Optimizing Room Size / Mode Calculation

A series of articles on how to design and plan the acoustic treatment of rooms for recording music and for critcal listening.

Optimizing Room Size / Mode Calculation

Postby Scott R. Foster » Fri May 09, 2008 3:28 pm

Optimizing Room Size / Mode Calculation

Modes are the driving forces behind a room’s reverberant field.  A mode is one of the resonances of an enclosed space; a mode can be interpreted by its dimension.  All rooms have modes, but not all rooms, or modes, are equal.   Some physical shapes are more likely than others to reflect sound waves along a repetitive path causing a modal resonance.  In a fairly bare rectangular room, the surfaces form three pairs of large parallel surfaces; each surface will bounce many waves back and forth with its parallel counterpart.  The modes that are generated between two parallel surfaces are called axial modes. There are other paths around a rectangular room besides back and forth between just two walls (as there is more than one way to bounce a pool ball around a pool table and back to where it started) but the paths between the parallel surfaces, the axial modes, are easier to picture and understand, and easier to calculate in your head... so for illustration purposes it helps to discuss solely the axial modes.   This does not mean you can ignore the other modes!  Do not be misled by trying to use a mode calculator that only looks at axial modes. Use a tool like this online calculator:

Bob Golds Online Mode Calculator

Or this Excel spreadsheet:

Eric's Calculators & Files

Think of a room's modes as its "tuning"... like an organ pipe having a pitch arising from its length.  You want your room to have en even distribution of resonances so that it will be as "uncolored "as possible.  

As an illustration, picture a spherical room 11.30’ in diameter it will have a single resonance of 50 Hz (and the harmonics of 50 Hz). Sound moves in air (at standard temperature and pressure) at 1,130 fps, thus a wave of sound with a frequency of 50 Hz (or cycles per second) in size fits perfectly bouncing between the surfaces of a spherical room 11.3’ in diameter (1,130 fps / 11.30’ / 2).  This means that sounds at this frequency (50 Hz) will echo very efficiently in this room.  Additionally, all of the harmonics of this wave will fit perfectly (50 Hz, 100 Hz, 150 Hz, 200 Hz, etc.).  This is not good. This room, if the walls are left reflective, will be highly “colored” at these frequencies.  To use such room for musical purposes it would need to be made very dead, otherwise everything we did in it would whistle a tune back at us in the key of 50 Hz.

Even in large, less reflective rooms one must pay attention to modes especially at the low end of the sound spectrum where modal density is low (generally below 200 Hz – but this threshold depends on the volume and reverb time of the room).  Modal density gets higher as you climb the band for the simple reason that you have more multiples (harmonics) from all the different resonances of the room mixing together and coming across more evenly. Nonetheless even at high frequencies modes can create problems if the harmonics of a number of different modes all line up at or very near a single frequency.

Assume a more typical room with parallel walls, floor and ceiling, having an 11.30’ ceiling, a width of 22.60’ and a depth of 45.20’.  The fundamental axial resonances of the three pairs of parallel surfaces in this room are not all 50 Hz like the single resonance of a sphere. But these dimensions still cause a similar problem.  They are no longer all 50Hz, instead only the ceiling has a 50 Hz fundamental resonance, but the first width mode is 25 Hz, and the first depth mode is 12.5 Hz.  These modes will support a string of harmonics up to 200 Hz as follows:

Ceiling 50, 100, 150, 200
Width, 25, 50, 75, 100, 125, 150, 175, 200
Length, 12.5, 25, 37.5, 50, 62.5, 75, 87.5, 100, 112.5, 125, 137.5, 150, 162.5, 175, 187.5, 200,

Sorting all of the above by size we get:

12.5, 25, 25, 37.5, 50, 50, 50, 62.5, 75, 75, 87.5, 100, 100, 100, 112.5, 125, 125, 137.5, 150, 150, 150, 162.5, 175, 175, 182.5, 200, 200, 200

Oops! Four triple redundancies at or below 200 Hz (at 50, 100, 150 and 200 Hz). We have radically changed the shape and dimensions of the room but we haven’t solved the problem of spacing the modes out so that they don’t line up.  Because these dimensions are small whole number multiples of 11.3, they still line up with disturbing regularity, for example, if we run these strings long enough, we will see that all of the floor to ceiling modes will be resonances of both the width and length dimensions.  Yikes! – this aint much better than the sphere, what do we do?  

The short lo-tech answer is to take into account the practical constraints of the budget and the practical potential wall locations we have to work with, and then use a spreadsheet, or calculator to string these numbers out for sets of dimensions we are capable of achieving.  With each of our iterations of dimensional choice, we get a snapshot of how the modes will interact.  In essence we go hunting for a set of dimensions which we can (i) afford and fit into the space available; and (ii) does a reasonable job of spreading the modes out (particularly at the low end of the spectrum).  Tools for automating this search have been developed (the links above for examples) and using these tools you can improve significantly the spread of the modes generated by a room like the one given in the example above by simple trial and error adjustment of the available practical dimensions.

Better still read up on how different acousticians have classed a number of room ratios into a "known good" zone... and how you might go about changing your room's dimensions to fall into the "good" area.

Acceptable Room Ratios (an overview)
ROOM RATIOS: M.M. LOUDEN ANALYSIS

To understand [in a crude way] how the idea of "known good" ratios work, assume we have a large empty space (perhaps a basement) with finished walls measuring 20’ wide 40’ long with an open ceiling with a 12’ clear height available to the bottom of the joists.  Further assume we want to divide the space up into a recording room and a control room and add a finished ceiling to both rooms.  If our preliminary design specified a recording room with the dimensions of 20’ wide by 20’ long, and our planned ceiling was going to reduce the finished ceiling to 10’ in height, we can quickly see by reducing these dimensions to a ratio (1.00 : 2.00 : 2.00) that with this design we are going to create many coincident modal harmonics and that some adjustment is necessary if we want to spread the modes out evenly.

In trying to address this problem, we could take a look at increasing the long dimension to 21” and raising the ceiling to 11’. To quickly check what this does to our room’s modes we look at the ratio these dimensions generate (1.00 : 1.82 : 2.10).   Immediately this looks better. To refine our analysis, we could plug the original dimensions into a mode calculator so we can look at axial modal components, and then do the same with the proposed changes.  The results would look something like the following

Original:

28, 28, 57, 57, 57, 85, 85, 113, 113, 113, 141, 141, 170, 170, 170, 198, 198, 226, 226, 226, 254, 254, 283, 283, 283

Proposed Changes:

24, 28, 49, 51, 57, 73, 85, 98, 103, 113, 122, 141, 147, 154, 170, 171, 196, 198, 205, 220, 226, 245, 254, 257, 269, 283, 294

Running our eyes across the two lists of axial modes it is obvious that the proposed changes in the room's dimensions will yield much improved the smoothness of the distribution in the room's modal harmonics. Refining the dimensions further we would look at the tangential and oblique modes to see what problems they are apt to cause.  

Just as we wish to avoid building a room with modes bunched up together, we also want to avoid generating a string of modes that are too far apart, and in essence, leave a hole in the reverberant field of the room. Some rules of thumb for simple modal analysis are (i) avoid exact coincidences (modes that are exactly equal) (ii) try and keep every mode at least 5% larger than its predecessor; (iii) try to keep from creating mode strings with large holes [modes more than 25 Hz apart will probably leave an audible hole]; (iv) focus on the part of the sound spectrum below 200 Hz as problems here will be more troublesome.

If you study the links provided above you will see that much thought has evolved as what makes a "good room" - from the simple idea of balancing the dimensions of a room.  The concepts and methods outlined above are dirt simple but they will guide you in avoiding the deepest pitfalls.

Important! Building funny shaped rooms with canted walls and ceilings will not obviate the negative potentials of bad modal spacing - this practice only makes it impossible for non-experts to predict what the resulting modal structure will be [in fact such predictions are non-trivial even for experts]. Also, building funny shaped rooms usually ends up reducing the room volume from what it would be if a rectilinear room where built instead - and this makes all the problems of recording in a small room worse - because you made your small room even smaller! In short, there is no problem in small room acoustics that cannot be lessened by the simple expedient of building a larger room... making your room smaller so it can be funny shaped is a bad idea because you are making the fundamental problems increase in severity, and unpreditable in scope.
Scott R. Foster
 
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