The Innovative ۲۴-Note Numerical System: A Computational Method and Model of the Relationships Between Notes, Elements, and Planets

This research introduces a ۲۴-note numerical framework and a computational method to examine the connections between music, mathematics, the theory of the four temperaments, and the constellations of the zodiac. Although previous studies have acknowledged these associations, they have lacked formal mathematical structures and systematic validation. The ۲۴-note system is significant in two respects: first, it corroborates the existing ۱۷-note system; and second, it establishes a vital theoretical foundation for microtones. By providing such a robust infrastructure, this research challenges the conventional Western musical framework—limited to whole tones and semitones—and paves the way for the practical application of microtones in fields such as music therapy, where their potential remains largely untapped. Despite the difference in the number of notes, the results demonstrate complete agreement between the ۲۴-note and ۱۷-note systems regarding the analysis of the scales associated with the seven primary *Dastgahs* of Iranian music. The study reveals that structural symmetry can identify symmetrical scales, but only when the sum of the "value numbers" (positions) in one tetrachord equals the sum in the other. Furthermore, the research introduces a highly innovative and valuable relational model linking musical notes to the four classical elements and the planets. The findings indicate that the note "A" (La) serves as the fundamental reference point, with the positions of other notes calibrated and determined in relation to it. Finally, the study presents the "Cyclic Boundary Theorem" (CBT); A novel mathematical concept that proposes two complementary computational approaches for the systematic numerical evaluation of segmented rotational systems.

کلیدواژه ها:

note-۲۴ ، music ، temperament ، zodiac ، octave ، Scale ، Numerical