
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 harpsichord26
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 harpsichord27
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 harpsichord28
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 harpsichord29
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 harpsichord30
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 harpsichord31
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

physics 1251 unit 2 session 22 strings guitar harp piano harpsichord32

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 harpsichord33
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 harpsichord34
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 harpsichord35
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 harpsichord36
Physics 1251 Unit 2 Session 22 Strings: Guitar, Harp, Piano & Harpsichord

Piano hammers are made of felt and wood.

Hammer Head

Back Check

Wood

Felt

Pivot hinge

physics 1251 unit 2 session 22 strings guitar harp piano harpsichord37
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 harpsichord38

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

Broadcast

physics 1251 unit 2 session 22 strings guitar harp piano harpsichord39
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 harpsichord40
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.