Bell Curve Envelope — User Guide

A dual-read time-colour transform shaped by a Gaussian-style bell envelope, with nine presets ranging from broad resonances to narrow bright or dark grains.

Author: Shai Cohen Affiliation: Department of Music, Bar-Ilan University, Israel Version: 0.5 (2026) License: MIT License Repo: https://github.com/ShaiCohen-ops/Praat-plugin_AudioTools

Contents:

What this does What is a bell curve?Quick start Presets Controls Dual-read colour transform Processing pipeline Channels and level Visualization Historical / compositional context Further reading

What this does

Bell Curve Envelope is not simply an amplitude-envelope generator. It first creates a new colour from two differently time-scaled reads of the pristine source, subtracts those reads, and only then shapes the result with a symmetric Gaussian-style envelope.

colour(n) = source(round(n / L)) - source(round(n × H))\noutput(t) = colour(t) × exp(-((t - centre) / r)^2)

For a stationary tone, the first read tends toward approximately f/L while the second tends toward approximately H×f. Their subtraction therefore creates a time-scaling-derived spectral colour rather than a conventional low-pass/high-pass filter.

What is a bell curve?

A Gaussian or bell-shaped curve is a smooth symmetric function that reaches its maximum at a centre point and falls away on both sides. In this script the shape is borrowed as a time envelope:

g(t) = exp(-((t - μ) / r)^2)\nμ = duration × Bell_center\nr = duration / Bell_width

Higher Bell_width values produce a narrower event because they make r smaller. Bell_center moves the maximum through the file from 0 (start) to 1 (end).

This is not a normalized probability distribution. The script's internal legacy variable is called sigma, but mathematically it is the e^-1 radius r, not the standard deviation σ of a normal-distribution PDF. The separate raised-cosine edge fade is also part of the final effective envelope.

Quick start

Select exactly one Sound.
Run Bell_curve_envelope.praat.
Choose a preset or Custom.
Use Bell_width for narrow/wide shaping and Bell_center to place the event in time.
Use the low/high factors for colour; values farther from 1 create stronger time-scaling contrast.
Run and inspect the four-panel visualization.

Presets

Preset	L	H	Bell width	Centre
Custom	1.1	1.1	4	0.5
Narrow Bell	1.1	1.1	6	0.5
Wide Bell	1.1	1.1	2	0.5
Low Freq Emphasis	1.5	1.0	4	0.5
High Freq Emphasis	1.0	1.5	4	0.5
Bright Grain	1.0	1.3	8	0.5
Dark Grain	1.3	1.0	8	0.5
Metallic Bell	1.2	1.2	5	0.5
Soft Resonance	1.05	1.05	2.5	0.5

Controls

Control	Default	Meaning
Low_freq_factor	1.1	Source read index `round(col/L)`; a stationary tone tends toward roughly `f/L`.
High_freq_factor	1.1	Source read index `round(col×H)`; a stationary tone tends toward roughly `Hf`.
Bell_width	4	Inverse width control: higher = narrower.
Bell_center	0.5	Normalized centre position, 0…1.
Edge_fade_ms	5 ms	Raised-cosine closure at file edges and at any branch that reaches the source end early.
Scale_peak	0.99	Target peak applied to every non-silent result.
Draw_visualization	On	Draws read geometry, envelope, spectrum and final waveform.
Play_result	On	Plays the result.

Dual-read colour transform

The two source reads are deliberately nearest-neighbour sample-index reads. They are not anti-aliased resamplers. This gives the tool a rougher, more digital colour and preserves the established sound of the processor.

If L < 1, the low branch can exhaust the source early.
If H > 1, the high branch can exhaust the source early.
When a branch ends before the file, the script closes only that branch with the same short raised-cosine logic used at the file edges.
If L = H = 1, the two pristine reads cancel exactly and the output is silence.

Names such as "Low Freq Emphasis" and "High Freq Emphasis" describe the stationary-tone consequence of the read law. They do not mean that the script performs conventional low/high shelving or band filtering.

Processing pipeline

Copy the original Sound; all channel rows are processed with the same index law.
Read the pristine source twice using round(col/L) and round(col×H).
Subtract the second read from the first.
Apply the Gaussian-style bell envelope.
Apply short raised-cosine fades at the file edges.
If the result is non-silent, target-normalize to Scale_peak.

Channels and level

Channel count: preserved; the Formula operates row-by-row.
Duration: preserved.
Sample rate: preserved.
Randomness: none.
Normalization: target peak, not a ceiling. Every non-silent output is scaled to Scale_peak.

Output name: <source>_bell_<preset>.

Visualization

A — Dual-read geometry: maps output time/sample position back to the two source-read positions.
B — Effective bell envelope: Gaussian law plus actual raised-cosine edge closure.
C — Measured spectrum: pristine source against the dual-read colour stage before the bell envelope.
D — Measured waveform: source and final output on one shared amplitude scale.

Historical / compositional context

The familiar bell-shaped Gaussian curve comes from the history of probability and measurement error; it became strongly associated with Carl Friedrich Gauss, although earlier work by Abraham de Moivre also contributed to the normal distribution. In this tool that mathematical shape is repurposed as a compositional envelope, not as a statistical model.

Musically, the bell shape creates a bounded event with a clear centre and smooth shoulders. With wide settings it behaves like a long resonance; with narrow settings it approaches a grain-like gesture. The important compositional distinction is that the envelope shapes a pre-transformed dual-read colour, so timbre and amplitude evolve together.