Why does equal temperament sound slightly out of tune?

Because it is, very slightly, on purpose. Equal temperament splits the octave into twelve identical steps of exactly 100 cents each, which lets an instrument play in every key. But the intervals the ear hears as "pure" are simple frequency ratios (a perfect fifth is 3:2, a major third is 5:4), and those do not line up with the equal steps. Equal temperament's major third lands 13.69 cents sharp of the pure 5:4 third, so a held major chord has a faint restless beating that just intonation would not.

How far off is an equal-tempered major third?

A pure (just) major third is the frequency ratio 5:4, which is 386.3 cents above the root. Equal temperament puts its major third at exactly 400 cents — so it is 13.69 cents sharp. For reference, a trained ear can hear a difference of about 5 cents, so 13.7 cents is clearly audible as the slightly bright, beating quality of a tempered third.

Is the equal-tempered perfect fifth in tune?

Almost. A pure fifth is 3:2, which is 701.96 cents; equal temperament uses 700 cents, so the fifth is only 1.96 cents flat — small enough that most listeners never notice. The fifth is the interval equal temperament preserves best. The thirds and sixths are where it compromises most, which is why they carry the characteristic "tempered" beating.

What frequency does a guitar tune to?

Standard tuning targets equal-tempered frequencies from A4 = 440 Hz: the six strings are E2 82.41 Hz, A2 110 Hz, D3 146.83 Hz, G3 196 Hz, B3 246.94 Hz and E4 329.63 Hz. A digital tuner listens to the string and shows how many cents you are above or below those exact numbers.

Why a Tuner and Your Ear Disagree

By Lucian — builder & engineer, LK Forge

Tune a guitar to a digital tuner, play a big open major chord, and something is faintly off — a slow shimmer in the sound. The tuner isn't wrong, and neither are your ears. They're using two different definitions of "in tune," and the gap between them is a fixed, measurable number. Our tuners target one definition; your ear prefers the other. Here's exactly how far apart they are.

Two definitions of "in tune"

A digital tuner uses equal temperament: the octave is chopped into twelve exactly equal steps of 100 cents each (a cent is 1/100 of a semitone). That evenness is what lets one piano play in all 24 keys without retuning. Your ear, though, calls an interval "pure" when the two frequencies form a simple whole-number ratio — a perfect fifth is 3:2, a major third is 5:4 — because then the waves lock together and stop beating. The catch: those pure ratios don't fall on the equal 100-cent grid. So the two systems disagree, note by note, by an amount you can compute exactly.

Key finding: equal temperament's major third is +13.69 cents off a pure third and its major sixth +15.64 cents; the perfect fifth is only -1.96 cents. The worst offender is the minor 7th at -17.6 cents. A trained ear notices about 5 cents — so the thirds and sixths are audibly tempered, the fifths and fourths barely.

Each bar is one chromatic interval; the height is how far equal temperament sits from the pure whole-number ratio. Only the octave, fourth and fifth land close. Computed from A4 = 440 Hz.

The full table

Interval	Pure ratio	Equal temp.	Pure (cents)	Off by
Unison	1/1	0	0	0
Minor 2nd	16/15	100	111.73	-11.73
Major 2nd	9/8	200	203.91	-3.91
Minor 3rd	6/5	300	315.64	-15.64
Major 3rd	5/4	400	386.31	+13.69
Perfect 4th	4/3	500	498.04	+1.96
Tritone	45/32	600	590.22	+9.78
Perfect 5th	3/2	700	701.96	-1.96
Minor 6th	8/5	800	813.69	-13.69
Major 6th	5/3	900	884.36	+15.64
Minor 7th	9/5	1000	1017.6	-17.6
Major 7th	15/8	1100	1088.27	+11.73
Octave	2/1	1200	1200	0

This is why a piano can never be perfectly in tune and perfectly playable in every key at once — the two goals are mathematically incompatible, and equal temperament is the elegant surrender that spreads the error evenly so no key sounds worse than any other. A string quartet or a barbershop group, with no frets or keys, can slide each note to the pure ratio and get thirds with no beating at all. That's the sound your ear is quietly asking a fretted, tempered instrument for and not quite getting.

What your tuner is actually listening for

A digital tuner takes the equal-tempered target frequency for each string and shows how many cents your string is above or below it. Every one of these comes from the same formula the tuners use — f = 440 × 2⁽⁽ᵐ⁻⁶⁹⁾ᐟ¹²⁾, where m is the MIDI note number:

Guitar

E2 — 82.41 Hz
A2 — 110 Hz
D3 — 146.83 Hz
G3 — 196 Hz
B3 — 246.94 Hz
E4 — 329.63 Hz

Bass

E1 — 41.2 Hz
A1 — 55 Hz
D2 — 73.42 Hz
G2 — 98 Hz

Violin

G3 — 196 Hz
D4 — 293.66 Hz
A4 — 440 Hz
E5 — 659.26 Hz

Ukulele

G4 — 392 Hz
C4 — 261.63 Hz
E4 — 329.63 Hz
A4 — 440 Hz

Reproduce this

Every number comes from scripts/gen-temperament-study.mjs — two one-line functions, no dependencies:

// cents in an interval given its frequency ratio
const cents = (ratio) => 1200 * Math.log2(ratio)

// equal-tempered note frequency (A4 = 440 Hz) — the tuners' formula
const midiToFreq = (m) => 440 * 2 ** ((m - 69) / 12)

// deviation of the equal-tempered interval from the pure one:
//   major third → 100*4 - cents(5/4) = 400 - 386.31 = +13.69 cents

Run it for each interval's just ratio (5:4, 3:2, …) and you get the whole table; feed each string's MIDI number to midiToFreq and you get every tuner target above.

Hear it for yourself

Tune up with a tuner, then play a major third and a fifth and listen for the difference in how "settled" they sound. Sweep two pitches close together on the tone generator to hear beating appear and vanish, and see how the intervals sit on the circle of fifths. All run in your browser with the Web Audio API — nothing to install.