Equal Temperament vs Just Intonation: The Comma Nobody Can Escape

A scale is a compromise, and someone has to choose which cracks show

Musical intervals sound most consonant when the frequencies involved form small whole-number ratios: an octave is exactly 2:1, a perfect fifth is exactly 3:2, a major third is exactly 5:4. A tuning system built entirely from these pure ratios is called just intonation, and it produces intervals that are genuinely, mathematically beat-free. The catch is that just intonation only works in the one key it was tuned for — the pure ratios do not stay pure when you transpose, because stacking them does not close back onto a clean octave.

Stack twelve pure perfect fifths (ratio 3/2) and you should, in principle, arrive back at the starting pitch class seven octaves higher. You do not. Twelve fifths of 3/2 each multiply out to (3/2)^12 approx 129.75, while seven octaves is 2^7 = 128 exactly. The mismatch, a ratio of about 1.0136, is the Pythagorean comma, and it is not a rounding error — it is a genuine mathematical fact that the group generated by pure fifths and pure octaves does not close.

Equal temperament: spread the comma out evenly

12-tone equal temperament (12-TET) resolves the comma by refusing to keep any interval perfectly pure except the octave. It divides the octave into twelve exactly equal steps on a logarithmic frequency scale, so every semitone is the same ratio:

semitone ratio  =  2^(1/12)  ~  1.059463
n semitones above a base frequency f0:   f = f0 * 2^(n/12)
perfect fifth (7 semitones):   2^(7/12) ~ 1.49831   (pure 3/2 = 1.5 exactly)
major third (4 semitones):     2^(4/12) ~ 1.25992   (pure 5/4 = 1.25 exactly)

Every 12-TET fifth is off from pure by only about 2 cents (2/100 of a semitone) — small enough that most listeners do not consciously notice a slow beat, though sensitive ears and sustained tones can. The major third fares far worse: it is nearly 14 cents sharp of pure 5/4, a difference easily audible as a distinct, faster beating, which is exactly why barbershop quartets and unaccompanied choirs, who can tune each chord freely rather than being locked to a keyboard, routinely sing thirds and fifths noticeably closer to the pure just-intonation ratios than any piano ever could.

Why equal temperament won anyway

The entire benefit of spreading the comma evenly across all twelve intervals is transposability: every key sounds equally (im)pure, so a piece can modulate freely through any key, an instrument can be built with fixed pitches (a piano's 88 keys, a fretted guitar neck) rather than requiring the player to bend pitch on the fly, and any two fixed-pitch instruments tuned to the same standard fully agree with each other. Just intonation, meanwhile, forces a hard choice: it delivers perfect consonance in the key it was set up for, and audibly wrong, unusable intervals in almost every other key, because the pure ratios that make one key beat-free are precisely the ratios that make a different key badly out of tune.

Historically, keyboard tuning went through several compromise systems before settling on equal temperament — meantone temperaments narrowed some fifths to widen thirds toward purity, well temperaments (associated with Bach's Well-Tempered Clavier) made each key sound subtly different in character while keeping all keys playable, and only in the 19th and 20th centuries did strict 12-TET become the near-universal default for pianos, guitars, and fixed-pitch instruments generally.

What the beating actually sounds like

When two tones are close to but not exactly at a simple ratio, their pressure waveforms slip in and out of phase at a rate equal to the difference between how far each overtone deviates from that ratio, producing an audible slow throb in loudness — the same beating phenomenon covered elsewhere on this site, just applied to the small mistuning equal temperament deliberately introduces rather than to two arbitrary close frequencies. A pure 3/2 fifth beats at 0 Hz by definition (or matches immediately with no throb); the tempered fifth beats slowly because it is not quite 3/2. The demo lets you switch a fifth or a third between its pure just-intonation ratio and its 12-TET equivalent and hear that throb appear.

Frequently asked questions

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