Equal temperament and the twelfth root of two
Why modern tuning splits the octave into twelve equal steps, how the twelfth root of two works, and where it differs from pure just intonation.
Splitting the octave into twelve equal steps
An octave is a doubling of frequency, and equal temperament divides that doubling into twelve identical ratios called semitones. Because the steps multiply rather than add, each one is the twelfth root of two, roughly 1.059463. Multiply a frequency by that number twelve times and you land exactly one octave higher. This is why 440 Hz reaches 880 Hz after twelve semitones and 220 Hz after twelve down.
Why a multiplying ratio, not a fixed step
Pitch is perceived logarithmically, so equal musical intervals correspond to equal frequency ratios, not equal frequency gaps. A semitone near the bottom of the range spans only a few hertz, while the same semitone two octaves up spans many times more. Using a constant ratio keeps every semitone sounding the same size to the ear, which is exactly what the 2^(semitones / 12) formula delivers.
Equal temperament versus just intonation
Just intonation builds intervals from simple whole-number ratios, such as 3:2 for a perfect fifth or 5:4 for a major third. Those ratios sound pure but only in one key at a time. Equal temperament nudges each interval slightly so that all twelve keys are equally usable, at the cost of small beating in chords. A tempered fifth is about 1.498307 rather than the pure 1.5, a difference of roughly two cents.
Using it to pitch shift and retune
The same math drives pitch shifters, samplers and tuners. To move a recorded sample up a whole tone, transpose it by two semitones and its playback frequency multiplies by about 1.122462. To match a track recorded slightly flat, enter a small fractional semitone shift until the frequencies line up. Because the ratio is exact, you can chain shifts, and the calculator shows the target frequency directly rather than leaving you to work it out.