one reed · one small side tube · two stable answers
The Hole That Jumps a Twelfth
A clarinet's register key does not open a new end to the instrument. It opens a narrow side tube, and the reed-driven air column must decide whether to leave one self-sustained oscillation for another. Here the pressure is computed, not drawn to order.
Hold the note. Open the little hole.
The nonlinear run is ready. Sound starts only after you press a button.
The jump is close to three times the frequency. In equal temperament that is a perfect twelfth, one octave plus a fifth. The obvious explanation is a stopped cylinder: the reed acts approximately like a closed pressure end, the bell like an open pressure end, so only odd quarter-wave modes fit.
That gives the available resonances. It does not say which stable oscillation survives after the vent opens. In the run above, a loss-free open vent leaves the pressure near its first register. Put the published amplitude-dependent reflection law back at the vent and the same geometry can let go into the second.
K̂ is dimensionless and model-specific. It is not a measured constant of brass. At K̂ = 0 the boundary is an ideal open-hole reflection; increasing acoustic speed through the small hole makes the quadratic loss matter.
The spectrum is not the selection
Linear spectrum
Predicts where pressure resonances sit and how vent geometry shifts their tuning.
Nonlinear reed
Feeds energy back into the bore. More than one oscillation can remain stable.
Localized loss
Changes the open vent's reflection with amplitude and can destabilize the first regime.
The 2026 study compared player tests with this sparse waveguide. Removing localized nonlinear loss strongly suppresses second-register production and fails to reproduce the experimental transition pattern, but the published sweep does not fall to zero. After the paper's pre-opening first-register filter, its displayed F3 cases still contain 83 second-register outcomes at 2 mm and 370 at 3 mm, with mean open-to-closed frequency ratios of 3.011928 and 3.028135. The deterministic nine-point control below is narrower and returns zero loss-free outcomes. The model still under-represents second-register production compared with players, especially at low pressure, so this is not a complete clarinet or a player.
One vent, three incompatible jobs
Move the geometry, then run two nonlinear opening tests. The tuning figures update immediately from the paper's linear length-correction equation and a lossless transfer-matrix root. The opening outcomes come from the time-domain model.
- overblow at one control point
- not run E3 and F4, γ = 0.8, ζ = 0.3
- linear twelfth error
- 26 / 32 cents E3 / F4, smaller is nearer 3:1
- linear throat B-flat shift
- 113 cents target: 95 to 105 cents
The linear gauges are live. Run the geometry to ask the reed whether it changes register.
The study found reliable extreme-note transitions only for diameters strictly between 1.5 and 4.0 mm. Requiring the same vent to make throat B-flat narrowed the viable diameter to roughly 3 mm. When that requirement was relaxed, its reported optimum used a 2.0 mm diameter, 13 mm chimney, 110 mm from the mouthpiece. In the full model that design put throat B-flat 69 cents flat but left B4 only 10 cents sharp. Those last two values are published full-model results, not outputs of the linear gauges here.
The check
The page ports the authors' 44,100 samples/s delay-line synthesis. Each run lasts 2.5 s, opens the vent at 0.8 s, estimates frequency from positive zero crossings, checks RMS amplitude and zero-crossing periodicity, and assigns the second register within the paper's tolerance of 200 cents.
| vent diameter | K̂ = 0, second | K̂ = 0.2, second | runs per cell |
|---|---|---|---|
| computing the 54-run control sweep… | |||
The sweep is running after the first drawing frame.
Uncertainty, free choices and refusal paths
- The page uses the authors' sparse waveguide and their minimal-example geometry, not a measured clarinet waveform. Audio is the computed dimensionless input pressure.
- K̂ = 0.2 is the paper's model choice for sharp edges, moderate reed closing pressure and mouthpiece aperture. It is not independently measured here.
- The nine-point check is our deterministic sample of γ = 0.55, 0.8, 1.05 and ζ = 0.12, 0.25, 0.38 with fixed random seeds. It is not the paper's 3,000-point Latin hypercube and makes no probability estimate.
- The paper varies opening time by one first-mode period and adds 5% control noise. This page fixes the opening at 0.8 s and uses deterministic 5% noise so results can be reproduced.
- The model neglects reed mass and damping, bell radiation and series impedance at the register hole. Real tone holes, bell flare, reed compliance, air temperature and the player's vocal tract perturb the frequencies.
- The design bench refuses a time-domain test if the vent would lie less than 20 mm before the tube end. Refusals are printed instead of silently classified.
- The paper's model produces fewer second-register cases than players and overproduces some high-amplitude regimes. No outcome here is a claim about every clarinet or player.