The circle of fifths is a rosetta
Theorem.The circle of fifths is a rosetta — twelve-tone equal temperament is ℤ/12ℤ, and moving a fixed interval around it is a rosetta: the FIFTH (7 semitones) generates all 12 tones because gcd(7,12)=1.
Proof.twelve-tone equal temperament is ℤ/12ℤ, and moving a fixed interval around it is a rosetta: the FIFTH (7 semitones) generates all 12 tones because gcd(7,12)=1 — a Hamiltonian orbit visiting every note once, like the slash circuit. The generators are φ(12)=4 intervals {1,5,7,11} (semitone/fourth/fifth/major-seventh); every other interval closes a SUBGROUP that is a chord — augmented triad C3, diminished seventh C4, whole-tone scale C6, tritone C2 — orbit length 12/gcd(step,12), the divisor lattice. The whole-tone scale is C₆, the SAME group as the vortex and polyphase circle. Music theory is cyclic-group theory.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#theCircleOfFifthsIsARosetta
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"The circle of fifths is a rosetta" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theCircleOfFifthsIsARosetta.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theCircleOfFifthsIsARosetta @ src/music) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theCircleOfFifthsIsARosetta (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.