Tuning: just intonation, temperament, harmonics and microtones

Tones are frequency ratios

Beneath the tone names hides physics. The octave is a 2:1 ratio (double the frequency), the perfect fifth a 3:2 ratio, the pure major third 5:4. The simpler the ratio, the more 'pure' and consonant the interval sounds.

Harmonics

Every tone of a real instrument contains harmonics - faint higher frequencies vibrating at whole-number ratios above the fundamental. These determine the timbre (tone colour) and explain why a guitar and a flute sound different on the same tone. On a guitar you can play them by lightly touching the string above the 12th fret.

The problem with just intonation

If we tuned every interval to its simple ratio (just intonation), a single key would sound divine - but the moment we modulated to another, the tones would be 'wrong'. Stacking pure fifths and pure octaves simply does not add up; the gap even has a name - the Pythagorean comma.

The compromise: equal temperament

Western music therefore uses equal temperament: it divides the octave into 12 exactly equal semitones. Every fifth is a hair too narrow, every third slightly too wide - but all keys are equally (out of) tune, so you can modulate freely. This is the compromise on which the piano, and all harmonic music of recent centuries, stands.

Microtones

Beyond the 12 semitones lie microtones - intervals smaller than a semitone. Many non-European traditions (such as Arabic and Indian music) use them deliberately, and modern composers explore the spaces between the piano keys.