Innovative ۲۴-Note numerical system: computational method and relationship model between notes, elements, and planets

This study introduces a ۲۴-note numerical framework and computational method to explore connections among music, mathematics, the theory of four temperaments, and zodiac constellation. While earlier research has acknowledged these links, it lacked formal mathematical structures and systematic validation. The ۲۴-note system is significant in two respects. First, it validates the existing ۱۷-note system. Second, it establishes a crucial theoretical foundation for microtones. By providing this solid groundwork, it challenges the conventional Western musical framework, which is confined to tones and semitones, and paves the way for the practical application of microtones in fields such as music therapy, where their potential has remained largely untapped. Despite differences in numbers, results show complete agreement between the ۲۴-note and ۱۷-note systems in analyzing scales within Iran’s seven main musical modes (Dastgāh). The study reveals that structural symmetry can identify symmetrical scales, but only when the sum of the "Valued Numbers" (Manzeli) within one tetrachord equals the sum within the other tetrachord. It also introduces an entirely novel and invaluable relational model that connects musical notes to the four classical elements and planets. Research demonstrates that note the "LA'' functions as the foundational reference, with the position of all other notes calibrated relative to it. Finally, it presents the Cyclic Boundary Theorem (CBT), a novel mathematical concept that offers two complementary computational approaches for the systematic numerical evaluation of partitioned cyclic systems.

کلیدواژه ها:

۲۴-Note Numerical System ، ۱۷-Note Numerical System ، Four Temperaments (Music) ، Persian Radif ، music therapy