# Octave (electronics)

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In electronics, an octave (symbol oct) is a logarithmic unit for ratios between frequencies, with one octave corresponding to a doubling of frequency. For example, the frequency one octave from (or above) 40 Hz is 80 Hz. The term is derived from the Western musical scale where an octave is a doubling in frequency.[note 1] Specification in terms of octaves is therefore common in audio electronics.

Along with the decade, it is a unit used to describe frequency bands or frequency ratios.[1][2]

## Ratios and slopes

A frequency ratio expressed in octaves is the base-2 logarithm (binary logarithm) of the ratio:

${\displaystyle {\text{number of octaves}}=\log _{2}\left({\frac {f_{2}}{f_{1}}}\right)}$

An amplifier or filter may be stated to have a frequency response of ±6 dB per octave over a particular frequency range, which signifies that the power gain changes by ±6 decibels (a factor of 4 in power), when the frequency changes by a factor of 2. This slope, or more precisely 10 log10(4) ≈ 6.0206 decibels per octave, corresponds to an amplitude gain proportional to frequency, which is equivalent to ±20 dB per decade (factor of 10 amplitude gain change for a factor of 10 frequency change). This would be a first-order filter.

## Example

The distance between the frequencies 20 Hz and 40 Hz is 1 octave. An amplitude of 52 dB at 4 kHz decreases as frequency increases at −2 dB/oct. What is the amplitude at 13 kHz?

${\displaystyle {\text{number of octaves}}=\log _{2}\left({\frac {13}{4}}\right)=1.7}$
${\displaystyle {\text{Mag}}_{13{\text{ kHz}}}=52{\text{ dB}}+(1.7{\text{ oct}}\times -2{\text{ dB/oct}})=48.6{\text{ dB}}.\,}$

## Notes

1. ^ The prefix octa-, denoting eight, refers to the eight notes of a diatonic scale; the association of the word with doubling is solely a matter of customary usage.

## References

1. ^ Levine, William S. (2010). The Control Handbook: Control System Fundamentals, p. 9–29. ISBN 9781420073621/ISBN 9781420073669.
2. ^ Perdikaris, G. (1991). Computer Controlled Systems: Theory and Applications, p. 117. ISBN 9780792314226.