Just intonation

Just intonation is a tuning approach that uses frequency ratios expressed as ratios of whole numbers. Its intervals are derived from relationships among harmonics, producing acoustically simple but context-dependent pitches.

Frequency ratio: A quotient comparing the frequencies of two periodic signals. Just intonation specifies intervals by assigning exact ratios to pairs of note frequencies.
Equal temperament: A tuning system that divides an octave into equal logarithmic steps. Unlike just intonation, it slightly alters most intervals so transposition preserves the same pitch relationships.
Barbershop seventh chord: A dominant seventh chord often tuned with unusually pure harmonic relationships in barbershop singing. Barbershop singers use flexible intonation to align chord partials and produce a strong blended sonority.
Pythagoras: An ancient Greek philosopher traditionally associated with early mathematical accounts of musical intervals. Pythagorean tradition connected consonant intervals with simple numerical ratios.
Harmonic series: The sequence of integer multiples of a fundamental frequency, present in the spectrum of many musical tones. Simple just intervals align partials in the harmonic series, often making them blend smoothly.
Meantone temperament: A family of temperaments that narrows fifths to improve the tuning of major thirds. It compromises pure fifths to make a practical compromise among just intervals.
Adaptive tuning: A tuning method that adjusts pitches in response to the notes sounding at a given moment. It can select just intervals for each chord while changing pitches as harmony moves.
Pythagoreanism: An ancient Greek philosophical and religious movement associated with number, harmony, and cosmic order. Its ideas helped frame musical ratios as evidence of mathematical order in nature.
Pythagorean tuning: A tuning system that builds pitches chiefly from perfect fifths with frequency ratio 3:2. It is a historically influential just-intonation system built primarily from one consonant ratio.
Well temperament: A family of temperaments that permits playing in all keys while giving each key a distinct interval character. It distributes tuning compromises across the scale rather than retaining only pure ratios.

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