How to Find the Frequency of Any Musical Note (A4 = 440 Hz)

Every note you hear has a number behind it: a frequency, measured in hertz, that tells you how many times per second the air pressure swings back and forth. A guitar's open A string vibrates 110 times a second. The A an octave above it vibrates 220 times. Knowing that number matters the moment you stop trusting your ears and start setting an oscillator, hand-editing a MIDI file, or checking whether a recording is actually in tune. This guide walks through where note frequencies come from, why A4 = 440 Hz is the anchor, and how a single formula generates the entire keyboard.

The Reference: Why A4 = 440 Hz

You can't assign frequencies to notes without first agreeing on one fixed point. The international standard, codified in ISO 16, sets the A above middle C — written A4 in scientific pitch notation — to exactly 440 Hz. That single decision pins down every other pitch, because the rest of the scale is defined relative to it.

440 isn't sacred, it's just the one almost everyone agreed on. Many European orchestras tune sharper, to 442 or 443 Hz, for a brighter sound. Baroque period ensembles often drop to around 415 Hz. And there's a vocal crowd that prefers 432 Hz, which lowers the whole keyboard by about 31.8 cents. None of these is "more natural" than the others — they're just different anchors for the same mathematical scale. The important part is that once you fix the reference, the math is identical no matter which number you pick.

The Formula: The Twelfth Root of Two

Western music splits the octave into twelve equal steps called semitones. "Equal" here is precise: each semitone multiplies the frequency by the same ratio, and twelve of those multiplications have to land you exactly one octave up — a doubling. So the ratio of one semitone is the twelfth root of two, roughly 1.059463. Climb twelve semitones and you've multiplied by 1.059463 twelve times, which equals exactly 2.

That gives the core identity for any note's frequency:

frequency = 440 × 2^(n / 12)

where n is the number of semitones above (positive) or below (negative) A4. An octave up is twelve semitones, so n = 12 makes the exponent 1 and doubles the frequency. An octave down is n = −12, which halves it. The same expression is often written against MIDI note numbers as f = reference × 2^((MIDI − 69) / 12), since A4 sits at MIDI 69 — the two forms are the same equation with a different way of counting steps.

A Worked Example

Let's run a few notes by hand. Start at the anchor:

A4 is our reference, n = 0, so frequency = 440 × 2^(0/12) = 440 Hz.
A5 is one octave up, n = +12, so frequency = 440 × 2^(12/12) = 440 × 2 = 880 Hz. The octave doubling falls straight out of the formula.
C5 is three semitones above A4 (A → A# → B → C), so n = +3. That gives 440 × 2^(3/12) = 440 × 2^(0.25) ≈ 440 × 1.18921 ≈ 523.25 Hz.

Drop down instead and A3 comes out at 440 × 2^(−12/12) = 220 Hz, exactly half of 440. Middle C (C4) is nine semitones below A4, landing at about 261.63 Hz — the number every piano tuner knows by heart. You can check all of these instantly with the Note Frequency Calculator, which runs both directions: type a note and read the Hz, or paste a measured frequency and get the nearest note back with its MIDI number and cents offset.

Cents, MIDI, and Reading Tuning Accuracy

Two extra numbers make the raw frequency useful. The first is the cent, defined as 1/100 of a semitone, so an octave spans 1200 cents. Cents are how you talk about being slightly out of tune: a pitch that reads +6 cents is sharp by six hundredths of a semitone. Most listeners can't hear anything under about ±5 cents; a trained ear starts catching errors around ±10 to 15 cents. One trap worth flagging — cents are logarithmic, not linear. Ten cents at A4 (440 Hz) is roughly 2.5 Hz, but ten cents down at A2 (110 Hz) is only about 0.6 Hz. The same cents value means a different Hz gap depending on where you are on the keyboard.

The second number is the MIDI note number, an integer from 0 to 127 where 60 is middle C and 69 is A4. Every semitone adds one, so C#4 is 61 and A3 is 57. This is the value your DAW, synth, and sequencer actually store and transmit, which makes it the safest way to identify a note across gear that disagrees on octave labels — some synths call middle C "C3," others "C5," but the MIDI number is never ambiguous.

When I Reach for This

The case where this clicked for me was matching a 432 Hz recording. A collaborator sent stems tuned to A4 = 432, and against my project's 440 default everything sounded subtly wrong — not a melody mistake, just a constant low-grade clash. I switched the reference to 432, looked up the new middle C (256.87 Hz instead of 261.63), and set my synth's master tune to the −31.8 cent offset the calculator reported. The clash vanished. Without that number I'd have nudged the pitch knob by ear for twenty minutes; with it, the fix took one read and one entry.

Putting It to Work

Once you understand that one formula drives the whole scale, a lot of small jobs get easier. Tuning by ear: pull up A2 at 110 Hz, play the sine preview, and match your string until the beating stops. Sound design: drop an oscillator onto C1 at 32.70 Hz instead of guessing a sub frequency. MIDI editing: convert a measured 196 Hz back to G3, MIDI 55, two cents flat, and you know exactly which note byte to write.

If you'd rather see the same logarithmic ratios in another context, the Scientific Calculator handles the 2^(n/12) powers directly, and the Unit Converter is handy when you're juggling sample rates and periods alongside frequencies. But for note work specifically, having the note name, the Hz, the MIDI number, and the cents offset on screen at once is what turns a vague "that sounds flat" into a precise correction.

Made by Toolora · Updated 2026-06-13