# One-third octave

A one-third octave is a logarithmic unit of frequency ratio equal to either one third of an octave (1200/3 = 400 cents: major third) or one tenth of a decade (3986.31/10 = 398.631 cents: M3  ).[1] An alternative (unambiguous) term for one tenth of a decade is a decidecade.[2][3]

## Definitions

### Base 2

ISO 18405:2017 defines a "one-third octave" (or "one-third octave (base 2)") as one third of an octave, corresponding to a frequency ratio of ${\displaystyle 2^{1/3}}$ . A one-third octave (base 2) is precisely 400 cents.

### Base 10

IEC 61260-1:2014 and ANSI S1.6-2016 define a "one-third octave" as one tenth of a decade, corresponding to a frequency ratio of ${\displaystyle 10^{1/10}}$ . This unit is referred to by ISO 18405 as a "decidecade" or "one-third octave (base 10)".[4]

One decidecade is equal to 100 savarts (approximately 398.631 cents).

4. ^ (This makes sense as, if we want one third of an octave, the ratio will be ${\displaystyle f2/f1=2^{1/3}}$ , and if we log10 both members of equation we have, ${\displaystyle log{(f2/f1)}=log{(2^{1/3})}->log(f2/f1)=log(2)*1/3}$ , which is approximately 0,1.