One-third octave

Augmented chord in the chromatic circle Play (help·info)

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 Play (help·info)).^[1] An alternative (unambiguous) term for one tenth of a decade is a decidecade.^[2][3]

One octave is a factor of 2, so ${\displaystyle \log _{10}(2)=0.301}$ decades per octave, while a third would be ${\displaystyle 0.{\overline {33}}}$.

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)". (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.

One decidecade is equal to 100 savarts (398.63 cents).

See also[]

• Decibel
• Octave band
• Pseudo-octave
• Tritonic scale

References[]