Infinite Canons: Maximally Self-Similar Melodic Lines and Canons with Infinite Solutions

What the math gives to ML

The paper constructs melodic objects whose response to any time-scale change is governed by a homomorphism from multiplicative tempo ratios to additive pitch intervals: composing tempo ratios corresponds to adding intervals, and the interval is independent of absolute time. The discrete version decomposes rational ratios into prime-factor exponents, while the continuous version uses a logarithmic map, yielding a clean representation of multiplicative scale changes. This structure can transfer to temporal neural networks as an exactly compositional tempo or dilation embedding, rather than asking a model to learn unrelated embeddings for different rates. The most promising test is a multi-rate sequence model conditioned on this homomorphism and evaluated on unseen rational and irrational time scales.