Two axes, two sine terms each, plotted against each other instead of against time. Each axis is axis(t) = a₁·sin(2π(f₁·t+p₁)) + a₂·sin(2π(f₂·t+p₂)) — the same harmonic-sum engine as the Additive Wave Sculptor, just two harmonics per axis instead of eight. A shared damping knob shrinks the amplitude over the length of the trace, turning a closed Lissajous loop into a spiralling harmonograph rosette.
A real harmonograph is a pendulum-driven drawing machine: two or more pendulums swing at slightly different rates, their motion combined through linkages to move a pen over paper, while friction slowly bleeds off their amplitude. Trace the pen's path and you get a rosette that starts wide and spirals inward as the pendulums lose energy. This instrument is that machine's mathematics without the hardware — each axis sums two sine terms (its own two-pendulum pair), and the damping knob applies the exponential decay e^(−damping·t) that friction would, over the length of the trace set by duration.
Set damping to zero and two clean integer-ratio frequencies (3:2, 5:4, 1:2) and you get a classic closed Lissajous figure instead of a harmonograph — the same equations, just without decay. Detune the second term on each axis very slightly away from the first (say 3 vs. 3.01) and, even with zero damping, the whole figure slowly precesses — that's the Animate button, drifting those second-term phases live.
Third instrument in the visual-synth family started by Palette Synth: same principle — a small set of periodic-sum coefficients driving a live picture — applied to a path in the plane instead of a color gradient or a single waveform. The trail itself is colored by Palette Synth's own cosine-gradient formula, reused rather than reinvented: it sweeps once through the gradient from the oldest to the newest point on the path, so a rosette's outer, older loops sit visibly in a different part of the gradient than its inner, newer ones instead of the whole trace reading as one flat color. The Audio-Reactive Reaction-Diffusion instrument and the Complex-Plane Fractal Mapper close out the family.