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The Bohemian Boolean

QRADS Explanation

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What does Quadratic Residue Acoustic Diffuser Simulator mean?

Well, an acoustic diffuser is a typically-wooden device placed on a wall or ceiling that disperses or “diffuses” sound evenly, when sound bounces off of it. This is not to be confused with a sound absorber, which absorbs sound and preventing it from spreading throughout a room, although music producers, acoustic engineers and others in the acoustic industry use both throughout a room to make sure that sound is at its highest quality.  

Why is this useful? An acoustic diffuser spreads out the sound randomly, ensuring that sound emanating from anywhere in the room is heard at an equal volume throughout the room. Under normal circumstances, for example, a person standing closer to a wall may hear a louder sound than someone standing an equivalent distance away from the source of sound but further away from the wall, as that person would hear both the original sound and the echo, while the person standing further away would hear the original sound, but an echo at a lower frequency and volume. 

However, if an acoustic diffuser is on the wall, the sound scatters off of the uneven shape of the acoustic diffuser randomly. Therefore, sound no longer behaves in its predictable manner with uneven frequencies and volumes filling the room, with a viewer’s perception of the sound “colored” by the walls. Instead, sound waves fill the room with an overall even frequency. This helps ensure that listeners get the best auditory experience anywhere in the room, which is why music producers listen to music in rooms with acoustic diffusers to ensure that they are hearing music “accurately” and perceiving sounds without the “walls” coloring the music. Similarly, some movie theaters and auditoriums have acoustic diffusers on the wall (or even just rough-textured walls) to disperse sound throughout the theater.

Whew! That was a lot! Now to tackle the “quadratic residue” part. In order to create acoustic diffusers (or at least the kind I created), one can attach wooden columns with different heights to a flat surface. Some acoustic diffusers also use wells with different depths – same idea. If the columns have randomly-decided heights, the sound should bounce off and spread evenly. But how does one ensure that the columns have random heights? One way is with quadratic residues.

What’s a quadratic residue? To discuss that, it is necessary to delve into modular arithmetic. If you have ever translated military time (23:00) to standard time (11:00 PM), good news! You are already familiar with the concepts behind modular arithmetic. All you had to do to find out standard time is subtract 12 hours from 23:00.

Let’s pretend that military time went beyond 23:59. What would 34:00 represent? Hint: Try subtracting 12 hours.

Would it be 22:00? Well, no, because standard time ranges from (12:00 to 11:59). Hint: Try subtracting 12 again! The answer is 10:00!

Basically, to find any military time in standard time, you have to subtract the highest multiple of 12 possible from military time and what we are left with is standard time. Put another way, find the remainder after military time is divided by 12. Represented in modular arithmetic, we could say that

22 ≡ 10 (mod 12) – read as 22 mod 12 is congruent to 10.
 

This means that if we use 12 as our divisor, the remainder after we divide 22 is 10.

Now, with quadratic residues, we are using the same idea, except instead of 12, we can use any prime number and instead of 22, we cycle through the squares of all the natural numbers less than that prime number. 
 
For example for the number 17, we create a list of all the moduli from 1² to 16² (mod 17). This means we start out with: 

0² = 0 ≡ 0 (mod 17)
1² = 1 ≡ 1 (mod 17)
2² = 4 ≡ 4 (mod 17)
3² = 9 ≡ 9 (mod 17)
4² = 16 ≡ 16 (mod 17)
 
Boring! So far, all of our squares were less than 17, so the remainders were as well. But here is where it starts getting interesting. Our squares are now greater than 17, ensuring our remainders will be different from the starting square. Each of the squares will correspond to a number between 1-16 (mod 17), but in an almost completely unpredictable order.
 
5² = 25 ≡ 8 (mod 17)
6² = 36 ≡ 2 (mod 17)
7² = 49 ≡ 15 (mod 17)
8² = 64 ≡ 13 (mod 17)
.
.
.
(and so on)
 
16² = 256 ≡ 1 (mod 17)
 
This is one of the closest ways we can get to generating a “truly random” sequence. This is how generated the heights of the wooden columns in the first row of my acoustic diffuser. Column 0 had a height of 0, Column 1 had a height of 1, Column 2 had a height of 4…. and Column 16 had a height of 1 again. Alternatively, I could have also used these numbers as “well depths” to create a different kind of acoustic diffuser.
 
For the next row, I just shifted all the residues up by 1 – Column 0 had a height of 1, Column 1 had a height of 4 and now Column 17 had a height of 0. I continued this pattern for all 17 rows (same number of rows as squares from 0 – 17). 
 
This same technique can be used for any prime number to create the Quadratic Residue Acoustic Diffuser. Hope this helps!

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