Harmonograph
Make harmonograph drawings: the looping, slowly shrinking lines a pen draws when swinging pendulums move it and the paper. Set up to four pendulums, each swinging side to side or in a circle, with its own frequency, phase, swing and damping, or start from one of 14 presets or a random design from a seed number. Watch it draw, then download the whole pen path as one continuous line in an SVG at real size for a pen plotter, a 300 DPI PNG, or a PDF on US Letter or A4. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). The SVG is the whole pen path as one unbroken line at the size you typed, for a pen plotter.
Design
Pendulums
Line
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. The designs are worked out from the pendulum formula by our own code. You may use what you make here, including commercially, with no credit needed.
How to make a harmonograph drawing
- Pick a preset, or choose Random design and type a seed (the same seed always gives the same design).
- Edit the pendulums: what each one swings, its swing, frequency, phase and damping.
- Set the drawing time: how many seconds of swing the pen records.
- Pick the colour and line width, the size and the paper.
- Download the PDF and print at 100% (Actual size), or the one-line SVG for a plotter.
What you can use it for
- Pen plotter and generative art
- Physics lessons: damped oscillation and resonance
- Music lessons: intervals (octave, fifth, fourth) as drawings
- Posters, cards and wall art
- Laser engraving and vinyl cutting
- Planning a real harmonograph build: try settings first
The 14 presets
| Preset | Pendulums | Frequencies (Hz) | Damping (per second) | Drawing time |
|---|---|---|---|---|
| Damped circle (decaying spiral, 1:1) | 2 | x 1, y 1 | 0.02, 0.02 | 70 s |
| Unison drift (1.000 : 1.005 Hz) | 2 | x 1, y 1.005 | 0.02, 0.02 | 70 s |
| Octave (2:1) with drift | 2 | x 1.003, y 0.5 | 0.018, 0.018 | 80 s |
| Fifth (3:2) with drift | 2 | x 1.5, y 1.002 | 0.02, 0.02 | 70 s |
| Fourth (4:3) with drift | 2 | x 2.002, y 1.5 | 0.025, 0.025 | 60 s |
| Major third (5:4) with drift | 2 | x 2.5, y 2.003 | 0.03, 0.03 | 50 s |
| Three pendulums (2 on x), near 1:1 | 3 | x 1, x 3.005, y 1.002 | 0.02, 0.04, 0.02 | 70 s |
| Four pendulums, near unison | 4 | x 1, x 1.004, y 1.001, y 0.998 | 0.015, 0.025, 0.015, 0.025 | 80 s |
| Four pendulums, fifth (3:2) | 4 | x 1.5, x 1.502, y 1.001, y 1 | 0.02, 0.03, 0.02, 0.03 | 70 s |
| Rotary + lateral, near 1:1 | 3 | rot 1, x 1.004, y 0.998 | 0.02, 0.02, 0.02 | 70 s |
| Rotary turning against, near 1:1 | 3 | rot -1, x 1.003, y 1.001 | 0.02, 0.02, 0.02 | 70 s |
| Rotary 3:1 | 3 | rot 1, x 3.003, y 3 | 0.02, 0.03, 0.03 | 60 s |
| Octave, slow decay | 2 | x 1.0015, y 0.5 | 0.008, 0.008 | 160 s |
| Fifth, quick decay | 2 | x 1.503, y 1 | 0.05, 0.045 | 30 s |
Interval names give the frequency ratio: octave 2:1, fifth 3:2, fourth 4:3, major third 5:4. The damped circle is exact: our tests check that its radius is A e^(−d t) at every point.
Damping
| Damping d | Swing left after 30 s | After 60 s | After 120 s |
|---|---|---|---|
| 0.005 per second | 86 % | 74 % | 55 % |
| 0.01 per second | 74 % | 55 % | 30 % |
| 0.02 per second | 55 % | 30 % | 9 % |
| 0.04 per second | 30 % | 9 % | 1 % |
Swing left = e^(−d t), the same formula the drawing uses.
Pendulum types
| Type | What it adds |
|---|---|
| Lateral on x | A sin(2π f t + p) e^(−d t) to x (side to side) |
| Lateral on y | The same to y (up and down) |
| Rotary | A sin(2π f t + p) e^(−d t) to x and A cos(2π f t + p) e^(−d t) to y: a circle that shrinks; a negative frequency turns the other way |
What prints
| Download | What is in it |
|---|---|
| The page at real size, vector: the pen path and a size bar | |
| PNG 300 DPI | The same page as a picture |
| SVG (one line) | Only the pen path, one unbroken line at the size you typed (longest side), for a pen plotter |
Every page has a 1 in (or 2 cm) bar at the bottom to check the print is at 100%. For a cutter that needs DXF, use SVG to DXF.
Which tool to use
| You want | Use |
|---|---|
| Damped pendulum drawings | Harmonograph (this page) |
| Undamped sine-wave figures (a:b ratio) | Lissajous Curve |
| Rolling-wheel curves | Spirograph Generator |
| A photo turned into thread lines on nails | String Art Generator |
| A photo turned into one spiral line | Spiral Art Generator |
Different machines: the Spirograph Generator rolls a wheel in a ring, the String Art Generator plans a thread path from a photo, and the Spiral Art Generator draws a photo as one spiral line.
More free tools
Curves, patterns and generative art?
Drawing and print tools that run privately in your browser, with no sign-up and no watermark.
Frequently asked questions
What is a harmonograph?
A drawing machine from the 19th century. Pendulums swing a pen and a table, and as their swing dies away the pen draws loops that slowly shrink towards the centre. Lateral pendulums swing in a straight line; a rotary one, on a gimbal, swings in a circle or ellipse.
Who invented the harmonograph?
Harmonographs appeared in the mid-19th century. Hugh Blackburn, a mathematics professor at the University of Glasgow, is commonly named as the inventor, and the machines were most popular in the 1890s.
What is the formula?
Each lateral pendulum adds A sin(2π f t + p) e^(−d t) to x or to y: A is the swing, f the frequency in swings per second, p the phase and d the damping. A rotary pendulum adds the sine to x and the cosine to y, so it moves in a circle. The pen is at the sum of all the pendulums.
What does damping do?
It makes the swing die away: after t seconds a pendulum swings e^(−d t) as far as at the start. With d = 0.01, it is down to 55 % after a minute. Our tests check that every pendulum peaks at exactly A e^(−d t).
Why are the frequencies slightly off whole ratios?
A real harmonograph is never perfectly tuned. A small drift, like 2.00 and 2.01 swings per second, makes the figure turn slowly as it shrinks, which gives the woven look. With exact ratios the loops lie on top of each other.
What is the seed?
A number that picks a random design. The same seed always gives the same pendulums and the same SVG, byte for byte, so you can share a seed and get the same drawing back.
How is this different from a spirograph?
A spirograph draws a wheel rolling inside a ring, so the curve repeats forever at the same size. A harmonograph adds damped swings, so the figure turns and shrinks inwards and never repeats exactly.
Can I draw it on a pen plotter?
Yes. The SVG download is the curve as one unbroken path (one start point, then straight segments), at the real size you typed, so the pen goes down once and lifts once. For a cutter or CNC program that wants DXF, convert the SVG with our SVG to DXF tool.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. The drawing is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.