Musical temperament

A system for tuning musical intervals by adjusting pitch relationships so instruments can play in multiple keys. Temperaments differ in how they distribute tuning compromises across intervals.

Just intonation: A tuning system that uses frequency ratios drawn from the harmonic series to produce acoustically pure intervals. It preserves pure ratios but makes many intervals unusable when music changes key.
Musical interval: The perceived and measured distance between two pitches. Temperament specifies the precise pitch relationships assigned to intervals.
Pythagoras: An ancient Greek philosopher traditionally associated with mathematical study of musical consonance. Pythagorean tuning takes its name from the ancient ratio-based account of musical intervals.
Piano tuning: The adjustment of a piano's strings to establish usable pitches and interval relationships. Fixed-pitch keyboards require a temperament before they can serve music in multiple keys.
Microtonal music: Music that uses pitch intervals smaller than a semitone or otherwise outside a standard twelve-tone framework. It may replace the familiar twelve-note pitch framework that common temperaments tune.
Equal temperament: A tuning system that divides an octave into equal logarithmic steps. Its equal semitones distribute tuning discrepancies evenly across all keys.
Frequency ratio: The quotient of two sound frequencies, often used to describe the acoustic size of an interval. A temperament changes the ratios assigned to familiar musical intervals.
Renaissance music: European art music composed and performed roughly from the fifteenth through the sixteenth centuries. Its emphasis on thirds encouraged tunings that improved those intervals over pure fifths.
Harpsichord: A keyboard instrument whose strings are plucked by quills or similar mechanisms. Historical harpsichords often used temperaments suited to the keys and repertoire performed.
Adaptive tuning: A tuning method that changes pitches in response to the notes sounding at a given moment. Unlike a fixed temperament, it can retune intervals as the harmony changes.

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