Cymatics Lab
Sprinkle sand on a vibrating plate and it collects on the lines that don't move — the Chladni pattern of that pitch. For one drum the pitches themselves are decided by the primes: a square membrane. Its standing waves are labelled by two whole numbers m, n and its pitches are m² + n². So a pitch exists exactly when the number is a sum of two squares, and how many patterns share it is the number of ways to write it that way — which Fermat showed is governed by the primes, sorted by whether they leave remainder 1 or 3 on division by 4. Choose a pitch, see the pattern, and read the arithmetic beside it.
Pitch N = m² + n²
Which pitches exist, 2 … 100 — and how many patterns share each
The rule is Fermat's: a number is a sum of two squares exactly when every prime in it with remainder 3 on division by 4 appears an even number of times. So 5, 13, 17, 29 (remainder 1) are pitches; 3, 7, 11, 19 (remainder 3) are silent; 9 = 3² is a pitch again because the 3 appears twice; and 65 = 5 × 13 has two pattern pairs because each of its primes contributes one. The number of patterns at pitch N is a formula in N's divisors — the same kind of formula, from the same family of functions, as zeta. Counting all the pitches below a height is counting grid points inside a quarter circle: the Gauss circle problem, with its unproven square-root error, from the Dimension Lab. When several patterns share a pitch, any mix of them is also a standing wave at that pitch; the "mix" slider rotates through those combinations and the sand lines reshape, which is what happens on a real plate too. This is the one drum whose cymatics are literally arithmetic. It is not the Riemann drum — its pitches grow like a surface's, and its primes act by deciding which pitches exist, not where each one sits — but it is the honest answer to "have people looked at cymatic patterns and primes": yes, and this is what they found.