Models · Model 005 · Math art
Sound Plates
Play a tone, and the sand draws it.
In 1787 Ernst Chladni drew a violin bow along the edge of a brass plate sprinkled with sand. The plate sang, and the sand ran to the lines that stayed still, drawing a figure for each note. Russell rebuilt the experiment in NetLogo, on a speaker tuned to A = 432. Here it is in your browser. Pick a tone, press Play, and watch the sand find its lines.
01 · The model
Sand on a singing plate
The gold grains are sand on a plate seen from above. Choose a tone and the plate shakes in a pattern of its own: the sand hops off the places that move and comes to rest along the lines that stay still. Press Play to hear the tone. As long as it plays, the plate keeps shaking.
396 Hz · Round plate
The sand is finding the still lines of 396 Hz.
The tone
Nearest note at A = 432: G4, 49 cents sharp. The arrow keys step from note to note.
The plate
Clamped at its rim, like the cone of a loudspeaker.
The sound
The tone with soft overtones above it. The sand follows only the tone itself.
02 · Seven tones
The chakra plates
Russell's first run of sound plates uses seven tones that a popular chart pairs with the chakras, from the root to the crown. Each figure below is drawn in your browser by the same model, on the round plate. Press one to hear it.
The chart, at 432hz.world, gives each of the seven main chakras one of the solfeggio frequencies, from 396 Hz for the root to 963 Hz for the crown. None of them falls on a note of the A = 432 scale. The root's 396 Hz lies almost halfway between G4 and A♭4, and the nearest to a note, the crown's 963 Hz, sits 12 cents below B5. The plate doesn't mind. It answers only to pitch.
As the tones climb, the figures grow finer. The root shakes the plate most in a mode with five still lines through the center, and the crown in one with four lines and two rings. Each tone also shakes the plate a little in a neighboring mode or two, which bend the lines into the figure the sand draws. Only 89 cents separate the root from the sacral, less than a semitone, yet the sacral's plate has a different mode at its heart: three lines and a ring.
The numbers
| Tone | Hz | Nearest note at A = 432 | Wavenumber | Round plate, lines × circles | Square plate, modes |
|---|---|---|---|---|---|
| I · Root | 396 | G4, 49 cents sharp | 9.21 | 5 × 1 (100%), 0 × 3 (28%), 3 × 2 (−26%) | (1, 5)+ (100%), (1, 5)− (26%), (0, 5)+ (15%) |
| II · Sacral | 417 | A♭4, 39 cents sharp | 9.45 | 3 × 2 (100%), 6 × 1 (22%), 5 × 1 (−17%) | (2, 5)− (100%), (2, 5)+ (24%), (1, 5)+ (−9%) |
| III · Solar plexus | 528 | C5, 47 cents sharp | 10.63 | 4 × 2 (100%), 1 × 3 (−34%), 7 × 1 (33%) | (0, 6)− (100%) |
| IV · Heart | 639 | E5, 22 cents flat | 11.70 | 2 × 3 (100%), 0 × 4 (−30%), 8 × 1 (−5%) | (3, 6)− (100%), (3, 6)+ (30%), (4, 5)+ (−22%) |
| V · Throat | 741 | F♯5, 34 cents sharp | 12.60 | 5 × 2 (100%), 8 × 1 (24%), 3 × 3 (−21%) | (1, 7)+ (100%), (5, 5)+ (35%), (1, 7)− (18%) |
| VI · Third eye | 852 | A5, 24 cents flat | 13.51 | 6 × 2 (100%), 9 × 1 (−18%), 1 × 4 (−15%) | (3, 7)− (100%), (3, 7)+ (−19%), (5, 6)− (−1%) |
| VII · Crown | 963 | B5, 12 cents flat | 14.36 | 4 × 3 (100%), 10 × 1 (3%) | (1, 8)+ (100%), (4, 7)+ (35%), (1, 8)− (22%) |
Each tone shakes the plate in the mode nearest its own pitch at full strength, and in the next two in proportion to how near they are. A minus sign means a mode swings the other way. A round mode's circles count the rim. The square plate's modes are named as Chladni's are, by the waves across each side.
03 · How it works
A plate, a tone, and sand
The plate. A plate rings in shapes of its own, called modes. In each one, some parts swing up and down while others, along lines, stay still. The round plate here is clamped at its rim, like the cone of the loudspeaker in Russell's NetLogo model. Its modes have straight still lines through the center and still circles, the rim among them. The square plate is free at its edges, like the brass plates Chladni bowed, and its modes cross it in lattices.
The tuning. Notes follow equal temperament with the A above middle C at 432 Hz instead of the usual 440, so middle C is 256.87 Hz. The plate is tuned too. Its lowest mode sounds at 27 Hz, the A four octaves below 432. A plate's pitch rises with the square of how many waves fit across it, so a tone two octaves higher fits twice as many, and its figure is twice as fine.
The tone. A tone shakes the plate hardest in the mode whose pitch lies nearest its own, and a little in the next two, more the nearer they are. Between modes, the blend bends the still lines into curves.
The sand. The grains start evenly scattered. At each step every grain hops a random distance that grows with how hard the plate shakes under it, then slides downhill toward the nearest still line. Grains on a still line barely move, so the sand piles up there. For each of the seven chakra tones, after 600 steps at least 97 grains in 100 sit where the plate moves less than a tenth as much as it does where it moves most, though those places cover only about a quarter of the round plate.
The sound. The tone you hear comes from your browser at its exact frequency. Russell's NetLogo model could play only the nearest key of a keyboard tuned to A = 440, so playing the true tone was one of the extensions he listed for it.
- Plates
- Round, clamped at the rim, with 311 modes. Square, free at the edges, with 637.
- Tuning
- A = 432 Hz. The plate's lowest mode sounds at 27 Hz.
- Tone
- The nearest mode, with a little of the next two
- Sand
- 40,000 grains, or 24,000 on a phone. They settle in about 600 steps.
- Sound
- The exact frequency, made by your browser
- Code
- About 240 lines of JavaScript, running in this page
04 · What to try
Six experiments
Each button sets up the plate and starts the tone. Turn your sound on, or watch the sand alone.
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Climb the chakras
The seven tones in turn, from the root to the crown, seven seconds each. The figures grow finer as the pitch rises, from the root's five still lines to the crown's four lines and two rings.
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Less than a semitone
The root sounds at 396 Hz and the sacral at 417 Hz, 89 cents higher. Set it up, then press II · Sacral. The five lines give way to three lines and a ring.
-
Square it
The solar plexus tone, 528 Hz, on the square plate. It lands almost exactly on one of the plate's modes, so no other mode joins in, and the sand draws a clean lattice of diamonds.
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Glide an octave
Russell's notes on his NetLogo model suggest letting the tone glide. This one slides from 216 to 432 Hz over 30 seconds. On the way, the sand dissolves and redraws itself through six figures.
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Watch the shaking
The heart tone, 639 Hz, with the shaking lit up and the sand scattered fresh. The grains leave the bright places, where the plate moves most, and come to rest in the dark, on the still lines.
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Seven octaves up
At 3,456 Hz, the A seven octaves above the plate's lowest tone, the round plate rings almost purely in one mode, and the sand draws a target: four lines and six rings.
Sources
Reading and code
- The modelRussell Weaver, Cymatics Speaker (432 Hz), a NetLogo model, 2026. This page recodes it in JavaScript.
- Chladni's figuresErnst Florens Friedrich Chladni, Entdeckungen über die Theorie des Klanges (Discoveries in the theory of sound), Leipzig, 1787. The plates of sand figures at the back of the book are the first of their kind.
- The platesThomas D. Rossing, Chladni's Law for Vibrating Plates, American Journal of Physics 50(3): 271–274, 1982.
- The mathMilton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, chapter 9, 1964. The Bessel functions that shape the round plate's modes, and the backward recurrence that computes them.
- The tonesAppum Studios, Chakra Frequencies: The Complete 7 Chakra Hz Chart, 432hz.world, June 3, 2026.
- NetLogoUri Wilensky, NetLogo, Center for Connected Learning and Computer-Based Modeling, Northwestern University, 1999.
This is a picture of the physics, not a model of any particular speaker or plate. The modes are textbook approximations, and the sand follows a simple rule, chosen as in Russell's NetLogo model to carry the grains to the still lines. The code is plain JavaScript and runs in this page.
The seven chakra plates are prints from Russell's studio, alongside seven contour plates. See all the prints.