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

## TL;DR

This paper explains how 1/f fluctuations, common in nature and music, arise from amplitude modulation through synchronization and resonance.

## Contribution

A new physical mechanism for 1/f fluctuations using amplitude modulation and demodulation in oscillatory systems.

## Key findings

- Stochastic synchronization and frequency-selective resonance generate 1/f spectra through amplitude modulation.
- 1/f fluctuations in music can be modeled without requiring the classical Kuramoto critical point.
- The mechanism applies to both musical performance and the structure of musical scores.

## Abstract

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.