Why does the sand only move at certain notes?

A plate responds strongly only at its resonant frequencies, when the note fits a natural way the plate can bend. Between them it barely vibrates, so the sand just shivers.

Why does sand collect on the lines and not on the moving parts?

The moving parts throw sand into the air, and it lands in a random spot nearby. The nodal lines don’t move, so sand that lands there stays. Give it a few seconds and nearly all of it has wandered onto the lines.

Are the frequencies realistic?

I scaled the pitches so they all fit on one ruler, but they keep the same order: more nodal lines, higher note. A real brass plate of about 20 to 30 cm has resonances from around a hundred hertz to several thousand.

What is the nut in the middle?

It’s the bolt the plate is clamped to, just as on Chladni’s plates. Because the centre is held still, every figure in the game has a nodal line passing through it.

Make Sand Sing

Play a note. Watch the sand draw it. Back to the game

How to play

Slide the brass pointer along the ruler to change the note the plate is played at, or drag the violin bow up and down (along the bottom edge on a phone) to keep the plate ringing. Most notes only make the sand shiver. When you hit one of the plate’s resonant frequencies, the plate rings, the pointer clicks into place, and the sand rushes into a pattern. Hold it there until the figure is sharp: it gets traced and pressed into your sketchbook, and a red notch marks the note on the ruler. There are 16 to find, from 104 Hz to 2,314 Hz. Fill the sketchbook and a round plate opens on the next page, with 16 more.

Tap the plate to sprinkle more sand, and swipe quickly across it to sweep sand away. Tap a red notch or a figure in your sketchbook to play it again, or tap an empty page for a hint. On a keyboard, the left and right arrows tune the plate (hold Shift for fine steps), holding Space draws the bow, S sprinkles sand and P switches plates.

Where the sand goes

A vibrating plate doesn’t move all over. It splits into patches that bend up and down in opposite directions, and between them are lines that hardly move at all: the nodal lines. Sand on a moving patch is thrown into the air and lands somewhere a little different each time, while sand that reaches a nodal line stays put. After a few seconds almost all of it sits on the lines, and you are looking at a drawing of the vibration. Higher notes divide the plate into more, smaller patches, so the figures get more complicated.

The square plate uses a classic approximation for a square plate with free edges. Each figure is two simple wave patterns added together, of the form cos(nπx)·cos(mπy) ± cos(mπx)·cos(nπy), and the pitch rises with n² + m². Real plates are messier. The figures are still recognisably the same family.

Round plates

On a round plate the nodal lines are straight diameters through the centre and circles around it. Chladni noticed a rule of thumb for them, now called Chladni’s law: on a flat round plate the pitch rises roughly with the square of (diameters + 2 × circles), so a ring raises the note more than a spoke does. The game’s round plate uses the shapes of a vibrating round membrane, built from Bessel functions: J_n(kr)·sin(nθ). A real plate’s figures differ in detail, but they are drawn from the same diameters and circles.

Chladni, Germain and the violin makers

The German physicist and musician Ernst Chladni published these figures in 1787 in Discoveries in the Theory of Sound. He clamped a brass plate at its centre, sprinkled sand on it and stroked its edge with a violin bow. He toured Europe demonstrating them, and in 1808 he showed them to Napoleon, who funded a prize from the French Academy of Sciences for a mathematical theory to explain them.

The problem was so hard that most mathematicians didn’t try, and in the end the only entrant was Sophie Germain, who had largely taught herself mathematics because women could not study at the École Polytechnique. Her first two entries were turned down, but on her third try, in 1816, she became the first woman to win a prize from the Paris Academy of Sciences. Her theory of elastic plates was incomplete, and later work by Kirchhoff and others finished it, but she laid the foundations of elasticity theory.

Instrument makers still do this. Violin and guitar makers sprinkle glitter or fine particles on a carved, unassembled top or back plate and vibrate it with a loudspeaker, then shave the wood until the figures appear at the right notes. The American violin maker and scientist Carleen Hutchins turned this into a method called free-plate tuning.

Questions people ask

Why does the sand only move at certain notes?: A plate responds strongly only at its resonant frequencies, when the note fits a natural way the plate can bend. Between them it barely vibrates, so the sand just shivers.
Why does sand collect on the lines and not on the moving parts?: The moving parts throw sand into the air, and it lands in a random spot nearby. The nodal lines don’t move, so sand that lands there stays. Give it a few seconds and nearly all of it has wandered onto the lines.
Are the frequencies realistic?: I scaled the pitches so they all fit on one ruler, but they keep the same order: more nodal lines, higher note. A real brass plate of about 20 to 30 cm has resonances from around a hundred hertz to several thousand.
What is the nut in the middle?: It’s the bolt the plate is clamped to, just as on Chladni’s plates. Because the centre is held still, every figure in the game has a nodal line passing through it.