Difference-Tone Cascade Theory: Formation of Consonance Order via Recursive Difference-Tone Sequences and Their Correspondence with Auditory Phenomena

Existing models of musical consonance often assign a single scalar value based on ratio simplicity, spectral roughness, periodicity, or template matching. While such measures correlate with averaged perceptual ratings, they do not explicitly describe how consonance-related structure can emerge from interactions between sounds across frequency-ratio space. This preprint formulates the Difference-Tone Cascade (DTC) as a recursive mapping on frequency-ratio space and introduces the Shannon entropy of recursive difference-tone sequences as a quantitative order measure. By iteratively generating difference tones from pure-tone inputs while sweeping frequency ratios continuously (without separating rational and irrational values), we observe a hierarchical organization: stable regions centered around low-order rational ratios with continuous transition patterns between them. These results suggest that structured “consonance order” patterns may be embedded in the frequency-ratio continuum as a property of the recursive cascade. We apply the proposed index to representative musical intervals, triads, and multiple tuning systems, obtaining evaluations broadly consistent with consonance hierarchies reported in prior work. As a potential extension toward closer auditory correspondence, we note that perceptual weighting based on envelope clarity could be incorporated. Scope / limitation. This work proposes a structural generative framework and does not claim a causal account of consonance experience. The causal role of difference tones, if any, should be tested in future controlled listening experiments.Planned follow-up: A full paper will compare DTC-based measures against existing consonance models on published datasets. Version: v1.0 (2026-02-10)