FM-1 firmware · oscillator study

CHUA, a synthesizer made of one chaotic circuit

α – · β –x – y – z –

Two capacitors, an inductor, a resistor and one nonlinear part. That is enough for the state to spiral around one equilibrium, get thrown to the other, and never repeat. Each voice of the FM-1 firmware runs its own copy of this circuit; the knobs below are the same knobs, mapped through the same tables.

DOUBLE SCROLLboth equilibria, chaotic

Chaosknob 1 · α 85

Shapeknob 2 · β 24.4

Outputknob 3 · mix X

Releaseknob 4 40

Play with the mouse or the computer keys A W S E D F T G Y H U J K O L P. Drag the scope to turn it.

The equations

Three state variables, one bent line

Made dimensionless, the circuit is three coupled equations. x is the voltage over C1, y the voltage over C2, z the current through the inductor. All the chaos comes from g(x), the Chua diode's current: a negative resistance that pumps energy in near zero.

dx/dt = α (y − x − g(x))

dy/dt = x − y + z

dz/dt = −β y

g(x) slope −8/7 for |x| < 1, −5/7 to |x| = 3, +2 beyond (the op-amps saturate)

pitch: the solver advances h = f_note · T_rotation / 44 100 per sample

Fig. 1 · the circuit

C2 and L ring like any LC tank. R couples them to C1, and the diode N_R across C1 is the only part that is not linear. Without it the circuit would just decay.

Fig. 2 · the diode

Five straight segments. The middle ones slope downward: negative resistance, energy in. The outer ones slope up where the op-amps saturate, which keeps every setting bounded. The dot is the circuit's x right now.

Fig. 3 · the route to chaos, at the current Shape

Every peak of x the circuit reaches, for each position of the Chaos knob. One dot per column is a clean cycle; two is period doubling; a smear is chaos. Where the dots jump to both signs the double scroll begins, and past the crisis the saturated circuit settles on one large loop. The amber line is the knob.

Fig. 4 · what you hear

x switches between the two scrolls: the deep, noisy part. y is the spiral itself, bright and closest to a pitch. z is the inductor's slow, round swing. Output blends them.

On the FM-1

The same circuit, eight times, in fixed point

Each voice integrates its own circuit with the midpoint method in Q20 fixed point, at most four substeps per sample. The regime landmarks, rotation times and loudness per setting were measured offline; this page uses the same numbers. The screen draws the newest voice's trajectory, like the scope above.

Control	On the FM-1
KNOB 1	Chaos: cycle, period 2, spiral, double scroll, outer
KNOB 2	Shape: β from 10 to 34
KNOB 3	Output: X, X+Y, Y, Y+Z, Z
KNOB 4	Release
SELECT	Attack
ALGORITHM	View: 3D, X-Y, X-Z, Y-Z, time
PRESETS	The eight presets above
SAVE	Everything into flash