Extended Karplus-Strong Guitar Synthesizer

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Abstract

This project implements Julius O. Smith III's Extended Karplus-Strong (EKS) physical modeling algorithm for plucked string synthesis, realized in the Faust programming language. The system features a 32-step sequencer with progressive rock-style arpeggiated patterns, stereo width modulation, LFO-based auto-panning, and high-quality Zita reverb. The implementation extends the original Karplus-Strong algorithm with frequency-dependent damping, pick position modeling, and dynamic-level compensation for realistic string behavior.

Key Features:

• Full EKS algorithm with linear-phase FIR damping filters
• 16-step sequencer with Hz-based tempo control
• LFO modulation for spatial movement
• Zita reverb integration
• Real-time parameter control for live performance

Based on: Smith, J. O. (2010). "Physical Audio Signal Processing". CCRMA, Stanford University.

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1. Introduction

1.1 Physical Modeling Synthesis

Physical modeling recreates instrument behavior through mathematical simulation of acoustic physics. The Extended Karplus-Strong algorithm models plucked string instruments by simulating:

• Wave propagation: Delay line representing string length
• Energy dissipation: Frequency-dependent damping filters
• Excitation: Initial displacement from plucking action
• Boundary conditions: Fixed endpoints (bridge and nut)

1.2 Extended Karplus-Strong Algorithm

The EKS extends the basic Karplus-Strong (1983) with:

1. Linear-phase damping: FIR3 filter for frequency-dependent decay
2. Pick position modeling: Comb filter for pluck location effects
3. Dynamic-level compensation: Adjustable high-frequency content
4. Brightness control: Independent tuning invariance parameter

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2. Equation Key & Symbol Reference

2.1 General Symbols

Symbol	Name	Description	Units/Range
$z$	Z-transform variable	Complex frequency variable in discrete-time domain	$z = e^{j\omega T}$
$z^{-1}$	Unit delay operator	Delays signal by one sample	-
$n$	Sample index	Discrete time index	$n \in \mathbb{Z}$
$x[n]$	Input signal	Discrete-time input at sample $n$	-
$y[n]$	Output signal	Discrete-time output at sample $n$	-
$T$	Sample period	Time between samples = $1/f_s$	seconds
$f_s$	Sample rate	Samples per second (typically 44.1 or 48 kHz)	Hz
$f$	Frequency	Fundamental frequency of string	Hz
$f_1$	Fundamental frequency	Same as $f$ (used in filter equations)	Hz

2.2 Transfer Functions & Filters

Symbol	Name	Description	Range
$H_d(z)$	Damping filter	Z-domain transfer function for string damping	-
$H_\beta(z)$	Pick-position filter	Comb filter modeling pluck location	-
$H_L(z)$	Level lowpass filter	Dynamic-level compensation filter	-
$H_{L,\omega_1}(s)$	Analog lowpass	Continuous-time lowpass prototype	-
$\rho$	Loop gain	Controls overall decay rate of string	$(0, 1]$
$S$	Stretching factor	Controls frequency-dependent damping	$[0, 1]$
$B$	Brightness	High-frequency content parameter	$[0, 1]$

2.3 Filter Coefficients

Symbol	Name	Description	Derivation
$b_0$	Feedforward coefficient 0	Current sample weight	$b_0 = 1 - S$
$b_1$	Feedforward coefficient 1	Delayed sample weight	$b_1 = S$
$g_0$	Center tap coefficient	FIR3 center tap	$g_0 = h_0 = \frac{1+B}{2}$
$g_1$	Side tap coefficient	FIR3 side taps (symmetric)	$g_1 = h_1 = \frac{1-B}{4}$
$h_0$	Impulse response (center)	Same as $g_0$	-
$h_1$	Impulse response (sides)	Same as $g_1$	-
$a_1$	Recursive coefficient	IIR feedback coefficient	$a_1 = \frac{1-\tilde{\omega}_1}{1+\tilde{\omega}_1}$

2.4 String Physical Parameters

Symbol	Name	Description	Typical Range
$P$	Period	String fundamental period in samples	$P = f_s / f$
$N$	Period (alternate)	Same as $P$ in pick-position equations	-
$t_{60}$	Decay time	Time for -60 dB amplitude decay	0.1 - 10 seconds
$\beta$	Pick position	Normalized pluck location (0=bridge, 0.5=center)	$(0, 0.5)$
$\eta$	Fractional delay	Fine-tuning fractional delay	$[0, 1)$
$L$	Dynamic level	Playing dynamics / velocity	$[0, 1]$

2.5 Angular Frequency

Symbol	Name	Description	Units
$\omega$	Angular frequency	$\omega = 2\pi f$	rad/s
$\omega_1$	Fundamental angular freq	$\omega_1 = 2\pi f_1$	rad/s
$\tilde{\omega}_1$	Prewarped frequency	$\tilde{\omega}_1 = \frac{\omega_1 T}{2}$	radians
$s$	Laplace variable	Complex frequency in continuous time	$s = \sigma + j\omega$

2.6 Special Functions & Operators

Symbol	Name	Description
$\mathbb{1}_{{condition}}$	Indicator function	1 if condition true, 0 otherwise
$\lfloor x \rfloor$	Floor function	Largest integer ≤ $x$
$\lfloor x + 0.5 \rfloor$	Round function	Nearest integer to $x$
$\cdot$	Multiplication	Scalar or element-wise multiplication
$+$	Addition	Summation operator

2.7 Sequencer & Modulation

Symbol	Name	Description	Range
$k$	Step index	Current sequencer step	$0$ to $31$
$g$	Gate/trigger	Binary trigger signal	${0, 1}$
$g_{\text{trigger}}(n)$	Trigger function	Gate at sample $n$	-

2.8 Key Relationships

Period-Frequency relationship: $$P = \frac{f_s}{f}$$

Loop gain from decay time: $$\rho = 0.001^{\frac{1}{f \cdot t_{60}}} = 10^{-3/(f \cdot t_{60})}$$

Brightness to filter coefficients: $$h_0 = \frac{1+B}{2}, \quad h_1 = \frac{1-B}{4}$$

Pick-position delay (samples): $$\text{delay} = \lfloor \beta \cdot P + 0.5 \rfloor$$

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3. Mathematical Foundations

3.1 One-Zero String Damping Filter

The simplest damping filter uses a single zero:

$$ H_d(z) = \rho[(1-S) + S \cdot z^{-1}] $$

Difference equation:

$$ y[n] = \rho[b_0 \cdot x[n] + b_1 \cdot x[n-1]] $$

where:

• $S \in [0,1]$ is the stretching factor
• $b_0 = 1 - S$ (feedforward coefficient)
• $b_1 = S$ (delayed feedforward coefficient)
• $\rho \in (0,1)$ is loop gain controlling decay rate

Brightness mapping:

$$ S = \frac{B}{2}, \quad b_1 = 0.5 \cdot B, \quad b_0 = 1.0 - b_1 $$

Decay time calculation:

$$ \rho = 0.001^{\frac{P \cdot T}{t_{60}}} $$

where $P$ is period (samples), $T$ is sample interval, $t_{60}$ is -60 dB decay time.

Characteristics:

• DC gain = $\rho$ (infinite decay at DC when $\rho=1$)
• Fastest decay at $S = 0.5$ (original Karplus-Strong)
• Higher frequencies decay faster than lower frequencies

Perceptual Effect:

• Low $\rho$ (0.5-0.7): String dies out quickly (plucked staccato, muted guitar)
• High $\rho$ (0.95-0.99): Long, ringing sustain (open strings, sitar-like drone)
• Low $S$ (0.0-0.3): Dull, dark tone (damped with palm, jazz tone)
• Medium $S$ (0.5): Classic Karplus-Strong pluck (vintage digital synth)
• High $S$ (0.7-1.0): Bright, metallic timbre (new steel strings, twangy)

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3.2 Two-Zero String Damping Filter (Linear Phase)

For improved tuning invariance, a symmetric FIR filter is used:

$$ H_d(z) = \rho[g_1 + g_0 \cdot z^{-1} + g_1 \cdot z^{-2}] $$

$$ = \rho \cdot z^{-1}[g_0 + g_1(z + z^{-1})] $$

Impulse response:

$$ h_d = [g_1, g_0, g_1, 0, 0, \ldots] $$

Symmetric around time $n=1$ (linear phase property).

Coefficients from brightness:

$$ h_0 = \frac{1 + B}{2} \quad \text{(center tap)} $$

$$ h_1 = \frac{1 - B}{4} \quad \text{(side taps)} $$

Implementation:

$$ y_{\text{damp}}[n] = \rho \cdot [h_0 \cdot x[n-1] + h_1 \cdot (x[n] + x[n-2])] $$

Loop gain:

$$ \rho = 0.001^{\frac{1}{f \cdot t_{60}}} $$

Advantages:

• Linear phase (no phase distortion)
• Better tuning invariance than one-zero
• Symmetric impulse response reduces computational complexity

Perceptual Effect:

• High $B$ (0.7-1.0): Bright, clear attack; harmonics ring clearly (fresh strings, aggressive pick attack)
• Medium $B$ (0.4-0.6): Balanced warmth and presence (well-worn strings, fingerstyle)
• Low $B$ (0.0-0.3): Dark, mellow sound; emphasizes fundamental (old flatwound strings, thumb picking)
• vs. One-Zero: Sounds more "in-tune" across the frequency range; less detuning artifacts

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3.3 Pick-Position Comb Filter

Models the effect of plucking at position $\beta$ along the string:

$$ H_\beta(z) = 1 - z^{-\lfloor \beta N + \frac{1}{2} \rfloor} $$

where:

• $\beta \in (0, 0.5)$ is normalized pick position ($0$ = bridge, $0.5$ = center)
• $N = P$ is period in samples
• Delay = $\lfloor \beta N + 0.5 \rfloor$ (rounded to nearest integer)
• Feedforward gain = $-1$ (inverts and subtracts delayed signal)

Spectral effect: Creates nulls at frequencies $f_k = \frac{k}{\beta \cdot T}$ for integer $k$.

Physical interpretation: Plucking at position $\beta$ cannot excite harmonics with nodes at that location.

Perceptual Effect:

• $\beta$ = 0.02-0.05 (near bridge): Thin, nasal, "honky" tone; emphasizes high harmonics (mandolin, banjo)
• $\beta$ = 0.13 (default): Realistic guitar pluck position; balanced harmonic content (standard guitar)
• $\beta$ = 0.25: Rounder, fuller tone; some harmonics missing (classical guitar, neck pickup)
• $\beta$ = 0.4-0.5 (near center): Very soft, hollow sound; many harmonics cancelled (12th fret harmonic area)
• Spectral nulls: Certain harmonics disappear completely, creating distinctive timbral "holes" in the sound

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3.4 Dynamic-Level Lowpass Filter

Compensates for high-frequency loss based on playing dynamics.

Continuous-time (analog):

$$ H_{L,\omega_1}(s) = \frac{\omega_1}{s + \omega_1} $$

where $\omega_1 = 2\pi f_1$ (fundamental angular frequency).

Discrete-time (bilinear transform):

$$ H_{L,\omega_1}(z) = \frac{\tilde{\omega}_1}{1+\tilde{\omega}_1} \cdot \frac{1+z^{-1}}{1-\frac{1-\tilde{\omega}_1}{1+\tilde{\omega}_1} \cdot z^{-1}} $$

where $\tilde{\omega}_1 = \frac{\omega_1 T}{2}$ (prewarped frequency).

Simplified IIR form:

$$ H_L(z) = g \cdot \frac{1+z^{-1}}{1-a_1 \cdot z^{-1}} $$

Dynamic blending:

$$ y_{\text{out}}[n] = L \cdot L_0(L) \cdot x[n] + (1-L) \cdot y_{\text{LP}}[n] $$

where:

• $L \in [0,1]$ is level parameter (from dynamic_level slider)
• $L_0(L) = L^{1/3}$ attenuates low levels
• $y_{\text{LP}}[n]$ is lowpass-filtered signal

Characteristics:

• Unity DC gain
• -3 dB at fundamental frequency $f_1$
• -6 dB/octave rolloff above $f_1$
• Bypassed at $L=1$ (maximum dynamics)

Perceptual Effect:

• High $L$ (0 dB, near 1.0): Bright, snappy attack; full harmonic spectrum (hard pick attack, forte)
• Medium $L$ (-20 dB, ~0.1): Softer attack; slightly rolled-off highs (normal playing, mezzo-forte)
• Low $L$ (-40 to -60 dB, near 0): Very soft, muffled pluck; mostly fundamental (gentle finger pluck, pianissimo)
• Musical analogy: Mimics how real strings sound duller when plucked softly vs. brightly when plucked hard
• Dynamic response: Creates velocity-sensitive timbre, not just volume

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3.5 Fundamental Period and Frequency Relationship

Basic relationship:

$$ P = \frac{f_s}{f} $$

where $f_s$ is sample rate and $f$ is fundamental frequency.

With fractional delay (for fine tuning):

$$ \text{Total Delay} = P - 2 + \eta $$

where $\eta \in [0,1)$ is fractional delay for precise tuning.

Perceptual Effect:

• Without fractional delay ($\eta = 0$): Slight pitch inaccuracies; noticeable detuning on certain notes
• With fractional delay: Perfect pitch tracking; all notes sound in-tune across the range
• Musical importance: Essential for realistic instrument simulation; prevents "digital" detuning artifacts

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3.6 Excitation Signal (Noise Burst)

Trigger function:

$$ g_{\text{trigger}}(n) = g \cdot \mathbb{1}_{{\text{release}(P) > 0}} $$

where $\mathbb{1}$ is the indicator function.

Release envelope:

$$ \text{release}(n) = \sum_{k} \text{decay}(n, x_k) $$

$$ \text{decay}(n,x) = x - \frac{\mathbb{1}_{{x>0}}}{n} $$

Creates exponential decay with time constant proportional to period $P$.

Physical interpretation: Simulates initial displacement of string at pluck time, with energy proportional to plucking force.

Perceptual Effect:

• Noise burst: Creates the initial "pluck" transient; sounds like the pick or finger hitting the string
• Broader burst: Softer, gentler attack (fingerstyle)
• Narrow burst: Sharp, percussive attack (hard pick, slap)
• White noise source: Provides full-spectrum excitation that the resonant delay line filters into pitched tone
• Without excitation: No sound (like a string that was never plucked)

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4. System Architecture

4.1 EKS Physical Model Diagram

graph TB
    subgraph "EKS Physical Model"
        N[White Noise] --> G{Gate?}
        G -->|Trigger| E[Excitation<br/>Burst Generator]
        E --> S[Smooth<br/>pick_angle]
        S --> PP[Pick Position<br/>Comb Filter<br/>β·P delay]
        PP --> LF[Level Filter<br/>HF Compensation]
        LF --> DL[Delay Line<br/>P-2 samples]

        DL --> OUT[Output]
        DL --> DF[Damping Filter<br/>ρ · FIR3 LPF]
        DF --> FB((+))
        LF --> FB
        FB --> DL

        P[Period P<br/>= SR/freq] -.-> DL
        P -.-> PP
        T60[Decay T60] -.-> DF
        BR[Brightness B] -.-> DF
    end

    classDef exciteStyle stroke:#f96,stroke-width:3px
    classDef delayStyle stroke:#4ecdc4,stroke-width:3px
    classDef dampStyle stroke:#95e1d3,stroke-width:3px
    classDef outStyle stroke:#ffd93d,stroke-width:3px

    class E exciteStyle
    class DL delayStyle
    class DF dampStyle
    class OUT outStyle

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4.2 Sequencer Implementation

graph TB
    UI[User Interface] --> NR[note_rate Hz]
    UI --> RUN[run_sequencer]

    NR --> CLK[Clock Generator<br/>os.lf_imptrain]
    RUN --> CLK

    CLK -->|Impulse| SEQ[Step Counter<br/>ba.pulse_countup_loop<br/>0-31]
    SEQ --> NOTE[Note Lookup<br/>note_at_step]
    NOTE --> MIDI[MIDI Note<br/>root + offset]
    MIDI --> FREQ[ba.midikey2hz]

    CLK -->|Gate| GATE[Gate Signal]
    FREQ --> EKS[EKS Synth]
    GATE --> EKS

    EKS --> OUT[Stereo Audio Output]

    classDef uiStyle stroke:#f96,stroke-width:3px
    classDef clockStyle stroke:#4ecdc4,stroke-width:3px
    classDef noteStyle stroke:#95e1d3,stroke-width:3px
    classDef eksStyle stroke:#f38181,stroke-width:3px
    classDef outStyle stroke:#ffd93d,stroke-width:3px

    class UI,NR,RUN uiStyle
    class CLK,SEQ clockStyle
    class NOTE,MIDI,FREQ noteStyle
    class EKS,GATE eksStyle
    class OUT outStyle

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4.3 Modulation and Stereo Processing

graph LR
    subgraph "Modulation & Stereo Processing"
        LFO[LFO Oscillator<br/>Sine Wave] --> MOD[×]
        MR[mod_rate] -.-> LFO
        MD[mod_depth] --> MOD
        LFO --> MOD

        PA[pan_angle<br/>Base Position] --> ADD((+))
        MOD --> ADD
        ADD --> CLIP[Clip 0-1]

        ML[Mono L] --> SP[Stereo Panner]
        MR2[Mono R<br/>+width delay] --> SP
        CLIP --> SP

        SP --> OL[Out L]
        SP --> OR[Out R]
    end

    classDef lfoStyle stroke:#c77dff,stroke-width:3px
    classDef panStyle stroke:#4ecdc4,stroke-width:3px

    class LFO,MOD lfoStyle
    class SP panStyle

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5. Parameters

5.1 Sequencer

Parameter	Range	Default	Description
run_sequencer	{0,1}	0	Start/stop sequencer
note_rate	1-30 Hz	12 Hz	Clock speed (12 Hz = 16th notes at 180 BPM)
root_note	MIDI 36-72	64 (E3)	Base pitch

5.2 EKS Synthesis

Parameter	Range	Default	Description	Perceptual Effect
gain	0-10	1.0	Output level	Overall volume control
pick_angle	0-0.9	0.9	Pick sharpness (higher = brighter attack)	0.0-0.3: soft/round attack; 0.5-0.7: normal; 0.8-0.9: sharp/percussive
pick_position	0.02-0.5	0.13	Pluck position ($\beta$)	0.02-0.05: bridge (thin/bright); 0.13: guitar (balanced); 0.25-0.5: neck (mellow/hollow)
decaytime_T60	0-10 s	1.0 s	String decay time	0.1-0.5s: muted/staccato; 1-2s: normal guitar; 5-10s: sustained/drone
brightness	0-1	0.7	High-frequency content ($B$)	0.0-0.3: dull/warm; 0.4-0.6: balanced; 0.7-1.0: bright/metallic
dynamic_level	-60 to 0 dB	-10 dB	Nyquist-limit level ($L$)	-60 to -40: soft/muffled; -20 to -10: normal; 0: hard/bright attack

5.3 Spatial

Parameter	Range	Default	Description	Perceptual Effect
spatial_width	0-1	0.5	Stereo width	0: mono (centered); 0.5: moderate width; 1.0: wide stereo (immersive)
pan_angle	0-1	0.5	Base stereo position (0=L, 0.5=C, 1=R)	0: hard left; 0.5: center; 1.0: hard right
mod_rate	0.01-10 Hz	0.5 Hz	LFO speed for auto-panning	0.01-0.1: slow drift; 0.5-2: gentle movement; 5-10: fast tremolo-like
mod_depth	0-1	0.5	LFO modulation depth	0: static position; 0.5: moderate sweep; 1.0: full L-R sweep

5.4 Reverb

Parameter	Range	Default	Description	Perceptual Effect
reverb_mix	0-1	0.3	Dry/wet balance	0: completely dry; 0.3: subtle ambience; 0.5-0.7: hall reverb; 1.0: wet cathedral

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6. Installation and Usage

6.1 Faust Web IDE (Quick Start)

1. Navigate to https://faustide.grame.fr/
2. Copy contents of eks_guitar_sequencer_final.dsp
3. Click "Run" to compile
4. Enable run_sequencer checkbox
5. Adjust note_rate slider for tempo control

6.2 Local Compilation

# Compile to C++
faust eks_with_mod.dsp -o eks_with_mod.cpp

# Compile to various targets
faust2jaqt eks_with_mod.dsp        # JACK Qt application
faust2alsa eks_with_mod.dsp        # ALSA standalone
faust2vst eks_with_mod.dsp         # VST plugin
faust2daisy eks_with_mod.dsp       # Electrosmith Daisy
faust2bela eks_with_mod.dsp        # Bela platform

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7. Project Structure

EKS/
├── eks_guitar_sequencer_final.dsp   # Main implementation (recommended)
├── eks_with_mod.dsp                 # Version with modulation + reverb
├── eks_guitar.dsp                   # Original EKS (fixed syntax)
├── eks_sequencer_test.dsp           # Sequencer test version
├── bass_sequencer.dsp               # Bass sequencer example
├── experiments/                     # Development versions
└── README.md

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8. Sequencer Pattern (E Minor Pentatonic)

The 32-step pattern creates cascading arpeggios through E minor pentatonic scale:

Steps 0-15 (main pattern): E, G, B, E+octave, with ascending/descending motion

Steps 16-31 (variation): Introduces A and F# passing tones

Musical context: Inspired by progressive rock keyboard arpeggios (e.g., Rick Wakeman's style in Yes - "Roundabout").

Tempo mapping:

• 6-8 Hz: Slower, deliberate
• 12 Hz: 16th notes at 180 BPM (sweet spot)
• 20-24 Hz: Rapid, virtuosic runs

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9. References

Foundational Papers

• Karplus, K., & Strong, A. (1983). "Digital Synthesis of Plucked-String and Drum Timbres". Computer Music Journal, 7(2), 43-55.
• Smith, J. O. (2010). "Physical Audio Signal Processing for Virtual Musical Instruments and Audio Effects". CCRMA, Stanford University. https://ccrma.stanford.edu/~jos/pasp/

Digital Waveguide Theory

• Smith, J. O. (1992). "Physical Modeling Using Digital Waveguides". Computer Music Journal, 16(4), 74-91.
• Välimäki, V., et al. (2006). "Discrete-Time Modeling of Musical Instruments". Reports on Progress in Physics, 69(1), 1-78.

Faust Language

• Orlarey, Y., Fober, D., & Letz, S. (2009). "FAUST: an Efficient Functional Approach to DSP Programming". New Computational Paradigms for Computer Music, Editions Delatour, France.
• Faust Programming Language

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10. License

This project uses the STK-4.3 license (Synthesis Toolkit), following the original EKS implementation by Julius O. Smith III. The code is free to use for research and educational purposes.

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11. Author

George Redpath (Ziforge)

• GitHub: @Ziforge
• Based on: Julius O. Smith III's EKS algorithm

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12. Acknowledgments

• Julius O. Smith III - Original Extended Karplus-Strong algorithm and implementation
• CCRMA, Stanford University - Physical audio signal processing research
• Faust Community - DSP programming language and compiler
• STK Project - Synthesis Toolkit framework

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13. Citation

@misc{redpath2025eks,
  author = {Redpath, George},
  title = {Extended Karplus-Strong Guitar Synthesizer},
  year = {2025},
  publisher = {GitHub},
  url = {https://github.com/Ziforge/EKS},
  note = {Based on Smith, J.O. (2010) Physical Audio Signal Processing}
}