the circle of fifths does not close
Theorem.the circle of fifths does not close — (3/2)¹²/2⁷ = 531441/524288 ≠ 1; 1200·log₂(531441/524288) = 23.46 cents; 1200·log₂(2^(7/12)·(2/3)) = −1.955 cents.
Proof.twelve perfect fifths are not seven octaves: (3/2)¹² / 2⁷ = 531441/524288, the Pythagorean comma, 23.4600 cents of excess. Equal temperament does not resolve this by making the fifth perfect — 2^(7/12) is 1.9550 cents flat of 3/2 — it distributes the comma across twelve steps. Both figures are exact consequences of the two ratios, so the whole of tuning practice is a response to an arithmetic fact rather than a matter of taste.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/stats/index.ts#circleOfFifthsDoesNotClose
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 does not close" 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 circleOfFifthsDoesNotClose.
- 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 (circleOfFifthsDoesNotClose @ src/stats) 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 circleOfFifthsDoesNotClose (src/stats/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.