Computational Mathematics in Music: Optimizing Frequency Modulation

This chapter presents an application of computational mathematics in music with particular focus on optimization of frequency modulation. We present the theoretical foundation, implementation details, and experimental evaluation of the method, using a set of acoustic instrument recordings as test inputs. The effectiveness of the proposed approach is assessed in terms of both auditory quality and expressive control, offering a promising direction for future research in hybrid sound synthesis systems. Recent advancements in computational capabilities have opened new avenues for innovation in sound synthesis techniques. Frequency modulation (FM) synthesis has long been recognized for its efficiency in producing musically engaging tones. However, conventional FM synthesis approaches often fall short in generating natural-sounding spectra and in mapping expressive, gestural input to synthesis parameters – largely due to the rigid separation between control and synthesis stages. In response to these limitations, this chapter explores a modified FM synthesis framework informed by adaptive audio-processing techniques. The proposed method involves analyzing datasets of real-world acoustic signals using Fast Fourier Transform (FFT) and developing a mathematical model driven by genetic algorithms to replicate the spectral and temporal characteristics of natural sounds. By integrating these insights into a modified FM synthesis structure, the technique allows for more realistic sound reproduction and seamless transitions between synthesized and acoustic-like audio textures.