What a mode is, in one image
Pluck a guitar string: it does not vibrate at any frequency, only at those its length allows. A room does the same with the air it contains. Between two parallel surfaces, a wave that returns on itself in phase reinforces; the others cancel.
The result is a set of preferred frequencies, the room modes, each with a fixed geometry in the room: antinodes where pressure is maximal, nodes where it cancels. That pattern depends neither on the subwoofer nor on the amplifier: it depends only on the dimensions.
The consequence is the one described in the page on subwoofer placement: at one spot a frequency will be too loud, fifty centimetres away almost absent.
Three families, and only one that really counts
A mode is designated by three integers, one per dimension. The frequency follows from the speed of sound and the three dimensions of the room:
f = (c / 2) × √( (nx/L)² + (ny/W)² + (nz/H)² )
Depending on how many of those integers are non-zero, three families are distinguished.
| Family | Non-zero integers | Surfaces involved | Count | Energy |
|---|---|---|---|---|
| Axial | 1 | 2, opposing | the lowest | the highest |
| Tangential | 2 | 4 | intermediate | distinctly weaker |
| Oblique | 3 | 6 | the highest | the weakest |
Hence a useful rule for reading a measurement: when looking for the cause of a bass problem, start with the axial modes.
What that gives in a real room
Take the 5.00 × 4.00 × 2.50 m room used as the example in the other pages. Below 120 Hz it has 18 modes, of which only 6 are axial. Here are the first twelve:
| Frequency | (nx, ny, nz) | Family |
|---|---|---|
| 34.3 Hz | (1, 0, 0) | axial · length |
| 42.9 Hz | (0, 1, 0) | axial · width |
| 54.9 Hz | (1, 1, 0) | tangential |
| 68.6 Hz | (0, 0, 1) | axial · height |
| 68.6 Hz | (2, 0, 0) | axial · length, order 2 |
| 76.7 Hz | (1, 0, 1) | tangential |
| 80.9 Hz | (0, 1, 1) | tangential |
| 80.9 Hz | (2, 1, 0) | tangential |
| 85.8 Hz | (0, 2, 0) | axial · width, order 2 |
| 87.9 Hz | (1, 1, 1) | oblique |
| 92.4 Hz | (1, 2, 0) | tangential |
| 97.0 Hz | (2, 0, 1) | tangential |
Look at the two rows at 68.6 Hz.
The trap of simple ratios between dimensions
Those two modes land on exactly the same frequency, and that is no accident: the height is 2.50 m, precisely half the 5.00 m length. The first-order height mode and the second-order length mode therefore coincide.
This is called a modal coincidence, and it concentrates the energy of two modes into a single frequency instead of spreading it. In this room, three such coincidences occur below 120 Hz: at 68.6 Hz, at 80.9 Hz and at 109.8 Hz.
That is the real content of the rule about avoiding simple ratios, and the comparison with the extreme case, the cube, is telling:
| Room | Coincidences below 120 Hz | Modes stacked each time | Cause |
|---|---|---|---|
| 5.00 × 4.00 × 2.50 m apparently well proportioned | 68.6 · 80.9 · 109.8 Hz | 2 | 2.50 m is half of 5.00 m |
| 3.00 m cube | 57.2 · 80.8 · 114.3 Hz | 3 | all three dimensions are equal |
The cube stacks three modes per coincidence instead of two, so each resonance there is correspondingly sharper. But it is the first row that should worry you: it only takes one dimension being a simple multiple of another, and a domestic room with a 2.50 m ceiling is very often exposed to it.
The frequency above which the question stops mattering
Modes stop being counted once they become too numerous and too closely spaced: they overlap, and the field becomes statistically diffuse. The boundary has a name, the Schroeder frequency, estimated from the volume of the room and its reverberation time.
| Room RT60 | Schroeder frequency (50 m³ volume) |
|---|---|
| 0.3 s | ≈ 155 Hz |
| 0.4 s | ≈ 179 Hz |
| 0.6 s | ≈ 219 Hz |
Two lessons. First, the widely quoted order of magnitude of 200 Hz is not arbitrary: it comes out of this calculation for a typical domestic room. Second, the boundary moves up when the room is more reverberant: a poorly treated room stays modal higher in frequency.
What a mode does to the ear: the ringing, not just the level
A mode is often described by its effect on the frequency response, a peak or a null. That is incomplete, and it hides what bothers listening most.
A mode is a resonance: once excited, it keeps vibrating after the signal has stopped. That ringing in time is what gives bass a heavy, muddled character, where one note masks the next. Two rooms can show the same response curve and sound very different, because their modes do not decay at the same rate.
It is also why measured reverberation time is almost always longer in the bass than in the midrange, and why a waterfall plot says things a plain response curve does not. The guide on measuring with REW covers reading that.
The only three levers, and what each can really do
Here is the practical conclusion, and it is easy to remember because there is nothing else.
| Lever | What it changes | What it does not change | What it costs |
|---|---|---|---|
| 1. Geometry | the frequency of the modes: the only lever that reaches it | nothing else | building or repartitioning. At the drawing stage, avoiding a coincidence costs a few centimetres |
| 2. Position | where you sit within the mode: near an antinode or a node | no mode is moved | nothing. Free, immediate, often the most effective |
| 3. Damping | the duration of the ringing, and the rounding of the peak | the frequency of the mode | thickness and surface area |
| Equalisation is not a lever on the mode | a level at the measurement point; it can attenuate a peak | neither the geometry nor the duration. It does not fill a node | it comes last, on what the other three have left |
1. Geometry. Changing a dimension moves the modal frequencies. It is the only lever that acts on the frequency itself, and it means building or repartitioning. It is also why a project is drawn before the building work: at that stage, avoiding a modal coincidence costs a few centimetres.
2. Position. Moving the source or the seat changes no mode, but changes where you sit within it. You choose to be near an antinode or a node. It is free, immediate, and often the most effective: that is the subject of the subwoofer page.
3. Damping. Absorbent treatment in the bass does not move a mode's frequency, it shortens its ringing and rounds off its peak. It is the only lever that acts on time, and therefore on the muddling described above.
And what is not a lever on the mode: equalisation. It corrects a level at the measurement point and can attenuate a peak, but it changes neither the geometry of the mode nor its duration, and it does not fill a node. It comes last, on whatever the other three levers have left.
Where the answer is different
The room is not a rectangular box. The formula assumes six flat surfaces, parallel in pairs. With an L-shaped room, a sloping ceiling or a loft, the modes still exist but no longer compute that way: a numerical solution is needed.
The surfaces are not rigid. A lightweight partition vibrates and absorbs in the bass, which damps the modes while letting energy escape. The calculation assumes perfectly reflective surfaces, so it is pessimistic on amplitude and optimistic on isolation.
The room opens onto another. The volume to consider is no longer that of the listening room alone, and the modes of the whole partly replace those of the room.
Temperature changes. The speed of sound depends on it, so the frequencies do too, in the order of a few tenths of a percent per degree. Negligible in practice, but it explains small differences between two measurements.
What HTM computes
HTM builds the list of your room's modes from the geometry you entered, flags the coincidences, and displays the pressure field map at a given frequency, which makes antinodes and nodes visible at the seats you declared.
For non-rectangular shapes the calculation is numerical rather than analytical, which is exactly the point: that is the case where the formula no longer suffices.
These are predictions, dependent on the dimensions entered and on assumptions about how the surfaces behave. The full approach is to predict and then verify by measurement, which HTM can compare against its own calculations.
Sources
- The room mode formula for a rectangular volume is classical in room acoustics; it follows from the wave equation with rigid boundaries.
- The Schroeder frequency is estimated by
2000 × √(RT60 / V), with reverberation time in seconds and volume in cubic metres. It is an order of magnitude, not a sharp boundary. - Every frequency on this page was computed for a speed of sound of 343 m/s, the usual value at 20 °C, on a 5.00 × 4.00 × 2.50 m room and a 3.00 m cube.
Further reading
The page on subwoofer placement makes use of the second lever, position, and the glossary defines the terms used here. The guide on measuring with REW shows how to observe all of this in your own room.