Why does a piano have 12 notes per octave? Because the irrational number log2(3/2) = 0.5849625… has a continued fraction whose convergents land on 2, 5, 12, 41, 53 — the equal divisions of the octave that best approximate a perfect fifth. Explore the number theory, then hear the difference between locked just intonation and drifting equal temperament.
The Lissajous figure plots the two tones against each other. When the frequency ratio is a simple integer (just intonation) the curve closes into a fixed knot and freezes. The 12-EDO fifth is 27/12, an irrational number just off 3/2, so its curve never closes — it slowly precesses.
Ear check: with a harmonic-rich tone, the just fifth's coincident partials (3f₁ = 2f₂) lock silently. In 12-EDO they miss by a hair and you hear slow beating at 0.75 Hz — the same number shown as “Beating” above. Toggle Just / 12-EDO while it plays.
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Expand log2(3/2) as a continued fraction
[0; 1, 1, 2, 2, 3, 1, 5, ...]. Its convergents p/q are the best rational
approximations of a fifth-per-octave. The denominator q is an EDO; the best q-EDO fifth uses
p steps (p = round(q·log2(3/2))). The just fifth is 701.955 cents.
| Convergent p/q | EDO (q) | Fifth = p steps | Fifth (cents) | Error vs 3:2 |
|---|
12-EDO (7/12) is the sweet spot: only -1.955 cents off, with few enough notes to play. 53-EDO (31/53) is near-perfect (-0.068 c) but unwieldy. This is exactly why keyboards settled on 12.
cents = 1200·log2(ratio). Error = 12-EDO − JI. The current interval is highlighted.
| Interval | JI ratio | JI cents | 12-EDO | Error |
|---|
How far each 12-EDO note sits from its pure ratio. Bars right = sharp (amber), left = flat (coral). The major third (+13.7 c) is the famous offender.
Pythagorean comma = 312219 = 531441524288 ≈ 23.460 cents.
Stack 12 pure fifths (3/2) and you overshoot 7 octaves by this much — you cannot tile the octave with pure fifths, so a keyboard must compromise.
Syntonic comma = 8180 ≈ 21.506 cents.
Four pure fifths (up two octaves) vs one pure major third (5/4) differ by this. It is the gap meantone temperament spreads out to sweeten thirds.