Interactive audio experiments on tuning: pure, equal-tempered, and meantone fifths, the Pythagorean comma, and scales in four temperaments.
Small experiments where we take instrument building apart and put the theory back together. Turn your sound on and press play.
After the octave, the fifth is the most consonant interval in music. But there is no single correct fifth. The one you get depends on the theory you build your tuning from. Press each card to hear the same two notes tuned three different ways.
Straight from the overtone series, a 3:2 frequency ratio. The two notes lock together with no beating. This is the fifth your ear calls in tune.
The octave split into twelve identical steps, the tuning behind most keyboards. This fifth sits two cents narrow, close enough to sound clean with only a faint shimmer.
An older keyboard tuning that narrows every fifth to buy sweeter thirds. The fifth is noticeably flat and beats faster.
The triad that opens a thousand songs, tuned four ways. The fifth barely shifts between systems. The third is where they split. Just and meantone set it low and pure, so the chord sits still and rings. Pythagorean and equal stretch it wide, adding a brighter, more restless edge. Tap each to hear the same chord move.
Stack twelve pure fifths and you should land back on the note you started from, seven octaves higher. You do not. Walk the circle and listen: the last note sits just above the first.
Twelve pure fifths overshoot seven octaves by the Pythagorean comma, about 23.5 cents. The circle of fifths never actually closes.
Every tuning is just a way of spending that comma.
Equal temperament shaves about two cents off every fifth so the circle closes perfectly. Every key sounds the same, and none is truly pure.
Meantone spends the comma on pure thirds instead, which leaves one unusable wolf fifth where all the leftover piles up.
Just intonation keeps every interval pure for a single key, then drifts out of tune as soon as you move away from home.
The same C major scale, tuned four ways. Play each one and listen to the melody bend as the thirds and sixths shift. The table below shows exactly where every note lands.
Pure whole-number ratios. Sweetest inside its home key.
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