Why the shape of a pulse matters

If a session pulses, something has to decide what one pulse looks like — how fast the volume rises, whether it holds, how it falls away. That is the envelope, and it is the most underrated parameter in this entire field.

Two sessions with an identical carrier and an identical beat rate can be pleasant or intolerable depending on it alone.

The six shapes

The engine implements six, and the catalog uses all of them.

• Trapezoidal — eased edges, flat top. The workhorse. 131 sessions.
• Raised cosine — no flat top at all; a continuous rise and fall. Softer. 191 sessions.
• Sigmoidal — a logistic rise and fall, slower to leave the floor and slower to arrive. The most-used shape here, 275 sessions.
• Smooth bell — gaussian. The gentlest onset of the six. 132 sessions.
• Sine — a single half-cycle. 25 sessions.
• Square — a hard gate. 23 sessions, deliberately few.

Those six cover 777 of the 777 sessions, because six is all there are.

The catalog used to claim otherwise. Forty-six sessions carried an envelope named "Custom PAC", "Dynamic PAC" or "Dynamic Multi-Stage", and a further 206 differed from a real shape only in capitalisation — "Raised Cosine" against "Raised cosine". None of the three exotic names was a shape the engine implements; every one of them specified shape 2 and rendered as plain sigmoidal. The names promised bespoke behavior that did not exist.

The label is now derived from the shape index rather than written alongside it, so it cannot say one thing while the engine does another. The build fails if it ever disagrees.

Every shape starts and ends at exactly zero across its on-period. That constraint is not aesthetic: a discontinuity into the silent part of the cycle is what a click is. Even the square shape has edges about a millisecond long rather than truly vertical, because a genuinely instantaneous edge produces an audible tick on every pulse.

Why square is rare

A hard gate is the strongest modulation — the most abrupt change in amplitude, and in principle the most vigorous drive.

It is also percussive, and at 23 sessions out of 777 you can see the judgement made. Over forty minutes a hard pulse becomes something you brace against. A bell-shaped envelope at the same rate delivers a slightly weaker stimulus that you can actually sit inside.

That trade — maximum drive against tolerability over time — is the whole design conversation about envelopes.

Depth: how far it falls

Alongside the shape sits depth: how much quieter the sound actually gets between pulses. Across this catalog it ranges from 0.5 to 0.98.

At 0.5 the tone merely dips — a gentle swell, easy to ignore, appropriate for long or sleep-directed sessions. At 0.98 it very nearly disappears and returns, which is a much stronger stimulus and much harder to ignore.

Shallow is not worse. For a 39-minute sleep session, shallow is the point.

Duty: how long the sound lasts

Duty cycle is the fraction of each cycle containing sound. Every session in this catalog runs at 50%, so the pulse and the gap are equal.

That uniformity is deliberate. A 50% duty at a given rate is the most neutral choice — shorter duties produce a clickier, sparser character that starts to feel like a metronome rather than a field.

How this shows up when you listen

• A session that feels jittery or nagging is usually a steep envelope at high depth, not a bad rate.
• A session that feels like it is not doing anything is often a bell envelope at low depth. That may be exactly right for sleep and wrong for focus.
• The same rate at high beta feels far rougher than at theta, because at 20+ Hz you stop hearing separate pulses and start hearing a texture in the tone. Faster sessions therefore use gentler envelopes.