A room is a three-dimensional resonator. Its three dimensions pick out a discrete set of standing-wave frequencies — the eigenfrequencies — that make bass boom in one corner and vanish at the next seat over. They are computable: f = (c/2)·√((nx/Lx)² + (ny/Ly)² + (nz/Lz)²). Down the center, the room goes in, the eigen-solver runs, the spectrum comes out. The blue team builds and defends it; the red team tries to break it.
source Lord Rayleigh (J. W. Strutt), The Theory of Sound, Vol. 2, 1877 — normal modes of a rectangular chamber. archive.org/details/theoryofsound02raylrich. Rendered, not quoted.
A rigid-walled rectangular room Lx×Ly×Lz supports a standing wave for every triple of non-negative integers (nx,ny,nz) — not all zero:
f = (c/2)·√((nx/Lx)² + (ny/Ly)² + (nz/Lz)²)
Three kinds by how many indices are non-zero: axial (one — between two walls), tangential (two — four walls), oblique (three — all six). An axial mode reduces to an organ pipe: n·c/(2L). For the current room, the live distribution of the lowest modes:
| mode | type | f (Hz) |
|---|
Set two indices to zero and √((n/L)²) collapses to n/L — the axial mode is exactly the-organ-pipe laid along one dimension: f = n·c/(2L). A room is three open pipes crossed, plus the Pythagorean obliques that use all three at once.
Each such mode is a fixed pattern of pressure nodes and antinodes — a frozen the-standing-wave in space, not time. A cube stacks its three axial modes at one frequency (degeneracy); unequal dimensions spread them. Rayleigh, 1877 → each sphere is the next one's premise.
The blue team's live check: re-solve the invariants — axial reduction to n·c/(2L), fundamental on the longest wall, cube degeneracy, and the Pythagorean identity f111² = f100² + f010² + f001². If red drops the squares, this badge is where it shows.
Feed the eigen-solver a box — three dimensions in metres — and a mode index (nx,ny,nz). The speed of sound is fixed at c = 343 m/s (air, 20 °C). The three mode families:
| type | non-zero | reflects off |
|---|---|---|
| axial | 1 index | 2 walls |
| tangential | 2 indices | 4 walls |
| oblique | 3 indices | 6 walls |
Axial modes are strongest (all the energy on one axis) and are the ones that boom. That box and that index are what you feed the panel below.
Change any dimension or index — the frequency and the whole spectrum are solved on the spot, never looked up.
Spectrum of every mode up to order 4 below 250 Hz — axial, tangential, oblique. Watch them pile up when the room goes cubic.
What the machine produces, proven: the discrete eigen-spectrum of the box. The fundamental is the axial mode along the longest wall; the modes cluster unevenly, and a cube collapses its three axials onto one frequency — the worst case. Every value below is computed from the Pythagorean law, not stored.
The blue team's witness (left) re-solves these invariants live; the red team (right) tries to make them wrong.
And modes only rule the low end. Above the Schroeder frequency (≈ 2000·√(T60/V)) they overlap so densely that the field is statistical — that is Sabine's regime, not Rayleigh's. This sphere is the wave theory below Schroeder; it does not describe the whole room.
"A bigger room has fewer mode problems." Cut. A bigger room has more modes, but lower and denser — it pushes the Schroeder frequency down. The problem moves; it does not vanish.
"A cube is an efficient room shape." Cut. Acoustically it is the worst: Lx=Ly=Lz stacks the three axial modes at one frequency, tripling that peak. Non-integer (e.g. golden) ratios spread them.
"Room modes are a bass thing, so treble is fine." Kept, corrected. True that modal booming is low-frequency — but the reason is sparse modal density below Schroeder, not the pitch itself.
The red team's move: drop the squares — replace the Pythagorean √(sum of squares) with a plain linear sum (c/2)·(nx/Lx + ny/Ly + nz/Lz). The obliques come out wrong. The blue team's witness (window 7) is watching.
Axial modes survive the swap (one term, so linear = Pythagorean), which is exactly why it is sneaky — but f111² = f100²+f010²+f001² breaks, the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.