To ring in the new school year, I have frequently listened to perhaps the most unexciting artist on Spotify: White Noise Radiance. They have several “albums” that last about 10 hours, and all the tracks are continuous types of noise. For instance, I am proud to say I have made my way through all 100 parts of their Brown Noise album. (Pt. 100 has under 600,000 streams, compared to Pt. 01’s almost 8 million – the former is truly slept on!).
There are many other types of noise this artist offers, from waterfalls to bacon sizzling, as well as an incredible 63-volume set of “Ambient birds,” but I want to focus on the “colors” of noise that are offered by many apps nowadays to supposedly aid in sleep or focus. This will lead us down a bit of a mathematical rabbit hole, although one that I promise will not get too technical (we’ve only just started school after all).
Let us start with the artist’s namesake: white noise. A signal is referred to as “white noise” if the intensity of the noise, measured in decibels (dB), is equal across all frequencies in a given range. For humans, we tend to be interested in the band between 20 Hz and 20,000 Hz, since this is what the average human ear can hear.
This constant intensity is a property also held by white light: if a light source is emitting all frequencies in the visible spectrum equally, then we will see it as white. The other colors of noise are named in this spirit: blue or violet noise is so named because the intensity goes up as the frequency increases, while red or pink noise has the most intensity in lower frequency bands.
But brown is not a color in the visible spectrum, so what about Brown Noise? This noise gets its name not from the color, but from a man named Robert Brown. A Scottish botanist from the 1800s, Brown noticed how certain pollen particles moved in a jittery fashion when immersed in liquid. Later, mathematicians and scientists, including Albert Einstein, developed a mathematical way to describe such movement, although it is still predominantly called Brownian motion. Being close enough to the color theme, this is how “brown noise” stuck.
Mathematically, Brownian motion is an example of a stochastic process, which is a sequence of random events – like the location of a small particle in a fluid – at specific points in time. Brown noise signals are constructed from Brownian motions: one can think of particles bouncing off air molecules at different points in time in a special way to create the sound vibrations that lead to the noise. The intensity of Brown noise is proportional to one over the square of the frequency.
These “power-law” spectra, where the intensity is related to the frequency raised to a specific power, is how colors of noise are rigorously defined. But what’s more, one can miraculously obtain white noise from a Brownian motion, by taking a special type of derivative of the motion. This is different from the derivatives you will see or have seen in calculus courses, but one can think of it as an “instantaneous rate of change” for the stochastic process. In part because the derivative of Brownian motion is white noise, stochastic integration, which is used in the stochastic calculus courses offered at Stevens, becomes possible.
But enough math for now! On a sentimental note, for new readers, this marks the beginning of my third year of writing For Math’s Sake, and my sixth year at Stevens. From my experience, college is a place of much noise, ranging from delightful (especially if you’re in DeBaun PAC for a concert!) to jarring. But it is the time for all of us to make that noise, freely and proudly, while keenly listening for the important signals in the mix.
