Music FIELD NOTES · 4 MIN READ
Equal Temperament and A440: How Notes Get Their Frequency
One anchor note, one irrational root: how A4 = 440 Hz plus the twelfth root of two assigns a frequency to every key on the piano.
Every frequency an instrument can play descends from a single convention: A4, the A above middle C, is 440 Hz. From that one anchor, equal temperament generates every other pitch on the piano with nothing more exotic than the twelfth root of two — an irrational number that divides the octave into twelve exactly equal steps and, in doing so, shaped two centuries of music.
The formula and why the twelfth root
The math is compact: frequency = 440 × 2^((midi − 69) / 12). The 69 is A4's MIDI note number, and the exponent counts semitones away from it. Each semitone multiplies the previous frequency by 2^(1/12) ≈ 1.05946, so after twelve steps the frequency has doubled — an octave. That is the whole design goal of equal temperament: an octave is a frequency ratio of exactly 2:1 (a string twice as long, a pipe twice as short, sounds an octave up), and the twelve semitones inside it should all feel the same size.
They feel the same size because they are the same size — in ratio terms. Human pitch perception is logarithmic: the distance between 220 and 440 Hz sounds identical to the distance between 440 and 880 Hz, even though the second pair spans twice the hertz. Equal temperament embraces that by making every step a multiplication rather than an addition, which is also why the cent — a hundredth of a semitone — is the natural unit for describing small pitch differences.
MIDI numbers are the map
The MIDI standard numbers every key: middle C (C4) is 60, A4 is 69, and each semitone is +1. The octave arithmetic is mechanical — MIDI = (octave + 1) × 12 + semitone — so C4 sits at (4+1)×12 = 60. Scientific pitch notation (C4, A5, F#2) and MIDI numbers carry the same information, and the frequency formula only needs the number. This is why tuning software, DAWs and analysers all speak MIDI under the hood: it turns the note world into a linear ruler the formula consumes directly.
Two spellings share each key: D#4 and Eb4 are the same 311.13 Hz. Equal temperament flattened the ancient distinction between sharps and flats — a compromise that let keyboards play in every key at the cost of tiny impurities everywhere, a trade J.S. Bach famously celebrated in the Well-Tempered Clavier.
Why 440, and what 442 or 432 change
The anchor itself is a social contract, not physics. Before 1939, orchestral A varied wildly — region to region, era to era, sometimes a semitone apart — and touring ensembles spent more energy re-tuning than rehearsing. An international conference fixed A4 at 440 Hz, later enshrined as ISO 16. Orchestras often drift above it anyway (442–444 Hz is common in Europe, prized for brightness), and period-instrument ensembles sit lower, near 415 Hz — a deliberate semitone down.
Change the anchor and the entire scale moves with it. At 442 Hz every note sits 7.85 cents sharp of standard; at 432 Hz, every note sits 31.8 cents flat. Nothing about the tuning's internal structure changes — only its absolute placement. The cents column of a frequency calculator captures exactly this: how far your reference sits from the 440 world that tuners, sample libraries and backing tracks assume.
Where reality bends the numbers
Real instruments negotiate with the formula rather than obeying it. Piano tuners apply "stretch tuning": the octaves are widened slightly because real strings are a little stiffer than the ideal string the math assumes, and the deviation grows toward both ends of the keyboard. Choirs and string sections drift toward just intonation on sustained chords because beating partials sound dirty; that is why a held major third sung by four voices can sound both sweeter and flatter than the equal-tempered 400 cents. None of this makes the calculator wrong — the equal-tempered values are the shared reference everything else is measured against.
Using the numbers in practice
The values earn their keep across the studio and the bench: synthesiser tuning and oscilloscope checks, verifying a sampler's pitch table, building microtonal experiments, science demonstrations, and hearing-test sweeps across octaves. Wherever "which key is which hertz" needs an answer a tuner can verify, the formula delivers one number, every time.
Try it yourself
The Note Frequency Calculator implements the formula directly: pick any note and octave from −1 to 10 and get the exact Hz, MIDI number and semitone distance from A4, with an adjustable A4 reference that shows the cents offset for 442, 432 or any pitch in between. To hear the number you generate, feed it to the Tone Generator, which plays any note or frequency through your browser; and to put frequency on a tempo grid, the BPM Delay Calculator handles the time-domain twin of this math.