−3 dB per octave means the power of a noise falls by about half every time the frequency doubles. It is the slope of pink noise, and it works out to about −10 dB per decade. The rest of this page takes the phrase apart: what an octave is, what a decibel is, and why the two together give a slope.
What is an octave?
An octave is a doubling of frequency. One octave corresponds to a doubling of frequency, so 440 Hz and 880 Hz are one octave apart, and so are 100 Hz and 200 Hz, or 5,000 Hz and 10,000 Hz.
That is why octaves are a useful way to read a spectrum. Each step up covers twice as many hertz as the one before it. The number of octaves between two frequencies is the base-2 logarithm of their ratio.
What is a decibel?
A decibel is a way to compare two levels, not an amount on its own. For power, a level is ten times the base-10 logarithm of the ratio. A 3 dB change is a power ratio of about 2, and a 10 dB change is a tenfold change in power.
Amplitude works differently. Because power goes with the square of amplitude, the amplitude formula uses 20 instead of 10. Doubling the amplitude is about 6 dB, and it quadruples the power.
The same "3 dB means double" idea shows up in hearing safety. NIOSH uses a 3-dB exchange rate, also called the equal-energy rule: for every 3 dB increase in noise level, the allowable exposure time is cut in half. That limit is for the workplace, not for the sound you play at home.
What is a slope in dB per octave?
A slope tells you how the level changes as you move up the spectrum. Read it left to right: every time you step up one octave, the level changes by that many decibels. A slope of −3 dB per octave means each octave is about 3 dB lower than the one below it.
You will also see slopes quoted per decade, which is a tenfold change in frequency. A decade is about 3.3 octaves, so −3 dB per octave is about −10 dB per decade. Both numbers describe the same line. A slope of 6 dB per octave, for comparison, is about 20 dB per decade.
Why does equal energy per octave give −3 dB?
Pink noise is defined by one idea: each octave carries an equal amount of noise energy. Each octave is twice as wide in hertz as the one below it. To keep the total energy the same in a wider band, the energy per hertz has to be lower. Twice the width means half the density per hertz, and half the power is about −3 dB.
That is where the 1/f comes from. The power spectral density of pink noise is inversely proportional to frequency, so doubling the frequency halves the power. On the amplitude scale the same noise falls as 1/√f, which is the same −3 dB per octave, just read on a different scale.
A quick check with numbers helps. Say the power density at 100 Hz is 1. At 200 Hz it is 1/2, at 400 Hz it is 1/4, and at 800 Hz it is 1/8. Each doubling of frequency halves the density, and each halving is about 3 dB, so the line drops steadily instead of flattening out. Over the range of 100 Hz to 800 Hz that is three octaves, or about 9 dB in total.
Which colours have which slope?
Every noise colour is a different slope. Flat is white; falling is pink and brown, and rising is blue and violet. These are the values from the standard reference for noise colours:
| Colour | Power density | Slope per octave | Slope per decade |
|---|---|---|---|
| White | Flat | 0 dB | 0 dB |
| Pink | 1/f | −3 dB | −10 dB |
| Brown | 1/f² | −6 dB | −20 dB |
| Blue | f | +3 dB | +10 dB |
| Violet | f² | +6 dB | +20 dB |
Brown noise falls twice as steeply as pink, which is why it sounds deeper. If you want the colours side by side, see how white, pink and brown noise differ, or read about brown noise on its own. More articles like this one live under sound science.
Is the slope the same as volume?
No. The slope is the shape of the sound, not how loud it is. Two pink noises can be quiet or loud and still have exactly the same −3 dB per octave. The slope only says how the level at low pitches compares with the level at high pitches.
A steeper fall means less of the high end and more of the low end, so the sound feels deeper. A flat or rising slope puts more weight on the high end, so it sounds brighter and hissier. Flat white noise sits in the middle: it has no slope at all, which is why it can sound like a bright hiss next to pink.
Try it
The easiest way to hear a slope is to switch between colours at the same volume. Start with white, which is flat. Then move to pink noise and notice how the top end softens. Go one step further to brown and the sound turns deeper again.
Hush Machine, coming to iPhone, puts that on one dial from white to deep brown. The noise is made live on the phone, so nothing loops and nothing repeats.
Bottom line
−3 dB per octave means the power halves each time the frequency doubles, which is pink noise. Per decade the same slope is about −10 dB. It describes the shape of a sound, not its volume.
Questions
- What does −3 dB per octave mean?
- Each time the frequency doubles, the noise power drops by about 3 dB, which is roughly half. It is the slope of pink noise.
- Is −3 dB per octave the same as −10 dB per decade?
- Yes. A decade is a tenfold change in frequency, about 3.3 octaves, so 3.01 dB per octave comes to about 10 dB per decade.
- Why does brown noise fall 6 dB per octave and pink only 3?
- Brown noise power density follows 1/f squared, so it falls twice as steeply as pink noise, which follows 1/f.
- Does the slope tell me how loud a sound is?
- No. The slope describes the shape of the spectrum, how the level changes from low pitch to high pitch. Volume is a separate setting.