Cymatics — scales and frequencies
The vocabulary is not chosen, it is generated. One operation suffices — doubling a line's offset about the centre of the square — and the origin is not free either: a level the doubling leaves in place has no preimage outside itself, so it cannot be generated and must be posited. There are exactly two, 0° and 120°, the second existing because 3 divides 12. Starting from those two and running the doubling backwards, the orthogonal offsets close in two steps on all 24, and the diagonal ones in three — including the inscribed diamond, which belongs to the vocabulary without being drawn.
And this generated vocabulary coincides, line for line, with the level curves of a single function: cos(πx/3), cut at seven heights. That function is an exact eigenfunction — not an approximation — of a square plate with guided edges: exact as such for the orthogonal families, and by degenerate pair (sum or difference of modes (m, n) and (n, m), in Colwell's sense) for the diagonal families. The correspondence is therefore no longer a resemblance measured by Fourier transform.
What remains out of reach is stated further down, and it should be read: no measurement has been made, and of the seven levels only one is a nodal line.
The detail, with the verification →
One wave, seven cuts
The eight orthogonal families are the lines where cos(πx/3) equals cos θ, for θ a multiple of 30°:
Mutation is the reflection of the angle about 90°, that is, the exchange of the wave's two phases. It is not a naming convention: since the doubling is two-to-one, every antecedent comes in pairs, and the base and its mutant are the two branches of the inverse.
Two exact systems, two laws
The product cos(πx/3)·cos(πy/3) is an exact eigenfunction of two different problems:
- the Kirchhoff plate with guided edges, biharmonic equation, whose spectrum gives f ∝ m² + n²;
- the rectangular acoustic cavity with rigid walls, Helmholtz equation with Neumann condition, whose spectrum gives f ∝ √(m² + n²).
The boundary condition is satisfied identically in both cases. The guided-edge plate is not the one from classical demonstrations, with free edges, where the cosine product is only an approximation; the cavity, for its part, is any rectangular box. The cavity result assumes a zero vertical index: a flat box, or excitation below the first vertical cutoff — beyond that, a third term enters the spectrum.
The two problems share their eigenfunctions but not their spectra, which gives a test on a single figure:
On the three values that are perfect squares, the plate law gives 128, 512, 1152 — ratios 1 : 4 : 9. The cavity law gives 128, 256, 384, that is, C, its octave, and the fifth above: ratios 1 : 2 : 3, the harmonic series.
The 128 Hz reference pitch is a calibration in both cases: it fixes the size of the system, not its geometry. What is not a calibration is the choice between the two laws — and that is measurable.
This page counts whole periods over the grid's twelve cells, which gives k² = 8 for the pattern with gaps 0 and 3. The plate convention counts half-wavelengths along the side, which gives mode (4,4) and k² = 32 for the same pattern. The two are related by k²(grid) = (m² + n²) ⁄ 4.
Two modes are at play, and they are not at the same frequency: the carrier (4,0), which is one-dimensional bending and carries the seven levels, and the figure (4,4), whose nodal set is the zero level and which sand would draw. Under the plate law their ratio is 2, under the cavity law √2. That is a second discriminating test.
Listening
Pure tone generator
To test without a microphone or equipment — the chosen frequency drives the pattern directly.
The fifteen scales
The reference pitch used here, for both scales, is C at 128 Hz — one octave below the C at 256 Hz that Joseph Sauveur proposed in 1713 before the Royal Academy of Sciences, for a simple reason: doubling 1 repeatedly places every octave on a whole number. It became known as the philosophical pitch. Each scale carries two values, measured separately on its two generations — T1 and bandes — whose dominant spatial frequencies overlap substantially rather than forming two separate registers. The sixteen measured values (the squared spatial frequencies, combined from both generations) are themselves whole numbers, but that is a property of the measurement, not of the pattern: a Fourier transform on a twelve-cell grid can only return whole numbers. What is not empty, however, is that these whole numbers are the m² + n² of an exact eigenmode — and that coincidence is checked line by line, not by a transform. On the proportional scale, each value then lands exactly on 128 times a whole number, with no remainder. Another reference pitch would work just as well; this one makes the structure legible.
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What is not established
Only one of the seven cuts is a nodal line — the one at 90°. The other six are iso-amplitude lines: sand would not show them. They are not invisible for that: time-average holographic interferometry, and ESPI, produce fringes that are precisely iso-amplitude curves. And in a water basin, the surface elevation is the mode: the seven levels are iso-elevation lines there, visible under raking light. A water basin adds a third system, but not a third test: sloshing there is dispersive, ω² = g·k·tanh(k·h), and all three systems share the figure but not the spectrum — the ratio that discriminates plate from cavity does not carry over to it.
The diagonal families are not the cuts of a guided mode. On the diagonals, the vocabulary takes offsets modulo 6, and the function is defined only up to sign: there are only four diagonal levels, |φ| = 1, √3/2, ½ and 0. Two of them are nodal sets of the degenerate combinations cos(4πx/a) ± cos(4πy/a) — the mechanism Colwell described in 1933: T1 YANG MUT (level 0) for the sum, T1 YANG (level 1, offsets 0 and ±3) for the difference. The other two diagonal levels have no plate mode behind them.
No physical measurement has been made. Everything advanced here is calculation on the published plates and symbolic verification. The cheapest experiment is a rectangular tank, a loudspeaker underneath, and raking light; it discriminates between the two frequency laws. A Chladni figure depends on the vessel's geometry, its depth, surface tension; nothing here replaces that.
