Physics 1251 The Science and Technology of Musical Sound Unit 3 Session 33 MWF Percussion...

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Unit 3Unit 3

Session 33 MWFSession 33 MWF

Percussion InstrumentsPercussion Instruments

Unit 3Unit 3

Session 33 MWFSession 33 MWF

Percussion InstrumentsPercussion Instruments

Physics 1251Physics 1251 Unit 3 Session 33Unit 3 Session 33 PercussionPercussion

In the video (In the video (AmadeusAmadeus), why did the ), why did the soprano smile so broadly when soprano smile so broadly when she sang the high notes?she sang the high notes?

She was deliberately raising the She was deliberately raising the pitch of the formant in order to pitch of the formant in order to match the pitch of the notes she match the pitch of the notes she was singing by changing the was singing by changing the shape of her vocal tract.shape of her vocal tract.


Q&A:Q&A:

• Throat Singers? Throat Singers? • Dogs barking?Dogs barking?• Smoking? Smoking? • Whistling?Whistling?• Hyoid bones?Hyoid bones?


11′ Lecture:′ Lecture:

• Percussion instruments are instruments Percussion instruments are instruments that are struck. that are struck.

• The timbre of their sound is determined The timbre of their sound is determined by their vibration recipe.by their vibration recipe.

• Their vibration recipe is determined by Their vibration recipe is determined by the modes of oscillation that are excited.the modes of oscillation that are excited.

• Often percussion instruments do not Often percussion instruments do not have pitch.have pitch.


The Rain StickThe Rain StickBy Seamus HeaneyBy Seamus Heaney

Upend the rain stick and what happens nextUpend the rain stick and what happens next

Is a music that you never would have knownIs a music that you never would have known

To listen for. In a cactus stalkTo listen for. In a cactus stalk

Downpour, sluice-rush, spillage and backwashDownpour, sluice-rush, spillage and backwash

Come flowing through. You stand there like a pipeCome flowing through. You stand there like a pipe

Being played by water, you shake it again lightlyBeing played by water, you shake it again lightly

And diminuendo runs through all its scalesAnd diminuendo runs through all its scales

Like a gutter stopping trickling. And now here comesLike a gutter stopping trickling. And now here comes

A sprinkle of drops out of the freshened leaves,A sprinkle of drops out of the freshened leaves,


The Rain StickThe Rain StickBy Seamus HeaneyBy Seamus Heaney

Then subtle little wets off grass and daises;Then subtle little wets off grass and daises;

Then glitter-drizzle, almost-breaths of air.Then glitter-drizzle, almost-breaths of air.

Upend the stick again. What happens nextUpend the stick again. What happens next

Is undiminished for having happened once,Is undiminished for having happened once,

Twice, ten, a thousand times before.Twice, ten, a thousand times before.

Who cares if all the music that transpiresWho cares if all the music that transpires

Is fall of grit or dry seeds through cactus?Is fall of grit or dry seeds through cactus?

You are like a rich man entering heavenYou are like a rich man entering heaven

Through the ear of a raindrop. Listen now again.Through the ear of a raindrop. Listen now again.

From From The Spirit LevelThe Spirit Level (New York: Noonday Press, 1996) p3. (New York: Noonday Press, 1996) p3.


The Percussion InstrumentsThe Percussion Instruments

Percussion – strikingPercussion – striking

PianoPianoHammer Hammer dulcimerdulcimer

Cymbals, Gongs, Cymbals, Gongs, PansPans

Xylophones, chimesXylophones, chimes OthersOthers

DrumsDrums

MembranesMembranes

Blocks, Blocks, bells, bells, shellsshells

PlatesPlates BarsBars

StringsStrings


80/2080/20The timbre of an instrument’s The timbre of an instrument’s sounds depends on its vibration sounds depends on its vibration recipe.recipe.

FrequencyFrequency

Am

plitu

de

Am

plitu

de

Am

plitu

de

Am

plitu

de

ff11 2f2f11 3f3f11 4f4f11

ff00

11

ffnn = n f = n f11

ffn mn m = = xxn mn m ff11

PitchedPitched

UnpitchedUnpitched


The Oscillation of a Clamped The Oscillation of a Clamped MembraneMembrane

Mode: Mode: (0,1)(0,1)ff0 10 1 = v/ = v/λ; v = λ; v = √(S/ √(S/ σ)σ)

ff0 10 1 = = xx0 10 1 /( /(π d) π d) ‧ √(S/ ‧ √(S/ σ)σ)

xx0 1 0 1 = 2.405= 2.405

Surface density Surface density σσ

Surface Surface Tension STension S

Surface density Surface density σσ= = mass/area mass/area σσ= density = density ‧ ‧ thicknessthicknessSurface Tension Surface Tension SS= force/length= force/length

d d


Clamped Membrane vs StringClamped Membrane vs String

ffn mn m = = xxn mn m /( /(π d) π d) ‧ √(S/ ‧ √(S/ σ) σ) xx0 1 0 1 = 2.405= 2.405


• • Surface Surface density density σσ= mass/area= mass/area• • Surface Tension Surface Tension SS= = force/lengthforce/length

Tension TTension T

Linear density Linear density μμ

• • Linear Linear density density μμ= = mass/lengthmass/length• • Tension Tension TT= force= force

ffn n = n /( = n /(2 L) 2 L) ‧ √(T/ ‧ √(T/ μ) μ) nn = 1, 2, 3, 4, 5, = 1, 2, 3, 4, 5, 6, 7….6, 7….

dd


LL


The Oscillation of a Clamped The Oscillation of a Clamped MembraneMembraneMode: Mode:

(0,1)(0,1) ff0 10 1 = v/ = v/λ; v = λ; v = √(S/ √(S/ σ)σ)

ff0 10 1 = = xx0 10 1 /( /(π d) π d) ‧ √(S/ ‧ √(S/ σ)σ)

xx0 1 0 1 = 2.405= 2.405Surface Surface Tension STension S


Example: Example: d = 0.30 m; m = 58 gm; T = 474 d = 0.30 m; m = 58 gm; T = 474 NN • • C = C = π d = .94 m π d = .94 m • • Area= 0.73 mArea= 0.73 m22 • • σσ= = 0.058 kg/0.073 m0.058 kg/0.073 m22=0.8 kg/m=0.8 kg/m2 2 • • S = T/C = 503. N/m S = T/C = 503. N/m ff0 10 1 = = xx0 10 1 /( /(π d) π d) ‧ √(S/ ‧ √(S/ σ)σ) = = 2.405/(0.94)2.405/(0.94)√(503/0.8)√(503/0.8)= = 64 Hz64 Hz

d d


The Modes of Oscillation The Modes of Oscillation of an (Ideal) of an (Ideal) Clamped MembraneClamped MembraneMode: Mode:

(0,1)(0,1)ff0 10 1 = = xx0 10 1 /( /(π d) π d) ‧ √(S/ ‧ √(S/ σ)σ)

xx0 1 0 1 = 2.405= 2.405Mode: Mode: (1,1)(1,1)ff1 11 1 = ( = (xx1 11 1 / / xx0 10 1) f) f0 10 1

xx1 11 1 / / xx0 10 1 = 1.594 = 1.594

Mode: Mode: (2,1)(2,1)ff2 12 1 = ( = (xx2 12 1 / / xx0 10 1) f) f0 10 1

xx2 12 1 / / xx0 10 1 = 2.136 = 2.136




The Modes of Oscillation The Modes of Oscillation of a Clamped of a Clamped MembraneMembrane

Mode: Mode: (0,1) (0,1) xxn m n m / /

xx0 1 0 1 : 1: 1

(1,1(1,1))1.51.59494

(2,1(2,1))2.12.13636

(0,2(0,2))2.22.29696

(3,1(3,1))2.62.65353

(1,2(1,2))2.92.91818

(4,1(4,1))3.13.15656

(2,2(2,2))3.53.50101

(0,3(0,3))3.63.60000

(5,1(5,1))3.63.65252


80/2080/20Membrane Acoustics:Membrane Acoustics:

• The overtones of a circular membrane The overtones of a circular membrane clamped at the edge are not harmonic clamped at the edge are not harmonic and, therefore, they have no pitch. and, therefore, they have no pitch.

ffn mn m = ( = (xxn mn m / /xx0101)f)f0101

• The frequencies The frequencies ffnmnm of a membrane are of a membrane are (1) proportional to the square root of the (1) proportional to the square root of the ratio of surface tension of the head to the ratio of surface tension of the head to the surface density surface density ∝√(S / ∝√(S / σ)σ) and (2) inversely and (2) inversely proportional to its diameter proportional to its diameter ∝1/d∝1/d..


Demonstration:Demonstration:

Normal Modes of a Oscillation of a Normal Modes of a Oscillation of a Clamped MembraneClamped Membrane


Ideal vs Real Membranes:Ideal vs Real Membranes:80/2080/20Real membranes have a lower Real membranes have a lower

frequencies than predicted for ideal frequencies than predicted for ideal membranes because of air loading; membranes because of air loading; the lowest frequencies are lowered the lowest frequencies are lowered the most.the most.


Mode Excitation:Mode Excitation:80/2080/20Only those frequencies for which Only those frequencies for which

the modes were excited will appear the modes were excited will appear in the vibration recipe.in the vibration recipe.

80/2080/20The highest frequency that can be The highest frequency that can be excited by a mallet that is in contact excited by a mallet that is in contact with the surface for a period of Twith the surface for a period of Tcontactcontact is is

f f maxmax= 2/T= 2/Tcontactcontact

Mode Excitation:Mode Excitation:

80/2080/20The highest frequency that can be The highest frequency that can be excited by a mallet that is in excited by a mallet that is in contact with the surface for a period contact with the surface for a period of Tof Tcontactcontact is is

f f maxmax= 2/T= 2/Tcontactcontact


TTcontactcontact = ½ T = ½ Tperiodperiod= = 1/(2f1/(2fmaxmax ) )


Demonstration:Demonstration:

Longitudinal Modes vs Transverse Longitudinal Modes vs Transverse Modes for a RodModes for a Rod


Longitudinal Wave (Sound Wave) Longitudinal Wave (Sound Wave)

•• DDensity ensity ρ ρ= = mass/volumemass/volume• • Young’s ModulusYoung’s Modulus EE= = stress/elongation stress/elongation

=stiffness=stiffness

• • vvLL = √E/ = √E/ ρρ

h: thicknessh: thicknessvvLL

ρ: ρ: densitydensity

E: Young’s E: Young’s ModulusModulus

ffnn= n v= n vLL/(2L)/(2L)


Bending Wave in a PlateBending Wave in a Plate

•• DDensity ensity ρ ρ= = mass/volumemass/volume• • Young’s ModulusYoung’s Modulus EE= = stress/elongation stress/elongation

=stiffness=stiffness

• • vvLL = √E/(.91 = √E/(.91 ρ)ρ)

vvbendbend = √[1.8 f h v = √[1.8 f h vL L ] ]

h: thicknessh: thicknessvvbendbend

ρ: ρ: densitydensity

E: Young’s E: Young’s ModulusModulus

ffnmnm = 0.0459 h v = 0.0459 h vLL( ( yynm nm /d)/d)22


The Modes of Oscillation The Modes of Oscillation of a Flat of a Flat CymbalCymbal

Mode: Mode: (2,0) (2,0) ffn m n m / /

ff0 1 0 1 : 1: 1

(0,1(0,1))1.71.73030

(3,0(3,0))2.32.32828

(1,1(1,1))3.93.91010

(4,0(4,0))4.14.11010

(5,0(5,0))6.36.3

00

(2,1(2,1))6.76.7

11

(0,2(0,2))3.63.60000


80/2080/20 Plate Acoustics: Plate Acoustics:

• The overtones of a circular plate clamped The overtones of a circular plate clamped in the center are not harmonic and, in the center are not harmonic and, therefore, have no pitch. therefore, have no pitch.

ffn mn m = ( = (yyn mn m / /yy2020))22 f f2020

• The frequencies The frequencies ffnmnm of a circular plate are of a circular plate are (1) proportional to the thickness (1) proportional to the thickness ∝h∝h and and (2) to the square root of the ratio of the (2) to the square root of the ratio of the stiffness and the density stiffness and the density ∝√E/∝√E/ρρ and (3) and (3) inversely proportional to the inversely proportional to the squaresquare of the of the diameter diameter ∝1/d∝1/d22 . .


Summary:Summary:• Percussion instruments are instruments Percussion instruments are instruments

that are struck.that are struck.• Their vibration recipe is often not Their vibration recipe is often not

harmonic and, therefore, they do not harmonic and, therefore, they do not have a definite pitch.have a definite pitch.

• For ideal circular edge-clamped For ideal circular edge-clamped membranes: fmembranes: fnmnm ∝( ∝(xxnmnm /d)√(S//d)√(S/σ)σ)..

• For circular plates free at the edge: For circular plates free at the edge: f fnmnm ∝h ‧ ( ∝h ‧ (yynmnm /d) /d) 2 2 √(E/√(E/ρ)ρ)..

• The maximum frequency excited by a The maximum frequency excited by a mallet is f mallet is f maxmax= 2/T= 2/Tcontactcontact..

Physics 1251 The Science and Technology of Musical Sound Unit 3 Session 33 MWF Percussion...

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