Musical Structure as Torus Geometry: Cyclic Permutations on 𝕋³ as a Generative Framework for Phrase Closure, Polyphony, and Scale Selection

We show that a cyclic permutation on the three-torus 𝕋³ encodes core structural elements of music — phrase closure, polyphonic voice decomposition, intervallic logic, melodic contour, rhythmic variation, and scale selection — within a five-parameter construction. The framework, which we call the ResonanT⁴ (RT⁴) musical mapping, requires only the integers (n, k) and frequencies (ω, ξ, ψ) to generate complete musical phrases with mathematically guaranteed properties. The orbit of the permutation is the phrase; phrase closure is a theorem, not a heuristic. When gcd(n, k) > 1, the orbit decomposes into identical sub-orbits offset in pitch — canonic voices, derived from number theory rather than compositional convention. We prove that quantization of an irrational winding ratio k/n to chromatic pitch classes produces a Sturmian sequence with exactly two interval types. We show that the golden ratio φ yields the most consonant interval pair in 12-TET — the perfect fifth and minor sixth. The circle of fifths emerges as the specific orbit (n = 12, k = 7). Exactly 23 distinct interval pair types exist across all admissible orbits. Computational verification across n = 2 to 10,000 confirms all theoretical predictions.