Fresnelcut

Diffuser

1D Primitive-Root Diffuser (PRD)

A primitive-root diffuser looks like a QRD but is built from a different number sequence. Its defining trick: it suppresses the specular (straight-back) reflection at its design frequency, which is why it exists as a separate design.

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How it differs from a QRD

Instead of `n² mod N`, a PRD uses a primitive-root sequence `g^n mod N`, which visits every non-zero residue exactly once — so there is no zero-depth (flat) well. The payoff is a dip in energy reflected straight back toward the source at the design frequency and its harmonics. See QRD vs PRD vs skyline for a side-by-side.

Geometry, fins, and the usable band are otherwise the same as a 1D QRD. A PRD has N−1 wells (no null well) rather than N.

Worked example: N = 7, g = 3 at 500 Hz

For the prime N = 7, the least primitive root is g = 3, so `s_n = 3^n mod 7` for n = 1…6 gives the sequence `3, 2, 6, 4, 5, 1` — every one of 1…6 appears exactly once. At a 500 Hz design frequency the same per-well step as a QRD applies, u = λ₀/(2N) ≈ 49.03 mm:

Well n123456
Residue sₙ326451
Depth (mm)147.198.1294.2196.1245.149.0

The deepest well is 294.2 mm — notably deeper than an equivalent QRD's 196.1 mm at the same N and f₀ (compare the 1D QRD worked example), because the primitive-root sequence's residues run 1…N−1 with no zero well, so its largest residue is bigger than a QRD's. Open this exact panel in the designer.

Stock, kerf, and what you'll need to build it

Because a PRD's residues top out higher than a same-N QRD's, budget more stock thickness for the same design frequency — the trade you're making for suppressing the straight-back echo. Otherwise the build is identical to a QRD: one sheet deep enough for the deepest well, fins thin relative to the well width, and kerf-aware spacing baked into the cut list.

PRD vs. a plain QRD

Reach for a PRD specifically when a direct front-to-back slap echo — not general harshness — is the complaint; a QRD is the simpler, better-documented default for everything else. For scatter in both axes with the same specular-suppression trick, see the skyline diffuser, which folds this same 1-D sequence into a 2-D grid.

Diagnosing whether you actually need a PRD

A true specular echo is directional and repeatable — clap or play a sharp transient from your listening position and you'll hear a distinct, single-direction slap back, usually off a flat rear or side wall roughly the same distance away on both sides. A generally 'live' or 'echoey' room without that one dominant slap is more often a QRD problem, since the PRD's specular notch targets exactly that single reflection rather than raising overall diffuseness. Confusing the two is the most common reason people upgrade from a QRD to a PRD and hear no real difference.

Mounting orientation

Like a QRD, a PRD's fins set the scatter axis: mount them vertically to spread reflections left-to-right, horizontally to spread them up-and-down. Because the specular notch is a property of the well-depth sequence rather than the mounting orientation, it holds either way — orientation only decides which axis gets the two-axis-independent benefits of a 1-D panel, not whether the straight-back suppression works.

The math
s_n = g^n mod N, d_n = s_n · λ₀ / (2N)

g = least primitive root of the prime N. The sequence is a permutation of 1…N−1, so no well has zero depth.

Best for

  • Setups where a strong straight-back echo is the problem
  • Anyone who wants a QRD alternative with a different scatter signature

Not the tool for

  • Low-frequency problems (use an absorber)
  • When you specifically want the simplest, most-documented panel (QRD)

Start from a common size

N is the number of wells (a prime). More wells scatter more smoothly but make a wider, deeper panel. Pick one to open the designer pre-set:

Frequently asked questions

How is a PRD's well sequence actually different from a QRD's?

A QRD uses `n² mod N`, which includes a zero (flat) well and repeats some residues. A PRD uses `g^n mod N` with g a primitive root of N — that sequence is a permutation of every non-zero residue, so it visits 1…N−1 exactly once with no flat well.

Will a PRD fix a general echoey or harsh room?

It will, but so will the simpler QRD. A PRD earns its keep specifically when a distinct straight-back slap echo is the complaint — that's the one thing it suppresses that a QRD doesn't.

Does a PRD need a different stock thickness than a QRD?

Usually a bit more at the same N and design frequency — see the worked example above. Its permutation of residues runs up to N−1, which is larger than a same-N QRD's largest quadratic residue.

Can I use any prime for N, or does g change?

Every prime has at least one primitive root g, so any of the presets works; Fresnelcut finds the least primitive root for you automatically when you pick N.

How do I tell if a straight-back slap echo is actually my problem?

Clap or play a sharp transient from your listening position — a true specular echo comes back as one distinct, repeatable slap from a specific direction, usually a flat rear or opposite wall. General liveliness or harshness without that one dominant reflection is more of a plain-QRD problem.

Related reading

Related panel types

See all diffuser & absorber types.