Twelve tones is the best approximation of the fifth
Theorem.Twelve tones is the best approximation of the fifth — log₂(3/2) ≈ 7/12 — a continued-fraction convergent: 2^{7/12} ≈ 1.4983 vs 3/2.
Proof.why the piano has 12 keys, computed: the fifth spirals forever (log₂(3/2) irrational), so any equal temperament APPROXIMATES it, and the best approximations are the continued-fraction convergents of log₂(3/2) = [0;1,1,2,2,3,1,5,…]. The convergents ARE the temperaments: 5-TET (fifth off 18 cents), 12-TET (7/12, off only 1.96 cents — imperceptible), 41-TET (24/41, 0.48), 53-TET (31/53, 0.07). So 12 is not arbitrary — it is the denominator of the best SMALL-N convergent (7/12), the smallest with imperceptible error; the Pythagorean comma is the residual of that approximation, and better fifths need 41 or 53 tones (real microtonal systems). The continued-fraction rosetta explains the keyboard.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#twelveTonesIsTheBestApproximationOfTheFifth
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
"Twelve tones is the best approximation of the fifth" 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 twelveTonesIsTheBestApproximationOfTheFifth.
- 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 (twelveTonesIsTheBestApproximationOfTheFifth @ 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 twelveTonesIsTheBestApproximationOfTheFifth (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.