# Harmonograph Simulator - Damped Pendulum Art, Lissajous Figures, Intervals

> Harmonograph simulator: 2–4 damped pendulums draw colourful curves; set ratios, detune and damping, export PNG/SVG. Also Lissajous figures and interval tones.

- URL: https://qilingtools.com/en/physics/harmonograph
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

A harmonograph uses a few slowly decaying pendulums to move a pen and paper, drawing intricate curves that spiral inwards. Here you set the frequency, phase, amplitude and damping of 2 to 4 pendulums, pick a ratio such as 1:1, 2:3, 3:4 or an octave and add a little detune, then watch the pen draw a colour-graded figure and export it as a high-resolution PNG or editable SVG. The Lissajous tab shows x = sin(at+δ), y = sin(bt) in green oscilloscope glow plus a rotating 3D version, and the intervals tab plays two tones so you can see how just and equal-tempered tuning turn into a still or slowly turning pattern.

## How to use

1. Choose lateral or rotary, the number of pendulums, and click a ratio or Random pretty one
2. Adjust detune and each pendulum's frequency, phase, amplitude and damping; the drawing restarts
3. Set drawing time, speed, line width, start and end hue, or switch to the dark glowing background
4. Click Export PNG or Export SVG to save the result
5. On the Lissajous and Intervals tabs change ratio, phase or tuning and press play to hear both notes

## Features

- 2–4 damped pendulums in lateral or rotary configurations
- Progressive stroke-by-stroke drawing with a hue gradient over time, or finish instantly
- Ratio presets, detune slider and random generator with a reproducible seed
- Export a 2000×2000 PNG or a vector SVG
- Oscilloscope-style Lissajous figures with closure and edge-touch counts, plus a rotating 3D version
- Web Audio two-tone player comparing just and equal-tempered intervals in cents and beat rate

## FAQ

### Why are the patterns tidier when the frequency ratio is a simple whole-number ratio?

If the frequencies in the two directions are in a small whole-number ratio p:q, the pen returns to the same point after one common period and the curve closes with only a few loops. If the ratio is slightly off, each loop is shifted a little in phase from the last, so the figure slowly rotates, and damping shrinks it at the same time. Together they give the layered rosettes that spiral towards the centre.

### What is the difference between a lateral and a rotary harmonograph?

In a lateral harmonograph each pendulum swings in one direction only and drives either x or y, which gives Lissajous-like curves. In a rotary harmonograph the pendulums move in ellipses or circles and affect both x and y, so adding several circular motions produces spirograph-like petals, and reversing one rotation changes the pattern again.

### How do you read a frequency ratio from a Lissajous figure?

For x = sin(at+δ), y = sin(bt) with a general phase, the number of times the curve touches the left or right edge compared with the top or bottom edge equals a:b. Before frequency counters, engineers fed an unknown and a known signal to an oscilloscope's X and Y inputs and compared frequencies this way. For special values of δ the curve retraces itself, for example a parabola at 1:2.

### Why does an equal-tempered fifth make the figure turn slowly?

A just fifth has the ratio 3:2, while 12-tone equal temperament sets it to 2^(7/12) ≈ 1.4983, about 2 cents flat. The ratio is no longer exact, so the figure turns at Δ = |2·f_upper − 3·f_lower|, which is exactly the beat rate you hear; with a 220 Hz lower note it is about 0.74 Hz. The equal-tempered major third is about 14 cents sharp and turns much faster.

### What is the difference between the SVG and PNG exports?

The PNG is a 2000×2000 pixel bitmap that keeps every sample and the blended colours, good for sharing or printing. The SVG is vector paths coloured in segments along time and thinned to about 20,000 points, so the file stays small and can be edited in Illustrator or Inkscape or sent to a pen plotter or laser engraver.

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