Cent (music) - Piecewise Linear Approximation

Piecewise Linear Approximation

As x increases from 0 to 1/12, the function 2x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a piecewise linear function which is numerically correct at semitones. That is, n cents for n from 0 to 100 may be approximated as 1+0.0005946n instead of 2n/1200. The rounded error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50, where the correct value of 21/24 = 1.02930 is approximated by 1 + 0.0005946*50 = 1.02973. This error is well below anything humanly audible, making this piecewise linear approximation suitable for most practical purposes.