A citation, an identity, and a sign: ondes.h

Three classes — triode, detector, voice — and three things worth recording about how they got here. One stage turned out to need no design decisions at all. One approximation turned out to be an exact identity. And one sign error made a distortion knob run backwards.

The circuit is Najnudel, Hélie, Roze & Boutin, "Simulation of an ondes Martenot circuit", IEEE/ACM TASLP 28, 2651–2660, 2020, modelling instrument No. 169 as five port-Hamiltonian stages. This file is not that: their full solve runs at 768 kHz and their plugin costs 85 % of a laptop core. What it takes from them is their own published reductions plus their published component values, and the header says which is which.

The tube is a citation, not a design

The plan framed the valve stage as a choice: a published grid-conduction curve, or the tanh family with an asymmetry bias voiced by ear. It is neither, and finding that out took nothing more than reading the paper properly.

The paper names a tube model — the enhanced Norman Koren model (Koren, Glass Audio 8(5), 1996, with Cohen & Hélie's grid-current branch, AES 129, 2010) — writes out its three equations, and publishes parameter sets in Table II fitted to the actual valves in ondes No. 169, together with each stage's supply voltage, cathode resistor and plate load.

So there was nothing to voice. k_6f5, k_6c5, k_2a3, k_op_demod, k_op_preamp and k_op_power are Table II transcribed, and the header says they are the citation.

A stage is then the static solution of ipc(vpc, vgc) = (Vbias − Vk − vpc)/Rp on the load line, with cathode bias Vk = Rk·Ipc found at the quiescent point. That is a memoryless nonlinearity in exactly the DAFx-07 sense, which matters for a practical reason: tabulating it is not an approximation of the model, it is the model. The table is rebuilt on a tube or operating-point change and read with linear interpolation, so the audio path costs a lookup rather than a root find.

The published points bias sanely — the 6C5 demodulator lands at Vk 2.70 V, Vp 86.5 V, Ip 2.70 mA, gain 4.86 — which is its own small confirmation that the transcription is right.

The sign that made the drive knob run backwards

The stage must invert, as a real common-cathode stage does, and this is load-bearing rather than cosmetic. The valve's asymmetry acts on whichever side of the waveform reaches its grid. An early cut normalized the output by the signed small-signal gain, which quietly un-inverted the stage, so the curve's lopsidedness landed on the wrong half of the waveform.

The symptom was unambiguous once measured: turning drive up reduced total harmonic content. A distortion control that gets cleaner as you push it is not a subtle bug, but it is only visible in a sweep — at any single setting the object sounded like a valve.

Two changes fixed it. The curve is now the true (inverting) plate swing, and normalization is by the gain's magnitude. And voice::core applies the demodulator's own grid-leak inversion explicitly — a growing envelope drives that grid toward cutoff — so the two inversions put the demodulator's plate in phase with the envelope while the curve has meanwhile acted on the underside. drive now sweeps harmonic content 0.221 → 0.344, monotonically.

The gain-staging lesson from fuzz.h was applied here from the start rather than learned again: each stage is normalized by its own small-signal gain, so drive changes the distortion and not the level.

The detector is an identity, not a simplification

The plan's instruction for this stage was "synthesize the difference tone directly as a sinusoid", and catching that as a mistake is the most valuable thing this build did.

The paper's 0.03 % distortion figure and its licence to replace oscillators with a sinewave generator apply to the oscillators. The demodulator is not a mixer handing you a difference tone; it is an envelope detector, and the envelope of cos(Φ) + cos(Φ − φ) is 2|cos(φ/2)|, whose Fourier series puts H2 at −14.0 dB, H3 at −21.3 dB and H4 at −26.4 dB. Synthesizing a sinusoid would have discarded the instrument's largest single source of harmonics before any of the modelled stages ran.

What replaces the carrier is better than a simplification. For amplitudes 1 and depth, the envelope is exactly

sqrt(1 + depth² + 2·depth·cos(2π f t))

so the 80 kHz carrier drops out of the arithmetic rather than being approximated away. Running the published RC detector on that closed form — instant attack through the diode, 200 µs decay through R4·C21 — reproduces a full heterodyne-plus-diode-plus-RC simulation to within 0.10 dB on every harmonic at every pitch tried (ondes.ipynb §2).

There is one systematic difference, and it is worth knowing it is systematic rather than noise: the closed form sits a uniform 3.0–3.2 % high, because a follower chasing real carrier half-cycles never quite reaches the peak between them. On a synthesizer with a level control, that is a constant.

The detector's characteristic pitch dependence comes along free, out of the same 200 µs: H2 runs −14.0 dB at A2 to −19.3 dB at A6, and the level falls 2.0 dB across those five octaves.

And a bonus nobody planned: because the closed form is parameterized by the two oscillator amplitudes, oscillator balance becomes a physical timbre control. depth is a real mismatch between two real oscillators, not an invented knob.

Three measurements that lied, and what they were doing

All three were committed to a notebook or a header before being caught.

Too few periods. The first measurement of the detector's harmonics at low pitch used a window holding about 2.75 periods of the fundamental. Spectral leakage at that resolution dominated everything, and it produced a confident, wrong claim in the header: "−9.8 dB at A2, level falls 9.7 dB". Redone with 60 cycles, the real answer is −14.0 dB and 2.0 dB. Both numbers were in a shipped header before the recheck.

Probing where the answer is exactly zero. The aliasing scenario probed half-integer harmonics of a tone that was exactly periodic in the analysis window. Those bins are analytically zero, so it measured −281 dB and passed triumphantly. Fixed by computing the actual fold frequencies for a tone at 2637 Hz — deliberately not a submultiple of 48 kHz — and skipping folds that land near real harmonics. This is the same family of error fuzz.h records under "choosing a tone that divides the sample rate", committed again in a different disguise.

Stopping the sweep at the first plateau. The header initially claimed "4× is the knee, then flat". The notebook's own more careful run — settled state, 131072-point Hann — showed 8× continuing to improve in the top octave. Corrected to "never worse", with the full table in the header, the test comment and the notebook.

The evidence that closed an open question in fuzz.h

fuzz.h measured its oversampling sequence going the wrong way — 4× worse than 2× — and had left an untested hypothesis behind: that the culprit is imaging, since zero-stuffing by N leaves N−1 images for one filter to suppress, and residual images entering a nonlinearity intermodulate into products that are not harmonics of the input.

This file runs the same 8th-order Butterworth chain around a comparably hard nonlinearity, and its sequence never reverses:

The difference between the two files is exactly the hypothesis: this object is a source. Nothing is zero-stuffed on the way up — the detector simply runs fast — so there are no images at all.

That was evidence, not proof — the nonlinearities differ too, and one confounded comparison does not settle a question. But it was the first evidence either way, and it pointed somewhere specific enough to act on.

Acting on it settled it. fuzz.h now cascades one 2× stage per doubling instead of zero-stuffing by N once, each stage filtering at a corner that never tightens however deep the cascade goes. Its reversal is gone — worst step-up past 2× is a ratio of 1.017 — and its 4× and 8× improved by two to four orders of magnitude, for about 5 % more CPU. This file needed no change, having no upsampler to fix.

Worth naming the shape of it, because it is not the usual one: the evidence that resolved a two-wave-old open question in one file came from building a different file that happened to differ in exactly the right variable. It was not designed as an experiment. It was noticed, written down in both headers as evidence rather than proof, and left where the next person would trip over it.

A wrapper test that found a kernel bug

tap.ondes~'s Min-level test asserts something a patcher would otherwise file as a bug report: with the key at rest, the object is exactly silent. It failed.

voice::set_smooth_ms set the voice's own ramps but never forwarded to touche::key, which keeps its own slew. So a key sitting at zero with @smooth 0 still sounded for 20 ms after every parameter touch.

This is the two-layer split working the way it is supposed to. The kernel suite tests DSP promises; the wrapper suite tests what a patcher will actually observe, and those are not the same set. The fix landed in the kernel with its own scenario, not in the wrapper.

What this file will not do

The real instrument has switchable waveform registers. Their filter shapes are in none of the sources obtained. Adding them from imagination is the one thing this file is careful not to do, and the omission is stated in the header, the object description and the reference page rather than left as a gap someone might charitably fill later.

Two controls are choices — where the intensity key sits (keyplacement) and the coupling transformer's winding sense (polarity) — because the paper's five stages do not settle either. Both are labelled as choices, and both were measured to confirm they are audible ones: about 0.09 and 0.12 of total harmonic content respectively.

Checkpoint

A stage that required no design because the paper published the model and its fitted parameters. A detector that is exact rather than approximate, and cheaper than the thing it replaces. One sign error that inverted the meaning of a distortion knob and was invisible at any single setting. Three measurements that lied in three different ways, all recorded. The evidence that closed fuzz.h's oversampler question, from a file that happened to differ in exactly the right variable and was not built as an experiment. And a wrapper test that found a kernel bug, which is the split doing its job.