Acoustics and Psychoacoustics, 5th Edition

 

Appendix 3: Converting between Frequency Ratios and Cents

The cent is defined as one hundredth of an equal tempered semitone, which is equivalent to one twelve-hundredth of an octave since there are 12 semitones to the octave. Thus one cent can be expressed as:

$1200\sqrt{2}or{2}^{\left[\frac{1}{1200}\right]}$

The frequency ratio of any interval (F1/F2) can therefore be calculated from that interval in cents (c) as follows:

$\frac{\text{F}1}{\text{F}2}=2^{\left|\frac{c}{1200}\right|}$

and the number of cents can be calculated from the frequency ratio by rearranging to give:

${\log }_2\left[\frac{F1}{F2}\right]=\left[\frac{c}{1200}\right]$

Therefore:

$c=1200{\log }_2\left[\frac{F1}{F2}\right]$

(A3.1)

For calculation convenience, a logarithm to base 2 can be expressed as a logarithm ...