Best room shape for 1,333 ft³–2,894 ft³
One shape serves every room between 1,333 ft³ and 2,894 ft³: length 2.247 and width 1.281 times the ceiling height. Nothing in the band is more than 4.72% off its own optimum.
Shape (L : W : H)
2.247 : 1.281 : 1
scale-free — multiply by the ceiling height you have
Covers
1,333 ft³–2,894 ft³
1,562 cubic-foot values
Worst case in band
4.72%
flat across 1,333 ft³–2,550 ft³
At 2,114 ft³
11.56 × 20.27 × 9.02 ft
widest axial gap 20.91 Hz, 14 axial modes below 200 Hz
Dimensions across the band
The same proportions at a spread of volumes inside the band. Only the absolute sizes change — they scale as the cube root of the volume — so find the row nearest the space you have and build to it. The last column is what that room gives up against the best shape known at its own exact volume.
| Volume | Width × Length × Height | Widest axial gap | vs optimal |
|---|---|---|---|
| 1,333 ft³ | 9.91 × 17.38 × 7.74 ft | 24.39 Hz | +4.72% |
| 1,556 ft³ | 10.44 × 18.30 × 8.15 ft | 23.16 Hz | +4.72% |
| 1,779 ft³ | 10.91 × 19.14 × 8.52 ft | 22.15 Hz | +4.72% |
| 2,002 ft³ | 11.35 × 19.90 × 8.86 ft | 21.30 Hz | +4.72% |
| 2,225 ft³ | 11.76 × 20.62 × 9.18 ft | 20.56 Hz | +4.72% |
| 2,448 ft³ | 12.14 × 21.28 × 9.47 ft | 19.91 Hz | +4.72% |
| 2,671 ft³ | 12.50 × 21.91 × 9.75 ft | 19.34 Hz | optimal |
| 2,894 ft³ | 12.84 × 22.51 × 10.02 ft | 18.83 Hz | optimal |
Why one shape covers the whole range
The objective is the widest gap between adjacent axial modes below 200 Hz. Under pure scaling every axial frequency moves as the cube root of the volume, so the best proportions do not change with volume at all — until a mode crosses the 200 Hz window edge and the maximum jumps to a different pair. Across this band the optimal proportions are not constant — they jump as the volume grows — so no single shape is optimal throughout it. The one published here is the single shape that stays within the tolerance of all of them. The worst any room does is 4.72%, and that figure is flat across 1,333–2,550 ft³ rather than peaking at one unlucky volume — where neither shape's spectrum changes which modes sit under 200 Hz, both objectives scale as the cube root of volume and their ratio does not move.
The shape sits strictly inside the Bolt area, so the criterion is not what chose it — the objective is.
Questions this band answers
What is the best room shape for 1,333 ft³–2,894 ft³?
A ratio of 2.247 : 1.281 : 1 — length : width : height. At the middle of the band, 2,114 ft³, that is 11.56 × 20.27 × 9.02 ft, and the widest gap between adjacent axial modes below 200 Hz is 20.91 Hz. The same ratio applies at every volume in the band; only the absolute dimensions scale, as the cube root of the volume.
How much worse than optimal is this shape for my exact volume?
At most 4.72%, and that figure holds flat across 1,333 ft³–2,550 ft³ rather than at one unlucky volume — above that range it is smaller. The optimal proportions are not constant across this band; they jump as the volume grows, and this shape is the single one that stays within the published tolerance of all of them.
Why is this a range rather than a page per volume?
Because a recommended room gets built to a finite tolerance. A ±1 inch error in framing moves this objective by a median 6.86%, so splitting the range more finely than that publishes distinctions nobody can construct. Three shapes cover 800–5,000 ft³ at that tolerance, and the covering is provably minimal there rather than merely an upper bound, because each shape's coverage set is a single unbroken interval.
Is this shape inside the Bolt area?
Yes — every shape published here is. The shape sits strictly inside the Bolt area, so the criterion is not what chose it — the objective is. Bolt (1946) is a citable constraint set rather than a sweep range someone typed: it caps length at 3.2 times the height, which puts the corridor outside the feasible set, and forces height ≤ width < length. Without it the answer to "what shape?" at a fixed volume is "build a longer corridor", which is a tautology rather than advice.
What does the widest axial gap actually tell me?
It is the largest hole in the low-frequency response you cannot fix with absorption. Axial modes are the strongest room resonances, and a wide gap between two adjacent ones is a band where the room supports very little. Minimising the widest gap spreads the modes as evenly as the volume allows. It says nothing about tangential or oblique modes, absorption, speaker placement or anything above 200 Hz.
The neighbouring bands
A different range, and a genuinely different shape — a boundary is where this shape stops being within tolerance, not an arbitrary cut.
Method and limits. Solved by minimising the widest gap between adjacent axial modes below 200 Hz, at fixed volume, over the Bolt (1946) ratio area. Axial modes are n·c/(2L) per axis with c = 343 m/s. The band is a covering: one shape is published for a range of volumes, with the range chosen so that no volume inside it pays more than 7% against the best shape known at that volume — the first tolerance above the 6.86% a ±1 inch framing error costs. "Best known" is the minimum over a live solve and every shape the solver returns anywhere on the range, which matters because the search is itself imperfect: it is beaten at 2.43% of volumes by a shape it finds elsewhere. Worst-case figures are measured across the band, not assumed. Tangential and oblique modes, absorption and placement are out of scope. Pure computation — nothing is fetched and nothing is recalled.