Modeling Physical Sound Synthesis
Julius Smith CCRMA, Stanford University AES-2006 Masterclass
October 7, 2006
Julius Smith
AES-2006 Masterclass 1 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Overview
Julius Smith
AES-2006 Masterclass 2 / 101
Outline
Overview
Outline
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
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
Julius Smith
AES-2006 Masterclass 3 / 101
Outline
Overview
Outline
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
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)
Julius Smith
AES-2006 Masterclass 3 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Voice Models
Julius Smith
AES-2006 Masterclass 4 / 101
Kelly-Lochbaum Vocal Tract Model
Overview Voice Models
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
e(n)
1 + k1 z 1 2 k1 k1 R1 z 1 2 1 k1
y(n)
Glottal Pulse Train or Noise e(n)
kM
1 + kM z 1 2 kM RM z 1 2 1 kM
Speech Output y(n)
(Unused Allpass Output)
Kelly-Lochbaum Vocal Tract Model (Piecewise Cylindrical)
John L. Kelly and Carol Lochbaum (1962)
Julius Smith
AES-2006 Masterclass 5 / 101
Sound Example
Overview Voice Models
Bicycle Built for Two: (WAV) (MP3)
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Vocal part by Kelly and Lochbaum (1961)
Julius Smith
AES-2006 Masterclass 6 / 101
Sound Example
Overview Voice Models
Bicycle Built for Two: (WAV) (MP3)
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Vocal part by Kelly and Lochbaum (1961) Musical accompaniment by Max Mathews
Julius Smith
AES-2006 Masterclass 6 / 101
Sound Example
Overview Voice Models
Bicycle Built for Two: (WAV) (MP3)
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Vocal part by Kelly and Lochbaum (1961) Musical accompaniment by Max Mathews Computed on an IBM 704
Julius Smith
AES-2006 Masterclass 6 / 101
Sound Example
Overview Voice Models
Bicycle Built for Two: (WAV) (MP3)
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
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 Fants book
Julius Smith
AES-2006 Masterclass 6 / 101
Sound Example
Overview Voice Models
Bicycle Built for Two: (WAV) (MP3)
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
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 Fants book Probably the rst digital physical-modeling synthesis sound example by any method
Julius Smith
AES-2006 Masterclass 6 / 101
Sound Example
Overview Voice Models
Bicycle Built for Two: (WAV) (MP3)
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
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 Fants book Probably the rst digital physical-modeling synthesis sound example by any method Inspired Arthur C. Clarke to adapt it for 2001: A Space Odyssey the computers rst song
Julius Smith
AES-2006 Masterclass 6 / 101
Shiela Sound Examples by Perry Cook (1990)
Overview Voice Models
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Diphones: (WAV) (MP3) Nasals: (WAV) (MP3) Scales: (WAV) (MP3) Shiela: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 7 / 101
Linear Prediction (LP) Vocal Tract Model
Overview Voice Models
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Can be interpreted as a modied Kelly-Lochbaum model In linear prediction, the glottal excitation must be an
impulse, or white noise
This prevents LP from nding a physical vocal-tract model
A more realistic glottal waveform e(n) is needed before the vocal
tract lter can have the right shape
How to augment LPC in this direction without going to a
full-blown articulatory synthesis model?
Jointly estimate glottal waveform e(n) so that the vocaltract lter converges to the right shape
Julius Smith
AES-2006 Masterclass 8 / 101
Klatt Derivative Glottal Wave
Two periods of the basic voicing waveform
Overview Voice Models
0.05
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
0
Amplitude
0.05
0.1
0.15
0.2
0.25
0
25
50
75
100
125 150 Time (samples)
175
200
225
250
Good for estimation:
Truncated parabola each period Coefcients easily t to phase-aligned inverse-lter output
Julius Smith
AES-2006 Masterclass 9 / 101
Sequential Unconstrained Minimization
(Hui Ling Lu, 2002)
Overview Voice Models
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Klatt glottal (parabola) parameters are estimated jointly with vocal tract lter coefcients
Formulation resembles that of the equation error method for
system identication
For phase alignement, we estimate
pitch (time varying) glottal closure instant each period
Optimization is convex in all but the phase-alignment dimension
Julius Smith
AES-2006 Masterclass 10 / 101
Liljencrantz-Fant Derivative Glottal Wave Model
LF glottal wave and LF derivative glottal wave
40
Overview Voice Models
Uo
Amplitude
30
glottal wave
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong
20
10
0
Tp
0
Tc
0.005
To
0.01 0.015
Time (sec) Ta
Amplitude
0
Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
Tp
derivative glottal wave To
Te Tc
0.5
1
Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Ee
0 0.005 0.01 0.015
Time (sec)
Julius Smith
AES-2006 Masterclass Better for intuitively parametrized expressive synthesis 11 / 101 LF model parameters are t to inverse lter output
Parametrized Phonation Types
Overview Voice Models
1 0 1 100 200 300 400 500 600 700 800 900 1000 pressed
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
1 0 1 100 200 300 400 500 600 700 800
normal
900
1000
1 0 1 100 200 300 400 500 600 700 800
breathy
900
1000
Julius Smith
AES-2006 Masterclass 12 / 101
Sound Examples by Hui Ling Lu
Overview Voice Models
Original: (WAV) (MP3) Synthesized:
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
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 estimated
KLGLOT88 Derivative Glottal Wave (Pressed) Synthesis 2 = Estimated Vocal Tract driven by the tted LF Derivative Glottal Wave (Pressed)
Google search: singing synthesis Lu
AES-2006 Masterclass 13 / 101
Julius Smith
Voice Model Estimation
Overview Voice Models
(Pamornpol (Tak) Jinachitra 2006)
Noise/Error v(n) g(n) Derivative glottal waveform Vocal tract 1 A(z) x(n) Clean speech Noise w(n) y(n) Noisy speech
KL Music Daisy Shiela Linear Prediction Glottal Model Source Estimation LF Glottal Model Phonation Variation Lu Sounds Current Work
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Parametric source-lter 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 parameters using EM algorithm with Kalman smoothing Reconstruct a clean voice using Kelly-Lochbaum and estimated parameters
Julius Smith
AES-2006 Masterclass 14 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Karplus-Strong (KS) Algorithm
Julius Smith
AES-2006 Masterclass 15 / 101
Karplus-Strong (KS) Algorithm (1983)
Overview Voice Models Karplus Strong
Output y (n)
+
N samples delay 1/2 1/2
y (n-N)
+
Karplus Strong
Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
z -1
Discovered (1978) as self-modifying wavetable synthesis Wavetable is preferably initialized with random numbers
Julius Smith
AES-2006 Masterclass 16 / 101
Karplus-Strong (KS) Algorithm (1983)
Overview Voice Models Karplus Strong
Output y (n)
+
N samples delay 1/2 1/2
y (n-N)
+
Karplus Strong
Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
z -1
Discovered (1978) as self-modifying wavetable synthesis Wavetable is preferably initialized with random numbers
Julius Smith
AES-2006 Masterclass 16 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Digital Waveguide Modeling
Julius Smith
AES-2006 Masterclass 17 / 101
Digital Waveguide Models (1985)
Overview Voice Models Karplus Strong Digital Waveguides
Lossless digital waveguide
= bidirectional delay line
at some wave impedance R
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
zN
R
zN
Useful for efcient models of
strings bores plane waves conical waves
AES-2006 Masterclass 18 / 101
Julius Smith
Horn Filter Design
Signal Scattering
Overview Voice Models Karplus Strong Digital Waveguides
Signal scattering is caused by a change in wave impedance R:
R R1 k1 = 2 R2 + R1 1 + k1 zN R1 zN 1 k1 k1 k1 zN R2 zN
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
If the wave impedance changes every spatial sample, the Kelly-Lochbaum vocal-tract model results.
Julius Smith
AES-2006 Masterclass 19 / 101
Horn Filter Design
Moving Termination: Ideal String
Overview Voice Models Karplus Strong Digital Waveguides
y(t,x)
v0 Position at rest: y 0 c x=0 x=c t0 x=L x
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Moving rigid termination for an ideal string.
Left endpoint moved at velocity v0 External force f0 = Rv0 R = K 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/
AES-2006 Masterclass 20 / 101
Julius Smith
Horn Filter Design
Digital Waveguide Equivalent Circuits
Overview Voice Models Karplus Strong Digital Waveguides
v0
a)
-1
-1
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
(x = 0)
(x = L)
f 0 = R v0
f(n)
b)
(x = 0)
(x = L)
a) Velocity waves. b) Force waves. (Animation)
Julius Smith
AES-2006 Masterclass 21 / 101
Horn Filter Design
Ideal Plucked String (Displacement Waves)
Overview Voice Models Karplus Strong Digital Waveguides
y (n) -1
+
y (n-N/2) Nut y-(n+N/2) (x = L)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Bridge y-(n)
-1
(x = Pluck Position)
(x = 0)
Load each delay line with half of initial string displacement Sum of upper and lower delay lines = string displacement
Julius Smith
AES-2006 Masterclass 22 / 101
Horn Filter Design
Ideal Struck String (Velocity Waves)
Overview Voice Models Karplus Strong Digital Waveguides
v+ (n)
c
+ v (n-N/2)
-1 Nut v-(n+N/2) (x = L)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Bridge v-(n)
-1 c
(x = Hammer Position)
(x = 0)
Hammer strike = momentum transfer = velocity step:
mh vh (0) = (mh + ms )vs (0+)
Julius Smith
AES-2006 Masterclass 23 / 101
Horn Filter Design
Digital Waveguide Interpretation of Karplus-Strong
Overview Voice Models Karplus Strong Digital Waveguides
Begin with an ideal damped string model:
Output (non-physical) y (n)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
+
y (n-N/2) g N/2 samples delay, N/2 loss factors g -1
+
N/2
Bridge Rigid Termination y-(n)
-1
Nut Rigid Termination y-(n+N/2) g
-N/2
N/2 samples delay, N/2 loss factors g
(x = 0)
(x = L)
Julius Smith
AES-2006 Masterclass 24 / 101
Horn Filter Design
Digital Waveguide Interpretation of Karplus-Strong
Overview Voice Models Karplus Strong Digital Waveguides
Begin with an ideal damped string model:
Output (non-physical) y (n)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
+
y (n-N/2) g N/2 samples delay, N/2 loss factors g -1
+
N/2
Bridge Rigid Termination y-(n)
-1
Nut Rigid Termination y-(n+N/2) g
-N/2
N/2 samples delay, N/2 loss factors g
(x = 0)
(x = L)
...
y (n)
+
g
g
g
z -1
z -1
z -1
...
y (nT,0)
y (nT,2cT)
...
y-(n)
z -1
g
z -1
g
z -1
g
...
Julius Smith
AES-2006 Masterclass 24 / 101
Horn Filter Design
Equivalent System: Gain Elements Commuted
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
All N loss factors g have been pushed through delay elements and combined at a single point.
Julius Smith
AES-2006 Masterclass 25 / 101
Horn Filter Design
Equivalent System: Gain Elements Commuted
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
All N loss factors g have been pushed through delay elements and combined at a single point. Computational Savings
fs = 50kHz, f1 = 100Hz delay = 500 Multiplies reduced by two orders of magnitude
Julius Smith
AES-2006 Masterclass 25 / 101
Horn Filter Design
Equivalent System: Gain Elements Commuted
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
All N loss factors g have been pushed through delay elements and combined at a single point. Computational Savings
fs = 50kHz, f1 = 100Hz delay = 500 Multiplies reduced by two orders of magnitude Input-output transfer function unchanged
Julius Smith
AES-2006 Masterclass 25 / 101
Horn Filter Design
Equivalent System: Gain Elements Commuted
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
All N loss factors g have been pushed through delay elements and combined at a single point. Computational Savings
Multiplies reduced by two orders of magnitude Input-output transfer function unchanged Round-off errors reduced
fs = 50kHz, f1 = 100Hz delay = 500
Julius Smith
AES-2006 Masterclass 25 / 101
Horn Filter Design
Frequency-Dependent Damping
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
Loss factors g should really be digital lters Gains in nature typically decrease with frequency Loop gain may not exceed 1 (for stability) Such lters also commute with delay elements (LTI) Typically only one gain lter used per loop
Julius Smith
AES-2006 Masterclass 26 / 101
Horn Filter Design
Frequency-Dependent Damping
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
Loss factors g should really be digital lters Gains in nature typically decrease with frequency Loop gain may not exceed 1 (for stability) Such lters also commute with delay elements (LTI) Typically only one gain lter used per loop
Julius Smith
AES-2006 Masterclass 26 / 101
Horn Filter Design
Frequency-Dependent Damping
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
Loss factors g should really be digital lters Gains in nature typically decrease with frequency Loop gain may not exceed 1 (for stability) Such lters also commute with delay elements (LTI) Typically only one gain lter used per loop
Julius Smith
AES-2006 Masterclass 26 / 101
Horn Filter Design
Frequency-Dependent Damping
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
Loss factors g should really be digital lters Gains in nature typically decrease with frequency Loop gain may not exceed 1 (for stability) Such lters also commute with delay elements (LTI) Typically only one gain lter used per loop
Julius Smith
AES-2006 Masterclass 26 / 101
Horn Filter Design
Frequency-Dependent Damping
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
gN
Loss factors g should really be digital lters Gains in nature typically decrease with frequency Loop gain may not exceed 1 (for stability) Such lters also commute with delay elements (LTI) Typically only one gain lter used per loop
Julius Smith
AES-2006 Masterclass 26 / 101
Horn Filter Design
Simplest Frequency-Dependent Loop Filter
Overview Voice Models Karplus Strong Digital Waveguides
G(z) = b0 + b1 z 1
Uniform delay b0 = b1 ( delay = 1/2 sample) Zero damping at dc b0 + b1 = 1
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
b0 = b1 = 1/2 G(ejT ) = cos (T /2) ,
|| fs
This is precisely the Karplus-Strong loop lter!
Julius Smith
AES-2006 Masterclass 27 / 101
Horn Filter Design
Simplest Frequency-Dependent Loop Filter
Overview Voice Models Karplus Strong Digital Waveguides
G(z) = b0 + b1 z 1
Uniform delay b0 = b1 ( delay = 1/2 sample) Zero damping at dc b0 + b1 = 1
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
b0 = b1 = 1/2 G(ejT ) = cos (T /2) ,
|| fs
This is precisely the Karplus-Strong loop lter!
Julius Smith
AES-2006 Masterclass 27 / 101
Horn Filter Design
Simplest Frequency-Dependent Loop Filter
Overview Voice Models Karplus Strong Digital Waveguides
G(z) = b0 + b1 z 1
Uniform delay b0 = b1 ( delay = 1/2 sample) Zero damping at dc b0 + b1 = 1
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
b0 = b1 = 1/2 G(ejT ) = cos (T /2) ,
|| fs
This is precisely the Karplus-Strong loop lter!
Julius Smith
AES-2006 Masterclass 27 / 101
Horn Filter Design
Karplus-Strong Algorithm
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay 1/2 1/2
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
z -1
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 random
amounts along the entire length of the string! (splucked string?)
Julius Smith
AES-2006 Masterclass 28 / 101
Horn Filter Design
Karplus-Strong Algorithm
Overview Voice Models Karplus Strong Digital Waveguides
Output y (n)
+
N samples delay 1/2 1/2
y (n-N)
+
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
z -1
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 random
amounts along the entire length of the string! (splucked string?)
Julius Smith
AES-2006 Masterclass 28 / 101
Horn Filter Design
EKS Algorithm (Jaffe-Smith 1983)
Overview Voice Models Karplus Strong Digital Waveguides
Hp (z)
H (z)
z N
H (z) Hs (z) Hd (z)
HL (z)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
H (z) = 1 z N = pick-position comb lter, (0, 1) Hd (z) = string-damping lter (one/two poles/zeros typical) Hs (z) = string-stiffness allpass lter (several poles and zeros) H (z) = HL (z) = (N ) z 1 = rst-order string-tuning allpass lter 1 (N ) z 1 1 RL = dynamic-level lowpass lter 1 RL z 1
AES-2006 Masterclass 29 / 101
= pitch period (2 string length) in samples 1p = pick-direction lowpass lter Hp (z) = 1 1 pz
N
Julius Smith
Horn Filter Design
STK EKS Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides
Synthesis Tool Kit (STK) by Perry Cook, Gary Scavone, and
others distributed by CCRMA: Google search: STK ToolKit
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
STK Plucked String: (WAV) (MP3)
Plucked String 1: (WAV) (MP3) Plucked String 2: (WAV) (MP3) Plucked String 3: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 30 / 101
Horn Filter Design
EKS Sound Example (1988)
Overview Voice Models Karplus Strong Digital Waveguides
Bach A-Minor ConcertoOrchestra Part: (WAV) (MP3)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Executed in real time on one Motorola DSP56001
(20 MHz clock, 128K SRAM)
Julius Smith
AES-2006 Masterclass 31 / 101
Horn Filter Design
EKS Sound Example (1988)
Overview Voice Models Karplus Strong Digital Waveguides
Bach A-Minor ConcertoOrchestra Part: (WAV) (MP3)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Executed in real time on one Motorola DSP56001
(20 MHz clock, 128K SRAM)
Developed for the NeXT Computer introduction at Davies
Symphony Hall, San Francisco, 1988
Julius Smith
AES-2006 Masterclass 31 / 101
Horn Filter Design
EKS Sound Example (1988)
Overview Voice Models Karplus Strong Digital Waveguides
Bach A-Minor ConcertoOrchestra Part: (WAV) (MP3)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Executed in real time on one Motorola DSP56001
(20 MHz clock, 128K SRAM)
Developed for the NeXT Computer introduction at Davies
Symphony Hall, San Francisco, 1988
Solo violin part was played live by Dan Kobialka of the San
Francisco Symphony
Julius Smith
AES-2006 Masterclass 31 / 101
Horn Filter Design
Example EKS Extension
Overview 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 again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Several of the Karplus-Strong algorithm extensions were based on its physical interpretation.
Originally, transfer-function methods were used (1982) Below is a digital waveguide derivation
Julius Smith
AES-2006 Masterclass 32 / 101
Horn Filter Design
String Excited Externally at One Point
Overview Voice Models Karplus Strong Digital Waveguides
f (n) Agraffe Rigid Termination f(n) (x = 0)
+
Example Output Delay Delay Bridge Yielding Termination
Hammer Strike f(t)
Filter
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Delay
Delay (x = striking position) (x = L)
Waveguide Canonical Form (1986)
Julius Smith
AES-2006 Masterclass 33 / 101
Horn Filter Design
String Excited Externally at One Point
Overview Voice Models Karplus Strong Digital Waveguides
f (n) Agraffe Rigid Termination f(n) (x = 0)
+
Example Output Delay Delay Bridge Yielding Termination
Hammer Strike f(t)
Filter
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Delay
Delay (x = striking position) (x = L)
Waveguide Canonical Form (1986) Equivalent System by Delay Consolidation:
String Output Hammer Strike f(t)
Delay
Delay
Filter
Julius Smith
AES-2006 Masterclass 33 / 101
Horn Filter Design
String Excited Externally at One Point
Overview Voice Models Karplus Strong Digital Waveguides
f (n) Agraffe Rigid Termination f(n) (x = 0)
+
Example Output Delay Delay Bridge Yielding Termination
Hammer Strike f(t)
Filter
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Delay
Delay (x = striking position) (x = L)
Waveguide Canonical Form (1986) Equivalent System by Delay Consolidation:
String Output Hammer Strike f(t)
Delay
Delay
Filter
Finally, we pull out the comb-lter component:
Julius Smith
AES-2006 Masterclass 33 / 101
Horn Filter Design
EKS Pick Position Extension
Overview Voice Models Karplus Strong Digital Waveguides
Equivalent System: Comb Filter Factored Out
String Output Hammer Strike f(t) Delay g(t) Delay
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Filter
1 + z 2M = H(z) = z N (2M +2N ) 1z
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)
Julius Smith
AES-2006 Masterclass 34 / 101
Horn Filter Design
EKS Pick Position Extension
Overview Voice Models Karplus Strong Digital Waveguides
Equivalent System: Comb Filter Factored Out
String Output Hammer Strike f(t) Delay g(t) Delay
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Filter
1 + z 2M = H(z) = z N (2M +2N ) 1z
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)
Julius Smith
AES-2006 Masterclass 34 / 101
Horn Filter Design
EKS Pick Position Extension
Overview Voice Models Karplus Strong Digital Waveguides
Equivalent System: Comb Filter Factored Out
String Output Hammer Strike f(t) Delay g(t) Delay
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Filter
1 + z 2M = H(z) = z N (2M +2N ) 1z
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)
Julius Smith
AES-2006 Masterclass 34 / 101
Horn Filter Design
EKS Pick Position Extension
Overview Voice Models Karplus Strong Digital Waveguides
Equivalent System: Comb Filter Factored Out
String Output Hammer Strike f(t) Delay g(t) Delay
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Filter
1 + z 2M = H(z) = z N (2M +2N ) 1z
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)
Julius Smith
AES-2006 Masterclass 34 / 101
Horn Filter Design
PLPC Cello (1982)
Overview Voice Models Karplus Strong Digital Waveguides
Bandlimited Impulse Train
Comb Filter
(bow position)
String Loop
(Extended Karplus Strong)
Body Filter
(40-pole LPC)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Periodic LPC used to estimate string-loop lter Normal LPC used for body model (40 poles) Excitation = Bandlimited impulse train (Moorer 1975):
K
cos(k0 t) =
k=1
sin[(K + 1/2)0 t] 1 2 sin(0 t/2) 2
Bow-position simulation = variable-delay differencing comb lter
(direct from physical interpretation) Sound Example: Moving Bow-Stroke Example: (WAV) (MP3) (Bowing point moves toward the bridge)
Julius Smith
AES-2006 Masterclass 35 / 101
Horn Filter Design
PLPC Cello (1982)
Overview Voice Models Karplus Strong Digital Waveguides
Bandlimited Impulse Train
Comb Filter
(bow position)
String Loop
(Extended Karplus Strong)
Body Filter
(40-pole LPC)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Periodic LPC used to estimate string-loop lter Normal LPC used for body model (40 poles) Excitation = Bandlimited impulse train (Moorer 1975):
K
cos(k0 t) =
k=1
sin[(K + 1/2)0 t] 1 2 sin(0 t/2) 2
Bow-position simulation = variable-delay differencing comb lter
(direct from physical interpretation) Sound Example: Moving Bow-Stroke Example: (WAV) (MP3) (Bowing point moves toward the bridge)
Julius Smith
AES-2006 Masterclass 35 / 101
Horn Filter Design
PLPC Cello (1982)
Overview Voice Models Karplus Strong Digital Waveguides
Bandlimited Impulse Train
Comb Filter
(bow position)
String Loop
(Extended Karplus Strong)
Body Filter
(40-pole LPC)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Periodic LPC used to estimate string-loop lter Normal LPC used for body model (40 poles) Excitation = Bandlimited impulse train (Moorer 1975):
K
cos(k0 t) =
k=1
sin[(K + 1/2)0 t] 1 2 sin(0 t/2) 2
Bow-position simulation = variable-delay differencing comb lter
(direct from physical interpretation) Sound Example: Moving Bow-Stroke Example: (WAV) (MP3) (Bowing point moves toward the bridge)
Julius Smith
AES-2006 Masterclass 35 / 101
Horn Filter Design
PLPC Cello (1982)
Overview Voice Models Karplus Strong Digital Waveguides
Bandlimited Impulse Train
Comb Filter
(bow position)
String Loop
(Extended Karplus Strong)
Body Filter
(40-pole LPC)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Periodic LPC used to estimate string-loop lter Normal LPC used for body model (40 poles) Excitation = Bandlimited impulse train (Moorer 1975):
K
cos(k0 t) =
k=1
sin[(K + 1/2)0 t] 1 2 sin(0 t/2) 2
Bow-position simulation = variable-delay differencing comb lter
(direct from physical interpretation) Sound Example: Moving Bow-Stroke Example: (WAV) (MP3) (Bowing point moves toward the bridge)
Julius Smith
AES-2006 Masterclass 35 / 101
Horn Filter Design
PLPC Cello (1982)
Overview Voice Models Karplus Strong Digital Waveguides
Bandlimited Impulse Train
Comb Filter
(bow position)
String Loop
(Extended Karplus Strong)
Body Filter
(40-pole LPC)
Digital Waveguide Signal Scattering Moving Termination Waveguide Model Plucked String Struck String Damped String Commuted Gains Damping Filters Simplest Damping Karplus-Strong again EKS Algorithm Physical Excitation Pick Position FFCF PLPC Cello
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
Periodic LPC used to estimate string-loop lter Normal LPC used for body model (40 poles) Excitation = Bandlimited impulse train (Moorer 1975):
K
cos(k0 t) =
k=1
sin[(K + 1/2)0 t] 1 2 sin(0 t/2) 2
Bow-position simulation = variable-delay differencing comb lter
(direct from physical interpretation) Sound Example: Moving Bow-Stroke Example: (WAV) (MP3) (Bowing point moves toward the bridge)
Julius Smith
AES-2006 Masterclass 35 / 101
Horn Filter Design
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Single-Reed Instruments
Julius Smith
AES-2006 Masterclass 36 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Embouchure p ( n ) Mouth Pressure pm ( n ) Reed p+ ( n ) Bore Tone-Hole Lattice Bell
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Main control variable = air pressure applied to reed Secondary control variable = reed embouchure Pressure waves = natural choice for simulation Bell power-complementary cross-over lter:
Radiation Omni at LF, more directional at HF
Low frequencies reect (inverted) High frequencies transmit Cross-over frequency 1500 Hz for clarinet (where wavelength bore diameter)
Julius Smith
AES-2006 Masterclass 37 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Single-Reed Digital Waveguide Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
h+
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Bore = bidirectional delay line (losses lumped) Bore length = 1/4 wavelength in lowest register
Bell reection -1 at low frequencies Mouthpiece reection +1
Reection lter depends on rst few open toneholes In a simple implementation, the bore is cut to a new length for
each pitch
Julius Smith
AES-2006 Masterclass 38 / 101
Simplied Single-Reed Theory
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
pm um
p b ub p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 39 / 101
Simplied Single-Reed Theory
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
pm um
p b ub p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
pm = mouth pressure (constant)
Julius Smith
AES-2006 Masterclass 39 / 101
Simplied Single-Reed Theory
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
pm um
p b ub p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
pm = mouth pressure (constant) pb = bore pressure (dynamic)
Julius Smith
AES-2006 Masterclass 39 / 101
Simplied Single-Reed Theory
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
pm um
p b ub p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
pm = mouth pressure (constant) pb = bore pressure (dynamic) p = pm pb = pressure drop across mouthpiece
Julius Smith
AES-2006 Masterclass 39 / 101
Simplied Single-Reed Theory
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
pm um
p b ub p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
pm = mouth pressure (constant) pb = bore pressure (dynamic) p = pm pb = pressure drop across mouthpiece um = resulting ow into mouthpiece
Julius Smith
AES-2006 Masterclass 39 / 101
Simplied Single-Reed Theory
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
pm um
p b ub p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
pm = mouth pressure (constant) pb = bore pressure (dynamic) p = pm pb = pressure drop across mouthpiece um = resulting ow into mouthpiece Rm (p ) = reed-aperture impedance (measured)
Julius Smith
AES-2006 Masterclass 39 / 101
Toward a Computational Reed Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Given:
pm = Mouth pressure p+ = Incoming traveling bore pressure b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 40 / 101
Toward a Computational Reed Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Given:
pm = Mouth pressure p+ = Incoming traveling bore pressure b
Find:
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
p = Outgoing traveling bore pressure b
Julius Smith
AES-2006 Masterclass 40 / 101
Toward a Computational Reed Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Given:
pm = Mouth pressure p+ = Incoming traveling bore pressure b
Find:
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
p = Outgoing traveling bore pressure b
such that:
p+ p p b 0 = um + ub = , + b Rm (p ) Rb p = pm pb = pm (p+ + p ) b b
Julius Smith
AES-2006 Masterclass 40 / 101
Toward a Computational Reed Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Given:
pm = Mouth pressure p+ = Incoming traveling bore pressure b
Find:
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
p = Outgoing traveling bore pressure b
such that:
p+ p p b 0 = um + ub = , + b Rm (p ) Rb p = pm pb = pm (p+ + p ) b b
Solving for p is not immediate because b Rm depends on p which depends on p . b
Julius Smith
AES-2006 Masterclass 40 / 101
Graphical Solution Technique (1983)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Graphically solve:
G(p ) = p+ p ,
where
p+ = pm 2p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
G(p ) = Rb um (p ) = Rb p /Rm (p )
Analogous to nding the operating point of a transistor by
intersecting its operating curve with the load line determined by the load resistance.
Outgoing wave is then p = pm p+ p (p+ ) b b Published by McIntyre Schumacher & Woodhouse 1983,
adapting the Friedlander-Keller method for bowed strings (1953)
Julius Smith
AES-2006 Masterclass 41 / 101
Graphical Solution Technique (1983)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Graphically solve:
G(p ) = p+ p ,
where
p+ = pm 2p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
G(p ) = Rb um (p ) = Rb p /Rm (p )
Analogous to nding the operating point of a transistor by
intersecting its operating curve with the load line determined by the load resistance.
Outgoing wave is then p = pm p+ p (p+ ) b b Published by McIntyre Schumacher & Woodhouse 1983,
adapting the Friedlander-Keller method for bowed strings (1953)
Julius Smith
AES-2006 Masterclass 41 / 101
Graphical Solution Technique (1983)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Graphically solve:
G(p ) = p+ p ,
where
p+ = pm 2p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
G(p ) = Rb um (p ) = Rb p /Rm (p )
Analogous to nding the operating point of a transistor by
intersecting its operating curve with the load line determined by the load resistance.
Outgoing wave is then p = pm p+ p (p+ ) b b Published by McIntyre Schumacher & Woodhouse 1983,
adapting the Friedlander-Keller method for bowed strings (1953)
Julius Smith
AES-2006 Masterclass 41 / 101
Graphical Solution Technique (1983)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Graphically solve:
G(p ) = p+ p ,
where
p+ = pm 2p+ b
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
G(p ) = Rb um (p ) = Rb p /Rm (p )
Analogous to nding the operating point of a transistor by
intersecting its operating curve with the load line determined by the load resistance.
Outgoing wave is then p = pm p+ p (p+ ) b b Published by McIntyre Schumacher & Woodhouse 1983,
adapting the Friedlander-Keller method for bowed strings (1953)
Julius Smith
AES-2006 Masterclass 41 / 101
Graphical Solution Technique Illustrated
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
p+ x p+
Active Region
(Negative Resistance)
Reed Shut Inhaling p Exhaling x
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
G ( x ) = p+ x
p+ = pm 2p+ b
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 (p+ )
AES-2006 Masterclass 42 / 101
Julius Smith
Digital Waveguide Single Reed, Cylindrical Bore Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
h+
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Digital waveguide clarinet
Control variable = mouth half-pressure Total reed cost = two subtractions, one multiply, and one table
lookup per sample
Julius Smith
AES-2006 Masterclass 43 / 101
Digital Waveguide Single Reed, Cylindrical Bore Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
h+
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Digital waveguide clarinet
Control variable = mouth half-pressure Total reed cost = two subtractions, one multiply, and one table
lookup per sample
Julius Smith
AES-2006 Masterclass 43 / 101
Digital Waveguide Single Reed, Cylindrical Bore Model (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
Mouth Pressure pm ( n ) 2
hm Reed Table
p ( n ) b Reed to Bell Delay
-
Output Filter Reflection Filter
*
-
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
h+
Embouchure Offset Reed p+ ( n ) b
Bell to Reed Delay
Bore
Bell
Digital waveguide clarinet
Control variable = mouth half-pressure Total reed cost = two subtractions, one multiply, and one table
lookup per sample
Julius Smith
AES-2006 Masterclass 43 / 101
Digital Waveguide Wind Instrument Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
STK Clarinet: (WAV) (MP3)
Google search: STK clarinet
Synthesis Tool Kit (STK) by Perry Cook, Gary Scavone, and
others distributed by CCRMA: Google search: STK ToolKit
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Staccato Systems Slide Flute
(based on STK ute, ca. 1995): (WAV) (MP3) Yamaha VL1 Virtual Lead synthesizer demos (1994):
Shakuhachi: (WAV) (MP3) Oboe and Bassoon: (WAV) (MP3) Tenor Saxophone: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 44 / 101
Digital Waveguide Wind Instrument Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
STK Clarinet: (WAV) (MP3)
Google search: STK clarinet
Synthesis Tool Kit (STK) by Perry Cook, Gary Scavone, and
others distributed by CCRMA: Google search: STK ToolKit
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Staccato Systems Slide Flute
(based on STK ute, ca. 1995): (WAV) (MP3) Yamaha VL1 Virtual Lead synthesizer demos (1994):
Shakuhachi: (WAV) (MP3) Oboe and Bassoon: (WAV) (MP3) Tenor Saxophone: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 44 / 101
Digital Waveguide Wind Instrument Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds
STK Clarinet: (WAV) (MP3)
Google search: STK clarinet
Synthesis Tool Kit (STK) by Perry Cook, Gary Scavone, and
others distributed by CCRMA: Google search: STK ToolKit
Schematic Model Waveguide Model Simplied Reed Reed Computation Graphical Solver Example Clarinet Wind Examples
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Staccato Systems Slide Flute
(based on STK ute, ca. 1995): (WAV) (MP3) Yamaha VL1 Virtual Lead synthesizer demos (1994):
Shakuhachi: (WAV) (MP3) Oboe and Bassoon: (WAV) (MP3) Tenor Saxophone: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 44 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bowed Strings
Julius Smith
AES-2006 Masterclass 45 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Velocity (Primary Control) v+ l s, -1 String v l s, Nut or Finger Lowpass
vb
v r s, String v+ r s, Bridge Body
Bow
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow Force Bow Position
A schematic model for bowed-string instruments.
Bow divides string into two sections Primary control variable = bow velocity
Bow junction = nonlinear two-port Must nd velocity input to string (injected equally to left and right)
such that friction force = string reaction force.
velocity waves = natural choice of wave variable
Julius Smith
AES-2006 Masterclass 46 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Velocity (Primary Control) v+ l s, -1 String v l s, Nut or Finger Lowpass
vb
v r s, String v+ r s, Bridge Body
Bow
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow Force Bow Position
A schematic model for bowed-string instruments.
Bow divides string into two sections Primary control variable = bow velocity
Bow junction = nonlinear two-port Must nd velocity input to string (injected equally to left and right)
such that friction force = string reaction force.
velocity waves = natural choice of wave variable
Julius Smith
AES-2006 Masterclass 46 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Velocity (Primary Control) v+ l s, -1 String v l s, Nut or Finger Lowpass
vb
v r s, String v+ r s, Bridge Body
Bow
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow Force Bow Position
A schematic model for bowed-string instruments.
Bow divides string into two sections Primary control variable = bow velocity
Bow junction = nonlinear two-port Must nd velocity input to string (injected equally to left and right)
such that friction force = string reaction force.
velocity waves = natural choice of wave variable
Julius Smith
AES-2006 Masterclass 46 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Velocity (Primary Control) v+ l s, -1 String v l s, Nut or Finger Lowpass
vb
v r s, String v+ r s, Bridge Body
Bow
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow Force Bow Position
A schematic model for bowed-string instruments.
Bow divides string into two sections Primary control variable = bow velocity
Bow junction = nonlinear two-port Must nd velocity input to string (injected equally to left and right)
such that friction force = string reaction force.
velocity waves = natural choice of wave variable
Julius Smith
AES-2006 Masterclass 46 / 101
Schematic Physical Model
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Velocity (Primary Control) v+ l s, -1 String v l s, Nut or Finger Lowpass
vb
v r s, String v+ r s, Bridge Body
Bow
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow Force Bow Position
A schematic model for bowed-string instruments.
Bow divides string into two sections Primary control variable = bow velocity
Bow junction = nonlinear two-port Must nd velocity input to string (injected equally to left and right)
such that friction force = string reaction force.
velocity waves = natural choice of wave variable
Julius Smith
AES-2006 Masterclass 46 / 101
Bow-String Contact Physics
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Applied Force
= Friction Curve Differential Velocity = String Wave Impedance Velocity Change
Reaction Force
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 47 / 101
Friedlander-Keller
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Break-Away / Capture Bow and String Slipping (Reduced Friction)
String "Load Line" Solution = Graphical Intersection "Incoming" differential velocity v v v+
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Normalized Friction Rb / Rs times differential velocity v
Bow and String Slipping (Reduced Friction) Bow and String Stuck Together
Overlay of normalized bow-string friction curve Rb (v )/Rs with the + string load line v v .
Julius Smith
AES-2006 Masterclass 48 / 101
vc vc
Bow-String Scattering Junction
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Friedlander-Keller diagram is solved when
+ Rb (v ) v = Rs v v
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 49 / 101
Bow-String Scattering Junction
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Friedlander-Keller diagram is solved when
+ Rb (v ) v = Rs v v
which implies
+ + vsr = vsl + (v ) v +
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
+ + vsl = vsr + (v ) v +
Julius Smith
AES-2006 Masterclass 49 / 101
Bow-String Scattering Junction
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Friedlander-Keller diagram is solved when
+ Rb (v ) v = Rs v v
which implies
+ + vsr = vsl + (v ) v +
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
+ + vsl = vsr + (v ) v +
where
vsr = transverse string velocity on the right of the bow
Julius Smith
AES-2006 Masterclass 49 / 101
Bow-String Scattering Junction
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Friedlander-Keller diagram is solved when
+ Rb (v ) v = Rs v v
which implies
+ + vsr = vsl + (v ) v +
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
+ + vsl = vsr + (v ) v +
where
vsr = transverse string velocity on the right of the bow vsl = string velocity left of the bow (vsl = vsr )
Julius Smith
AES-2006 Masterclass 49 / 101
Bow-String Scattering Junction
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Friedlander-Keller diagram is solved when
+ Rb (v ) v = Rs v v
which implies
+ + vsr = vsl + (v ) v +
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
+ + vsl = vsr + (v ) v +
where
vsr = transverse string velocity on the right of the bow vsl = string velocity left of the bow (vsl = vsr )
+ + + v = vb (vsr + vsl ) = incoming differential velocity
Julius Smith
AES-2006 Masterclass 49 / 101
Bow-String Scattering Junction
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Friedlander-Keller diagram is solved when
+ Rb (v ) v = Rs v v
which implies
+ + vsr vsl = + (v ) v +
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
+ + vsl = vsr + (v ) v +
where
vsr = transverse string velocity on the right of the bow vsl = string velocity left of the bow (vsl = vsr )
+ + + v = vb (vsr + vsl ) = incoming differential velocity
vb = bow velocity, and (v ) is given by . . . +
Julius Smith
AES-2006 Masterclass 49 / 101
Bow-String Reection Coefcient
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
+ r v (v ) (v ) = + + 1 + r v (v )
where
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 50 / 101
Bow-String Reection Coefcient
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
+ r v (v ) (v ) = + + 1 + r v (v )
where
r(v ) = 0.25Rb (v )/Rs
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 50 / 101
Bow-String Reection Coefcient
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
+ r v (v ) (v ) = + + 1 + r v (v )
where
r(v ) = 0.25Rb (v )/Rs v = vb vs
bow velocity minus string velocity
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 50 / 101
Bow-String Reection Coefcient
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
+ r v (v ) (v ) = + + 1 + r v (v )
where
r(v ) = 0.25Rb (v )/Rs v = vb vs
bow velocity minus string velocity
+ + vs = vsl + vsl = vsr + vsr = transverse string velocity
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 50 / 101
Bow-String Reection Coefcient
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
+ r v (v ) (v ) = + + 1 + r v (v )
where
r(v ) = 0.25Rb (v )/Rs v = vb vs
bow velocity minus string velocity
+ + vs = vsl + vsl = vsr + vsr = transverse string velocity
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Rs = wave impedance of string
Julius Smith
AES-2006 Masterclass 50 / 101
Bow-String Reection Coefcient
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
+ r v (v ) (v ) = + + 1 + r v (v )
where
r(v ) = 0.25Rb (v )/Rs v = vb vs
bow velocity minus string velocity
+ + vs = vsl + vsl = vsr + vsr = transverse string velocity
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Rs = wave impedance of string Rb (v ) = friction coefcient for the bow against the string, i.e., Fb (v ) = Rb (v ) v
Julius Smith
AES-2006 Masterclass 50 / 101
Simplied, Piecewise Linear Bow Table
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow and String Stuck Together Break-Away / Capture ( v+ ) Break-Away / Capture
1
Total Reflection Total Transmission
0 -1
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow and String Slipping Frictionlessly vc
0
vc
v+
1
Flat center portion corresponds to a xed reection coefcient
seen by a traveling wave encountering the bow stuck against the string Outer sections give a smaller reection coefcient corresponding to the reduced bow-string interaction force while the string is slipping under the bow Hysteresis is neglected
AES-2006 Masterclass 51 / 101
Julius Smith
Simplied, Piecewise Linear Bow Table
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow and String Stuck Together Break-Away / Capture ( v+ ) Break-Away / Capture
1
Total Reflection Total Transmission
0 -1
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow and String Slipping Frictionlessly vc
0
vc
v+
1
Flat center portion corresponds to a xed reection coefcient
seen by a traveling wave encountering the bow stuck against the string Outer sections give a smaller reection coefcient corresponding to the reduced bow-string interaction force while the string is slipping under the bow Hysteresis is neglected
AES-2006 Masterclass 51 / 101
Julius Smith
Simplied, Piecewise Linear Bow Table
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow and String Stuck Together Break-Away / Capture ( v+ ) Break-Away / Capture
1
Total Reflection Total Transmission
0 -1
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow and String Slipping Frictionlessly vc
0
vc
v+
1
Flat center portion corresponds to a xed reection coefcient
seen by a traveling wave encountering the bow stuck against the string Outer sections give a smaller reection coefcient corresponding to the reduced bow-string interaction force while the string is slipping under the bow Hysteresis is neglected
AES-2006 Masterclass 51 / 101
Julius Smith
Simplied, Piecewise Linear Bow Table
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow and String Stuck Together Break-Away / Capture ( v+ ) Break-Away / Capture
1
Total Reflection Total Transmission
0 -1
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Bow and String Slipping Frictionlessly vc
0
vc
v+
1
Flat center portion corresponds to a xed reection coefcient
seen by a traveling wave encountering the bow stuck against the string Outer sections give a smaller reection coefcient corresponding to the reduced bow-string interaction force while the string is slipping under the bow Hysteresis is neglected
AES-2006 Masterclass 51 / 101
Julius Smith
Digital Waveguide Bowed Strings (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Force Bow Velocity Nut to Bow Delay vb -1
+ vs, l
vs, r
Bow to Bridge Delay
Body Filter Reflection Filter
-
v+
Bow Table
*
Bow to Nut Delay
+ vs, r vs, l
Bridge to Bow Delay
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Nut
String
Bow
String
BridgeBody
Air
Reection lter summarizes all losses per period
(due to bridge, bow, nger, etc.) Bow-string junction = memoryless lookup table (or segmented polynomial)
Julius Smith
AES-2006 Masterclass 52 / 101
Digital Waveguide Bowed Strings (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Force Bow Velocity Nut to Bow Delay vb -1
+ vs, l
vs, r
Bow to Bridge Delay
Body Filter Reflection Filter
-
v+
Bow Table
*
Bow to Nut Delay
+ vs, r vs, l
Bridge to Bow Delay
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Nut
String
Bow
String
BridgeBody
Air
Reection lter summarizes all losses per period
(due to bridge, bow, nger, etc.) Bow-string junction = memoryless lookup table (or segmented polynomial)
Julius Smith
AES-2006 Masterclass 52 / 101
Digital Waveguide Bowed Strings (1986)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Bow Force Bow Velocity Nut to Bow Delay vb -1
+ vs, l
vs, r
Bow to Bridge Delay
Body Filter Reflection Filter
-
v+
Bow Table
*
Bow to Nut Delay
+ vs, r vs, l
Bridge to Bow Delay
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Nut
String
Bow
String
BridgeBody
Air
Reection lter summarizes all losses per period
(due to bridge, bow, nger, etc.) Bow-string junction = memoryless lookup table (or segmented polynomial)
Julius Smith
AES-2006 Masterclass 52 / 101
Electric Cello Sound Examples (Peder Larson)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Staccato Notes: (WAV) (MP3)
(short strokes of high bow pressure, as from a bouncing bow) Bachs First Suite for Unaccompanied Cello: (WAV) (MP3)
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 53 / 101
Electric Cello Sound Examples (Peder Larson)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Staccato Notes: (WAV) (MP3)
(short strokes of high bow pressure, as from a bouncing bow) Bachs First Suite for Unaccompanied Cello: (WAV) (MP3)
Schematic Model Bow-String Contact Bow-String Junction Bow-String Reection Simplied Bow Table Bowed Strings
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 53 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Electric Guitar with Overdrive and Feedback
Julius Smith
AES-2006 Masterclass 54 / 101
Soft Clipper
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
0.8
f (x) =
2 3, x
2 3,
x3 3 ,
x 1 1 x 1 x1
3
x=1:0.01:1; plot([(2/3)*ones(1,100), xx. /3, (2/3)*ones(1,100)])
Distortion Guitar
Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
f(x(n))
0.6
0.4
0.2
Finite Differences
0
0.2
0.4
0.6
0.8 2
1.5
1
0.5
0 x(n)
0.5
1
1.5
2
Julius Smith
AES-2006 Masterclass 55 / 101
Amplier Distortion + Amplier Feedback
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
Sullivan 1990
Pre-distortion output level String 1 Nonlinear Distortion
Distortion Guitar
Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
. . .
Pre-distortion gain String N Amplier Feedback Gain Amplier Feedback Delay
Output Signal Distortion output level
Distortion output signal often further ltered by an amplier cabinet lter, representing speaker cabinet, driver responses, etc.
Julius Smith AES-2006 Masterclass 56 / 101
Distortion Guitar Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
(Stanford Sondius Project, ca. 1995)
Distortion Guitar: (WAV) (MP3) Amplier Feedback 1: (WAV) (MP3) Amplier Feedback 2: (WAV) (MP3)
Distortion Guitar
Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 57 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Commuted Waveguide Synthesis
Julius Smith
AES-2006 Masterclass 58 / 101
Commuted Synthesis of Acoustic Strings (1993)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
e(t) Trigger Excitation String
s(t) Resonator
y(t) Output
Schematic diagram of a stringed musical instrument.
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 59 / 101
Commuted Synthesis of Acoustic Strings (1993)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
e(t) Trigger Excitation String
s(t) Resonator
y(t) Output
Schematic diagram of a stringed musical instrument.
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Trigger
Excitation
Resonator
String
Output
Equivalent diagram in the linear, time-invariant case.
Julius Smith
AES-2006 Masterclass 59 / 101
Commuted Synthesis of Acoustic Strings (1993)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
e(t) Trigger Excitation String
s(t) Resonator
y(t) Output
Schematic diagram of a stringed musical instrument.
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Trigger
Excitation
Resonator
String
Output
Equivalent diagram in the linear, time-invariant case.
Aggregate Excitation a(t) String x(t) Output
Trigger
Use of an aggregate excitation given by the convolution of original excitation with the resonator impulse response.
Julius Smith
AES-2006 Masterclass 59 / 101
Commuted Components
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Trigger
Aggregate Excitation
a(t) String
x(t) Output
Plucked Resonator driving a String.
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
s(t)
Bridge Coupling
Guitar Body
Air Absorption
Room Response
y(t) Output
Possible components of a guitar resonator.
Julius Smith
AES-2006 Masterclass 60 / 101
Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
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)
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 61 / 101
Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
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)
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
STK Mandolin
STK Mandolin 1: (WAV) (MP3) STK Mandolin 2: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 61 / 101
Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
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
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 62 / 101
Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
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
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
1
http://www.acoustics.hut.fi/
AES-2006 Masterclass 62 / 101
Julius Smith
Commuted Synthesis of Linearized Violin
a)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Amplitude(n) Frequency(n)
Impulse Train
e(n) String
s(n)
Output x(n) Resonator Output x(n) String
b)
Amplitude(n) Frequency(n) Impulse Train
e(n) Resonator
a(n)
c)
Amplitude(n) Frequency(n) Impulse-Response Train
a(n) String
Output x(n)
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Assumes ideal Helmholtz motion of string Sound Examples (Stanford Sondius project, ca. 1995): Violin 1: (WAV) (MP3) Bass: (WAV) (MP3) Violin 2: (WAV) (MP3) Cello: (WAV) (MP3) Ensemble: (WAV) (MP3) Viola 1: (WAV) (MP3) Viola 2: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 63 / 101
Commuted Synthesis of Linearized Violin
a)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Amplitude(n) Frequency(n)
Impulse Train
e(n) String
s(n)
Output x(n) Resonator Output x(n) String
b)
Amplitude(n) Frequency(n) Impulse Train
e(n) Resonator
a(n)
c)
Amplitude(n) Frequency(n) Impulse-Response Train
a(n) String
Output x(n)
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Assumes ideal Helmholtz motion of string Sound Examples (Stanford Sondius project, ca. 1995): Violin 1: (WAV) (MP3) Bass: (WAV) (MP3) Violin 2: (WAV) (MP3) Cello: (WAV) (MP3) Ensemble: (WAV) (MP3) Viola 1: (WAV) (MP3) Viola 2: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 63 / 101
Commuted Synthesis of Linearized Violin
a)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Amplitude(n) Frequency(n)
Impulse Train
e(n) String
s(n)
Output x(n) Resonator Output x(n) String
b)
Amplitude(n) Frequency(n) Impulse Train
e(n) Resonator
a(n)
c)
Amplitude(n) Frequency(n) Impulse-Response Train
a(n) String
Output x(n)
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Assumes ideal Helmholtz motion of string Sound Examples (Stanford Sondius project, ca. 1995): Violin 1: (WAV) (MP3) Bass: (WAV) (MP3) Violin 2: (WAV) (MP3) Cello: (WAV) (MP3) Ensemble: (WAV) (MP3) Viola 1: (WAV) (MP3) Viola 2: (WAV) (MP3)
Julius Smith
AES-2006 Masterclass 63 / 101
Commuted Piano Synthesis (1995)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Force 0.5 0.4 0.3 0.2 0.1 5 10 15 Time 20
Hammer-string interaction pulses (force):
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 64 / 101
Synthesis of Hammer-String Interaction Pulse
Overview
Impulse
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Impulse Response Lowpass Filter Time Time
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Faster collisions correspond to narrower pulses
(nonlinear lter )
For a given velocity, lter is linear time-invariant Piano is linearized for each hammer velocity
Julius Smith
AES-2006 Masterclass 65 / 101
Synthesis of Hammer-String Interaction Pulse
Overview
Impulse
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Impulse Response Lowpass Filter Time Time
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Faster collisions correspond to narrower pulses
(nonlinear lter )
For a given velocity, lter is linear time-invariant Piano is linearized for each hammer velocity
Julius Smith
AES-2006 Masterclass 65 / 101
Synthesis of Hammer-String Interaction Pulse
Overview
Impulse
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Impulse Response Lowpass Filter Time Time
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Faster collisions correspond to narrower pulses
(nonlinear lter )
For a given velocity, lter is linear time-invariant Piano is linearized for each hammer velocity
Julius Smith
AES-2006 Masterclass 65 / 101
Synthesis of Hammer-String Interaction Pulse
Overview
Impulse
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Impulse Response Lowpass Filter Time Time
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Faster collisions correspond to narrower pulses
(nonlinear lter )
For a given velocity, lter is linear time-invariant Piano is linearized for each hammer velocity
Julius Smith
AES-2006 Masterclass 65 / 101
Multiple Hammer-String Interaction Pulses
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Superimpose several individual pulses:
Force Impulse 1 Impulse 2 1 2 3 LPF1 LPF2 LPF3 0 Time
Distortion Guitar Commuted Synthesis
+
String Input
Impulse 3
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 66 / 101
Multiple Hammer-String Interaction Pulses
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
Superimpose several individual pulses:
Force Impulse 1 Impulse 2 1 2 3 LPF1 LPF2 LPF3 0 Time
Distortion Guitar Commuted Synthesis
+
String Input
Impulse 3
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
As impulse amplitude grows (faster hammer strike), output pulses become taller and thinner, showing less overlap.
Julius Smith
AES-2006 Masterclass 66 / 101
Complete Piano Model
Natural Ordering:
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
vc
Impulse 1 Tapped 1 GenerDelay Trigger ator Line 2 3 LPF1 LPF2 LPF3
+
String
Sound Board & Enclosure Output
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Soundboard and enclosure are commuted Only need a stored recording of their impulse response An enormous digital lter is otherwise required
AES-2006 Masterclass 67 / 101
Julius Smith
Complete Piano Model
Natural Ordering:
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
vc
Impulse 1 Tapped 1 GenerDelay Trigger ator Line 2 3 LPF1 LPF2 LPF3
+
String
Sound Board & Enclosure Output
Commuted Ordering:
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
vc
Sound Board & Enclosure Trigger Impulse Response Tapped Delay Line LPF1 LPF2 LPF3
+
String
Output
Soundboard and enclosure are commuted Only need a stored recording of their impulse response An enormous digital lter is otherwise required
AES-2006 Masterclass 67 / 101
Julius Smith
Complete Piano Model
Natural Ordering:
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
vc
Impulse 1 Tapped 1 GenerDelay Trigger ator Line 2 3 LPF1 LPF2 LPF3
+
String
Sound Board & Enclosure Output
Commuted Ordering:
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
vc
Sound Board & Enclosure Trigger Impulse Response Tapped Delay Line LPF1 LPF2 LPF3
+
String
Output
Soundboard and enclosure are commuted Only need a stored recording of their impulse response An enormous digital lter is otherwise required
AES-2006 Masterclass 67 / 101
Julius Smith
Complete Piano Model
Natural Ordering:
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
vc
Impulse 1 Tapped 1 GenerDelay Trigger ator Line 2 3 LPF1 LPF2 LPF3
+
String
Sound Board & Enclosure Output
Commuted Ordering:
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
vc
Sound Board & Enclosure Trigger Impulse Response Tapped Delay Line LPF1 LPF2 LPF3
+
String
Output
Soundboard and enclosure are commuted Only need a stored recording of their impulse response An enormous digital lter is otherwise required
AES-2006 Masterclass 67 / 101
Julius Smith
Complete Piano Model
Natural Ordering:
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
vc
Impulse 1 Tapped 1 GenerDelay Trigger ator Line 2 3 LPF1 LPF2 LPF3
+
String
Sound Board & Enclosure Output
Commuted Ordering:
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
vc
Sound Board & Enclosure Trigger Impulse Response Tapped Delay Line LPF1 LPF2 LPF3
+
String
Output
Soundboard and enclosure are commuted Only need a stored recording of their impulse response An enormous digital lter is otherwise required
AES-2006 Masterclass 67 / 101
Julius Smith
Piano and Harpsichord Sound Examples
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
(Stanford Sondius Project, ca. 1995)
Piano: (WAV) (MP3) Harpsichord 1: (WAV) (MP3) Harpsichord 2: (WAV) (MP3)
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 68 / 101
More Recent Harpsichord Example
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis
Harpsichord Soundboard Hammer-Response Musical Commuted Harpsichord Example More examples
Reference: Sound Synthesis of the Harpsichord Using a Computationally Efcient Physical Model, by Vesa Valimaki, Henri Penttinen, Jonte Knif, Mikael Laurson, and Cumhur Erkut JASP-2004
Acoustic Strings Sound Examples Linearized Violin Commuted Piano Pulse Synthesis Complete Piano Sound Examples
Digital Waveguide Mesh Horn Filter Design Finite Differences
Google search: Harpsichord Sound Synthesis
Julius Smith
AES-2006 Masterclass 69 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Digital Waveguide Mesh
Julius Smith
AES-2006 Masterclass 70 / 101
2D Waveguide Mesh
At each junction:
VJ
outk
z1 z1 z1 z1 z1 z1 z1 z1 z1 z1 z1 z1 z1 z1 4-port Scattering Junction z1 z1 z1 z1 z1 z1 4-port Scattering Junction z1 z1 z1 z1 4-port Scattering Junction z1 z1 z1 z1
=
in1 + in2 + in3 + in4
2 k = 1, 2, 3, 4
= VJ ink ,
Sound example:
Gongs (lossless nonlinear rim by
4-port Scattering Junction z1 z1
4-port Scattering Junction z1 z1
4-port Scattering Junction z1 z1
Pierce and Van Duyne JASA1997)
z1 z1
4-port Scattering Junction z1 z1
z1 z1
4-port Scattering Junction z1 z1
z1 z1
4-port Scattering Junction z1 z1
z1 z1
Julius Smith
AES-2006 Masterclass 71 / 101
Recent Mesh Topic: Virtual Quadratic Residue Diffusers
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
Manfred Schroeders Quadratic Residue Diffuser for N = 17:
dn
N
Commuted Synthesis
w
Digital Waveguide Mesh
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
Horn Filter Design Finite Differences
Thin vertical lines = rigid separators between wells. Well depths are N sn , where
sn = n2 (mod N ),
is a quadratic residue sequence
nZ
Julius Smith
AES-2006 Masterclass 72 / 101
Recent Mesh Topic: Virtual Quadratic Residue Diffusers
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
Manfred Schroeders Quadratic Residue Diffuser for N = 17:
dn
N
Commuted Synthesis
w
Digital Waveguide Mesh
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
Horn Filter Design Finite Differences
Thin vertical lines = rigid separators between wells. Well depths are N sn , where
sn = n2 (mod N ),
is a quadratic residue sequence
nZ
Julius Smith
AES-2006 Masterclass 72 / 101
Recent Mesh Topic: Virtual Quadratic Residue Diffusers
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
Manfred Schroeders Quadratic Residue Diffuser for N = 17:
dn
N
Commuted Synthesis
w
Digital Waveguide Mesh
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
Horn Filter Design Finite Differences
Thin vertical lines = rigid separators between wells. Well depths are N sn , where
sn = n2 (mod N ),
is a quadratic residue sequence
nZ
Julius Smith
AES-2006 Masterclass 72 / 101
Example QRD
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
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, . . .]
Reection magnitude equal in N equally spaced directions at the design wavelength
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 73 / 101
Example QRD
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
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, . . .]
Reection magnitude equal in N equally spaced directions at the design wavelength (and not bad in between)
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 73 / 101
QRD Termination of a 2D Digital Waveguide Mesh
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
(Lee-Smith ICMC-2004)
J J J J
J J J J
J J J J J J J J J J
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
Horn Filter Design Finite Differences
Julius Smith
AES-2006 Masterclass 74 / 101
Scattering levels from a Schroeder diffuser (solid) and a straight boundary (dashed)
90
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh
10 20
30 60
30
2D Waveguide Mesh QR Diffuser Example QRD QRD Mesh Boundary Scattering Levels
0
Horn Filter Design Finite Differences
(Animations)
Julius Smith
AES-2006 Masterclass 75 / 101
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Horn Filter Design
Julius Smith
AES-2006 Masterclass 76 / 101
Bore Prole Reconstruction from Measured Trumpet Reectance
10
Overview
9
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
radius (mm)
8 7 6 5 4 3 0 0.2 0.4 0.6 distance (m) 0.8 1 1.2 empirical model
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Inverse scattering applied to pulse-reectometry data to t
piecewise-cylindrical model (like LPC model) Bore prole 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
AES-2006 Masterclass 77 / 101
Julius Smith
Trumpet-Bell Impulse Response Computed from Estimated Piecewise-Cylindrical Model
Overview Voice Models
Bell Reflection Impulse Response 0.015 0.01
Karplus Strong Digital Waveguides Single Reeds
0.005 0
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
Amplitude
-0.005 -0.01 -0.015 -0.02 -0.025 0
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
1
2
3
4 5 time (ms)
6
7
8
9
From pulse reectometry on trumpet with no mouthpiece Bore prole is reconstructed, smoothed, and segmented Impulse response of bell segment = ideal lter
AES-2006 Masterclass 78 / 101
Julius Smith
Trumpet-Bell Filter Design Problem
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
At fs = 44.1 kHz, impulse-response length 400 600
samples A length 400 FIR bell lter 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
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 79 / 101
Measured Trombone Bell Reectance
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings
trombone bell 0.005
0
0.005
Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
0.01
0.015 0 1 2 3 time (ms) 4 5 6 7
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 80 / 101
Idea! (1998)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
0.02
Break up impulse response into exponential or polynomial
segments Exponential and polynomial impulse-responses can be designed using Truncated IIR (TIIR) Filters Google search: TIIR Horns
Bell Impulse Response Segmentation
Commuted Synthesis
0.015
Digital Waveguide Mesh
0.01
Horn Filter Design
0.005
seg1 seg2 ncs3 ncs4
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Amplitude
0
0.005
0.01
0.015
0.02
0.025
0
50
100
150
200 250 Time (samples)
300
350
400
450
Julius Smith
AES-2006 Masterclass 81 / 101
Four-Exponential Fit to Estimated Trumpet-Bell Filter (Exp-4)
Overview
0.02 TimeDomain Fit with Four OffsetExponential Segments
Voice Models
0.015
Karplus Strong Digital Waveguides
0.01
0.005
ideal exp1 exp2 exp3 exp4
Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
Amplitude
Single Reeds
0
0.005
0.01
0.015
0.02
0.025
0
50
100
150
200 250 Time (samples)
300
350
400
450
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Exp-4 Impulse Response Fit
Julius Smith
AES-2006 Masterclass 82 / 101
Two Exponentials Connected by a Cubic Spline Measured Trumpet Data (Exp2-S3)
Overview
0.02 Response Segmentation
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
0.015
seg1 seg2 seg3
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
0
50
100
150 Time (samples)
200
250
300
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 83 / 101
Two Exponentials Followed by a 6th-Order IIR Filter Designed by Steiglitz McBride Algorithm (Exp2-SM6)
Overview
0.02 Response Segmentation
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
0.015
seg1 seg2 tail
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
0
50
100
150
200 250 Time (samples)
300
350
400
450
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 84 / 101
Exp2-SM6 Amplitude Response Fit
Overview
0 Magnitude Fit over Entire Nyquist Band
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
Magnitude (dB)
10
Ideal Approximation
20
30
40
50
60
70
80
90
100
0
0.05
0.1
0.15
0.2 0.25 0.3 Normalized Frequency
0.35
0.4
0.45
0.5
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 85 / 101
Exp2-SM6 Low-Frequency Zoom
Overview
0 Magnitude Fit
Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
Magnitude (dB)
5
Ideal Approximation
10
15
20
25
30
35
40
45
50
0
0.01
0.02
0.03 0.04 Normalized Frequency
0.05
0.06
0.07
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 86 / 101
Exp2-SM6 Group Delay Fit
Overview
700 Group Delay Fit
Voice Models
600
Ideal approximation
Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar
200 500
Group Delay (samples)
400
300
Commuted Synthesis
100
Digital Waveguide Mesh Horn Filter Design
0 0 0.01 0.02 0.03 0.04 Normalized Frequency 0.05 0.06 0.07
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 87 / 101
Exp2-SM6 Phase Delay Fit
Overview
200 Phase Delay Fit
Voice Models Karplus Strong Digital Waveguides
Phase Delay (samples)
180
160
140
Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
120
100
80
60
40
20
Ideal Approximation
0 0.01 0.02 0.03 0.04 Normalized Frequency 0.05 0.06 0.07
0
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Julius Smith
AES-2006 Masterclass 88 / 101
A One-Pole (Almost) TIIR Filter
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design
Warm(n) & Switch(n) Warm(n) & Switch(n)
1 0 1
Clear(n) Clear(n) 1P 1P
1
y(n)
0
...
x(n)
+
0
N Shared Delay Line
-
... ...
Switch(n)
(rising edge active) Clear(n) ... 1P = 1 pz1
1
Clear and halt filter 1 Start filter 1 on direct signal Switch to using filter 1
...
Bore Prole Impulse Response Bell Filter Design Bell Reectance Idea! (1998) Four Exponentials Exponentials & Cubic Exponentials & SM Example One-Pole TIIR
Finite Differences
Warm(n)
... Switch(n) ...
N
Clear & halt filter 2
... etc. ...
n
Time(samples)
Generates truncated exponentials or constants Filter complexity on average one pole Higher orders give truncated polynomial impulse responses
AES-2006 Masterclass 89 / 101
Julius Smith
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Finite Differences
Julius Smith
AES-2006 Masterclass 90 / 101
Lumped Modeling Elements
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
The foregoing models were distributed-parameter systems Distributed systems supporting wave propagation We also need lumped-parameter systems, such as for
Piano hammer Brass-players lips Vocal-fold models Reeds with mass Helmholtz resonators
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
Instead of sampled traveling waves, we employ nite difference
schemes to model lumped systems
Julius Smith
AES-2006 Masterclass 91 / 101
Finite Difference Approximation (FDA)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Consider the simple differential equation relating velocity and force for an ideal mass:
v(t) f(t) m 0 x(t)
dv F (s) = msV (s) f (t) = m dt
Assume f (t) = input signal, and v(t) = output. We need to discretize this continuous time equation. Input signal = fn = f (nT ), n = 0, 1, 2, . . .,
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
where T is the sampling interval. The FDS will compute vn given fn m for n = 0, 1, 2, . . ..
AES-2006 Masterclass 92 / 101
Julius Smith
Finite Difference Approximation (Contd)
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
The Finite Difference Approximation replaces differentiation by a nite difference, e.g.,
dv dt
vn vn1 T vn+1 vn1 2T
(backward difference) (centered difference)
Using the backwards-difference approximation, we obtain
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
vn = vn1 +
T fn , m
n = 0, 1, 2, . . . .
We see that v1 must be specied as an initial condition.
Julius Smith
AES-2006 Masterclass 93 / 101
Bilinear Transform of the Ideal Mass
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Starting with the driving point impedance
R(s) =
F (s) = ms V (s)
the bilinear transform gives the digital impedance
Fd (z) 1 z 1 = Rd (z) = R c Vd (z) 1 + z 1
1 z 1 = mc 1 + z 1
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
Julius Smith
AES-2006 Masterclass 94 / 101
Bilinear Transform of the Ideal Mass, Contd
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Multiplying out
Fd (z) + z 1 Fd (z) = mcVd (z) mcz 1 Vd (z)
and taking the inverse z transform gives
fn + fn1 = mc (vn vn1 )
or
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
1 vn = vn1 + (fn + fn1 ) mc
(The fn1 term is new relative to the FDA.) Equivalent to the trapezoid rule for numerical integration.
Julius Smith
AES-2006 Masterclass 95 / 101
Accuracy
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
Recall that the backward-difference approximation is rst-order accurate in T :
2 a = d + O(d T )
For the trapezoid rule we get
3 a = d + O(d T 2 )
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
Trapezoid rule (bilinear transform) is second-order accurate in T . Higher order accuracy obtainable using more neighboring grid
points. How should these extra grid points be brought in?
Julius Smith
AES-2006 Masterclass 96 / 101
Observations
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
The FDA gave us
T vn = vn1 + fn , m
n = 0, 1, 2, . . .
which is a one-pole digital lter having a pole at z = 1 and a zero at z = 0.
Similarly, the BLT gave us
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
T vn = vn1 + (fn + fn1 ) 2m
which has the same pole, but a zero at z = 1.
Question: How do we use more poles and zeros to obtain a
more accurate FDS?
Julius Smith
AES-2006 Masterclass 97 / 101
Digital Filter Design Approach
Overview Voice Models Karplus Strong Digital Waveguides Single Reeds Bowed Strings Distortion Guitar Commuted Synthesis Digital Waveguide Mesh Horn Filter Design Finite Differences
The driving-point impedance R(s) = ms of an ideal mass is an ideal differentiator (scaled by m):
R(j) = mj.
It is therefore natural to dene the ideal digital differentiator as
H(ejT ) = j,
T [, )
Ideal Mass Bilinear Transform Accuracy Observations Digital Filter Design Differentiator
Julius Smith
AES-2006 Masterclass 98 / 101
Frequency Response of the Ideal Digital Differentiator
Overview Voice Models Karplus Strong Digital Waveguides
Im Gain Re
Single Reeds Bowed Strings Dis...