Harmonics, Geometry, and the Prime Structure of Musical Scales

We establish a geometric foundation for musical consonance and scale structure based on integer subdivision. Four results are derived: (1) a characterization of musical consonance as nodal coincidence, measured by the least common multiple of the mode numbers, consistent with the empirical rankings of Plomp and Levelt and with Euler's gradus suavitatis; (2) a formula for natural scale sizes as the continued fraction convergent denominators of log_p(q) for prime pairs {p, q}, extending Carey and Clampitt's well-formed scale theorem by providing a selection principle for canonical generators; (3) a combinatorial formula (2k+1)^n for the number of distinct intervals in an n-prime system at harmonic depth k; and (4) a structural correspondence between the crystallographic restriction theorem and Pythagorean tuning, both selecting the prime set {2, 3}. Five testable predictions are proposed, including a new prediction relating multi-frequency Chladni figure complexity to interval consonance via lcm(p, q).