Transcript Physics 1251 The Science and Technology of Musical Sound

Physics 1251
The Science and Technology
of Musical Sound
Unit 3
Session 34 MWF
Percussion with Pitch
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.
Physics 1251
Unit 3 Session 34
Percussion with Pitch
1′ Lecture:
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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.
Physics 1251
Unit 3 Session 34
Percussion with Pitch
The Percussion Instruments
Strings
Membranes
Drums
Piano
Hammer dulcimer
Percussion – striking
Plates
Cymbals, Gongs, Pans
Bars
Xylophones, chimes
Blocks, bells,
shells
Others
Physics 1251
Unit 3 Session 34
Percussion with Pitch
task of producing pitch in a percussion
instrument is an exercise in manipulating the
overtones into a harmonic series.
Amplitude Amplitude
80/20The
fn m = xn m f10 Unpitched
f01
fn = n f1
f1
2f1
Frequency
3f1
Pitched
4f1
Physics 1251
Unit 3 Session 33
Percussion
The Modes of vibration of an ideal string are
harmonic.
Tension T
L
Linear density μ
• Linear density
μ= mass/length
• Tension
T= force
fn = n /(2 L) ‧ √(T/ μ)
n = 1, 2, 3, 4, 5, 6, 7….
The stiffness of the wire
increases the frequency of
the higher frequency
harmonics.
₧ = 3986¢ Log(nf1 /440) + I(₧)
I(₧) = Inharmonicity
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Inharmonicity
Inharmonicity of Piano
40¢
20¢
-20¢
Pitch (¢)
Because of the inharmonicity of strings the octaves are
“stretched” in a piano.
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Tympani and Tabla
Physics 1251
Unit 3 Session 33
Percussion
Orchestral Percussion
Tympani
Physics 1251
Unit 3 Session 33
Percussion
Tympani are tuned by adjusting
the tension on the head.
Tension device
Tension pedal
Physics 1251
Unit 3 Session 34
Percussion with Pitch
The Modes of Oscillation
of an (Ideal) Clamped Membrane
Surface density σ
Mode: (0,1)
f0 1 = 0.7655/ d ‧ √(S/ σ)
Surface Tension S
Mode: (1,1)
Mode: (2,1)
f1 1 = 1.594 f0 1
f2 1 = 2.136 f0 1
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Air Loading of a Clamped Membrane
Surface density σ
Surface Tension S
The mass of air moved by
the membrane adds to the
effective surface density,
lowering the frequency.
Air mass
Physics 1251
80/20The
Unit 3 Session 34
Percussion with Pitch
kettle of Tympani modifies the
membrane frequencies by the interaction
of the air resonances with the surface
modes.
Modes of air vibration
Physics 1251
Unit 3 Session 34
Percussion with Pitch
The Modes of Oscillation
Strike point
Mode: (0,1)
fn m/f01 : 1
(1,2)
2.918
(1,1)
(2,1)
of Tympani
(0,2)
(3,1)
1.594
2.136
2.296
2.653
(4,1)
(2,2)
(0,3)
(5,1)
3.156
3.501
3.600
3.652
Physics 1251
Unit 3 Session 34
Percussion with Pitch
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
80/20Tympani
f0
(0,1)
2f0
3f0
4f0
5f0
6f0
(1,1) (2,1)
(3,1)
(4,1) (0,3)
(3,2)
(0,3)
(3,2)
(0,2)
(1,2)
(2,2)
(2,2)
(5,1)
(0,2)
(1,2)
Frequency
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Metalophones:
Glockenspiels, Xylophones, Marimbas and Vibes
Xylo: wood
Phone: sound
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Metalophones:
Glockenspiels, Xylophones and Marimbas
Bar
h thickness
L Length
w width
Density ρ = mass/volume
Young’s Modulus E= Force/elongation
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Metalophones:
Glockenspiels, Xylophones and Marimbas
Longitudinal Waves in a Bar
vL = √E/ ρ
Anti-node
node
Anti-node
Longitudinal
Wave Velocity
fn = n/2L√E/ ρ like an open pipe
Density ρ = mass/volume
Young’s Modulus E= Stress/Elongation
Physics 1251
Unit 3 Session 33
Percussion
Bending Wave in a Bar
vbend
h: thickness
ρ: density
E: Young’s Modulus
• Density
ρ= mass/volume
• Young’s Modulus
E= stress/elongation
=stiffness
• vL = √E/ ρ
Longitudinal Wave Velocity
fnm = ynm h vL /L2
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Bending Modes in Bars:
End Clamped
f1= 0.1782 fo
f2= 1.116 fo
f3=3.125 fo
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Bending Modes in Bars:
Free Ends
f1= 1.133 fo
.224 L
f2= 3.125 fo
f3=6.125 fo
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Glockenspiel, Orchestra Bells:
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Orchestral Chimes
Free Ends
End Plug
f1= 1.133 fo
f2= 3.125 fo
f3=6.125 fo
Physics 1251
Marimba
Unit 3 Session 34
Percussion with Pitch
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Mode Frequencies in Undercut Bar:
Undercut Bar in Xylophone, Marimba and Vibraphone
Xylophone
f1/f1 = 1.00
f2 /f1 = 3.00
f3 /f1 =6.1
λ/4
Marimba/Vibes
f1 /f1 = 1.00
f2 /f1 = 4.00
f3 /f1 =6.5
Vibraphone
Physics 1251
Unit 3 Session 34
Percussion with Pitch
What is the different between a
Xylophone, a Marimba and a
Vibraphone?
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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.
Physics 1251
Unit 3 Session 34
Percussion with Pitch
Summary:
•
•
•
•
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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.