Cells

The magic circle

One cell is one rotation in phase space, and its orbit is the circle you see. Per sample,

$$x \mathrel{-}= \varepsilon\,y,\qquad y \mathrel{+}= \varepsilon\,x \ \text{(the updated } x\text{)}, \qquad \varepsilon = 2\sin\!\left(\pi f / f_s\right).$$

This is the Gordon–Smith oscillator (1985; the “magic circle” of the synthesis folklore). The two half-steps are each a shear — $\det = 1$ each, $\det = 1$ together — and the quantity $x^2 + y^2 - \varepsilon x y$ is conserved exactly, so the orbit keeps its size for ever: the oscillator is exact in pitch (within 0.000 cent across MIDI 21–108 at 44.1 k and 48 k, because $\varepsilon$ is the exact rotation’s coefficient, not a Taylor series) and incapable of drift. A strike sets the orbit — $x = A$, $y = \varepsilon A / 2$, the amplitude the velocity’s (0.25 at 127) — and the release is a declared contraction per sub-step, a T60 the patch names. The output is $x$ through an attack ramp (a mixer stage, not a map on the cell) and the press’s level.

The MPE correction. A glide retunes $\varepsilon$ every frame; a retune alone would change the orbit’s size (the conserved quantity has $\varepsilon$ in it), so the cell is re-based onto the new $\varepsilon$ with its amplitude kept: ±48 semitones over two seconds costs 0.097 dB against the naive form’s 0.63 for a one-block jump.

The cell and the shears, live. The strip under the orbit tracks x² + y² − εxy, the conserved quantity; with the decay at zero it does not move. The lab mode's red control is the naive update — x and y from the same old values — which multiplies the orbit's area by 1 + ε² every step: at A3 the orbit grows twenty-thousandfold a second, the worklet mutes full scale and restarts the engine. That is why the engine ships the other update.open the lab page ↗

The shears

A shear of one coordinate by a function of the other is still symplectic: $x \mathrel{-}= \varepsilon\,(y + s(y))$ with a cubic $s$ bends the circle and the bend is the harmonics the spectrum shows arriving. A symplectic cubic is a gentle waveshaper — the third harmonic 33 dB under the fundamental at a shear of 0.3, 23 dB at full gain — and it detunes the cell; the cell calibrates the shear’s detune at start-up and compensates it, so every note stays within a fifth of a cent at full gain (the calibration within 0.19 cent for the cubic, 0.02 for the fold). The two shear kinds are the cubic and the fold; the hard timbres belong to the Duffing cell and the kicked rotor. The shears run in a 2× section; the fold’s sharper corners would want a 4× budget with a half-band filter, an open item carried rather than spent.

The filter

The state-variable filter is the thematic form — the Chamberlin core at twice the rate, $q = 2 - 1.9\cdot$resonance — a declared dissipation: its $q$ is the one place in the cell where $\det < 1$, and it is the same everywhere. CC 74 scales the cutoff by $2^{(t - 0.5)\cdot 6}$, three octaves either way about the patch’s. The exact tuning of the Chamberlin core holds for the undamped ring only; with damping the ring sits 2.3–2.5 cents sharp at C8 at $Q = 50$, so the coefficient is solved per block for the damped ringing frequency in double precision — the ring within 0.008 cent across the keyboard.

Denormals, and the rest

Sixty-four released voices’ tails stay flat to the end: this Apple CPU shows no denormal cliff with flush-to-zero off, and the x86 boxes have theirs to show. Sixteen cells with the filter and a shear take 3 % of the audio callback (0.09 ms worst of 2.67 at 128 frames and 48 k).

The profile

A magic-circle cell per voice: a pure sine at the same amplitude on every note — flat to a tenth of a decibel from A0 to C8 — with no harmonics (the faint second and third are the measurement’s own leakage). Every voice’s profile →