Scales are necklaces on the rosetta
Theorem.Scales are necklaces on the rosetta — a scale = a binary necklace on 12 beads — subsets of ℤ/12 up to rotation (Burnside counts them).
Proof.if the 12-tone rosetta is a bead circle, a scale is a subset and two are the same under rotation (transposition) or reflection (inversion), so the count of essentially-different scales is a Burnside/Pólya orbit-count over C₁₂ and D₁₂ — landing exactly on Forte's documented tables: 352 pitch-class sets up to transposition, 224 set classes up to transposition+inversion. The 4096 raw subsets collapse to these few because the rosetta is symmetric. Music's whole catalogue of chords and scales is a group-orbit count on the moving rosetta.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/music/index.ts#scalesAreNecklacesOnTheRosetta
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
"Scales are necklaces on the 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 scalesAreNecklacesOnTheRosetta.
- 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 (scalesAreNecklacesOnTheRosetta @ 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 scalesAreNecklacesOnTheRosetta (src/music/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.