In particular, so-called 1/fα noise series with α close to 1 (also called pink noise) occur in sound, music and countless human artifacts or natural events, from the fluctuations of the flood levels of the Nile to movements of the stock market. As a consequence, many generative models for 1/f noise have been designed to produce series that look or sound "natural" or "human". In this paper, we formulate the generation of 1/f series as a hard constraint satisfaction problem, so that 1/f noise generation can be used as an add-on to arbitrary sequence generation problems. We take inspiration from a simple yet beautiful stochastic algorithm invented by Voss and introduce the Voss constraint. We show that Voss' algorithm can be modeled as a tree of ternary sum constraints, leading to efficient filtering. We illustrate our constraint with a melody generation problem, and show that the addition of the Voss constraint tends indeed to produce sequences whose spectrum have a 1/f distribution, regardless of the other constraints of the problem. We discuss the advantages and limitations of this approach and possible extensions.

Many natural phenomena exhibit power law spectra. In particular, so-called 1/f α noise series with α close to 1 (also called pink noise) occur in sound, music and countless human artifacts or natural events, from the fluctuations of the flood levels of the Nile to movements of the stock market. As a consequence, many generative models for 1/f noise have been designed to produce series that look or sound “natural” or “human”. In this paper, we formulate the generation of 1/f series as a hard constraint satisfaction problem, so that 1/f noise generation can be used as an add-on to arbitrary sequence generation problems. We take inspiration from a simple yet beautiful stochastic algorithm invented by Voss and introduce the Voss constraint. We show that Voss’ algorithm can be modeled as a tree of ternary sum constraints, leading to efficient filtering. We illustrate our constraint with a melody generation problem, and show that the addition of the Voss constraint tends indeed to produce sequences whose spectrum have a 1/f distribution, regardless of the other constraints of the problem. We discuss the advantages and limitations of this approach and possible extensions.