How To Build A Quadratic Residue Diffuser

acoustic treatement bedroom recording studio control room design diffusion frequencies qrd 13 quadratic residue diffuser room acoustics schroeder diffuser Dec 18, 2023

One of the most effective and efficient types of diffusers are reflection phase grating diffusers otherwise known as Schroeder diffusers. A type of Schroeder diffuser is the Quadratic Residue Diffuser (QRD) and in this article I will teach you how the diffusers work, the theory behind them, and how to build one. 


1) How Does a QRD work? 

Manfred R. Schroeder developed the quadratic residue diffuser by using a series of wells of different depths with a constant width that are separated by thin dividers to evenly disperse sound back into the room. (Everest, 265). He found that a number sequence based on prime numbers can be used to calculate the depth of each well. This number sequence can also be repeated over and over (in periods) to expand diffusion across a wall or ceiling. 

When designing a QRD the maximum frequency for diffusion is determined by the well width and the minimum frequency is determined by the well depth. For example a QRD that needs to cover most of the audible spectrum would need to have very deep wells up to several feet and very thin well widths to cover say from 60 Hz to 10,000 Hz. This is one of the design limitations of QRD and phase grating diffusers in general. 


2) The Math Behind QRD?

So don't get scared, but I am going to talk about math. I am not a big math genius, so I always need to break down the numbers in simple terms, ideally, this makes me a better teacher for you! So, the equation behind the number sequence used to build the QRD is: 

Well depth proportionality factor = n^2 modulo p

where n = integer greater than or equal to 0 

p = prime number

If you are like me, you had no idea what the heck modulo meant. Modulo is a calculation in number theory. It is a short cut for the following equation

n^2 / p = x + remainder or residue

n^2 modulo p = remainder or residue

The modulo operation says take the n squared in our equation and divide so there is a whole number and a remainder. It says that the modulo will = the remainder not the whole number. If that still seems confusing let's do an example. 

Let's say we want to build a QRD with a prime number of 11. Now lets pick an integer of 5 in our number sequence from 0 to 11 for the diffuser. Again, I just chose 5 randomly for this example, when building the diffuser you would do 0 then 1 then 2 and so on up to 11. 

So, in our equation: 

Well depth = 5^2 modulo 11 

Well depth = 25/11 = 2 with a remainder of 3 

Well depth = 3 

11 goes into 25 two times, which equals 22 and to get to 25 we have 3 left over. The remainder and residue terminology is interchangeable. Confusing, I know! 

So, the QRD are usually made from the following prime number sequences: 5, 7, 11, 13, 17, 19 and 23. 

Below is a chart from the Master Handbook of Acoustics that shows the well depth ratios for all of those prime number diffusers. 

You will also notice the diagram showing one period or one diffuser length. Notice this period goes from 0 to 1 in the prime 17 sequence. This shows how diffusers can be placed side by side to continue the sequence across a wall or ceiling. 


3) How To Design Your QRD

So, first we must choose a prime number to build our QRD. Each prime number has some benefits and limitations and ideally in a room design you might use a combination of different prime numbers to achieve a diffuse sound field across as much of the audible spectrum as possible. All that is a bit beyond this article. For this example, I will be building a QRD 13 based on the prime number 13 sequence. 

Now, the chart above is great in theory, but it is hard to imagine translating all those ratios to our actual physical diffuser. So, I recommend using a handy QRD calculator. There are a lot on the web, but this one does the trick:

I will be doing this example in metric since my design is for a client in New Zealand, but you can do yours in imperial and all the concepts stay the same. 

My design is limited by space, as is the case for many home studios, so I will be working with a theoretical depth of 13 cm. I chose 13 cm because it will make the measurements whole numbers not decimals. I could have done a multiple of 13 like 26 cm, but my deepest well would be too deep for my room limitations. The width of my design should also be a multiple of 13. The reason is that this will make all of my well depths a whole number, so my design will be much easier to measure and build. 

With a theoretical depth of 13cm and a diffuser width of 65cm here are my well depths and well widths for my diffuser. 

Notice, I get nice whole numbers. My well width is 5cm and my well depth are all whole numbers as well. We also get my low frequency cutoff and high frequency cut off. Ideally, I would diffuse from 20 Hz all the way to 20 kHz, but that is not really possible, so instead I am happy with a nice low mid to upper mid diffusion on the back wall of my design. 

This diffuser will help to widen and deepen the sound field when mixing and it will add back some of the life to the room, since I am using a lot of absorption to help control the low end and early reflections in my design. 

4) How To Build Your QRD

Now that you have all the well depths, and well widths you can design your diffuser. Use cabinet grade or marine AA grade plywood so the diffuser will look nice. In my design I am using 12mm marine AA plywood for my back, top, bottom and sides and 6mm marine AA grade plywood for my fins to separate the wells. 


For my spacers I used 50 mm x 50 mm blocks on both the top and bottom of my diffuser to create the right well depth. To fill the wells you can use 12mm plywood to lay on top of your blocks. Here is a diagram to show how to do this. 

Notice the well depth is the distance from the top of the diffuser fin to the to of the 12 mm plywood on top of your block. Each well is 1-13 from the chart above moving from left to right. 

It is good practice to dado out the well dividers on the back, top,  and bottom so that you can simply slide your fins into the grooves. You can then attach the fins with wood glue. To build the top, bottom, back and sides you can use wood glue and some small screws to adhere the box together. Your spacer blocks and spacer faces can be attached using wood glue as well to keep the design clean and free of nails and screws. 



You now know what a quadratic residue diffuser is, how the number sequence is derived, how to calculate your own design and how to build one. I hope this article was helpful in demystifing one of the more complex acoustic tools and allowing you to experiment with your own designs for your home studio. 


Work Cited: 

Everest, Frederick A., and Ken C. Pohlmann. “Schroeder Diffusers.” Master Handbook of Acoustics, McGraw-Hill, New York, 2015.





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