The Great Tuning Compromise
For 2,500 years, musicians have wrestled with a mathematical impossibility: you cannot tune all intervals to be perfectly pure in all keys simultaneously. Stacking twelve perfect fifths (3:2 ratio) overshoots seven octaves by about 23.5 cents — the Pythagorean comma. This tiny discrepancy forced the invention of temperament — systems that distribute the error across different intervals and keys to make practical music possible.
Equal Temperament: The Modern Standard
Equal temperament, which became standard in the 19th century, solves the problem by making every semitone exactly equal — each with a ratio of 2^(1/12). No interval except the octave is mathematically pure, but the errors are small enough to be acceptable. The perfect fifth is only 2 cents flat; the major third is about 14 cents sharp. This uniformity lets pianists play in all 12 keys without retuning.
Just Intonation: Mathematical Purity
Just intonation tunes each interval to its simplest frequency ratio: 3:2 for a fifth, 5:4 for a major third, 4:3 for a fourth. These pure intervals produce no acoustic beating — they literally vibrate in sync. Barbershop quartets, a cappella groups, and string quartets naturally drift toward just intonation when sustaining chords, because the locked-in resonance is unmistakable and physically satisfying.
Hearing the Difference
The most audible difference between tuning systems is the major third. In equal temperament, it's 400 cents; in just intonation, it's 386.3 cents — a 13.7-cent gap that creates a noticeable beating when two notes are sustained. This simulator lets you visualize the waveform beating and measure the cent differences for every interval, revealing the precise trade-offs that underlie all of Western music's tuning history.