cmpt 318: lecture 12 physical modeling synthesistamaras/physmodeling/physmodeling.pdf · physical...

CMPT 318: Lecture 12Physical Modeling Synthesis

Tamara Smyth, [email protected] of Computing Science,

Simon Fraser University

March 22, 2006

1

Physical Modeling

• While the synthesis techniques we’ve been studying todate (AM, FM, waveshaping, additive and subtractivesynthesis etc), simulate sounds by reproducing asignal’s spectral content (analysis and thenresynthesis), physical models are based onmathematical descriptions of acoustic systemsthemselves.

• Though there is a strong tendency to model existingmusical instruments, physical modeling techniquescan be used to simulate any acoustic system.

• Modeling musical instruments can give us moreinsight in how the instrument functions, and allows usto extend the instrument in ways that would not bepossible “in the real world”.

• Once a system is digitally simulated, we are lessconstrained by scientific reality.

CMPT 318: Fundamentals of Computerized Sound: Lecture 12 2

Why physical modeling?

Control Parameters

• One of the strength of physical modeling lies in themore intuitive control parameters. Because they arebased on real instruments, the control parameterstend to be physical, and often more easily related tothe effect on the produced sound.

• If you try create a horn sound that has the variationin spectral content when played with different blowingpressures, you may have a very hard time figuring outwhich parameters to change in an FM synthesisalgorithm. With a good physical model and a windcontroller, all you have to do is increase blowingpressure at the input, and the sound produced willrespond accordingly.

Less Data Requirement

• Physical modeling is quite flexible and a singlealgorithm can achieve a variety of sounds, withoutrequiring more memory—unlike many other synthesistechniques which are really only as good as the datathey have.


• Once a physical model is complete, it’s range ofsounds is produced by changing the parameters—notmy adding more complexity.

• If we have a physical model of a musical instrument,it’s very easy to extend the instrument in ways thatwould either not be possible or convenient in a realworld (like for instance, changing the length or shapeof the bore on a wind instrument to dimensions whichwould not be possible for a human to handle).

• Sound approaches acoustic quality


Virtual Musical Instruments

Plucked and Struck Wind Instruments

Classical guitar Trumpet

Harpsichord Tenor Saxophone

Piano Oboe and bassoon

Bowed Strings Electric Guitars

Double bass Bass pickup

Viola Distortion + feedback

Violin Wah-wah effect

A complete colloection of physical modelling soundexamples is available at http://www-ccrma.stanford.edu/̃jos/waveguide/Sound Examples.html


Physical vs. Signal Based Models

Physical Modelling Signal-Based Models

Case by case Same approach for all sounds

Represents sound source Represents sound receiver

Compact descriptions Large memory requirements

Attacks are natural Attacks are more difficult

Efficient expressivity Limited expressivity,(requires mapping“layer”)


A Very Brief Introduction to MusicalAcoustics

• Recall that sound is a wave which is created byvibrating objects and propagated through a mediumfrom one location to another.

• The basis for understanding physical modeling usingwaveguide synthesis is the physics of waves—and inparticular, mechanical waves.


Mechanical Waves

• A mechanical wave can be described as a disturbancethat travels through a medium, transporting energyfrom one location to another location.

• The medium is simply the material through which thedisturbance is moving; it can be thought of as a seriesof interacting particles and can take the form of agas, liquid or solid.

• We will focus on the simplest case which is thetraveling wave.

• A traveling wave is any kind of wave whichpropagates in a single direction, i.e. along onedimension, with negligible change in shape.

• There are two basic types of wave motion fortravelling waves:

1. Longitudinal waves

2. Transverse waves


Traveling Waves

Longitudinal Wave

Particle displacement is parallel to the direction of wavepropagation.

Figure 1: Longitudinal wave. Animation, courtesy of Dr. Dan Russell, Kettering University,available on class website.

Transverse Wave

Particle displacement is perpendicular to the direction ofwave propagation.

Figure 2: Transverse Wave, animation courtesy of Dr. Dan Russell, Kettering University.


Vibration in Musical Instruments

• Nearly all objects, when hit or struck or plucked orstrummed or somehow disturbed, will vibrate.

• If you drop a pencil on the floor, it will begin tovibrate.

• If you pluck a guitar string, it will begin to vibrate.

• If you blow over the top of a bottle, the air inside willvibrate.

• When each of these objects vibrate, they tend tovibrate at a particular frequency or a set offrequencies known as their natural frequency(ies).

• The natural frequency of a string can be changed bychanging its length (as we do with our fingers whenwe apply pressure on the string).

• Likewise in a trombone, we can make the tube longerand shorter, effectively changing the soundingfrequency.


Standing Waves in Musical Instruments

• Very few objects vibrate at a single frequency.

• In many cases, the vibrations produce more complexwaveforms with a set of frequencies having a wholenumber mathematical relationships between them.That is, they are harmonics.

• Sometimes however, objects that vibrate at a set offrequencies that are not related by an integer and aretherefore less likely to produce a sounding pitch.

• The key to this phenomenon is understandingstanding wave patterns.


Reflection and the Formation ofStanding Waves

• In a one dimensional medium, waves can propagate intwo directions.

• When a wave is confined to a given space along themedium, it will soon reach it’s end point, reflect andtravel back in the opposite direction.

• Depending on the end (boundary) condition, thereflection will either be inverted or remain unchanged.

Figure 3: Reflection of an impulse wave from a fixed end causes an inverse of the impulse.


Interference and the Formation ofStanding Waves

• Reflection will cause both constructive and

destructive interference with the oncoming waves.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2

Figure 4: Perfect construction and destructive interference causes the production of standingwaves.

• A standing wave is the pattern resulting from theperfectly timed interference of two waves (sometimesmore) of the same frequency with different directionsof travel within the same medium.


Nodes and Antinodes

• The point at which there is no displacement is calleda node.

• Where there is maximum displacement is called anantinode.

Node Node Node Node Node

Antinode Antinode Antinode Antinode

Figure 5: Node-Antinode pattern.

• A standing waves is actually a pattern of nodes andantinodes.


Standing Waves and Harmonics

• The addition of a node will likely require the additionof one or several antinode(s) to maintain the pattern.

• The shortes pattern with a hard-reflection at bothends is given by node-antinode-node (see firstharmonic).

• If we add another node to this pattern we will alsohave to add an antinode just before to obtain,node-antinode-node-antinode-node (see secondharmonic).

First Harmonic (fundamental frequency)Node Node

Antinode

Second HarmonicNode Node Node

Antinode Antinode

Third HarmonicNode Node Node Node

Antinode Antinode Antinode

Figure 6: Formation of Harmonics.


Guitar String

• A guitar string, fixed at both ends (the nut and thebridge), has a number of frequencies at which it willnaturally vibrate, corresponding to the harmonics.

• If the length of a guitar string is known, thewavelength λ associated with each of the harmonicscan be found.

• The figure below gives the relationship between thewavelength of the harmonics and the string length.

L = 1/2 λ

L = λ

L = 3/2 λ

Figure 7: Guitar string.


Open End Acoustic Tubes

• Wind instruments typically make use of an acoustictubes of varying lengths which often curve up onthemselves to conserve space (or other reasons). Thisdoesn’t, however, change their lengths!

• If both ends of the tube are open, then the tube issaid to contain an open end air column.

• The open ends are always pressure anti-nodes sinceair is free to oscillate.

L = 1/2 λ

L = λ

L = 3/2 λ

Figure 8: Open-open tube (eg. flute and wind chimes).


Closed End Acoustic Tubes

• The closed end acts as a fixed end which prevents theoscillation of air, and thus are always pressure nodes.

L = 1/4 λ

L = 3/4 λ

L = 5/4 λ

Figure 9: Closed-open tube (eg. clarinet) having first, third and fifth harmonics.


Wave Equation

• Digital waveguide synthesis begins with D’Alembert’ssolution to the wave equation. The wave equation,given by

∂2y

∂t2= c2∂

2y

∂x2,

describes the displacement of the perfectly flexiblestring (without stiffness).

• D’Alembert observed that in a medium such as astring (or wind instrument), we find wave traveling tothe left and right along one dimension. He came upwith the generalized solution (or d’Alembert’ssolution) to the lossless, 1D, second order waveequation

y(t, x) = yr(t − x/c) + yl(t + x/c),

which means there are waves are traveling to the rightand left with velocity c.


Digital Waveguides

• D’Alembert’s solution is given by

y(t, x) = yr(t − x/c) + yl(t + x/c),

• In sampling D’Alembert’s solution, we make thefollowing substitution

x −→ mX

x −→ nT

to obtain the equation for the digital waveguide

y(n,m) = yr(n − m) + yl(n + m).


Traveling Wave Simulation

Delay lines model acoustic propagation delay.

y(n−M)y(n)Delay by M samples

input sequence: 1, 2, 3, 4, 5, etc.

0 0 0 0 0 0 0 01 0

outin

M=3

0 0 0 0 0 0 0 01 M=3

in out

2

0 0 0 0 0 0 01 M=32 3

in out

0 0 0 0 0 01 M=32 3 4

inout

0 0 0 0 01 M=32 3 4 5

inout

output sequence: 0, 0, 0, 1, 2, etc.


Delay line in C

static double D[M]; /* initialized to zero */

static long ptr=0; /* read-write offset */

double delayline(double x)

{

double y = D[ptr]; /* read operation */

D[ptr++] = x; /* write operation */

if (ptr >= M) { ptr -= M; } /* wrap ptr */

return y;

}


Ideal Waveguide Simulation

• A digital waveguide is a bidirectional delay-linemodelling left and right travelling waves.

y(nT, 0) y(nT, ξ)

y+(n) y+(n-M)

y−(n-M)y−(n)

M sample delay

M sample delay

x=0 x=ξ x=L

y(nT,mX) = y+(n − M) + y−(n + M)

where T is the sampling period (time period betweensamples), and X is the spatial rate in m/s (X = cTwhere c is the wave velocity).


Waves on a String

• Simulating Displacement Traveling Waves on a String

-1output-1

H(ω)

String displacement


Practical Example

Pressure waves in an acoustic tube where L = 1.Sampling rate:

fs = 44100

Temporal sampling period:

T =1

44100

Spatial sampling period:

X = cT =340

44100= 7.7mm

Number of samples delay:

M =L

X=

1

cT=

1

0.0077= 130


cmpt 318: lecture 12 physical modeling synthesistamaras/physmodeling/physmodeling.pdf · physical...

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