Dynamic Synchronization and Resonance as the Origin of 1/f Fluctuations—Amplitude Modulation Across Music and Nature

In natural systems, astrophysics, biological physics, and social physics, 1/f fluctuations are observed across a wide range of systems. Focusing on the case of music, we propose and verify a physical mechanism for generating these fluctuations. This mechanism is based on amplitude modulation (AM) and demodulation (DM), where the 1/f spectral law appears not in the raw waveform but in its demodulated amplitude envelope. Two distinct yet complementary processes generate the required AM: (i) stochastic synchronization among oscillators, modeled via an extended Kuramoto framework that captures perpetual synchronization-desynchronization cycles, and (ii) frequency-selective resonance, modeled by spectral accumulation of eigenmodes in acoustic or structural environments. Numerical simulations demonstrate that both mechanisms, acting alone or in combination, robustly generate 1/f spectra spanning several digits when demodulation is applied and that the classical Kuramoto critical point is not essential for its emergence. While this analysis focuses on 1/f fluctuations in musical performance and acoustics, we also note that 1/f fluctuations inherent in musical scores may be similarly described by the AM/DM mechanism.