Physics 1251 The Science and Technology of Musical Sound

1
Physics 1251The Science and Technology of
Musical Sound

• Unit 3
• Session 34 MWF
• Percussion with Pitch

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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

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

The larger cymbal must be about 15 thicker than
the smaller one, since the frequency is
proportional to the thickness and inversely
proportional to the square of the diameter.
3
Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 1' Lecture
• Piano strings exhibit inharmonicity because of
  the stiffness of the wire.
• Some percussion instruments have pitch.
• Pitch results from a harmonic series of
  overtones.
• Tympani and Tabla are pitched drums.
• Orchestra Chimes, Glockenspiel, Xylophone,
  Marimba and Vibraphone have intonation.

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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• The Percussion Instruments

Membranes
Strings
Drums
Piano
Hammer dulcimer
Percussion striking
Blocks, bells, shells
Bars
Plates
Cymbals, Gongs, Pans
Others
Xylophones, chimes
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 80/20The task of producing pitch in a percussion
  instrument is an exercise in manipulating the
  overtones into a harmonic series.

Frequency
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Physics 1251 Unit 3 Session 33 Percussion

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

Linear density µ mass/length
The stiffness of the wire increases the frequency
of the higher frequency harmonics.
Tension T force
fn n /(2 L) ? v(T/ µ) n 1, 2, 3, 4,
5, 6, 7.
P 3986 Log(nf1 /440) I(P) I(P)
Inharmonicity
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Inharmonicity of Piano

40
20
Inharmonicity
-20
Pitch ()
Because of the inharmonicity of strings the
octaves are stretched in a piano.
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Tympani and Tabla

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Physics 1251 Unit 3 Session 33 Percussion

• Orchestral Percussion

Tympani
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Physics 1251 Unit 3 Session 33 Percussion

• Tympani are tuned by adjusting the tension
  on the head.

Tension device
Tension pedal
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• The Modes of Oscillation
  of an (Ideal) Clamped Membrane

Mode (0,1)
f0 1 0.7655/ d ? v(S/ s)
Mode (1,1)
Mode (2,1)
f1 1 1.594 f0 1
f2 1 2.136 f0 1
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Air Loading of a Clamped Membrane

The mass of air moved by the membrane adds to the
effective surface density, lowering the frequency.
Air mass
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• 80/20The kettle of Tympani modifies the membrane
  frequencies by the interaction of the air
  resonances with the surface modes.

Modes of air vibration
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• The Modes of Oscillation
  of Tympani

Mode (0,1) fn m/f01 1
(1,1)1.594
(2,1)2.136
(0,2)2.296
(3,1)2.653
(1,2)2.918
(4,1)3.156
(2,2)3.501
(0,3)3.600
(5,1)3.652
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

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

Amplitude
Frequency
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Metalophones
• Glockenspiels, Xylophones, Marimbas and Vibes

Xylo wood Phone sound
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Metalophones
• Glockenspiels, Xylophones and Marimbas

Bar
h thickness
w width
L Length
Density ? mass/volume Youngs Modulus E
Force/elongation
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Metalophones
• Glockenspiels, Xylophones and Marimbas

Longitudinal Waves in a Bar
vL vE/ ? Longitudinal Wave Velocity
node
Anti-node
Anti-node
fn n/2LvE/ ? like an open pipe
Density ? mass/volumeYoungs Modulus E
Stress/Elongation
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Physics 1251 Unit 3 Session 33 Percussion

• Bending Wave in a Bar

vbend
h thickness
? density
E Youngs Modulus
Density ? mass/volume
vL vE/ ? Longitudinal Wave Velocity
Youngs Modulus E stress/elongation
stiffness
fnm ynm h vL /L2
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Bending Modes in Bars

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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Bending Modes in Bars

Free Ends

f1 1.133 fo f2 3.125 fo f36.125 fo
.224 L
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Glockenspiel, Orchestra Bells

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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Orchestral Chimes

Free Ends
End Plug
f1 1.133 fo f2 3.125 fo f36.125 fo
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Marimba

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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Mode Frequencies in Undercut Bar

Undercut Bar in Xylophone, Marimba and Vibraphone
Xylophone f1/f1 1.00f2 /f1 3.00f3 /f1 6.1
Marimba/Vibes f1 /f1 1.00f2 /f1 4.00f3 /f1
6.5
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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• What is the different between a Xylophone, a
  Marimba and a Vibraphone?
• The depth of the undercut a marimba is undercut
  more than a xylophone.
• The first harmonic of a xylophone is 3x the
  fundamental, for a marimba and vibe it is 4x.
• The xylophone sounds brighter and the marimba
  more mellow.
• Vibes have a tremolo mechanism.

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Physics 1251 Unit 3 Session 34 Percussion
with Pitch

• Summary
• Piano strings exhibit inharmonicity because of
  the stiffness of the wire.
• Some percussion instruments have pitch.
• Pitch results from a harmonic series of
  overtones.
• Tympani and Tabla are pitched drums.
• Orchestra Chimes, Glockenspiel, Xylophone,
  Marimba and Vibraphone have intonation.
• Marimba are undercut more than xylophones.