Hong Life CalculatorsUnderstanding the octave and third-octave band calculator — concept and principle
Acoustic measurements, noise specifications and equalisers often split the spectrum into octave or one-third-octave bands. To filter, compare or report band data correctly you need each band's edges, not just its centre. This calculator gives both edges and the bandwidth in hertz for a chosen centre frequency.
How it is calculated
For a band of 1/b octave (b = 1 for octaves, b = 3 for third-octaves):
- Lower edge: f₁ = f_c / 2^(1/(2b))
- Upper edge: f₂ = f_c × 2^(1/(2b))
- Bandwidth: Δf = f₂ − f₁ = f_c × (2^(1/(2b)) − 2^(−1/(2b)))
All frequencies are in Hz. For an octave the edge factor is √2 ≈ 1.4142; for a third-octave it is 2^(1/6) ≈ 1.1225.
The principle
Human pitch perception is roughly logarithmic: going up an octave always doubles the frequency, whether from 100 to 200 Hz or from 1,000 to 2,000 Hz. Bands of constant relative width therefore look equally wide on a logarithmic axis. The centre frequency is the geometric mean of the edges, f_c = √(f₁ f₂), so the edges sit the same ratio below and above it. A one-octave band has f₂/f₁ = 2; three third-octave bands stacked together make one octave. The bandwidth in hertz grows in proportion to f_c, which is why such filters are called constant-percentage bandwidth filters.
Worked example
- Inputs: f_c = 1,000 Hz, bandwidth = 1 octave (b = 1).
- Edge factor: 2^(1/2) = 1.414214.
- Lower edge: 1,000 / 1.414214 = 707.1067812 Hz.
- Upper edge: 1,000 × 1.414214 = 1,414.213562 Hz.
- Bandwidth: 1,414.2136 − 707.1068 = 707.1067812 Hz.
For the 1/3-octave option, the same centre gives edges of about 890.9 Hz and 1,122.5 Hz and a bandwidth of about 231.6 Hz.
| Centre frequency (Hz) | Lower edge (Hz) |
|---|---|
| 250 | 176.7766953 |
| 500 | 353.5533906 |
| 750 | 530.3300859 |
| 1,000 | 707.1067812 |
| 1,250 | 883.8834765 |
| 1,500 | 1,060.660172 |
| 2,000 | 1,414.213562 |
How to read the result
An octave band is about 70.7% of its centre frequency wide; a third-octave band about 23.2%. Standard octave centres are 31.5, 63, 125, 250, 500, 1k, 2k, 4k, 8k and 16k Hz, and third-octave series add values such as 800, 1k, 1.25k and 1.6k. These nominal values are rounded, so exact calculated centres differ slightly from the label.
Common mistakes
- Taking the arithmetic mean of the edges as the centre; the centre is the geometric mean.
- Assuming octave edges are f_c/2 and 2f_c; that would be a two-octave band.
- Adding band levels in decibels arithmetically; they must be summed on an energy basis.
Frequently asked questions
Why do sound level meters use third-octave bands?
They are close to the critical bands of human hearing over much of the audible range, giving useful detail without the scatter of a narrowband spectrum.
How do I get octave levels from third-octave levels?
Combine the three bands energetically: L = 10 log₁₀(10^(L₁/10) + 10^(L₂/10) + 10^(L₃/10)).
Is 1,000 Hz always a band centre?
Yes. Both octave and third-octave series are anchored at the 1 kHz reference frequency.
Sources: IEC 61260-1:2014, Electroacoustics — Octave-band and fractional-octave-band filters — Part 1: Specifications · ANSI/ASA S1.11, Specification for Octave-Band and Fractional-Octave-Band Analog and Digital Filters