Physics 1251 The Science and Technology of Musical Sound Unit 3 Session 34 MWF Percussion with Pitch...

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 34 MWFSession 34 MWF

Percussion with PitchPercussion with Pitch

Unit 3Unit 3

Session 34 MWFSession 34 MWF

Percussion with PitchPercussion with Pitch

Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch

A percussionist has two nearly A percussionist has two nearly identical cymbals. They have identical cymbals. They have identical fundamental frequencies, identical fundamental frequencies, but one is 15 inches in diameter but one is 15 inches in diameter while the other is 14 inches in while the other is 14 inches in diameter. What must be true diameter. What must be true about the two?about the two?The larger cymbal must be about The larger cymbal must be about

15% thicker than the smaller one, 15% thicker than the smaller one, since the frequency is proportional to since the frequency is proportional to the thickness and inversely the thickness and inversely proportional to the square of the proportional to the square of the diameter.diameter.


11′ Lecture:′ Lecture:• Piano strings exhibit inharmonicity Piano strings exhibit inharmonicity

because of the stiffness of the wire.because of the stiffness of the wire.• Some percussion instruments have pitch. Some percussion instruments have pitch.

• Pitch results from a harmonic series of Pitch results from a harmonic series of overtones.overtones.

• Tympani and Tabla are pitched drums.Tympani and Tabla are pitched drums.• Orchestra Chimes, Glockenspiel, Orchestra Chimes, Glockenspiel,

Xylophone, Marimba and Vibraphone have Xylophone, Marimba and Vibraphone have intonation.intonation.


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 task of producing pitch in a The task of producing pitch in a percussion instrument is an percussion instrument is an exercise in manipulating the exercise in manipulating the overtones into a harmonic series.overtones into a harmonic series.

FrequencyFrequency

Am

plitu

de

Am

plitu

de

ff00

11

ffn mn m = = xxn mn m ff1010

UnpitchedUnpitched

Am

plitu

de

Am

plitu

de

ff11 2f2f11 3f3f11 4f4f11

ffnn = n f = n f11 PitchedPitched

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

The Modes of vibration of an ideal The Modes of vibration of an ideal string are harmonic. string are harmonic.

• • 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….

Tension TTension T

Linear density Linear density μμLL

The stiffness of the The stiffness of the wire increases the wire increases the frequency of the frequency of the higher frequency higher frequency harmonics.harmonics.₧ ₧ = 3986= 3986¢ Log(nf¢ Log(nf1 1 /440) /440)

+ + II((₧) ₧)

II((₧) = Inharmonicity₧) = Inharmonicity


Inharmonicity of PianoInharmonicity of Piano

Pitch Pitch ((¢)¢)

Inh

arm

on

iIn

harm

on

icit

ycit

y

Because of the inharmonicity of strings Because of the inharmonicity of strings the octaves are “stretched” in a piano.the octaves are “stretched” in a piano.

2020¢¢

4040¢¢

-20-20¢¢


Tympani and TablaTympani and Tabla


Orchestral PercussionOrchestral Percussion

TympaniTympani


Tympani are tuned by adjusting Tympani are tuned by adjusting the tension on the the tension on the

head.head.Tension Tension devicedevice

Tension Tension pedalpedal


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 = 0.7655/ = 0.7655/ d d ‧ √(S/ ‧ √(S/ σ)σ)

Mode: Mode: (1,1)(1,1)ff1 11 1 = 1.594 f = 1.594 f0 10 1

Mode: Mode: (2,1)(2,1)ff2 12 1 = 2.136 f = 2.136 f0 10 1

Surface Surface Tension STension S

Surface density Surface density σσ


Air Loading of a Clamped Air Loading of a Clamped MembraneMembrane

Surface Surface Tension STension S

Surface density Surface density σσ

Air Air massmass

The mass of air The mass of air moved by the moved by the membrane adds to membrane adds to the effective surface the effective surface density, lowering the density, lowering the frequency.frequency.


80/2080/20The kettle of Tympani modifies the The kettle of Tympani modifies the membrane frequencies by the membrane frequencies by the interaction of the air resonances interaction of the air resonances with the surface modes.with the surface modes.Modes of air vibrationModes of air vibration


The Modes of Oscillation The Modes of Oscillation of of TympaniTympani

Mode: Mode: (0,1) (0,1) ffn n

mm/f/f01 01 : 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

Strike Strike pointpoint


80/2080/20Tympani achieve pitch by (1) Tympani achieve pitch by (1) suppression of “radial” modes; (2) suppression of “radial” modes; (2) modification of other mode modification of other mode frequencies by air loading and the frequencies by air loading and the effect of the kettle ; (3) attenuation effect of the kettle ; (3) attenuation of the lowest mode.of the lowest mode.

FrequencyFrequencyAm

plitu

de

Am

plitu

de

(0,1(0,1)) (1,1(1,1

))(2,1(2,1)) (0,2(0,2

))

(3,1(3,1)) (1,2(1,2

))

(4,1(4,1)) (2,2(2,2

))

(0,3(0,3)) (5,1(5,1

))

(3,2(3,2))

(0,1(0,1)) (1,1(1,1

))(2,1(2,1)) (0,2)(0,2)

(3,1(3,1)) (1,2)(1,2)

(4,1(4,1)) (2,2)(2,2)

(0,3)(0,3)(5,1(5,1))

(3,2)(3,2)

5f5f

00

2f2f00 3f3f00 4f4f00 6f6f00ff00


Metalophones: Metalophones: Glockenspiels, Xylophones, Marimbas and Glockenspiels, Xylophones, Marimbas and

VibesVibes

XyloXylo: wood: wood

PhonePhone: : soundsound


Metalophones: Metalophones:

Glockenspiels, Xylophones and Glockenspiels, Xylophones and MarimbasMarimbas

h h thickness thickness

L LengthL Length

Density Density ρ = mass/volumeρ = mass/volume

Young’s Modulus E= Young’s Modulus E= Force/elongationForce/elongation

BarBar

w widthw width


Metalophones: Metalophones:

Glockenspiels, Xylophones and Glockenspiels, Xylophones and MarimbasMarimbas

vvLL = √E/ = √E/ ρρ

LongitudinLongitudinal Wave al Wave VelocityVelocity

Density Density ρ = mass/volumeρ = mass/volumeYoung’s Modulus E= Young’s Modulus E= Stress/ElongationStress/Elongation

Longitudinal Waves in a Longitudinal Waves in a BarBar

ffnn = n/2L = n/2L√E/ √E/ ρ like an open ρ like an open pipepipe

nodenodeAnti-nodeAnti-node Anti-nodeAnti-node


Bending Wave in a BarBending Wave in a Bar

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

=stiffness=stiffness

• • vvLL = √E/ = √E/ ρρ

Longitudinal Wave VelocityLongitudinal Wave Velocity

h: thicknessh: thicknessvvbendbend

ρ: ρ: densitydensity

E: Young’s E: Young’s ModulusModulus

ffnmnm = = yynm nm h vh vLL /L/L22


Bending Modes in Bars:Bending Modes in Bars:

ff11= 0.1782 = 0.1782 ffoo

ff22= 1.116 = 1.116 ffoo

ff33=3.125 =3.125 ffoo

End ClampedEnd Clamped


Bending Modes in Bars:Bending Modes in Bars:

ff11= 1.133 = 1.133 ffoo

ff22= 3.125 = 3.125 ffoo

ff33=6.125 =6.125 ffoo

.224 L.224 L

Free EndsFree Ends


Glockenspiel, Orchestra Bells:Glockenspiel, Orchestra Bells:


Orchestral ChimesOrchestral Chimes

ff11= 1.133 = 1.133 ffoo

ff22= 3.125 = 3.125 ffoo

ff33=6.125 =6.125 ffoo

Free EndsFree Ends

End PlugEnd Plug


MarimbaMarimba


Mode Frequencies in Undercut Bar:Mode Frequencies in Undercut Bar:

XylophonXylophonee

ff11/f/f11 = 1.00 = 1.00ff2 2 /f/f11 = 3.00 = 3.00ff3 3 /f/f11 =6.1 =6.1

Undercut Bar in Xylophone, Marimba and Undercut Bar in Xylophone, Marimba and VibraphoneVibraphone

Marimba/Marimba/VibesVibes

ff1 1 /f/f11 = 1.00 = 1.00ff2 2 /f/f11 = 4.00 = 4.00ff3 3 /f/f11 =6.5 =6.5

λ/4λ/4

VibraphoneVibraphone


What is the different between a What is the different between a Xylophone, a Marimba and a Xylophone, a Marimba and a Vibraphone?Vibraphone?

• The depth of the undercut: a marimba The depth of the undercut: a marimba is undercut more than a xylophone.is undercut more than a xylophone.

• The first harmonic of a xylophone is 3x The first harmonic of a xylophone is 3x the fundamental, for a marimba and the fundamental, for a marimba and “vibe” it is 4x.“vibe” it is 4x.

• The xylophone sounds “brighter” and The xylophone sounds “brighter” and the marimba more “mellow.”the marimba more “mellow.”

• Vibes have a tremolo mechanism.Vibes have a tremolo mechanism.


Summary:Summary:• Piano strings exhibit inharmonicity Piano strings exhibit inharmonicity

because of the stiffness of the wire.because of the stiffness of the wire.• Some percussion instruments have pitch. Some percussion instruments have pitch.

• Pitch results from a harmonic series of Pitch results from a harmonic series of overtones.overtones.

• Tympani and Tabla are pitched drums.Tympani and Tabla are pitched drums.• Orchestra Chimes, Glockenspiel, Orchestra Chimes, Glockenspiel,

Xylophone, Marimba and Vibraphone Xylophone, Marimba and Vibraphone have intonation.have intonation.

• Marimba are undercut more than Marimba are undercut more than xylophones.xylophones.

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Physics 1251 The Science and Technology of Musical Sound Unit 3 Session 34 MWF Percussion with Pitch...

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