Pythagorean tuning

Pythagorean tuning is a musical tuning system built by stacking pure fifths and reducing them to a single octave. Its intervals use ratios of powers of 2 and 3, but its circle of fifths does not close exactly.

Perfect fifth: A musical interval whose simplest just-intonation frequency ratio is 3:2. Stacking this interval supplies the system’s basic tuning operation.
Pythagoras: An ancient Greek philosopher traditionally associated with early mathematical accounts of musical intervals. Later tradition connected the system’s numerical account of consonance to his school.
Pythagorean tuning in medieval music: The use of fifth-generated pitches in medieval Western musical practice and theory. It shows how the system served actual repertories before meantone tuning spread.
Quarter-comma meantone: A meantone temperament that narrows each fifth by one quarter of the syntonic comma. It improves major thirds by making fifths less pure than Pythagorean fifths.
Pythagorean comma: The small discrepancy between twelve pure fifths and seven octaves, with ratio 531441:524288. It quantifies why repeated fifths cannot produce a perfectly closed circle.
Pythagoreanism: An ancient Greek philosophical and religious movement that treated number as fundamental to reality. The tuning system became part of this movement’s account of cosmic numerical harmony.
Organum: An early form of polyphony in which one voice moves alongside a sustained or parallel melody. Parallel fourths and fifths made the tuning of early polyphony especially consequential.
Well temperament: A family of unequal temperaments that permits music in all keys with different interval characters. It offers broader key access without making every semitone equal.
Pythagorean scale: A diatonic scale whose pitches are derived from pure fifths and octave equivalence. It is a familiar scale structure generated by this tuning method.
String length: The vibrating length of a string, which determines pitch when other properties remain constant. Simple string-length divisions demonstrate the ratios underlying Pythagorean intervals.

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