"Room ratios" is a whole major subject in studio design. Room ratios are important, to a certain extent, but many people get confused about the issue, and make wrong decisions based on wrong information.
This article is quite long, but it needs to be: There are actually quite a few myths, mysteries, and misunderstandings associated with room ratios and modal response (and even some plain old snake-oil silliness), all over the internet.
Thus, first-time studio builders often get very confused about it all, and struggle with their designs, trying to get a "perfect" room ratio (there is no such thing), or wondering how they can "get rid of the modes" (a really bad idea, and only achievable with a bulldozer...), or wondering about other fancy terms associated with this subject, such as "Standing Waves" and "the Schroeder frequency": it all adds up to confusion.
So I wrote this article to try to help set things straight. Hopefully, it will be useful for you, and help clarify a complicated issue.
Firstly, we need to look at what room modes are, then we can look at what they do to you, then we can figure out what to do about them.
It works like this: The walls of your studio (or any other room) create natural resonances in the air inside the room. (This is totally different from the MSM resonance of the walls themselves: this is all about what happens within the ROOM, not what happens inside the walls. Two totally different things.)
So, you have resonant waves inside the room. We call those "standing waves" or "room modes". Those "modes" (resonances) occur at very specific frequencies that are directly related to the distances between the room boundaries (walls, floor, ceiling). They are called "standing waves" because they appear to be stationary inside the room: they are not REALLY stationary, since the energy is still moving through the room. But the pressure peaks and nulls always fall at the exact same points in the room each time the wave energy passes, so the "wave" seems to be fixed, static, and unmoving inside the room. If you could see the way the sound energy bounces off a wall, forming a standing wave, it would look like this: (For the acoustic purists among us: Yes, I'm well aware that this isn't an accurate analogy for sound waves, for all those technical reasons you have in mind, but it is a simple, easy way to understand "standing waves", and that's the point).
You can see that one single wave starts out approaching from the left, the hits the wall, and bounces back... but then the "bounced" version interferes with the "original" version of itself, and together they form that "standing wave" pattern, where the peaks and nulls always occur at the same places. So it's actually TWO waves that are interacting with each other: the original wave, and the same one that bounced back.
Here's a different view that might make it a bit more clear, showing each wave individually, then the final wave (which is the sum of the other two): The top one is the wave approaching from the left, the second one is the bounced wave that came back again, moving back to the left. The third one is what you get whey you add those two together.
This is sometimes a hard concept to grasp, but it's important to understand that a "standing wave" is not really "standing still" at all! Maybe that's not such a good name for it...
Now, the animations above are a bit confusing too, because they are not showing room modes at all! They are just showing the concept of standing waves. Those diagrams show a wave bouncing off only one wall, but room modes need at least two walls to form. The difference is that a simple standing wave bouncing off one wall does not resonate, but a room mode DOES resonate, because there's another wall that sends it right back where it came from, to repeat the whole sequence in an endless loop. In other words, in the simple case of the wave coming in from the left, bouncing off the wall on the right, and heading back out to the left, the energy "disappears" after it runs off the left edge of the diagram, and never comes back again, but if you put another wall over there on the left, then the energy DOES come back again, ... and again ... and again ... and again... in fact, if the walls were perfectly reflective, it would keep on bouncing between them forever!
So, if you mentally imagine that there's another wall on the left side of that second diagram that is doing exactly the same thing as the wall on the right, reflecting the wave back again, then you have a proper "room mode". (There's one more thing that would need to happen to form a room mode, and we'll get to that later, but for now, try to get that mental picture in your head of the two waves bouncing back and forth across the room, interacting with each other, and creating a resonant standing wave in between the walls.)
Now think about this: As you can see with the diagrams above, every time the two waves form a peak at one spot in the room, the intensity (loudness) of the resulting final wave is the SUM of those two waves... so the amplitude (=loudness) is TWICE as big as one wave by itself, at the peaks. And at the dips, the sum of both waves is "zero": they cancel each other out. So that specific tone would not be heard at all at that "null" spot in the room. This is why you can walk around the room when you have a mode present, and hear how the sound level increases and decreases as you move your ear from place to place: you are moving from the peak to the dip.
Now for the "resonant" part: if there really was a wall on the left of that diagram, reflecting the wave back again, then on the next trip that it took across the room you would have THREE waves adding up on top of each other, then when that one bounced off the right wall and came back again, it would be FOUR waves adding up, then five, six, seven, eight... etc. (Yes, they do really add up like that. You might think that it would then become infinitely loud after a few hundred trips, but in fact there are losses involved, and other factors that have to do with the air itself, so there's a limit to how loud it can get.): In simple terms: the room "resonates" at that frequency. Even worse, that build-up of energy is sort of "stored" in the standing wave itself, and if you then cut off the note that you were playing to cause it, the room does NOT cut-off! It carries on resonating at that frequency for a while, with the note slowly dying away: So, it "rings". The room carries on playing that tone even after you stop playing it. Clearly, that's not a good thing for a studio! We'll get back to that later...
So, to recap: a "mode" is just a sound wave that is bouncing back and forth between the walls, creating a "standing wave". It resonates because the energy in each "bounce" is added to the energy from the other "bounces", so it gets louder and louder, but it does reach a limit fairly fast. Then it can carry on "ringing" after the original note that caused it has already stopped.
Above, I mentioned that you need "one other thing" to get a standing wave / room mode to form: the right distance. For this to happen, the distance between the two walls must be exactly the same as the wavelength of that wave. If the distance is NOT the same, then the wave does not "stand", and there is no resonance! The wave can still bounce around between the two walls, sure, but it won't resonate, because the peaks and dips will form at a DIFFERENT place in the room each time around. It's only when the peaks and dips form at the SAME place that you get resonance, and build-up. If the wave peak is at a different location in the room every time, then there is no build up: so no mode.
In other words: you can only have a room mode for frequencies where the wavelength matches the distance between walls. For all other frequencies, there will be no mode.
If you could see the actual air pressure variations in the room caused by such a standing wave, it would look like this: I'll get back to explaining that in more detail later...
So a simple definition of a mode is this: A "mode" is one manner in which a room can resonate. You can think of it as a path that a sound wave can take around the room, bouncing off the walls, floor and ceiling, then arriving back at the starting point, in phase with itself, and going in the same direction it originally was.
Now, for the interesting part: All rooms have several such "modes", regardless of the shape! You might think that the walls have to be parallel for this to work, but that actually isn't true. Rooms of any shape, even with drastically non-parallel walls, still have modes. The inside of your car, for example, does have modes, just like any other room: (thanks for that, Andre!) Even inside a cylinder, where the walls are round, not flat, and not parallel, you still have modes. There are still ways that a sound wave can bounce around and form a standing wave, in ANY shape or size of room.
Its a common misconception to think that by changing the shape of the room you can "get rid" of the modes: you can't. A different shape might have different modes, but they will still be there. Here's a diagram from the famous "Master Handbook of Acoustics" that demonstrates this very clearly. It shows one specific mode in two rooms that have the exact same floor area, but very different shapes:
So, another brief summary of everything up to this point: if you play a pure tone in your room, and it happens to be at the exact same frequency as one of the "modes" in the room (any room, any shape), then you can physically walk around inside the room and experience the "standing" nature of the wave: you will hear that tone grossly exaggerated at some points in the room, greatly amplified, while at other points it will sound normal, and at yet other points it will practically disappear: you won't be able to hear it at all, or you hear it but greatly attenuated, very soft.
Next important point: a room doesn't just have one "mode" of vibration: it has many. You have modes that form across the room between the two side walls, and modes that form along the length of the room, between the front an back wall, and modes that form vertically, between the floor and ceiling. These are called "axial" modes, because they form along one of the three axes of the room (here "axes" is the plural of "axis", not something you use to chop down trees...). But there are other types of modes too: you can have a mode that forms between FOUR surfaces of the room, instead of just two, and those are called "tangential" modes. And you can also have modes that form between all six surfaces of the room; those are called "oblique" modes. This graphic shows all three types of modes, and how they form: To complicate things even more, if a mode can form at one specific frequency that coincides with the distance between surfaces, then a different mode will form at TWICE that frequency, because now TWO wavelengths will fit in... and yet another mode will form at three times the frequency, because three wavelengths will now fit in between the surfaces... also at four times, and five times, and .... blah blah blah.... fifty times, and etc. In musical terms: there will be an infinite series of harmonics of that mode, all the way up the musical spectrum.
Next point to consider: The peaks and nulls fall at different places in the room for all of these different modes. So the spot in the room where one mode was deafening might turn out to be the null for a different mode. There are equations for figuring it all out and predicting every single possible mode for a given room, but mostly we don't bother too much about even trying to do that! It can get very complicated, very fast, when you try to predict all of the modes for a room... There's no point. If you are interested, here's the actual equation (taken from the Master Handbook of Acoustics, page 327): If you wanted to use that to figure out your room modes, what you would need to do is start by setting P, and Q to zero, and set "R" to 1, then calculate. That will give you the frequency for the 0.0.1 mode of your room. Then change R back to 0 and set Q to 1. Calculate. That's the frequency of the 0.1.0 mode. Then change Q back to aero and set P to 1. Calculate. 1.0.0.. Then keep R at 1 and change Q to 1. Calculate. Then change P to 1. The change R to 2. Calculate. And so on, for all possible combinations of P, Q, and R, with all the modal numbers you want to know about. This will produce a table showing you the modal frequencies for each of the modes. That's what the calculators do for you (I'll talk about those later). As you can imagine, there's a huge number of modes that you could put in such a table! An infinite series, in fact. Here's part of such a table, generated by a room mode calculator app: That shows the first 30 modes for a typical room. The first column on the left of that table is just the number of the mode, the second column is the frequency, the third one is the closest musical note, the forth one is the mode "name" or "index" (more about that in the paragraph below), and the last column tells you what type of mode it is: axial, tangential, or oblique.
The first mode on that list is identified as mode "1-0-0", which means it is the mode that runs only along the lengthwise axis of the room, and is not involved with the ceiling, floor, or side walls. (It's important to note that there are two different methods for identifying or "naming" modes: 1-0-0 and 0.0.1.... Those both refer to the same mode! The first system has them in the order Length - Width - Height, the second system has them in the order Height - Width - Length. So be careful, and make sure you know which system you are looking at...)
Anyway, if you really wanted to, you could calculate all the modes for your room all the way up the entire spectrum, and you would fill a small book with that list.... You don't need to do that! It really is pointless to look at thousands of modes for your room. You only need to look at a few of them, not all of them. Mostly, for typical home studios, we are only interested in the modes at the bottom end of the musical scale: below about 500 Hz, and especially below about 200 Hz. There are simple tools that will do that for you, and I'll mention those later.
Here's the modal prediction from one such tool for a typical small room: That shows the predicted modes for a typical size home studio, measuring 8 feet high (2.4m), 10 feet wide (3m) and 13 feet long (4m). The grey-and-white background of that diagram is actually a piano keyboard, so you can see the relationship between the musical scale and the modes. Each vertical line shows the position of one mode. The longer red lines are axial modes, the intermediate length purple lines are tangential modes, and the short blue lines are oblique modes. There's also a frequency scale across the bottom, so you can see what frequency each mode is at. I won't go into more details about that yet.... later...
Now for the flip side of this modal thing: If you happen to play a note at a frequency where the room does NOT have any modes, then there is NO resonance for that note: no standing wave, no ringing, no peaks, no dips: the note sounds normal, even, smooth, and the same all over the room.
Obviously, it's not good if you play a song, and the room "sings along" with some notes but not with others! You might think that the ideal situation is that the room should not "sing along" with ANY notes, but that is not possible: as I mentioned above, there are ALWAYS modes in any room. The only way to get rid of all the modes, is to drive a bulldozer through the room, removing all the walls, the floor, and the ceiling... The only possible way to get rid of all the modes in your room
So this brings us to another misconception: People sometimes want to get rid of modes because they think modes are a bad thing, and that their room has "too many" of them. Actually, it's the other way around: In a small room, it's not that you have too many modes: the problem is that you don't have enough!
Let's back-track a bit, and start putting some numbers to it: if you have a mode (standing wave) that forms at a specific frequency, then playing at a slightly different frequency might show no mode at all: for example, if a tone of exactly 78 Hz (D# on your bass guitar) creates a standing wave that is clearly identifiable as you walk around the room, with major nulls and peaks, then a tone of 82 Hz (E on the bass) might show no modes at all: it sounds the same at all points in the room. Because there are no natural resonances, no "room modes" associated with that frequency. So modes are very "tight" and "narrow". But those two notes are only 4 Hz apart! They are VERY close. Yet, each mode only forms at one specific frequency, plus maybe a very small range to either side, but not for adjacent frequencies: in other words, modes they are very "narrow bandwidth", or very high Q.
That's the problem. A BIG problem.
Of course, you don't want that to happen in a control room, because it implies that you would hear different things at different places in the room, for any given song! At some places in the room, some bass notes would be overwhelming, while at other places the same notes would be muted. As you can imagine, if you happen to have your mix position (your ears) located at such a point in the control room, you'd never be able to mix anything well, as you would not be hearing what the music REALLY sounds like: you would be hearing the way the room "colors" that sound instead. As you subconsciously compensate for the room modes while you are mixing, you could end up with a song that sounds great in that room at the mix position: the best ever! But it would sound terrible when you played it at any other location, such as in your car, on your iPhone, in your house, on the radio, at a club, in a church, etc. Your mix would not "translate". Because your mix was compensating for modes that only happen in YOUR room: they do not happen in the other places...
And you also don't want major modal issues in a tracking room / live room / rehearsal room / music class room, for similar reasons: As an instrument plays up and down the scale, some notes will sound louder than others, and will "ring" longer. The instrument won't sound even and balanced.
OK, so now I have painted the scary-ugly "modes are terrible monsters that eat your mixes and music" picture. Now lets look at that a bit more in depth, to get the real picture, and understand why they look bad, but aren't so bad in reality, and what you should (and should not) do about them.
So let's go back to thinking about those room modes (also called "eigenmodes" sometimes): remember I said that they occur at very specific frequencies, and they are very narrow in bandwidth? This implies that if you played that D# on your bass guitar, it might trigger a massive modal resonance, but then you play either a C or an E and there is no mode, so they sound normal. Clearly, that's a bad situation. But what if there was a room mode at every single frequency? What if there was one mode for C, a different mode for D and yet another one for E, and one more for F? In that case, there would be no problem, since all notes would still sound the same! Each note would trigger its own mode, and things would be happy again. If there were modes for every single frequency on the spectrum, and they all sounded the same, then you could mix in there with no problems!
And that's exactly what happens at higher frequencies. Just not at low frequencies. Because of "wavelength" once again...
It works like this: remember I said that modes are related to the distance between walls? It's a very simple relationship. Remember I said the waves are "standing" because the peaks and nulls occur at the same spot in the room? In simple terms, for every frequency where a wave fits in exactly between two walls, then there will be a standing wave. And also for exactly twice that frequency, since two wavelengths of that note will now fit. And the same for three times that frequency, since three full waves will now fit in between the same walls. Etc. All the way up the scale.
So if you have a room mode at 98 Hz in your room, then you will also have modes at 196 Hz (double), 294 (triple), 392 (x4), 490(x5), 588(x6), 686(x7) etc., all the way up. If the very next mode in your room after that original 98 Hz one, happened to be at 131 Hz, then there would also be modes at 262 Hz(x2), 393(x3), 524(x4), 655(x5), etc.
That's terrible, right? There must be thousands of modes at higher frequencies!!! That must be awful!
Actually, no, it's not awful at all! In fact, it is fantastic! Yes there are thousands of modes at higher frequencies, and that's a GOOD thing. You WANT lots of modes, for the reasons I gave above: If you have many modes for each note on the scale, then the room sounds the same for ALL notes, which is what you want. It's good, not bad. In simple terms: "Many modes close together on the spectrum" = good. "Just a few modes, far apart" = bad.
But now let's use a bit of music theory and math and common sense here, to see what the real problem is.
If your room has a mode at 98Hz, and the next mode is at 131 Hz, that's a difference of 32%! 98 Hz is a "G2". So you have a mode for "G2". but your very next mode is a "C3" at 131Hz. That's five notes higher on the scale: your modes completely skip over G2#, A2, A2#, and B2. No modes for them! So those four notes in the middle sound perfectly normal in your room, but the G2 and C3 are loud and long.
However, move up a couple of octaves: ...
There's a harmonic of your 98Hz mode at 588 Hz, and there's a harmonic of your 131 Hz mode at 524 Hz. 524 Hz is C5 on the musical scale, and 588 Hz is a D5. They are only two notes apart! Not five, as before.
Go up a bit more, and we have one mode at 655 and another at 686. 655 Hz. is an E5, and 686 is an F5. they are adjacent notes. Nothing in between! We have what we want: a mode for every note.
The further up you go, the closer the spacing is. You can see that on the room mode prediction graph, above: the lines get closer and closer together as you go up the scale. In fact, as you move up the scale even higher, you find several modes for each note. Wonderful!
So at high frequencies, there is no problem: plenty of modes to go around and keep the music sounding good.
The problem is at low frequencies, where the modes are few and far between.
The reason there are few modes at low frequencies is very simple: wavelengths are very long compared to the size of the room. At 20 Hz (the lower limit of the audible spectrum, and also E0 on the organ keyboard), the wavelength is over 56 feet (17m)! So your room would have to be 56 feet long (17 meters long) in order to have a mode for 20 Hz.
Actually, I lied: Up to this point, I've been simplifying a bit to make it easy to understand, but now it's time to be more accurate: it turns out that what matters is not the full wave, but the half wave: the full wave has to exactly fit into the "there and back" distance between the walls, so the distance between the walls needs to be half of that: the half-wavelength. So to get a mode for 20 Hz, your room needs to be 56 / 2 = 28 feet long (8.5M) . Obviously, most home studios do not have modes at 20 Hz, because there's no way you can fit a 28 foot (eight meter) control room into most houses!
So clearly, the longest available distance defines your lowest mode. If we take the dimensions from the hypothetical case in that room mode calculator above as an example, the length of the control room is 13 feet (4m), the width is 10 feet (3m), and the height is 8 feet. (2.4M) (typical of a very small home studio). So the lowest mode you could possibly have in that room, would be at about 43 Hz (fits into 13 feet or 4M perfectly). That's an "F1" on your bass guitar.
The next highest mode that this room could support is the one related to the next dimension of the room: In this case, that would be width, at 10 feet / 3M. That works out to 56.5 Hz. That's an "A1#" on your bass guitar. Five entire notes up the scale.
And the third major mode would be the one related to the height of the room, which is 8 feet /2.5M, which that works out to 71 Hz, or C2# on the bass guitar. Another four entire notes up the scale.
There are NO other fundamental modes in that room. So as you play every note going up the scale on your bass guitar (or keyboard), you get huge massive ringing at F, A# and C#, while all the other notes sound normal. As you play up the scale, it goes "tink.tink.tink.BOOOOM.tink.tink.tink.tink.BOOOOOM.tink.tink.tink.BOOOOOM.tink.tink...."
Not a happy picture.
Sure, there are harmonic modes of all those notes higher up the scale. But in the low end, your modes are very few, and very far between.
As I mentioned before, some people think: "If modes are bad, then we have to get rid of them". Wrong! What you need is MORE modes, not less. Ideally, you need a couple of modes at every single possible note on the scale, such that all notes sound the same in your room. In other words, the reverberant field would be smooth and even. Modes would be very close together, and evenly spread. And this is where Mr. Schroeder comes into the picture (I mentioned his name before, right at the start of this article): many years ago, Mr. Schroeder did some research on this, and discovered that if you have three modes for any given note on the scale, you are fine (sort of... he phrased it a bit different, but that's the idea). So, as you go up the scale, the modes get closer and closer, more and more dense, until eventually there are at least 3 for each note. That point, where you first get three modes per note, is the "Schroeder Frequency." That' it! There's nothing complicated about Mr. Schroeder and his frequency: It's just the point on the musical scale, where there are "enough" modes above that point for each note on the scale, but "not enough" modes for all the notes below it. That's all. In the predicted modal response graphic above, you can see the text over to the right, saying "Schroeder-Fq" next to a vertical line at 285 Hz. That's the "Schroeder Frequency" for that hypothetical room. Below 285 Hz, there are not enough modes to go around. Above 285 Hz, there are plenty.
The ideal situation would be to have a room where the Schroeder frequency is below 20 Hz, since that would mean that there are at least 3 modes for every possible audible tone. However, the room would have to be very large to get the Schroeder frequency down that low.
So, recapping one more time: trying to "get rid of modes" is a bad idea. And even if it were a good idea, it would still be impossible! Because modes are related to walls, we get back to the bulldozer! Knock down the walls...
The only other way to get a control room that has no modes at all, is to have no walls at all! Go mix in the middle of a big empty field, sitting on top of a 56 foot (17 M) ladder, and you'll have no modes to worry about....
Since that isn't feasible, we have to learn to live with modes.
Or rather, we have to learn to live with the LACK of modes in the low end. As I said, the problem is not that we have too many modes, but rather that we don't have enough of them in the low frequencies. That is clearly evident on the modal prediction graph. There are huge empty gaps in the low frequencies, with no modes.
To recap some more: As Mr. Schroeder discovered, for any give room there is a point on the spectrum where there are "enough" modes. Above that point, there are several modes per note, but below it there are not. There's a mathematical method for determining where that point is: Since Mr. Schroeder was the guy who figured it out, years ago, this is now known as the "Schroeder frequency" for the room. Above the Schroeder frequency for a room, modes are not a problem, because there are are lots of them spaced very close together. Below the Schroeder frequency, there's a problem: the modes are spaced far apart, and unevenly. (The Schroeder frequency is a bit more complex than just that, since it also considers treatment, but this gives you an idea...)
There's a very simple equation for calculating what the Schroeder frequency is for any give room. Not surprisingly, this is often called the Schroeder equation:
f(sch) >= 2000 * √(T60/V)
T60 is the 60 dB decay time for the room,
V is the cubic volume of the room.
It's that simple.
To show you more graphically what that means in real life, here's a frequency response diagram for a typical real room: You can see what I'm talking about, visually there. At about 300 Hz, the peaks and valleys stop. Below that, it's all "mountains of the moon", but above 300 Hz, the line gets a hell of a lot flatter, with only minor variations. Thus, you could assume that the Schroeder frequency for this room is somewhere around 300 Hz. ... and you can see why it isn't necessary to worry about predicting the modal response above that, because it's mostly flat. The remaining variations are not even modal in nature, but rather from things like reflections, comb filtering, and suchlike, so even trying to predict them is non-trivial. It's easier to measure the actual response of the room after it is built, then just treat the issues that are really there.
So, that's all you ever wanted to know about modes, and standing waves, and Schroeder frequency!
End of story....
Ummmm.... not really!
But it still doesn't tell you what to DO about it! And we still haven't talked about "room ratios" much.
So, this next part is mostly meant for people who are able to design their room from scratch, where they can change their dimensions a bit for the walls and ceiling height. It's not really relevant for people who are trying to make a studio inside an existing room, where the walls, floor, and ceiling are fixed, and cannot be moved. (It might still be very interesting for them to understand it, though).
Alrighty then: if modes are such a big problem, what can we do about that?
In reality, not much!
The ONLY tool at your disposal to help with modes, is deciding how far apart you will place the walls, floor, and ceiling. In other words, the dimensions of the room. Since modes form at frequencies related to the distance between the room boundary surfaces (walls, floor, ceiling), then by playing around with those distances, you can indeed have some control of the modes..... but even then, not much. What matters here is not so much the actual dimensions themselves (although they are important for other reasons), but rather the relationship between the dimensions:
That's where "room ratios" come in: The entire point of looking for a good room ratio, is to find a set of dimensions for your room that spreads out the available modes as evenly as possible among the notes in the low end of the scale. The idea is to try to make the room have at least one mode for each note, down as low as you can. And the other big point is to NOT have modes that line up with each other, as that would make that note ring twice as loud and twice as long. So "room ratios" is all about fiddling with the dimensions to get the modes into good places on the low end of the spectrum: keep them spread out evenly and smoothly, without being bunched up in some places, or having large gaps between them in other places.
Room ratios are normally reference to the height of the room: In other words, it answers the question: "How long is the room relative to the height, and how wide is it relative to the height?" The ratio is often expressed in the form H:W:L (Height : Width : Length), as a set of three numbers where the first one represents the height (and is almost always "1"), the second one is the width, and the third one is the length. So a "room ratio" of 1:2:3 would mean that the room is twice as wide as it is high, and three times as long as it is high. The actual dimensions might be 8 feet high by 16 feet wide by 24 feet long, or 10 feet high by 20 feet wide by 30 feet long, or 12 feet high by 24 feet wide by 36 feet long... in all three cases the RATIO is the same: 1:2:3... so the modal spread would be the same for all three of these rooms, even though the frequencies would be different. But in fact it's the modal spread that matters more: The relationship between the modes.
In fact, I did it for you: here are the modal analysis results for those three cases from above, for three different rooms:
You can see that the relationship between the modes has not changed at all: the lines are still in the same places relative to each other. But the frequencies have changed: they are now at different spots on the musical scale.
So, if you are lucky enough to be able to change the dimensions of your room (by moving the walls and ceiling around), then you can choose a "room ratio" that has the modes spaced out sort of evenly, and NOT choose a ratio where the modes are bunched up together. For example, if your room is 10 feet long and 10 feet wide and 10 feet high (3m x 3m x 3m), then all of the modes will occur at the exact same frequency: 56.5 Hz. So the resonance when you play an A1 on the bass, or cello, or hit an A1 on the keyboard, will by tripled! It will be three times louder and very disgusting. The nulls will be three times deeper. That's a bad situation, so don't ever choose room dimensions that are the same as each other, if you can avoid it.
You get the same problem for dimensions that are multiples of each other: a room 10 feet high (3m) by 20 feet wide (6m) by 30 feet long (9m) is also terrible. That would be the ratio I mentioned above as an example: 1:2:3: Bad! All of the second harmonics of 10 feet will line up with the 20 foot modes, and all of the third harmonics will line up with the 30 foot modes, so you get the same "multiplied" effect. Bad. Bad. Bad.
In other words, you want a room where the dimensions are mathematically different from each other, with no simple relationship to each other. It turns out that as long as they are different by 5%, you are fine. So that room with the 10 x 10 x 10 dimensions would be much better if the width was 5% narrower, at about 9'6", and the ceiling were 5% higher, at about 10'6". That's a better ratio.
That brings up the obvious question: What ratio is best?
Answer: there isn't one!
Over the years, many scientists have tested many ratios, both mathematically and also in the real world, and come up with some that are really good. The ratios they found are named after them: Sepmeyer, Louden, Boner, Volkmann, etc. Then along came a guy called Bolt, who drew a graph showing all possible ratios, and he highlighted the good ones found by all the other guys, and predicted by mathematical equations, plus a few of his own: If you plot your own room ratio on that graph, and it falls inside the "Bolt area", then likely it is a good one, and if it falls outside the "Bolt area", then likely it is a bad one. Sort of. But not really! This is the "Bolt Region": As I said: If you figure out your room ratio, and plot it on that graph, and it is inside the curvy thing in the middle, then you are probably OK (but not guaranteed). And if your ratio is outside the curvy shape, then it is probably not OK (but it still might be). The reason for those caveats is simple: Later research has show that the Bolt region isn't really carved in stone at all: there are some good ratios outside of it, and there are some bad ratios inside of it. In fact, a few years ago Cox published a new diagram, which includes the bolt area, and also adds other areas. His diagram looks like this: The red triangle outlines the set of ratios recommended by the IEC, the orange triangles outline the set of ratios recommended by the EBU, and the green one is the original Bolt diagram, the white areas in the background are predicted by computer simulation to be really bad, the light gray areas are predicted to be somewhat good (fairly even spread), and the dark gray areas are predicted to be very good. Perhaps the best conclusion you can draw from this, is that if your ratio is within one of the areas where all four sets of "good" overlap (Bolt, IEC, EBU and dark gray), then you should be great! But as you can see, there aren't many of those! In fact, there is just one very small patch where they all coincide. Here's a link to the actual calculator show in that diagram above: http://www.acoustic.ua/forms/rr.en.html
And one more related measurement: Bonello.
Along with all the other researchers mentioned above, in the late '70s/early '80s a guy called Oscar Bonello came along, and presented a theory that, if you plot the number of modes in each one-third octave band for any given room, then the resulting graph should show a continuous rise with increasing frequency: in other words, the number of modes in each 1/3 octave band should be greater than the number in the band just below it, and less than the band just above it. Worst case, it should be the same as the one below it, but not lower. He postulates that if there is a decrease in the number of modes as you go up the scale, then that's bad. Thus, many room mode calculators today will also generate a "Bonello plot" to show you that information, and help you make a decision about your room ration. Here's an example: That one shows the actual set of room modes across the top for a room that measures 17 feet long, 13 feet wide, and 9 feet high (I chose those numbers arbitrarily). Ir also has several smaller plots below that main graph. The second one from the left is the "Bonello chart", and the one on the right is the Bolt plot. The red "X" marks the location of this specific room ration within the Bolt area. Also, you can see that the Bonello chart shows that the numbers stay the same or rise as you go from left to right, and visually you can see on the top modal plot, that the modes are fairly evenly spread out. This room would probably be good, from the modal point of view. Now here's another one, where I made the room two feet narrower but kept the length and height the same: As you can see from the Bonello plot, the numbers go up, then down, then up again, then down, then up again, and the ratio is well outside the Bolt area. You can even see from the actual modal plot across the top that they are not spread evenly now: there's a big gap between 31 Hz and 51 Hz with no modes at all, then several bunched together around 85 Hz... etc. So this room would probably not be good, from the modal point of view.
So, in summary: there are no perfect ratios, only good ratios and bad ratios.
It is impossible to have a "perfect" ratio in a small room, simply because that would require that there are enough modes so you can have three of them for every note on the musical scale, (remember Mr. Schroeder?). In other words, the Schroeder frequency would have to be lower than 20 Hz. ... but that's the entire problem with small rooms! They are too small to have many modes at low frequencies! There just are not enough modes in the low end. So you can choose a ratio that spreads them a bit more this way or a bit more that way, but all you are doing is re-arranging deck chairs on the Titanic, in pleasant-looking patterns. The problem is not the location of the deck chairs; the problem is that your boat is sunk! Likewise for your studio: the problem is not the locations of the modes: the problem is that your room is sunk. No matter what you do with the dimensions, you cannot put a mode at every note, unless you make the room bigger. It is physically impossible.
But that does not mean that your room will be bad. That's yet another common perception, and it is dead wrong. People think that if they can't get a good ratio for their room, then they might as well just give up and forget about it. Wrong! Carry on reading to find out why...
A word of warning here: Every now and then I have to answer questions about way-far-out-ultra-crazy methods for determining the "perfect" room ratio: every few years, somebody new comes out with a frankly nutty new "method" for making a perfect room. Fortunately, most of those mysteries fade quickly. But one of those, which stubbornly refuses to die, is the infamous "Cardas Golden Cuboid" ratio. Supposedly, you can use this mystical mathematical method for everything from building a perfect room, to laying out your speakers in any room, to making speaker wires, and even to making tiny little wooden blocks that will isolate your gear from the room (I kid you not... you can buy a set of six of those for "just" US$ 20 each! Imagine that... And you can even buy a set of 2 meter long speaker wires manufactured using the same mystical ratio, for just US$ 2,480... yup. That's not an error: they really do cost two thousand four hundred and eighty United States dollars, and they are 2 meters long (about 6 feet) There really is one born every minute...) Anyway, getting back on point: supposedly, there is a magical mathematical property to this "golden cuboid ratio" that is absolute perfection, and will make your room and your gear and even your cables incredibly perfectly something-or-other... except of course that there is no such thing! Check with any mathematician you might happen to know, and ask him what the "golden cuboid ratio" is: you'll get a blank stare. It does not exist. There is no mathematical theory behind this... it is pure fantasy, and meaningless. Stay away. The advice given for setting up speakers using this snake-oil ratio, is really, really bad. If somebody suggest that you should consider the "golden cuboid" for your room, then suggest that they should see a shrink... In fact, any place that sells simple wire for a thousand dollars per meter, is clearly out to scam you, so you'd be silly to believe anything else they tell you.
OK, so modes are bad, they eat your mixes, ratios can help, but there's no such thing as a good one, and snake-oil sellers are still around today.
All of this leads to the question you didn't ask yet, but are probably heading for: What can I do about it?
Here's the thing: Modes are only a big problem if they "ring". The standing wave is only a problem if the energy builds up and up and up, with each passing cycle, until it is screaming, and then that "built up" energy carries on singing away, even after the original note stops. That's the problem. If you stop playing that A1 on your guitar, and the room keeps on playing an A1 for a couple of seconds more, because it "stored" the resonant energy and is now releasing it, then that's a BIG problem! The room is playing tunes that never were in the original music!
If a mode doesn't ring like that, then it is no longer a major issue. (It is still an issue for other reasons, just not a major one....)
So how do you stop a mode? Well, you can't stop it from being there. But you CAN stop it from "ringing". You can "damp" the resonance sufficiently that the mode dies away fast, and does not ring. You can remove the resonant energy and convert it into heat: no more problem! This is where bass traps come in.
A simple analogy: it's not good if you own a large angry dog that barks all the time and bites your visitors, but it's fine to own a large angry dog with a muzzle on his mouth, so he cannot bark and cannot bite! That's what bass trapping is all about: it puts a muzzle on your modes. A bass trap doesn't get rid of the problem, but it does keep it under control. You use strategically placed acoustic treatment devices inside the room that absorb the ringing of the mode, then it cannot ring. There are several ways to do that, with different strategies, but the good news is that in most rooms it is possible to get significant damping on the worst modes, so that they don't ring badly, and don't cause problems. Note that bass trapping does not absorb the mode: it just absorbs the ringing. Some people don't understand this, and think that the bass trapping makes the modes go away: it doesn't. All it does is to damp them. The modes are still there, and still affect the room acoustics in other ways, but with good damping, at least they don't "ring" any more.
And that is the secret to making a control room good in the low end! Choose a good ratio to keep the modes spread around evenly, then damp the hell out of the low end, so modes cannot ring. It's that simple.
The smaller the room, the more treatment you need. And since those waves are huge (many feet long), you need huge bass trapping (many feet long/wide/high/deep). It takes up lots of space, and the best place for it is in the corners of the room, because that's where all modes terminate. If you want to find a mode in your room, go look for it in the corner: it will be there. All modes have a pressure node in two or more corners, so by treating the corners, you are guaranteed of hitting all the modes. Modes also affect the walls between the corners, of course (sometimes people forget this simple fact), but corners are the best place to "trap" them.
As I said, there is no single "best" ratio, but there are good ones. You can use a "Room Mode Calculator" such as the one I used to make the graphs above, to help you figure out which "good ones" are within reach of the possible area you have available, then choose the closest good one, and go with that. And stay away from the bad ones.
Arguably, Sepmeyer's first ratio (1 : 1.14 : 1.39) is the "best", since it can have the smoothest distribution of modes... but only if the room is already within a certain size range. Other ratios might be more suitable if your room has a different set of possible dimensions. So there is no "best".
As I mentioned earlier, the complete set of modes in your room consists of the axial modes, plus the tangential modes, plus the oblique modes. That's what a good room mode calculator (a.k.a. "room ratio calculator") will show you. There are bad calculators that only show you the axial modes, which is pretty pointless, and good ones show you all three types. Yes, it is true that tangential modes are lower in intensity than axials, and that oblique modes are lower still, but they can still cause you trouble. So you can't ignore those. If you find a calculator on the internet that only calculates axial modes, forget it: its no use.
Use one of these Room Ratio calculators to figure out the best dimensions for your room:
Both of those are very good, and will help you to decide how best to build your room. They give you tons of information that is really useful to help figure out the best dimensions. I used the second one above ("Amroc", by Andy Mel) to generate the graphs that I used earlier.
Now, after having said all of that humongous rant above about modes, there's one more thing I have to say about them, and this is the most important of all:
It is that simple.
For the majority of home studios, modal response isn't that important, and you can't change it anyway, since changing it means knocking down walls! So why worry about things that aren't important, when there are much more important things to worry about!
Modal response is just one of many aspects that I take into account when designing a room, but there are many more. For example, if I would have to bring down the height of the ceiling in a room that already had a low ceiling, just to get a better ratio, I would not do that. Also, if I would have to reduce the volume of a room that already had very little volume, I would not do that either. Ditto for changing the width or height, if it would create reflection problems at the listening position, or changing the length, if it would make the rear wall SBIR dip worse. You can treat problematic modes inside a room to get them under control, but you can't "treat" a room to make it have a higher ceiling or more volume! Those two things are more important than room modes, in most situations.
So, when you design your room, it is wise to choose a ratio that is close to one of the good ones, or inside the Bolt area, but you do NOT need to go nuts about it! Occasionally I even hear stories about "acoustic experts" who go ballistic about ratios, and will totally reject a room, outright, without even considering it at all, if it does not meet their rigid agenda about modal response, the Bonello plot, the Bolt diagram or some other criteria they have come up with. That makes no sense. Modes really are not that big of an issue, if the room is treated correctly! (I sometimes wonder if those "experts" just don't know how to treat a room correctly, and are scared to even try! So they have to reject any room that doesn't fit their agenda... They only take on rooms that are simple to treat, that doesn't need a good understanding of the acoustic issues, Hmmm....). In reality, almost ANY room can benefit tremendously from good acoustic treatment, including rooms with "terrible" ratios and "awful" modal response.
There's no need to nudge things around by millimeters or smalls fractions of an inch, hoping to get a "better" ratio. Just stay away from the bad ones, get close to a good one, and you are done. End of story.
If you are still awake after reading through all of that, then hopefully you found something useful in there! And if you are asleep, then I'll shut up and let you carry on sleeping, peacefully ....
- Stuart -