physical modeling sound synthesis - ccrmajos/pdf/aes-masterclass.pdf · 2020. 10. 30. · early...

Julius Smith AES-2006 Masterclass – 1 / 101

Physical Modeling Sound Synthesis

Julius Smith

CCRMA, Stanford University

AES-2006 Masterclass

October 7, 2006

Overview

Overview

Early Voice Models

Digital Voice Models

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis

Digital Waveguide Mesh

Horn Filter Design

Finite Differences

Virtual Analog

Wave Digital Filters


Outline

Overview

• Outline

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Synthesis approaches in approximate historical order, including

selected research updates, for

• Voice• Plucked Strings• Woodwinds• Bowed Strings• Piano and Harpsichord• Membranes and Plates• Horns• Related Topics

Reference: Physical Audio Signal Processing (online book)

http://ccrma.stanford.edu/~jos/pasp/

Early Voice Models

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Early Talking Machines (Virtual Talking Heads)


Joseph Faber’s Euphonia

https://irrationalgeographic.wordpress.com/2009/06/24/joseph-fabers-talking-euphonia/

Early Brazen Heads (Wikipedia)


• ≈ 1125: First talking head description:William of Malmesbury’s History of the English Kings:

◦ Pope Sylvester II said to have traveled to al-Andalus and stolen a tomeof secret knowledge

◦ Only able to escape through demonic assistance◦ Cast the head of a statue using his knowledge of astrology◦ It would not speak until spoken to, but then answered any yes/no

question put to it

• Early devices were deemed heretical by the Church and often destroyed• 1599: Albertus Magnus had a head with a human voice and breath:

◦ ”A certain reasoning process” bestowed by a cacodemon (evil demon)◦ Thomas Aquinas destroyed it for continually interrupting his ruminations

(not everyone wants a yacking cacodemon around all the time)

• By the 18th century, talking machines became acceptable as“scientific pursuit”

Wolfgang Von Kempelin’s Speaking Machine (1791)

Overview

Early Voice Models

• Virtual Heads

• Von Kempelin

• Euphonia

• Voder Keyboard

• Voder Schematic

• Voder Demos


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Replica of Wolfgang Von Kempelin’s Speaking Machine

http://en.wikipedia.org/wiki/Wolfgang_von_Kempelen's_Speaking_Machine

Joseph Faber’s Euphonia (1846)

Overview

Early Voice Models

• Virtual Heads

• Von Kempelin

• Euphonia

• Voder Keyboard

• Voder Schematic

• Voder Demos


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog




17 levers, a bellows, and a telegraphic line — sang “God Save the Queen”


Joseph Faber’s Euphonia (1846)

Overview

Early Voice Models

• Virtual Heads

• Von Kempelin

• Euphonia

• Voder Keyboard

• Voder Schematic

• Voder Demos


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog





The Voder (Homer Dudley — 1939 Worlds Fair)


http://davidszondy.com/future/robot/voder.htm>

http://ccrma.stanford.edu/~jos/wav/voder.wav

Voder Keyboard


http://www.acoustics.hut.fi/publications/files/theses/

lemmetty mst/chap2.html — (from Klatt 1987)

Voder Schematic


http://ptolemy.eecs.berkeley.edu/~eal/audio/voder.html

Voder Demos

Overview

Early Voice Models

• Virtual Heads

• Von Kempelin

• Euphonia

• Voder Keyboard

• Voder Schematic

• Voder Demos


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



• Video• Audio

http://www.youtube.com/watch?v=0rAyrmm7vv0http://www.youtube.com/watch?v=5hyI_dM5cGo


Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Kelly-Lochbaum Vocal Tract Model

(Discrete-Time Transmission-Line Model)

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”

• Linear Prediction

• Glottal Model

• Source Estimation

• LF Glottal Model

• Phonation Variation

• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


John L. Kelly and Carol Lochbaum (1962)

Sound Example

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


“Bicycle Built for Two”: (WAV) (MP3)

• Vocal part by Kelly and Lochbaum (1961)• Musical accompaniment by Max Mathews• Computed on an IBM 704• Based on Russian speech-vowel data from Gunnar Fant’s book• Probably the first digital physical-modeling synthesis sound

example by any method

• Inspired Arthur C. Clarke to adapt it for “2001: A Space Odyssey”— the computer’s “first song”

http://ccrma.stanford.edu/~jos/wav/daisy-klm.wavhttp://ccrma.stanford.edu/~jos/mp3/daisy-klm.mp3

“Shiela” Sound Examples by Perry Cook (1990)

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


• Diphones: (WAV) (MP3)• Nasals: (WAV) (MP3)• Scales: (WAV) (MP3)• “Shiela”: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/diphs.wavhttp://ccrma.stanford.edu/~jos/mp3/diphs.mp3http://ccrma.stanford.edu/~jos/wav/nasals.wavhttp://ccrma.stanford.edu/~jos/mp3/nasals.mp3http://ccrma.stanford.edu/~jos/wav/vocaliz.wavhttp://ccrma.stanford.edu/~jos/mp3/vocaliz.mp3http://ccrma.stanford.edu/~jos/wav/shiela.wavhttp://ccrma.stanford.edu/~jos/mp3/shiela.mp3

Linear Prediction (LP) Vocal Tract Model

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


• Can be interpreted as a modified Kelly-Lochbaum model• In linear prediction, the glottal excitation must be an

• impulse, or• white noise

This prevents LP from finding a physical vocal-tract model

• A more realistic glottal waveform e(n) is needed before the vocaltract filter can have the “right shape”

• How to augment LPC in this direction without going to afull-blown articulatory synthesis model?

Jointly estimate glottal waveform e(n) so that the vocal-tract filter converges to the “right shape”

Klatt Derivative Glottal Wave

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


0 25 50 75 100 125 150 175 200 225 250−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Time (samples)

Ampl

itude

Two periods of the basic voicing waveform

Good for estimation:

• Truncated parabola each period• Coefficients easily fit to phase-aligned inverse-filter output

Sequential Unconstrained Minimization

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


(Hui Ling Lu, 2002)

Klatt glottal (parabola) parameters are estimated jointly with vocal

tract filter coefficients

• Formulation resembles that of the equation error method forsystem identification (also used in invfreqz in matlab)

• For phase alignment, we estimate• pitch (time varying)• glottal closure instant each period

• Optimization is convex in all but the phase-alignment dimension⇒ one potentially nonlinear line search

Liljencrantz-Fant Derivative Glottal Wave Model

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


0 0.005 0.01 0.0150

10

20

30

40

Am

plitu

de

Time (sec)

To

Uo

TcTp

LF glottal wave and LF derivative glottal wave

0 0.005 0.01 0.015

−1

−0.5

0

Am

plitu

de

Time (sec)

ToTp TcTe

−Ee

glottal wave

derivative glottal wave Ta

• Better for intuitively parametrized expressive synthesis• LF model parameters are fit to inverse filter output• Use of Klatt model in forming filter estimate yields a “more

physical” filter than LP

Parametrized Phonation Types

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


100 200 300 400 500 600 700 800 900 1000

−1

0

1 pressed

100 200 300 400 500 600 700 800 900 1000

−1

0

1 normal

100 200 300 400 500 600 700 800 900 1000

−1

0

1 breathy

Sound Examples by Hui Ling Lu

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


• Original: (WAV) (MP3)• Synthesized:

• Pressed Phonation: (WAV) (MP3)• Normal Phonation: (WAV) (MP3)• Breathy Phonation: (WAV) (MP3)

• Original: (WAV) (MP3)• Synthesis 1: (WAV) (MP3)• Synthesis 2: (WAV) (MP3)

where

• Synthesis 1 = Estimated Vocal Tract driven by estimatedKLGLOT88 Derivative Glottal Wave (Pressed)

• Synthesis 2 = Estimated Vocal Tract driven by the fitted LFDerivative Glottal Wave (Pressed)

Google search: singing synthesis Lu

http://ccrma.stanford.edu/~jos/wav/norm_a_1.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1.mp3http://ccrma.stanford.edu/~jos/wav/norm_a_1_1VT_Rd03_0825.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1_1VT_Rd03_0825.mp3http://ccrma.stanford.edu/~jos/wav/norm_a_1_1VT_Rd12_0825.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1_1VT_Rd12_0825.mp3http://ccrma.stanford.edu/~jos/wav/norm_a_1_breathy.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1_breathy.mp3http://ccrma.stanford.edu/~jos/wav/norm_a_1.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1.mp3http://ccrma.stanford.edu/~jos/wav/norm_a_1_KL.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1_KL.mp3http://ccrma.stanford.edu/~jos/wav/norm_a_1_LF.wavhttp://ccrma.stanford.edu/~jos/mp3/norm_a_1_LF.mp3http://www.google.com/search?q=singing+synthesis+Lu

Voice Model Estimation

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


(Pamornpol (Tak) Jinachitra 2006)

Noisy

1A(z)

v(n) w(n)

y(n)x(n)

g(n)

Vocal tract

Derivativeglottal waveform

NoiseNoise/Error

Cleanspeech speech

• Parametric source-filter model of voice + noise• State-space framework with derivative glottal waveform as

input and A model for dynamics

• Jointly estimate AR parameters and glottal source parametersusing EM algorithm with Kalman smoothing

• Reconstruct a clean voice using Kelly-Lochbaum andestimated parameters

Online 2D Vowel Demo (2014)


Jan Schnupp, Eli Nelken, and Andrew King

http://auditoryneuroscience.com/topics/two-formant-artificial-vowels

http://auditoryneuroscience.com/topics/two-formant-artificial-vowels

Pink Trombone Voical Synthesis (March 2017)

Overview

Early Voice Models


• KL Voice

• “Daisy”

• “Shiela”


• Glottal Model




• Lu Sounds

• Tak

• 2D Vowels

• Pink Trombone

Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences


Neil Thapen

Institute of Mathematics of the Academy of Sciences

Czech Republic

http://dood.al/pinktrombone/

http://dood.al/pinktrombone/

Karplus-Strong (KS) Algorithm

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Karplus-Strong (KS) Algorithm (1983)

Overview

Early Voice Models


Karplus Strong

• Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



N samples delayOutput y (n)+

z 1-

1/2

1/2

y (n-N)+

• Discovered (1978) as “self-modifying wavetable synthesis”• Wavetable is preferably initialized with random numbers• No physical modeling at this point

Digital Waveguide Modeling

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Digital Waveguide Models (1985)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

• Digital Waveguide

• Signal Scattering

• Moving Termination

• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters

• Simplest Damping

• Karplus-Strong Aha

• EKS Algorithm

• Physical Excitation

• Pick Position FFCF

• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Lossless digital waveguide∆= bidirectional delay line

at some wave impedance R

z−N

z−N

R

Useful for efficient models of

• strings• bores• plane waves• conical waves

Signal Scattering

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Signal scattering is caused by a change in wave impedance R:

If the wave impedance changes every spatial sample, the

Kelly-Lochbaum vocal-tract model results (also need reflecting

terminations)

Moving Termination: Ideal String

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Moving rigid termination for an ideal string

• Left endpoint moved at velocity v0• External force f0 = Rv0• R =

√Kǫ is the wave impedance (for transverse waves)

• Relevant to bowed strings (when bow pulls string)• String moves with speed v0 or 0 only• String is always one or two straight segments• “Helmholtz corner” (slope discontinuity) shuttles back and forth at

speed c =√

K/ǫ

Digital Waveguide “Equivalent Circuits”

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



(x = 0) (x = L)

vRf 00 =

(x = 0) (x = L)

-1-1

v0

a)

b)

f(n)

a) Velocity waves. b) Force waves.

(Animation)

(Interactive Animation)

../swgt/movet.htmlhttps://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html

Ideal Plucked String (Displacement Waves)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



(x = 0) (x = L)

y (n+N/2)

-1“Bridge”

y (n)+

“Nut”

-y (n)-

-1

y (n-N/2)+

(x = Pluck Position)

• Load each delay line with half of initial string displacement• Sum of upper and lower delay lines = string displacement

Ideal Struck String (Velocity Waves)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



(x = 0) (x = L)

v (n+N/2)

-1“Bridge”

v (n)+

“Nut”

-v (n)-

-1

v (n-N/2)+

(x = Hammer Position)

c

c

Hammer strike = momentum transfer = velocity step:

mhvh(0−) = (mh +ms)vs(0+)

Digital Waveguide Interpretation of Karplus-Strong

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Begin with an ideal damped string model:

(x = 0) (x = L)

N/2 samples delay, N/2 loss factors g

y (n-N/2)+

y (n+N/2)

Output (non-physical)

-1 “Bridge”Rigid Termination

y (n)+

“Nut”Rigid Termination

N/2 samples delay, N/2 loss factors g -

gN/2

g-N/2

y (n)-

-1

. . .

. . .. . .

. . .

z 1-

z 1-

z 1-

z 1-z 1-

z 1-

y (nT,2cT)

y (n)-

y (n)+

y (nT,0)

g g

g g g

g

Equivalent System: Gain Elements Commuted

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



N samples delayOutput

g N

y (n-N)+y (n)+

All N loss factors g have been “pushed” through delay elements andcombined at a single point.

Computational Savings

• fs = 50kHz, f1 = 100Hz ⇒ delay = 500• Multiplies reduced by two orders of magnitude• Input-output transfer function unchanged• Round-off errors reduced

Frequency-Dependent Damping

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



N samples delayOutput

g N

y (n-N)+y (n)+

• Loss factors g should really be digital filters• Gains in nature typically decrease with frequency• Loop gain may not exceed 1 (for stability)• Such filters also commute with delay elements (LTI)• Typically only one gain filter used per loop

Simplest Frequency-Dependent Loop Filter

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Ĝ(z) = b0 + b1z−1

• Uniform delay ⇒ b0 = b1 (⇒ delay = 1/2 sample)

• Zero damping at dc ⇒ b0 + b1 = 1⇒ b0 = b1 = 1/2⇒

Ĝ(ejωT ) = cos (ωT/2) , |ω| ≤ πfs

• This is precisely the Karplus-Strong loop filter!

Karplus-Strong Algorithm Revisited

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



N samples delayOutput y (n)+

z 1-

1/2

1/2

y (n-N)+

Physical Interpretation

• Delay line is initialized with noise (random numbers)• Therefore, assuming a displacement-wave simulation:

◦ Initial string displacement = sum of delay-line halves◦ Initial string velocity ⇐ difference of delay-line halves

• The Karplus-Strong “string” is thus plucked and struck by randomamounts along the entire length of the string!

(“splucked string”?)

EKS Algorithm (Jaffe-Smith 1983)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



z−N

Hs(z)

Hp(z) Hβ(z) HL(z)

Hρ(z) Hd(z)

N = pitch period (2× string length) in samples

Hp(z) =1− p

1− p z−1 = pick-direction lowpass filter

Hβ(z) = 1− z−βN = pick-position comb filter, β ∈ (0, 1)Hd(z) = string-damping filter (one/two poles/zeros typical)

Hs(z) = string-stiffness allpass filter (several poles and zeros)

Hρ(z) =ρ(N)− z−11− ρ(N) z−1 = first-order string-tuning allpass filter

HL(z) =1−RL

1−RL z−1= dynamic-level lowpass filter

Karplus-Strong Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



• “Vintage” 8-bit sound examples:• Original Plucked String: (WAV) (MP3)• Drum: (WAV) (MP3)• Stretched Drum: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/pluck.wavhttp://ccrma.stanford.edu/~jos/mp3/pluck.mp3http://ccrma.stanford.edu/~jos/wav/ksdrum.wavhttp://ccrma.stanford.edu/~jos/mp3/ksdrum.mp3http://ccrma.stanford.edu/~jos/wav/ksdrumst.wavhttp://ccrma.stanford.edu/~jos/mp3/ksdrumst.mp3

EKS Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Plucked String: (WAV) (MP3)

• Plucked String 1: (WAV) (MP3)• Plucked String 2: (WAV) (MP3)• Plucked String 3: (WAV) (MP3)

(Computed using Plucked.cpp in the C++ Synthesis Tool Kit

(STK) by Perry Cook and Gary Scavone)

http://ccrma.stanford.edu/~jos/wav/plucked.wavhttp://ccrma.stanford.edu/~jos/mp3/plucked.mp3http://ccrma.stanford.edu/~jos/wav/karplus2.wavhttp://ccrma.stanford.edu/~jos/mp3/karplus2.mp3http://ccrma.stanford.edu/~jos/wav/karplus1.wavhttp://ccrma.stanford.edu/~jos/mp3/karplus1.mp3http://ccrma.stanford.edu/~jos/wav/ks44k.wavhttp://ccrma.stanford.edu/~jos/mp3/ks44k.mp3

EKS Sound Example (1988)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Bach A-Minor Concerto—Orchestra Part: (WAV) (MP3)

• Executed in real time on one Motorola DSP56001(20 MHz clock, 128K SRAM)

• Developed for the NeXT Computer introduction at DaviesSymphony Hall, San Francisco, 1988

• Solo violin part was played live by Dan Kobialka of the SanFrancisco Symphony

http://ccrma.stanford.edu/~jos/wav/bachfugue.wavhttp://ccrma.stanford.edu/~jos/mp3/bachfugue.mp3

Example EKS Extension

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Several of the Karplus-Strong algorithm extensions

were based on its physical interpretation.

• Originally, transfer-function methods were used (1982)• Later, digital waveguide methods were used (1986)

String Excited Externally at One Point

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



(x = 0) (x = L)

f (n)+

“Agraffe”Rigid

Termination

f (n)-

Del M

Del M

Delay N

Delay N

(x = striking position)

Hammer Strike f(t)

ExampleOutput

Filter“Bridge”Yielding

Termination

“Waveguide Canonical Form (1986)”

Equivalent System by Delay Consolidation:

Del 2M Delay 2N

String Output

Filter

Hammer

Strike f(t)

Finally, we “pull out” the comb-filter component:

EKS “Pick Position” Extension

Overview

Early Voice Models


Karplus Strong

Digital Waveguides




• Waveguide Model

• Plucked String

• Struck String

• Damped String

• Commuted Gains

• Damping Filters



• EKS Algorithm



• PLPC Cello

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



Equivalent System: Comb Filter Factored Out

Delay 2M+2NHammer

Strike f(t)

Filter

Del 2M

g(t)

Out (from Del N)

H(z) = z−N1 + z−2M

1− z−(2M+2N) =(

1 + z−2M) z−N

1− z−(2M+2N)

• Excitation Position controlled by left delay-line length• Fundamental Frequency controlled by right delay-line length• “Transfer function modeling” based on a physical model (1982)

PLPC Cello (1982)


Body Filter

(bow position) (EKS)

Comb FilterBandlimitedImpulseTrain

(40-pole LPC)

String Loop

• Periodic LPC used to estimate string-loop filter• Normal LPC used for body model (40 poles)• Excitation = Bandlimited impulse train (Moorer 1975):

K∑

k=1

cos(kω0t) =sin[(K + 1/2)ω0t]

2 sin(ω0t/2)− 1

2

• Bow-position simulation = variable-delay differencing comb filter (directfrom physical interpretation)

• Sound Example:Moving Bow-Stroke Example: (WAV) (MP3)

(Bowing point moves toward the “bridge”)

http://ccrma.stanford.edu/~jos/wav/cello82.wavhttp://ccrma.stanford.edu/~jos/mp3/cello82.mp3

Single-Reed Instruments

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Schematic Physical Model

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed

• Reed Computation

• Graphical Solver

• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog

Wave Digital FiltersJulius Smith AES-2006 Masterclass – 50 / 101

Reed Bore

Mouth

Pressure

Embouchure

Tone-Hole Lattice Bell

( )np+

( )np−

( )npm

• Main control variable = air pressure applied to reed• Secondary control variable = reed embouchure• Pressure waves = natural choice for simulation• Radiation ≈ “Omni” at LF, more directional at HF

Bell ≈ power-complementary “cross-over” filter:

• Low frequencies reflect (inverted)• High frequencies transmit• Cross-over frequency ≈ 1500 Hz for clarinet

(where wavelength ≈ bore diameter)

Single-Reed Digital Waveguide Model (1986)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


• Bore = bidirectional delay line (losses lumped)• Bore length = 1/4 wavelength in lowest register

• Bell reflection ≈ -1 at low frequencies• Mouthpiece reflection ≈ +1

• Reflection filter depends on first few open toneholes• In a simple implementation, the bore is “cut to a new length” for

each pitch

Simplified Single-Reed Theory

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


pm∆= mouth pressure (constant)

pb∆= bore pressure (dynamic)

p∆∆= pm − pb ∆= pressure drop across mouthpiece

um∆= resulting flow into mouthpiece

Rm(p∆)∆= reed-aperture impedance (measured)

Toward a Computational Reed Model

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


Given:

pm = Mouth pressure

p+b = Incoming traveling bore pressure

Find:

p−b = Outgoing traveling bore pressure

such that:

0 = um + ub =p∆

Rm(p∆)+

p+b − p−bRb

,

p∆∆= pm − pb = pm − (p+b + p−b )

Solving for p−b is not immediate becauseRm depends on p∆ which depends on p

−

b .

Graphical Solution Technique (1983)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


Graphically solve:

G(p∆) = p+∆ − p∆, p+∆

∆= pm − 2p+b

where

G(p∆)∆= Rb um(p∆) = Rb p∆/Rm(p∆)

• Analogous to finding the “operating point” of a transistor byintersecting its “operating curve” with the “load line”

determined by the load resistance.

• Outgoing wave is then p−b = pm − p+b − p∆(p+∆)

Graphical Solution Technique Illustrated

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


Iteratively solve:

p+∆ − p∆ = Rb p∆/Rm(p∆), where p+∆∆= pm − 2p+b

Solution can be pre-computed and stored in a look-up

table ρ(p+∆)

Digital Waveguide Single Reed, Cylindrical Bore Model (1986)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


Digital waveguide clarinet

• Control variable = mouth half-pressure• Total reed cost = two subtractions, one multiply, and one table

lookup per sample

Digital Waveguide Wind Instrument Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

• Schematic Model

• Waveguide Model

• Simplified Reed



• Example

• Clarinet

• Wind Examples

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog


• STK Clarinet: (WAV) (MP3)Google search: STK clarinet

• See also Faust-STK Clarinet (new)• Staccato Systems Slide Flute

(based on STK flute, ca. 1995): (WAV) (MP3)

• Yamaha VL1 “Virtual Lead” synthesizer demos (1994):• Shakuhachi: (WAV) (MP3)• Oboe and Bassoon: (WAV) (MP3)• Tenor Saxophone: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/clarinet.wavhttp://ccrma.stanford.edu/~jos/mp3/clarinet.mp3http://www.google.com/search?q=STK+clarinethttp://ccrma.stanford.edu/~jos/wav/slideflute.wavhttp://ccrma.stanford.edu/~jos/mp3/slideflute.mp3http://ccrma.stanford.edu/~jos/wav/shakuhachi.wavhttp://ccrma.stanford.edu/~jos/mp3/shakuhachi.mp3http://ccrma.stanford.edu/~jos/wav/oboe-bassoon.wavhttp://ccrma.stanford.edu/~jos/mp3/oboe-bassoon.mp3http://ccrma.stanford.edu/~jos/wav/tenor-sax.wavhttp://ccrma.stanford.edu/~jos/mp3/tenor-sax.mp3

Bowed Strings

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Schematic Physical Model

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model

• Bow-String Contact

• Bow-String Junction

• Bow-String Reflection

• Simplified Bow Table

• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



A schematic model for bowed-string instruments.

• Bow divides string into two sections• Primary control variable = bow velocity

⇒ velocity waves = natural choice of wave variable• Bow junction = nonlinear two-port• Must find velocity input to string (injected equally to left and

right) such that friction force = string reaction force.

Bow-String Contact Physics

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Applied Force = Friction Curve × Differential Velocity‖

Reaction Force = String Wave Impedance × Velocity Change

Friedlander-Keller

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Overlay of normalized bow-string friction curve Rb(v∆)/Rs with thestring “load line” v+∆ − v∆.

Bow-String Scattering Junction

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Friedlander-Keller diagram is solved when

Rb(v∆) · v∆ = Rs ·[

v+∆ − v∆]

which implies

v−s,r = v+s,l + ρ̂(v

+∆) · v+∆

v−s,l = v+s,r + ρ̂(v

+∆) · v+∆

where

vs,r = transverse string velocity on the right of the bow

vs,l = string velocity left of the bow (vs,l = vs,r)

v+∆∆= vb − (v+s,r + v+s,l) = “incoming differential velocity”

vb = bow velocity, and ρ̂(v+∆) is given by . . .

Bow-String Reflection Coefficient

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



ρ̂(v+∆) =r(

v∆(v+∆)

)

1 + r(

v∆(v+∆)

) where

r(v∆) = 0.25Rb(v∆)/Rs

v∆ = vb − vs bow velocity minus string velocityvs = v

+s,l + v

−

s,l = v+s,r + v

−

s,r = transverse string velocity

Rs = wave impedance of string

Rb(v∆) = friction coefficient for the bow against the string, i.e.,

Fb(v∆) = Rb(v∆) · v∆

Simplified, Piecewise Linear Bow Table

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



• Flat center portion corresponds to a fixed reflection coefficient“seen” by a traveling wave encountering the bow stuck against

the string

• Outer sections give a smaller reflection coefficientcorresponding to the reduced bow-string interaction force while

the string is slipping under the bow

• Hysteresis is neglected

Digital Waveguide Bowed Strings (1986)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



• Reflection filter summarizes all losses per period(due to bridge, bow, finger, etc.)

• Bow-string junction = memoryless lookup table(or segmented polynomial)

Bowed and Plucked Sound Examples by Esteban Maestre

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

• Schematic Model





• Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



• Synthetically Bowing STK’s Bowed.cpp• Finite-Width Thermal Friction Model for Bowed String• Waveguide Strings Coupled to Modal-Synthesis Bridge

http://ccrma.stanford.edu/~esteban/demos/syntheticBowingSTK_1.wavhttp://ccrma.stanford.edu/~esteban/demos/violinThermal_1.mp4http://ccrma.stanford.edu/~esteban/demos/pluckedGuitar_1.wav

Electric Guitar with Overdrive

and Feedback

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Amplifier Distortion + Amplifier Feedback

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Sullivan 1990

Post-distortion gain

GainFeedbackAmplifier

Nonlinear Distortion

Amplifier Feedback Delay

String 1

String N

Output...

Direct-signal gain

Pre-distortion gain

Distortion output signal often further filtered by an amplifier cabinet

filter, representing speaker cabinet, driver responses, etc.

Distortion Guitar Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



(Stanford Sondius Project, ca. 1995)

• Distortion Guitar: (WAV) (MP3)• Amplifier Feedback 1: (WAV) (MP3)• Amplifier Feedback 2: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/gtr-dist-jimi.wavhttp://ccrma.stanford.edu/~jos/mp3/gtr-dist-jimi.mp3http://ccrma.stanford.edu/~jos/wav/ElectricGuitar.wavhttp://ccrma.stanford.edu/~jos/mp3/ElectricGuitar.mp3http://ccrma.stanford.edu/~jos/wav/gtr-dist-yes.wavhttp://ccrma.stanford.edu/~jos/mp3/gtr-dist-yes.mp3

Commuted Waveguide Synthesis

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Commuted Synthesis of Acoustic Strings (1993)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis

• Acoustic Strings

• Sound Examples

• Linearized Violin

• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Trigger Outpute(t) s(t) y(t)

ResonatorStringExcitation

Schematic diagram of a stringed musical instrument.

Trigger OutputResonator StringExcitation

Equivalent diagram in the linear, time-invariant case.

Aggregate

Excitation

a(t)String

x(t)OutputTrigger

Use of an aggregate excitation given by the convolution of original

excitation with the resonator impulse response.

Commuted Components

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Aggregate

Excitation

a(t)String

x(t)OutputTrigger

“Plucked Resonator” driving a String.

y(t)Bridge

Coupling

s(t)Guitar

Body

Air

Absorption

Room

Response Output

Possible components of a guitar resonator.

Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Electric Guitar (Pick-Ups and/or Body-Model Added) (Stanford

Sondius Project → Staccato Systems, Inc. → ADI, ca. 1995)

• Example 1: (WAV) (MP3)• Example 2: (WAV) (MP3)• Example 3: (WAV) (MP3)• Virtual “wah-wah pedal”: (WAV) (MP3)

STK Mandolin

• STK Mandolin 1: (WAV) (MP3)• STK Mandolin 2: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/gtr-jazz.wavhttp://ccrma.stanford.edu/~jos/mp3/gtr-jazz.mp3http://ccrma.stanford.edu/~jos/wav/gtr-jaz-2.wavhttp://ccrma.stanford.edu/~jos/mp3/gtr-jaz-2.mp3http://ccrma.stanford.edu/~jos/wav/gtr-jazz-3.wavhttp://ccrma.stanford.edu/~jos/mp3/gtr-jazz-3.mp3http://ccrma.stanford.edu/~jos/wav/gtr-wah.wavhttp://ccrma.stanford.edu/~jos/mp3/gtr-wah.mp3http://ccrma.stanford.edu/~jos/wav/mandolin1.wavhttp://ccrma.stanford.edu/~jos/mp3/mandolin1.mp3http://ccrma.stanford.edu/~jos/wav/mandolin2.wavhttp://ccrma.stanford.edu/~jos/mp3/mandolin2.mp3

Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



More Recent Acoustic Guitar

• Bach Prelude in E Major: (WAV) (MP3)• Bach Loure in E Major: (WAV) (MP3)• More examples• Yet more examples

Virtual performance by Dr. Mikael Laurson, Sibelius Institute

Virtual guitar by Helsinki Univ. of Tech., Acoustics Lab1

1http://www.acoustics.hut.fi/

http://ccrma.stanford.edu/~jos/wav/silence.wavhttp://ccrma.stanford.edu/~jos/wav/Prelude.wavhttp://ccrma.stanford.edu/~jos/mp3/Prelude.mp3http://ccrma.stanford.edu/~jos/wav/silence.wavhttp://ccrma.stanford.edu/~jos/wav/Loure.wavhttp://ccrma.stanford.edu/~jos/mp3/Loure.mp3http://ccrma.stanford.edu/~jos/wav/silence.wavhttp://www2.siba.fi/soundingscore/guitar.htmlhttp://www.acoustics.hut.fi/demos/dafx2000-synth/http://www2.siba.fi/soundingscore/MikaelsHomePage/MikaelsHomepage.htmlhttp://www.acoustics.hut.fi/

Commuted Synthesis of Linearized Violin

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



e(n) x(n)s(n)Amplitude(n)

Frequency(n)

x(n)Amplitude(n)

Frequency(n)

a(n)

a(n) x(n)

Output

Impulse-Response

Train

Amplitude(n)

Frequency(n)

a)

b)

c)

Output

Output

String

String

e(n)

String Resonator

Resonator

Impulse

Train

Impulse

Train

• Assumes ideal Helmholtz motion of string• Sound Examples (Stanford Sondius project, ca. 1995):

• Bass: (WAV) (MP3)• Cello: (WAV) (MP3)• Viola 1: (WAV) (MP3)• Viola 2: (WAV) (MP3)

• Violin 1: (WAV) (MP3)• Violin 2: (WAV) (MP3)• Duet: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/bass.wavhttp://ccrma.stanford.edu/~jos/mp3/bass.mp3http://ccrma.stanford.edu/~jos/wav/cello.wavhttp://ccrma.stanford.edu/~jos/mp3/cello.mp3http://ccrma.stanford.edu/~jos/wav/viola.wavhttp://ccrma.stanford.edu/~jos/mp3/viola.mp3http://ccrma.stanford.edu/~jos/wav/viola2.wavhttp://ccrma.stanford.edu/~jos/mp3/viola2.mp3http://ccrma.stanford.edu/~jos/wav/violin.wavhttp://ccrma.stanford.edu/~jos/mp3/violin.mp3http://ccrma.stanford.edu/~jos/wav/violin2.wavhttp://ccrma.stanford.edu/~jos/mp3/violin2.mp3http://ccrma.stanford.edu/~jos/wav/vln-lin-cs.wavhttp://ccrma.stanford.edu/~jos/mp3/vln-lin-cs.mp3http://ccrma.stanford.edu/~jos/wav/silence.wav

Commuted Piano Synthesis (1995)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Hammer-string interaction pulses (force):

5 10 15 20Time

0.1

0.2

0.3

0.4

0.5

Force

Synthesis of Hammer-String Interaction Pulse

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Time Time

LowpassFilter

Impulse Impulse Response

• Faster collisions correspond to narrower pulses(nonlinear filter )

• For a given velocity, filter is linear time-invariant

• Piano is “linearized” for each hammer velocity

Multiple Hammer-String Interaction Pulses

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Superimpose several individual pulses:

As impulse amplitude grows (faster hammer strike), output pulses

become taller and thinner, showing less overlap.

Complete Piano Model

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



Natural Ordering:

Commuted Ordering:

LPF1

LPF2

LPF3

+ String OutputTappedDelayLineTrigger

vc

Sound Board& Enclosure

Impulse Response

• Soundboard and enclosure are commuted• Only need a stored recording of their impulse response• An enormous digital filter is otherwise required

Piano and Harpsichord Sound Examples

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



(Stanford Sondius Project, ca. 1995)

• Piano: (WAV) (MP3)• Harpsichord 1: (WAV) (MP3)• Harpsichord 2: (WAV) (MP3)

http://ccrma.stanford.edu/~jos/wav/pno-cs.wavhttp://ccrma.stanford.edu/~jos/mp3/pno-cs.mp3http://ccrma.stanford.edu/~jos/wav/harpsi-cs.wavhttp://ccrma.stanford.edu/~jos/mp3/harpsi-cs.mp3http://ccrma.stanford.edu/~jos/wav/Harpsichord.wavhttp://ccrma.stanford.edu/~jos/mp3/Harpsichord.mp3

More Recent Harpsichord Example

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• Sound Examples


• Commuted Piano

• Pulse Synthesis

• Complete Piano

• Sound Examples


Horn Filter Design

Finite Differences

Virtual Analog



• Harpsichord Soundboard Hammer-Response: (WAV) (MP3)• Musical Commuted Harpsichord Example: (WAV) (MP3)• More examples

References:

• “Sound Synthesis of the Harpsichord Using a ComputationallyEfficient Physical Model”,

• by Vesa Välimäki, Henri Penttinen, Jonte Knif, Mikael Laurson,and Cumhur Erkut, JASP-2004

• Recent dissertation by Jack Perng (Stanford, Physics/CCRMA)

http://ccrma.stanford.edu/~jos/wav/Cembalo-Body-Response.wavhttp://ccrma.stanford.edu/~jos/mp3/Cembalo-Body-Response.mp3http://ccrma.stanford.edu/~jos/wav/frobergergigue.wavhttp://ccrma.stanford.edu/~jos/mp3/frobergergigue.mp3http://www.acoustics.hut.fi/publications/papers/jasp-harpsy/


Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



2D Waveguide Mesh


z−1 z−1

4-portScatteringJunction

z−1

z−1

z−1 z−1


z−1

z−1

z−1 z−1


z−1

z−1

z−1

z−1

z−1 z−1


z−1

z−1

z−1 z−1


z−1

z−1

z−1 z−1


z−1

z−1

z−1

z−1

z−1 z−1


z−1

z−1

z−1 z−1


z−1

z−1

z−1 z−1


z−1

z−1

z−1

z−1

z−1 z−1 z−1 z−1 z−1 z−1

At each junction:

VJ =in1 + in2 + in3 + in4

2outk = VJ − ink, k = 1, 2, 3, 4

Sound example:

• Gongs (lossless nonlinear rim byPierce and Van Duyne

JASA-1997)

http://ccrma.stanford.edu/~jos/wav/gongs.wav

Recent Mesh Topic: Virtual Quadratic Residue Diffusers

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


• 2D Waveguide Mesh

• QR Diffuser

• Example QRD

• QRD Mesh Boundary

• Scattering Levels

Horn Filter Design

Finite Differences

Virtual Analog



Manfred Schroeder’s Quadratic Residue Diffuser for N = 17:

N

dn

w

• Thin vertical lines = rigid separators between wells.• Well depths are N − sn, where

sn = n2

(mod N ), n ∈ Z

is a quadratic residue sequence

Example QRD

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



• QR Diffuser

• Example QRD



Horn Filter Design

Finite Differences

Virtual Analog



For N = 17, we have

s = [0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1; 0, 1, . . .]

Reflection magnitude equal in N equally spaced directions at the“design wavelength” (and “not bad” in between)

QRD Termination of a 2D Digital Waveguide Mesh

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



• QR Diffuser

• Example QRD



Horn Filter Design

Finite Differences

Virtual Analog



(Lee-Smith ICMC-2004)

J

J

J J J J J J

J J

JJ J

JJJ J

J

Scattering levels from a Schroeder diffuser (solid) and a

straight boundary (dashed)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis



• QR Diffuser

• Example QRD



Horn Filter Design

Finite Differences

Virtual Analog



10

20

30

30

60

90

0

Animations: https://ccrma.stanford.edu/˜kglee/2dmesh QRD/

https://ccrma.stanford.edu/~kglee/2dmesh_QRD/index.html

Horn Filter Design

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

Finite Differences

Virtual Analog



Bore Profile Reconstruction from Measured Trumpet Re-

flectance

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile

• Impulse Response

• Bell Filter Design

• Bell Reflectance

• Idea! (1998)

• Four Exponentials

• Exponentials & Cubic

• Exponentials & SM

• Example

• One-Pole TIIR

Finite Differences

Virtual AnalogJulius Smith AES-2006 Masterclass – 89 / 101

0 0.2 0.4 0.6 0.8 1 1.23

4

5

6

7

8

9

10

distance (m)

radiu

s (

mm

)

empiricalmodel

• Inverse scattering applied to pulse-reflectometry data to fitpiecewise-cylindrical model (like LPC model)

• Bore profile reconstruction is reasonable up to bell• The bell is not physically equivalent to a piecewise-cylindrical

acoustic tube, due to

• complex radiation impedance,• conversion to higher order transverse modes

Trumpet-Bell Impulse Response Computed from Estimated

Piecewise-Cylindrical Model

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 1 2 3 4 5 6 7 8 9-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time (ms)

Am

plit

ud

e

Bell Reflection Impulse Response

• From pulse reflectometry on trumpet with no mouthpiece• Bore profile is reconstructed, smoothed, and segmented• Impulse response of “bell segment” = “ideal filter”

Trumpet-Bell Filter Design Problem

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


• At fs = 44.1 kHz, impulse-response length ≈ 400−−600samples

• A length 400 FIR bell filter is expensive!• Convert to IIR? Hard because

◦ Phase (resonance tunings) must be preserved◦ Magnitude (resonance Q) must be preserved◦ Rise time ≈ 150 samples◦ Phase-sensitive IIR design methods perform poorly

Measured Trombone Bell Reflectance

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 1 2 3 4 5 6 7

−0.015

−0.01

−0.005

0

0.005

time (ms)

trombone bell

Idea! (1998)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


• Break up impulse response into exponential or polynomialsegments

• Exponential and polynomial impulse-responses can be designedusing Truncated IIR (TIIR) Filters

• Google search: TIIR Horns

0 50 100 150 200 250 300 350 400 450−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (samples)

Ampli

tude

Bell Impulse Response Segmentation

seg1seg2ncs3ncs4

http://www.google.com/search?q=TIIR+Horns

Four-Exponential Fit to Estimated Trumpet-Bell Filter (Exp-4)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 50 100 150 200 250 300 350 400 450−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (samples)

Ampli

tude

Time−Domain Fit with Four Offset−Exponential Segments

idealexp1exp2exp3exp4

Exp-4 Impulse Response Fit

Two Exponentials Connected by a Cubic Spline Measured

Trumpet Data (Exp2-S3)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 50 100 150 200 250 300−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (samples)

Response Segmentation

seg1seg2seg3

Two Exponentials Followed by a 6th-Order IIR Filter Designed

by Steiglitz McBride Algorithm (Exp2-SM6)

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 50 100 150 200 250 300 350 400 450−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (samples)

Response Segmentation

seg1seg2tail

Exp2-SM6 Amplitude Response Fit

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Normalized Frequency

Mag

nitu

de (d

B)

Magnitude Fit over Entire Nyquist Band

IdealApproximation

Exp2-SM6 Low-Frequency Zoom

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

Finite Differences


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Normalized Frequency

Mag

nitu

de (d

B)

Magnitude Fit

IdealApproximation

Exp2-SM6 Group Delay Fit

Overview

Early Voice Models


Karplus Strong

Digital Waveguides

Single Reeds

Bowed Strings

Distortion Guitar

Commuted Synthesis


Horn Filter Design

• Bore Profile




• Idea! (1998)




• Example

• One-Pole TIIR

physical modeling sound synthesis - ccrmajos/pdf/aes-masterclass.pdf · 2020. 10. 30. · early...

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