Equal Temperament & The Mathematical Physics of Pitch
In standard twelve-tone equal temperament (12-TET), an octave is divided into 12 semitones on a logarithmic frequency scale, where the ratio between each successive semitone is the twelfth root of two (\(\sqrt[12]{2} pprox 1.059463\)).
$$f(n) = f_0 imes 2^{ rac{n - 69 + rac{c}{100}}{12}}$$
Scientific frequency formula where \(f_0\) is reference pitch (440 Hz), \(n\) is MIDI number, and \(c\) is cents offset
The physical wavelength of the sound wave traveling through room air at 20°C (where the speed of sound \(c pprox 343 ext{ m/s}\)) is calculated as:
$$\lambda = rac{c}{f} = rac{343 ext{ m/s}}{f ext{ (Hz)}}$$
Physical wavelength formula in meters