Calculate Quadratic Residue Diffuser
( given size )

Calculate here QRD ( estimated frequency )

The Quadratic Residue Diffuser

Schroeder presented methods of designing concert hall ceilings that could avoid direct reflections into the audience. In 1975, he provided a way of designing highly diffusing surfaces based on binary maximum-length sequences, and showed that these periodic sequences have the property that their harmonic amplitudes are all equal.
He later extended his method and proposed surface structures that give excellent sound diffusion over larger bandwidths.
This is based on quadratic residue sequences of elementary number theory, investigated by A. M. Legendre and C. F. Gauss. These sequences are defined by:

sn = n2 mod(p)

i.e. n2 is taken as the least nonnegative remainder modulo N, and N is an odd prime number.
For p = 17, the quadratic residue sequence reads as follows (starting with ):

sn = 0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1; 0, 1...

How does it work?

  • Pick a theoretical depth you want to use for your diffuser.
  • Now let's pick a non-even prime number: 5, 7, 11, 13, 17, 19 or 43.
  • How deep should each compartment be? To figure that out we need to do a couple of calculations, the first of which is the step size, and that's pretty simple. Take your theoretical depth and divide it by the prime number and you've got your step size. So, that's 21 cm divided by 7. So our step size is 3 cm.
  • The formula is n squared modulo p. N is the number in the sequence and p is the prime number we're using.
  • If the prime number you're using is 7, then set up a little table counting up from 0 to 7: 0, 1, 2, 3, 4, 5, 6, 7.
  • We picked a theoretical depth of 21 cm for this 7-compartment diffusor, so each step, which will be 1/7 of 21 cm, will be 3 cm. Let's multiply each of our residues by the step size:

For the 0 in our sequence, which has a residue of 0, 0 x 3 cm = 0 cm
For the 1 in our sequence, which has a residue of 1, 1 x 3 cm = 3 cm
For the 2 in our sequence, which has a residue of 4, 4 x 3 cm = 12 cm
For the 3 in our sequence, which has a residue of 2, 2 x 3 cm = 6 cm
For the 4 in our sequence, which has a residue of 2, 2 x 3 cm = 6 cm
For the 5 in our sequence, which has a residue of 4, 4 x 3 cm = 12 cm
For the 6 in our sequence, which has a residue of 1, 1 x 3 cm = 3 cm

How wide should we make each of these compartments?
The high frequency limit of the diffusor is half the wavelength of the compartment width, so the narrower the compartment, the higher the high frequency limit.

7 is a lucky number for a diffusor because the greatest step size is only 4/7 of the theoretical depth, so the diffusor only has to actually be 12 cm deep to reap the benefit of being theoretically 21 cm deep.

Early Reflections

Room reflections arriving at the listening seat within a time window of 10 milliseconds not only color timbre but can also confuse image outlines. In most cases, the major offender is early sidewall reflections whose path length is less than 3.4 meters delayed relative to the direct sound. As a rule of thumb, sound is delayed 1 mS for every 34 cm of travel. Early specular reflections off hard surfaces are particularly nasty because they can be almost as intense as the direct sound.

The most effective in controlling sound dispersion in the 2 kHz to 4 kHz octave is aiming the radiation lobe of the tweeter away from the sidewall toward the listening position.

A light absorptive treatment along critical sidewall spots may further improve stereo performance. This maybe achieved by means of ordinary domestic items such as wall hangings and book cases. Alternatively, the absorptive treatments maybe replaced by diffusive surfaces.


Diffusor width:   cm
theoretical depth:   cm
Prime seed: 
Show sn  
Speed of sound:  344 m/s or 1125 ft/s.

t.akustik Diffusor SD-1

Diagram for SD-1

Here some pictures of the building from my first Quadratic Residue Diffusor with

Diffusor width: 49 cm,
theoretical depth: 21 cm,
Amount of wells: 7

Using RPG Diffusors to Enhance Room Acoustics

YouTube: Make This Skyline Diffuser

Ideal Acoustic Diffuser Placement Guide

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