Twelfth root of two

Octaves (12 semitones) increase exponentially when measured on a linear frequency scale (Hz).

Octaves are equally spaced when measured on a logarithmic scale (cents).

The twelfth root of two or ${\displaystyle {\sqrt[{12}]{2}}}$ (or equivalently ${\displaystyle 2^{1/12}}$) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone (Play^ⓘ) in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones).^[a] A semitone itself is divided into 100 cents (1 cent = ${\displaystyle {\sqrt[{1200}]{2}}=2^{1/1200}}$).

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