phy-2464 physical basis of music · 2005-03-16 · phy2464 -the physical basis of music phy-2464...

PHY2464 - The Physical Basis of Music

PHY-2464Physical Basis of Music

PHYPHY--24642464Physical Basis of MusicPhysical Basis of Music

Presentation 19Characteristic Sound (Timbre) of

Wind InstrumentsAdapted from Sam Matteson’s

Unit 3 Session 30 and Unit 1 Session 10Sam Trickey

Mar. 15, 2005

Presentation 19Presentation 19Characteristic Sound (Timbre) of Characteristic Sound (Timbre) of

Wind InstrumentsWind InstrumentsAdapted from Sam Matteson’s

Unit 3 Session 30 and Unit 1 Session 10Sam Trickey

Mar. 15, 2005

PHYPHY--2464 2464 Pres. 19 Sound of Wind InstrumentsPres. 19 Sound of Wind Instruments

REMINDERS:REMINDERS:Brass instruments are characterized by “buzzing” Brass instruments are characterized by “buzzing” the lips in a mouth piece. the lips in a mouth piece.

•• KEY FACTS KEY FACTS –– Pipe is Pipe is closedclosed atat mouthpiecemouthpiece endend to to an excellent approximation. The buzzing is stabilized an excellent approximation. The buzzing is stabilized by feedback from the open pipe end which matches by feedback from the open pipe end which matches impedance peaks.impedance peaks.

••There are There are valvedvalved and and valvelessvalveless brass instrumentsbrass instruments••There are “brasses” made with little or no metal.There are “brasses” made with little or no metal.



What is the key physical difference among a Cornet, What is the key physical difference among a Cornet, Trumpet, and Trumpet, and FlFlüügelhorngelhorn??

Cornet

TrumpetTrumpet

Flügelhorn

The fraction of the horn that is The fraction of the horn that is cone/cylinder/flare.cone/cylinder/flare.•• Trumpet Trumpet –– most cylindricalmost cylindrical•• Cornet Cornet ---- more conicalmore conical•• Flugel Horn Flugel Horn –– most conicalmost conical


Transverse Flute Transverse Flute –– open pipe instrumentopen pipe instrumentDriven by air flow against the edge of the Driven by air flow against the edge of the embochureembochure hole. A hole. A

pressure node exists there. Pitch is controlled by holes in pressure node exists there. Pitch is controlled by holes in the tube.the tube.

EmbrochureEmbrochure holehole

Air flowAir flow



Single Reed instruments Single Reed instruments –– stopped pipe stopped pipe The reed opens and closes like a valve, pressurizing the pipe The reed opens and closes like a valve, pressurizing the pipe

when open, closing due to the Bernoulli effect when the when open, closing due to the Bernoulli effect when the air flows. A pressure antiair flows. A pressure anti--node exists at the reed. Pitch is node exists at the reed. Pitch is controlled by holes in the pipe.controlled by holes in the pipe.

Air flowAir flow

ReedReedTonguingTonguing


Double Reed instrumentsDouble Reed instruments –– stopped pipe stopped pipe As with single reeds, the reed opens and closes like a valve, As with single reeds, the reed opens and closes like a valve,

pressurizing the pipe when open, closing due to the pressurizing the pipe when open, closing due to the Bernoulli effect when the air flows. A pressure antiBernoulli effect when the air flows. A pressure anti--node node exists at the reed. Pitch is controlled by holes in the pipe.exists at the reed. Pitch is controlled by holes in the pipe.

Air flowAir flow

Reed TipReed Tip

Pressure PulsesPressure Pulses



Brass instruments Brass instruments –– stopped pipestopped pipeBrass instruments are played by the playerBrass instruments are played by the player’’s lips that form a lip s lips that form a lip

valve. The lip valve admits pressure pulses into the pipe. valve. The lip valve admits pressure pulses into the pipe. The frequency is determined by the breath air pressure, The frequency is determined by the breath air pressure, the lip tension and the resonances of the pipe.the lip tension and the resonances of the pipe. A pressure A pressure antianti--node exists at the playernode exists at the player’’s lips. Pitch is controlled by s lips. Pitch is controlled by valves and extra tubing, and/or slides, and/or valves and extra tubing, and/or slides, and/or overblowingoverblowing..

Louis Armstrong Louis Armstrong –– trumpet trumpet (1901(1901--1971)1971)


Wind instrument comparisonWind instrument comparison

BrassBrass

ffPedal Pedal ToneTone

Other Other Woodwinds Woodwinds ClarinetClarinetFluteFlute

ff11

2f2f11

3f3f11

4f4f11

5f5f11

ff11

3f3f11

5f5f11 5f5f11

ff11

2f2f11

3f3f11

4f4f11

6f6f11

ff11 = v/2L= v/2L ff11 = v/4L= v/4L ff11 = v/2(L+c)= v/2(L+c)LL

cc

ffoo = = (1+(1+ξξ)v/4(L+c))v/4(L+c)

ff11

ffOO

2f2fOO

3f3fOO

4f4fOO

5f5fOO

6f6fOO



More wind instrument comparisonMore wind instrument comparison

BrassBrassOther Other WoodwindsWoodwindsClarinetClarinetFluteFlute

ff11 = v/2L= v/2L ff11 = v/4L= v/4L ff11 = v/2(L+c)= v/2(L+c)LL

cc

ffoo = = (1+(1+ξξ)v/4(L+c))v/4(L+c)

Open Open

Cylinder Cylinder

NNpp –– NNpp

ffnn = nf= nf1 1

ff11 = v/2L= v/2L

Stopped Stopped

Cylinder Cylinder

AApp –– NNpp

ff2n2n--11 = (2n= (2n--1)f1)f11

ff11= v/4L= v/4L

Stopped Stopped

Cone Cone

AApp –– NNpp

ffnn = nf= nf11

ff11= v/2(L+c)= v/2(L+c)

Stopped Stopped Combination Combination AApp –– NNpp

ffnn = nf= nf00

ff00= (1+= (1+ξξ))v/4(L+c)v/4(L+c)


In the flute, feedback from the acoustic standing In the flute, feedback from the acoustic standing wave locks the frequency of the oscillation if the wave locks the frequency of the oscillation if the edge tone is near the fundamental frequency.edge tone is near the fundamental frequency.Matches impedance Matches impedance minima.minima.Displacement waveDisplacement wave

ffedgeedge = 0.2 = 0.2 vvjetjet /b/b

ffnn = n v/ 2L; = n v/ 2L; ffedgeedge ≈≈ ffnn



In reed instruments, pressure standing wave In reed instruments, pressure standing wave feedback locks the frequency of the oscillation of feedback locks the frequency of the oscillation of the reed; matches impedance the reed; matches impedance maxima.maxima.

Pressure wavePressure waveff2n2n--11 = (2n= (2n--1) v/ 4L1) v/ 4L′′

Pressure invertsPressure invertsLL′′ = L + 0.6 r= L + 0.6 r

0.6 r0.6 r


Brass Instruments Brass Instruments are stopped pipes.are stopped pipes.

•• The playerThe player’’s lips s lips produce a displaceproduce a displace--mentment node (pressure node (pressure antinodeantinode) at the mouth) at the mouth--piecepiece. . A displacement A displacement antianti--node (pressure node (pressure node) exists at the bell.node) exists at the bell.

Winton Marsalis Winton Marsalis TrumpetTrumpet



Feedback from ResonancesFeedback from ResonancesKEY FACT: Wind instrument pitch is KEY FACT: Wind instrument pitch is

determined by the influence on the determined by the influence on the jet/reed/lipjet/reed/lip--valve of feedback from the valve of feedback from the pressure/displacement standing waves in pressure/displacement standing waves in the pipe. The near matching of impedance the pipe. The near matching of impedance maxima or minima determines the stable maxima or minima determines the stable ((““playableplayable””) pitches.) pitches.


Interlude regarding basics of SPECTRA:Interlude regarding basics of SPECTRA:•• The energy of an oscillator or of sound dissipates The energy of an oscillator or of sound dissipates

in an exponential decay.in an exponential decay.•• An oscillator can be caused to vibrate in An oscillator can be caused to vibrate in

“sympathy” when the driving frequency is close “sympathy” when the driving frequency is close to that of a natural mode of oscillation.to that of a natural mode of oscillation.

•• The timbre of an instrument is determined by its The timbre of an instrument is determined by its spectrumspectrum

•• The spectrum of an instrument changes with time The spectrum of an instrument changes with time because of transients.because of transients.



Exponential Decay (pressure as example)Exponential Decay (pressure as example)

1/ 2/0, / 2t t

envelope envp p=


A A FourierFourier DecompositionDecomposition is a representation of all is a representation of all the components that comprise a waveform, the components that comprise a waveform, amplitudeamplitude versus frequency and versus frequency and phasephase versus versus frequency.frequency.



Example: Synthesis of a Square waveExample: Synthesis of a Square wave--The Fourier SpectrumThe Fourier Spectrum

FrequencyFrequencyffoo 3f3foo 5f5foo 7f7foo

11stst Overtone Overtone –– 33rdrd HarmonicHarmonic

22ndnd Overtone Overtone –– 55thth HarmonicHarmonic

Fundamental Fundamental –– 11stst HarmonicHarmonic

Am

plitu

deA

mpl

itude


Waveform SynthesisHarmonic n = 1 2 3 4 5 6 7 8 9Overtone f0 f1 f2 f3 f4 f5 f6 f7 f8Amplitude = 1 0.5 0.333 0.25 0.2 0.167 0.143 0.125 0.111 Amplitude (Pa)

all modes =1/odd=2: 1 0 0 0 0 0 0 0 0 Phase

Frequency (Hz) = 440 880 1320 1760 2200 2640 3080 3520 3960 1000439 440 441 879 880 881 1319 1320 1321 1759

0 1 0 0 0.5 0 0 0.333333 0 0Composite Waveform

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8

Time (ms)

Am

plitu

de (P

a)

Fourier Spectrum

0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000Frequency (Hz)

Com

pone

nt A

mpl

itude

(Pa)



ADSR: Attack, Decay, Sustain, ReleaseADSR: Attack, Decay, Sustain, Release

The envelope of the amplitude of all musical soundThe envelope of the amplitude of all musical soundis described by ADSR .is described by ADSR .

AttackAttack DecayDecay SustainSustainReleaseRelease

timetime

ampl

itude

ampl

itude


Wind InstrumentsWind Instruments•• A jet produces a fluctuating air flow, while a A jet produces a fluctuating air flow, while a

reed or the lips produce pressure pulsations, the reed or the lips produce pressure pulsations, the frequencies of which are controlled by frequencies of which are controlled by impedance matching feedback from standing impedance matching feedback from standing waves in the horn.waves in the horn.

♩♩ ♪♪ ♫♫ ff1 1 ff2 2 ff3 3 ff44fn

~~ ~~

Flow fluctuations Flow fluctuations or Pressure or Pressure pulsationspulsations

Standing waves in hornStanding waves in horn

FeedbackFeedback



Effect of ExcitationEffect of Excitation

•• The mode of excitation of the horn The mode of excitation of the horn significantly influences the significantly influences the ““harmonic harmonic reciperecipe”” (spectrum) of the air column.(spectrum) of the air column.

•• The relative strengths (amplitudes) of The relative strengths (amplitudes) of the harmonics are determined by the the harmonics are determined by the excitation of the jet/reed/lipexcitation of the jet/reed/lip--valve. valve.

•• The spectrum of the radiated sound The spectrum of the radiated sound determines the timbre.determines the timbre.


Driven Pipe Vibrational Spectrum (Driven Pipe Vibrational Spectrum (““RecipeRecipe””))

AA

AA

AA

Pipe SpectrumPipe Spectrum

Mouthpiece SpectrumMouthpiece Spectrum

Driven Pipe SpectrumDriven Pipe Spectrum

FrequencyFrequency



Effect of the PipeEffect of the Pipe

•• A pipe is three dimensional; therefore, 3A pipe is three dimensional; therefore, 3--D D modes of oscillation are possible in the pipe.modes of oscillation are possible in the pipe.

•• Only those modes with frequency above a cutOnly those modes with frequency above a cut--off frequency foff frequency fcc (low frequency cut(low frequency cut--off) will off) will exist in the pipe.exist in the pipe.

f > ff > fcc for propagation.for propagation.

Transverse Modes of Vibration of an Air ColumnTransverse Modes of Vibration of an Air Column


(0,0)(0,0)

(1,0)(1,0)

(2,0)(2,0)

DD

CutCut--off Frequencyoff Frequencyffcc = = qqnn mm v/D; v/D; for f < ffor f < fcc no propagationno propagationqq0000 = 0; q= 0; q1010 = 0.59; q= 0.59; q2020 = 0.97= 0.97



Effect of Transverse Modes on SpectrumEffect of Transverse Modes on Spectrum

•• More transverse modes implies more intensity.More transverse modes implies more intensity.

•• Most influential in high frequency harmonics.Most influential in high frequency harmonics.

•• Shape and relative diameter of pipe influence Shape and relative diameter of pipe influence transverse modes.transverse modes.

•• Thus, a square, triangular, and circular crossThus, a square, triangular, and circular cross--section organ pipes have different timbres.section organ pipes have different timbres.


Reflections from the array of holes in a woodwind Reflections from the array of holes in a woodwind affect the relative strength of the high frequency affect the relative strength of the high frequency harmonics in the pipe.harmonics in the pipe.Displacement waveDisplacement wave

Reflections from holes Reflections from holes (closed and open)(closed and open)



Effect of Holes on TransmissionEffect of Holes on Transmission

•• Larger holes have greater effect.Larger holes have greater effect.•• A A ““high pass filter:high pass filter:”” Low frequencies tend to be Low frequencies tend to be

reflected more and high frequencies transmitted reflected more and high frequencies transmitted more.more.

•• Thus the holes make a Thus the holes make a ““brighterbrighter”” sounding sounding instrument.instrument.


Reflections from joints and imperfections affect the Reflections from joints and imperfections affect the relative amplitude of the high frequency relative amplitude of the high frequency harmonics in the pipe.harmonics in the pipe.ReflectionsReflections



Filtering of Wind Instrument Sound Filtering of Wind Instrument Sound •• The specifics of transmission of the various The specifics of transmission of the various

frequency components in the pipe produce a frequency components in the pipe produce a filtering effect on the frequency spectrum of the filtering effect on the frequency spectrum of the sound.sound.

♩♩ ♪♪ ♫♫ ff1 1 ff2 2 ff3 3 ff44fn

~~ ~~

Transmission through hornTransmission through horn


Radiation of Sound from Wind Instruments Radiation of Sound from Wind Instruments •• The radiation characteristics of the bell alter The radiation characteristics of the bell alter

((““shapeshape””) the harmonic recipe, hence strongly ) the harmonic recipe, hence strongly influence the timbre of the instrument.influence the timbre of the instrument.

♩♩ ♪♪ ♫♫ ff1 1 ff2 2 ff3 3 ff44fn

~~ ~~

Radiation CharacteristicsRadiation Characteristics



The diameter of the mouth and the flare rate of the bell The diameter of the mouth and the flare rate of the bell determine the radiation characteristics of brass determine the radiation characteristics of brass instruments.instruments.

Cornet

TrumpetTrumpet

Flügelhorn

••The The larger the borelarger the bore diameter, the more diameter, the more intense the intense the low frequencylow frequency harmonics.harmonics.••The more The more rapid the flarerapid the flare, the more the , the more the low frequencies are reflected, and thus, low frequencies are reflected, and thus, the more the more high frequencyhigh frequency harmonics are harmonics are radiated.radiated.


The BellThe Bell

a = a = aaoo exp(m x)+ bexp(m x)+ b

m = “flare constant.”m = “flare constant.”

Larger m means more Larger m means more rapid flare rapid flare → → accentuate accentuate high frequencieshigh frequencies

xx

Exponential HornExponential Horn



French Horn BellFrench Horn BellBessel HornBessel Horn

xx

aa

a = a = aaoo ee--((εεxx)) +b+b

Called “Bessel Horns” because Called “Bessel Horns” because the standing wave follows a the standing wave follows a Bessel FunctionBessel Function..


MutesMutes•• The French Horn playerThe French Horn player’’s hand modifies the s hand modifies the

radiation characteristics of the horn, as well as radiation characteristics of the horn, as well as the effective flare.the effective flare.

•• Mutes reduce the effective area of the horn and, Mutes reduce the effective area of the horn and, therefore, reduce the intensity.therefore, reduce the intensity.

•• Mutes tend to reduce more the first and second Mutes tend to reduce more the first and second harmonic of the pipe than higher frequency harmonic of the pipe than higher frequency harmonics due to their internal modes of harmonics due to their internal modes of oscillation.oscillation.

•• Mutes make brass sound Mutes make brass sound ““thin and reedy.thin and reedy.””



Summary Summary --

•• The pitch of a wind instrument is determined by the length The pitch of a wind instrument is determined by the length and shape of its air column.and shape of its air column.

•• The effective length of the air column is controlled with The effective length of the air column is controlled with holes, valves and slides.holes, valves and slides.

•• Feedback from the resonances of the pipe select the Feedback from the resonances of the pipe select the frequency of oscillation of the jet, reed or lipfrequency of oscillation of the jet, reed or lip--valve.valve.

•• The excitation, transmission and The excitation, transmission and emittanceemittance of the sound in of the sound in the horn determine the timbre of the instrument.the horn determine the timbre of the instrument.

phy-2464 physical basis of music · 2005-03-16 · phy2464 -the physical basis of music phy-2464...

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