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## Cents

A cent is the 100th part of a halftone in the Equal Temperament in a logarithmic scale. Because our ears hear ratios rather then distances on a frequency scale, a logarithmic scale is much more natural than a linear one, allowing us to express intervals as distances.

The definition of a cent is: Given a frequency ratio r, its distance expressed in cents is

$LC(r) = \frac{\log(r)}{\log(\sqrt[1200]{2})}$

As an example, the perfect fifth has a ratio of 3/2, and taking the above formula, this is LC(1.5) = 701.9550. A fifth in the Equal Temperament is $$\left( \sqrt[12]{2} \right) ^7$$ (7 halftones), in cents $$LC\left( \sqrt[12]{2}^7 \right) = 700$$ exactly.