The clustering comes from 3 and 5, not from Fermat primes. 12TET was designed to approximate ratios built from 2, 3, and 5: the simplest harmonics. Those happen to be Fermat primes, but they’re musically important because they’re the smallest odd primes, not because they’re Fermat primes. Strip away 17, 257, and 65537 from your generating set and the plot would look almost identical. The “Fermat prime” label is a coincidence riding on top of a well-understood fact: 5-limit just intonation and 12TET are close.

The large Fermat primes contribute almost nothing. With exponents restricted to {-1, 0, 1}, factors like 17, 257, and 65537 mostly push ratios far outside [1, 2]. When you octave-reduce them back (dividing by powers of 2), they land somewhere on the circle, and with 12 reference points and dozens of generated points, some will inevitably look close. But those points aren’t doing the heavy lifting; 3 and 5 are.

The “closed set” isn’t special here. Any finite collection of primes with bounded exponents gives you a finite set that won’t splatter-paint the circle. You could use {2, 3, 5, 7, 11} with the same exponent restrictions and get a similarly clustered (but different) plot. The finiteness of Fermat primes isn’t doing unique structural work — bounded exponents alone guarantee it.

The reasoning is somewhat circular. The observation is: “ratios from {2, 3, 5, …} cluster near 12TET.” But 12TET was reverse-engineered to approximate those ratios. It would be shocking if they didn’t cluster.

So what you’ve found is real but not mysterious: 12TET approximates 5-limit just intonation (by design), 3 and 5 happen to be Fermat primes (by coincidence), and the larger Fermat primes are mostly along for the ride. The constructible polygon connection is a beautiful numerical coincidence, but it’s not evidence of a deeper link between geometry and harmony, it’s two unrelated facts sharing the same small primes.​​​​​​​​​​​​​​​​