The continuous rosetta beneath the scales
Theorem.The continuous rosetta beneath the scales — pitch class = log₂ f (mod 1) — the real line covering every finite scale.
Proof.log₂(frequency) is the real-line cover of every finite scale — the ear hears ratios as intervals, so log turns multiplicative pitch additive and the octave circle is ℝ/ℤ; C₁₂/C₆ are lattices in it. The day's illusion/invariant diagnostic sorts music: the OCTAVE (log₂2 = 1) is the INVARIANT — exact in every tuning, the wall that never moves — while the circle of fifths CLOSING at twelve is the ILLUSION, since log₂(3/2) is irrational (else 2^{p+q}=3^q) and the just fifths SPIRAL FOREVER (dense orbit, Weyl); only equal temperament forces closure, paying the Pythagorean comma ((3/2)^12/2^7 ≈ 1.0136, 23.46 cents). a432 seeds the ladder 432·2^x. The discrete rosettas lift into one logarithm.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#theContinuousRosettaBeneathTheScales
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 continuous rosetta beneath the scales" 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 theContinuousRosettaBeneathTheScales.
- 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 (theContinuousRosettaBeneathTheScales @ 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 theContinuousRosettaBeneathTheScales (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.