Tuning Systems
The octave mapped to a circle, with twelve notes placed by four different tuning systems. Each solves the impossible problem of dividing the octave differently — and each compromise leaves a different set of intervals pure.
About this lab
The fundamental problem of tuning is arithmetic: no power of 3/2 is ever a power of 2. Stacking twelve perfect fifths (each a frequency ratio of 3:2) produces a note that overshoots seven octaves by about 23.46 cents — the Pythagorean comma. This means you cannot have an instrument where every fifth is pure and every octave is pure simultaneously. Every tuning system is a different strategy for distributing this unavoidable error across the twelve notes of the chromatic scale.
Pythagorean tuning, attributed to the ancient Greeks, keeps all fifths pure (3:2) and dumps the entire comma into a single dissonant "wolf fifth." This works beautifully for monophonic melody and simple harmony in one key but becomes intolerable when you try to modulate. Just Intonation goes further, tuning thirds to the pure 5:4 ratio as well, producing triads of extraordinary clarity — but only in one key. Transpose and the intervals collapse into unusable dissonance. Just Intonation is the tuning of barbershop quartets and well-trained choirs, who adjust continuously by ear.
Quarter-comma Meantone, dominant from the Renaissance through the Baroque, splits the difference: it narrows each fifth by a quarter of a syntonic comma (the difference between a Pythagorean and a pure major third), giving eight pure major thirds at the cost of one very bad "wolf fifth" and some rough keys. It sounds glorious in C, F, and G major, but D-flat major is nearly unplayable. This is why Baroque keyboard music tends to stay in certain key signatures — the tuning literally could not support others.
Equal Temperament, proposed in various forms since the 16th century and universally adopted by the 20th, solves the problem by making every semitone exactly 100 cents (a frequency ratio of 2^(1/12)). No interval except the octave is pure: every fifth is about 2 cents flat, every major third is about 14 cents sharp. But every key sounds equally good — or equally compromised. This is the democratic tuning, the one that enables Chopin's key-wandering nocturnes and Coltrane's Giant Steps. What it sacrifices in purity it gains in freedom. The circle of fifths closes, but only because we bent every fifth to make it fit.