Circle of sound: mathematics as a key to discovering new frequencies in musical composition techniques

This study presents the circle of sound, a mathematically grounded framework for generating new microtonal frequencies in musical composition. The model is defined by the function πr2 + f, where r represents radius and f denotes a fundamental tone frequency. In this study, a fixed radius of 1.5 was selected, corresponding to a minor third interval and producing a consistent 7 Hz differential across a single octave register. This differential was applied in both compositional sketches and a vocal duet experiment to explore the perceptual and harmonic impact of micro-dissonances. The calculation results show that the new frequencies generated for the C4 pitch are 268.70 Hz, C#4 = 284.25 Hz, D4 = 300.73 Hz, D#4 = 318.20 Hz, E4 = 336.70 Hz, F4 = 356.30 Hz, F#4 = 377.06 Hz, G4 = 399.07 Hz, G#4 = 422.37 Hz, A4 = 447.07 Hz, A#4 = 473.23 Hz and B4 = 500.95 Hz. While listener perception data remain preliminary, qualitative feedback from musicians suggests that the resulting harmonic textures diverge from conventional tonal systems. These findings contribute to ongoing discourse in spectral music, computational musicology and psychoacoustics and propose new directions for adaptive and generative composition system.