The Mathematics of Harmony
Why does a piano play in every key, but a medieval organ could only handle a few? The answer lies in mathematics. Pure intervals use simple ratios—3:2 for a fifth, 5:4 for a major third—but they can’t all fit into 12 notes per octave without compromise. For centuries, tuners have battled the Pythagorean comma, tamed the wolf fifth, and invented ingenious temperaments. Explore Pythagorean tuning, just intonation, equal temperament, gamelan scales, Arabic maqam, and the alien beauty of the Bohlen-Pierce scale—all with real audio so you can hear the difference.
The building blocks: harmonics, ratios, and the fundamental problem of tuning.
Click any of the first 16 harmonics to hear them. The natural overtone series is the foundation of all music—every note contains this invisible ladder of frequencies.
Build a scale by stacking perfect fifths (3:2). Walk the spiral of fifths and hear the Pythagorean comma—the 23-cent gap that proves you can’t close the circle.
Compare every interval in both tuning systems. Listen for beating in equal temperament—the wavering that pure ratios eliminate. Toggle between just, equal, and both.
Two sliders, two frequencies, real-time visualization. Watch and hear how close frequencies create beating—the wavering piano tuners use to detect imperfect intervals.
How musicians solved the tuning problem across centuries—from wolf-howling meantone to Bach’s well-tempered revolution.
In meantone temperament, 11 fifths are pure but the 12th—the wolf—howls at 737 cents. Click around the circle to hear consonance dissolve into dissonance.
Werckmeister III, Kirnberger III, and equal temperament compared. Play major triads in every key and see how each temperament gives keys their unique color.
Play I–vi–ii–V–I in just intonation and watch the pitch drift down by a syntonic comma each cycle. After 5 cycles, you’ve lost nearly a semitone. This is why we temper.
Beyond Western 12-TET: gamelan metallophones, Arabic quarter tones, and scales from other dimensions.
Play a virtual metallophone in Javanese slendro (5-tone) and pelog (7-tone) tunings. Hear the shimmering, non-Western intervals and interlocking kotekan patterns.
13 equal steps per tritave (3:1) instead of 12 per octave. Its consonant chord is 3:5:7. Click the nodes to explore this alien-yet-beautiful tuning system.
Arabic music’s 24 quarter tones per octave enable maqams like Rast, Bayati, Saba, and Hijaz—each with distinct emotional character. Play them on a virtual oud.