13 × 20 × 9 ft room
Every standing wave this room supports below 300 Hz, solved from its geometry alone. Lowest mode 28.1 Hz on the length; 20 axial, 101 tangential and 129 oblique modes in total.
Mode distribution to 300 Hz
Every mode this room supports, at its exact frequency. Even spacing reads as a well-behaved room; a wide gap followed by a clump is where it will sound hollow and then boom.
Volume
66.3 m³
2340 ft³, 103.5 m² of surface
Worst axial gap
28.1 Hz
widest hole between adjacent axial modes under 200 Hz
Densest cluster
10 modes at 191 Hz
modes within 5 Hz of each other, below 200 Hz
Bolt criterion
inside
ratio 1 : 1.44 : 2.22 (height : width : length)
The three fundamentals
One per pair of parallel surfaces, at c/2L with c = 343 m/s. These are the loudest and most audible resonances in the room; everything else is built on them.
length
28.1Hz
20 ft · half-wavelength fit
width
43.3Hz
13 ft · half-wavelength fit
height
62.5Hz
9 ft · half-wavelength fit
Axial modes in full20 modes
The modes that run between one pair of parallel surfaces. They carry the most energy and are the ones worth treating first.
| Frequency | Axis | Order | Indices (nx, ny, nz) |
|---|---|---|---|
| 28.1 Hz | length | 1 | 1, 0, 0 |
| 43.3 Hz | width | 1 | 0, 1, 0 |
| 56.3 Hz | length | 2 | 2, 0, 0 |
| 62.5 Hz | height | 1 | 0, 0, 1 |
| 84.4 Hz | length | 3 | 3, 0, 0 |
| 86.6 Hz | width | 2 | 0, 2, 0 |
| 112.5 Hz | length | 4 | 4, 0, 0 |
| 125.0 Hz | height | 2 | 0, 0, 2 |
| 129.8 Hz | width | 3 | 0, 3, 0 |
| 140.7 Hz | length | 5 | 5, 0, 0 |
| 168.8 Hz | length | 6 | 6, 0, 0 |
| 173.1 Hz | width | 4 | 0, 4, 0 |
| 187.6 Hz | height | 3 | 0, 0, 3 |
| 196.9 Hz | length | 7 | 7, 0, 0 |
| 216.4 Hz | width | 5 | 0, 5, 0 |
| 225.1 Hz | length | 8 | 8, 0, 0 |
| 250.1 Hz | height | 4 | 0, 0, 4 |
| 253.2 Hz | length | 9 | 9, 0, 0 |
| 259.7 Hz | width | 6 | 0, 6, 0 |
| 281.3 Hz | length | 10 | 10, 0, 0 |
Questions this room answers
What is the lowest room mode in a 13 × 20 × 9 ft room?
28.1 Hz, set by the length (20 ft) dimension. A half-wavelength has to fit between the two parallel surfaces, so the first mode is c/2L — with c = 343 m/s at 20 °C. Below this frequency the room cannot support a resonance at all, and bass output falls off regardless of the loudspeaker.
Where will a 13 × 20 × 9 ft room sound uneven?
The densest clustering below 200 Hz is 10 modes within 5 Hz of each other around 191 Hz — expect that region to sound louder and to ring longer. The largest gap between adjacent axial modes below 200 Hz is 28.1 Hz, and wide gaps are heard as a hole, not as neutrality.
Are 13 × 20 × 9 ft proportions good for a listening or control room?
The ratio (1 : 1.44 : 2.22, height : width : length) falls inside the Bolt area — the region Bolt published in 1946 within which low-frequency modes distribute comparatively evenly. That is one criterion about proportions only; it says nothing about absorption, symmetry or speaker placement.
Above what frequency do modes stop mattering here?
The Schroeder frequency for this 66.3 m³ volume is 135 Hz if RT60 is 0.3 s, and 190 Hz if RT60 is 0.6 s. Above it the modes overlap densely enough to behave statistically rather than individually. RT60 is a property of the materials in the room, not of its shape, so this page can only bracket it — measure your own to place the crossover exactly.
One dimension different
The comparison worth making before you build: change one dimension by a foot and every mode moves.
Method and limits. Frequencies solve f = (c/2)·√((nx/Lx)² + (ny/Ly)² + (nz/Lz)²) for a rectangular room with rigid boundaries, at c = 343 m/s (20 °C). Real rooms have windows, doors, soffits and non-rigid walls, all of which shift and damp these frequencies — treat this as where to look, not as a measurement. The Schroeder crossover additionally depends on RT60, which is a property of the room's materials and not of its shape, so it is quoted here against two assumed values rather than as a fact about this room. Nothing on this page is fetched or estimated; it is solved from the three dimensions in the URL.