Tool · Coupling
Sidechain Simulator
The Kuramoto coupling strength κ is not a metaphor — in music production it has a direct analogue: sidechain compression. Slide κ up and hear the bass phase-lock to the kick.
Paused
An 8-bar kick + bass loop at 120 BPM. Loops continuously while playing.
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κ = 0 — Drift
No coupling. The kick and the bass play at full levels simultaneously — they fight for the same low-frequency space. Muddy, undifferentiated. Each oscillator drifts on its own phase.
What this is demonstrating
In the Pulse as Coupling appendix, the Kuramoto model describes how coupled oscillators transition from independent drift to collective phase-lock above a critical coupling strength κc. Below κc, each oscillator runs on its own trajectory. Above it, they synchronize.
Sidechain compression is the music-production rendering of that same knob. The compressor on the bass is triggered by the kick — every time the kick hits, the bass is briefly compressed. The result is that the bass’s rhythm is forced into phase-lock with the kick’s rhythm. Two oscillators that were drifting independently are coupled through the compressor’s gain-reduction signal.
As you slide κ from 0 to 1 above, you are hearing the Kuramoto coupling strength rendered directly. At 0 there is no coupling — the two voices fight. At 1 the bass fully ducks to every kick — they are locked. The intermediate values are the approach to critical coupling.
Note. The audio loops on this page are demonstration placeholders synthesized programmatically. The production version will replace them with music Alex actually makes.