Sculpting the Spectrum

A sawtooth wave is the block of marble: every harmonic is in there, the n-th at 1/n amplitude, a comb of partials stretching to the top of hearing. A filter is the chisel. Subtractive synthesis is exactly what the name says — you start with everything and carve away what you don’t want. Below, the carving happens in front of you: the ghost comb is the raw saw, the bold curve is the filter, and the solid comb is what survives. Drag directly on the plot and press ▶ hear it — the sound you hear is the literal buffer behind the solid curve.

The ghosted comb is the input: a sawtooth at A2 · 110 Hz, straight from Quiver’s bandlimited Vco. The bold curve is the filter’s magnitude response — not a textbook sketch but the exact discrete response of Quiver’s Svf, evaluated from the same math that runs in Svf::tick. The cutoff knob sits at CV 0.50; the module maps that knob exponentially, \( f_c = 20 \cdot 1000^{\text{cv}} \) Hz, so CV 0 is 20 Hz, CV 0.5 is ~632 Hz, and CV 1 is 20 kHz — every hundredth of knob travel is the same musical distance, about a ninth of an octave. That is why the dot glides evenly across the log axis instead of spending 95% of its travel above 1 kHz. Resonance is at 0.20, which sets the damping \( k = 2 - 2\cdot\text{res} \) — the sharpness of the peak at the cutoff. The solid comb is the output: the same saw pushed sample-by-sample through the filter. Watch it as you drag — every harmonic lands exactly where the ghost comb meets the response curve, because the output is the input times the response.

The slim strip under the spectrum replays the same story in time: three cycles of the raw saw (ghosted) with the filtered output drawn over it. A sawtooth’s razor edge is nothing but high harmonics in phase, so as the lowpass takes them away the corners visibly melt — the ramp survives, the snap rounds off. And the ◠ sweep button performs the most famous gesture in synthesis for you: the cutoff rides from closed to open and back over a couple of seconds, each position leaving a fading ghost of the response curve behind — a long-exposure photograph of a filter sweep. Your own cutoff setting is untouched and the plot returns to it when the sweep lands.

Things to try

1. Sweep the cutoff down in LP mode, slowly, from 1.00 toward 0. The harmonics don’t fade together — they disappear one at a time, from the top down, and the sound darkens while the pitch never moves. That ordered demolition is subtractive synthesis.
2. Isolate a single harmonic: switch to BP, push res to 0.95+, and drag the cutoff dot onto the third partial (330 Hz for A2). One tooth of the comb survives at full height; the saw becomes a near-sine. A filter this sharp is a harmonic selector.
3. Find the self-oscillation: set res to 1.00. The damping floors at \( k \approx 0 \) and the peak explodes off the top of the plot — the filter rings at the cutoff frequency even between harmonics, adding a pitch of its own. Press ▶ hear it and drag the cutoff: you’re playing the filter. (The Rust module soft-clips its two integrator states, so this near-lossless resonator sustains a bounded whistle instead of diverging.)
4. Notch = LP + HP: flip between LP, HP, and Notch at the same cutoff. The notch curve is the other two summed — and in Svf::tick it literally is: notch = low + high. Everything cancels only where the two responses overlap in antiphase, right at the cutoff.
5. Feel the knob law: scrub the cutoff CV 0.25 → 0.50 → 0.75. Each equal step slides the curve an equal distance on the log axis — equal octaves per knob-degree. A linear-in-Hz knob would cram all the musically useful range into its first few degrees.
6. Remove the fundamental: in HP mode, sweep the cutoff up past 110 Hz. The lowest partial vanishes first, yet the ear still hears the same pitch — the surviving harmonics imply the missing fundamental.
7. Watch the corners melt: in LP mode, drag the cutoff from 0.9 down to about 0.3 while watching the time strip. The saw’s edges are made of high harmonics, so they soften first; by the time only a few partials survive, the razor has become a wave. Flip to HP and the opposite happens — the body drains away and only the snap of the edge remains.
8. Take a long exposure: press ◠ sweep with res around 0.6. The trail of fading curves is the whole family of responses one knob can draw — the resonant bump gliding along the log axis at constant width, which is exactly why a filter sweep sounds like a vowel morph and not a volume change. (With reduced motion enabled, five static snapshots appear instead.)

What you just saw

The state-variable filter computes all four responses at once from one two-integrator core. With \( g \) the integrator gain and \( k \) the damping, the transfer functions are

\[ H_{lp}(s) = \frac{g^2}{D(s)}, \qquad H_{bp}(s) = \frac{g s}{D(s)}, \qquad H_{hp}(s) = \frac{s^2}{D(s)}, \]

all sharing the denominator

\[ D(s) = s^2 + k g s + g^2, \qquad k = \frac{1}{Q} = 2 - 2\cdot\text{res}. \]

The notch is the sum of the extremes, \( H_{notch} = (g^2 + s^2)/D \), and the three primary outputs obey the identity \( H_{lp} + k H_{bp} + H_{hp} = 1 \) — the core splits the input, it never invents energy. At res = 0 the damping is \( k = 2 \) (Q = 0.5, no peak); as res → 1, \( k \to 0 \) and Q → ∞: the poles slide onto the unit circle and the filter becomes a sine oscillator at \( f_c \).

Quiver’s Svf is a Zavalishin topology-preserving-transform (zero-delay feedback) discretization of that analog prototype, with the prewarped coefficient

\[ g = \tan\left(\frac{\pi f_c}{f_s}\right). \]

The bilinear transform squeezes the analog frequency axis into the digital one, and the tangent prewarp bends \( f_c \) in advance by exactly the amount the squeeze will undo — so the cutoff lands where you asked all the way toward Nyquist, where the older Chamberlin core (coefficient \( 2\sin(\pi f_c/f_s) \)) froze above roughly \( f_s/6 \). It also makes the bold curve honest: evaluating the prototype at \( s = j\tan(\pi f / f_s) \) gives the exact response of the digital filter, and that is precisely what the plot draws.

The Quiver code

The widget is this patch. Svf takes the audio at in, the knob CVs at cutoff and res (plus fm, keytrack, and keytrack_amt for modulation), and produces lp, bp, hp, and notch simultaneously — patch whichever mode you want:

use quiver::prelude::*;

let sample_rate = 44100.0;
let mut patch = Patch::new(sample_rate);

// Sound source and filter — the ghost comb and the bold curve.
let vco = patch.add("vco", Vco::new(sample_rate));
let vcf = patch.add("vcf", Svf::new(sample_rate));

// The two knobs, as constant CVs (0-1, exponential cutoff law inside).
let cutoff = patch.add("cutoff", Offset::new(0.5)); // 20·1000^0.5 ≈ 632 Hz
let res = patch.add("res", Offset::new(0.2))        // k = 2 - 2·0.2 = 1.6

let output = patch.add("output", StereoOutput::new());

// Saw → filter → lowpass tap → out. Swap "lp" for "bp"/"hp"/"notch".
patch.connect(vco.out("saw"), vcf.in_("in")).unwrap();
patch.connect(cutoff.out("out"), vcf.in_("cutoff")).unwrap();
patch.connect(res.out("out"), vcf.in_("res")).unwrap();
patch.connect(vcf.out("lp"), output.in_("left")).unwrap();
patch.connect(vcf.out("lp"), output.in_("right")).unwrap();

patch.set_output(output.id());
patch.compile().unwrap();

let (left, _right) = patch.tick(); // one filtered sample, in volts

Go deeper

• Tutorial: Filter Modulation — drive the cutoff with an LFO and the static curve above starts to move.
• Reference: Filters — the full Svf and DiodeLadderFilter port maps.
• Next explorable: Shaping Time — the filter sculpts the spectrum; the envelope sculpts when you hear it.

Next: Shaping Time