Quantum Walk Telemetry of Incommensurate Musical Lattices: Topological Localization and Spectral Fractal Dimensions in Tuning Topologies

We present an open-source quantum simulation framework evaluating Continuous-Time Quantum Walks (CTQW) across one-dimensional tight-binding lattices modulated by spatial potentials derived from historical tuning architectures. By treating non-periodic, untempered pitch intervals as unique solid-state crystal analogs, we map the irrational frequency steps of the pure Spiral of Fifths (α = log2(1.5)) alongside symmetric and alternative quasiperiodic modulations. Telemetry indicates that the specific spatial geometry of the pentatonic structural parameter suppresses wavepacket diffusion by over a full order of magnitude compared to standard periodic baselines (α = 0.0). We apply a 1D box-counting regression to demonstrate that this trapping behavior is governed by a heavily fractured, low-dimensional Cantor-dust energy spectrum (Df = 0.5744) at the Aubry-André critical phase boundary (λ = 2J). Finally, by introducing randomized kinetic hopping noise (δJ = ±0.3), we confirm that this fractal state configuration provides robust topological shielding against structural defects. These results decouple acoustic relationships from human auditory biology, establishing a novel computational link between multi-scale mathematical sequences and quantum transport properties.