Musical tuning systems
No tuning system can make all intervals pure. This is a mathematical certainty: log2(3) is irrational, so no stack of perfect fifths ever exactly reaches an octave. Every tuning system is a different compromise with this impossibility. Play the keyboard below and hear three different solutions.
12-TET: f = 440 × 2(n−69)/12 · Just: simple ratios (3:2, 5:4, 6:5) · Pythagorean comma: (3/2)12 / 27 ≈ 23.46 cents
Select an interval. It plays in all three systems sequentially so you can hear the difference.
The impossible compromise
No tuning system can simultaneously make all intervals pure. This is not a limitation of technology or craftsmanship — it is a mathematical certainty. The fundamental frequency ratio of a perfect fifth is 3/2. Stack twelve perfect fifths and you should return to the starting note, seven octaves higher. But (3/2)12 = 531441/524288 ≈ 129.746, while 27 = 128. The difference — the Pythagorean comma, roughly 23.46 cents — is the gap between the geometry of fifths and the geometry of octaves. Log2(3) is irrational: the spiral of fifths and the ladder of octaves never land on the same point. Every tuning system in the history of music is a different way of distributing this unavoidable error.
Equal temperament: democratic imperfection
Twelve-tone equal temperament divides the octave into twelve identical semitones, each with a frequency ratio of 21/12 ≈ 1.05946. No interval is pure except the octave itself. The fifth is 1.96 cents flat. The major third is 13.69 cents sharp — noticeably so to a trained ear. But every key sounds identically “wrong,” which means you can modulate freely from C major to F# major without any interval becoming worse than any other. Equal temperament won not because it sounds best — it does not — but because it is a coordination equilibrium. Once keyboards were mass-produced during the Industrial Revolution and pianos needed to play in every key, the compromise that treats every key equally became the only one manufacturers could agree on. We are all still living inside that agreement.
Just intonation: the sound of simple ratios
When two frequencies stand in a simple integer ratio — 3:2 for a fifth, 5:4 for a major third, 6:5 for a minor third — their combined waveform is periodic, and the ear perceives pure consonance with zero beating. This is just intonation: intervals built from the harmonic series itself. A major triad in just intonation (4:5:6) has a luminous clarity that equal temperament cannot match. But the purity is local. Set up just intervals from C, and the D-to-A fifth (which should be 3/2) comes out as 40/27 — noticeably flat. Modulate to another key and the whole system falls apart. Barbershop quartets and a cappella choirs naturally drift toward just intonation because their pitch is flexible; they are not constrained by frets or keys. They solve the problem by never needing to modulate far.
Pythagorean tuning: the spiral of fifths
Build everything from the perfect fifth. Start on C, go up a fifth to G, up a fifth to D, and keep going: A, E, B, F#, C#, G#, D#, A#, E#. Twelve fifths later you have visited all twelve notes of the chromatic scale. Each fifth is pure (3/2), and each major second is a sweet 9/8. But the major third comes out as 81/64 ≈ 1.2656, audibly sharper than the just 5/4 = 1.2500. And the twelfth fifth does not close the circle: it overshoots by the Pythagorean comma. Medieval musicians absorbed this error into a single rarely-used interval — the wolf fifth, usually G#–Eb — making eleven fifths pure and one terrible. Most music stayed in keys that avoided the wolf. The system worked beautifully until composers started wanting to use all twelve keys.
Hearing the difference
The difference between tuning systems is subtle but unmistakable once you know what to listen for. Play a major third in this lab: 12-TET gives 24/12 ≈ 1.2599, just intonation gives 5/4 = 1.2500. That one percent difference creates audible beating — a gentle amplitude wobble whose frequency equals the difference between the two notes' frequencies. In equal temperament a major third at concert pitch produces beats at roughly 10 Hz — fast enough to hear as a subtle roughness. In just intonation the same interval produces zero beats: perfect stillness. Pipe organ tuners set temperament by counting beats. Piano technicians learn to hear intervals that are 1–2 cents off. Barbershop singers lock chords by eliminating beats instinctively. For most listeners, the difference between tuning systems takes training to notice — but once you hear it, you cannot unhear it.
Mathematics and music
The twelve-note chromatic scale is not an arbitrary cultural choice. The continued fraction expansion of log2(3/2) — the size of a fifth measured in octaves — gives the convergents 1/2, 3/5, 7/12, 10/17, 24/41, 31/53... The denominators are the numbers of equally spaced notes that best approximate a perfect fifth. The number 12 appears early in the sequence, which is why twelve notes per octave “work”: twelve equal steps approximate a 3/2 ratio to within 2 cents. The next substantially better approximation is 53 notes per octave, which the Chinese music theorist Jing Fang calculated in 45 BC — more than two thousand years before European mathematicians arrived at the same result. Some Turkish makam music uses 53 divisions of the octave. The mathematics of rational approximation explains both why twelve notes dominate world music and why some traditions chose differently.