The pitch is only half a note. The other half is its shape in time — the swell of a pad, the snap of a pluck, the long tail of a bell. Four numbers draw that shape: Attack rises to the peak, Decay falls to a held level, Sustain holds it while the key is down, Release fades it back to silence. Down the center, data flows: the four numbers go in, the generator draws the curve, the amplitude comes out. The blue team proves the curve never jumps; the red team tries to make it click.
source Robert A. Moog, Voltage-Controlled Electronic Music Modules, J. Audio Eng. Soc. 13(3):200–206, 1965 — the Moog modular 911 envelope generator (T1/T2/E‑SUS/T3, after Ussachevsky, 1965), later normalised to A‑D‑S‑R. preprint (Bob Moog Foundation). Rendered, not quoted.
The envelope is a piecewise function of time a(t), peak amplitude 1. With attack A, decay D, sustain level S, release R, and key-up at tu:
attack 0→1 over [0,A] · decay 1→S over [A,A+D] · sustain hold S over [A+D, tu] · release S→0 over [tu, tu+R].
Live values at every phase boundary — adjacent segments must agree (no jump):
| boundary | left | right |
|---|
The oscillator gives the raw, endless tone: pitch and timbre with no beginning and no end. The envelope gives it a life in time.
Multiply them — a(t) × osc(t) — and you have a note that starts, breathes, and dies. The same four numbers shape every synthesized sound; change only A and R and a pluck becomes a pad. Each sphere is the next one's premise; the neighbours live in THE FOLD.
The blue team's live check: re-measure the jump across the key-up boundary and confirm the curve is continuous. If red skips the release, the click shows up here.
An ADSR generator takes exactly four inputs, plus how long the key is held:
| knob | what it sets | units |
|---|---|---|
| A | time to rise from 0 to the peak | seconds |
| D | time to fall from peak to sustain | seconds |
| S | level held while the key is down | 0…1 |
| R | time to fall from that level to 0 | seconds |
A, D, R are times; S alone is a level. That asymmetry is the whole shape — and it is what you feed the generator below.
Move any slider — the curve is drawn from the piecewise function on the spot, never looked up. Key-up sits at A+D+hold, so the sustain phase is always real.
What the machine produces, proven: a single continuous curve from 0 → peak → sustain → 0. Every phase boundary agrees to <1e‑9; the envelope is exactly 0 before the note and after the release completes; and a zero sustain collapses it to a percussive attack‑decay spike — all self‑checked at boot.
The blue team's witness (left) re-measures the boundary jump live; the red team (right) tries to make it click.
And the segments are drawn linear here for a clean proof — the Moog 911 and every analog generator charge and discharge an RC network, so the real curves are exponential. Linear ADSR is the teaching idealisation, not the hardware. The envelope is not the sound; it is the first proof that a sound's shape in time is a computable object.
"Sustain is a time, like the others." Cut. S is a level (0…1); A, D, R are times. Sustain has no duration of its own — it lasts exactly as long as the key is held.
"Release starts from the sustain level." Corrected. Release starts from whatever the level is at key-up — if the key lifts mid-decay, the release ramps down from there. It equals S only when the key outlives the decay.
"Cutting a note off is instant and harmless." Kept, corrected. Cutting amplitude to 0 with no release leaves a step discontinuity — a broadband click. The whole point of R is to avoid it. That is the tamper.
The red team's move: skip the release. Cut the amplitude to 0 the instant the key lifts instead of ramping it down. The blue team's witness (window 7) is watching the key-up boundary.
Skip the release and a step appears at key-up: the amplitude drops from the sustain level straight to 0. The witness re-measures the jump, sees it is no longer near zero, and turns red. Press play — you can hear the click. Nothing is faked; the attack is real and it is caught.