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:

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

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

Therefore:

For calculation convenience, a logarithm to base 2 can be expressed as a logarithm to base 10. Suppose:

Then by definition:

x = 2^y

Taking logarithms to base 10:

log₁₀[x] = log₁₀[2^y] = y log₁₀[2]

Substituting in Equation A3.2 for y:

log₁₀[x] = log₂[x] log₁₀[2]

Rearranging:

Substituting Equation A3.3 into Equation A3.1:

Evaluating the constants to give the equation for calculating the cents value of a frequency ratio:

In semitones (s), this is equivalent to:

Rearranging Equation A3.4 to give the equation for calculating the frequency ratio from a cent value: