GALLERY NO. 011 / SOUND & SYMMETRY
An experiment you can hear

The drum that
splits a note.

A square drum has two different ways of vibrating that make exactly the same pitch. Stretch the square ever so slightly—and the single note becomes two. Strike it, listen, and watch symmetry break.

01 / The membrane● Perfect square
motion slowed for the eye
STRIKE POINT

Try this: tap anywhere on the drum. Then strike the exact center and see what disappears.

02 / Break the symmetry
1.000 : 1
tallerperfect squarewider
Which notes are we hearing?

Audio starts on your first strike.

03 / Two ways to ring

Identical notes, different shapes

Tap either pattern to hear and see that one mode alone. Tap it again to return to your chosen ensemble.

Pitch separation0.00 Hz
220 Hzmidpoint310 Hz

Exactly coincident. A quarter-turn of the square exchanges the patterns without changing their pitch.

04 / A pair becomes a pulse

Listen for the beating

When the pitches are almost—but not quite—equal, they interfere. The sound swells and fades.

Predicted loudness envelope, from just the twin modes
strike / 0 s0.8 s1.6 s2.4 s
Symmetry intact. With exactly matching pitches there is no beating. Try “Tiny stretch” to make a slow pulse appear.

The animation is slowed down; the audio uses the actual mode frequencies shown at left. Headphones make the beating easier to hear.

Why a single note divides in two

For the mathematically curious
↘ no approximations needed

Fix the area of a rectangular membrane at one square unit. If its width-to-height ratio is r, its sides are √r and 1/√r. A mode with m horizontal and n vertical half-waves has shape

umn(x,y) = sin(mπx / a) sin(nπy / b)

The boundary stays motionless. The mode's pitch is proportional to the square root of its Dirichlet Laplacian eigenvalue:

fmn = (c/2) √(m²/r + n²r)

Our sound synthesizer fixes f₁₁ at 160 Hz when the drum is square; changing the shape keeps its area, not its fundamental pitch, fixed.

For the two patterns displayed above, the squared frequencies differ by an exact amount:

f²12 − f²21 = (3c²/4)(r − 1/r)

Thus they coincide precisely when r = 1. More mysteriously, if you strike the center, both twins vanish from the sound: each has a nodal line there, so the strike excites neither one.

✧ The shape changes continuously, but an exact symmetry can disappear all at once.

Model: an ideal membrane with fixed edges, uniform tension and density; audio is a synthesized sum of its standing-wave modes, not a physical recording. Finite damping and modal weights are aesthetic approximations. Actual displacement is a superposition of sine-wave eigenfunctions and depends on the strike point. All calculations run locally in this file.