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Physics 1251 The Science and Technology of Musical Sound. Unit 3 Session 34 MWF Percussion with Pitch. Physics 1251Unit 3 Session 34 Percussion with Pitch. - PowerPoint PPT Presentation
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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.
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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.
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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¢¢
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
Tympani and TablaTympani and Tabla
Physics 1251Physics 1251 Unit 3 Session 33Unit 3 Session 33 PercussionPercussion
Orchestral PercussionOrchestral Percussion
TympaniTympani
Physics 1251Physics 1251 Unit 3 Session 33Unit 3 Session 33 PercussionPercussion
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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.
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
Metalophones: Metalophones: Glockenspiels, Xylophones, Marimbas and Glockenspiels, Xylophones, Marimbas and
VibesVibes
XyloXylo: wood: wood
PhonePhone: : soundsound
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 33Unit 3 Session 33 PercussionPercussion
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
Glockenspiel, Orchestra Bells:Glockenspiel, Orchestra Bells:
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
MarimbaMarimba
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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.
Physics 1251Physics 1251 Unit 3 Session 34Unit 3 Session 34 Percussion with PitchPercussion with Pitch
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.