How it Works
A fixed-pitch instrument like a piano is tuned once and must sound acceptable in every key, so Western music settled on 12-tone equal temperament (12-TET): the octave is divided into twelve identical steps, each a frequency ratio of 2^(1/12) ≈ 1.05946. This makes transposition and modulation between keys completely seamless — a major third above any note is always exactly 400 cents — but except for the octave itself, every interval is a slightly impure approximation of the simple whole-number ratios that vibrating strings and air columns naturally produce.
Just intonation instead tunes intervals to those small-integer ratios directly — a fifth as exactly 3/2, a major third as exactly 5/4 — which makes chords ring with zero beating in the key they were tuned for. The catch is that a just-intonation scale only works cleanly in one key; transpose it and the same "pure" ratios land on the wrong pitches. Equal temperament trades that purity for universal playability, and the size of the trade is measured in cents: the small gap between the ET version of an interval and its just counterpart.
ET frequency: f(n semitones) = f₀ · 2^(n/12)
Just ratios: unison 1/1 · m3 6/5 · M3 5/4 · P4 4/3 · P5 3/2 · m6 8/5 · M6 5/3 · octave 2/1
Just frequency: f = f₀ · (num/den)
Cents: cents = 1200 · log₂(f_ET / f_just)
Beat rate: Δf = |f_ET − f_just| (Hz)
Frequently Asked Questions
Why do pianos use equal temperament instead of just intonation?
A piano's strings are tuned once and can't be adjusted per key while playing, so every interval must work reasonably well in every key. 12-tone equal temperament (12-TET) divides the octave into twelve identical semitone ratios of 2^(1/12), making every key transpose perfectly — at the cost of every interval except the octave being very slightly impure compared to the simple whole-number ratios of just intonation.
Why do a cappella singers and string quartets naturally drift toward just intonation?
Voices and unfretted strings can adjust pitch continuously in real time, so performers instinctively lock onto the pure, beat-free ratios (3/2, 5/4, 4/3, etc.) that make chords ring with minimal roughness. Ensembles without a fixed-pitch instrument routinely tune chords a few cents away from equal temperament toward these just ratios, especially on sustained final chords.
What do "cents" mean in music theory?
A cent is 1/100 of an equal-tempered semitone, so an octave equals exactly 1200 cents. Cents let you compare tiny tuning differences on a linear scale: cents = 1200 · log2(f1/f2). Most listeners can detect pitch differences of 5-10 cents in a sustained tone, which is why the deviations between ET and just intonation (a few cents to about 16 cents) are audible, especially as beating.
Why is the major third the most out-of-tune interval in 12-TET?
The just major third is the simple ratio 5/4 (386.3 cents), while the equal-tempered major third is exactly 400 cents — a gap of about 13.7 cents sharp. No other common consonant interval in 12-TET deviates as far from its just counterpart; the fifth and fourth are within about 2 cents of pure, which is why thirds historically caused the most tuning controversy.
What were meantone and well temperament, and why were they replaced by equal temperament?
Meantone temperament (popular from the Renaissance) tuned most major thirds nearly pure by narrowing the fifths, which sounded beautiful in a handful of keys but produced a horribly out-of-tune "wolf" interval in others. Well temperaments (like those associated with Bach's era) unevenly distributed the compromise so every key was usable, each with a subtly different color. Equal temperament finally made every key sound identical, trading that variety for universal playability.
Why do guitars use an equal-ish temperament with compromises?
Guitar frets are straight bars that must serve every string at once, so they're cut to true 12-TET semitone spacing. But the physics of pressing a string against a fret slightly raises its pitch (extra string stretch), and no single compensation setting is exact for every string and fret, so guitars carry small, unavoidable tuning compromises on top of equal temperament itself.
What is the harmonic series and how does it relate to just intonation ratios?
The harmonic series is the sequence of frequencies at integer multiples of a fundamental (f, 2f, 3f, 4f...) that naturally occur in vibrating strings and air columns. Just intonation ratios like 3/2 (fifth) and 5/4 (major third) come directly from low-numbered harmonics — the 3rd and 5th harmonics — which is why they sound so consonant and beat-free when in tune.
Can you actually hear a 14-cent deviation?
Yes — 14 cents is well above the roughly 5-10 cent detection threshold for a held tone, and when two tones close in frequency are sounded together, the beating they produce is audible even for much smaller deviations, often under 1 Hz of difference. That's exactly what this simulator's Play button lets you compare directly.
Why does this simulator show a "beat rate" for equal temperament?
When two tones are only a few Hz apart, the ear hears a periodic rise and fall in loudness called beating, at a rate equal to the frequency difference between them. Piano tuners historically used exactly this — counting beats per second between a note and a pure reference — to set equal temperament by ear before electronic tuners existed.