Arnol'd lock

Lineage. The twist is the Lamb–Oseen swirl of the swirl page — Professor Jaffer’s extension of the comb to the vortex, whose rotation by a function of the radius alone is exact at any angle. The kick is a canonical transformation written from a generating function, the form the Chirikov map and the spark shear already use; the picture it draws is Arnol’d’s: the tongues of the circle map, where a driven rotation locks to a rational winding number and islands stand still in the stroboscopic frame.

What it does. Under Kessho a pressed note no longer feeds or stirs: it locks to the bass. The lowest held note is the reference; for every other voice the engine reads the interval $f/f_{\text{bass}}$ (the bend included) and takes its fractional part $\Omega_0$ — a fifth $3{:}2$ gives $1/2$, a major third $5{:}4$ gives $1/4$ — and each frame applies one iteration of $M = T \circ \text{Kick}$ about the voice’s drop, in the action-angle frame $\rho = r^2/2$, $\theta$:

$$\rho_1 = \rho + K\,W(\rho_1)\sin q(\theta - \varphi), \qquad \theta_1 = \theta + \tfrac{K}{q}\,W'(\rho_1)\cos q(\theta - \varphi),$$

the kick from the generating function $F_2 = \theta\rho_1 + \tfrac{K}{q}W(\rho_1)\cos q(\theta-\varphi)$ (a canonical transformation: $\det J = 1$), followed by the twist $T$ — the Lamb–Oseen rotation with its core at the ring, scaled so the ring turns exactly $\Omega_0$ turns per frame and windowed to zero by two and a half radii. $q$ is the tongue’s denominator: the nearest continued-fraction convergent $p/q$ of $\Omega_0$ with $2 \le q \le 6$ inside its tongue $|\Omega_0 - p/q| < 1/(2q^2)$, else six; an integer ratio is $q = 1$, a full turn. The press is $K$, as a fraction of $K_{\max} = 0.8/\max|W'|$ — the bound that keeps the kick a bijection. The window $W$ is one across the ring and falls to zero by $1.5\,\rho_{\text{drop}}$ and at the centre.

The lookup is one Newton. The shader needs $M^{-1}$: undo the twist (a rotation keeps $r$, so $\rho_1$ is the texel’s own), then solve $\theta$ from $\theta_1$ — one scalar Newton, five iterations — and take $\rho$ explicitly. The inverse pass (the soak’s pairs) looks $M$ up instead: one Newton on $\rho_1$. Exact, never sub-stepped.

Locked and beating. At an exact ratio the $q$ islands stand still frame after frame — the ink inside them librates, the islands do not move. Off the ratio by $\delta = \Omega_0 - p/q$ the kick’s frame precesses by $2\pi\delta$ a frame and the pattern turns with it: the beat, in bass periods per frame. A MIDI fifth is equal temperament’s $2^{7/12}$, two cents under $3{:}2$, and its two islands make one slow turn in six hundred frames; a tempered major third, fourteen cents over $5{:}4$, turns six times faster — the temperament’s beats drawn on the water. Bend the fifth ten cents sharp and the islands turn at $\delta = 0.0070$ turns a frame. Crossing a tongue boundary (a bend from the fifth down to the third) crossfades $K$ through zero over fifty milliseconds and switches $q$ at the zero, so the petals never pop.

The bass stirs. The reference voice has no interval to lock to; under the lock its press is the swirl, as before — a solo note under pressure still stirs.

Live, under Kessho: the bass at the lower left and a tempered fifth above it, pressed — its drop locks to the bass, and the two islands of q = 2 turn once in six hundred frames, the temperament's two cents drawn on the water. c = +2 is the just fifth: they stand still. i = 4 with c = −14 is the just major third, four petals; i = 4 at 0 beats six times faster. ◇ picks the mineral.open full size ↗

Ownership

• Channel pressure under Kessho’s default table (press_mode 3; the Arnol’d tongue lock choice in the settings): the press is the kick, the pitch against the bass the twist. The Arnol’d K control (CC) is the alternative, the larger of the two driving.
• Gesture route: sumi_add_arnold(x, y, radius, winding_omega, k, q, twist) — one iteration, with the twist switchable off (the bench’s red control: without it the kick alone makes no islands, the ink collecting on the kick’s nodal lines).

In the bench

--arnold-test: the pass against its closed form and its exact inverse, the $3{:}2$ and $5{:}4$ islands standing over six hundred iterations, the tempered fifth’s beat and the ten-cent precession, the twist’s red, the tongue crossing’s flicker; --soak arnold the four-part gate (the pairs through the inverse pass); --arnold-render <dir> the locked and the beating renders in Sumi and Kessho.