Formula
Formula used
\( L_{\text{octave}} = 10 \log_{10} \left( \displaystyle\sum_{i=1}^{n} 10^{L_{\text{1/3 octave},i}/10} \right) \)
Inputs
Fill the three third-octave bands of an octave to compute it. Results update as you type.
Understand the theory — physics concept and calculationsinteractive
A sound level meter often measures in third-octave bands (3 fine bands per octave), while a standard or a datasheet works in octaves (wide bands). Converting means grouping the bands — but as always in acoustics, you add energy, not decibels. Edit the spectrum below: each octave band is the energetic sum of its three thirds.
Drag the thin bars (third-octaves). Each octave band (wide) is recomputed live by energetic summation.
But first: why split sound into bands? A fine analysis (FFT, frequency by frequency) shows every detail — ideal to hunt a tonal peak — but it is heavy and fluctuating. Third-octave and octave bands group the energy into standardized bands: more readable, comparable to standards, and close to how the ear analyses sound.
The same spectrum, increasingly 'grouped'. The tonal peak, sharp in the fine analysis, dilutes in third-octaves then vanishes in octaves.
Octaves and third-octaves
An octave is a doubling of frequency (ratio 2 between edges). A third-octave splits each octave into 3 bands of ratio \(2^{1/3} \approx 1.26\). The standardized centre frequencies (IEC 61260) are shared by everyone: 125, 250, 500, 1000 Hz… for octaves, and 100, 125, 160, 200, 250, 315… for thirds.
Each octave contains exactly 3 thirds: for example the 1000 Hz octave groups the 800, 1000 and 1250 Hz thirds. These filters have a constant relative bandwidth: every band always covers the same percentage of its centre frequency (≈ 23 % for a third, ≈ 70 % for an octave). Bands are therefore narrow at low frequencies and wide at high ones — which is why frequency is drawn on a logarithmic axis.
The conversion formula
The level of an octave band is the energetic sum of the levels \(L_i\) of its three third-octaves:
Convert each third to energy (\(10^{L_i/10}\)), add, then return to decibels. It is exactly the same operation as level addition — applied band by band.
Why energy, not an average?
Because a level in dB is a logarithmic quantity tied to acoustic energy. Grouping three bands means merging their energetic contributions into one wider band: the energies add up. An arithmetic average of the dB would systematically underestimate the result. For instance three bands at 60, 63 and 66 dB combine to 68.4 dB — clearly above the loudest one, and never their mean.
When do you convert?
The direction is always fine → wide: you can sum thirds into octaves, but you cannot go back from an octave to its three thirds (the distribution is lost). You typically convert because the measurement is in thirds while the regulatory limit, the NR/NC curve or the sound reduction index are given in octaves. NR/NC curves, the sound reduction index \(R\) of partitions and regulatory spectra are nearly always tabulated in octaves, so converting a third-octave measurement to octaves is an everyday step in a consultancy.
What about A-weighting?
If you want a dB(A) spectrum, apply the A-weighting band by band (one term \(A_i\) per frequency) before summing. You may weight then group into octaves, or group then weight with the octave's A value — both are fine as long as you stay consistent. In short, weighting is a per-band correction while the grouping into octaves is an energy sum — keep the two operations separate in your head. See the other tools →
What you gain and what you lose
Moving to octaves simplifies reading (8–10 values instead of 30) but smooths the spectrum: a narrow tonal peak, clearly visible in thirds, can be diluted inside a wide octave band. To diagnose a precise frequency, keep the thirds (or even a fine FFT analysis). Good practice: measure and archive in third-octaves (richer), then convert to octaves only when you need to compare against a limit or report a summary.
Worked examples
| Third-octaves | Octave | Reading |
|---|---|---|
| 3 thirds at 50 dB | 54.8 dB | +10·log 3 = +4.77 dB |
| 60 / 63 / 66 dB | 68.4 dB | energetic summation |
| 70 / 60 / 60 dB | 70.8 dB | the loud third dominates |
| 3 thirds at 0 dB | 4.8 dB | the rule holds at any level |
- Averaging the three thirds: (60+63+66)/3 = 63 dB. Wrong → 68.4 dB.
- Adding the dB arithmetically (50+50+50 = 150 dB). Wrong → 54.8 dB.
- Trying to "un-convert" an octave back into three thirds: impossible.
- Mixing thirds in dB and dB(A) in the same sum.
Related tools
Sources : J.-C. Pascal, Vibrations et Acoustique (ENSIM, Le Mans Université); IEC 61260 (octave and fractional-octave band filters); D. A. Bies & C. H. Hansen, Engineering Noise Control.