A simple model for pink noise from amplitude modulations

We propose a simple model for the origin of pink noise (or 1/f fluctuation) based on the beat of cooperative waves. These cooperative waves arise spontaneously in a system with synchronization, resonance, and infrared divergence. Many cooperative waves with close frequencies can produce signals of arbitrary small frequencies from a system of small size. This beat mechanism can be understood as amplitude modulation. The pink noise can appear after the demodulation process, which produces a variety of pink noise in many fields. The pink noise thus formed from the beat has nothing to do with dissipation or long-time memory. We also suggest new ways of looking at pink noise in shallow earthquakes, solar flares, and stellar activities.

Low-Frequency Characterization of Music Sounds -- Ultra-Bass Richness from the Sound Wave Beats

The orchestra performance is full of sublime rich sounds. In particular, the unison of violins sounds different from the solo violin. We try to clarify this difference and similarity of unison and solo numerically analyzing the beat of `violins` with timbre, vibrato, melody, and resonance. Characteristic properties appear in the very low-frequency part in the power spectrum of the wave amplitude squared. This ultra-buss richness (UBR) can be a new characteristic of sound on top of the well-known pitch, loudness, and timbre, although being inaudible directly. We find this UBR is always characterized by a power-law at low-frequency with the index around -1 and appears everywhere in music and thus being universal. Furthermore, we explore this power-law property towards much smaller frequency regions and suggest possible relation to the 1/f noise often found in music and many other fields in nature.