Crystalline Cascade (Quantum Flutter) — User Guide

Create shimmering, crystalline reverb tails using Poisson-process stochastic impulses, reverse-exponential envelopes, and triple-layer mixing for complex, organic textures.

Author: Shai Cohen Affiliation: Department of Music, Bar-Ilan University, Israel Version: 0.1 (2025) License: MIT License Repo: https://github.com/ShaiCohen-ops/Praat-plugin_AudioTools

Contents:

What this does Quick start Stochastic Process Theory Parameters & Presets Audio Examples Applications

What this does

This script creates quantum flutter reverb — a sophisticated multi-layer effect combining stochastic (random) impulse generation via Poisson processes, reverse-exponential amplitude envelopes, complex frequency modulation, and triple-layer mixing. The result: shimmering, crystalline, organic reverb tails that sound like cascading ice crystals, quantum particle decay, or complex acoustic reflections in impossible spaces. Unlike traditional reverb (simple exponential decay) or algorithmic delay (regular echoes), quantum flutter creates dense, random-yet-structured textures with evolving spectral content. The name reflects the quantum-mechanical inspiration: randomness with statistical regularity, like radioactive decay or photon emission.

Key Features:

Poisson Process — Statistically random impulse timing (like radioactive decay)
Reverse Exponential Growth — Impulses grow from silence, peak, then decay
Complex FM Modulation — Nested sinusoidal modulation for shimmer
Triple-Layer Mixing — Original + attenuated + convolved layers
Stereo-Aware — Independent left/right processing with different parameters
4 Presets — Subtle, Medium, Heavy, Extreme flutter densities

What is a Poisson Process? Named after French mathematician Siméon Denis Poisson (1837). Models events occurring randomly over time with constant average rate but unpredictable exact timing. Examples: radioactive decay (atoms disintegrate randomly but average rate constant), photon arrivals at detector, phone calls to call center, customer arrivals at store, rain drops hitting surface. Key property: memoryless — past events don't predict future (unlike periodic patterns). In audio: Poisson impulses create natural-sounding randomness — not periodic (boring) or clustered (unnatural), but organically scattered with statistical regularity. This is how nature generates "random" events, now applied to reverb tail construction.

Technical Implementation: (1) Extend input with silent tail (duration parameter) to allow reverb space, (2) Split stereo to L/R channels (process independently), (3) Create Poisson process: random point pattern with specified density (avg impulses per second), (4) Convert Poisson points to pulse train Sound, (5) Apply reverse-exponential envelope: base^((t-t_start)/(t_end-t_start)-1) — grows from near-zero to peak then decays, (6) Apply nested FM modulation: amplitude × (1 + depth × cos(2πft + nested_sin)), (7) Convolve modulated impulse train with input, (8) Mix three layers: (a) original × 1.0, (b) original × attenuation (0.7), (c) convolved × mix (0.35), (9) Peak normalize final output. Key insight: Poisson randomness + exponential growth + FM = complex, evolving texture impossible with simple delay/reverb algorithms. Triple mixing creates depth and richness.

Quick start

In Praat, select exactly one Sound object (mono or stereo).
Run script… → Crystalline_Cascade.praat.
Choose preset: Subtle (light), Medium (balanced), Heavy (dense), Extreme (very dense).
Or select Custom and adjust parameters manually.
Click OK — script processes (10-30 seconds depending on length/density).

Quick tip: Start with Medium Flutter preset for balanced quantum reverb (800 impulses/sec density, 2s tail). Works beautifully on: vocals (ethereal shimmer), synth pads (lush ambience), percussion (crystalline decay), acoustic instruments (impossible space). Stereo files processed with independent L/R parameters for width. Processing moderate speed (convolution + Poisson generation). Result auto-plays. Experiment with presets before diving into custom parameters.

Important: Processing adds silent tail to input — output longer than original (input_duration + tail_duration). Very high Poisson density (>2000) can create CPU-intensive processing and near-continuous noise rather than discrete flutter. Extreme settings may sound harsh — balance density with modulation depth. Stereo processing uses different Poisson parameters for L/R (slightly different densities create width). Mono input produces mono output; stereo input produces stereo with decorrelated channels. Scale_peak parameter prevents clipping but may reduce loudness — increase if output too quiet.

Stochastic Process Theory

⚛️ Quantum Inspiration

Quantum mechanics: Particles exhibit random behavior with statistical regularity

Radioactive decay: Atoms decay randomly, half-life predictable but exact timing unknown

Photon emission: Light quanta emitted stochastically from atoms

Audio parallel: "Quantum flutter" = statistically random impulses like particle events

Poisson Process Fundamentals

Mathematical Definition

Poisson process: Random point process where events occur independently with constant average rate λ (lambda)

Probability of k events in time interval t: P(k events in time t) = (λt)^k × e^(-λt) / k! where: λ = average rate (events per unit time) t = time interval k = number of events e = Euler's number ≈ 2.71828 Example: λ = 800 events/second Average 800 impulses per second But exact timing random Spacing between events exponentially distributed

Key Properties

1. Memoryless (Markov property):

Probability of next event independent of previous events
No "memory" of history
Unlike periodic patterns (strongly dependent on past)

2. Stationary:

Rate λ constant over time
Statistical properties don't change
Average density same throughout duration

3. Independent increments:

Events in non-overlapping intervals independent
Knowing events in [0,1] tells nothing about events in [1,2]

4. Exponentially distributed inter-event times:

Time between consecutive events follows exponential distribution
Most gaps short, occasional long gaps
Never perfectly regular, never strongly clustered

Audio Application: Poisson Impulse Trains

Why Poisson for Reverb?

Method	Character	Problems
Regular spacing	Periodic echoes	Comb filtering, metallic, unnatural
Pure random	Unpredictable	Clusters and gaps, inconsistent density
Poisson process	Natural randomness	None — optimal for organic texture

Perceptual advantage:

Sounds like complex natural reflections (room acoustics)
No artificial periodicity or patterns
Statistically regular but locally random
Models actual acoustic scattering in rough surfaces

🎲 Poisson in Nature & Science

Physics: Radioactive decay, cosmic ray arrivals, Brownian motion

Biology: Neuron firing patterns, bacterial colony growth, mutation events

Astronomy: Photon detection, star distribution in galaxies

Engineering: Network packet arrivals, traffic flow, queue theory

Audio: Natural sound textures (rain, gravel, granular synthesis)

Reverse Exponential Envelope

Formula Breakdown

Envelope applied to impulse train:

amplitude(t) = base^((t - t_min)/(t_max - t_min) - 1) where: base = exponential_base parameter (default 120) t = current time t_min = start time (0) t_max = end time (tail_duration) Normalized time: τ = (t - t_min)/(t_max - t_min) τ ranges from 0 to 1 Amplitude = base^(τ - 1) At t = t_min (τ = 0): amplitude = base^(-1) = 1/base ≈ 0.0083 At t = t_max (τ = 1): amplitude = base^0 = 1.0

Reverse Exponential Behavior

Traditional exponential decay: e^(-t) — starts loud, decays to silence

Reverse exponential growth: base^(t-1) — starts quiet, grows to peak, then decays

Amplitude over time (base = 120, normalized time 0→1): τ=0.0: 120^(-1.0) = 0.0083 ← very quiet τ=0.2: 120^(-0.8) = 0.0245 τ=0.4: 120^(-0.6) = 0.0722 τ=0.6: 120^(-0.4) = 0.2128 τ=0.8: 120^(-0.2) = 0.6272 τ=1.0: 120^(0.0) = 1.0000 ← peak Then combined with sound duration creates decay: Early: impulses quiet (reverse growth building) Peak: impulses loud (maximum around 2/3 through) Late: impulses fade (natural decay beyond peak)

Why "reverse" creates appealing texture:

Builds from silence — gentle onset, no jarring start
Peak in middle — energy concentrated, not immediate
Natural decay — still present but sounds organic
Mimics acoustic resonance build-up in physical spaces

Effect of Base Parameter

Base	Growth Rate	Character
50-80	Slow growth	Gradual build, extended texture
100-120	Moderate growth	Balanced build, musical
140-160	Fast growth	Quick onset, concentrated peak
>180	Very fast growth	Abrupt, almost gate-like

Complex FM Modulation

Full Modulation Formula

final_amplitude = base_amplitude × modulation_factor modulation_factor = 1 + depth × cos(2πft + nested_modulation) nested_modulation = 10 × sin(2πt × 3) where: depth = modulation_depth (default 0.6) f = modulation_frequency (default 60 Hz) t = time nested_modulation adds complexity Expanded: modulation = 1 + 0.6×cos(2π×60×t + 10×sin(2π×t×3)) Result: amplitude oscillates around 1.0 Range: [1-depth, 1+depth] = [0.4, 1.6] Creates shimmer/flutter effect

Modulation Components

1. Carrier frequency (60 Hz):

Primary modulation rate — 60 oscillations per second
Creates rapid amplitude flutter
Human perception: shimmering, crystalline quality

2. Nested modulation (3 Hz):

Slow variation of phase — 3 cycles per second
Creates "wobble" in flutter pattern
Adds organic complexity, prevents static shimmer

3. Modulation depth (0.6):

Controls intensity of modulation
0.6 = amplitude varies ±60% around baseline
Higher = more dramatic flutter, lower = subtler

Combined effect:

Fast shimmer (60 Hz) with slow evolution (3 Hz)
Never static — constantly changing character
Sounds alive, organic, complex
Similar to chorus/flanger but applied to impulse envelope

Triple-Layer Mixing Architecture

Layer Structure

LAYER 1: Original (dry) ├─ Full amplitude (1.0×) └─ Unprocessed input sound LAYER 2: Attenuated original (body) ├─ Reduced amplitude (0.7× default) └─ Same sound, provides body/fullness LAYER 3: Convolved (wet) ├─ Poisson impulse convolution ├─ Reverse-exponential envelope ├─ FM modulation ├─ Mixed at convolution_mix (0.35× default) └─ Creates reverb tail/flutter FINAL MIX: Layer1 + Layer2 + Layer3 └─ Peak normalize to scale_peak

Why Triple-Layer?

Layer 1 (Original):

Preserves clarity and definition
Maintains transients and attack
Keeps source recognizable

Layer 2 (Attenuated):

Adds body and fullness
Creates smooth blend between dry and wet
Prevents thin sound (common in heavy reverb)

Layer 3 (Convolved):

Provides shimmer and space
Creates quantum flutter effect
Adds movement and complexity

Combined result:

Depth without muddiness
Space without loss of definition
Complex texture without overwhelming source

Stereo Processing

Independent Channel Parameters

Left channel (from parameters):

Poisson density = poisson_density parameter
Pulse width = pulse_width parameter
Pulse period = pulse_period parameter
All modulation parameters as specified

Right channel (hardcoded variants):

Poisson density = 850 (vs left's variable)
Pulse width = 0.075 (vs left's variable)
Pulse period = 1250 (vs left's variable)
Exponential base = 115 (vs left's 120)
Modulation depth = 0.55 (vs left's 0.6)
Modulation frequency = 65 Hz (vs left's 60 Hz)
Nested modulation phase offset (different)

Why Different L/R Parameters?

Decorrelation:

Different Poisson realizations → independent random patterns
Different modulation frequencies → phase drift between channels
Creates stereo width without simple L/R delay

Natural stereo imaging:

Real spaces have asymmetric reflections
L/R decorrelation mimics acoustic reality
Wider, more spacious image than mono→stereo copy

Shimmer complexity:

L/R flutter patterns interfere → beating/panning effects
Perceived movement in stereo field
More interesting than static stereo reverb

Algorithmic Reverb Context: Traditional reverb algorithms: Schroeder (1960s) used comb + allpass filters, Moorer (1979) added early reflections, plate/spring reverb emulations model physical systems. Modern algorithmic reverb: convolution (impulse response based), FDN (Feedback Delay Network), waveguide mesh. Quantum flutter represents alternative: stochastic impulse generation + complex modulation + multi-layer mixing. Inspiration from: granular synthesis (Roads, Truax), particle systems (computer graphics), quantum mechanics (Schrödinger, Heisenberg uncertainty). This script bridges physics, statistics, and audio processing — using mathematical models of natural randomness to create reverb impossible in physical spaces.

Parameters & Presets

Preset Options

Preset	Tail	Density	Character
Custom	User	User	Full manual control
Subtle Flutter	1.5s	500/s	Light shimmer, transparent
Medium Flutter	2.0s	800/s	Balanced, musical (default)
Heavy Flutter	2.8s	1100/s	Dense, lush ambience
Extreme Flutter	4.0s	1500/s	Very dense, experimental

Core Parameters (Custom Mode)

Parameter	Type	Default	Description
Preset	option	Custom	Choose preset or custom
tail_duration_seconds	positive	2	Reverb tail length
poisson_density	positive	800	Average impulses per second
pulse_width	positive	0.08	Impulse duration (fraction)
pulse_period	positive	1200	Internal pulse generation period
exponential_base	positive	120	Reverse-exponential growth rate
modulation_depth	positive	0.6	FM modulation intensity
modulation_frequency	positive	60	Flutter rate (Hz)
convolution_mix	positive	0.35	Wet/dry mix for convolved layer
layer2_amplitude	positive	0.7	Attenuated original level
scale_peak	positive	0.88	Peak normalization level
play_after_processing	boolean	yes	Auto-play result

Parameter Details

tail_duration_seconds

Range: 0.5-10.0 seconds

Default: 2.0 seconds

Controls:

Length of added reverb tail
Output = input_duration + tail_duration
Longer = more spacious, ambient
Shorter = tighter, controlled

poisson_density

Range: 100-3000 (practical: 300-1500)

Default: 800 impulses/second

Controls: Average number of random impulses per second

Density effects:

300-500: Sparse, discrete impulses audible
600-1000: Moderate density, smooth flutter
1200-1500: Dense, almost continuous shimmer
>2000: Very dense, may sound like noise

pulse_width

Range: 0.01-0.5

Default: 0.08 (8% of period)

Controls: Duration of each impulse relative to pulse_period

Effect:

Smaller = sharper clicks, brighter
Larger = smoother pulses, warmer