Tuning and Equal Temperament

Every note a jazz piano plays is slightly out of tune — and that compromise is exactly what makes jazz harmony work. Equal temperament divides the octave into 12 identical semitones so that every key sounds equally good (and equally imperfect), which is the only way a fixed-pitch instrument can modulate freely and let C♯ and D♭ be the same key on the keyboard. Understanding this trade-off explains why Tritone Substitution even makes sense, and why horn players and singers constantly bend away from the piano’s grid.

The problem: pure intervals don’t add up to a closed system

The overtone series gives us naturally pure intervals as simple frequency ratios — a perfect fifth is 3:2, a major third is 5:4. The trouble is that stacking pure fifths never gets you back to a clean octave: twelve pure fifths overshoot the equivalent seven octaves by about 23.5 cents, an error called the Pythagorean comma. Any tuning built entirely from pure ratios ends up with 12 unique, non-interchangeable pitches, so a piano tuned that way would sound gorgeous in one key and painfully sour in another.

Interval	Pure ratio	Pure (cents)	12-TET (cents)	Deviation
Perfect fifth	3:2	701.96	700	about 2 cents flat — inaudible to most ears
Major third	5:4	386.31	400	about 14 cents sharp — audible as “beating”

The fix: divide the octave into 12 equal pieces

Equal temperament solves the closure problem by ignoring pure ratios altogether and just splitting the octave (2:1) into 12 mathematically identical half steps. Each semitone becomes a frequency ratio of 2^(1/12) ≈ 1.05946, and musicians measure these tiny differences in cents, where 100 cents equal one semitone and 1200 cents equal an octave. The comma that used to break the system is instead spread evenly across all twelve fifths — about 1.95 cents shaved off each one — so no single key suffers more than any other.

1 semitone = 2^(1/12) ≈ 1.0594631 frequency ratio = 100 cents
1 octave = 1200 cents = frequency ratio 2:1
Standard reference pitch: A = 440 Hz (concert pitch, sets absolute frequencies for everything else)

Why this makes jazz harmony possible at all

Because every semitone is identical, C♯ and D♭ land on the exact same key and the exact same frequency — something that is literally false in just intonation tuning, where each spelling implies a different pure pitch. That symmetry is what lets a dominant chord be reinterpreted a tritone away: G7 and D♭7 share the same tritone (B–F, or equivalently E–B♭), so Tritone Substitution only works because equal temperament has erased the distinction between enharmonic spellings. The same symmetry underwrites the circle of fifths, free modulation and transposition to any of the twelve keys, and the entire logic of Key Signatures and Accidentals as notational conveniences rather than acoustic necessities.

The catch: it’s a compromise, not perfection

It’s worth being honest that 12-TET is a deliberate mistuning of nearly everything except the octave — the major third is roughly 14 cents sharp of pure, the minor third about 16 cents flat, and only the octave itself is exact. That sharpness is why sustained major triads on a well-tuned piano have an audible shimmer or “beat” that a barbershop quartet singing pure just intervals does not. This is also the backdrop against which Blue Notes make sense: a horn or vocalist bending the ♭3 or ♭7 toward a pitch that lies between the 12-TET grid points isn’t playing “out of tune” — they’re using dissonance and inflection expressively, treating the equal-tempered grid as a reference point to depart from, not a cage. Fixed-pitch instruments like piano and vibraphone hold down that grid; horns and voices float around it.

♫ Listen

Coleman Hawkins — “Body and Soul” (RCA Bluebird, 1939): Hawkins’ tenor winds around Gene Rodgers’ fixed piano chords in the intro, phrasing with vibrato and pitch inflection that sits outside the equal-tempered grid the piano is locked into.
Billie Holiday — “Billie’s Blues” (1936): her vocal line bends and slides around the piano’s fixed pitches, treating Blue Notes as expressive departures from 12-TET rather than errors.