From Pythagoras to Equal Temperament

Why a scale built from pure fifths gave way to twelve equal semitones.

Related

MedievalBaroqueIntervalOrgan

The question

If octaves and fifths stay pure, one note has nowhere to sit, and thirds and sixths drift from what the ear expects. Once a keyboard is fixed, tuning must decide which intervals to keep pure and where to hide error.

Historical arc

In the sixth century BCE, the Pythagoreans defined the octave, fifth and fourth by string length. Medieval theorists wrote these ratios into Boethius and derived scales by cycling fifths. Medieval organs loved pure fifths, but thirds were impure, so just intonation and meantone appeared. Renaissance keyboards needed free modulation; meantone sweetened the third but lost remote keys. In the eighteenth century, twelve equal semitones divided the octave, and Bach's Well-Tempered Clavier displayed a workable temperament across all twenty-four keys. In the late nineteenth century, cents turned deviations into numbers, and twentieth-century composers returned to just intonation and microtones.

The turning point

The problem was the wolf fifth. Cycle twelve fifths and the last one is narrower than pure by a Pythagorean comma. Meantone hid the wolf in rarely used keys; equal temperament spread the error across every semitone. Equal temperament is not purer, only usable everywhere. It frees modulation while leaving every third slightly away from pure.

What to listen for

Begin with a chant whose melody rests on modal steps. Then hear Renaissance polyphony, where thirds sound unusually pure, a trace of just intonation or meantone. Finally hear a prelude from Bach's Well-Tempered Clavier, where every key works on one keyboard. The library versions are synth sounds, so notice the sweetness or roughness of thirds.

Related

Tuning history runs from Medieval into Baroque, with Interval and Organ.

Sources

• The New Grove Dictionary of Music and Musicians, 2nd ed., entry "Tuning" — historical outline checked, not quoted
• J. Murray Barbour, Tuning and Temperament (Michigan State College Press) — data checked, not quoted

Used for fact-checking only; all prose is original.