An algorithmic approach to the harmonic sets of Just Intonation

This paper investigates the mathematical foundations of Just Intonation through the lens of Renaissance architectural proportions. While Pythagorean tuning and Equal Temperament are go\-ver\-ned by transparent, recursive algorithms, the 5-limit system of Just Intonation is often perceived as a non-systematic collection of local optimizations. By shifting the analytical focus from frequency ratios to the classical harmonic mean - a cornerstone of Palladian and Albertian architectural theo\-ry - we propose a generative algorithmic approach to pitch selection. Operating on a foundational nucleus of perfect consonances, the algorithm iteratively ``harvests'' new pitches only if they establish rigorous harmonic proportions with the existing set. Our results demonstrate that this procedure does not merely replicate the modern diatonic scale; instead, it identifies a finite, algebraically closed set of sounds that mirrors the structural logic of the Renaissance hexachord and the modal finales. This finding suggests that the "natural" scale of the 16th century was not an arbitrary cultural construct, but a mathematically perfectum design - a ``rational soul'' where internal consistency precludes the infinite drift of Pythagorean fifths. The study concludes by drawing a parallel between this closed harmonic system and the transition from the mystic infinity of Gothic architecture to the unified, proportional space of the Renaissance temple.