The transistor ladder

Some filters are tools; this one is a character actor. The four-stage transistor ladder — the Moog circuit — colors everything it touches: the resonance pushes back against the bass, the stages saturate into one another, and at the top of the resonance range it stops filtering and starts singing. tap.ladder~ is a zero-delay-feedback model of that circuit with a tanh saturator in every stage. This chapter is what each control trades, and what the measurements say the model actually delivers.

Companion material: the reference page and help patcher in the TapTools-Max package, and the verification notebook — every number below is an executed measurement. For the linear ladder — the cheap, polite Stilson/Smith model — see tap.fourpole~; this object is its nonlinear sibling.

What the model gets right

Two things separate a serious ladder model from a filter with a "Moog" label:

• Tuning that survives the top octaves. The classic digital shortcut goes audibly flat as the cutoff rises. This model is prewarped ZDF: measured self-oscillation lands at 1000.2 Hz for a 1 kHz cutoff (0.02 % error) — and, the part that's actually hard, 8009 Hz for an 8 kHz cutoff (0.11 %). You can play the resonance like an oscillator anywhere on the keyboard.
• Nonlinearity inside the loop, not bolted on. Each stage saturates, and the feedback fights the saturation the way the hardware does. That is where the compression, the "sag," and the bounded self-oscillation come from.

The whole filter: four stages, one loop. The red tap sets resonance, the amber paths are the comp bargain and the Xpander mode taps.

The knobs, one by one

frequency and the right inlet

Cutoff in Hz; a signal in the right inlet drives it with true per-sample resolution. Like everything here it rides the smooth ramp when set by message.

resonance — up to and past the edge

0 to 1.1. At 1.0 the loop gain reaches the oscillation threshold; above it the filter sings at the cutoff, amplitude-limited by the tanh stages (ping it to start — silence is a fixed point). Under the edge, resonance does the authentic ladder thing: it eats your passband (see comp).

drive — how hard to lean on the stages

Input gain (dB) into the saturating ladder. Measured THD on a 100 Hz tone: 0.5 % at 0 dB, 3.5 % at 8, 16.5 % at 16, 33 % at 24 — a smooth walk from "slightly thick" to "fuzz pedal's cousin." All odd harmonics, because tanh is symmetric — which is exactly why asym exists.

asym — the even harmonics of real hardware

Real transistors don't match; their operating points sit slightly off-center, and that asymmetry is where a hardware ladder's even-harmonic warmth lives. asym (0..1) models the mismatch. Measured on a driven tone: the 2nd harmonic sits at −156 dB (numerically absent) at asym 0 and rises to −18.6 dB relative to the fundamental at 0.6. One honest warning from the reference page: an asymmetric saturator can produce slight signal-dependent DC — follow with tap.dcblock~ if something downstream cares.

comp — the passband bargain

A real ladder trades passband level for resonance: the feedback subtracts from the input. Measured at resonance 0.9: the passband sits at −13.2 dB with comp 0 (the authentic droop) and at 0.0 dB with comp 1 (fully restored). Vintage behavior or modern behavior — your call, continuously.

mode — pole mixing, the Xpander trick

lp24, lp12, bp12, bp24, hp12, hp24: mixing the ladder's stage taps yields whole families of responses from the same four poles (the Oberheim Xpander's famous trick). Measured small-signal slopes: 23.4 dB/oct for lp24, 11.7 for lp12. The resonance and saturation behavior carries into every mode — a resonant bp24 through drive is a very different animal from tap.svf~'s clean bandpass.

oversample — paying for the saturation honestly

The tanh stages generate harmonics past Nyquist that fold back as inharmonic alias tones. Measured on a hard-driven 5 kHz tone: going from 1× to 4× oversampling drops the non-harmonic (alias) energy by 13.5 dB. The default 2× is the working compromise; use 4× when you drive high notes hard, 1× when you're filtering bass and counting CPU.

solver — fast or exact

The nonlinear loop can be solved with one predictor-corrector pass (fast, the default) or by Newton iteration to convergence (exact, circuit-simulation accuracy). They are audibly identical until drive and resonance are both pushed hard; exact is there for when you want to know, and for renders where CPU is free.

Recipes

• The bass patch: tap.vco~ saw stack (see the oscillator chapter's Moog recipe) → @mode lp24 @resonance 0.35 @drive 9 @asym 0.45 @comp 0.25. Keep comp low; the droop is the vintage glue.
• The acid line: @resonance 0.85 @drive 15, envelope into the frequency inlet, and let the resonance fight the saturation.
• The kick synthesizer: @resonance 1.05, ping it with a click, and ride frequency down fast — a self-oscillating ladder is a sine with attitude.

When it is not the right tool

• Transparent filtering. Every pole of this filter has an opinion. For surgical work use tap.svf~ (clean circuit) or tap.filter~.
• Morphing responses. The pole-mix modes switch; they don't glide. Continuous response morphing is tap.svf~'s morph.
• CPU-constrained patches that just need "4-pole lowpass." tap.fourpole~ is the linear ladder at a fraction of the cost — no saturation, no oversampling, no opinions.

Checkpoint

A prewarped ZDF four-stage ladder with tanh in every stage: self-oscillation in tune within 0.11 % even at 8 kHz, drive that walks THD from 0.5 % to 33 %, asym switching on the even harmonics of mismatched transistors, comp choosing between authentic passband droop and modern flatness, pole-mixed multimode outputs, and oversampling that measurably pays down the saturation's aliasing. The character filter — spend your tone budget here.