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:
(A3.1) |
For calculation convenience, a logarithm to base 2 can be expressed as a logarithm to base 10. Suppose:
log2 [X] = y | (A3.2) |
Then by definition:
X = 2y
Taking logarithms to base 10:
Substituting in Equation A3.2 for y:
Rearranging:
(A3.3) |
Substituting Equation A3.3 into Equation A3.1:
Evaluating the constants to give the equation for calculating the cents value of a frequency ratio:
(A3.4) |
In semitones (s), this is equivalent to:
(A3.5) |
Rearranging Equation A3.4 to give the equation for calculating the frequency ratio from a cent value:
(A3.6) |
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