Suzu — the synth

Suzu (鈴, the small bell) is the synth inside the sound engine, a source beside the sampler. It exists because the visual engine suggested it: midi-sink composes $\det J = 1$ maps on a canvas, and the same mathematics, one octave down, composes $\det J = 1$ maps on phase space. Each oscillator is a 2-D cell $(x, y)$ in which area preservation means amplitude stability by construction instead of ink incompressibility. Dissipation exists — filters resonate, notes decay — but it is declared, never accidental, and the one rule that is never broken is the canvas’s own: no element may create area it was not handed.

The chapters mirror the operator book. Where the canvas says area-preserving, the synth says amplitude-stable; where an operator declares exact or sub-stepped, a DSP element declares its class below; where every operator ships with its live demo, every voice ships with its sound profile and its lab page, the engine itself compiled to WebAssembly.

The DSP class table

Every element of Suzu declares one row of this table; its header comment, its test and its lab page name the row.

Class	det of the per-sample map	Members as shipped	Discipline
Symplectic	= 1 exactly	the cell’s leapfrog; the shears (cubic, fold; the Duffing cubic on the position; the bell’s brass shear); the lattice’s coupling kick; the Verlet chain’s and the bore’s leapfrog; the kicked rotor’s kick and re-base	amplitude confined by construction; the drift test is the gate
Dissipative (conformal)	< 1, declared, uniform	the filter’s $q$; the cell’s release (a T60); per-mode decay $\gamma_k$; the chain’s and the bore’s per-node damping; the hybrid’s one-zero loss and round-trip factor; the bridge cells’ decays; the radiation ports (a non-negative real part at every frequency); the valve’s $Q$; the reed’s stop at the lay	every T60 within 0.1 % of the declared
Drive	injection, not a map	the strike (the orbit set at $A$); the pluck added to the chain’s state; the Duffing drive; the bow’s seed on a resting mode	bounded per sample
Self-excited	⋛ 1, state-dependent, bounded by construction	the breath bow (an energy servo per mode); the winds’ exciters — the jet, the reed and the lips	the bow’s limit cycle gated; the winds’ power-limited port and the mouth-power ledger — the energy held ≤ ∫P·Q dt, by arithmetic
Lossless transport	= 1	the hybrid string’s ring with its Thiran allpass; the jet’s Hermite-interpolated delay	every coupling into a transport passive: a scattering junction balanced exactly per sample, gated at load and by the soak
Projected (a declared correction)	a projection after each control step	the chaotic modulator’s energy	measures a no-op (within $9\cdot10^{-8}$ of 1) and bounds by construction

Unconditional anti-dissipation — $\det > 1$ regardless of state — is banned; the self-excited class is admitted because its injection is servo-bounded or power-limited.

The voices

Kind	Voice	Chapter
the cell	a magic-circle oscillator with the shears and the filter — the sine that cannot drift	Cells
the modal lattice	up to sixteen coupled cells on placed partials — string, bar, bell, glass, plucked string — and the breath bow	The modal voice
the Verlet chain	masses on springs, a string’s body	Strings and chaos
the hybrid string	a lossless delay line into a bridge of cells, a scattering junction that balances	Strings and chaos
the Duffing cell	the magic circle with a hardening spring: the clang	Strings and chaos
the kicked rotor	Chirikov’s standard map as an oscillator, the mod wheel its K	Strings and chaos
the flute	an open-open bore blown by a jet; the overblow is the physics	Winds
the saxophone	a reed on a truncated cone, Benade’s mouthpiece	Winds
the trumpet	lips on a flared bore, CC 74 the embouchure	Winds

A chaotic modulator (a double pendulum at control rate) rides beside any of them, and the orbit trace lets any of them draw its own phase portrait into the water.

How MPE plays it

Through the normalizer, every dimension has one consumer:

dimension (MPE)	Suzu’s consumer
strike (velocity)	the kick across the modes by the preset’s profile — the plucked string’s pluck in closed form, the chain’s triangle, the cell’s orbit at $A$, the modulator’s energy
press (channel pressure)	the output gain $0.35 + 0.65\,p$; the Duffing drive’s amplitude; and, with the press blows, the mouth of the bow and the winds. The thesis’s “sustained small drive into damped modes” is not built: where there is breath, the bow is the press consumer
breath (CC 2; CC 11 as alias)	the bow’s target per mode; the winds’ mouth pressure
glide (pitch bend)	every cell’s $\varepsilon$ retuned per frame with the orbit re-based; the chain re-derived; the hybrid retuned; the bores re-cut
slide (CC 74)	the filter’s cutoff, ±3 octaves about the patch’s; the trumpet’s embouchure
swirl (poly pressure)	the lattice’s coupling, $\kappa$ += 0.5·swirl per block
mod wheel (CC 1)	the rotor’s K, up to 2.5
lift (note-off)	the release’s declared contraction; a released wind’s mouth goes to zero

The gates

Every claim above is a test in the headless suite, the desktop bench or the lab, and every one has a red control — the naive form that fails:

• Drift. The bare cell at $A = 0.5$ for ten minutes: +0.0001 dB, +0.000 cent. The naive simultaneous update ($\det = 1 + \varepsilon^2$) is past 0.1 dB within a second and +690 dB by the tenth minute.
• Glide. ±48 semitones over 2 s — 0.097 dB re-based (the gate 0.5).
• The energy ledger. The bell, 8 modes, every decay zero, ten minutes: −0.0026 dB; decays restored, every T60 within 0.1 % of the declared.
• Tuning. The oscillator within 0.000 cent and the filter’s ring within 0.008 across MIDI 21–108 at 44.1 k and 48 k; the lattice’s partials within 0.08 cent at any admitted coupling; the chain and the bores exact by construction; the hybrid within 0.144 cent.
• The bow’s limit cycle. From silence within 5 % in 4.9 τ, from twice the target in 1.7 τ, the two orbits agreeing to 0.00 %; zero breath → −66 dB in 2.2 s.
• The stability bound. $k\,dt^2 > 1$ rejected with its message; 1.05 bypassed goes non-finite within a second, 0.95 stays bounded.
• Loop passivity. The hybrid undamped for ten minutes: −0.018 dB; a junction gain of 1.05 bypassed goes non-finite in 31 s; the load-time probe rejects over 1.03.
• The mouth-power ledger. The jet’s stored energy ≤ 0.06 % of ∫P·Q over a phrase; the valves’ probe E ≤ 1.05·∫P·Q at 30 ms (the naive junction: $10^{300}$×); the ledger live in the lab.
• The overblow. The breath ramp jumps the octave within 7 cents, autonomous, on the desktop and in Chrome.
• The web gate. The wasm against a native build of the same source: bit for bit for the cell, Verlet, the hybrid, Duffing, the rotor and the sax; within −80 dB for the lattice, the flute and the trumpet (the math libraries’ last bits); the negative control red.
• The combined stress. The fifteen-channel storm with a heavy library layered under the chain and under the hybrid: 0 XRuns, 0 dropped, the visuals at or above 94 fps.

Where it runs

Suzu plays on every native platform — the desktop’s Sound section carries every knob of the current voice kind (Patches); the tablets offer five patches on their Sound page — and in the browser as the lab, deviceless, driven by an AudioWorklet. It costs CPU by design, as the thesis argued: sixteen cells with the filter and a shear take 3 % of the audio callback, sixteen lattices 3.5 %, ten 80-node chains 21 %.

Deferred, and said so

Physical stick-slip (Helmholtz) bowing — the servo bow is its honest predecessor; 2-D membranes and gongs (a 32×32 drumhead is ~50 M node updates a second per voice — a single-instance resonator when a budget comes, and the one place compute-shader audio becomes plausible); the brass shock itself (the brass is a declared shear at the bell); and user-editable patches beyond the knobs.