📻 Noise Colors — White, Pink & Brown

Power spectral density describes how the average power of a random signal is distributed across frequency, S(f). Noise "colors" are simply names for characteristic PSD shapes: flat S(f) is white, S(f) ∝ 1/f is pink, and S(f) ∝ 1/f² is brown. Two signals with the same PSD shape sound the same "color" even though the exact random samples differ every time you generate them.

The name is an analogy to white light, which contains roughly equal intensity across all visible wavelengths. White noise likewise has equal power per hertz at every audible frequency — its power spectral density is flat, S(f) ∝ f⁰. It is generated here simply by drawing a new independent random sample every tick, with no correlation between samples.

Human hearing groups frequencies into octave bands, and each octave spans a doubling of frequency, so higher octaves contain far more individual hertz than lower ones. White noise has equal energy per hertz, so equal-width octaves at high frequencies collect much more total energy than low ones — it sounds hiss-heavy and bright. Pink noise instead has equal energy per octave, matching how we perceive pitch, which removes that high-frequency emphasis and makes it sound warmer and duller despite carrying real energy at every hertz.

The Voss-McCartney method keeps a small bank of independent random "rows," each updated at a different rate: row 0 changes every sample, row 1 every 2 samples, row 2 every 4 samples, and so on in octave steps. Summing all the rows produces a signal whose spectrum falls off close to 1/f, because slower rows contribute more low-frequency content and faster rows contribute more high-frequency content in octave-balanced proportions. A cheap way to pick which row to refresh each step is to count the trailing zero bits of an incrementing counter, which naturally updates row 0 half the time, row 1 a quarter of the time, and so on.

White noise's flat, broadband spectrum covers essentially all audible frequencies at once, so it is very effective at raising the auditory masking threshold across the whole hearing range — quieter, more variable sounds like a door closing or traffic get buried under a constant, unchanging hiss. Because it has no distinctive pitch or rhythm, the brain also habituates to it quickly instead of continually flagging it as a novel, attention-grabbing sound.

1/f spectra are extremely common in physical and biological time series: heartbeat interval variability, neural firing rates, river flow, and even music show approximately pink spectra. This seems to arise whenever a system has many interacting processes operating across a wide range of timescales, each contributing roughly equally per octave rather than per hertz — no single mechanism fully explains it, but the ubiquity of multi-scale, weakly coupled processes in nature is the leading explanation.

Brown noise sounds like a deep, rumbling roar — like a waterfall or heavy rain heard from a distance — because its power falls off steeply (1/f²) as frequency rises, so almost all of its energy sits in the low end of the spectrum and very little reaches the highs. It is named for Brownian motion, not the color brown; the name is a pun on Robert Brown, the botanist who described the random jittering of pollen grains in water.

Brown noise is literally the running sum (discrete integral) of white noise, which is exactly the definition of a one-dimensional random walk or discrete Brownian motion: at every step you add a new independent random increment to your current position. Integrating in time is equivalent to dividing by frequency in the spectral domain, so integrating white noise (flat spectrum) once produces a 1/f² spectrum — the extra factor of f compared to pink noise's single 1/f comes from that doubled effect of integration.

Blue noise has power that rises with frequency, S(f) ∝ f, the mirror image of pink noise. It is generated here by differencing successive white noise samples, the discrete equivalent of differentiation, which boosts high frequencies while suppressing low ones. It sounds thin, hissy, and sharp, and is used in fields like dithering and stippling where evenly spread high-frequency energy without low-frequency clumping is desirable.