Chladni

Sprinkle sand on a plate and bow its edge: the sand gathers on the nodal lines of the plate’s standing wave. For a mode $W = \cos(k_x x)\cos(k_y y)$ those lines are a grid — the cell boundaries. midi-sink cannot gather ink — every operator here preserves area exactly, and Liouville’s theorem says an area-preserving map never changes density — but it can stretch, and the Chladni operator stretches the ink along the edges of the keys.

The cells are the eddies. The layout’s keys are circles — the ones the tablets draw — and every one of them is an eddy. A texel inside a key turns about the key’s centre by

$$\theta \,\big(4\rho^2(1-\rho^2)\big)^2,\qquad \rho = r/R,$$

zero at the centre, full at $\rho = 1/\sqrt 2$, zero again at the rim with zero slope, so the note’s drop rests where it fell and is wound from its edge while the water between the keys never moves. A rotation preserves $r$, so the map has $\det J = 1$ inside every key and is continuous across the touching rims: class exact, and the negative pass is the exact inverse. Layouts that draw no keys — the circle of fifths, the rolls, the theremin — take the largest circle at each note that touches no neighbour’s.

Two senses. Neighbouring keys counter-rotate on the layout’s own checkerboard; the balance control turns the odd keys’ sense from opposite through still to the same.

The inverse Chladni. A second mode sums every key’s ring into one divergence-free flow instead of turning each key alone. The keys may then grow past their circles and overlap, and the water between them is stirred by both neighbours — the burst’s class, area-preserving to first order, a pass at a time.

Two clusters, then a chord on the chromatic grid: every key winds its ink from the edge while the water between the keys rests. A is the stir, B the odd keys' sense, c the eddy's size within its key, t how long the plate is bowed.open full size ↗

Invariants

• Exact. Measured on the desktop harness after 150 stirred frames: the key centres and the corners between keys move less than a quarter texel while the rings move sixteen; a ring round a centre turns 1.45 rad and stretches 0.09, a ring round a corner does nothing.
• The eddies are the keys. The stir’s discs are the layout probe’s cells, same centres and radius — the geometry the tablets draw and the fingers touch.
• It bakes, and only bakes. The flow writes into the field while the stir is held, a pass whenever it carries two quanta of displacement, never an absolute. The dip is the undo.
• The ingress rule applies where a key is clipped by the canvas edge.

Ownership

• The stir (CC 106 on the desktop, Stir (A) in the settings) and the balance (CC 107, Balance (B)), unmapped in the core; Cell size, the eddy’s disc as a fraction of the key (capped at the key in the exact mode); the mode, Discs (exact) or Inverse Chladni. Under the Anod medium’s binding table the per-note bend plays the stir: its distance from centre sets the rate, its sign the sense — a bend down after a bend up is the inverse pass, so a vibrato stirs back and forth and comes home; the last bend written wins across voices, and a CC mapped to the stir overrides.
• sumi_add_chladni(psi, balance, s_x, x_0, s_y, y_0) as a gesture: one step of the two-wave cellular flow on a lattice of its own; a negative psi applies the exact inverse.

The cellular flow, kept as a gesture

The first Chladni was the Taylor–Green vortex, the cellular flow whose stream function is the plate’s mode shape itself,

$$\psi = \Psi\,\cos\big(k_x(x - x_0)\big)\cos\big(k_y(y - y_0)\big),$$

an exact solution of the Navier–Stokes equations: eddies in the cells, neighbours turning opposite ways, separatrices exactly on $\psi = 0$ — the nodal lines. The product is not separable, but $\cos u\cos v = \tfrac12[\cos(u - v) + \cos(u + v)]$ is a sum of two waves, each a function of one diagonal coordinate, and the flow of such a wave is a pure shear along the direction where that coordinate is constant. One step of the flow is two exact shears — the kick-drift splitting of a symplectic integrator — and its exact inverse is both shears negated in reversed order (a sign flip alone is not: crossed shears do not commute). That flow lives on as the gesture above; the layout’s keys replaced it as the stirred figure because the plate the player sees is made of circles, not of a lattice’s separatrices.

Lineage

Ernst Chladni’s figures (1787) are the picture; the exact rotation inside a disc is Rankine’s vortex with a smooth profile, the operator the vortex family already used; Taylor and Green’s cellular vortex (1937) is the two-wave gesture, split into two exact shears; and Aref and Ottino’s chaotic advection is why iterating either winds the ink.