accordion — an air-powered physically-modeled free reed #

A single-file C engine (like pop/bell/c/) whose sound comes from a free reed driven into self-oscillation by bellows pressure — not an oscillator with an envelope on it. Below a threshold pressure the reed is silent; above it, airflow feeds the swing each cycle and the note grows out of turbulence noise. The attack breathes because the air takes time to build and the reed takes time to speak.

Zero dependencies beyond libm.

pop/accordion/
  c/accordion.c   the engine
  c/build.sh      cc -O2 -std=c11 -Wall -Wextra -o accordion accordion.c -lm
  out/            demo renders

The model #

Reed — a van-der-Pol-type nonlinear oscillator (the standard reduction of the Fletcher/Rossing free-reed treatment):

x'' = -w(p)^2 (x - x_off(p))  +  w·drive·( p·(1 - x²/X(p)²) - ζ₀ )·x'  + noise(p)

• The airflow damping term goes negative when p exceeds ζ₀ — energy flows into the swing and the oscillation grows exponentially from the turbulence seed until the amplitude-dependent factor saturates it. Growth rate scales with pressure: soft notes speak slower (measured ~145 ms to 90 % at --vel 0.3, ~125 ms at 0.8, on top of the bellows RC).
• The intrinsic loss ζ₀ is kept outside the amplitude factor so it is always dissipative — folding it into one van-der-Pol mu explodes at release, when mu < 0 and |x| > X flips the saturation into anti-damping.
• Saturation amplitude X(p) grows with pressure, so bellows dynamics are heard in level, not just in the flow factor: the tremor and the swell actually modulate loudness.
• x_off(p): static pressure deflection — pumping harder skews the pulse and feeds the even harmonics.
• w(p): reeds flatten under pressure — ~10 cents measured at full pumping (a ~7-cent programmed detent plus a real nonlinear-damping shift), which also turns the tremor into a faint coupled vibrato.

Flow gate (the buzz) — the radiated sound is the derivative of the volume flow through the slot, U = A(x)·√p. The aperture A has a kink at the frame plane and is asymmetric: above the plane the gap opens freely; below it the tongue sits inside the slot and the gap barely reopens. One real closure per cycle ⇒ dense odd+even comb with the fundamental on top (a symmetric |x| fold puts the octave above the fundamental — measured, wrong, fixed). Measured sustain spectra: 25–32 harmonics above −50 dB at A4, comb to Nyquist at A2; bass reeds carry pitch on the comb (A2 fundamental ~4 dB under h2, like the real thing). Output is leveled by 1/w0 so pitch doesn't change loudness.

Bellows — ONE pressure signal shared by every reed in the render:

• attack RC (air builds behind the pallets; pull opens slower than push),
• hand tremor: a drifting 4–6 Hz LFO plus a slow random-walk unsteadiness, depth = --tremor (measured ~12 % RMS level wobble at 0.8),
• shapes: push (firm, small sag), pull (gentle open, slight crescendo), swell (hairpin < > peaking ~55 % in), shake (fast ±strokes ~3.6 Hz — pressure dwells near zero at each turnaround so the reeds re-speak every stroke: measured 7.3 Hz throb, level dipping to ~0.2 of peak),
• release: the pallet closes, pressure bleeds off, and the residual air leaves as a little filtered-noise exhale.

Musette — --voices 2|3 reeds per note at ±--detune cents (2 = on + sharp, 3 = flat/on/sharp), each with small random spreads in drive, gate and flattening so they start and drift independently. A 12 ¢ pair at A4 measures a 2.8–3 Hz amplitude beat at ~27 % depth — the wet shimmer.

Body — four fixed 2-pole resonances (box thump 215 Hz, wooden mid 540, grille presence 1350, air 2900) over a 55 % dry path, DC blocker, grille-cloth rolloff at 6.5 kHz. Plus a key-click + valve-thump transient at note start and pressure-scaled breath noise throughout.

Chord — --chord "A3,C#4,E4" renders every note off the same bellows: one attack, one tremor, one release for the whole handful of reeds. That shared air is what makes it an accordion and not three reeds in a row.

Integration: semi-implicit Euler at 4× oversampling.

CLI #

./build.sh
./accordion --note A4 --dur 4 --out note.wav
./accordion --note D4 --voices 3 --detune 14 --bellows pull --out musette.wav
./accordion --chord "A3,C#4,E4" --bellows swell --dur 7 --out chord.wav
./accordion --note G4 --voices 2 --bellows shake --dur 5 --out shake.wav

Demos (out/) #

./accordion --note A4  --voices 1 --tremor 0.35 --dur 4 --out ../out/single-a4.wav
./accordion --note D4  --voices 3 --detune 14 --tremor 0.4 --bellows pull --dur 5 --out ../out/musette-d4.wav
./accordion --chord "A3,C#4,E4" --voices 2 --bellows swell --tremor 0.45 --dur 7 --out ../out/swell-chord-a.wav
./accordion --note G4  --voices 2 --bellows shake --dur 5 --out ../out/shake-g4.wav

Notes / honest limits #

• The reed is a reduced model (one bending mode + phenomenological flow gate), not a full aeroelastic simulation: torsional modes, reed-frame contact chatter and the two-reed push/pull switch of a real box are absent — push/pull here differ in gesture (attack, sag/crescendo), not in reed pair.
• Body resonances are a fixed EQ, not a measured accordion impulse response; swap the body_init table to chase a specific box.
• The flow-gate kink is not band-limited, so very high notes fold a little HF hash past Nyquist even at 4× reed oversampling (inaudible under the grille rolloff in practice, but it's there).
• note_to_freq and the WAV writer follow pop/bell/c/bell.c conventions.